text stringlengths 4 2.78M |
|---|
---
abstract: 'The link of two concepts, indistinguishability and entanglement, with the energy-time uncertainty principle is demonstrated in a system composed of two strongly coupled bosonic modes. Working in the limit of a short interaction time, we find that the inclusion of the antiresonant terms to the coupling Hamiltonian leads the system to relax to a state which is not the ground state of the system. This effect occurs passively by just presence of the antiresonant terms and is explained in terms of the time-energy uncertainty principle for the simple reason that at a very short interaction time, the uncertainty in the energy is of order of the energy of a single excitation, thereby leading to a distribution of the population among the zero, singly and doubly excited states. The population distribution, correlations and entanglement are shown to be substantially depend on whether the modes decay independently or collectively to an exterior reservoir. In particular, when the modes decay independently with equal rates, entanglement with the complete distinguishability of the modes is observed. The modes can be made mutually coherent if they decay with unequal rates. However, the visibility in the single-photon interference cannot exceed $50\%$. When the modes experience collective damping, they are indistinguishable even if decay with equal rates and the visibility can, in principle, be as large as unity. We find that this feature derives from the decay of the system to a pure entangled state rather than the expected mixed state. When the modes decay with equal rates, the steady-state values of the density matrix elements are found dependent on their initial values.'
author:
- Smail Bougouffa
- Zbigniew Ficek
title: 'Evidence of indistinguishability and entanglement determined by the energy-time uncertainty principle in a system of two strongly coupled bosonic modes'
---
Introduction {#sec1}
============
There has recently been a great interest in the realization of quantum networks of coupled qubits formed by spatially periodic structures of trapped atoms [@st02; @hm13; @gh15], arrays of coupled optical cavities [@hb06; @as07; @zd08; @tf10; @kb12; @sl12; @m13] or superconducting electrical circuits [@lf13; @fg11; @ma14; @ht12]. Quantum networks provide an experimental platform for spatial transport of quantum states required for quantum cryptography, quantum teleportation, simulation of many-body systems, quantum information processing and quantum computation. Optical cavities are ideally suited for the implementation of quantum networks where the inter-cavity coupling might be realized through the output cavity fields which could be focused and transmitted by optical elements, for example, short optical fibers. The primary objective is to achieve strong and lossless couplings. Therefore, different coupling schemes have been proposed to accomplish an efficient transfer of photons between adjacent cavities including overlapping evanescent field modes, optical fibers or waveguides, and hopping fields, the tunneling of photons between cavities [@bq14]. Exchange of information between the cavities is often affected by dissipation and decoherence induced by the unavoidable coupling to the environment. For coupling via fibers or waveguides, major obstacles are losses inside the fiber or waveguide material.
A number of theoretical and experiments studies were carried out on the simplest quantum network composed of only two cavities, and several schemes have been proposed in which an efficient transmission between the cavities could be achieved [@ZOR00; @HBP08; @MC15]. In most treatments the cavities contained two-level atoms, and the creation of entanglement between the atoms and its transfer to the cavity modes was considered [@cz97; @tp97; @ek99; @mb04; @io08; @bek08; @ye10; @zl10; @bb10; @sy11; @xf12; @bf13]. It has also been demonstrated that effective quantum gates between atoms located in distant cavities can be realized even in the presence of losses and imperfections in coupling strengths [@sm06; @bk08]. In addition, the interaction of the cavities with an injected squeezed field or with a squeezed reservoir has been studied [@kc04; @ti10; @ab15].
The previous work on quantum networks of coupled cavities was limited to the weak coupling regime described by the coupling Hamiltonian containing only resonant terms, the photon hopping between the modes. In general, the coupling Hamiltonian also contains antiresonant terms such that the creation of an excitation in a given mode is accompanied by the creation of a negative energy quantum in the other mode. In the weak coupling regime the antiresonant terms make much smaller contributions and therefore are often omitted, under the rotating-wave approximation [@ae75]. However, in the strong coupling regime in which the magnitude of the coupling strength is comparable to the frequency of the modes, the antiresonant terms make notable contributions leading to novel features [@bs40; @wh97; @bk00; @nb08; @jl09; @zr09; @cy10; @lz12; @yl13].
In this paper, we consider a pair of coupled bosonic modes represented by two single-mode cavities coupled by a short waveguide. In studying the interaction between the cavities, we include both resonant and antiresonant terms in the interaction Hamiltonian. To say this another way, we permit for two types of the interactions, linear and non-linear to contribute simultaneously to the coupling between the cavities. Notice that the inclusion of the antiresonant terms is equivalent to take into account the energy non-conserving terms in the interaction between the cavities. These terms are known to produce virtual photons which can survive only for a time $\Delta t\sim 1/\omega$, where $\omega$ is the frequency of the modes. According to the energy-time uncertainty principle, at such short times the virtual photons fail to conserve energy by an amount $\Delta E$, which is of order of the energy of a single excitation, $\Delta E\sim \hbar\omega$. This fact can lead to a redistribution of the population among states differing in energy by $\hbar\omega$. Of particular interest is the stationary limit the system attains over this short time. This requires a strong coupling of the modes and a fast damping of the modes if one would like to achieve a stationary state over such a short time. Therefore, our results apply to a short observation time and the ultra-strong coupling regime. Some results are also presented for the so-called deep strong coupling regime, corresponding the coupling strengths larger than the field frequency [@cr10; @br12; @dl14; @sz14].
We show that the system exhibits features, in particular coherence and entanglement features that are not present in the weak coupling regime. Two cases are studied: (i) the modes decay independently, and (ii) the modes decay collectively to an external reservoir. We find that the modes decaying independently with equal rates can be found entangled and simultaneously behaving as mutually incoherent. We calculate the visibility of the interference fringes and show how the “which-path” information is made possible when the modes decay with equal rates. The “which-way” information, however, is not possible when the modes decay with unequal rates, so a mutual coherence can be established resulting in single-photon interference between the modes. We find an upper bound that the visibility cannot exceed $50\%$ when the modes decay independently.
The modes can, however, be made entangled and simultaneously exhibiting quantum interference with $100\%$ visibility if they decay collectively. We find that in this case, the modes are always indistinguishable independent of whether they decay with equal or unequal rates. In addition, we find that the collective damping can lead to the steady-state values of the density matrix which depends on initial conditions.
The paper is organized as follows. In Sec. \[sec2\] we introduce the model and formulate the master equation for the density operator of the system. The equations of motion for the density matrix elements and their steady-state solutions are given in Sec. \[sec3\]. The equations of motion are simple enough that we can find their steady-state values analytically. In Sec. \[sec4\] we discuss the problem of distinguishability between the modes induced by the energy-time uncertainty principle and methods to make the modes indistinguishable. An upper bound is imposed on the visibility of the interference fringes when the modes decay independently and it can be overtaken if the modes decay collectively. In Sec. \[sec5\] we examine the conditions for entanglement. Some remarks are made about the connection between the one- and two-photon coherences. Finally, in Sec. \[sec6\], we summarize and conclude our results.
The model and approach {#sec2}
======================
We consider a pair of strongly coupled bosonic modes of equal frequencies $\omega$, labelled by the suffices $A$ and $B$. The modes are represented by the annihilation and creation operators, $\hat{a}_{j}, \hat{a}^{\dagger}_{j}\, (j=A,B)$, which satisfy the commutation relation $[\hat{a}_{i},\hat{a}^{\dagger}_{j}]=\delta_{ij}$. We assume that apart from the strong dynamical influence on each other through the direct coupling, the modes can also influence on each other through modes of the reservoir to which they are damped with rates $\gamma_{A}$ and $\gamma_{B}$, respectively. We will investigate two cases in which the modes decay independently or collectively. We will refer to these cases as the decay of the modes to either separate reservoirs or a common reservoir. In order to take into account contributions of the antiresonant (non-RWA) terms, we will require the coupling strengths and damping rates to be comparable to the frequency $\omega$. In other words, we will work in the ultra-strong coupling regime. We are interested in the steady-state characteristics of the system, in which the strong coupling processes counterbalance the decay process.
In practice this model could be realized in a circuit QED system where the ultra-strong coupling regime with the ratio of the coupling strength $g$ to the resonator frequency $\omega$ of order $g/\omega =0.1$ has been achieved [@nd10; @fl10; @rb12]. Ultra-strong couplings with a rate up to $g/\omega =0.58$ have been realized with two high-mobility two-dimensional electron gases coupled to a metamaterial [@sm12]. Recently, even higher coupling rates of up to $g/\omega =0.87$ have been reached in semiconductor heterostructures [@ms14]. The most relevant to the model considered in the present paper are experiments with photonic crystal nanocavities coupled to a short waveguide [@ks13]. Owing to its small optical loss and tight field confinement, waveguides are capable of mediating strong and long range couplings using photons propagating in their guided modes. Recently, it has been demonstrated experimentally that a strong coupling with a ratio $g/\omega \approx 0.1$ can be achieved between two single-mode cavities subject of a very short decay time of photons out of the cavities to a waveguide composed of discrete modes [@st11]. Schematic diagram of the experiment is shown in Fig. \[fig1\].
The properties of the coupled modes, including the damping of the modes due to their coupling to the reservoir, are determined by the density operator $\rho$ which satisfies the following master equation $$\begin{aligned}
\frac{d}{dt}\tilde{\rho} = -\frac{i}{\hbar}\left[\tilde{H}_{AB},\tilde{\rho}\right] + {\cal L}\tilde{\rho} ,\label{b1}\end{aligned}$$ where $\tilde{\rho}$ is the density operator in the interaction picture and $\tilde{H}_{AB}$ is the coupling Hamiltonian between the modes $$\tilde{H}_{AB} = \hbar g\!\left(\hat{a}_{A}^{\dag}\hat{a}_{B}\!+\!\hat{a}_{B}^{\dag}\hat{a}_{A}\!+\!\hat{a}_{A}\hat{a}_{B}e^{2i\omega t}\!+\!\hat{a}_{B}^{\dag}\hat{a}_{A}^{\dag}e^{-2i\omega t}\right) ,\label{b2}$$ Taking into account a very short decay time of photons to the waveguide, we have included into the coupling Hamiltonian the resonant (RWA) as well as antiresonant (non-RWA) terms which, as we will see, can have notable contributions at such short evolution times. The RWA terms represent the linear, a beam splitter type coupling between the cavities, whereas the non-RWA terms describes the nonlinear (parametric) type coupling. In order to distinguish between the contributions of the linear and nonlinear terms, we will work with the Hamiltonian of the form $$\begin{aligned}
\tilde{H}_{AB} &=\hbar\kappa\left(\hat{a}_{A}\hat{a}_{B}^{\dag}+\hat{a}_{A}^{\dag}\hat{a}_{B}\right) \nonumber\\
&+\hbar\epsilon\left(\hat{a}_{A}\hat{a}_{B}e^{2i\omega t} +\hat{a}_{B}^{\dag}\hat{a}_{A}^{\dag}e^{-2i\omega t}\right) ,\label{b4}\end{aligned}$$ where $\kappa$ determines the strength of the linear, whereas $\epsilon$ determines the strength of the nonlinear coupling. The term ${\cal L}\tilde{\rho}$, appearing in the master equation (\[b1\]), is an operator representing the damping of the modes to the external environment (reservoir). In general, it contains resonant and antiresonant terms. A recent investigation by Joshi [*et al.*]{} [@jo14] shows that the antiresonant terms present in the damping part of the master equation can modify the dynamics of strongly coupled modes. However, a further insight into the results reveals that the antiresonant terms change the results quantitatively, but not alter the qualitative behavior. Therefore, we retain only the resonant terms in ${\cal L}\tilde{\rho}$: $$\begin{aligned}
{\cal L}\tilde{\rho} &= -\frac{1}{2}\sum_{j=A,B}\gamma_{j}\!\left(\hat{a}_{j}^{\dag}\hat{a}_{j}\tilde{\rho} + \tilde{\rho} \hat{a}_{j}^{\dag}\hat{a}_{j} -2\hat{a}_{j}\tilde{\rho} \hat{a}_{j}^{\dag}\right) \nonumber\\
&-\frac{1}{2}\sum_{i\neq j=A,B}\gamma\!\left(\hat{a}_{i}^{\dag}\hat{a}_{j}\tilde{\rho} + \tilde{\rho}\hat{a}_{i}^{\dag}\hat{a}_{j} - 2\hat{a}_{j}\tilde{\rho} \hat{a}_{i}^{\dag}\right) ,\label{b3}\end{aligned}$$ where $\gamma_{j}$ is the damping rate of the mode $j$, and $\gamma$ is the cross damping rate at which the modes are coupled to each other through the interaction with the same reservoir. The coupling reflects the fact that, as a photon is emitted by the spontaneous decay of the mode $A$ it can be absorbed by the mode $B$, and vice versa. In other words, $\gamma$ describes a collective damping of the modes. The strength of the collective damping depends on the rates $\gamma_{A}$ and $\gamma_{B}$ and the polarization of the modes that $\gamma =\sqrt{\gamma_{A}\gamma_{B}}\cos\theta$, where $\theta$ is the angle between the polarization directions of the modes. If the polarizations are parallel then $\theta =0$ and the collective damping is maximal, $\gamma=\sqrt{\gamma_{A}\gamma_{B}}$, while if the polarizations are perpendicular, then $\gamma=0$.
An obvious question arises, under which conditions both terms in the Hamiltonian (\[b4\]) could simultaneously contribute to the dynamics of the system. In the presence of the antiresonant terms there are two time scales of the evolution of the system, one determined by the parameters $\kappa, g$ and $\gamma_{j}$ and the other determined by $\omega$. The resonant terms in the master equation (\[b1\]) experience a variation on a time scale $\Delta t_{r}\sim 1/\kappa, (\sim 1/g, 1/\gamma_{j})$, whereas the antiresonant terms experience a variation on a time scale of $\Delta t_{ar}\sim 1/\omega$. Therefore, these two time scales should be comparable $(\Delta t_{r}\approx \Delta t_{ar})$ in order the steady state be reached with the antiresonant terms participating fully in the dynamics. Thus, observation (detection) times should be comparable to $\Delta t_{ar}$.
In what follows, we explore the role of the resonant and antiresonant terms on the steady-state characteristics of the system. Analytic expressions are obtained for the density matrix elements which then are used to investigate the influence of the two kind of couplings between the modes on the population distribution, distinguishability and entanglement of the modes.
Steady-state solutions {#sec3}
======================
Given the master equation (\[b1\]), we can use the photon number representation for the density operator and derive equations of motion for the density matrix elements. Suppose that initially there is no excitation present in the modes, i.e., the initial state of the system was a vacuum state $\ket{0_{A}}\ket{0_{B}}$. Since there is no external excitation field present, one would expect that the modes would remain in their vacuum states for all times. However, we will demonstrate that the system evolves to a steady-state in which the singly and doubly excited states can have nonzero populations. To demonstrate this, we consider a basis set of low excitation states consisting of four states $$\begin{aligned}
\ket{1} &= \ket{0_{A}}\ket{0_{B}} ,\quad \ket{2} =\ket{0_{A}}\ket{1_{B}} , \nonumber\\
\ket{3} &= \ket{1_{A}}\ket{0_{B}} ,\quad \ket{4} =\ket{1_{A}}\ket{1_{B}} ,\label{9}\end{aligned}$$ where $\ket{0_{j}}$ and $\ket{1_{j}}$ are zero and one excitation states of the cavity $j$. The singly and doubly excited states have been included into the basis in order to fully account effects of the antiresonant terms $\hat{a}_{A}\hat{a}_{B}$ and $\hat{a}_{B}^{\dagger}\hat{a}_{A}^{\dagger}$, which couple the vacuum state to higher excitation states.
The reason for the inclusion of the low excitation states can be understood by noting that the inclusion of the antiresonant terms in the master equation (\[b1\]) leads to the steady-state to be achieved on a time scale of order $\Delta t \sim 1/\omega$. If the evolution time is of order $\Delta t$, the energy-time uncertainty principle, $\Delta E \Delta t \geq \hbar/2$, enforces that a precision $\Delta E$ of the energy of photons has to be at least of order of $\Delta E \approx \hbar \omega$, which is of order of the one-photon energy. Thus, over the evolution time $\Delta t \approx 1/\omega$, an excitation of the system to the states $\ket{1_{A}}\ket{0_{B}}$, $\ket{0_{A}}\ket{1_{B}}$, and $\ket{1_{A}}\ket{1_{B}}$ is possible.
In the basis (\[9\]) the density operator $\rho$ has fifteen independent matrix elements. The equations of motion for the density matrix elements which can have nonzero values in the steady state are $$\begin{aligned}
\dot{\rho}_{11} &=& \gamma_{B}\rho_{22} + \gamma_{A}\rho_{33} + \gamma\left(\rho_{23} + \rho_{32}\right) + i\epsilon\left(\rho_{14} - \rho_{41}\right) ,\nonumber\\
\dot{\rho}_{22} &=& \gamma_{A} -2\gamma_{0}\rho_{22} - \gamma_{A}\left(\rho_{11} +\rho_{33}\right) \nonumber\\
&& -\frac{1}{2}\left(\gamma - 2i\kappa\right)\rho_{23} -\frac{1}{2}\left(\gamma + 2i\kappa\right)\rho_{32} ,\nonumber\\
\dot{\rho}_{33} &=& \gamma_{B} -2\gamma_{0}\rho_{33} - \gamma_{B}\left(\rho_{11} + \rho_{22}\right) \nonumber\\
&& -\frac{1}{2}\left(\gamma + 2i\kappa\right)\rho_{23} -\frac{1}{2}\left(\gamma - 2i\kappa\right)\rho_{32} ,\nonumber\\
\dot{\rho}_{23} &=& \gamma -\gamma_{0}\rho_{23} - \gamma\left(\rho_{11} +\rho_{22} +\rho_{33}\right) \nonumber\\
&&-\frac{1}{2}\left(\gamma -2i\kappa\right)\rho_{22} -\frac{1}{2}\left(\gamma + 2i\kappa\right)\rho_{33} ,\nonumber\\
\dot{\rho}_{14} &=& -i\epsilon -\left(\gamma_{0} - 2i\omega\right)\rho_{14} +i\epsilon\left(2\rho_{11} + \rho_{22} + \rho_{33}\right) ,\label{9a}\end{aligned}$$ where $\gamma_{0} = (\gamma_{A}+\gamma_{B})/2$, and $\rho_{44}$ is found from the closure relation of the conservation of the total population, $\rho_{11} + \rho_{22} + \rho_{33} +\rho_{44} =1$. The set of coupled equations for the density matrix elements involves the populations and the one-photon $\rho_{23}$ and two-photon $\rho_{14}$ coherences.
The set of the differential equations can be written in a matrix form $$\begin{aligned}
\frac{d}{dt}\vec{Y} = M\vec{Y} + \vec{P} ,\label{v22}\end{aligned}$$ where the vector $\vec{Y}$ has the components $$\begin{aligned}
Y_{1} &= \rho_{11} ,\ Y_{2}=\rho_{22} ,\ Y_{3} =\rho_{33} ,\ Y_{4} = \rho_{23}+\rho_{32} ,\nonumber\\
Y_{5} &= i\!\left(\rho_{23}\!-\!\rho_{32}\right) ,\ Y_{6} = \rho_{14}\!+\!\rho_{41} ,\ Y_{7} = i\!\left( \rho_{14}\!-\!\rho_{41}\right) .\end{aligned}$$ Nonzero components of the vector $\vec{P}$ are $$\begin{aligned}
P_{2} = \gamma_{A}, \ P_{3} =\gamma_{B} ,\ P_{4} = 2\gamma ,\ P_{7} = 2\epsilon ,\end{aligned}$$ and $M$ is the $7\times 7$ matrix of real coefficients $$\begin{aligned}
M = \left(
\begin{array}{ccccccc}
0 & \gamma_{B} & \gamma_{A} & \gamma & 0 & 0 & \epsilon \\
-\gamma_{A} & -2\gamma_{0} & -\gamma_{A} & -\gamma/2 & \kappa & 0 & 0 \\
-\gamma_{B} & -\gamma_{B} & -2\gamma_{0} & -\gamma/2 & -\kappa & 0 & 0 \\
-2\gamma & -3\gamma & -3\gamma & -\gamma_{0} & 0 & 0 & 0 \\
0 & -2\kappa & 2\kappa & 0 & -\gamma_{0} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -\gamma_{0} & 2\omega \\
-4\epsilon & -2\epsilon & -2\epsilon & 0 & 0 & -2\omega & -\gamma_{0}
\end{array}\right) .\label{e19}\end{aligned}$$ The matrix $M$ describes the effects of the coupling terms $\kappa$ and $\epsilon$ as well as those of the dampings.
Solving Eq. (\[v22\]) for the steady-state, we find the diagonal matrix elements to be $$\begin{aligned}
\label{24}
\rho^{s}_{11} &= \frac{(\epsilon^{2}\!+\!4\omega^2\!+\!\gamma_{0}^2)}{D}\!\left[4\kappa^{2}\!\left(\gamma_{0}^{2} -\gamma^{2}\right)\!+\!\gamma_{0}^{2}\!\left(\gamma_{A}\gamma_{B} -\gamma^{2}\right)\right] ,\nonumber \\
\rho^{s}_{22} &= \frac{\epsilon^2}{D}\left[\left(4\kappa^2+\gamma_A^2\right)\left(\gamma_{0}^2- \gamma^2\right) +\frac{1}{4}\gamma^2\left(\gamma_A-\gamma_B\right)^2\right] , \nonumber\\
\rho^{s}_{33} &= \frac{\epsilon^2}{D}\left[(4\kappa^2+\gamma_B^2)\big(\gamma_{0}^2-\gamma^2\big)+\frac{1}{4}\gamma^2(\gamma_A-\gamma_B)^2\right] , \nonumber\\
\rho^{s}_{44} &= \frac{\epsilon^2}{D}\left[4\kappa^{2}\left(\gamma_{0}^{2} -\gamma^{2}\right) +\gamma_{0}^{2}\left(\gamma_{A}\gamma_{B} -\gamma^{2}\right)\right] ,\end{aligned}$$ and the off-diagonal elements $$\begin{aligned}
\label{24u}
\rho^{s}_{14} &= \frac{i\epsilon(\gamma_{0}+2i\omega)}{D}\left[4\kappa^{2}\left(\gamma_{0}^{2} -\gamma^{2}\right) +\gamma_{0}^{2}\left(\gamma_{A}\gamma_{B} -\gamma^{2}\right)\right] ,\nonumber \\
\rho^{s}_{23} &= \frac{i(\gamma_{A}\!-\!\gamma_B)\epsilon^2}{4D}\!\left[8\kappa\big(\gamma_{0}^2-\gamma^2\big) +i\gamma(\gamma_A^2-\gamma_B^2)\right] ,\end{aligned}$$ where $$\begin{aligned}
\label{25}
D &= \left(\gamma_{0}^{2}+4\omega^{2}\right)\left[4\kappa^{2}\left(\gamma_{0}^{2}-\gamma^{2}\right) +\gamma_{0}^{2}\left(\gamma_{A}\gamma_{B}-\gamma^{2}\right)\right] \nonumber\\
&+4\epsilon^{2}\left(\gamma_{0}^{2}-\gamma^{2}\right)\left(\gamma_{0}^{2} +4\kappa^{2}\right) .\end{aligned}$$ From Eq. (\[24\]), we see that the steady-state of the coupled modes is not the ground state $\ket 1 = \ket{0_{A}}\ket{0_{B}}$. The population is redistributed between the states including the doubly excited state $\ket 4$. There are no external sources of photons, like driving laser fields in the system. This effect occurs passively by just adding the antiresonant (non-RWA) terms determined by $\epsilon$. When these terms are ignored, the standard RWA result is obtained with $\rho_{11}=1$ and no population in the excited states.
In addition, the steady-state solution is strongly affected by the coupling of the modes to the reservoir. In particular, a coherence is generated in the process of spontaneous emission with unequal damping rates, $\gamma_{A}\neq \gamma_{B}$. Let us discuss in greater detail the cases of independent $(\gamma =0)$ and collective $(\gamma\neq 0)$ dampings, under unbalanced $(\gamma_{A}\neq \gamma_{B})$ and balanced $(\gamma_{A}=\gamma_{B})$ decays of the modes.
Unbalanced decay: $\gamma_{A}\neq \gamma_{B}$
---------------------------------------------
When the modes decay independently, $\gamma =0$, and then the steady-state solution (\[24\]) reduces to $$\begin{aligned}
\rho^{s}_{11} &=& \frac{\left(4\kappa^2+\gamma_A\gamma_B\right)}{D_{0}}\left(4\omega^2+\epsilon^2+\gamma_{0}^2\right) ,\nonumber\\
\rho^{s}_{22} &=& \frac{\epsilon^2\left(4\kappa^2+\gamma_A^2\right)}{D_{0}}, \quad
\rho^{s}_{33} = \frac{\epsilon^2\left(4\kappa^2+\gamma_B^2\right)}{D_{0}}, \nonumber\\
\rho^{s}_{44} &=& \frac{\epsilon^2\left(4\kappa^2+\gamma_A\gamma_B\right)}{D_{0}} ,\quad
\rho^{s}_{23} = \frac{2i(\gamma_A-\gamma_B)\kappa\epsilon^2}{D_{0}} ,\nonumber\\
\rho^{s}_{14} &=& \frac{i\epsilon\left(4\kappa^2+\gamma_A\gamma_B\right)}{D_{0}}(\gamma_{0}+2i\omega) ,\label{18}\end{aligned}$$ where $$\begin{aligned}
D_{0} = \left(4\kappa^2\!+\!\gamma_A\gamma_B\right)\!\left(4\omega^2\!+\!\gamma_{0}^2\right)
+ 4\epsilon^{2}\!\left(4\kappa^2\!+\!\gamma_{0}^2\right) .\label{19}\end{aligned}$$ Expression for $\rho_{23}^{s}$ shows that a coherence is generated by spontaneous decay of the modes even if the modes decay independently. It requires the modes to decay with unequal rates, $\gamma_{A}\neq \gamma_{B}$; that is, unbalanced decay plays a constructive role in the generation of the one-photon coherence. The unbalanced decay of the modes creates a population inversion between states $\ket 2$ and $\ket 3$ that $$\rho^{s}_{22} - \rho^{s}_{33} = \frac{\epsilon^2\left(\gamma_A^2 -\gamma_{B}^{2}\right)}{D_{0}} = \frac{2\epsilon^{2}\gamma_{0}\left(\gamma_A -\gamma_{B}\right)}{D_{0}} .$$ Then the coherence can be written as $$\rho^{s}_{23} = \frac{2i(\gamma_A-\gamma_B)\kappa\epsilon^2}{D_{0}} = \frac{i\kappa}{\gamma_{0}}\left(\rho^{s}_{22} - \rho^{s}_{33}\right) .$$ This shows the familiar fact that the coherence between two states is proportional to the product of the driving field strength and the population inversion. That is, the linear coupling $\kappa$ between the modes is a complete analog of a coherent driving of two quantum states.
Another interesting observation is that the unbalanced decay can lead to a population inversion between the doubly excited state $\ket 4$ and the singly excited states $\ket{2}$ and $\ket{3}$). Really, if we evaluate ratios $\rho^{s}_{22}/\rho^{s}_{44}$ and $\rho^{s}_{33}/\rho^{s}_{44}$, we find the result $$\begin{aligned}
\frac{\rho^{s}_{22}}{\rho^{s}_{44}} &= 1 + \frac{\gamma_{A}\left(\gamma_{A}-\gamma_{B}\right)}{\kappa^{2} +\gamma_{A}\gamma_{B}} ,\nonumber\\
\frac{\rho^{s}_{33}}{\rho^{s}_{44}} &= 1- \frac{\gamma_{B}\left(\gamma_{A}-\gamma_{B}\right)}{\kappa^{2} +\gamma_{A}\gamma_{B}} .\end{aligned}$$ We see that depending on whether $\gamma_{A}>\gamma_{B}$ or $\gamma_{A}<\gamma_{B}$, the population can be inverted between $\ket 4$ and either $\ket 2$ or $\ket 3$. It is interesting that the population can be inverted between $\ket 4$ and only one of the singly excited states.
Consider now the case when the modes decay collectively. If the collective damping rate is maximal, $\gamma =\sqrt{\gamma_{A}\gamma_{B}}$, the solution (\[24\]) simplifies to $$\begin{aligned}
\label{24a}
\rho^{s}_{11} &= \frac{\kappa^{2}\left(\epsilon^{2} + 4\omega^2 + \gamma_{0}^2\right)}{ \tilde{D}} ,\nonumber \\
\rho^{s}_{22} &= \frac{\epsilon^{2}\left(2\kappa^2+\gamma_{A}\gamma_{0}\right)}{2 \tilde{D}} ,\quad
\rho^{s}_{33} = \frac{\epsilon^{2}\left(2\kappa^2+\gamma_{B}\gamma_{0}\right)}{2 \tilde{D}} , \nonumber\\
\rho^{s}_{44} &= \frac{\kappa^{2}\epsilon^2}{ \tilde{D}} ,\quad
\rho^{s}_{14} = \frac{i\epsilon\kappa^{2}\left(\gamma_{0}+2i\omega\right)}{\tilde{D}} ,\nonumber \\
\rho^{s}_{23} &= \frac{i\epsilon^2}{2\tilde{D}} \left[\kappa\left(\gamma_{A} -\gamma_{B}\right) + i\gamma_{0}\sqrt{\gamma_{A}\gamma_{B}}\,\right] ,\end{aligned}$$ with $$\begin{aligned}
\label{25a}
\tilde{D} &= \kappa^{2}\left(\gamma_{0}^{2}+4\omega^{2}\right) + \epsilon^{2}\left(\gamma_{0}^{2}+4\kappa^{2}\right) .\end{aligned}$$ There are several important differences between Eq. (\[24a\]) and the result (\[18\]) for independent reservoirs.
First of all, the coherence $\rho_{23}^{s}$ is composed of two parts: the part proportional to $\kappa$ is driven directly by the linear coupling between the modes, while the part proportional to $\gamma$ results from an exchange of the excitation through the coupling of the modes to the same reservoir. This shows that a coherence between two states can be generated even if there is no population difference between the states. This property of the coherence can have an interesting effect on the redistribution of the population between the states. It is easily seen from Eq. (\[24a\]) that in the absence of the linear coupling $(\kappa =0)$ the entire population is redistributed (trapped) in the single excitation states with the populations of the states and the coherence between them given by $$\rho^{s}_{22} = \frac{\gamma_{A}}{2\gamma_{0}} ,\quad
\rho^{s}_{33} = \frac{\gamma_{B}}{2\gamma_{0}} , \quad
\rho^{s}_{23} = -\frac{\sqrt{\gamma_{A}\gamma_{B}}}{2\gamma_{0}} .\label{b25}$$ We may introduce symmetric and antisymmetric combinations of the singly excitation states $$\begin{aligned}
\ket{b} &=& \frac{1}{\sqrt{2\gamma_{0}}}\left(\sqrt{\gamma_{A}}\ket 3 + \sqrt{\gamma_{B}}\ket 2 \right) ,\nonumber\\
\ket{d} &=& \frac{1}{\sqrt{2\gamma_{0}}}\left(\sqrt{\gamma_{B}}\ket 3 - \sqrt{\gamma_{A}}\ket 2 \right) ,\label{b22}\end{aligned}$$ and find using Eq. (\[b25\]) that $\rho_{bb}=0$ and $\rho_{dd}=1$. Clearly, the steady-state of the modes is not a mixed state but a pure entangled state $\ket d$. Thus, despite the interaction with a dissipative reservoir, the system evolves to a pure entangled state rather than the expected mixed state.
In addition, there is no population inversion between the double excitation state $\ket 4$ and the single excitation states $\ket 2$ and $\ket 3$. It is easy to see, Eq. (\[24a\]) for the populations lead to ratios $$\frac{\rho^{s}_{22}}{\rho^{s}_{44}} = 1 + \frac{\gamma_{A}\gamma_{0}}{2\kappa^{2}} ,\quad
\frac{\rho^{s}_{33}}{\rho^{s}_{44}} = 1+ \frac{\gamma_{B}\gamma_{0}}{2\kappa^{2} } ,$$ which are always greater than $1$.
Balanced decay: $\gamma_{A} =\gamma_{B}$ {#sec3b}
----------------------------------------
Let us now discuss the steady-state solutions in the case of balanced decay of the modes, i.e., decay with equal damping rates, $\gamma_{A}=\gamma_{B}$. We will see that this leads to quite different features than those found for unbalanced decays. The most important difference is that it requires to consider separately the steady-state solutions for two regions of $\gamma$: $\gamma < \gamma_{0}$ and $\gamma =\gamma_{0}$. This is because the determinant of the matrix $M$, Eq. (\[e19\]), is equal to zero when $\gamma=\sqrt{\gamma_{A}\gamma_{B}}$ and $\gamma_{A}=\gamma_{B}$.
We first examine the steady-state solution for $\gamma <\gamma_{0}$. $$\begin{aligned}
\label{24p}
\rho^{s}_{11} &= \frac{\epsilon^{2} + \gamma_{0}^{2} + 4\omega^2}{D^{\prime}} ,\quad
\rho^{s}_{22} = \rho^{s}_{33} = \rho^{s}_{44} = \frac{\epsilon^2}{D^{\prime}} ,\nonumber\\
\rho^{s}_{14} &= \frac{i\epsilon(\gamma_{0}+2i\omega)}{D^{\prime}} ,\quad
\rho^{s}_{23} = 0 ,\end{aligned}$$ where $D^{\prime} = 4\epsilon^{2} +\gamma_{0}^{2}+4\omega^{2}$. We see that as long as $\gamma <\gamma_{0}$, the system relaxes to a mixed state which is independent of $\gamma$ and $\kappa$. Moreover, the populations of the singly and doubly excited states are exactly equal. In other words, when measuring the populations of the excited states, all measurement outcomes would occur with equal probability. Since $\rho^{s}_{23}=0$, no entangled states are created between the singly excited states. We can conclude that as long as $\gamma_{A}=\gamma_{B}\equiv \gamma_{0}$ and $\gamma <\gamma_{0}$, there is no difference in the decay of the modes into local reservoirs and into a common reservoir.
The fact that the result (\[24p\]) is independent of $\gamma$ may lead one to conclude that it is also valid in the limit of $\gamma =\gamma_{0}$. But this result is [*not*]{} correct in this limit since Det$[M]=0$ when $\gamma=\sqrt{\gamma_{A}\gamma_{B}}$ and $\gamma_{A}=\gamma_{B}\equiv \gamma_{0}$. In order to find the correct steady-state of the system, we rewrite the equations of motion (\[9a\]) in the basis, $\{\ket{1}, \ket{b}, \ket{d}, \ket{4}\}$ and find that the corresponding equations of motion are $$\begin{aligned}
\dot{\rho}_{dd} &=& 0 ,\nonumber\\
\dot{\rho}_{bb} &=& 2\gamma_{0}\!\left(1\!-\!\rho_{dd}\right) -2\gamma_{0}\rho_{bb} -2\gamma_{0}\rho_{11} ,\nonumber\\
\dot{\rho}_{bd} &=& -\left(\gamma_{0} + 2i\kappa\right)\rho_{bd} ,\nonumber\\
\dot{\rho}_{11} &=& 2\gamma_{0}\rho_{bb} + i\epsilon\left(\rho_{14} - \rho_{41}\right) ,\nonumber\\
\dot{\rho}_{14} &=& -i\epsilon -\left(\gamma_{0}\!-\!2i\omega\right)\!\rho_{14} + i\epsilon\!\left(2\rho_{11}\!+\!\rho_{bb}\!+\!\rho_{dd}\right) .\label{25e}\end{aligned}$$ Since $\dot{\rho}_{dd} = 0$, the state $\ket d$ is totally decoupled from the remaining states and does not evolve in time. In other words, an initial population of the state $\ket d$ will remain constant for all times.
With $\rho_{dd}$ constant, the steady-state solution of Eq. (\[25e\]) is of the form $$\begin{aligned}
\rho_{bd} &=& \rho_{db} = 0 ,\nonumber\\
\rho_{dd} &=& \rho_{dd}(0) ,\nonumber\\
\rho_{bb} &=& \frac{\epsilon^2}{3\epsilon^2+\gamma_{0}^2+4\omega^2}\left[1-\rho_{dd}(0)\right] ,\nonumber\\
\rho_{11} &=& \frac{\epsilon^2+\gamma_{0}^2+4\omega^2}{3\epsilon^2+\gamma_{0}^2+4\omega^2}\left[1-\rho_{dd}(0)\right] ,\nonumber\\
\rho_{14} &=& i\frac{\epsilon(\gamma_{0} +2i\omega)}{3\epsilon^2+\gamma_{0}^2+4\omega^2}\left[1-\rho_{dd}(0)\right] .\label{26a}\end{aligned}$$ In terms of the product state basis, the corresponding solution is $$\begin{aligned}
\rho_{11}^{s} &=& \frac{\epsilon^2+\gamma_{0}^2+4\omega^2}{3\epsilon^2+\gamma_{0}^2+4\omega^2}\left[1-\rho_{dd}(0)\right] ,\nonumber\\
\rho_{22}^{s} &=& \rho_{33}^{s} = \frac{1}{2}\left\{1 -\frac{2\epsilon^2+\gamma_{0}^2+4\omega^2}{3\epsilon^2+\gamma_{0}^2+4\omega^2}
\left[1-\rho_{dd}(0)\right]\right\} ,\nonumber\\
\rho_{44}^{s} &=& \frac{\epsilon^2}{3\epsilon^2+\gamma_{0}^2+4\omega^2}\left[1-\rho_{dd}(0)\right] ,\nonumber\\
\rho_{14}^{s} &=& i\frac{\epsilon(\gamma_{0}+2i\omega)}{3\epsilon^2+\gamma_{0}^2+4\omega^2}\left[1-\rho_{dd}(0)\right] ,\nonumber\\
\rho_{23}^{s} &=& -\frac{1}{2}\left\{1 -\frac{4\epsilon^2+\gamma_{0}^2+4\omega^2}{3\epsilon^2+\gamma_{0}^2+4\omega^2}\left[1-\rho_{dd}(0)\right]\right\} .\label{26b}\end{aligned}$$ We see that the physical consequences of the complete decoupling of the state $\ket d$ from the remaining states is the dependence of the steady-state values of the density matrix elements on initial conditions. Note that the system no longer evolves to a pure state $\it unless$ it is prepared initially in the state $\ket d$. Thus, depending on the way we prepare the system initially, we can realize different situations. Note also the steady-state of the system is independent of $\kappa$. Hence, if $\rho_{dd}(0)=1$, the only steady-state for Eq. (\[25e\]) is $\rho_{dd}=1$ with all other density matrix elements equal to zero. It means that if the system is initially prepared in the state $\ket d$, it will remain in this state for all times, i.e., $\rho_{dd}(t) = \rho_{dd}(0)$.
Distinguishability of the modes {#sec4}
===============================
The presence of the linear and nonlinear couplings between the cavities $A$ and $B$ may lead one to suspect that the modes of the cavities are indistinguishable. In particular, if we assume that only a single excitation is present that the system is in either $\ket{1_{A}}\ket{0_{B}}$ or $\ket{0_{A}}\ket{1_{B}}$ state, then the action of the linear coupling $\kappa$ generates a state which is a linear superposition of the one-photon states. As is well known, the probability of detecting a photon emitted from the superposition state exhibits interference effects. The interference is regarded as a signature of indistinguishability of the states.
Nevertheless, we will demonstrate that the modes can be distinguishable even in the presence of the couplings that “which-path” information is made possible due to the inclusion of the state $\ket{1_{A}}\ket{1_{B}}$ enforced by the energy-time uncertainty principle. However, the “which-way” information can be erased by allowing the cavities to decay with different rates. To show this, we consider electromagnetic fields $\hat{E}_{A}(\vec{r},t)$ and $\hat{E}_{B}(\vec{r},t)$ of the cavities $A$ and $B$ at position $\vec{r}$ at time $t$. Since fields of the cavities are treated as single-mode fields, the negative frequency parts of the fields can be written as $$\begin{aligned}
\hat{E}^{(-)}_{A}(\vec{r},t) &= {\cal E}\hat{a}_{A}e^{i\left(\vec{k}_{A}\cdot \vec{r} - \omega t\right)} ,\nonumber\\
\hat{E}^{(-)}_{B}(\vec{r},t) &= {\cal E}\hat{a}_{B}e^{i\left(\vec{k}_{B}\cdot \vec{r} - \omega t\right)} ,\end{aligned}$$ where $\vec{k}_{A}$ and $\vec{k}_{B}$ are wave vectors of the modes and ${\cal E}$ is a constant amplitude. Then the intensity of the field detected by a photodetector located at $\vec{r}$ at time $t$ is given by $$\begin{aligned}
I(\vec{r},t) &= \alpha \left\langle \left(\hat{E}^{(+)}_{A} + \hat{E}^{(+)}_{B}\right)\left(\hat{E}^{(-)}_{A} + \hat{E}^{(-)}_{B}\right)\right\rangle \nonumber\\
&= \alpha|{\cal E}|^{2}\left\{2\rho_{44} + \rho_{22} + \rho_{33}\right. \nonumber\\
&\left. +\, 2|\rho_{23}|\cos\!\left[\left(\vec{k}_{A}\!-\!\vec{k}_{B}\!\right)\!\cdot\!\vec{r} +{\rm arg}(\phi_{A}\!-\!\phi_{B})\right]\right\} ,\label{b37}\end{aligned}$$ where $\alpha$ is a constant characteristic of the detector, and we have written $$\begin{aligned}
\langle \hat{a}^{\dag}_{A}\hat{a}_{A}\rangle &= \rho_{44} + \rho_{22} ,\quad \langle \hat{a}^{\dag}_{B}\hat{a}_{B}\rangle = \rho_{44} + \rho_{33} ,\nonumber\\
\langle \hat{a}^{\dag}_{A}\hat{a}_{B}\rangle &= \langle \hat{a}^{\dag}_{B}\hat{a}_{A}\rangle^{\ast} = |\rho_{23}|e^{i(\phi_{A}-\phi_{B})} .\end{aligned}$$ We see from Eq. (\[b37\]) that the intensity varies periodically with position only if the coherence $|\rho_{23}|$ is different from zero. From the definition of the first-order visibility and Eq. (\[b37\]), we find that $$\begin{aligned}
{\cal V} = \frac{I_{max} - I_{min}}{I_{max}+I_{min}} = \frac{2|\rho_{23}|}{2\rho_{44} + \rho_{22}+\rho_{33}} ,\label{b39a}\end{aligned}$$ and then by using Eqs. (\[24\]) and (\[24u\]) we find $$\begin{aligned}
{\cal V} = \frac{|\gamma_{d}|\sqrt{4\kappa^{2}(\gamma_{0}^{2}-\gamma^{2})^{2} +(\gamma\gamma_{0}\gamma_{d})^{2}}}{(4\kappa^{2}+\gamma_{0}^{2})(\gamma_{0}^{2}-\gamma^{2})} ,\label{b39}\end{aligned}$$ where $\gamma_{d}=(\gamma_{A}-\gamma_{B})/2$. This simple result for the first-order visibility is strongly dependent on whether the modes are damped with equal $(\gamma_{A}=\gamma_{B})$ or unequal $(\gamma_{A}\neq \gamma_{B})$ rates. If the modes are damped with equal rates, $\gamma_{d}=0$, and then the interference pattern vanishes. Hence, independent of the presence of the couplings, the modes are distinguishable when are damped with the same rates. The reason of the distinguishability of the modes is the inclusion of the state $\ket{1_{A}}\ket{1_{B}}$ into the dynamics of the system enforced by the energy-time uncertainty principle. In physical terms, we may attribute this to the fact that the modes, each occupied by a photon, are resolved at the detector. For example, if a photon is detected in mode $A$ it must come from this mode since two occupied modes cannot exchange photons. An alternative explanation is that two decay channels from the state $\ket{1_{A}}\ket{1_{B}}$ exist: $\ket{1_{A}}\ket{1_{B}}\rightarrow \ket{0_{A}}\ket{1_{B}}$ and $\ket{1_{A}}\ket{1_{B}}\rightarrow \ket{1_{A}}\ket{0_{B}}$. Then, one can distinguish from which channel the detected photon came by measuring the population of the states $\ket{1_{A}}\ket{0_{B}}$ and $\ket{0_{A}}\ket{1_{B}}$.
One can notice from Eq. (\[b39\]) that the visibility is independent of $\epsilon$. Note also that for the visibility to be nonzero it is required that not only $\gamma_{d}\neq 0$ but also $\kappa\neq 0$ and/or $\gamma\neq 0$. Thus, in the case of the collective decay $(\gamma\neq 0)$ the visibility can be different from zero even if $\kappa =0$. Equation (\[b39\]) also shows that the visibility is maximal when either $\gamma_{A}\gg\gamma_{B}$ or $\gamma_{B}\gg\gamma_{A}$. Consequently, we can make the modes indistinguishable by erasing one of the photons through a fast spontaneous emission of one of the two modes. To put it another way, when one of the photons is erased by spontaneous emission then the remaining photon can produce the interference since in the presence of the coupling $\kappa$ it is impossible to determine from which mode the detected photon came. This restores the first-order interference which is a manifestation of the intrinsic indistinguishability of two possible paths of the detected photon.
It is worth emphasizing that there is an upper limit of $50\%$ for the first-order visibility ${\cal V}$ when the modes independently decay to the reservoirs. On the other hand, when the modes decay collectively the visibility can be close to unity and can be independent of $\kappa$. To show this, we introduce ratios $R\equiv \kappa/\gamma_{0}$ and $u\equiv \gamma_{d}/\gamma_{0}$, and then find that Eq. (\[b39\]) yields $$\begin{aligned}
{\cal V}\equiv {\cal V}_{s} = |u|\frac{2R}{4R^{2}+1} \label{b41}\end{aligned}$$ for the decay to separate reservoirs $(\gamma=0)$, and $$\begin{aligned}
{\cal V}\equiv {\cal V}_{c} = \frac{\sqrt{4R^{2}u^{2} + 1 -u^{2}}}{4R^{2}+1} \label{b42}\end{aligned}$$ for the decay to a common reservoir with $\gamma = \sqrt{\gamma_{A}\gamma_{B}}$.
Since $|u|\leq 1$, the visibility ${\cal V}_{s}$ can be no larger than $50\%$, and it is required $R\neq 0$ for ${\cal V}_{s}$ to be different from zero. It follows from Eq. (\[b41\]) that ${\cal V}_{s}$ has its largest value of ${\cal V}_{s}=1/2$ when $R=1/2$ and $|u|=1$. Clearly, there is an upper limit of $50\%$ for the visibility when the modes decay to separate reservoirs. In contrast, the visibility ${\cal V}_{c}$ can exceed $50\%$ and can approach $100\%$ even when $R=0\, (\kappa =0)$. It can happen when the linear coupling $\kappa$ is weak, $R\ll 1$, or even if it is absent, $R=0$. In this limit, ${\cal V}\approx\sqrt{1-u^{2}}$, which can be close to $1$ when $u\approx 0\, (\gamma_{A}\approx \gamma_{B})$. It should be noted that in this case the system is in the pure entangled state $\ket d$, Eq. (\[b22\]), which is not the maximally entangled state. We stress that it is impossible to put $u=1$ in Eq. (\[b42\]), at which the visibility would correspond to that of a maximally entangled state since we cannot assume $\gamma_{A}=\gamma_{B}$ in the expression (\[b42\]). The expression for the visibility given in Eq. (\[b39\]) is valid only for $\gamma_{A}\neq \gamma_{B}$.
To consider the limit $\gamma=\sqrt{\gamma_{A}\gamma_{B}}$ with $\gamma_{A}=\gamma_{B}$ in the evaluation of the visibility ${\cal V}$ given by Eq. (\[b39a\]), we must apply the steady-state solutions given in Eq. (\[26b\]). Thus, substituting Eq. (\[26b\]) into Eq. (\[b39a\]) we get $$\begin{aligned}
{\cal V} = \frac{|\epsilon^{2}-(4\epsilon^{2}+\gamma_{0}^{2}+4\omega^{2})\rho_{dd}(0)|}{3\epsilon^{2}+(\gamma_{0}^{2}+4\omega^{2})\rho_{dd}(0)} .\label{b40}\end{aligned}$$ In comparison with Eq. (\[b39\]), we see that the dependence of the visibility on $\kappa$ is absent. The most obvious difference is the dependence on the initial state $\rho_{dd}(0)$. For $\rho_{dd}(0)=0$, the visibility ${\cal V}=1/3$ irrespective of $\epsilon$ and $\gamma_{0}$. In the other extreme when $\rho_{dd}(0)=1$, the visibility reaches its maximal value of ${\cal V}=1$ also irrespective of $\epsilon$ and $\gamma_{0}$. This behavior can be explained as a result of the transition of the system from a mixed state involving three states $\ket 1, \ket b, \ket 4$ to a pure state involving the state $\ket d$, which is a maximally entangled state. These results suggest that the interference can be used to detect one-photon entangled states in the system.
Finally, we would like to comment about the connection between indistinguishability and the presence of two significantly different decay rates in the system. Although the modes $A$ and $B$ decay with the same rate it must not be thought that in this case the interference pattern is always absent. If the modes decay collectively there are two superposition states in the system $\ket b$ and $\ket d$ which decay with significantly different rates. According to Eq. (\[25e\]), the state $\ket b$ decays with a rate $2\gamma_{0}$ whereas the state $\ket d$ is metastable. Clearly, the decay rates of the superposition states are significantly different even when $\gamma_{A}=\gamma_{B}$. Therefore, we may conclude that the one-photon interference results from the presence of unequal decay rates in the system.
Entanglement between the modes {#sec5}
==============================
The strong dependence of the steady-state of the system on whether $\gamma_{A}\neq \gamma_{B}$ or $\gamma_{A}=\gamma_{B}$ may have a significant effect on entanglement between the modes. The question of the creation of entanglement between the modes is addressed by considering the concurrence, a measure of entanglement between two systems [@W98]. Since the evolution of the system is described by the density operator whose the matrix representation in the basis (\[9\]) is of the $X$ form, the concurrence can be calculated analytically and can be expressed as $$\label{10}
\mathcal{C}(t) = \max\left\{0,C_{1}(t),C_{2}(t)\right\} ,$$ where $$\begin{aligned}
C_{1}(t) &=& 2\left[|\rho_{23}(t)|-\sqrt{\rho_{11}(t)\rho_{44}(t)}\right]\label{11} ,\label{11}\\
C_{2}(t) &=& 2\left[|\rho_{14}(t)|-\sqrt{\rho_{22}(t)\rho_{33}(t)}\right] .\label{12}\end{aligned}$$ There are two quantities which determine a nonzero concurrence. Obviously, either $C_{1}(t)>0$ or $C_{2}(t)>0$ is required for the modes to be entangled. The quantity $C_{1}(t)$ determines an entanglement created by the coherence $\rho_{23}(t)$, whereas $C_{2}(t)$ determines an entanglement created by the coherence $\rho_{14}(t)$. It follows that $C_{1}(t)>0$ corresponds to an entangled state involving the one-photon states while $C_{2}(t)>0$ corresponds to an entangled state involving the zero and two-photon states. We have already seen that the coherence $\rho_{23}(t)$ can be created by the linear coupling $\kappa$ and also by the collective damping $\gamma$, while the coherence $\rho_{14}(t)$ can be created by the nonlinear coupling $\epsilon$.
Although there is no direct connection between the one- and two-photon coherences, we find that in the system considered here the modes exhibit an interesting coherence effect [@ow90; @mg96; @lm98]. Namely, the modes can be [*anticoherent*]{} that the one-photon coherence $\rho_{23}$ vanishes and at the same time $\rho_{14}$ is maximal. This is shown in Fig. \[fig2\] where we plot the variation of the absolute values of the coherences $|\rho_{23}|$ and $|\rho_{14}|$ with $\gamma_{d}$. When $\gamma_{d}=0$, the coherence $|\rho_{23}|$ vanishes whereas $|\rho_{14}|$ has a maximum. Thus, for $\gamma_{d}=0$ the modes are completely anticoherent. However, as soon as $\gamma_{d}\neq 0$, the coherence $\rho_{23}$ is different from zero. In this case, the modes are regarded as partially mutually coherent. It is interesting to note from Fig. \[fig2\] that an increase of $|\rho_{23}|$ results in a decrease of $|\rho_{14}|$ and vice versa. This “anticoherence” can be reflected in entanglement that an increase of $C_{1}(t)$ leads to a decrease of $C_{2}(t)$.
The case of independent decay, $\gamma =0$
------------------------------------------
Let us turn to detailed analysis of the concurrence for the case of independent unbalanced decays, $\gamma =0$ and $\gamma_{A}\neq \gamma_{B}$. Using the steady-state solution (\[18\]), the concurrence can be easily determined and is given by $$\label{20}
\mathcal{C}^{s} = \max\left(0,C_{1}^{s},C_{2}^{s}\right) ,$$ where $$\begin{aligned}
\label{21}
C_{1}^{s} = \frac{2\epsilon}{D_{0}}\!\left[4|\gamma_{d}|\kappa\epsilon -(4\kappa^{2}\!+\!\gamma_{0}^{2}-\gamma_{d}^{2})\sqrt{4\omega^2 +\epsilon^{2}+\gamma_{0}^2}\right] ,\end{aligned}$$ and $$\begin{aligned}
\label{22}
C_{2}^{s} &= \frac{2\epsilon}{D_{0}}\Bigg[(4\kappa^2+\gamma_A\gamma_B)\sqrt{4\omega^2+\gamma_{0}^2} \nonumber\\
&-\epsilon\sqrt{(4\kappa^2+\gamma_A^2)(4\kappa^2+\gamma_B^2)}\Bigg] .\end{aligned}$$ It is seen from Eqs. (\[21\]) and (\[22\]) that a nonzero $\epsilon$ is necessary for both quantities $C^{s}_{1}$ and $C^{s}_{2}$ to be nonzero. However, a nonzero $\kappa$ is needed for $C^{s}_{1}$ to be positive, while $C^{s}_{2}$ can be positive even for $\kappa =0$. Moreover, the damping rates should be different $(\gamma_{A}\neq \gamma_{B})$ for $C^{s}_{1}$ to be positive. This means that in the case of an unbalanced damping rates, entanglement between the modes can be determined by two criteria. These two criteria do not overlap that they determine two separate ranges of the parameters at which entanglement occurs.
**(a)**\
**(b)** ![(Color online) Stationary concurrence $\mathcal{C}^{s}$ as a function of the coupling strengths $\kappa$ and $\epsilon$ when the cavity modes decay to separate reservoirs, $\gamma =0$. The damping rate $\gamma_B$ is fixed at $\gamma_B = 0.01\omega$ and (a) $\gamma_A =0.01\omega$, (b) $\gamma_A =0.1\omega$ and (c) $\gamma_A =0.2\omega$. The red surface represents a contribution of $C_{1}^{s}$, while the green (light gray) part represents the contribution of $C_{2}^{s}$ to the entanglement created between the modes.[]{data-label="fig3"}](Fig2a "fig:"){width=".72\columnwidth"}\
**(c)** ![(Color online) Stationary concurrence $\mathcal{C}^{s}$ as a function of the coupling strengths $\kappa$ and $\epsilon$ when the cavity modes decay to separate reservoirs, $\gamma =0$. The damping rate $\gamma_B$ is fixed at $\gamma_B = 0.01\omega$ and (a) $\gamma_A =0.01\omega$, (b) $\gamma_A =0.1\omega$ and (c) $\gamma_A =0.2\omega$. The red surface represents a contribution of $C_{1}^{s}$, while the green (light gray) part represents the contribution of $C_{2}^{s}$ to the entanglement created between the modes.[]{data-label="fig3"}](Fig2b "fig:"){width=".72\columnwidth"}\
![(Color online) Stationary concurrence $\mathcal{C}^{s}$ as a function of the coupling strengths $\kappa$ and $\epsilon$ when the cavity modes decay to separate reservoirs, $\gamma =0$. The damping rate $\gamma_B$ is fixed at $\gamma_B = 0.01\omega$ and (a) $\gamma_A =0.01\omega$, (b) $\gamma_A =0.1\omega$ and (c) $\gamma_A =0.2\omega$. The red surface represents a contribution of $C_{1}^{s}$, while the green (light gray) part represents the contribution of $C_{2}^{s}$ to the entanglement created between the modes.[]{data-label="fig3"}](Fig2c "fig:"){width=".72\columnwidth"}
The concurrence given by Eq. (\[20\]) is plotted in Fig. \[fig3\] as a function of the coupling strengths $\kappa$ and $\epsilon$. For the balanced decay, Fig. \[fig3\](a), the entanglement is independent of $\kappa$ and occurs in a range of $\epsilon<\sqrt{\gamma_{0}^2+4\omega^2}$. For the unbalanced decay, Figs. \[fig3\](b) and (c), there are two separate ranges of the parameters where entanglement occurs. As discussed above, these two ranges are determined by $C_{1}^{s}>0$ and $C_{2}^{s}>0$, respectively. We see a gap between the $C_{2}^{s}>0$ and $C_{1}^{s}>0$ structures that entanglement created by the one- and two-photon coherences lies in separate ranges of the parameters. Moreover, the magnitude of $C_{2}^{s}$ is reduced in the range of $\kappa$ where $C_{1}^{s}$ emerges. Evidently, with an increasing asymmetry between the damping rates the entanglement shifts from $C_{2}^{s}$ to $C_{1}^{s}$. Thus, the creation of entanglement by the coherence $\rho_{23}$ occurs in expense of the entanglement created by the coherence $\rho_{14}$. One can also notice from Fig. \[fig3\] that the entanglement as determined by $C_{2}^{s}$ occurs in the parameters range $\kappa/\omega \ll1$ and $\epsilon/\omega \approx 1$. On the other hand, the entanglement as determined by $C_{1}^{s}$ occurs in the deep strong coupling regime of $\epsilon/\omega >1$.
It is interesting to examine which of the two quantities, $C_{1}^{s}$ or $C_{2}^{s}$, produces the largest degree of entanglement and whether the maximum corresponds to the case of distinguishable or indistinguishable modes. A quick inspection of Eq. (\[22\]) shows that $C_{2}^{s}$ achieves its maximum value at $\gamma_{d}=0$ and the corresponding maximum value is $$C_{2}^{s} = \frac{2\epsilon\left(\sqrt{4\omega^2+\gamma_{0}^2} -\epsilon\right)}{4\omega^2+4\epsilon^{2}+\gamma_{0}^2} .$$ Viewed as a function of $\epsilon$, $C_{2}^{s}$ is maximal at $\epsilon = \sqrt{4\omega^2+\gamma_{0}^2}/4$, in which case $C_{2}^{s}=3/10$.
An inspection of Eq. (\[21\]) reveals that the maximum value of $C_{1}^{s}$ occurs for $|\gamma_{d}|=\gamma_{0}$, corresponding to either $\gamma_{A}\gg\gamma_{B}$ or $\gamma_{B}\gg\gamma_{A}$, and when $\epsilon,\gamma_{0}\gg \kappa$. In these limits, $C_{1}^{s}$ is small, $C_{1}^{s} = 2\kappa/\gamma_{0}\ll 1$. This means that the largest degree of entanglement produced in the system is that determined by $C_{2}^{s}$. Since for $|\gamma_{d}|= 0$, at which $C_{2}^{s}$ attains its maximum value, the visibility ${\cal V}_{s}=0$, we conclude the largest degree of entanglement is achieved when the modes are completely distinguishable.
The case of collective decay, $\gamma\neq 0$
--------------------------------------------
Let now turn to the case of the decay of the modes to a common reservoir with $\gamma=\sqrt{\gamma_{A}\gamma_{B}}$. The corresponding steady-state solution for the density matrix elements are given by Eq. (\[24a\]). When applying Eq. (\[24a\]) to the concurrence, we find the following expressions for the quantities $C_{1}^{s}$ and $C_{2}^{s}$: $$\begin{aligned}
\label{25b}
C_{1}^{s} &=& \frac{2\epsilon}{\tilde{D}}\Bigg[\frac{1}{2}\epsilon\sqrt{\kappa^{2}(\gamma_{A}-\gamma_{B})^{2} +\gamma_{A}\gamma_{B}\gamma_{0}^{2}} \nonumber\\
&&-\kappa^{2}\sqrt{\epsilon^2+4\omega^2+\gamma_{0}^2}\Bigg] ,\end{aligned}$$ and $$\begin{aligned}
\label{25c}
C_{2}^{s} &= \frac{2\epsilon}{\tilde{D}}\Bigg[\kappa^{2}\sqrt{\gamma_{0}^{2} + 4\omega^2} \nonumber\\
&-\frac{1}{2}\epsilon\sqrt{(2\kappa^2+\gamma_{A}\gamma_{0})(2\kappa^2+\gamma_{B}\gamma_{0})}\Bigg] .\end{aligned}$$ We see that in the case of damping of the modes to a common reservoir, the role of $\kappa$ in the creation of entanglement reversed, a nonzero $\kappa$ is now required for $C_{2}^{s}$ to be positive, whereas $C_{1}^{s}$ can be positive even for $\kappa =0$.
If we set $\kappa=0$ in Eqs. (\[25b\]) and (\[25c\]), we find $C_{2}^{s}<0$ and $C_{1}^{s}=\sqrt{\gamma_{A}\gamma_{B}}/\gamma_{0}$. In this case, the concurrence is insensitive to $\epsilon$. Therefore, the modes can be entangled for all values of $\epsilon$. The reason is that now the mechanism responsible for the generation of the entanglement is in the trapping of the population in the state $\ket d$. Note that the concurrence depends only on the damping rates and therefore can be close to the optimum value of unity, which can be achieved for $\gamma_{A}\approx \gamma_{B}$.
**(a)**\
**(b)**![(Color online) Stationary concurrence $\mathcal{C}^{s}$ as a function of the coupling strengths $\kappa$ and $\epsilon$ when the cavity modes decay to a common reservoir. The plots are for $\gamma_A =0.2\omega$, $\gamma_B = 0.01\omega$ and different $\gamma$: (a) $\gamma = 0$, (b) $\gamma = \frac{1}{2}\sqrt{\gamma_{A}\gamma_{B}}$ and (c) $\gamma = \sqrt{\gamma_{A}\gamma_{B}}$. The red surface represents the contribution of $C_{1}^{s}$, while the right green (light gray) surface represents the contribution of $C_{2}^{s}$.[]{data-label="fig4"}](Fig3a "fig:"){width=".72\columnwidth"}\
**(c)**![(Color online) Stationary concurrence $\mathcal{C}^{s}$ as a function of the coupling strengths $\kappa$ and $\epsilon$ when the cavity modes decay to a common reservoir. The plots are for $\gamma_A =0.2\omega$, $\gamma_B = 0.01\omega$ and different $\gamma$: (a) $\gamma = 0$, (b) $\gamma = \frac{1}{2}\sqrt{\gamma_{A}\gamma_{B}}$ and (c) $\gamma = \sqrt{\gamma_{A}\gamma_{B}}$. The red surface represents the contribution of $C_{1}^{s}$, while the right green (light gray) surface represents the contribution of $C_{2}^{s}$.[]{data-label="fig4"}](Fig3b "fig:"){width=".72\columnwidth"}\
![(Color online) Stationary concurrence $\mathcal{C}^{s}$ as a function of the coupling strengths $\kappa$ and $\epsilon$ when the cavity modes decay to a common reservoir. The plots are for $\gamma_A =0.2\omega$, $\gamma_B = 0.01\omega$ and different $\gamma$: (a) $\gamma = 0$, (b) $\gamma = \frac{1}{2}\sqrt{\gamma_{A}\gamma_{B}}$ and (c) $\gamma = \sqrt{\gamma_{A}\gamma_{B}}$. The red surface represents the contribution of $C_{1}^{s}$, while the right green (light gray) surface represents the contribution of $C_{2}^{s}$.[]{data-label="fig4"}](Fig3c "fig:"){width=".72\columnwidth"}
Figure \[fig4\] shows the effect of the collective damping $\gamma$ on the concurrence of the modes. We see that the collective decay results in an entanglement which is associated mostly with the quantity $C_{1}^{s}$. Hence, it is mostly associated with the presence of the coherence $\rho_{23}$. Moreover, the concurrence although the most positive in the strong coupling regime of the antiresonant terms, it seen to be positive in the weak coupling regime of both resonant $(\kappa/\omega \ll 1)$ and antiresonant $(\epsilon/\omega \ll 1)$ terms.
Comparing $C_{1}^{s}$ with the visibility, Eq. (\[b42\]), we easily find that in the case of $\kappa =0$, where the system evolves to the pure state $\ket d$, $C_{1}^{s}$ is equal to ${\cal V}_{c}$. Thus, in the case of pure states there is a direct connection between indistinguishability and entanglement [@jb03; @sm10]. Otherwise, when the system is in a mixed state, one can observe entanglement with the complete distinguishability of the modes and vice versa.
Finally, we consider the case of the balanced decay of the modes $(\gamma_{A}=\gamma_{B}\equiv \gamma_{0})$ with the collective damping rate $\gamma =\gamma_{0}$. We have shown in Sec. \[sec3b\] that in this special case the steady-state values of the density matrix elements depend on initial conditions. Moreover, it is independent of $\kappa$. Therefore, it is straightforward, using the results given in Eq. (\[26b\]), to show that entanglement between the modes is related to the initial state. Specifically, if initially the system is prepared in the maximally entangled state $\ket d$, it will remain in this state for all times. If the initial state is different for $\ket d$ then the system can decay to an entangled state created by $\epsilon$. This is illustrated in Fig. \[fig5\], where we plot the concurrence as a function of $\epsilon$ and the initial population of the state $\ket d$.
![(Color online) Stationary concurrence in terms of $\epsilon$ and the initial condition $\rho_{dd}(0)$ for $\gamma_a=\gamma_B=\gamma=0.01\omega$.[]{data-label="fig5"}](Fig4f){width="0.8\columnwidth"}
Similarly to the case of the unbalanced decay presented in Fig. \[fig3\], the entanglement is mostly associated with the presence of the coherence $\rho_{23}$. Only for initial states at which $\rho_{dd}(0)\approx 0$, the entanglement created is associated with the coherence $\rho_{14}$. Moreover, the entanglement, which is independent of the coupling strength $\kappa$ of the resonant terms, is present in all ranges of the coupling strength $\epsilon$ of the antiresonant terms.
Comparing the concurrence with the visibility, we see that in the case of the collective decay of the modes, the maximum entanglement is achieved when the modes are indistinguishable, and the maximum possible entanglement of $C^{s}=1$ is achieved when the modes are completely indistinguishable, ${\cal V}=1$. Therefore, we may conclude that in the case the collective decay of the modes, more entanglement is achieved with more indistinguishability and the maximum possible entanglement is achieved with completely indistinguishable modes.
Second-order correlations
-------------------------
We have seen that the creation of entanglement is determined by two criteria $C_{1}^{s}$ and $C_{2}^{s}$ which do not overlap. In other words, these two criteria determine two distinct ranges of the parameters at which entanglement occurs. We may relate these criteria to the normalized second-order photon-photon correlation function $g^{(2)}(0)$ which is directly measurable in coincidence counting schemes and provides a test of whether the photons are correlated (bunched) or anti-correlated (antibunched) [@zz16]. For this purpose, we consider the normalized second-order correlation function, which for the two modes $A$ and $B$ is [@pa82] $$\begin{aligned}
g^{(2)}(0) \equiv \frac{\langle\hat{a}_{A}^{\dag}\hat{a}_{B}^{\dag}\hat{a}_{A}\hat{a}_{B}\rangle}{\langle\hat{a}_{A}^{\dag}\hat{a}_{A}\rangle\langle \hat{a}_{B}^{\dag}\hat{a}_{B}\rangle} =\frac{\rho_{44}}{(\rho_{44}+\rho_{22})(\rho_{44}+\rho_{33})} .\end{aligned}$$ If we compare this expression with the expressions for $C_{1}^{s}$ and $C_{2}^{s}$, Eqs. (\[11\]) and (\[12\]), we find that there is no direct connection here between entanglement and the second-order photon correlations. The quantities $C_{1}^{s}$ and $C_{2}^{s}$ are given in terms of the coherences and populations, while $g^{(2)}(0)$ is given entirely in term of the populations. Nevertheless, we can demonstrate that entanglement determined by $C_{1}^{s}>0$ occurs in the range of the parameters at which $g^{(2)}(0)<1$, whereas entanglement determined by $C_{2}^{s}>0$ occurs in the range at which $g^{(2)}(0)>1$.
Let us examine the relations for the case of independent reservoirs $(\gamma =0)$. To do this, let us first assume that $\gamma_{A}=\gamma_{B}$. Then using Eqs. (\[18\]) and (\[19\]) we readily find $$\begin{aligned}
g^{(2)}(0) = 1 + \frac{4\omega^{2}+\gamma_{0}^{2}}{4\epsilon^{2}} ,\end{aligned}$$ from which it is clear that $g^{(2)}(0)$ is always greater than one. This means that emitted photons exhibit bunching effect when $\gamma_{A}=\gamma_{B}$.
On the other hand, when $\gamma_{A}\neq \gamma_{B}$ and in the limit of $\kappa< \epsilon$, it can be shown that $g^{(2)}(0)$ is of the form $$\begin{aligned}
g^{(2)}(0)\approx 1 - \frac{4\kappa^{2}\gamma_{d}^{2}}{(4\kappa^{2}+\gamma_{0}^{2})^{2}-\gamma_{0}^{2}\gamma_{d}^{2}} .\end{aligned}$$ Here we see that $g^{(2)}(0)$ is always less than one. It follows that for $\gamma_{A}\neq \gamma_{B}$ and $\kappa <\epsilon$, the emitted photons exhibit antibunching effect.
**(a)**\
**(b)**![(Color online) Stationary second-order correlation function $g^{(2)}(0)$ plotted as a function of the coupling strengths $\kappa$ and $\epsilon$ for the same parameters as in Fig. \[fig3\].[]{data-label="fig6"}](Fig2d "fig:"){width=".72\columnwidth"}\
**(c)**![(Color online) Stationary second-order correlation function $g^{(2)}(0)$ plotted as a function of the coupling strengths $\kappa$ and $\epsilon$ for the same parameters as in Fig. \[fig3\].[]{data-label="fig6"}](Fig2e "fig:"){width=".72\columnwidth"}\
![(Color online) Stationary second-order correlation function $g^{(2)}(0)$ plotted as a function of the coupling strengths $\kappa$ and $\epsilon$ for the same parameters as in Fig. \[fig3\].[]{data-label="fig6"}](Fig2f "fig:"){width=".72\columnwidth"}
Figure \[fig6\] shows the variation of $g^{(2)}(0)$ with $\kappa$ and $\epsilon$ for the same parameters as in Fig. \[fig3\], where we illustrated the variation of ${\cal C}^{s}$ with $\kappa$ and $\epsilon$. It is seen that $g^{(2)}(0)$ decreases with an increasing $\epsilon$ and for $\gamma_{A}=\gamma_{B}$ attains a minimum value of $g^{(2)}(0)=1$ independent of $\kappa$. For $\gamma_{A}\neq \gamma_{B}$, there is a range of $\kappa\, (\kappa\ll\epsilon)$ at which $g^{(2)}(0)<1$. Comparing Fig. \[fig6\] with Fig. \[fig3\], we see that the positive values of $C_{2}^{s}$ lie within the parameters ranges permissible for photon bunching, $g^{(2)}(0)>1$, and the positive values of $C_{1}^{s}$ lie within the permissible ranges for photon antibunching, $g^{(2)}(0)<1$.
Summarizing, there is a connections between entanglement and photon statistics that the entanglement determined by $C_{1}^{s}>0$ is related to photon antibunching whereas the entanglement determined by $C_{2}^{s}>0$ is related to photon bunching effect.
Summary and conclusions {#sec6}
=======================
We have investigated two concepts of quantum mechanics, indistinguishability and entanglement, in a system composed of two strongly coupled bosonic modes. We have found that the use of both resonant (RWA) and antiresonant (non-RWA) terms in the interaction between the modes forms a natural link of the two concepts with the energy-time uncertainty principle. The inclusion of the antiresonant terms requires to work in an ultra-strong coupling regime and at a very short interaction time. We have found nonzero population distribution and coherences between the low energy states and have interpreted the distribution as the result of the uncertainty in energy which over a very short interaction time is of the order of the one-photon energy.
The analysis of the steady-state of the system has demonstrated the importance of the dissipation in the redistribution of the population and in the creation of coherences between the low energy states. To explore the role of the dissipation, we have calculated the steady-state of the system when the modes decay either independently or collectively. We have found that when the modes decay independently, the distinguishability and entanglement of the modes depend strongly on whether the modes decay with equal or unequal rates. In particular, when the modes decay with equal rates, entanglement with the complete distinguishability of the modes can be observed; the entangled cavity modes behave as mutually incoherent. When the modes decay with unequal rates, a single-photon coherence is induced between the modes resulting in indistinguishability, single-photon interference between the modes. We have found an upper bound of the single-photon visibility that the visibility cannot exceed $50\%$ when the modes decay independently. When the modes decay with equal rates we show that “which-path” information is made possible and the visibility in single-photon interference vanishes.
When the modes decay collectively, the single-photon coherence is created even if the modes decay with equal rates. The additional pathway induced by the collective decay rate results in nearly perfect visibility of the interference pattern even in the absence of the resonant coupling between the modes. We have shown that the collective damping creates superposition states and in the absence of the resonant coupling the steady-state is a pure entangled state rather than the expected mixed state. This can result in entanglement with the complete indistinguishability of the modes, that the modes entangled through the collective decay behave as mutually coherent. In addition, we have found that the collective damping with the maximal rate can lead to the steady-state values for the density matrix elements which depend on their initial values. This requires that the modes decay with equal rates. Then, depending on the initial state, the modes can be found in a mixed or in a pure maximally entangled state.
[100]{}
D. V. Strekalov, A. Turlapov, A. Kumarakrishnan, and T. Sleator, Phys. Rev A [**66**]{}, 023601 (2002).
C. L. Hung, S. M. Meenehan, D. E. Chang, O. Painter, and H. J. Kimble, New J. Phys. [**15**]{}, 083026 (2013).
A. Goban, C. L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, Phys. Rev. Lett. [**115**]{}, 063601 (2015).
M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, Nature Phys. [**2**]{}, 849 (2006).
D. G. Angelakis, M. F. Santos, and S. Bose, Phys. Rev. A **76**, 031805(R) (2007).
L Zhou, H. Dong, Y. X. Liu, C. P. Sun, and F. Nori, Phys. Rev. A [**78**]{}, 063827 (2008).
A. Tomadin and R. Fazio, J. Opt. Soc. Am. **27**, A130 (2010).
E. Kyoseva, A. Beige, and L. C. Kwek, New J. Phys. [**14**]{}, 023023 (2012).
L.-H. Sun and G. X. Li, Phys. Rev. A [**85**]{}, 065801 (2012).
I Mirza, S van Enk and H J Kimble, J. Opt. Soc. Am. B [**30**]{}(10), 2640 (2013).
A. F. van Loo, A. Fedorov, K. Lalumiere, B. C. Sanders, A. Blais, and A. Wallraff, Science **342**, 1494 (2013).
S. Filipp, M. Göppl, J. M. Fink, M. Baur, R. Bianchetti, L. Steffen, and A. Wallraff, Phys. Rev. A [**83**]{}, 063827 (2011).
J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, Nature Commun. [**5**]{}, 5186 (2014).
A. A. Houck, H. E. Tureci, and J. Koch, Nature Phys. **8**, 292 (2012).
F. Badshah, S. Qamar, and M. Paternostro, Phys. Rev. A [**90**]{}, 033813 (2014).
H. Zoubi, M. Orenstien, and A. Ron, Phys. Rev. A, **62**, 033801 (2000).
M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, Laser and Photon. Rev. **2**, 527 (2008).
A. Metelmann and A. A. Clerk, Phys. Rev. X [**5**]{}, 021025 (2015).
J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, Phys. Rev. Lett. [**78**]{}, 3221 (1997).
T. Pellizzari, Phys. Rev. Lett. [**79**]{}, 5242 (1997).
S. J. van Enk, H. J. Kimble, J. I. Cirac, and P. Zoller, Phys. Rev. A [**59**]{}, 2659 (1999).
S. Mancini and S. Bose, Phys. Rev. [**70**]{}, 022307 (2004).
E. K. Irish, C. D. Ogden, and M. S. Kim, Phys. Rev. A [**77**]{}, 033801(2008).
J. Busch, E. S. Kyoseva, M. Trupke, and A. Beige, Phys. Rev. A [**78**]{}, 040301(R) (2008).
M. Yonac and J. H. Eberly, Phys. Rev. A [**82**]{}, 022321 (2010).
K. Zhang and Z.-Y. Li, Phys. Rev. A [**81**]{}, 033843 (2010).
J. Busch and A. Beige, Phys. Rev. A [**82**]{}, 053824 (2010).
W. Shu and T. Yu, J. Phys. B: At. Mol. Opt. Phys. [**44**]{}, 225501 (2011).
P. Xue, Z. Ficek, and B. C. Sanders, Phys. Rev. A [**86**]{}, 043826 (2012).
S. Bougouffa and Z. Ficek, Phys. Rev. A **88**, 022317 (2013).
A. Serafini, S. Mancini, and S. Bose, Phys. Rev. Lett. [**96**]{}, 010503 (2006).
M. Benyoucef, S. Kiravittaya, Y. F. Mei, A. Rastelli, and O. G. Schmidt, Phys. Rev. B [**77**]{}, 035108 (2008).
B. Kraus and J. I. Cirac, Phys. Rev. Lett. [**92**]{}, 013602 (2004).
R. Tahira, M. Ikram, S. Bougouffa, and M. S. Zubairy, J. Phys. B: At. Mol. Opt. Phys. **43**, 035502 (2010).
K. Aloufi, S. Bougouffa, and Z. Ficek, Phys. Scr. [**90**]{}, 074020 (2015).
L. Allen and J. H. Eberly, [*Optical Resonance and Two-Level Atoms*]{}, (Wiley, New York, 1975).
F. Bloch and A. Siegert, Phys. Rev. [**57**]{}, 522 (1940).
C. J. Wei, S. A. Holmstrom, M. J. Sellars, N. B. Manson, and Z. Ficek, J. Phys. B: At. Mol. Opt. Phys. [**30**]{}, 2735 (1997).
D. E. Browne and C. H. Keitel, J. Mod. Opt. [**47**]{}, 1307 (2000).
H. T. Ng and K. Burnett, New J. Phys. [**10**]{}, 123014 (2008).
J. Jing, Z.-G. Lü, and Z. Ficek, Phys. Rev. A [**79**]{}, 044305 (2009).
D. Zueco, G. M. Reuther, S. Kohler, and P. Hänggi, Phys. Rev. A [**80**]{}, 033846 (2009).
Q.-H. Chen, Y. Yang, T. Liu, and K.-L. Wang, Phys. Rev. A [**82**]{}, 052306 (2010).
Z. Lü and H. Zheng, Phys. Rev. A [**86**]{}, 023831 (2012).
Y. Yan, Z. Lü, and H. Zheng, Phys. Rev. A [**88**]{}, 053821 (2013).
J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, Phys. Rev. Lett. [**105**]{}, 263603 (2010).
D. Ballester, G. Romero, J. J. García-Ripoll, F. Deppe, and E. Solano, Phys. Rev. X [**2**]{}, 021007 (2012).
S. De Liberato, Phys. Rev. Lett. [**112**]{}, 016401 (2014).
E. Sanchez-Burillo, D. Zueco, J. J. Garcia-Ripoll, and L. Martin-Moreno, Phys. Rev. Lett. [**113**]{}, 263604 (2014).
T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. García-Ripoll, D. Zueco, T. H¬ummer, E. Solano, A. Marx, and R. Gross, Nat. Phys. [**6**]{}, 772 (2010).
P. Forn-Diaz, J. Lisenfeld, D. Marcos, J. J. Garcia-Ripoll, E. Solano, C. J. P. M. Harmans, and J. E. Mooij, Phys. Rev. Lett. [**105**]{}, 237001 (2010).
G. Romero, D. Ballester, Y. M. Wang, V. Scarani, and E. Solano, Phys. Rev. Lett. [**108**]{}, 120501 (2014).
G. Scalari, C. Maissen, D. Turciková, D. Hagenmüller, S. De Liberato, C. Ciuti, C. Reichl, D. Schuh, W. Wegscheider, M. Beck, and J. Faist, Science [**335**]{}, 1323 (2012).
C. Maissen, G. Scalari, F. Valmorra, M. Beck, J. Faist, S. Cibella, R. Leoni, C. Reichl, C. Charpentier, and W. Wegscheider, Phys. Rev. B [**90**]{}, 205309 (2014).
R. Konoike, Y. Sato, Y. Tanaka, T. Asano, and S. Noda, Phys. Rev. B [**87**]{}, 165138 (2013).
Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, Nature Photonics [**6**]{}, 56 (2012).
C. Joshi, P. Öhberg, J. D. Cresser, and E. Andersson, Phys. Rev. A [**90**]{}, 063815 (2014).
W. K. Wootters, Phys. Rev. Lett. **80**, 2245 (1998).
Z. Y. Ou, L. J. Wang, X. Y. Zou, and L. Mandel, Phys. Rev. A [**41**]{}, 1597 (1990).
C. H. Monken, A. Garuccio, D. Branning, J. R. Torgerson, F. Narducci, and L. Mandel, Phys. Rev. A [**53**]{}, 1782 (1996).
L. Mandel, Pure Appl. Opt. [**7**]{}, 927 (1998).
M. Jakob and J. A. Bergou, arXiv:0302075 (2003).
J. Suzuki, C. Miniatura, and K. Nemoto, Phys. Rev. A [**81**]{}, 062307 (2010).
Y. Zhang, J. Zhang, and C.-S. Yu, Scientific Rep. [**6**]{}, 24098 (2016).
H. Paul, Rev. Mod. Phys. [**54**]{}, 1061 (1982).
|
---
author:
- |
Artur Czerwi[ń]{}ski\
1. Institute of Physics\
Nicolaus Copernicus University\
87-100 Toru[ń]{}\
2. Center for Theoretical Physics\
Polish Academy of Sciences\
02-668 Warszawa\
title: Some Remarks on Quantum Tomography in Laser Cooling
---
Introduction
============
By the term *quantum tomography* one understands a wide variety of methods which aim to reconstruct an accurate representation of a quantum system by taking a series of measurements. One of the most fundamental models of quantum tomography, the so-called static tomography model, enables to reconstruct the density matrix of a quantum system provided one can measure $N^2 - 1$ distinct observables (where $N = dim \mathcal{H}$). This approach can be found in many papers and books, such as [@altepeter04; @alicki87; @genki03]. A completely new approach to quantum tomography originated in 2011 in the paper [@bamber], where it was shown that a wave function can be obtained directly if one employs the idea of weak masurement. Later, it was proved that this approach can be generalized so that it can also be applied to mixed states identification [@wu].
The most important property that all tomography models should possess is practicability, which means that a theoretical model can be in the future implemented in an experiment. Therefore, in this article we employ the stroboscopic approach to quantum tomography, which for the first time appeared in [@jam83] and then it was developed in other research papers such as [@jam00] and [@jam04]. One can also look up a very well-written review paper[@jam12]. Recently some new ideas concerning the stroboscopic approach has been presented in [@czerwin15].
The stroboscopic tomography concentrates on determining the optimal criteria for quantum tomography of open quantum systems. The data for reconstruction is provided by mean values of some hermitian operators $\{ Q_1, \dots, Q_p\}$, where obviously $Q_i = Q_i^*$. The underlying assumption behind this approach claims that if one has the knowledge about the evolution of the system, each observable can be measured a certain number of times at different instants. Although there are many possible aspects concerning this problem, in this article we are mainly interested in the minimal number of distinct observables and moments of measurement required for quantum tomography. One can recall the theorem concerning the miniaml number of observables [@jam00].
For a quantum system with dynamics given by a master equation of the form $$\label{eq:kossakowski}
\dot{\rho} = \mathbb{L} \rho$$ one can calculate the minimal number of distinct observables for quantum tomography from the formula $$\eta := \max \limits_{\lambda \in \sigma (\mathbb{L})} \{ dim Ker (\mathbb{L} - \lambda \mathbb{I})\},$$ which means that for every generator $\mathbb{L}$ there exists a set of observables $\{Q_1, \dots, Q_{\eta}\}$ such that their expectation values determine the initial density matrix. Consequetly, they also determine the complete trajectory of the state.
The operator $\mathbb{L}$ that appears in the equation shall be called the generator of evolution. The number $\eta$ is usually refered to as *the index of cyclicity of a quantum system*.
One can also recall the theorem on the upper limit of moments of measurement [@jam04].
In order to reconstruct the density matrix of an open quantum system the number of times that each observable from the set $\{ Q_1, ..., Q_{\eta} \}$ should be measured (denoted by $M_i$ for $i=1,\dots,\eta$) fulfills the inequality $$M_i \leq \mu(\lambda,\mathbb{L}),$$ where by $\mu(\lambda, \mathbb{L})$ one denotes the degree of the minimal polynomial of $\mathbb{L}$.
The above theorem gives the upper boundary concerning the number of measurements of each distinct observable. Naturally, another problem relates to the choice of the time instants. Some considerations about this issue can be found in [@jam04].
The knowledge about the stroboscopic tomography shall be applied in the next chapter to the evolution model known as laser cooling.
Quantum tomography in laser cooling - initial results
=====================================================
An example often studied in the area of laser spectroscopy is a quantum system subject to laser cooling with three energy levels ($dim \mathcal{H} = 3$). The evolution of the density matrix of such a three level system is given by $$\label{eq:1}
\frac{ d \rho}{d t} = - i [H(t), \rho] + \gamma_1 \left ( E_1 \rho E_1 ^* - \frac{1}{2} \{ E_1 ^* E_1, \rho \} \right ) + \gamma_2 \left ( E_2 \rho E_2^* - \frac{1}{2} \{ E_2^* E_2 , \rho \} \right ),$$ where $ E_1 = \left | 1 \right \rangle \left \langle 2 \right | $ and $ E_2 = \left | 3 \right \rangle \left \langle 2 \right |$.
For simplicity it will be assumed that $$\label{eq:2}
\left | 1 \right \rangle = \left [ \begin{matrix} 1 \\ 0 \\ 0 \end{matrix} \right] \text{ } \left | 2 \right \rangle = \left [ \begin{matrix} 0 \\ 1 \\ 0 \end{matrix} \right] \text{ } \left | 3 \right \rangle = \left [ \begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right].$$
Moreover in this analysis we take $ H(t) = [0] $, where $[0]$ denotes a $3-$dimensional matrix with all entries equal $0$. This assumption means that we shall analyze only the Lindbladian part of the equation of evolution.
Based on vectorization theory [@vector] the quantum Liouville operator of the system with dynamics given by can be explicitly expressed as $$\label{eq:3}
\mathbb{L} = \gamma_1 \left ( E_1 \otimes E_1 - \frac{1}{2} ( \mathbb{I} \otimes E_1 ^T E_1 + E_1 ^T E_1 \otimes \mathbb{I}) \right ) + \gamma_2 \left ( E_2 \otimes E_2 -\frac{1}{2} ( \mathbb{I} \otimes E_2 ^T E_2 + E_2^T E_2 \otimes \mathbb{I} ) \right ).$$
Taking into account the assumptions the matrix form of the quantum generator $\mathbb{L}$ can be obtained $$\label{eq:4}
\mathbb{L} = \left[ \begin{matrix} 0 & 0 & 0 & 0 & \gamma_1& 0 & 0 & 0 & 0 \\
0 & -\frac{1}{2} (\gamma_1 + \gamma_2 ) & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -\frac{1}{2} (\gamma_1 + \gamma_2 ) & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & - (\gamma_1 + \gamma_2 ) & 0 & 0 & 0 & 0 \\
0 & 0 & 0 &0 & 0 & -\frac{1}{2} (\gamma_1 + \gamma_2 ) & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 &0 &0 & 0 & 0 & -\frac{1}{2} (\gamma_1 + \gamma_2 ) & 0 \\
0 & 0 & 0 & 0 & \gamma_2 & 0 & 0 & 0 & 0 \end{matrix} \right ].$$
Having the explicit form of quantum generator $ \mathbb{L}$ one can easily calculate its eigenvalues $$\label{eq:5}
\sigma ( \mathbb{L}) = \{ 0, 0,0,0, -(\gamma_1 + \gamma_2) , -\frac{1}{2}(\gamma_1 + \gamma_2), -\frac{1}{2}(\gamma_1 + \gamma_2), -\frac{1}{2}(\gamma_1 + \gamma_2), -\frac{1}{2}(\gamma_1 + \gamma_2) \}.$$
Since in this case the operator $ \mathbb{L}$ is not self-adjoint, the algebraic multiplicity of an eigenvalue does not have to be equal to its geometric multiplicity. But one can qucikly determine that there are four linearly independent eigenvectors that correspond to the eigenvalue $0$. Therefore we can find the index of cyclicity for the operator in question $$\label{eq:6}
\max \limits_{\lambda \in \sigma (\mathbb{L})} \{ dim Ker (\mathbb{L} - \lambda I) \} = 4,$$ which means that we need at least four distinct observables to perform quantum tomography on the analyzed system. One can instantly notice that if the static approach was applied to laser cooling, one would have to measure 8 distinct observables. If one thinks of potential applications in experiments, then this result means that one would have to prepare 4 diffeent experimental set-ups instead of 8. This observation suggest that the stroboscopic approach has an obvious advantage over the static approach.
The next issue that we are interested in is the minimal polynomial for operator $ \mathbb{L} $. Assuming that this polynomial has the form $$\label{eq:7}
d_3 \mathbb{L} ^ 3 + d_2 \mathbb{L} ^ 2+d_1 \mathbb{L} + d_0 \mathbb{I} = 0,$$ one can get $$\label{eq:8}
d_3 = 1,\text{ } d_2 = \frac{3}{2} (\gamma_1 + \gamma_2),\text{ } d_1 = \frac{1}{2} (\gamma_1 + \gamma_2)^2,\text{ } d_0 = 0.$$
Thus we see that $ \mu = deg \text{ }\mu (\lambda, \mathbb{L}) = 3 $. This means that each obsrvable should be measured at most at three different time instants. Thus one can conclude that the total number of measurements for quantum tomography in laser cooling cannot exceed 12.
Having found these results we can conclude that in order to reconstruct the density matrix of the system in question we need 4 observables $Q_1, Q_2, Q_3, Q_4$ that fulfill the condition [@jam12] $$\label{eq:9}
\bigoplus \limits_{i=0}^4 K_3 (\mathbb{L},Q_i) = B_*(\mathcal{H}),$$ where by $ K_3 (\mathbb{L},Q_i)$ one should understand a Krylov subspace which is expressed as $$\label{eq:10}
K_3 (\mathbb{L},Q_i) = \{ Q_i, \mathbb{L}^* Q_i, (\mathbb{L}^*)^2 Q_i \}.$$
The equation gives the necessary condition that the set of observables $\{ Q_1, \dots, Q_4\}$ needs to fulfill in order for one to be able to perform quantum tomography in laser cooling. This condition can be used to determine the explicit forms of these observables.
Summary
=======
In this paper we presented some remarks concerning quantum tomography in laser cooling. The stroboscopic approach was applied to determine to optimal criteria for quantum observability. The results shall be developed in next paper towards the complete quantum tomography model for laser cooling.
[99]{}
Altepeter J. B., James D. F. V., Kwiat P. G.: 4 Qubit Quantum State Tomography. In: Paris, M. G. A., Rehacek, J. (eds.) Quantum State Estimation, pp. 111-145. Springer, Berlin (2004).
Alicki R., Lendi K.: Quantum Dynamical Semigroups and Applications, Springer, Berlin (1987).
Kimura G.: The Bloch vector for N-level systems. Phys. Lett. A **314**, 339-349 (2003).
Lundeen J. S., Sutherland B., Patel A., Stewart C., Bamber C.: Direct measurement of the quantum wavefunction. Nature **474**, 188 (2011).
Wu S.: State tomography via weak masurements. Scientific reports **3**, 1193 (2013).
Jamiołkowski A.: The minimal Number of Operators for Observability of N-level Quantum Systems. Int. J. Theor. Phys. **22**, 369-376 (1983).
Jamiołkowski A.: On complete and incomplete sets of observables, the principle of maximum entropy - revisited. Rep. Math. Phys. **46**, 469-482 (2000).
Jamiołkowski A.: On a Stroboscopic Approach to Quantum Tomography of Qudits Governed by Gaussian Semigroups. OSID **11**, 63-70 (2004).
Jamiołkowski A.: Fusion Frames and Dynamics of Open Quantum Systems, Quantum Optics and Laser Experiments, Dr. Sergiy Lyagushyn (Ed.), ISBN: 978-953-307-937-0, InTech, DOI: 10.5772/31317 (2012). Available from: http://www.intechopen.com/books/quantum-optics-and-laser-experiments/fusion-frames-and-dynamics-of-open-quantum-systems
Czerwi[ń]{}ski A.: A new approach to measurement in quantum tomography. arXiv:1504.01326 (2015).
Henderson, H. V., Searle, S. R.: The vec-permutation matrix, the vec operator and Kronecker products: A review. Linear and Multilinear A. [*[9]{}*]{}, 271-288 (1981).
|
---
abstract: 'Using molecular dynamic simulations we show that single-layers of molybdenum disulfide (MoS$_2$) and graphene can effectively reject ions and allow high water permeability. Solutions of water and three cations with different valence (Na$^+$, Zn$^{2+}$ and Fe$^{3+}$) were investigated in the presence of the two types of membranes and the results indicate a high dependence of the ion rejection on the cation charge. The associative characteristic of ferric chloride leads to a high rate of ion rejection by both nanopores, while the monovalent sodium chloride induces lower rejection rates. Particularly, MoS$_2$ shows 100% of Fe$^{3+}$ rejection for all pore sizes and applied pressures. On the other hand, the water permeation did not varies with the cation valence, having dependence only with the nanopore geometric and chemical characteristic. This study helps to understand the fluid transport through nanoporous membrane, essential for the development of new technologies for pollutants removal from water.'
author:
- Mateus Henrique Köhler
- José Rafael Bordin
- 'Marcia C. Barbosa'
title: '2D nanoporous membrane for cation removal from water: effects of ionic valence, membrane hydrophobicity and pore size'
---
Introduction
============
Centuries of misuse of natural resources has stressed available freshwater supplies throughout the world. With the rapid development of industries, chemical waste has been thrown deliberately in water to the point of making it difficult to clean. Particularly, direct or indirect discharge of heavy metals into the environment has increased recently, especially in developing countries [@ko-jhm2017]. Unlike organic contaminants, heavy metals are not biodegradable and tend to accumulate in living organisms. Many heavy metal ions are known also to be toxic or carcinogenic [@gumpu-sab2015]. Toxic heavy metals of particular concern in treatment of industrial waste-water include zinc, copper, iron, mercury, cadmium, lead and chromium.
As a result, filtration process that can acquire freshwater from contaminated, brackish water or seawater is an effective method to also increase the potable water supply. Modern desalination is mainly based on reverse osmosis (RO) performed through membranes, due to their low energy consumption and easy operation. Current RO plants have already operated near the thermodynamic limit, with the applied pressure being only 10 to 20% higher than the osmotic pressure of the concentrate [@li-des2017]. Meanwhile, advances in nanotechnology have inspired the design of novel membranes based on two-dimensional (2D) nanomaterials. Nanopores with diameters ranging from a few Angstroms to several nanometers can be drilled in membranes to fabricate molecular sieves [@wang-nn2017]. As the diameter of the pore approaches the size of the hydrated ions, various types of ions can be rejected by nanoporous membranes leading to efficient water desalination. Graphene, a single-atom-thick carbon membrane was demonstrated to have several orders of magnitude higher flux rates when compared with conventional zeolite membranes [@celebi-science2014]. In this way, graphene and graphene oxided are one of the most prominent materials for high-efficient membranes [@Xu15; @Kemp13; @huang-jpcl2015]. More recently, others 2D materials have also been investigated for water filtration. A nanoporous single-layer of molybdenum disulfide (MoS$_2$) has shown great desalination capacity [@kou-pccp2016; @weifeng-acsnano2016; @aluru-nc2015]. The possibility to craft the pore edge with Mo, S or both provides flexibility to design the nanopore with desired functionality. In the same way, boron nitride nanosheets also has been investigated for water purification from distinct pollutants [@Lei13; @Azamat15]. Therefore, not only the nanopore size matters for cleaning of water purposes but also the hydrophobicity and geometry of the porous.
For instance, the performance of commercial RO membrane is usually on the order of 0.1 L/cm$^{2}\cdot$day$\cdot$MPa (1.18 g/m$^{2}\cdot$s$\cdot$atm) [@pendergast-ees2011]. With the aid of zeolite nanosheets, permeability high as 1.3 L/cm$^{2}\cdot$day$\cdot$MPa can be obtained [@jamali-jpcc2017]. Recent studies has show that MoS$_2$ nanopore filters have potential to achieve a water permeability of roughly 100 g/m$^{2}\cdot$s$\cdot$atm [@weifeng-acsnano2016] – 2 orders of magnitude higher than the commercial RO. This is comparable with that measured experimentally for the graphene filter ($\sim$70 g/m$^{2}\cdot$s$\cdot$atm) under similar conditions [@surwade-nn2015]. These results have shown that the water permeability scales linearly with the pore density. Therefore, the water filtering performance of 2D nanopores can be even higher.
[@tanugi-nl2012; @aluru-nc2015], [@garaj-nature2010], [@yoon-acsnano2016] [@ohern-nl2015] [@feng-nl2015] [@liu-nl2017] [@wang-nl2017] [@jang-acsnano2017] [@Achtyl15; @Levita16; @Jijo17]
In this work, we address the issue of the selectivity of the porous. In order to do that, we compare the water filtration capacity of MoS$_2$ and graphene through molecular dynamics simulations. While graphene is a purely hydrophobic material, MoS$_2$ sheets have both hydrophobic (S) and hydrophilic (Mo) sites. Recent studies have shown that the water dynamics and structure inside hydrophobic or hydrophilic pores can be quite distinct regarding the pore size [@Mosko14; @kohler-pccp2017; @bordin-PhysA17] and even near hydrophobic or hydrophilic protein sites [@mateus_protein]. Three cations are considered: the standard monovalent sodium (Na$^+$), the divalent zinc (Zn$^{2+}$) and trivalent iron (Fe$^{3+}$). The study of sodium removal is relevant due to it applications for water desalination [@Corry08; @Das14; @Mah15]. Zinc is a trace element that is essential for human health. It is important for the physiological functions of living tissue and regulates many biochemical processes. However, excess of zinc can cause eminent health problems [@fu-jem2011]. The cation Zn$^{2+}$ is ranked 75th in the [*Comprehensive Environmental Response, Compensation and Liability Act*]{} (CERCLA) 2017 priority list of hazardous substances. In its trivalent form, ferric chloride Fe$^{3+}$Cl$_3^-$ is a natural flocculant, with high power of aggregation. It is also on the CERCLA list with recommended limit concentration of 0.3 mg/l. In this way, we explore the water permeation and cations rejection by nanopore with distinct radii. Our results shows that the hydrophilic/hydrophobic MoS$_2$ nanopore have a higher salt rejection in all scenarios, while the purely hydrophobic graphene have a higher water permeation. Specially, MoS$_2$ membranes shows the impressive capacity of block all the trivalent iron cations regardless the nanopore size.
Our paper is organized as follow. In the Section \[methods\] we introduce our model and the details about the simulation method. On Section \[water-results\] we show and discuss our results for the water permeation in the distinct membranes, while in the Section \[ion-results\] we show the ion rejection properties for each case. Finally, a summary of our results and the conclusions are shown in Section \[conclusions\].
Computational Details and Methods {#methods}
=================================
Molecular dynamics (MD) simulations were performed using the LAMMPS package [@plimpton1995]. A typical simulation box consists of a graphene sheet acting as a rigid piston in order to apply an external force (pressure) over the ionic solution. The pressure gradient forces the solution against the 2D nanopore: a single-layer of molybdenum disulfide or graphene. Figure \[fig1\] shows the schematic representation of the simulation framework.
![(a) Schematic representation of the simulation framework. The system is divided as follows: On the left side we can see the piston (graphene) pressing the ionic solution (in this case, water+NaCl) against the MoS$_2$ nanopore. For the case of a graphene nanopore the depiction is the same, but with a porous graphene sheet instead of the MoS$_2$ sheet. On the right side we have bulk water. (b) Definition of the pore diameter $d$. []{data-label="fig1"}](fig1.eps){width="12.5cm"}
A nanopore was drilled in both MoS$_2$ and graphene sheets by removing the desired atoms, as shown in Figure \[fig1\]. The accessible pore diameters considered in this work range from 0.26 - 0.95 nm for the MoS$_2$ and 0.17 - 0.92 nm for the graphene . [@aluru-nc2015] The system contains 22000 atoms distributed in a box with dimensions $5\times 5 \times 13$ nm in x, y and z, respectively. Although the usual salinity of seawater is $\sim0.6$M, we choose a molarity of $\sim1.0$M for all the cations (Na$^{+}$, Zn$^{2+}$ and Fe$^{3+}$) due the computational cost associated with low-molarity solutions.
The TIP4P/2005 [@abascal-jcp2005] water model was used and the SHAKE algorithm [@ryckaert1977] was employed to maintain the rigidity of the water molecules. The non-bonded interactions are described by the Lennard-Jones (LJ) potential with a cutoff distance of 0.1 nm and the parameters tabulated in Table 1. The Lorentz-Berthelot mixing rule were used to obtain the LJ parameters for different atomic species. The long-range electrostatic interactions were calculated by the [*Particle Particle Particle Mesh*]{} method [@hockney1981]. Periodic boundary conditions were applied in all the three directions.
\[t1\]
Interaction $\sigma$ (nm) $\varepsilon$ (kcal/mol) Charge
---------------------------- --------------- -------------------------- ---------
C$-$C [@farimani-jpcb2011] 3.39 0.0692 0.00
Mo$-$Mo [@liang-prb2009] 4.20 0.0135 0.60
S$-$S [@liang-prb2009] 3.13 0.4612 -0.30
O$-$O [@abascal-jcp2005] 3.1589 0.1852 -1.1128
H$-$H 0.00 0.00 0.5564
Na$-$Na [@raul-jpcb2016] 2.52 0.0347 1.00
Cl$-$Cl [@raul-jpcb2016] 3.85 0.3824 -1.00
Zn$-$Zn [@hinkle-jced2016] 0.0125 1.960 2.00
Fe$-$Fe [@hinkle-jced2016] 0.18 0.745 3.00
: The Lennard-Jones parameters and charges of the simulated atoms. The crossed parameters were obtained by Lorentz-Berthelot rule.
For each simulation, the system was first equilibrated for constant number of particles, pressure and temperature (NPT) ensemble for 1 ns at P = 1 atm and T = 300 K. Graphene and MoS$_2$ atoms were held fixed in the space during equilibration and the NPT simulations allow water to reach its equilibrium density (1 g/cm$^3$). After the pressure equilibration, a 5 ns simulation in the constant number of particles, volume and temperature (NVT) ensemble to further equilibrate the system at the same T = 300 K. Finally, a 10 ns production run were carried out, also in the NVT ensemble. The Nosé-Hoover thermostat [@nose1984; @hoover1985] was used at each 0.1 ps in both NPT and NVT simulations, and the Nosé-Hoover barostat was used to keep the pressure constant in the NPT simulations. Different external pressures were applied on the rigid piston to characterize the water filtration through the 2D (graphene and MoS$_2$) nanopores. For simplicity, the pores were held fixed in space to study solely the water transport and ion rejection properties of these materials. The external pressures range from 10 to 100 MPa. These are higher than the osmotic pressure used in the experiments. The reason for applying such high pressures at MD simulations with running time in nanosecond scale is because the low pressures would yield a very low water flux that would not go above the statistical error. We carried out three independent simulations for each system collecting the trajectories of atoms every picoseconds.
Water flux {#water-results}
==========
![Water flux as a function of the applied pressure for MoS$_2$ and graphene nanopores with similar pore areas. (a) monovalent Na$^+$, (b) divalent Zn$^{2+}$ and (c) trivalent Fe$^{3+}$ cations are considered for the ionic solution at the reservoir. (d) Water permeability through the pores as function of the pore diameter for the case of $\Delta$P = 50 MPa. The dotted lines are a guide to the eye.[]{data-label="fig2"}](fig2.eps){width="15.5cm"}
First, let us compare the flux performance of the graphene and the MoS$_2$ membranes. In the Figure \[fig2\], we show the water flux through 2D nanopores in number of molecules per nanosecond (MoS$_2$ and graphene) as a function of the applied pressure gradient for different pore diameters. The water is filtered from a reservoir containing an ionic solution of either monovalent sodium (Na$^+$), divalent zinc (Zn$^{2+}$) or trivalent iron cations (Fe$^{3+}$). In all cases, chlorine (Cl$^-$) was used as the standard anion. Four pore sizes for each material were investigated.
Our results indicates that for the smaller pore diameter, the black points in the Figure \[fig2\], both materials have the same water permeation. However, for the other values of pore diameter the graphene membrane shows a higher water flux, for all applied pressure gradient. While the flux at the purely hydrophobic graphene pore for a fixed pressure monotonically increases with the pore diameter, this is not the case for the MoS$_2$ pore for which the flows shows a minimum around pore diameter of $0.37$ nm probably due to the non uniform distribution of the hydrophobic and hydrophilic sites of the pore. The Figures \[fig2\](a), (b) and (c) show that this behavior of the water flux is not affected by the cation valence, only by the applied pressure, by geometric effects and by the pore composition. For instance, the 0.46 nm graphene pore shows enhanced water flux compatible with the 0.6 nm MoS$_2$ pore for all cations. Therefore, is clear that pore composition affects the water permeation properties more than the water-ion interaction.
This result agrees with the findings by Aluru and his group [@aluru-nc2015], were they showed that even a small change in pore composition can lead to enhanced water flux through a MoS$_2$ nanocavity. This is also consistent with our recent findings that the dynamics of water inside nanopores with diameter $\approx$ 1.0 nm is strongly affected by the presence of hydrophilic or hydrophobic sites [@kohler-pccp2017]. This investigation, over distinct cation valences and membranes, highlights the importance of the nanopore physical-chemistry properties for water filtration processes.
To quantify the water permeability through the pores, we compute the permeability coefficient, $p$, across the pore. For dilute solutions $$\begin{aligned}
\label{eq1}
p=\frac{j_{\mathrm{w}}}{ -V_{\mathrm{w}}\Delta C_{\mathrm{s}} +\frac{V_{\mathrm{W}}}{N_{A}k_{\mathrm{B}}T}\hspace{0.1cm} \Delta P }\end{aligned}$$ where $j_{\mathrm{w}}$ is the flux of water (H$_2$O/ns), $V_{\mathrm{w}}$ is the molar volume of water (19 ml/mol), $\Delta C_{\mathrm{s}}$ is the concentration gradient of the solute (1.0 M), $N_{A}$ is the Avogadro number, $k_{\mathrm{B}}$ is the Boltzmann constant, $T$ is the temperature (300 K) and $\Delta$ P is the applied hydrodynamic pressure (MPa).
The case of $\Delta$ P = 50 MPa is shown in Figure \[fig2\](d). The permeability coefficient of the MoS$_2$ range from approximately 33 to 55 H$_2$O/ns for the 0.26 and 0.95 nm diameters, respectively. The graphene nanopore presents a permeability coefficient of $\sim$ 34 - 63 H$_2$O/ns as the pore diameter is varied from 0.17 to 0.92 nm, respectively. For smaller pores the difference between MoS$_2$ and graphene is inside the error bars, whereas for the larger pores both materials exhibit high permeability rates, with a slight advantage in the case of graphene.
![Averaged axial distribution of water molecules inside the (a) graphene (Gra) and (b) MoS$_2$ nanopores with distinct diameters. Here, z = 0 is at the center of the pore, the external pressure is $\Delta$P = 10 MPa and the cation is the Na$^+$.[]{data-label="fig3"}](fig3.eps){width="14cm"}
The water structure and dynamics inside nanopores are strongly related [@kohler-pccp2017; @Bordin13a]. Therefore, distinct structural regimes can lead to different diffusive behaviors. In the Figure \[fig3\] we present the distribution of water molecules in the z-direction inside the MoS$_2$ (solid line) and graphene (dotted line) nanopores. As for the water flux, the water axial distribution is not affects by the cation valence. Therefore, for simplicity and since there are more studies about monovalent salts, we show only the Na$^+$ case. The nanopore length in the z-direction, considering the van der Walls diameter for each sheet, is 0.63 (-0.315 to 0.315) nm for the MoS$_2$ and 0.34 (-0.17 to 0.17) nm for the graphene. The structure inside both pores are considerably different. For the graphene nanopore, shown in Figure \[fig3\](a), there is no favorable positions for the water molecules to remain throughout the simulation. This can be related to the hydrophobic characteristic of the graphene sheet and the high slippage observed for water inside carbon nanopores [@Falk10; @Tocci14]. Since all the pore is hydrophobic, there is no preferable position for the water molecules, and the permeability is higher. On the other hand, along the MoS$_2$ cavity we can observe a high structuration in three sharp peaks, as shown in Figure \[fig3\](b). This structuration comes from the existence of hydrophilic (Mo) and hydrophobic sites (S atoms). This layered organization within the MoS$_2$ nanopore can be linked to the reduced flux compared with graphene, since it implies an additional term in the energy required for the water molecule to pass through the pore.
The higher water flux through graphene nanopores compared with MoS$_2$ imply that for a desired water flux, a smaller applied pressure is needed for graphene. Nevertheless, it is important to note that both fluxes are higher, specially when compared with currently desalination technologies [@aluru-nc2015; @azamat-cms2017]. Therefore, both materials are capable of providing a high water permeability. The question is whether these materials are also able to effectively clean the water by removing the ions.
Ion rejection efficiency {#ion-results}
========================
The other important aspect for the cleaning of water is the membrane ability to separate water and ions. In this way, we investigate how the cation valence and the pore size affects the percentage rejected ions. In the Figure \[fig4\] we show the percentage of total ions rejected by the 2D nanopores as a function of the applied pressure for the three cations. The pores diameters are the same from the discussed in the previous section.
The ion rejection by the smallest pores, 0.17 and 0.26 nm for graphene and MoS$_2$, respectively, was 100% for all applied pressures and cation solutions. This is expected since the pore size is much smaller than the hydration radii of the cations. Therefore, is more energetically favorable for the cation to remain in the bulk solution instead of strip off the water and enter the pore [@Bordin12a]. As the pore diameter increases this energetic penalty becomes smaller. As well, the valence plays a crucial role here, with the monovalent ions having a smaller penalty than divalent and trivalent cations. In this way, for the nanopores with diameter 0.37 nm and 0.46 nm for graphene and MoS$_2$, respectively, Na$^+$ and Cl$^-$ ions flow through the pore reducing the rejection efficiency for both materials, as we can see in the Figure \[fig4\](a). However, it is important to note that the ion rejection performance of molybdenum disulfide membranes is superior from the observed for graphene membranes for all ranges of pressure, sizes and cation valences. For instance, for the divalent case Zn$^{2+}$, shown in the Figure \[fig4\](b) and the smaller $\Delta$P the rejection is 100% for all pores sizes in the MoS$_2$ membrane, while for the graphene membrane we observe cation permeation for the bigger pores.
![Percentage of ion rejection by various pores as a function of the applied pressure. Pores with different diameters are considered. []{data-label="fig4"}](fig4.eps){width="15.5cm"}
The MoS$_2$ membrane shows a very good performance for the rejection of the trivalent cation Fe$^{3+}$. As the Figure \[fig4\](c) shows, for all nanopore size and applied pressure the rejection is 100%. Such efficiency was not observed in the graphene membranes, were only the case with small pore diameter as 100% of iron rejection. Here, we should address that not only the hydration shell plays an important role in the cations rejection. While sodium chloride is uniformly dispersed in water and we do not observe clusters at the simulated concentration, the iron cations tend to form large clusters of ferric chlorides in solution, as shown in Figure \[fig5\]. Moreover, we observe this structures throughout the whole simulation and even at high pressure regime the clusters remains too large to overcome the pore. In fact, ferric chlorides are effective as primary coagulants due to their associative character in solution. At controlled concentrations, it is excellent for both drinking and wastewater treatment applications, including phosphorus removal [@kim-ee2015], sludge conditioning and struvite control [@amuda-jhm2007; @sun-wr2015]. It also prevent odor and corrosion by controlling hydrogen sulfide formation. Additionally, our results indicates that the associative properties of ferric chlorides can be used to increase the efficiency of salt rejection by both MoS$_2$ and graphene nanopores, which may contribute in water cleaning devices.
Summary and conclusions {#conclusions}
=======================
We have calculated water fluxes through various MoS$_2$ and graphene nanopores and the respective percentage of total ions rejected by both materials as a function of the applied pressure gradient. Our results indicate that 2D nanoporous membranes are promising for water purification and salt rejection. The selectivity of the membranes was found to depend on factors such as the pore diameter, the cationic valence and the applied pressure. Nevertheless, our results shows that the ion valency do not affect the water permeation – this is only affected by the pore size and chemical composition.
Particularly, our findings indicate that graphene is a better water conductor than MoS$_2$, with a higher permeability coefficient. Although, both material have presented high water fluxes. On the other hand, MoS$_2$ nanopores with water accessible pore diameters ranging from 0.26 to 0.95 nm strongly reject ions even at theoretically high pressures of 100 MPa. Additionally, the rejection is shown to depend strongly on the ion valence. It reaches 100% for trivalent ferric chloride (Fe$^{3+}$Cl$_{3}^-$) for all MoS$_2$ pore sizes and applied pressures. This is a direct result of the ability of heavy metals to form agglomerates, eventually exhibiting long ionic chains. At the same time, this did not affected the water flux. Then, the ferric chloride properties can be used to improve the effectiveness of 2D material based nanofilters. New studies are been performed in this direction.
We thank the Brazilian agencies CNPq and INCT-FCx for the financial support, CENAPAD/SP and CESUP/UFRGS for the computer time.
[10]{}
D. Ko, J. S. Lee, H. A. Patel, M. H. Jakobsen, Y. Hwange, C. T. Yavuzan, H. B. Hansen, and H. R. Andersen, J. Hazard. Mater. [**332**]{}, 140 (2017).
M. B. Gumpu, S. Sethuraman, U. M. Krishnan, and J. B. B. Rayappan, Sens. Actuators B: Chem. [**213**]{}, 515 (2015).
M. Li, Desalination [**422**]{}, 124 (2017).
L. Wang, M. S. H. Boutilier, P. R. Kidambi, D. Jang, N. G. Hadjiconstantinou, and R. Karnik, Nat. Nanotech. [**12**]{}, 509 (2017).
K. Celebi, J. Buchheim, R. M. Wyss, A. Droudian, P. Gasser, I. Shorubalko, J.-I. Kye, C. Lee, and H. G. Park, Science [**344**]{}, 289 (2014).
Q. Xu, H. Xu, J. Chen, Y. Lv, C. Dong, and T. S. Sreeprasadc, Inorg. Chem. Front. [**2**]{}, 417 (2015).
K. C. Kemp, H. Seema, M. Saleh, N. H. Le, K. Mahesh, V. Chandra, and K. S. Kim, Nanoscale [**5**]{}, 3149 (2013).
L. Huang, M. Zhang, C. Li, and G. Shi, J. Phys. Chem. Lett. [**6**]{}, 2806 (2015).
J. Kou, J. Yao, L. Wu, X. Zhou, H. Lu, F. Wu, and J. Fan, Phys. Chem. Chem. Phys. [**18**]{}, 22210 (2016).
W. Li, Y. Yang, J. K. Weber, G. Zhang, and R. Zhou, ACS Nano [**10**]{}, 1829 (2016).
M. Heiranian, A. B. Farimani, and N. R. Aluru, Nat. Commun. [**6**]{}, 8616 (2015).
W. Lei, D. Portehault, D. Liu, S. Qin, and Y. Chen, Nature Comm. [**4**]{}, 1777 (2013).
J. Azamat, B. S. Sattary, A. Khataee, and S. W. Joo, J. Mol. Graph. Model. [**61**]{}, 13 (2015).
M. M. Pendergast and E. M. Hoek, Energy Environ. Sci. [**4**]{}, 1946 (2011).
S. H. Jamali, T. J. H. Vlugt, and L.-C. Lin, J. Phys. Chem. C [**121**]{}, 11273 (2017).
S. P. Surwade, S. N. Smirnov, I. V. Vlassiouk, R. R. Unocic, G. M. Veith, S. Dai, and S. M. Mahurin, Nat. Nanotech. [**10**]{}, 459 (2015).
D. Cohen-Tanugi and J. C. Grossman, Nano Lett. [**12**]{}, 3602 (2012).
S. Garaj, W. Hubbard, A. Reina, J. Kong, D. Branton, and J. A. Golovchenko, Nature [**467**]{}, 190 (2010).
K. Yoon, A. Rahnamoun, J. L. Swett, V. Iberi, D. A. Cullen, I. V. Vlassiouk, A. Belianinov, S. Jesse, X. Sang, O. S. Ovchinnikova, A. J. Rondinone, R. R. Unocic, and A. C. van Duin, ACS Nano [**10**]{}, 8376 (2016).
S. C. OHern, D. Jang, S. Bose, J.-C. Idrobo, Y. Song, T. Laoui, J. Kong, and R. Karnik, Nano Lett. [**15**]{}, 3254 (2015).
J. Feng, K. Liu, M. Graf, M. Lihter, R. D. Bulushev, D. Dumcenco, D. T. L. Alexander, D. Krasnozhon, T. Vuletic, A. Kis, and A. Radenovic, Nano Lett. [**15**]{}, 3431 (2015).
K. Liu, M. Lihter, A. Sarathy, S. Caneva, H. Qiu, D. Deiana, V. Tileli, D. T. L. Alexander, S. Hofmann, D. Dumcenco, A. Kis, J.-P. Leburton, and A. Radenovic, Nano Lett. [**17**]{}, 4223 (2017).
Z. Wang, Q. Tu, S. Zheng, J. J. Urban, S. Li, and B. Mi, Nano Lett. [**17**]{}, 7289 (2017).
D. Jang, J.-C. Idrobo, T. Laoui, and R. Karnik, ACS Nano [**11**]{}, 10042 (2017).
J. L. Achtyl, R. R. Unocic, L. Xu, Y. Cai, M. Raju, W. Zhang, R. L. Sacci, I. V. Vlassiouk, P. F. Fulvio, P. Ganesh, D. J. Wesolowski, S. Dai, A. C. T. [van Duin]{}, M. Neurock, and F. M. Geiger, Nature Communications [**6**]{}, 6539 (2015).
G. Levita, P. Restuccia, and M. Righi, Carbon [**107**]{}, 878 (2016).
J. Abrahamand, K. S. Vasu, C. D. Williams, K. Gopinadhan, Y. Su, C. T. Cherian, J. Dix, E. Prestat, S. J. Haigh, I. V. Grigorieva, P. Carbone, A. K. Geim, and R. R. Nair, Nat. Nanotech. [**12**]{}, 546 (2017).
I. Moskowitz, M. A. Snyder, and J. Mittal, J. Chem. Phys. [**141**]{}, 18C532 (2014).
M. H. Köhler, J. R. Bordin, L. B. da Silva, and M. C. Barbosa, Phys. Chem. Chem. Phys. [**19**]{}, 12921 (2017).
J. R. Bordin and M. C. Barbosa, Phys. A [**467**]{}, 137 (2017).
M. H. Köhler, R. C. Barbosa, L. B. da Silva, and M. C. Barbosa, Phys. A [**468**]{}, 733 (2017).
B. Corry, J. Phys. Chem. B [**112**]{}, 1427 (2008).
R. Das, M. E. Ali, S. B. A. Hamid, S. Ramakrishna, and Z. Z. Chowdhury, Desalination [**336**]{}, 97 (2014).
K. A. Mahmoud, B. Mansoor, A. Mansour, and M. Khraisheh, Desalination [**356**]{}, 208 (2015).
F. Fu and Q. Wang, J. Environ. Management [**92**]{}, 407 (2011).
S. Plimpton, J. Comput. Phys. [**117**]{}, 1 (1995).
J. Abascal and C. Vega, J. Chem. Phys. [**123**]{}, 234505 (2005).
J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. Comput. Phys. [**23**]{}, 327 (1977).
R. W. Hockney and J. W. Eastwood, , McGraw-Hill, New York, 1981.
A. Barati Farimani and N. R. Aluru, J. Phys. Chem. B [**115**]{}, 12145 (2011).
T. Liang, S. R. Phillpot, and S. B. Sinnott, Phys. Rev. B [**79**]{}, 245110 (2009).
R. Fuentes-Azcatl and M. C. Barbosa, J. Phys. Chem. B [**120**]{}, 2460 (2016).
K. R. Hinkle, C. J. Jameson, and S. Murad, J. Chem. Eng. Data [**61**]{}, 1578 (2016).
S. Nosé, Mol. Phys. [**52**]{}, 255 (1984).
W. G. Hoover, Phys. Rev. A. [**31**]{}, 1695 (1985).
J. R. Bordin, A. Diehl, and M. C. Barbosa, J. Phys. Chem. B [**117**]{}, 7047 (2013).
K. Falk, F. Sedlmeier, L. Joly, R. R. Netz, and L. Bocquet, Nano Lett. [**10**]{}, 4067 (2010).
G. Tocci, L. Joly, and A. Michaelides, Nano Lett. [**14**]{}, 6872 (2014).
J. Azamat and A. Khataee, Comp. Mat. Sci. [**137**]{}, 201 (2017).
J. R. Bordin, A. Diehl, M. C. Barbosa, and Y. Levin, Phys. Rev. E [**85**]{}, 031914 (2012).
B. Kim, M. Gautier, P. Molle, P. Michel, and R. Gourdon, Ecol. Eng. [**80**]{}, 53 (2015).
O. Amuda and I. Amoo, J. Hazard. Mater. [**141**]{}, 778 (2007).
J. Sun, I. Pikaar, K. R. Sharma, J. Keller, and Z. Yuan, Water Research [**71**]{}, 150 (2015).
|
---
abstract: |
Load balancing algorithms play a crucial role in delivering robust application performance in data centers and cloud networks. Recently, strong interest has emerged in Join-the-Idle-Queue (JIQ) algorithms, which rely on tokens issued by idle servers in dispatching tasks and outperform power-of-$d$ policies. Specifically, JIQ strategies involve minimal information exchange, and yet achieve zero blocking and wait in the many-server limit. The latter property prevails in a multiple-dispatcher scenario when the loads are strictly equal among dispatchers. For various reasons it is not uncommon however for skewed load patterns to occur. We leverage product-form representations and fluid limits to establish that the blocking and wait then no longer vanish, even for arbitrarily low overall load. Remarkably, it is the least-loaded dispatcher that throttles tokens and leaves idle servers stranded, thus acting as bottleneck.
Motivated by the above issues, we introduce two enhancements of the ordinary JIQ scheme where tokens are either distributed non-uniformly or occasionally exchanged among the various dispatchers. We prove that these extensions can achieve zero blocking and wait in the many-server limit, for any subcritical overall load and arbitrarily skewed load profiles. Extensive simulation experiments demonstrate that the asymptotic results are highly accurate, even for moderately sized systems.
author:
- Mark van der Boor
- Sem Borst
- Johan van Leeuwaarden
title: |
Load Balancing in Large-Scale Systems\
with Multiple Dispatchers
---
Introduction
============
[*Background and motivation.*]{} Load balancing algorithms provide a crucial mechanism for achieving efficient resource allocation in parallel-server systems, ensuring high server utilization and robust user performance. The design of scalable load balancing algorithms has attracted immense interest in recent years, motivated by the challenges involved in dispatching jobs in large-scale cloud networks and data centers with massive numbers of servers.
In particular, token-based algorithms such as the Join-the-Idle-Queue (JIQ) scheme [@BB08; @LXKGLG11] have gained huge popularity recently. In the JIQ scheme, idle servers send tokens to the dispatcher (or one among several dispatchers) to advertise their availability. When a job arrives and the dispatcher has tokens available, it assigns the job to one of the corresponding servers (and disposes of the token). When no tokens are available at the time of a job arrival, the job may either be discarded or forwarded to a randomly selected server. Note that a server only issues a token when a job completion leaves its queue empty. Thus at most one message is generated per job (or possibly two messages, in case a token is revoked when an idle server receives a job through random selection from a dispatcher without any tokens).
Under Markovian assumptions, the JIQ scheme achieves a zero probability of wait for any fixed subcritical load per server in a regime where the total number of servers grows large [@Stolyar15a]. Thus the JIQ scheme provides asymptotically optimal performance with minimal communication overhead (at most one or two messages per job), and outperforms power-of-$d$ policies as we will further discuss below.
The latter asymptotic optimality of the JIQ scheme prevails in a multiple-dispatcher scenario provided the job arrival rates at the various dispatchers are exactly equal [@Stolyar15b]. When the various dispatchers receive jobs from external sources it is difficult however to perfectly balance the job arrival rates, and hence it is not uncommon for skewed load patterns to arise.
[*Key contributions.*]{} In the present paper we examine the performance of the JIQ scheme in the presence of possibly heterogeneous dispatcher loads. We distinguish two scenarios, referred to as [*blocking*]{} and [*queueing*]{}, depending on whether jobs are discarded or forwarded to a randomly selected server in the absence of any tokens at the dispatcher. We use exact product-form distributions and fluid-limit techniques to establish that the blocking and wait no longer vanish for asymmetric dispatcher loads as the total number of servers grows large. In fact, even for an arbitrarily small degree of skewness and arbitrarily low overall load, the blocking and wait are strictly positive in the limit. We show that, surprisingly, it is the least-loaded dispatcher that acts as a bottleneck and throttles the flow of tokens. The accumulation of tokens at the least-loaded dispatcher hampers the visibility of idle servers to the heavier-loaded dispatchers, and leaves idle servers stranded while jobs queue up at other servers.
In order to counter the above-described performance degradation for asymmetric dispatcher loads, we introduce two extensions to the basic JIQ scheme. In the first mechanism tokens are not uniformly distributed among dispatchers but in proportion to the respective loads. We prove that this enhancement achieves zero blocking and wait in a many-server regime, for any subcritical overall load and arbitrarily skewed load patterns. In the second approach, tokens are continuously exchanged among the various dispatchers at some exponential rate. We establish that for any load profile with subcritical overall load there exists a finite token exchange rate for which the blocking and wait vanish in the many-server limit. Extensive simulation experiments are conducted to corroborate these results, indicating that they apply even in moderately sized systems.
In summary we make three key contributions:
1\) We show how the blocking scenario can be represented in terms of a closed Jackson network. We leverage the associated product-form distribution to express the blocking probability as function of the relevant load parameters.
2\) We use fluid-limit techniques to establish that in both the blocking and the queueing scenario the system performance depends on the aggregate load and the minimum load across all dispatchers. The fluid-limit regime not only offers analytical tractability, but is also highly relevant given the massive numbers of servers in data centers and cloud operations.
3\) We propose two enhancements to the basic JIQ scheme where tokens are either distributed non-uniformly or occasionally exchanged among the various dispatchers. We demonstrate that these mechanisms can achieve zero blocking and wait in the many-server limit, for any subcritical overall load and arbitrarily skewed load profiles.
[*Discussion of alternative schemes and related work.*]{} As mentioned above, the JIQ scheme outperforms power-of-$d$ policies in terms of communication overhead and user performance. In a power-of-$d$ policy an incoming job is assigned to a server with the shortest queue among $d$ randomly selected servers from the total available pool of $N$ servers. In the absence of memory at the dispatcher(s), this involves an exchange of $2 d$ messages per job (assuming $d \geq 2$).
In [@Mitzenmacher01; @VDK96] mean-field limits are established for power-of-$d$ policies in Markovian scenarios with a single dispatcher and identical servers. These results indicate that even a value as small as $d = 2$ yields significant performance improvements over a purely random assignment scheme ($d = 1$) in large-scale systems, in the sense that the tail of the queue length distribution at each individual server falls off much more rapidly. This is commonly referred to as the ‘power-of-two’ effect. At the same time, a small value of $d$ significantly reduces the amount of information exchange compared to the classical Join-the-Shortest-Queue (JSQ) policy (which corresponds to $d = N$) in large-scale systems. These results also extend to heterogeneous servers, non-Markovian service requirements and loss systems [@BLP10; @BLP12; @MKM15; @MKMG15; @XDLS15].
In summary, power-of-$d$ policies involve low communication overhead for fixed $d$, and can even deliver asymptotically optimal performance (when the value of $d$ suitably scales with $N$ [@MBLW16c; @MBLW16d; @MBLW16b]). In contrast to the JIQ scheme however, for no single value of $d$, a power-of-$d$ policy can achieve both low communication overhead and asymptotically optimal performance, which is also reflected in recent results in [@GTZ16]. The only exception arises in case of batch arrivals when the value of $d$ and the batch size grow large in a specific proportion, as can be deduced from the arguments in [@YSK15].
Scenarios with multiple dispatchers have hardly received any attention so far. The results for the JIQ scheme in [@LXKGLG11; @Mitzenmacher16; @Stolyar15b] all assume that the loads at the various dispatchers are strictly equal. We are not aware of any results for heterogeneous dispatcher loads. To the best of our knowledge, power-of-$d$-policies have not been considered in a multiple-dispatcher scenario at all. While the results in [@Stolyar15b] show that the JIQ scheme is asymptotically optimal for symmetric dispatcher loads, even when the servers are heterogeneous, it is readily seen that power-of-$d$ policies cannot even be maximally stable in that case for any fixed value of $d$.
[*Organization of the paper.*]{} The remainder of the paper is organized as follows. In Section \[mode\] we present a detailed model description, specify the two proposed enhancements and state the main results. In Section \[jacksonnetwork\] we describe how the blocking scenario can be represented in terms of a closed Jackson network, and leverage the associated product-form distribution to obtain an insightful formula for the blocking probability. We then turn to a fluid-limit approach in Section \[fluidlimitinblocking\] to analyze the two proposed enhancements in the blocking scenario. A similar analysis is adopted in Section \[fluidlimitinqueueing\] in the queueing scenario to obtain results for the basic model and the enhanced variants. Finally, in Section \[conc\] we make some concluding remarks and briefly discuss future research directions.
Model description, notation and key results {#mode}
===========================================
We consider a system with $N$ parallel identical servers and a fixed set of $R$ (not depending on $N$) dispatchers, as depicted in \[modelfigure\]. Jobs arrive at dispatcher $r$ as a Poisson process of rate $\alpha_r \lambda N$, with $\alpha_r > 0$, $r = 1, \dots, R$, $\sum_{r = 1}^{R} \alpha_r = 1$, and $\lambda$ denoting the job arrival rate per server. For conciseness, we denote $\alpha = (\alpha_1, \dots, \alpha_R)$, and without loss of generality we assume that the dispatchers are indexed such that $\alpha_1 \geq \alpha_2 \geq \dots \geq \alpha_R$. The job processing requirements are independent and exponentially distributed with unit mean at each of the servers.
When a server becomes idle, it sends a token to one of the dispatchers selected uniformly at random, advertising its availability. When a job arrives at a dispatcher which has tokens available, one of the tokens is selected, and the job is immediately forwarded to the corresponding server.
![Schematic view of the model with $R$ dispatchers and $N$ servers.[]{data-label="modelfigure"}](ModelFigure5.pdf){width="\scaling\columnwidth"}
We distinguish two scenarios when a job arrives at a dispatcher which has no tokens available, referred to as the [*blocking*]{} and [*queueing*]{} scenario respectively. In the blocking scenario, the incoming job is blocked and instantly discarded. In the queueing scenario, the arriving job is forwarded to one of the servers selected uniformly at random. If the selected server happens to be idle, then the outstanding token at one of the other dispatchers is revoked.
In the queueing scenario we assume $\lambda < 1$, which is not only necessary but also sufficient for stability. It is not difficult to show that the joint queue length process is stochastically majorized by a case where each job is sent to a uniformly at random selected server. In the latter case, the system decomposes into $N$ independent M/M/1 queues, each of which has load $\lambda<1$ and is stable.
Denote by $X_0(t)$ the number of busy servers and by $X_r(t)$ the number of tokens held by dispatcher $r$ at time $t$, $r = 1, \dots, R$. Note that $\sum_{r = 0}^{R} X_r(t) \equiv N$ for all $t$. Also, denote by $Y_i(t)$ the number of servers with $i$ jobs (including a possible job being processed) at time $t$, $i \geq 0$, so that $X_0(t) \equiv \sum_{i = 1}^{\infty} Y_i(t)$.
In the blocking scenario, no server can have more than one job, i.e. $Y_i(t)=0$ for all $i \geq 2$ and $X_0(t) = Y_1(t)$. Because of the symmetry among the servers, the state of the system can thus be described by the vector $X(t) = (X_0(t), X_1(t), \dots, X_R(t))$, and $\{X(t)\}_{t \geq 0}$ evolves as a Markov process, with state space $S := \{n \in \mathbb{N}^{R+1}: \sum_{i = 0}^{R} n_i = N\}$.
Likewise, in the queueing scenario, the state of the system can be described by the vector $U(t) = (Y(t), X_1(t), \dots, X_R(t))$ with $Y(t) = (Y_i(t))_{i \geq 0}$, and $\{U(t)\}_{t \geq 0}$ also evolves as a Markov process. Denote by $B(R, N, \lambda, \alpha)$ the steady-state blocking probability of an arbitrary job in the blocking scenario. Also, denote by $W(R, N, \lambda, \alpha)$ a random variable with the steady-state waiting-time distribution of an arbitrary job in the queueing scenario. In \[jacksonnetwork\] we will prove the following theorem for the blocking scenario.
\[maintheorem\] As $N\to\infty$, $$B(R,N,\lambda,\alpha) \to \max\{1-R\alpha_R,1-1/\lambda\}.$$
\[maintheorem\] shows that in the many-server limit the system performance in terms of blocking is either determined by the relative load of the least-loaded dispatcher, or by the aggregate load. This may be informally explained as follows. Let ${\bar{x}_0}$ be the expected fraction of busy servers in steady state, so that each dispatcher receives tokens on average at a rate ${\bar{x}_0}N/R$. We distinguish two cases, depending on whether a positive fraction of the tokens reside at the least-loaded dispatcher $R$ in the limit or not. If that is the case, then the job arrival rate $\alpha_R \lambda N$ at dispatcher $R$ must equal the rate ${\bar{x}_0}N / R$ at which it receives tokens, i.e., ${\bar{x}_0}/ R = \alpha_R \lambda$. Otherwise, the job arrival rate $\alpha_R \lambda N$ at dispatcher $R$ must be no less the rate ${\bar{x}_0}N / R$ at which it receives tokens, i.e., ${\bar{x}_0}/ R \leq \alpha_R \lambda$. Since dispatcher $R$ is the least-loaded, it then follows that ${\bar{x}_0}/ R \leq \alpha_r \lambda$ for all $r = 1, \dots, R$, which means that the job arrival rate at all the dispatchers is higher that the rate at which tokens are received. Thus the fraction of tokens at each dispatcher is zero in the limit, i.e., the fraction of idle servers is zero, implying ${\bar{x}_0}= 1$. Combining the two cases, and observing that ${\bar{x}_0}\leq 1$, we conclude ${\bar{x}_0}= \min\{R \alpha_R \lambda, 1\}$. Because of Little’s law, ${\bar{x}_0}$ is related to the blocking probability $B$ as ${\bar{x}_0}= \lambda (1 - B)$. This yields $1 - B = \min\{R \alpha_R \lambda, 1 / \lambda\}$, or equivalently, $B = \max\{1 - R \alpha_R, 1 - 1 / \lambda\}$ as stated in Theorem 1.
The above explanation also reveals that in the limit dispatcher $R$ (or the set of least-loaded dispatchers in case of ties) inevitably ends up with all the available tokens, if any. The accumulation of tokens hampers the visibility of idle servers to the heavier-loaded dispatchers, and leaves idle servers stranded while jobs queue up at other servers.
\[3db1\] illustrates \[maintheorem\] for $R=2$ dispatchers and $N=10^5$ servers, and clearly reflects the two separate regions in which the blocking probability depends on either $\alpha_R$ or $\lambda$. The line represents the cross-over curve $R \alpha_R = 2 \alpha_2 = 2 (1 - \alpha_1) = 1 / \lambda$.
![Blocking probability $B(2, N, \lambda, (\alpha_1, 1 - \alpha_1))$ obtained by \[generalformulaBlocking\], for $R=2$ dispatchers and $N=10^5$ servers as function of $\lambda$ and $\alpha_1$.[]{data-label="3db1"}](3DBlocking2a.pdf){width="\scaling\columnwidth"}
In Section \[fluidlimitinqueueing\] we will establish the following theorem for the queueing scenario.
\[maintheorem2\] For $\lambda < 1$ and $N\to\infty$,
$$\mathbb{E}[W(R, N, \lambda, \alpha)] \to
\frac{\lambda_2(R, \lambda, \alpha)}{1 - \lambda_2(R, \lambda, \alpha)},$$ where $$\lambda_2(R,\lambda,\alpha)=1-\frac{1-\lambda\sum_{i=1}^{r^*}\alpha_i}{1-\lambda r^* / R}$$ with $$\label{maintheorem2r}
r^*=\sup
\big\{r \big| \alpha_r > \frac{1}{R}\frac{1-\lambda\sum_{i=1}^{r}\alpha_i}{1-\lambda r/R}\big\}$$ and the convention that $r^*=0$ if $\alpha_1=\hdots=\alpha_R=1/R$.
$\lambda_2$ can be interpreted as the rate at which jobs are forwarded to randomly selected servers. Furthermore, dispatchers $1, \hdots, r^*$ receive tokens at a lower rate than the incoming jobs, and in particular $\lambda_2^*=0$ if and only if $r^*=0$. When $R=2$, \[maintheorem2\] simplifies to $$\label{maintheorem22}
\mathbb{E}[W(2, N, \lambda, (1 - \alpha_2, \alpha_2))] \rightarrow
\frac{\lambda (1-2\alpha_2)}{2-2\lambda (1-\alpha_2)}.$$
When the arrival rates at all dispatchers are strictly equal, i.e., $\alpha_r = 1/R$ for all $r = 1, \dots, R$, \[maintheorem,maintheorem2\] indicate that the stationary blocking probability and the mean waiting time asymptotically vanish in a regime where the total number of servers $N$ grows large, which is in agreement with the results in [@Stolyar15b]. However, when the arrival rates at the various dispatchers are not perfectly equal, so that $\alpha_R < 1 / R$, the blocking probability and mean wait are strictly positive in the limit, even for arbitrarily low overall load and an arbitrarily small degree of skewness in the arrival rates. Thus, the basic JIQ scheme fails to achieve asymptotically optimal performance when the dispatcher loads are not strictly equal.
In order to counter the above-described performance degradation for asymmetric dispatcher loads, we propose two enhancements.
\[alg1\] When a server becomes idle, it sends a token to dispatcher $r$ with probability $\beta_r$.
\[alg2\] Any token is transferred to a uniformly randomly selected dispatcher at rate $\nu$.
Note that the token exchange mechanism only creates a constant communication overhead per job as long as the rate $\nu$ does not depend on the number of servers $N$, and thus preserves the scalability of the basic JIQ scheme.
The above enhancements can achieve asymptotically optimal performance for suitable values of the $\beta_r$ parameters and the exchange rate $\nu$, as stated in the next proposition.
\[maintheorem3\] For any $\lambda<1$, the stationary blocking probability in the blocking scenario and the mean waiting time in the queueing scenario asymptotically vanish as $N\to\infty$, upon using \[alg1\] with $\beta_r=\alpha_r$ or \[alg2\] with $\nu\geq \frac{\lambda}{1-\lambda}(\alpha_1 R-1)$.
The minimum value of $\nu$ required in the blocking scenario may be intuitively understood as follows. Zero blocking means that a fraction $\lambda$ of the servers must be busy, and thus a fraction $1-\lambda$ of the tokens reside with the various dispatchers, while the heaviest loaded dispatcher 1 receives enough tokens for all incoming jobs: $\alpha_1 \lambda \leq \lambda/R + \nu(1-\lambda)/R$ which is satisfied by the given minimum value of $\nu$.
A similar reasoning applies to the queueing scenario, although in that case the number of servers with exactly one job no longer equals the number of busy servers, and a different approach is needed.
In order to establish \[maintheorem2,maintheorem3\], we examine in \[fluidlimitinblocking,fluidlimitinqueueing\] the fluid limits for the blocking and queueing scenarios, respectively. Rigorous proofs to establish weak convergence to the fluid limit are omitted, but can be constructed along similar lines as in [@HK94]. The fluid-limit regime not only provides mathematical tractability, but is also particularly relevant given the massive numbers of servers in data centers and cloud operations. Simulation experiments will be conducted to verify the accuracy of the fluid-limit approximations, and show an excellent match, even in small systems (small values of $N$).
Jackson network representation {#jacksonnetwork}
==============================
In this section we describe how the blocking scenario can be represented in terms of a closed Jackson network. We leverage the associated product-form distribution to express the asymptotic blocking probability as a function of the aggregate load and the minimum load across all dispatchers, proving \[maintheorem\].
We view the system dynamics in the blocking scenario in terms of the process $\{X(t)\}_{t \geq 0}$ as a fixed total population of $N$ tokens that circulate through a network of $R+1$ stations. Specifically, the tokens can reside either at station $0$, meaning that the corresponding server is busy, or at some station $r$, indicating that the corresponding server is idle and has an outstanding token with dispatcher $r$, $r = 1, \dots, R$. Let $s_i(k)$ denote the service rate at station $i$ when there are $k$ tokens present. Then $s_0(k)=k$ and $s_r(k)=1$ for $r=1,\dots,R$. The service times are exponentially distributed at all stations, but station $0$ is an infinite-server node with mean service time $\mu_0^{-1}=1$, while station $r$ is a single-server node with mean service time $\mu_r^{-1} = (\alpha_r \lambda N)^{-1}$, $r=1,\dots,R$. The routing probabilities $p_{ij}$ of tokens moving from station $i$ to station $j$ are given by $p_{r0}=1$ for $r=1,\dots,R$ and $p_{0r}=1/R$ for $r=1,\dots,R$. With $\gamma_i$ denoting the throughput of tokens at station $i$, the traffic equations $$\gamma_i = \sum_{j = 0}^{R} \gamma_j p_{ji}, \quad i=0,\dots,R$$ uniquely determine the relative values of the throughputs. Let $\pi(n) := \lim_{t \to \infty} \mathbb{P}(X(t) = n)$ be the stationary probability that the process $\{X(t)\}_{t \geq 0}$ resides in state $n \in S$. The theory of closed Jackson networks [@Kelly79] implies $$\pi(n_0,n_1,\hdots,n_R)=G^{-1} \prod_{i=0}^R \frac{(\gamma_i/\mu_i)^{n_i}}{\prod_{m=1}^{n_i} s_i(m)},$$ with $G$ a normalization constant.
The blocking probability can then be expressed by summing the probabilities $\pi(n)$ over all the states with $n_r = 0$ where no tokens are available at dispatcher $r$, and weighting these with the fractions $\alpha_r$, $r = 1, \dots, R$: $$\label{generalformulaBlocking}
B(R,N,\lambda,\alpha)=\sum_{r=1}^R \alpha_r \sum_{n\in \{n | n_r=0\}} \pi(n).$$ Despite this rather complicated expression, \[maintheorem\] provides a compact characterization of the blocking probability in the many-server limit $N\to \infty$, as will be proved in the Appendix. The proof uses stochastic coupling, for which we define a ‘better’ system and a ‘worse’ system. Both systems are amenable to analysis and have an identical blocking probability in the many-server limit $N\to \infty$.
The better system merges the first $R-1$ dispatchers into one super-dispatcher, which results in two dispatchers with arrival rates $(1-\alpha_R)\lambda N$ and $\alpha_R\lambda N$, respectively. However, in contrast to the original blocking scenario, when a job is completed and leaves a server idle, a token is not sent to either dispatcher with equal probability. Instead, tokens are sent to the super-dispatcher with probability $\frac{R-1}{R}$. To analyze this better system, we study in \[modelA\] the blocking scenario with $R = 2$ enhanced with non-uniform token allotment. The worse system thins the incoming rates of jobs at the dispatchers, so that some jobs are blocked, irrespective of whether or not the dispatcher has any tokens available. This thinning process is defined as follows: a job arriving at dispatcher $r$ is blocked with probability $$\frac{\alpha_r-\alpha_R}{\alpha_r} +
\frac{\max\left\{0, \alpha_R-1/(R\lambda)\right\}}{\alpha_r}.$$ This thinning process is designed in such a way that the system with admitted jobs behaves as a system with total arrival rate $\lambda\leq 1$ in which all arrival rates are equal ($\alpha_1=\hdots=\alpha_R$), which is analyzed in \[modelB\]. With coupling, one can show that the blocking probability of the ‘better system’ is lower and of the ‘worse system’ is higher, which completes the proof. Specifically, when the arrival moments, the service times and the token-allotment are coupled, the number of tokens used at each dispatcher by time $t$ is always lower in the worse system and higher in the better system. Due to page limitations, the detailed coupling arguments are omitted, but it is intuitively clear that the better system performs better and the worse system performs worse. Namely, the tokens at dispatchers 1 to $R-1$ are consolidated in the better system. If there is at least one token amongst these dispatchers, any job arriving at any of the dispatchers can make use of a token. In the original system, a job is blocked when the token amongst the first $R-1$ dispatchers, is not present at the dispatchers at which a job arrives. The worse system performs obviously worse, since blocking jobs beforehand has no benefits for the acceptance of jobs.
While we assumed exponentially distributed service times, the infinite-server node is symmetric and thus the product-form solution in \[generalformulaBlocking\] as well as \[maintheorem\] still hold for phase-type distributions [@Kelly79].
Fluid limit in the blocking scenario {#fluidlimitinblocking}
====================================
We now turn to the fluid-limit analysis and start with the blocking scenario. We consider a sequence of systems indexed by the total number of servers $N$. Denote by $x_0^N(t) = \frac{1}{N} X_0^N(t)$ the fraction of busy servers and by $x_r^N(t) = \frac{1}{N} X_r^N(t)$ the normalized number of tokens held by dispatcher $r$ in the $N$-th system at time $t$. Further define $x^N(t) = (x_0^N(t), x_1^N(t) \dots, x_R^N(t))$ and assume that $x^N(0) \to x^\infty$ as $N \to \infty$, with $\sum_{i = 0}^{R} x_i^\infty = 1$. Then any weak limit $x(t)$ of the sequence $\{x^N(t)\}_{t \geq 0}$ as $N \to \infty$ is called a fluid limit.
The fluid limit $x(t)$ in the blocking scenario with \[alg1,alg2\] in place satisfies the set of differential equations $$\label{fpexpr1}
\frac{{\text{d}}x_0(t)}{{\text{d}}t} = {\sum_{r=1}^R}z_r(t) - x_0(t),$$ $$\label{fpexpr2}
\frac{{\text{d}}x_r(t)}{{\text{d}}t} =
\beta_r x_0(t) + \nu\left(\frac{1-x_0(t)}{R}-x_r(t)\right) - z_r(t),$$ with $$\label{fpexpr2z}
z_r(t) = \alpha_r \lambda -
\left[\alpha_r \lambda - \beta_r x_0(t) - \nu \frac{1-x_0(t)}{R}\right]^+
\mathds{1}\left\{x_r(t)=0\right\},$$ where $[\cdot]^+=\max\{\cdot,0\}$ and initial condition $x(0) = x^\infty$.
The above set of fluid-limit equations may be interpreted as follows. The term $z_r(t)$ represents the (scaled) rate at which dispatcher $r$ uses tokens and forwards incoming jobs to idle servers at time $t$. \[fpexpr2z\] reflects that the latter rate equals the job arrival rate $\alpha_r \lambda$, unless the fraction of tokens held by dispatcher $r$ is zero ($x_r(t) = 0$), and the rate $\beta_r x_0(t) + \nu (1 - x_0(t)) / R$ at which it receives tokens from idle servers or through the exchange mechanism is less than the job arrival rate. \[fpexpr1\] states that the rate of change in the fraction of busy servers is the difference between the aggregate rate ${\sum_{r=1}^R}z_r(t)$ at which the various dispatchers use tokens and forward jobs to idle servers, and the rate $x_0(t)$ at which jobs are completed and busy servers become idle. \[fpexpr2\] captures that the rate of change of the fraction of tokens held by dispatcher $r$ is the balance of the rate $\beta_r x_0(t) + \nu (1 - x_0(t)) / R$ at which it receives tokens from idle servers or through the exchange mechanism, and the rate $z_r(t) + \nu x_r(t)$ at which it uses tokens and forwards jobs to idle servers or releases tokens through the exchange mechanism.\
![Fluid-limit trajectories $x_i(t)$ for $R=2$ dispatchers, $\lambda=0.9$ and $\alpha_1=0.8$. Averaged sample paths for $N=100$ servers are represented by lighter colors, and closely match the fluid-limit dynamics.[]{data-label="fwtp5"}](FluidLimits4a.pdf){width="\scaling\columnwidth"}
\[fwtp5\] shows the exact and simulated fluid-limit trajectories. We observe that the simulation results closely match the fluid-limit dynamics. We further note that in the long run only dispatcher $2$ with the lower arrival rate holds a strictly positive fraction of the tokens, corroborating Theorem 1.
Fixed-point analysis
--------------------
In order to determine the fixed point(s) $x^*$, we set ${\text{d}}x_i(t)/{\text{d}}t = 0$ for all $i = 0, 1, \dots, R$, and obtain $$\label{fpexpr3c}
x_0^* = {\sum_{r=1}^R}z_r^*,$$ $$\label{fpexpr3d}
z_r^* = \beta_r x_0^* + \nu\left(\frac{1-x_0^*}{R}-x_r^*\right),$$ and $$\label{fpexpr3b}
z_r^* = \alpha_r \lambda -
\left[\alpha_r \lambda - \beta_r x_0^* - \nu \frac{1-x_0^*}{R}\right]^+
\mathds{1}\left\{x_r^*=0\right\}.$$
Without proof, we assume that the many-server ($N\to\infty$) and stationary ($t\to\infty$) limits commute, so that $x_0^*$ is also the limit of the mean fraction of busy servers in stationarity. Because of Little’s law, the limit $B$ of the blocking probability satisfies $$\label{littlecons}
x_0^*=\lambda(1-B).$$ This in particular implies that $x_0^*=\lambda$ leads to $B=0$: vanishing blocking.
*Basic JIQ scheme.* We first consider the basic JIQ scheme, i.e., $\beta_r = 1 / R$ for all $r = 1, \dots, R$ and $\nu = 0$. \[fpexpr3d,fpexpr3b\] yield $$\frac{x_0^*}{R} = z_r^* = \alpha_r \lambda -
\left[\alpha_r \lambda - \frac{x_0^*}{R}\right]^+
\mathds{1}\left\{x_r^*=0\right\},$$ or equivalently, $$\label{basicjiq}
\alpha_r \lambda - \frac{x_0^*}{R} =
\left[\alpha_r \lambda - \frac{x_0^*}{R}\right]^+
\mathds{1}\left\{x_r^*=0\right\}.$$ Now let ${\mathcal I} = \{r: \alpha_r = \alpha_R\}$ be the index set of the least-loaded dispatchers. \[basicjiq\] forces $x_r^* = 0$ for all $r \notin {\mathcal I}$.
We now distinguish two cases, depending on whether or not $x_r^* = 0$ for all $r \in {\mathcal I}$ as well. If that is the case, then we must have $x_0^* = 1$, and $\alpha_R \lambda \geq x_0^*/R$, i.e., $\lambda \geq 1 / (R \alpha_R)$. Otherwise, we must have $\alpha_R \lambda = x_0^*/R$, i.e., $x_0^* = R \alpha_R \lambda$, so $x_0^* \leq 1$ forces $\lambda \leq 1 / (R \alpha_R)$.
In conclusion, we have $x_0^* = \min\{R \alpha_R \lambda, 1\}$. When $\lambda \geq 1 / (R \alpha_R)$ so that $x_0^* = 1$, it must be the case that $x_r^* = 0$ for all $r = 1, \dots, R$. When $\lambda < 1 / (R \alpha_R)$ so that $x_0^* < 1$, any vector $x^*$ with $x_r^* = 0$ for all $r \notin {\mathcal I}$ and $\sum_{r \in {\mathcal I}} x_r^* = 1 - x_0^*$ is a fixed point. In particular, for equal dispatcher loads, i.e., $\alpha_r = 1 / R$ for all $r = 1, \dots, R$, so that ${\mathcal I} = \{1, \dots, R\}$, we have $x_r^* = 0$ for all $r = 1, \dots, R$ when $\lambda \geq 1$, while any vector $x^*$ with ${\sum_{r=1}^R}x_r^* = 1 - \lambda$ is a fixed point when $\lambda < 1$.
We use \[littlecons\] to find $B = 1 - \frac{1}{\lambda} \min\{R \alpha_R \lambda, 1\} =
\max\{1 - R \alpha_R, 1 - 1 / \lambda\}$, which agrees with \[maintheorem\].
In \[table1\] we compare the fluid-limit approximations for the blocking probability with the exact formula from the Jackson network representation and simulation results for various numbers of servers.
-------------------- --------- ------------ --------- ------------
N Jackson simulation Jackson simulation
$10$ 0.6021 0.6032 0.3201 0.3205
$20$ 0.6000 0.6006 0.2545 0.2552
$50$ 0.6000 0.6004 0.2092 0.2095
$100$ 0.6000 0.6007 0.2006 0.2010
fluid ($N=\infty$) 0.6000 - 0.2000 -
-------------------- --------- ------------ --------- ------------
: Blocking probabilities for $\lambda=0.9$ and $\alpha_1=0.8$ or $\alpha_1=0.6$.
\[table1\]
\[table1\] shows that the Jackson network analysis agrees with the simulation results. Furthermore, the more symmetric the loads, the lower the blocking probability, which is consistent with \[maintheorem\]. Also, the fluid-limit approximation is highly accurate, even for a fairly small number of servers.
Enhancements
------------
We now examine the behavior of the system for \[alg1,alg2\], and show that they can achieve asymptotically zero blocking for any $\lambda\leq 1$ and suitable parameter values as identified in \[maintheorem3\]. In light of \[littlecons\] it suffices to show that $x_0^*=\lambda$ for both enhancements.
Consider \[alg1\]; $\beta_r=\alpha_r$ and $\nu=0$. \[fpexpr3b,fpexpr3d\] give $\lambda-x_0^*=[\lambda-x_0^*]^+\mathds{1}\{x_r^*=0\}$ for all $r$ (this shows $x_0^*\leq \lambda$). Assume that $x_0^*<\lambda$. Then, $x_r^*=0$ for all $r$, which results in $x_0^*=1$, since $\sum_{i=0}^R x_i^*=1$. This contradicts $x_0^*\leq \lambda<1$, so that $x_0^*=\lambda$.
Notice that any point for which $x_0^*=\lambda$ and for which $\sum_{i=0}^R x_i^*=1$, results in $z_r^*=\alpha_r\lambda$ which is in accordance with \[fpexpr3c,fpexpr3d,fpexpr3b\], and therefore is a fixed point.\
We next consider \[alg2\] with $\beta_r=1/R$ and $\nu\geq \frac{\lambda}{1-\lambda}(R\alpha_1-1)$. We use \[fpexpr3c,fpexpr3b\] twice, first they give $x_0^*\leq \lambda$, so that $x_0^*=\lambda-{\varepsilon}$ for some ${\varepsilon}\geq0$. Second, $z_r^*=\alpha_r\lambda$ if the term in brackets in \[fpexpr3b\] is non-positive. Otherwise, $$\begin{split}
z_r^* &\geq \alpha_r\lambda - \left[\alpha_r \lambda - \frac{x_0^*}{R}-\nu\frac{1-x_0^*}{R}\right]\\
&\geq \alpha_r\lambda - \alpha_r\lambda + \frac{\lambda-{\varepsilon}}{R} + \frac{\lambda}{1-\lambda}(R\alpha_1-1)
\frac{1-\lambda}{R}+\nu\frac{{\varepsilon}}{R}\\
&\geq \alpha_r \lambda - \frac{{\varepsilon}}{R}(1-\nu),
\end{split}$$ which by \[fpexpr3c\] gives $\lambda-{\varepsilon}=x_0^*={\sum_{r=1}^R}z_r^*\geq \lambda-{\varepsilon}(1-\nu)$, so that $x_0^*=\lambda$.\
\[fwtp3\] displays the blocking probability as $N\to\infty$ for the system with both enhancements. Since $\alpha_1=0.7$, we have that $\beta_1=0.7$ is optimal. The blocking probability decreases as $\beta_1$ approaches $\alpha_1$ and as $\nu$ increases. For $\nu>0$, it suffices to choose $\beta_1$ close to $\alpha_1$, which implies that it is not necessary to know the exact loads, for the enhancements to be effective.
![Blocking probability $B$ in the limit for $R=2$, $\lambda=0.9$ and $\alpha_1=0.7$, for different values of $\beta_1$ and $\nu$.[]{data-label="fwtp3"}](Alg12BlockingProb2a.pdf){width="\scaling\columnwidth"}
Fluid limit in the queueing scenario {#fluidlimitinqueueing}
====================================
We now proceed to the queueing scenario (with $\lambda < 1$ for stability). As before, we consider a sequence of systems indexed by the total number of servers $N$. Denote by $y_i^N(t) = \frac{1}{N} Y_i^N(t)$ the fraction of servers with $i$ jobs and by $x_r^N(t) = \frac{1}{N} X_r^N(t)$ the normalized number of tokens held by dispatcher $r$ in the $N$-th system at time $t$, $r = 1, \dots, R$. Further define $u^N(t) = (y^N(t), x_1^N(t) \dots, x_R^N(t))$, with $y^N(t) = (y_i^N(t))_{i \geq 0}$, and assume that $u^N(0) \to u^\infty$ as $N \to \infty$, with $\sum_{i = 1}^{\infty} y_i^\infty + \sum_{r = 1}^{R} x_r^\infty = 1$. Then any weak limit $u(t)$ of the sequence $\{u^N(t)\}_{t \geq 0}$ as $N \to \infty$ is called a fluid limit.
The fluid limit $u(t)$ in the queueing scenario with \[alg1,alg2\] in place obeys the set of differential equations $$\label{fluidlimitexpr1}
\frac{{\text{d}}y_0(t)}{{\text{d}}t} =
y_1(t) - \lambda_1(t) - \lambda_2(t) y_0(t),$$ $$\label{fluidlimitexpr2}
\begin{split}
\frac{{\text{d}}y_i(t)}{{\text{d}}t} &= \lambda_1(t)\mathds{1}\left\{i=1\right\} + \lambda_2(t) y_{i-1}(t) \\
&+ y_{i+1}(t) - y_i(t) - \lambda_2(t) y_i(t)
\mbox{ for all } i \geq 1,
\end{split}$$ $$\label{fluidlimitexpr3}
\frac{{\text{d}}x_r(t)}{{\text{d}}t} =
\beta_r y_1(t) + \nu\left(\frac{y_0(t)}{R}-x_r(t)\right) - z_r(t) -
\lambda_2(t) x_r(t),$$ with $$z_r(t) = \alpha_r \lambda -
\left[\alpha_r \lambda - \beta_r y_1(t) - \nu \frac{y_0(t)}{R}\right]^+
\mathds{1}\left\{x_r(t)=0\right\},$$$$\label{fluidlimitexpr4}
\lambda_1(t) = {\sum_{r=1}^R}z_r(t), \quad
\lambda_2(t) = \lambda - \lambda_1(t),$$ and initial condition $u(0) = u^\infty$.
The above set of fluid-limit equations may be interpreted as follows. Similarly as in the blocking scenario, the term $z_r(t)$ represents the (scaled) rate at which dispatcher $r$ uses tokens and forwards incoming jobs to idle servers at time $t$. Accordingly, $\lambda_1(t)$ is the aggregate rate at which dispatchers use tokens to forward jobs to (guaranteed) idle servers at time $t$, while $\lambda_2(t)$ is the aggregate rate at which jobs are forwarded to randomly selected servers (which may or may not be idle). \[fluidlimitexpr1\] reflects that the rate of change in the fraction of idle servers is the difference between the aggregate rate $y_0(t)$ at which jobs are completed by servers with one job, and the rate $\lambda_1(t)$ at which dispatchers use tokens to forward jobs to idle servers plus the rate $\lambda_2(t) y_0(t)$ at which jobs are forward to randomly selected servers that happen to be idle. \[fluidlimitexpr2\] states that the rate of change in the fraction of servers with $i$ jobs is the balance of the rate $\lambda_2(t) y_{i-1}(t)$ at which jobs are forwarded to randomly selected servers with $i - 1$ jobs plus the aggregate rate $y_{i+1}(t)$ at which jobs are completed by servers with $i + 1$ jobs, and the rate $\lambda_2(t) y_i(t)$ at which jobs are forwarded to randomly selected servers with $i$ jobs plus the aggregate rate $y_i(t)$ at which jobs are completed by servers with $i$ jobs. In case $i = 1$, the rate at which dispatchers use tokens to forward jobs to idle servers should be included as additional positive term.
\[fluidlimitexpr3\] is similar to \[fpexpr2\], where the additional fourth term captures the rate at which tokens are revoked when jobs are forwarded to randomly selected servers that happen to be idle.
Fixed-point analysis
--------------------
In order to determine the fixed point(s) $u^*$, we set ${\text{d}}y_i(t)/{\text{d}}t = 0$ for all $i \geq 0$, and ${\text{d}}x_r(t)/{\text{d}}t = 0$ for all $r = 1, \dots, R$. We obtain $$\label{mm}
y_1^* = \lambda_1^* + \lambda_2^* y_0^*, \hspace{.2cm} (1+\lambda_2^*) y_i^* = \lambda_2^* y_{i-1}^* + y_{i+1}^*
\text{ for all } i \geq 2,$$ Solving \[mm\] gives $$y_0^*=1-\lambda, \hspace{.2cm} y_k^* = \lambda (1 - \lambda_2^*) \left(\lambda_2^*\right)^{k-1}
\text{ for all } k \geq 1.$$ Thus the mean number of jobs at a server is $$\sum_{k = 1}^{\infty} k y_k^* = \sum_{k=1}^{\infty} k \lambda (1 - \lambda_2^*)
\left(\lambda_2^*\right)^{k-1} = \frac{\lambda}{1-\lambda_2^*}.$$
As for the blocking scenario, we assume that the many-server and stationary limits commute. Little’s law then gives $$\label{littlequeueing}
\sum_{k=1}^\infty k y_k^*=\lambda(\mathbb{E}[W]+1),$$ where the left-hand side represents the mean number of jobs at a server in the many-server limit. We use \[littlequeueing\] to obtain $$\label{meanwaitingtime}
\mathbb{E}[W] = \frac{\sum_{k = 1}^{\infty} k y_k^*}{\lambda} - 1 = \frac{\lambda_2^*}{1 - \lambda_2^*},$$ which shows vanishing wait in case $\lambda_2^*=0$. We also obtain the following equations for the fixed point: $$\label{zrexpr}
z_r^* = \beta_r \lambda (1 - \lambda_2^*) - \lambda_2^* x_r^* +
\nu\left(\frac{1-\lambda}{R}-x_r^*\right),$$ $$\label{zrexpr2}
z_r^* = \alpha_r \lambda - \left[\alpha_r \lambda -
\beta_r \lambda (1 - \lambda_2^*) - \nu \frac{1 - \lambda}{R}\right]^+
\mathds{1}\left\{x_r^*=0\right\},$$ and $$\label{zrexprl1}
\lambda_1^* = \sum_{r=1}^R z_r^*.$$ We define $$\label{fluidlimit2g}
q_r^*=z_r^* \Big|_{x_r^*=0}=\beta_r\lambda(1-\lambda_2^*)+\nu \frac{1-\lambda}{R}.$$
\[zrexpr\] implies $z_r^* \leq q_r^*$ and \[zrexpr2\] leads to $z_r^*\leq \alpha_r\lambda$, $x_r^*=0 \Rightarrow z_r^*=q_r^*$ and $x_r^*>0 \Rightarrow z_r^*=\alpha_r\lambda$, yielding $z_r^*=\min\{q_r^*,\alpha_r\lambda\}$ and thus
$$\label{exprl1}
\lambda_1^*=\sum_{r=1}^R \min\{q_r^*,\alpha_r\lambda\}.$$
*Basic JIQ scheme.* In case $\beta_r=1/R$ and $\nu=0$, \[exprl1\] can be rewritten to $$\lambda_2^*=\sum_{r=1}^R\left[\alpha_r\lambda-\frac{\lambda(1-\lambda_2^*)}{R}\right]^+,$$ and by further calculations, since $\alpha_r$ is decreasing in $r$, to the expression for $\lambda_2^*$ in \[maintheorem2\].
In \[table3\] we compare the fluid-limit approximations for the mean-waiting time with simulation figures for various numbers of servers.
-------------------- ----------------------------- -----------------------------
$\lambda=0.9, \alpha_1=0.8$ $\lambda=0.9, \alpha_1=0.6$
N simulation simulation
$10$ 2.5824 2.1234
$20$ 1.7349 1.1386
$50$ 1.1704 0.5001
$100$ 1.0173 0.2981
$200$ 0.9764 0.2207
$500$ 0.9631 0.1983
$1000$ 0.9599 0.1962
fluid ($N=\infty$) 0.9643 0.1957
-------------------- ----------------------------- -----------------------------
: Mean waiting time for $\lambda=0.9$ and $\alpha_1=0.8$ or $\alpha_1=0.6$.
\[table3\]
\[table3\] shows that the fluid-limit analysis agrees with the simulation results, although the number of servers needs to be larger than in the blocking scenario for extremely high accuracy to be observed. Similarly to \[table1\], the more symmetric the loads, the better the performance and the lower the mean waiting time, which is in line with \[maintheorem2\].
Enhancements
------------
We examine the behavior of the system for \[alg1,alg2\] and show that they can achieve asymptotically zero waiting for any $\lambda<1$ and suitable parameter values as identified in \[maintheorem3\]. In view of \[meanwaitingtime\] it suffices to show that $\lambda_2^*=0$ for both enhancements. We first consider \[alg1\] in which $\alpha_r=\beta_r$ for all $r$ and $\nu=0$. \[fluidlimit2g\] gives $q_r^* - \alpha_r\lambda = -\beta_r\lambda\lambda_2^*\leq 0$ for all $r$ and $q_r^*=\beta_r\lambda(1-\lambda+\lambda_1^*)$. We obtain $$\lambda_1^*=\sum_{r=1}^R q_r^* = \lambda(1-\lambda+\lambda_1^*),$$ which has a unique solution $\lambda_1^*=\lambda$, so that $\lambda_2^*=0$.\
Next, we turn to \[alg2\] where $\nu \geq \frac{\lambda}{1-\lambda}(R \alpha_1-1) $ and $\beta_r=1/R$. If the term in brackets in \[zrexpr2\] is non-positive, $z_r^*=\alpha_r\lambda$. Otherwise, $$\begin{split}
z_r^* &\geq \alpha_r\lambda - \left[ \alpha_r\lambda-\frac{\lambda(1-\lambda_2^*)}{R}-\nu\frac{1-\lambda}{R}\right]\\
&\geq\alpha_r\lambda-\alpha_r\lambda + \frac{\lambda - \lambda\lambda_2^*}{R}+\frac{\lambda}{R}(R\alpha_1-1)\geq \alpha_r\lambda - \frac{\lambda\lambda_2^*}{R},
\end{split}$$ which by \[zrexprl1\] gives $\lambda-\lambda_2^*=\lambda_1^* \geq \lambda-\lambda\lambda_2^*$, and since $\lambda<1$, we obtain $\lambda_2^*=0$.\
\[fwtp4\] displays the mean waiting time as $N\to\infty$ for the system with both enhancements. Similarly to \[fwtp3\], we can greatly improve the performance by tuning $\beta$ and $\nu$. Again $\alpha_1=0.7$, so that $\beta_1=0.7$ is the best choice. The mean waiting time decreases as $\beta_1$ approaches $\alpha_1$, or as the rate $\nu$ increases. Exact knowledge of the arrival rates is not required, and a rough approximation of $\beta_1$ and a small value of $\nu$ are sufficient for the mean waiting time to vanish.
![Mean waiting time $\mathbb{E}[W]$ in the limit for $R=2$, $\lambda=0.9$ and $\alpha_1=0.7$, for different values of $\beta_1$ and $\nu$.[]{data-label="fwtp4"}](Alg12WaitingTime2aa.pdf){width="\scaling\columnwidth"}
Conclusion {#conc}
==========
We examined the performance of the Join-the-Idle-Queue (JIQ) scheme in large-scale systems with several possibly heterogeneous dispatchers. We used product-form representations and fluid limits to show that the basic JIQ scheme fails to deliver zero blocking and wait for any asymmetric dispatcher loads, even for arbitrarily low overall load. Remarkably, it is the least-loaded dispatcher that throttles tokens and leaves idle servers stranded, thus acting as bottleneck.
In order to counter the performance degradation for asymmetric dispatcher loads, we introduced two extensions of the basic JIQ scheme where tokens are either distributed non-uniformly or occasionally exchanged among the various dispatchers. We proved that these extensions can achieve zero blocking and wait in the many-server limit, for any subcritical overall load and arbitrarily skewed load profiles. Extensive simulation experiments corroborated these results, indicating that they apply even in moderately sized systems.
It is worth emphasizing that the proposed enhancements involve no or constant additional communication overhead per job, and hence retain the scalability of the basic JIQ scheme. The algorithms do rely on suitable parameter settings, and it would be of interest to develop learning techniques for that.
While we allowed the dispatchers to be heterogeneous, we assumed all the servers to be statistically identical, and the service requirements to be exponentially distributed. As noted earlier, \[maintheorem\] in fact holds for non-exponential service requirement distributions as well. In ongoing work we aim to extend \[maintheorem2,maintheorem3\] to possibly non-exponential service requirement distributions.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported by the NWO Gravitation Networks grant 024.002.003, an NWO TOP-GO grant and an ERC Starting Grant.
[99]{}
R. Badonnel, M. Burgess (2008). Dynamic pull-based load balancing for autonomic servers. [*Proc. IEEE NOMS 2008*]{}, 751–754. M. Bramson, Y. Lu, B. Prabhakar (2010). Randomized load balancing with general service time distributions. [*ACM SIGMETRICS Perf. Eval. Rev.*]{} [**38 (1)**]{}, 275–286.
M. Bramson, Y. Lu, B. Prabhakar (2012). Asymptotic independence of queues under randomized load balancing. [*Queueing Systems*]{} [**71 (3)**]{}, 247–292.
D. Gamarnik, J. Tsitsiklis, M. Zubeldia (2016). Delay, memory and messaging tradeoffs in distributed service systems. [*Proc. ACM SIGMETRICS Perf. Eval. Rev.*]{} [**44 (1)**]{}, 1–12.
P. Hunt, T. Kurtz (1994). Large loss networks. [*Stoc. Proc. Appl.*]{} [**53 (2)**]{}, 363–378.
F. P. Kelly (2011). Reversibility and stochastic networks. [*Cambridge University Press*]{}, New York.
Y. Lu, Q. Xie, G. Kliot, A. Geller, J. Larus, A. Greenberg (2011). Join-idle-queue: A novel load balancing algorithm for dynamically scalable web services. [*Perf. Eval.*]{} [**68 (11)**]{}, 1056–1071.
M. Mitzenmacher (2001). The power of two choices in randomized load balancing. [*IEEE Trans. Par. Distr. Syst.*]{} [**12 (10)**]{}, 1094–1104.
M. Mitzenmacher (2016). Analyzing distributed Join-Idle Queue: A fluid limit approach. Preprint, arXiv:1606.01833.
D. Mukherjee, S.C. Borst, J.S.H. van Leeuwaarden, P.A. Whiting (2016). Asymptotic optimality of power-of-$d$ load balancing in large-scale systems. Preprint, arXiv:1612.00722.
D. Mukherjee, S.C. Borst, J.S.H. van Leeuwaarden, P.A. Whiting (2016). Universality of power-of-$d$ load balancing in many-server systems. Preprint, arXiv:1612.00723.
D. Mukherjee, S.C. Borst, J.S.H. van Leeuwaarden, P.A. Whiting (2016). Universality of power-of-$d$ load balancing schemes. [*ACM SIGMETRICS Perf. Eval. Rev.*]{} [**44 (2)**]{}, 36–38.
A. Mukhopadhyay, A. Karthik, R.R. Mazumdar (2015). Randomized assignment of jobs to servers in heterogeneous clusters of shared servers for low delay. [*Stoc. Syst.*]{} [**6 (1)**]{}, 90–131.
A. Mukhopadhyay, A. Karthik, R.R. Mazumdar, F. Guillemin (2016). Mean field and propagation of chaos in multi-class heterogeneous loss models. [*Perf. Eval.*]{} [**91**]{}, 117–131.
A.L. Stolyar (2015). Pull-based load distribution in large-scale heterogeneous service systems. [*Queueing Systems*]{} [**80 (4)**]{}, 341–361.
A.L. Stolyar (2015). Pull-based load distribution among heterogeneous parallel servers: the case of multiple routers. [*Queueing Systems*]{}, to appear.
N. Vvedenskaya, R. Dobrushin, F. Karpelevich (1996). Queueing system with selection of the shortest of two queues: An asymptotic approach. [*Prob. Inf. Trans.*]{} [**32 (1)**]{}, 20–34.
Q. Xie, X. Dong, Y. Lu, R. Srikant (2015). Power of $d$ choices for large-scale bin packing: A loss model. [*ACM SIGMETRICS Perf. Eval. Rev.*]{} [**43 (1)**]{}, 321–334. L. Ying, R. Srikant, X. Kang (2015). The power of slightly more than one sample in randomized load balancing. [*Proc. IEEE INFOCOM 2015*]{}, 1131–1139.
Proof of \[maintheorem\]
========================
We now present the analyses of the models required for the proof of \[maintheorem\], along the lines sketched in \[jacksonnetwork\].
Blocking model with two dispatchers (better system) {#modelA}
---------------------------------------------------
Consider the blocking scenario with $N$ servers, $R = 2$ dispatchers and arrival fractions $\alpha_1$ and $\alpha_2$. The probability of sending a token to dispatcher $i$ is $\beta_i$. Without loss of generality, assume $\frac{\alpha_1}{\beta_1} \geq \frac{\alpha_2}{\beta_2}$. Since this is a closed Jackson network (see \[jacksonnetwork\]), we have the stationary distribution $$\label{eqrp}
\pi(n_0,n_1,n_2)
= G^{-1} \frac{(\lambda N)^{n_1}}{(\frac{\alpha_1}{2\beta_1}\lambda N)^{n_1}}\frac{(\lambda N)^{n_2}}{(\frac{\alpha_2}{2\beta_2}\lambda N)^{n_2}} \frac{(2\lambda N)^{n_0}}{n_0!},$$ with $G$ the normalization constant. We use this to prove the following proposition.
\[propA\] $${\underset{N\rightarrow\infty}{\lim}}B(2, N,\lambda, \hdots) = \max\left\{\beta_1\left(\frac{\alpha_1}{\beta_1}-\frac{\alpha_2}{\beta_2}\right),1-1/\lambda\right\}.$$
Define ${{\gamma}}_i=\frac{\alpha_i}{2\beta_i}$. The proof of \[propA\] starts from an exact expression for the blocking probability that follows from \[eqrp\]: $$\label{blockingwithZ}
B(2, N, \lambda, \hdots)
=({{\gamma}}_1-{{\gamma}}_2)\frac{1+\frac{\alpha_2}{\alpha_1}\frac{{{\gamma}}_1}{{{\gamma}}_2}Z(N,\lambda,{{\gamma}}_1,{{\gamma}}_2)}{\frac{{{\gamma}}_1}{\alpha_1}-\frac{{{\gamma}}_1}{\alpha_1}Z(N,\lambda,{{\gamma}}_1,{{\gamma}}_2)},$$ with $$Z(N,\lambda,{{\gamma}}_1,{{\gamma}}_2)=\left(\frac{{{\gamma}}_2}{{{\gamma}}_1}\right)^{N+1}\frac{{\sum_{n=0}^N}\frac{\left(2{{\gamma}}_1\lambda N \right)^n}{n!}}{{\sum_{n=0}^N}\frac{\left(2{{\gamma}}_2\lambda N \right)^n}{n!}}.$$ Through tedious calculations, it may be shown that $$\underset{N\rightarrow \infty}{\lim} Z(N,\lambda,{{\gamma}}_1,{{\gamma}}_2)=
\begin{cases}
0 & 2{{\gamma}}_2\lambda\leq 1,\\
\frac{{{\gamma}}_2}{{{\gamma}}_1}\frac{2{{\gamma}}_2\lambda-1}{2{{\gamma}}_2\lambda-\frac{{{\gamma}}_2}{{{\gamma}}_1}} & 2{{\gamma}}_2\lambda>1.
\end{cases}$$ Substitution into \[blockingwithZ\] then yields \[propA\].
Symmetric blocking model (worse system) {#modelB}
---------------------------------------
Consider the blocking model with $N$ servers, $R$ dispatchers and $\alpha_1=\hdots =\alpha_R=1/R$, assume $\lambda\leq 1$ and use the stationary distribution $\pi$ and blocking probability $B(R,N,\lambda)$ provided in \[jacksonnetwork\].
\[recfor\] $$\label{reccc}
B(R+1,N,\lambda)=\frac{R}{N+R-\lambda N(1-B(R,N,\lambda))}.$$
\[propB\] For $\lambda < 1$, $\underset{N\rightarrow \infty}{\lim} B(R,N,\lambda)=0$.
The proofs again start from an exact expression for the blocking probability obtained from the stationary distribution: $$\label{blockingprobR}
B(R, N, \lambda)=\frac{{\sum_{n=0}^N}\binom{N+R-2-n}{R-2}\frac{(\lambda N)^n}{n!}}{{\sum_{n=0}^N}\binom{N+R-1-n}{R-1}\frac{(\lambda N)^n}{n!}}.$$ \[recfor\] follows from inserting \[blockingprobR\] at both sides of \[reccc\]. \[propB\] follows from $$\begin{aligned}
B(R+1, N,\lambda)&=\frac{R}{R+N(1-\lambda) +\lambda N B(R, N,\lambda)}\nonumber\\
&\leq \frac{R}{R+N(1-\lambda)}{\overset{N\rightarrow\infty }{\rightarrow}}0.\end{aligned}$$
|
---
abstract: 'The decay of correlations in ionic fluids is a classical problem in soft matter physics that underpins applications ranging from controlling colloidal self-assembly to batteries and supercapacitors. The conventional wisdom, based on analyzing a solvent-free electrolyte model, suggests that all correlation functions between species decay with a common decay length in the asymptotic far field limit. Nonetheless a solvent is present in many electrolyte systems. We show using an analytical theory and molecular dynamics simulations that multiple decay lengths can coexist in the asymptotic limit as well as at intermediate distances once a hard sphere solvent is considered. Our analysis provides an explanation for the recently observed discontinuous change in the structural force across a thin film of ionic liquid-solvent mixtures as the composition is varied, as well as reframes recent debates in the literature about the screening length in concentrated electrolytes.'
author:
- Fabian Coupette
- 'Alpha A. Lee'
- Andreas Härtel
date: 'published in: Physical Review Letters **121**, 075501 (2018), DOI: [10.1103/PhysRevLett.121.075501](https://doi.org/10.1103/PhysRevLett.121.075501)'
title: Screening lengths in ionic fluids
---
The study of ionic fluids and electrolytes has received significant interest in recent times due to its central relevance to a plethora of technological applications, ranging from controlling colloidal self-assembly [@evans_book_1999] to supercapacitors and batteries [@fedorov_cr114_2014]. The challenge deals with the rich physics that arise from competing long-ranged Coulomb interactions and the steric repulsion of particles. The arrangement of ions in bulk and near interfaces governs properties such as capacitance [@bozym_jpcl6_2015; @limmer_prl115_2015; @uralcan_jpcl7_2016] and effective forces between colloids [@zhang_nature542_2017]; thus a physics understanding of how ion-ion correlations decay and how electric fields are screened is central to designing fit for purpose electrolytes.
The decay of correlations in ionic fluids is a classical problem in soft matter and liquid state physics [@attard_inbook_2007; @levin_rpp65_2002]. According to the conventional wisdom, all correlation functions in a simple fluid mixture where particles interact via short-ranged and Coulomb interactions decay asymptotically in the same form, *i.e.*, $e^{-r/\lambda} \cos(\omega r - \tau)/r$, and, crucially, the decay length $\lambda$ – synonymously the screening length – and oscillation frequency $\omega$ are the same for all correlation functions [@evans_jcp100_1994]. This common decay has been explicitly verified for the restricted primitive model (RPM), a simple binary solvent-free electrolyte model that is paradigmatic in electrolyte physics – it has been shown that the cation-cation, cation-anion and anion-anion correlation functions all decay with the same decay length and oscillation frequency [@attard_pre48_1993; @leotedecarvalho_mp83_1994], which has also been used for the interpretation of experiments [@zeng_langmuir28_2012; @gebbie_pnas112_2015; @smith_prl118_2017]. However, in technological applications, ions are usually mixed with a solvent in order to enhance conductivity and reduce viscosity [@mcewen_jes146_1999; @zhu_science332_2011; @yang_science341_2013]. This raises the important question of how the presence of solvents influences ion-ion correlations.
Recent surface force balance experiments show that the disjoining force between charged surfaces across ionic liquid-solvent mixtures decays in an oscillatory manner with an exponentially decaying envelope [@moazzami-gudarzi_prl117_2016; @smith_prl118_2017; @schoen_bjnt9_2018]. However, as the ion concentration is increased, the oscillation frequency undergoes a steplike transition [@smith_prl118_2017] – at low ion concentration, it is comparable to the size of the solvent molecule, whereas for concentrated electrolytes it is comparable to the size of an ion pair. This is qualitatively reminiscent of structural crossover in a binary mixture of “big” and “small” colloids [@grodon_jcp121_2004; @baumgartl_prl98_2007; @statt_jcp144_2016]. However, an ion-solvent mixture is evidently at least a three component system and a corresponding mechanism in electrolyte-solvent mixtures is, perhaps surprisingly, hitherto unknown.
In this Letter, we demonstrate that the decay of correlation functions in a simple fluid mixture is not necessarily unique, *i.e.*, there is no common asymptotic decay length and oscillation wavelength. By considering a hard sphere electrolyte in a hard sphere solvent – one of the simplest possible extensions of the paradigmatic RPM model that includes the physics of electrolyte-solvent interactions – we show theoretically that ion-ion correlations and ion-solvent correlations can have different asymptotic decay lengths and support this result using simulations. These decays are either density- or charge-driven and related to the length scales of steric and Coulombic interactions. While ion-solvent correlations are not affected by charge correlations, ion-ion correlations decay according to a superposition of both effects. However, asymptotic decay is determined by the slowest decaying contribution, which strongly varies with the system composition. Our theory explains the experimentally observed switch of the structural force as the crossover from density-driven to charge-driven decay [@smith_prl118_2017]. Moreover, it illustrates the importance of space-filling solvent, an often overlooked piece of physics in the theoretical modeling of electrolytes.
To concretize ideas, we consider a hard sphere ion-solvent mixture (HISM) [@grimson_cpl86_1982; @tang_jcp97_1992; @boda_jcp112_2000; @rotenberg_jpcm30_2018] throughout this Letter: ions and solvent are modeled as hard spheres of the same diameter $d$. The ions (solvent) with number density $\rho$ ($\rho_0$) carry point charges $Z_{\pm} = \pm e$ ($Z_{0} = 0$). The dielectric nature of the solvent is modeled by a homogeneous dielectric background with a relative permittivity $\varepsilon$. The pair interaction potential between two particles of species $\nu,\nu' \in \{+,-,0\}$ at separation $r$ is given by $$\begin{aligned}
v_{\nu\nu'}(r) &=
\begin{cases}
\displaystyle{\infty } & r < d \\
\displaystyle{k_{\rm B}T \lambda_{\rm B} \frac{Z_{\nu} Z_{\nu'}}{r}} & r \geq d ,
\end{cases}
\label{eq:pair-interaction-potential}\end{aligned}$$ where $\lambda_{\rm B}=e^2/(4\pi\varepsilon_0\varepsilon k_{\rm B}T)$ denotes the Bjerrum length and $k_{\rm B}$ Boltzmann’s constant.
![\[fig:screening-length\] Total pair-correlation functions $h_{\nu\nu'}(r)$ for ion concentration $\rho=1$ M and concentration of neutral particles (a) $\rho_{0}=10$ M and (b) $\rho_{0}=40$ M. Symbols represent data from MD simulations and lines from our theory. For symmetry reasons, we only show data for the four given combinations of species. The insets show the same data but plotted on semilogarithmic scale. Dashed lines represent the predicted monotonic decay $\exp(-r/\lambda_{\nu\nu'})$ with screening length $\lambda_{\nu\nu'}$ from theory. ](./figure1){width="8.5cm"}
Figure \[fig:screening-length\] shows that the HISM model can have two distinguished coexisting screening lengths at finite range. We performed MD simulations of the HISM in an equilibrated bulk system using the ESPResSo package [@limbach_cpc174_2006; @arnold_book_2013]. Hard particle interactions are modeled using a shifted and truncated purely repulsive Lennard-Jones potential $4\epsilon[(\sigma/r)^{12}-(\sigma/r)^{6}+c_{\rm shift}]$ with $\epsilon=10^4 \, k_{\rm B}T$ and $\sigma=2^{-1/6} \, d$. The simulations are performed in a cubic box of volume $V=L\times L\times L$ with periodic boundaries and $L=30 \, d$. We used $d=0.3$ nm and $\lambda_{\rm B}=0.7$ nm, which corresponds to $\varepsilon\approx 80$ and $T\approx 300$ K. At ionic concentration $\rho=1$ M and solvent concentration $\rho_0=10$ M, Fig. \[fig:screening-length\](a) clearly shows two coexisting decay lengths with oscillatory and purely exponential decay, respectively, at intermediate separations. Figure \[fig:screening-length\](b) shows that at a higher solvent concentration $\rho_0=40$ M, both ion-ion and ion-solvent correlations share the same intermediate decay length and oscillation wavelength. Our theory (see below) predicts that this finite range decay is the same as the asymptotic decay.
To explain the origin of those coexisting decay lengths, we turn to a theoretical description of HISM based on the density functional theory (DFT) formalism [@hansen_book_2013]. Within DFT, the free energy is expressed as a functional of one-body densities [@hansen_book_2013]. For HISM, we can split the pair potential into hard core and electrostatic contributions, $v_{\nu \nu'} = v^{\mathrm{hs}}_{\nu \nu'} +v^{\mathrm{es}}_{\nu \nu'}$. The difference between ideal gas free energy and the exact free energy can be partitioned into three components [@hansen_book_2013], $$\begin{aligned}
F = F^{\rm hs} + F^{\rm es} + F^{\rm corr} ,
\label{eq:functional_expansion}\end{aligned}$$ where $F^{\rm hs}$ is the hard sphere contribution, $F^{\rm es}$ the electrostatic contribution, and $F^{\rm corr}$ a correlation term that contains remaining contributions. The splitting in Eq. (\[eq:functional\_expansion\]), although mathematically trivial, allows us to identify symmetries in the corresponding direct correlations $c_{\nu\nu'}^{\rm hs}$, $c_{\nu\nu'}^{\rm es}$, and $c_{\nu\nu'}^{\rm corr}$. The latter follow from a second functional derivative of the excess free energy with respect to the density, *i.e.*, $$\begin{aligned}
c_{\nu\nu'}(r)
&= - \frac{1}{k_{\rm B}T}
\frac{\delta^2 F}{\delta \rho_{\nu}({\bf r_1}) \delta \rho_{\nu'}({\bf r_2)}} , \end{aligned}$$ where the homogeneity of the bulk implies $r=|{\bf r_1}-{\bf r_2}|$. The hard sphere contribution depends only on the macroscopic packing fraction and the particle diameter $d$, and therefore, it scales equally with the number density of each component. From $F^{\rm es}$ given by [@hansen_book_2013] $$\begin{aligned}
F^{\rm es} &= \frac{1}{2} \sum_{\nu} \sum_{\nu'}
\int\int \rho_\nu( {\bf r}) \rho_{\nu'}({\bf r'}) v^{\rm es}_{\nu \nu'}({\bf r},{\bf r'}) {\; \mathrm{d}}{\bf r} {\; \mathrm{d}}{ \bf r'} , \end{aligned}$$ the electrostatic contribution follows with $$\begin{aligned}
c_{\nu\nu'}^{\rm es}(r) &= -\frac{v^{\rm es}_{\nu \nu'}(r)}{k_{\rm B}T} .
\label{eq:es-c}\end{aligned}$$ Hence, $c^{\rm es}:= c_{++}^{\rm es} = -c_{+-}^{\rm es}$. Finally, the correlation term underlies the fundamental symmetries of the system, *i.e.*, positive and negative ions are structurally equivalent such that $c_{++}^{\rm corr} = c_{--}^{\rm corr}$, $c_{+-}^{\rm corr} = c_{-+}^{\rm corr}$, and $c_{+0}^{\rm corr} = c_{-0}^{\rm corr} = c_{0-}^{\rm corr} = c_{0+}^{\rm corr}$.
The decomposition in Eq. (\[eq:functional\_expansion\]) entails that the most general form of the direct correlation matrix $\mathcal{C}$ in the species basis $\{+,-,0\}$ for the HISM model is given by $$\begin{aligned}
\mathcal{C} &=
\begin{pmatrix}
c^{\rm hs} + c^{\rm es} + c_{++}^{\rm corr} & c^{\rm hs} - c^{\rm es} + c_{+-}^{\rm corr} & c^{\rm hs} + c_{+0}^{\rm corr} \\
c^{\rm hs} - c^{\rm es} + c_{+-}^{\rm corr} & c^{\rm hs} + c^{\rm es} + c_{++}^{\rm corr} & c^{\rm hs} + c_{+0}^{\rm corr} \\
c^{\rm hs} + c_{+0}^{\rm corr} & c^{\rm hs} + c_{+0}^{\rm corr} & c^{\rm hs} + c_{00}^{\rm corr}
\end{pmatrix}.
\label{eq:hism-c}\end{aligned}$$ To proceed, we need to relate the direct correlation functions to the total correlation functions $h_{\nu\nu'}=(\mathcal{H})_{\nu\nu'}$, the observables in simulations and experiments. We use the Ornstein-Zernike relation in Fourier space $$\begin{aligned}
\hat{\mathcal{H}} &= \left( \mathbb{1}-\hat{\mathcal{C}} \varrho \right)^{-1} \hat{\mathcal{C}} ,
\label{eq:oz-fourier}\end{aligned}$$ where we introduced a number density matrix $\varrho = \mathrm{diag}(\rho,\rho,\rho_0)$, and $\hat{f}$ denotes the Fourier transformation of a function $f$. Substituting Eq. (\[eq:hism-c\]) into (\[eq:oz-fourier\]) yields an algebraic expression for the total correlation matrix $\hat{\mathcal{H}}$, the eigenvectors of which are given by ${\bf w}_{\rm cc}=(1,-1,0)$, ${\bf w}_{\rm dd}^{+}$, and ${\bf w}_{\rm dd}^{-}$. The former is equal to one of the eigenvectors of the RPM and gives rise to the well established charge-charge correlation $h_{\rm cc}=h_{++}-h_{+-}$ as an eigenvalue [@hansen_book_2013]. The eigenvectors ${\bf w}_{\rm dd}^{\pm}$ become stationary in the limit of vanishing $c_{\nu\nu'}^{\rm corr}$ with $(1,1,1)$ and $(-1,-1,2)$; the first of them gives rise to a density-density correlation, while the second corresponds to an ion-solvent correlation that has a vanishing eigenvalue.
In particular, the resulting total charge-charge correlation function reads $$\begin{aligned}
\hat{h}_{\rm cc} &=
\frac{\hat{c}^{\rm corr}_{ \rm cc} + 2 \hat{c}^{\rm es} }
{1 - \rho ( \hat{c}^{\rm corr}_{ \rm cc} + 2 \hat{c}^{\rm es}) } .
\label{eq:hcc}\end{aligned}$$ Transforming it back into real space yields the formal solution $$\begin{aligned}
h_{\rm cc}(r)
&= \frac{1}{2 \pi^2 r} \int_0^\infty k \sin(k r)
\frac{\hat{c}^{\rm corr}_{ \rm cc} + 2 \hat{c}^{\rm es} }
{1 - \rho ( \hat{c}^{\rm corr}_{ \rm cc} + 2 \hat{c}^{\rm es})}
{\; \mathrm{d}}k \nonumber \\
&= \frac{1}{2 \pi r}\sum_{q \in Q_{\rm cc}}
\Re \left[ \mathrm{Res}_q \left\{
\frac{ (\hat{c}^{\rm corr}_{ \rm cc} + 2 \hat{c}^{\rm es}) q \exp(i q r) }
{1 - \rho ( \hat{c}^{\rm corr}_{ \rm cc} + 2 \hat{c}^{\rm es}) }
\right\} \right] ,
\label{eq:residue}\end{aligned}$$ where $Q_{\rm cc}$ contains the roots of $$\begin{aligned}
1 - \rho ( \hat{c}^{\rm corr}_{ \rm cc} + 2 \hat{c}^{\rm es}) &= 0
\label{eq:root}\end{aligned}$$ with positive imaginary part. The second equality in Eq. (\[eq:residue\]) makes use of the residue theorem and does, therefore, only hold without further analysis if Eq. (\[eq:root\]) does not have any purely real solutions and the elements of $Q_{\rm cc}$ are isolated points in the upper complex half plane (we refer to Refs. [@kjellander_cpl200_1992; @attard_pre48_1993; @leotedecarvalho_mp83_1994; @evans_jcp100_1994] for similar derivations). The eigenvalues to ${\bf w}^\pm_{\rm dd}$ share a common denominator, *i.e.*, they are a set $Q_{\rm dd}$ of singularities corresponding to the roots with positive imaginary part of the generic equation $$\begin{aligned}
1 - \rho (2\hat{c}^{\rm hs} + \hat{c}_{\rm dd}^{\rm corr}) &~ \notag \\
- \rho_0 [ 2\rho(\hat{c}_{+0}^{\rm corr})^2 + \hat{c}_{00}^{\rm corr} (1-\rho\hat{c}_{\rm dd}^{\rm corr}) &~ \notag \\
+ \hat{c}^{\rm hs}(1 - \rho (\hat{c}_{\rm dd}^{\rm corr}-4\hat{c}_{+0}^{\rm corr}+2\hat{c}_{00}^{\rm corr}))] &= 0 ,
\label{eq:root2}\end{aligned}$$ where $c_{\rm dd}^{\rm corr}:= c_{++}^{\rm corr}+c_{+-}^{\rm corr}$. Note that Eq. \[eq:root2\] is independent of $c^{\rm es}$.
The dominant contribution to a total correlation function $h_{\nu\nu'}$ in the asymptotic long-range limit $r\to\infty$ is determined by the (leading) pole $\bar{q}_{\nu \nu'}=\Re[\bar{q}_{\nu \nu'}] +i \Im[\bar{q}_{\nu \nu'}] \in Q_{\nu\nu'}$ with the smallest imaginary part [@fisher_jcp50_1969]. It is convenient to introduce the decay length $\lambda_{\nu \nu'}=1/\Im[\bar{q}_{\nu \nu'}]$ and decay oscillation frequency $\omega_{\nu \nu'}=\Re[\bar{q}_{\nu \nu'}]$. This pole causes the asymptotic decay [@fisher_jcp50_1969; @henderson_jcp97_1992; @evans_jcp100_1994] $$\begin{aligned}
h_{\nu\nu'}(r \rightarrow \infty) \propto \frac{\exp(- r/\lambda_{\nu\nu'} ) \cos(\omega_{\nu\nu'} r-\tau_{\nu\nu'})}{r} ,
\label{eq:fit}\end{aligned}$$ where $\tau_{\nu\nu'}$ is a phase shift. The pole, however, could be suppressed on intermediate length scales by a small amplitude such that its contribution would become neglectable. If there are two poles with decay lengths $\lambda_1>\lambda_2$ but amplitudes $A_1<A_2$, pole 2 will dominate until $r\gtrsim\log(A_1/A_2)[\lambda_1^{-1}-\lambda_2^{-1}]^{-1}$, which is a long length scale if $A_1 \ll A_2$.
Importantly, two competing decay lengths arise from the solutions to Eqs. (\[eq:root\]) and (\[eq:root2\]). Switching back into the species basis yields the central result of this Letter, $$\begin{aligned}
\lambda_{++} =
\lambda_{+-} &= \max[\lambda_{\rm cc}, \lambda_{\rm dd}] , \label{eq:lambda-cc} \\
\lambda_{0+} = \lambda_{0-} = \lambda_{00} &= \lambda_{\rm dd} . \label{eq:lambda-dd}\end{aligned}$$ The charge-charge correlation does not affect solvent correlations, because ${\bf w}_{\rm cc}\perp(0,0,1)$. Notice that we only made use of the fundamental symmetries in HISM. In other words, in general, it is not true that all species correlations decay with the same decay length. Correlations involving solvent particles decay on a length scale $\lambda_{\rm dd}$ different from the charge-charge correlation length scale $\lambda_{\rm cc}$. If $\lambda_{\rm cc} > \lambda_{\rm dd}$, two distinct length scales coexist, as we have shown for intermediate ranges in Fig. \[fig:screening-length\](a). The same applies for the corresponding oscillation frequencies $\omega_{\rm cc}$ and $\omega_{\rm dd}$. Crucially, this implies that while the dominant decay length continuously changes, the oscillation wavelength of ion-ion correlations can rapidly shift.
![\[fig:screening-regimes\] Theoretical prediction for (a) decay length and (b) inverse oscillation frequency of the leading charge and density pole, respectively, shown against the ion concentration $\rho$, for $\rho_0=10$ M and $\rho_0=40$ M with $d=0.3$ nm and $\lambda_{\rm B}=0.7$ nm. Symbols in (a) mark the decay lengths that correspond to the data in Figs. \[fig:screening-length\](a) ($\square$) and \[fig:screening-length\](b) ($\bigcirc$). Vertical dashed lines mark points where the asymptotically leading pole changes from charge to density and *vice versa* for $\rho_0=40$ M; leading inverse oscillation frequencies for $\rho_0=40$ M are highlighted with bold lines. ](./figure2){width="8.5cm"}
To illustrate this effect, we proceed by specifying the functional $F$ in our theoretical framework: we use the White Bear mark II functional for the hard-sphere contribution [@hansen-goos_jpcm18_2006] and Eq. (\[eq:es-c\]) with $v_{\nu\nu'}^{\rm es}(r)=0$ for $r<d$ for the electrostatic term. By setting $c^{\rm corr}\equiv 0$ we obtain analytical correlation functions that are sufficient to illustrate the mechanism of the wavelength switch; for the observed systems deviations due to this approximation mainly occur at particle contact, as shown in Fig. \[fig:screening-length\].
Figure \[fig:screening-regimes\] shows quantitative predictions of our theory for the decay lengths and the oscillation wavelengths in HISM. The density-induced correlation length, $\lambda_{\rm dd}$, is a monotonic function of the macroscopic volume fraction because steric correlations are enhanced as the system becomes denser. However, the charge-induced correlation length, $\lambda_{\rm cc}$, is a nonmonotonic function of the ion density but independent of the solvent density. Further, this is the length scale of the decay of the effective electrostatic potential that an ion generates. $\lambda_{\rm cc}$ decreases for an increasing ion density in a dilute electrolyte because ions are surrounded by counterions and this arrangement progressively screens the electric field that an ion generates. However, past a threshold ion concentration, ion-ion correlations lead to a counterion solvation shell that overcompensates the ionic charge, which causes a second solvation shell to solvate the counterions, triggering an oscillatory decay [@attard_inbook_2007]. In this regime, increasing the ion concentration amplifies ion-ion correlations; thus the screening length grows. The situation, when the charge pole that determines $\lambda_{\rm cc}$ changes from purely imaginary to complex, *i.e.*, the decay changes from monotonic to oscillatory, is called a Kirkwood transition [@kirkwood_jcp7_1939], and here it coincides with the change between decreasing and increasing screening length. When $\lambda_{\rm cc} > \lambda_{\rm dd}$, which is the case for a large region of RPM’s parameter space, the ion-ion correlations decay with a decay length that is the electrostatic screening length but different from the ion-solvent and solvent-solvent correlations decay (Fig. \[fig:screening-regimes\]a). For a high solvent concentration, however, we find a regime $\lambda_{\rm dd} >\lambda_{\rm cc}$ where all species correlations decay with one common decay length $\lambda_{\rm dd}$ \[see also Fig. \[fig:screening-length\](b)\] but different from the charge-charge decay length. Thus, the electrostatic screening length must be distinguished from the decay length of species correlation functions that is typically observed in experiments.
Although the ion-ion decay length switches continuously from one pole to another in Fig. \[fig:screening-regimes\](a), Fig. \[fig:screening-regimes\](b) shows that the corresponding oscillation frequency exhibits a discontinuous jump that occurs when the two leading poles have equal imaginary but different real parts. This jump is precisely the effect observed in experimental studies of the surface force across ion-solvent mixtures [@smith_prl118_2017] – the oscillation wavelength switches abruptly. In the experiment, ions and solvent molecules are approximately of the same size, and the oscillation wavelength jumps from $d$ to $2d$, which agrees squarely with the prediction in Fig. \[fig:screening-regimes\](b) (see the Supplemental Material [@SM] for a detailed comparison). Note that the position of this discontinuous jump in the oscillation wavelength is different from the onset of charge oscillations at the Kirkwood transition when the real part of the charge pole first takes a finite nonvanishing value [@parrinello_rnc2_1979; @stell_prl37_1976]. Furthermore, the increase of the decay length in Fig. \[fig:screening-regimes\](a) accurately describes the decay of the structural force in experiments [@smith_prl118_2017]. However, the experiments show an additional much longer decay length at large separations, which is neither predicted in our theory and other recent theoretical studies of underscreening [@lee_prl119_2017; @rotenberg_jpcm30_2018] nor observed in our simulations on HISM (see Fig. S2 in [@SM]). This long-ranged decay length might arise from a set of additional poles induced by a mechanism that is not contained in the simplified HISM model. For instance, dipolar solvent-solvent interactions, as present in water, could lead to an additional long decay length. Since this long-ranged decay is experimentally only observed at long distances, the corresponding leading pole should have a small amplitude and, therefore, could be suppressed at intermediate distances (see Fig. \[fig:screening-length\]).
![\[fig:phase-diag\] (a) Different regimes of decay in the $\rho$ – $\rho_0$ plane of ion and neutral particle concentrations: (A) purely exponential, charge-dominated, (B) oscillatory, exponentially damped, density-dominated, and (C) oscillatory, exponentially damped, charge-dominated. The vertical dashed line represents the Kirkwood line (K) [@kirkwood_jcp7_1939; @leotedecarvalho_mp83_1994]. The right boundary of region A represents the Fisher-Widom line (FW) [@fisher_jcp50_1969; @evans_mp80_1993]. The tilted dashed line marks a total volume fraction $\eta=0.5$. Symbols are explained in Fig. \[fig:screening-regimes\]. (b) The same diagram in the $\rho$ – $\lambda_{\rm B}$ plane for $\rho_0=40$ M. ](./figure3){width="8.0cm"}
The three different regimes of asymptotic decay in HISM – purely exponential and charge-dominated decay (A), oscillatory exponentially damped and density-dominated decay (B), and oscillatory exponentially damped and charge-dominated decay (C) – are summarized in Fig. \[fig:phase-diag\]. While ion-ion correlations in regions A and C are dominated by the charge pole, ion-ion correlations couple to the solvent in region B. This region appears at high solvent concentrations between A and C such that the Fisher-Widom line [@fisher_jcp50_1969; @evans_mp80_1993] of the ions shifts towards lower ion concentrations (away from the Kirkwood line [@kirkwood_jcp7_1939; @leotedecarvalho_mp83_1994]). A second branch separates regions A and C at which the frequency jumps from $\omega_{\rm dd}$ to $\omega_{\rm cc}$.
Our conclusions are derived by assuming symmetry between positive and negative ions in Eq. (\[eq:hism-c\]). If this symmetry is broken by different ion sizes, all correlation functions couple and share the same set of poles; thus they all decay asymptotically in the same form. However, at intermediate range, simulations of asymmetric ions still exhibit the same coexistence of decay lengths and oscillation frequencies as shown here for the symmetric case [@SM]. Consequently, ion size asymmetry can be considered as a perturbation to the symmetric HISM model so that its predictions are still valid for decay lengths in asymmetric systems at (experimentally relevant) intermediate distances.
In summary, we demonstrated the possible coexistence of two asymptotic decay lengths for hard sphere ions in a hard sphere solvent. Our theory explains recent experimental findings concerning a jump of the wavelength of the structural force in ionic fluids [@smith_prl118_2017], and it sheds new light on the screening in dense electrolytes and the fitting of structural forces [@schoen_bjnt9_2018]. Our results are important for the interpretation of measurements and effective interactions [@gottwald_prl92_2004; @leger_jcp123_2005; @denton_pre96_2017; @schoen_bjnt9_2018], because they show that species correlation functions can be superpositions of charge contributions and density contributions of the same order of magnitude. A fit using the asymptotic form (\[eq:fit\]) hence cannot be expected to be accurate on intermediate length scales. Furthermore, the transition from monotonic to oscillatory decay underpins wetting phenomena [@chernov_prl60_1988; @henderson_pre50_1994]. The existence of multiple coexisting species-dependent decay lengths implies that addressable wetting could be achieved. Tuning the asymptotic correlations may also be used to control colloidal dispersions, for instance to prevent aggregation [@zhang_nature542_2017] and to switch effective potentials by tuning the salt concentration [@li_pnas114_2017]. It might be promising to construct complex interactions to achieve a rich crossover structure, for instance in complex plasmas [@morfill_rmp81_2009], colloid-polymer mixtures [@brader_pre63_2001], and colloidal fluids [@archer_jcp126_2007].
The authors would like to thank M. Oettel, R. Kjellander, and R. Evans for insightful discussions. A. A. L. acknowledges the support of the Winton Programme for the Physics of Sustainability.
[53]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [**]{} (, , ) [****, ()](\doibase 10.1021/cr400374x) [****, ()](\doibase
10.1021/acs.jpclett.5b00899) [****, ()](\doibase 10.1103/PhysRevLett.115.256102) [****, ()](\doibase 10.1021/acs.jpclett.6b00859) [****, ()](\doibase 10.1038/nature21041) “,” in [**](\doibase 10.1002/9780470141519.ch1), Vol. (, ) Chap. , pp. [****, ()](\doibase 10.1088/0034-4885/65/11/201) [****, ()](\doibase 10.1063/1.466920) [****, ()](\doibase 10.1103/PhysRevE.48.3604) [****, ()](\doibase 10.1080/00268979400101491) [****, ()](\doibase 10.1021/la2049822) [****, ()](\doibase 10.1073/pnas.1508366112) [****, ()](\doibase 10.1103/PhysRevLett.118.096002) [****, ()](\doibase 10.1149/1.1391827) [****, ()](\doibase 10.1126/science.1200770) [****, ()](\doibase
10.1126/science.1239089) [****, ()](\doibase
10.1103/PhysRevLett.117.088001) [****, ()](\doibase 10.3762/bjnano.9.101) [****, ()](\doibase
10.1063/1.1798057) [****, ()](\doibase
10.1103/PhysRevLett.98.198303) [****, ()](\doibase
10.1063/1.4945808) [****, ()](\doibase 10.1016/0009-2614(82)83119-9) [****, ()](\doibase 10.1063/1.463595) [****, ()](\doibase 10.1063/1.481507) [****, ()](\doibase 10.1088/1361-648X/aaa3ac) [****, ()](\doibase
10.1016/j.cpc.2005.10.005) in [**](\doibase 10.1007/978-3-642-32979-1_1), , Vol. , (, ) pp. @noop [**]{}, ed. (, , ) [****, ()](\doibase 10.1016/0009-2614(92)87048-T) [****, ()](\doibase 10.1063/1.1671624) [****, ()](\doibase 10.1063/1.463652) [****, ()](\doibase 10.1088/0953-8984/18/37/002) [****, ()](\doibase 10.1063/1.1750344) @noop [****, ()](\doibase 10.1007/BF02724355) [****, ()](\doibase 10.1103/PhysRevLett.37.1369) [****, ()](\doibase 10.1103/PhysRevLett.119.026002) [****, ()](\doibase 10.1080/00268979300102621) [****, ()](\doibase
10.1103/PhysRevLett.92.068301) [****, ()](\doibase 10.1063/1.1979480) [****, ()](\doibase 10.1103/PhysRevE.96.062610) [****, ()](\doibase 10.1103/PhysRevLett.60.2488) [****, ()](\doibase 10.1103/PhysRevE.50.4836) [****, ()](\doibase
10.1073/pnas.1713168114) [****, ()](\doibase 10.1103/RevModPhys.81.1353) [****, ()](\doibase 10.1103/PhysRevE.63.041405) [****, ()](\doibase
10.1063/1.2405355) [****, ()](\doibase 10.1021/acs.jpclett.6b00867) [****, ()](\doibase
10.1039/C7FD00168A) [****, ()](\doibase 10.1063/1.3569131) [****, ()](\doibase 10.1016/S0167-7322(01)00192-1) [****, ()](\doibase
10.1103/PhysRevE.82.051404)
|
---
abstract: |
Plant-pollinator associations are often seen as purely mutualistic, while in reality they can be more complex. Indeed they may also display a diverse array of antagonistic interactions, such as competition and victim–exploiter interactions. In some cases mutualistic and antagonistic interactions are carried-out by the same species but at different life-stages. As a consequence, population structure affects the balance of inter-specific associations, a topic that is receiving increased attention. In this paper, we developed a model that captures the basic features of the interaction between a flowering plant and an insect with a larval stage that feeds on the plant’s vegetative tissues (e.g. leaves) and an adult pollinator stage. Our model is able to display a rich set of dynamics, the most remarkable of which involves victim–exploiter oscillations that allow plants to attain abundances above their carrying capacities, and the periodic alternation between states dominated by mutualism or antagonism. Our study indicates that changes in the insect’s life cycle can modify the balance between mutualism and antagonism, causing important qualitative changes in the interaction dynamics. These changes in the life cycle could be caused by a variety of external drivers, such as temperature, plant nutrients, pesticides and changes in the diet of adult pollinators.
Keywords: *mutualism, pollination, herbivory, insects, stage-structure, oscillations*
author:
- 'Tomás A. Revilla'
- 'Francisco Encinas-Viso'
bibliography:
- 'plamodel.bib'
title: 'Dynamical transitions in a pollination–herbivory interaction: a conflict between mutualism and antagonism'
---
Introduction
============
> *Il faut bien que je supporte deux ou trois chenilles si je veux connaître les papillons*
>
> Le Petit Prince, Chapitre IX – Antoine de Saint-Exupéry
Mutualism entails costs in addition to benefits. Conflicting goals can lead to cheating where one party incurs the cost of providing energy to enable mutualism, while the other exploits, but does not reciprocate (e.g. nectar robbers). There can also be costs concerning other detrimental interactions that run in parallel with mutualism, such as predation, parasitism or competition involving the same parties. Moreover, some of these antagonistic interactions (e.g. competition) seem to be important for the evolution and stability of mutualism [@Jones_compmut_2012]. In general, these costs have important consequences at the population and community level, because the net outcome can turn out beneficial or detrimental, but perhaps more interestingly, variable [@bronstein-tree94]. Variable interactions challenge the view that ecological communities are structured by well defined interactions at the species level such as competition (–,–), victim-exploiter (–,+) or mutualism (+,+).
Pollination is one of the most important mutualisms occurring between plants and animals. This form of trading resources for services greatly explains the evolutionary success of flowering plants in almost all terrestrial systems. It is responsible for the well being of ecosystem services. During the larval stage of many insect pollinators, such as Lepidopterans (butterflies and moths), the larvae feed on plant leaves to mature and become adult pollinators [@adler_bronstein-ecology04; @wackers_etal-are07; @bronstein_etal-annbot09; @altermatt_pearse-amnat11]. These ontogenetic diet shifts [@rudolf_lafferty-ecolett11] are very common and important in understanding the ecological and evolutionary dynamics of plant–animal mutualisms. Interestingly, in some cases larvae feed on the same plant species that they will pollinate as adults [@irwin-cb10; @bronstein_etal-annbot09]. This shows that in several cases mutualistic and antagonistic interactions are exerted by the same species. and a potential conflict arises for the plant. between the benefits of mutualism and the costs of herbivory. One of the best known examples is the interaction between tobacco plants (*Nicotiana* *attenuata*) and the hawkmoth (*Manduca* *sexta*) [@baldwin-oecologia88; @kessler_etal-cb10], whose larva is commonly called the tobacco hornworm. There are other examples of this type of interaction in the genus *Manduca* (Sphingidae), such as between the tomato plant (*Lycopersicon esculentum*) and the five-spotted hawkmoth (*Manduca quinquemaculata*) [@Kennedy_2003]. These larvae have received a lot of attention due to their negative effects on agricultural crops [@Campbell_1991_pests].
The interaction between *Manduca sexta* and *Datura wrightii* (Solanacea) [@bronstein_etal-annbot09; @alarcon_etal-ecoent08] is another good example illustrating the costs and benefits of pollination mutualisms [@bronstein_etal-annbot09]. *D. wrightii* provides high volumes of nectar and seems to depend heavily on the pollination service by *M. sexta* adults [@alarcon_etal-ecoent08]. However, *M. sexta* larvae, which feed on *D. wrightii* vegetative tissue, can have severe negative effects on plant fitness [@McFadden_1968; @Barron-Gafford_2012]. We could assume that the benefits of pollination might outweigh the costs of herbivory for this mutualism to be relatively viable. The question is what are the conditions, in terms of benefits (pollination) and costs (herbivory), for this mutualistic interaction to be stable?
In the pollination–herbivory cases mentioned previously the benefits and costs for the plant are clearly differentiated. This is because the role of an insect as a pollinator or herbivore depends on the stage in its life cycle [@miller_rudolf-tree11]. Thus, whether mutualism or herbivory dominates the interaction is dependent on insect abundance and its population structure. In other words the *cost:benefit* ratio must be positively related with the insect’s *larva:adult* ratio. For a hypothetical scenario in which the costs of herbivory (–) and the benefits of pollination (+) are balanced for the plant (0), an increase in larval abundance relative to adults should bias the relationship towards a victim-exploiter one (–,+). Whereas an increase in adult abundance relative to larvae should bias the relationship towards mutualism (+,+). Under equilibrium conditions, one would expect transitions (bifurcations) from (–,+) to (0,+) to (+,+) and vice-versa as relevant parameters affecting the plant and the insect life-histories vary, such as flower production, mortalities or larvae maturation rates. However, under dynamic scenarios the outcome may be more complex: a victim–exploiter state (–,+) enhances larva development into pollinating adults, but this tips the interaction into a mutualism (+,+), which in turn contributes greater production of larva leading back to a victim–exploiter state (–,+). This raises the possibility of feedback between the plant–insect interaction and insect population structure, which can potentially lead to periodic alternation between mutualism and herbivory. Thus, when non-equilibrium dynamics are involved, questions concerning the overall nature (positive, neutral or negative) of mixed interactions may not have simple answers.
In this article we study the feedback between insect population structure, pollination and herbivory. We want to understand how the balance between costs (herbivory) and benefits (pollination) affects the interaction between plants (e.g. *D. wrightii*) and herbivore–pollinator insects (e.g. *M. sexta*)? Also what role does insect development have in this balance and on the resulting dynamics? We use a mathematical model which considers two different resources provided by the same plant species, nectar and vegetative tissues. Nectar consumption benefits the plant in the form of fertilized ovules, and consumption of vegetative tissues by larvae causes a cost. Our model predicts that the balance between mutualism and antagonism, and the long term stability of the plant–insect association, can be greatly affected by changes in larval development rates, as well as by changes in the diet of adult pollinators.
PLA (plant-larva-adult) model\[sec:Model\]
==========================================
Our model concerns the dynamics of the interaction between a plant and an insect. The insect life cycle comprises an adult phase that pollinates the flowers and a larval phase that feed on non-reproductive tissues of the same plant. Adults oviposit on the same species that they pollinate (e.g. *D. wrightii – M. sexta* interaction). Let denote the biomass densities of the plant, the larva, and the adult insect with $P,L$ and $A$ respectively. An additional variable, the total biomass of flowers $F$, enables the mutualism by providing resources to the insect (nectar), and by collecting services for the plant (pollination). The relationship is *facultative–obligatory*. In the absence of the insect, the plant’s vegetative biomass grows logistically, preventing its extinction. In the absence of the plant however, the insect always goes extinct because larval development relies exclusively on herbivory, even if the adults pollinate other plant species. This is based on the biology of *M. sexta* [@bronstein_etal-annbot09]. The mechanism of interaction between these four variables ($P,L,A,F$) is described by the following system of ordinary differential equations (ODE):
$$\begin{aligned}\frac{dP}{dt} & =rP(1-cP)+\sigma aFA-bPL\\
\frac{dF}{dt} & =sP-wF-aFA\\
\frac{dL}{dt} & =\epsilon aFA+gA-\gamma bPL-mL\\
\frac{dA}{dt} & =\gamma bPL-nA
\end{aligned}
\label{eq:pfla}$$
where $r$: plant intrinsic growth rate, $c$: plant intra-specific self-regulation coefficient (also the inverse its carrying capacity), $a$: pollination rate, $b$: herbivory rate, $s:$ flower production rate, $w$: flower decay rate, $m,n$: larva and adult mortality rates, $\sigma$: plant pollination efficiency ratio, $\epsilon$: adult consumption efficiency ratio. Like $\epsilon$, parameter $\gamma$ is also a consumption efficiency ratio, but we will call it the maturation rate for brevity since we will refer to it frequently. Our model assumes that pollination leads to flower closure [@primack-ares85], causing resource limitation for adult insects. Parameter $g$ represents a reproduction rate resulting from the pollination of other plants species, which we do not model explicitly. Most of our results are for $g=0$.
We now consider the fact that flowers are ephemeral compared with the life cycles of plants and insects. In other words, some variables $(P,L,A)$ have slower dynamics, and others $(F)$ are fast [@rinaldi_scheffer-ecosystems00]. Given the near constancy of plants and animals in the flower equation of (\[eq:pfla\]), we can predict that flowers will approach a quasi-steady-state (or quasi-equilibrium) biomass $F\approx sP/(w+aA)$, before $P,L$ and $A$ can vary appreciably. Substituting the quasi-steady-state biomass in system (\[eq:pfla\]) we arrive at:
$$\begin{aligned}\frac{dP}{dt} & =rP(1-cP)+\sigma\left[\frac{asA}{w+aA}\right]P-bPL\\
\frac{dL}{dt} & =\epsilon\left[\frac{asP}{w+aA}\right]A+gA-\gamma bPL-mL\\
\frac{dA}{dt} & =\gamma bPL-nA
\end{aligned}
\label{eq:upla}$$
In system (\[eq:upla\]) the quantities in square brackets can be regarded as functional responses. Plant benefits saturate with adult pollinator biomass, i.e. pollination exhibits diminishing returns. The functional response for the insects is linear in the plant biomass, but is affected by intraspecific competition [@schoener-tpb78] for mutualistic resources.
We non-dimensionalized this model to reduce the parameter space from 12 to 9 parameters, by casting biomasses with respect to the plant’s carrying capacity $(1/c)$ and time in units of plant biomass renewal time $(1/r)$. This results in a PLA (plant, larva, adult) scaled model:
$$\begin{aligned}\frac{dx}{d\tau} & =x(1-x)+\sigma\frac{\alpha z}{\eta+z}x-\beta xy\\
\frac{dy}{d\tau} & =\epsilon\frac{\alpha x}{\eta+z}z+\phi z-\gamma\beta xy-\mu y\\
\frac{dz}{d\tau} & =\gamma\beta xy-\nu z
\end{aligned}
\label{eq:pla}$$
Table \[tab:vars\_and\_pars\] lists the relevant transformations.
Symbol Description Value $c=0.01,r=0.05$
------------------ ---------------------------------------------- ---------- --------------------
$x=cP,y=cL,z=cA$ plant, larval and adult biomass variable
$\tau=rt$ time variable
$\alpha=s/r$ asymptotic pollination rate 5 $s=0.25$
$\eta=wc/a$ half-saturation constant of pollination 0.1 $w=0.5$ & $a=0.05$
$\beta=b/rc$ herbivory rate 0 to 100 $b=0$ to 0.05
$\mu=m/r$ larva mortality rate 1 $m=0.05$
$\nu=n/r$ adult mortality rate 2 $n=0.1$
$\phi=g/r$ insect intrinsic reproduction rate 0 or 1 $g=0$ or 0.05
$\sigma$ plant pollination conversion ratio 5
$\epsilon$ insect pollination conversion ratio 0.5
$\gamma$ maturation rate (herbivory conversion ratio) 0 to 0.1
: \[tab:vars\_and\_pars\]Variables and parameters of the scaled PLA model (\[eq:pla\]) and values used for numerical analyses. The last column shows a corresponding set of parameter values in the unscaled version of the same model (\[eq:upla\]), for plant carrying capacities of $c^{-1}=100$ biomass units, and $r^{-1}=20$ time units.
There is an important clarification to make concerning the nature and scales of the conversion efficiency ratios $\sigma,\epsilon$ involved in pollination, and $\gamma$ for herbivory and maturation. This has to do with the fact that flowers *per se* are not resources or services, but *organs* that enable the mutualism to take place, and they mean different things in terms of biomass production for plants and animals. For insects, the yield of pollination is thermodynamically constrained. First of all, a given biomass $F$ of flowers contains an amount of nectar that is necessarily less than $F$. More importantly, part of this nectar is devoted to survival, or wasted, leaving even less for reproduction. Similarly, not all the biomass consumed by larvae will contribute to their maturation to adult. *Ergo* $\epsilon<1,\gamma<1$. Regarding the returns from pollination for the plants, the situation is very different. Each flower harbors a large number of ovules, thus a potentially large number of seeds [@fagan_etal-theorecol14], each of which will increase in biomass by consuming resources not considered by our model (e.g. nutrients, light). Consequently, a given biomass of pollinated flowers can produce a larger biomass of mature plants, making $\sigma$ larger than 1.
Results\[sec:Results\]
======================
The PLA model (\[eq:pla\]) has many parameters, however here we focus on herbivory rates $(\beta)$ and larvae maturation $(\gamma)$, because increasing $\beta$ turns the net balance interaction towards antagonism, whereas increasing $\gamma$ shifts insect population structure towards the adult phase, turning the net balance towards mutualism. Both parameters also relate to the state variables at equilibrium (i.e. $z/y=\beta\gamma x/\nu$ in (\[eq:pla\] for $dz/d\tau=0$). In section \[sub:numerical\] we studied the joint effects of varying $\beta$ and $\gamma$ numerically (parameter values in Table \[tab:vars\_and\_pars\]). In section \[sub:analytical\] we present a simplified graphical analysis of our model, in order to explain how different dynamics can arise, by varying $\beta,\gamma$ and other parameters.
Numerical results\[sub:numerical\]
----------------------------------
Figure \[fig:beta\_vs\_gamma\_phi0\] shows interaction outcomes of the PLA model, as a function of $\beta$ and $\gamma$ for specialist pollinators $(\phi=0)$. This parameter space is divided by a decreasing $R_{o}=1$ line that indicates whether or not insects can invade when rare. $R_{o}$ is defined as (see derivation in Appendix A):
$$R_{o}=\frac{\epsilon\alpha\gamma\beta}{\eta\nu(\mu+\gamma\beta)}\label{eq:r0}$$
and we call it the *basic reproductive number*, according to the argument that follows. Consider the following in system (\[eq:pla\]): if the plant is at carrying capacity $(x=1)$, and is invaded by a very small number of adult insects $(z\approx0)$, the average number of larvae produced by a single adult in a given instant is $\epsilon\alpha x/(\eta+z)\approx\epsilon\alpha/\eta$, and during its life-time $(\nu^{-1})$ it is $\epsilon\alpha/\eta\nu$. Larvae die at the rate $\mu$, or mature with a rate equal to $\gamma\beta x=\gamma\beta$, per larva. Thus, the probability of larvae becoming adults rather than dying is $\gamma\beta/(\mu+\gamma\beta)$. Multiplying the life-time contribution of an adult by this probability gives the expected number of new adults replacing one adult per generation during an invasion ($R_{o}$). More formally, $R_{o}$ is the expected number of adult-insect-grams replacing one adult-insect-gram per generation (assuming a constant mass-per-individual ratio).
Below the $R_{o}=1$ line, small insect populations cannot replace themselves $(R_{o}<1)$ and two outcomes are possible. If the maturation rate is too low, the plant only equilibrium $(x=1,y=z=0)$ is globally stable and plant–insect coexistence is impossible for all initial conditions. If the maturation rate is large enough, stable coexistence is possible, but only if the initial plant and insect biomass are large enough. This is expected in models where at least one species, here the insect, is an obligate mutualist. In this region of the space of parameters, the growth of small insect populations increases with population size, a phenomenon called the Allee effect [@stephens_etal-oikos99].
Above the $R_{o}=1$ line the plant only equilibrium is always unstable against the invasion of small insect populations $(R_{o}>1)$. Plants and insects can coexist in a stable equilibrium or via limit cycles (stable oscillations). The zone of limit cycles occurs for intermediate values of the maturation rate ($\gamma$) and it widens with rate of herbivory ($\beta$).
Plant equilibrium when coexisting with insects can be above or below the carrying capacity $(x=1)$. When above carrying capacity the net result of the interaction is a mutualism (+,+). While in the second case we have antagonism, more specifically net herbivory (–,+). As it would be expected, increasing herbivory rates $(\beta)$ shifts this net balance towards antagonism (low plant biomass), while decreasing it shifts the balance towards mutualism (high plant biomass). The quantitative response to increases in the maturation rate $(\gamma)$ is more complex however (see the bifurcation plot in Appendix A).
Given that there is herbivory, we encounter victim–exploiter oscillations. However, the oscillations in the PLA model are special in the sense that the plant can attain maximum biomasses above the carrying capacity $(x>1)$. For an example see Figure \[fig:dynamics\_muther\]. Instead of a stable balance between antagonism and mutualism, we can say that the outcome in Figure \[fig:dynamics\_muther\] is a periodic alternation of both cases. This is not seen in simple victim–exploiter models, where oscillations are always below the victim’s carrying capacity [@rosenzweig_macarthur-amnat63; @rosenzweig-science71]. The relative position of the cycles along the plant axis is also affected by herbivory: if $\beta$ decreases (increases), plant maxima and minima will increase (decrease) in Figure \[fig:dynamics\_muther\] (see bifurcation plot in Appendix A). In some cases the entire plant cycle (maxima and minima) ends above the carrying capacity if $\beta$ is low enough (see Appendix C), but further decrease causes damped oscillations. We also found examples in which coexistence can be stable or lead to limit cycles depending on the initial conditions (see example in Appendix C), but this happens in a very restrictive region in the space of parameters (see bifurcation plot in Appendix A). Limit cycles can also cross the plant’s carrying capacity under the original interaction mechanism (\[eq:pfla\]), which does not assume the steady–state in the flowers (see an example in Appendix C, which uses the parameter of the last column in Table \[tab:vars\_and\_pars\]).
![\[fig:dynamics\_muther\]Limit cycles in the PLA model (\[eq:pla\]) with plant biomasses alternating above and below the carrying capacity (dotted line). Parameters as in Table \[tab:vars\_and\_pars\], with $\gamma=0.01,\beta=10$. Blue:plant, green:larva, red:adult.](dynamics_muther_tseries){width="0.6\paperwidth"}
Figure \[fig:beta\_vs\_gamma\_phi1\] shows the $\beta$ vs $\gamma$ parameter space of the model when the adults are more generalist. The relative positions of the plant-only, Allee effect, and coexistence regions are similar to the case of specialist pollinators (Figure \[fig:beta\_vs\_gamma\_phi0\]). However, the region of limit cycles is much larger. The $R_{0}=1$ line is closer to the origin, because the expression for $R_{0}$ is now (see derivation in Appendix A):
$$R_{0}=\frac{(\epsilon\alpha+\phi\eta)\gamma\beta}{\eta\nu(\mu+\gamma\beta)}\label{eq:r0gen}$$
In other words, this means that the more generalist the adult pollinators (larger $\phi$), the more likely they can invade when rare. There is also a small overlap between the Allee effect and limit cycle regions, i.e. parameter combinations for which the long term outcome could be insect extinction or plant–insect oscillations, depending on the initial conditions.
Graphical analysis\[sub:analytical\]
------------------------------------
The general features of the interaction can be studied by phase-plane analysis. To make this easier, we collapsed the three-dimensional PLA model into a two-dimensional plant–larva (PL) model, by assuming that adults are extremely short lived compared with plants and larvae (see resulting ODE in Appendix B). The closest realization of this assumption could be *Manduca sexta*, which has a larval stage of approximately 20-25 days and adult stages of around 7 days [@Reinecke_1980; @Ziegler_1991]. For a given parametrization (Table \[tab:vars\_and\_pars\]), the PL model has the same equilibria as the PLA model, but not the exact same global dynamics due to the alteration of time scales. Yet, this simplification provides insights about the outcomes displayed in Figures \[fig:beta\_vs\_gamma\_phi0\] and \[fig:beta\_vs\_gamma\_phi1\].
Figure \[fig:phasespace\] shows plant and larva isoclines (i.e. non-trivial nullclines) and coexistence equilibria (intersections). Isocline properties are analytically justified (Appendix B). These sketches are grossly exaggerated, but this facilitates the representation of features that are hard to notice by plotting them numerically (e.g. with parameters like in Table \[tab:vars\_and\_pars\]).
Plant isoclines take two main forms:
$$\begin{cases}
\gamma\sigma\alpha<\eta\nu & \textrm{the isocline lies entirely below (to the left of) the carrying capacity}\\
\gamma\sigma\alpha>\eta\nu & \textrm{parts of the isocline lie above (to the right of) the carrying capacity}
\end{cases}\label{eq:pcases}$$
In both cases, plants grow between the isocline and the axes, and decrease otherwise. Larva isoclines are simpler, they start in the plant axis and bend towards the right when insects tend towards specialization ($\phi<\nu$). When insects tend towards generalism ($\phi>\nu$), their isoclines increase rapidly upwards like the letter “J” (not shown here, see Appendix B) . Insects grow below and right of the larva isocline, and decrease otherwise.
The $\gamma\sigma\alpha<\eta\nu$ case in Figure \[fig:phasespace\]A covers scenarios in which pollination rates $(\alpha)$, plant benefits $(\sigma)$, adult pollinator lifetimes $(1/\nu)$ and larva-to-adult transition rates $(\gamma)$ are low. The plant’s isocline is a decreasing curve crossing the plant’s axis at its carrying capacity K $(x=1,y=0)$. Coexistence is unfavorable for the plant since its equilibrium biomass lies below the carrying capacity $(x<1)$. The local dynamics around the coexistence equilibrium indicates oscillations, and we can use the geometry of the intersection to infer that the equilibrium is stable (eigenvalue analysis is too difficult to perform for this model): Figure \[fig:phasespace\]A shows that if plants increase (or decrease) above the equilibrium while keeping the insect density fixed, they enter a zone of negative (or positive) growth; and the same holds for the insects while keeping the plants fixed. In ecological terms, both species are self-limited around the coexistence equilibrium, which as a rule of thumb is a strong indication of stability [@case2000]. Together with the fact that the trivial $(x=0,y=0)$ and carrying capacity equilibrium $(x=1,y=0)$ are saddle points, we conclude that plants and insects achieve an equilibrium after a period of transient oscillations (provided that insects are viable, e.g. $\beta,\gamma,\epsilon$ are large enough). Indeed, for extreme scenarios of negligible plant pollination benefits (i.e. $\alpha$ and/or $\sigma$ tend to zero), the plant’s isocline approximates a straight line with a negative slope, like the isocline of a logistic prey in a Lotka–Volterra model, which is well known to cause damped oscillations [@case2000].
The $\gamma\sigma\alpha>\eta\nu$ case in Figures \[fig:phasespace\]B,C,D covers scenarios in which pollination rates $(\alpha)$, pollination benefits $(\sigma)$, adult pollinator lifetimes $(1/\nu)$ and larva-to-adult (harm-to-benefit) transition rates $(\gamma)$ are high. One part of the plant’s isocline lies above the carrying capacity, which means that coexistence equilibria with plant biomass larger than the carrying capacity $(x>1)$ are possible; this is favorable for the plant. The isocline also displays a “hump” like in the classical victim–exploiter models [@rosenzweig_macarthur-amnat63]. Intersections at the right of the hump would lead to damped oscillations, like in Figure \[fig:phasespace\]A (for $\gamma\sigma\alpha<\eta\nu$). Intersections at the left of the hump, like in Figure \[fig:phasespace\]B, suggest oscillations that will increase in amplitude. This is because a small increase (decrease) along the plant’s axis leaves the plant at the growing (decreasing) side of its isocline, promoting further increase. This means that plants do not experience self-limitation, a rule-of-thumb indicator of instability [@case2000] and we infer that interactions would not dampen, leading to limit cycles. We have seen in Figure \[fig:dynamics\_muther\] that limit cycles can pass above the plant’s carrying capacity, which is implied in Figure \[fig:phasespace\]B, by picturing the maximum of the plant’s hump at the right of carrying capacity, and the intersection of isoclines between both points. Even if the hump lies at left of the carrying capacity, we cannot use this graphical analysis to discard the possibility of limit cycles overcoming the carrying capacity.
For $\gamma\sigma\alpha>\eta\nu$ the plant’s isocline also “folds” from its rightmost extent back towards the carrying capacity point. An intersection with this fold is shown in Figure \[fig:phasespace\]C, resulting in an equilibrium above the plant’s carrying capacity $(x>1)$, which is approached without oscillations. Intersections can also result in two equilibria, in which one of them is always unstable and belongs to a threshold above which insect invasion is possible (and not possible if below) (Figure \[fig:phasespace\]D). This explains the Allee effect, i.e. insect intrinsic growth rates increase from negative to positive as insect initial density increases.
When the second inequality of (\[eq:pcases\]) widens $(\gamma\sigma\alpha\gg\eta\nu)$, the plant’s isocline tends to take a mushroom-like shape (or “anvil” or letter “$\Omega$”), as in Figure \[fig:phasespace\]D. Indeed, as $\gamma,\sigma,\alpha$ increase and/or $\eta,\nu$ decrease more and more, the decreasing segment of the isocline approximates a decreasing line, while the rest of the isocline is pushed closer and closer to the axes. In other words, when pollination rates $(\alpha)$, benefits $(\sigma)$, adult lifetimes $(1/\nu)$ and larva development rates $(\gamma)$ increase, plant isoclines would resemble the isocline of a logistic prey, with a “pseudo” carrying capacity (the rightmost extent of the isocline) larger than the intrinsic carrying capacity ($x=1$). These conditions would promote stable coexistence with large plant equilibrium biomasses.
Discussion\[sec:Discussion\]
============================
We developed a plant–insect model that considers two interaction types, pollination and herbivory. Ours belongs to a class of models [@hernandez-rspb98; @holland_deangelis-ecology10] in which balances between costs and benefits cause continuous variation in interaction strengths, as well as transitions among interaction types (mutualism, predation, competition). In our particular case, interaction types depend on the stage of the insect’s life cycle, as inspired by the interaction between *M. sexta* and *D. wrightii* [@bronstein_etal-annbot09; @alarcon_etal-ecoent08] or between *M. sexta* and *N. attenuata* [@baldwin-oecologia88]. There are many other examples of pollination–herbivory in Lepidopterans, where adult butterflies pollinate the same plants exploited by their larvae [@wackers_etal-are07; @altermatt_pearse-amnat11]. We assign antagonistic and mutualistic roles to larva and adult insect stages respectively, which enable us to study the consequences of ontogenetic changes on the dynamics of plant–insect associations, a topic that is receiving increased attention [@miller_rudolf-tree11; @rudolf_lafferty-ecolett11]. Our model could be generalized to other scenarios, in which drastic ontogenetic niche shifts cause the separation of benefits and costs in time and space. But excludes cases like the yucca/yucca moth interaction [@holland_etal-amnat02], where adult pollinated ovules face larval predation, i.e. benefits themselves are deducted.
Instead of using species biomasses as resource and service proxies [@holland_deangelis-ecology10], we consider a mechanism (\[eq:pfla\]) that treats resources more explicitly [@encinas_etal-jtb14]. We use flowers as a direct proxy of resource availability, by assuming a uniform volume of nectar per flower. Nectar consumption by insects is concomitant with service exploitation by the plants (pollination), based on the assumption that flowers contain uniform numbers of ovules. Pollination also leads to flower closure [@primack-ares85], making them limiting resources. Flowers are ephemeral compared with plants and insects, so we consider that they attain a steady-state between production and disappearance. As a result, the dynamics is stated only in terms of plant, larva and adult populations, i.e. the PLA model (\[eq:pla\]). The feasibility of the results described by our analysis depends on several parameters. The consumption, mortalities and growth rates, and the carrying capacities (e.g. $a,b,m,n$ and $r,c$ in the fourth column of Table \[tab:vars\_and\_pars\]), have values close to the ranges considered by other models [@holland_deangelis-ecology10; @johnson_amarasekare-jtb13]. Oscillations, for example, require large herbivory rates, but this is usual for *M. sexta* [@McFadden_1968].
Mutualism–antagonism cycles
---------------------------
The PLA model displays plant–insect coexistence for any combination of (non-trivial) initial conditions where insects can invade when rare $(R_{o}>1)$. Coexistence is also possible where insects cannot invade when rare $(R_{o}<1)$, but this requires high initial biomasses of plants and insects (Allee effect). Coexistence can take the form of a stable equilibrium, but it can also take the form of stable oscillations, i.e. limit cycles.
Previous models combining mutualism and antagonism predict oscillations, but they are transient ones [@holland_etal-amnat02; @wang_deangelis-mbe12], or the limit cycles occur entirely below the plant’s carrying capacity [@holland_etal-theorecol13]. We have good reasons to conclude that the cycles are herbivory driven and not simply a consequence of the PLA model having many variables and non-linearities. First of all, limit cycles require herbivory rates $(\beta)$ to be large enough. Second, given limit cycles, an increase in the maturation rate $(\gamma)$ causes a transition to stable coexistence, and further increase in $\beta$ is required to induce limit cycles again (Figure \[fig:beta\_vs\_gamma\_phi0\]). This makes sense because by speeding up the transition from larva to adult, the total effect of herbivory on the plants is reduced, hence preventing a crash in plant biomass followed by a crash in the insects. Third, when adult pollinators have alternative food sources $(\phi>1)$, the zone of limit cycles in the space of parameters becomes larger (Figure \[fig:beta\_vs\_gamma\_phi1\]). This also makes sense, because the total effect of herbivory increases by an additional supply of larva (which is not limited by the nectar of the plant considered), leading to a plant biomass crash followed by insect decline.
The graphical analysis provides another indication that oscillations are herbivory driven. On the one hand insect isoclines (or rather larva isoclines) are always positively sloped, and insects only grow when plant biomass is large enough (how large depends on insect’s population size, due to intra-specific competition). Plant isoclines, on the other hand, can display a hump (Figure \[fig:phasespace\]B,C,D), and they grow (decrease) below (above) the hump. These two features of insect and plant isoclines are associated with limit cycles in classical victim–exploiter models (@rosenzweig_macarthur-amnat63). If there is no herbivory or another form of antagonism (e.g. competition) but only mutualism, the plant’s isocline would be a positively sloped line, and plants would attain large populations in the presence of large insect populations, without cycles. However, mutualism is still essential for limit cycles: if mutualistic benefits are not large enough $(\gamma\sigma\alpha<\eta\nu)$, plant isoclines do not have a hump (Figure \[fig:phasespace\]A) and oscillations are predicted to vanish. The effect of mutualism on stability is like the effect of enrichment on the stability in pure victim–exploiter models [@rosenzweig-science71], by allowing the plants to overcome the limits imposed by their intrinsic carrying capacity (e.g. the pseudo-carrying capacity K’ in Figure \[fig:phasespace\]D).
Classification of outcomes: mutualism or herbivory?
---------------------------------------------------
Interactions can be classified according to the net effect of one species on the abundance (biomass, density) of another (but see other schemes @abrams-oecologia87). This classification scheme can be problematic in empirical contexts, because reference baselines such as carrying capacities are usually not known and because stable abundances make little sense under the influence of unpredictable external fluctuations [@hernandez-jtb09].
Our PLA model illustrates the classification issue when non-equilibrium dynamics are generated endogenously, i.e. not by external perturbations. Since plants are facultative mutualists and insects are obligatory ones, one can say the outcome is *net mutualism* (+,+) or *net herbivory* (–,+), if the coexistence is stable, and the plant equilibrium ends up respectively above or below the carrying capacity [@hernandez-rspb98; @holland_deangelis-ecology10]. If coexistence is under non-equilibrium conditions and plant oscillations are entirely below the carrying capacity (e.g. for large $\beta$), the outcome is detrimental for plant and hence there is net herbivory (–,+); oscillations may in fact be considered irrelevant for this conclusion (or may further support the case of herbivory, read below). However, when the plant oscillation maximum is above carrying capacity and the minimum is below, like in Figure \[fig:dynamics\_muther\], could we say that the system alternates periodically between states of net mutualism and net herbivory? Here perhaps a time-based average over the cycle can help up us decide. The situation could be more complicated if plant oscillations lie entirely above the carrying capacity (see an example in Appendix C): one can say that the net outcome is a mutualism due to enlarged plant biomasses, but the oscillations indicates that a victim–exploiter interaction exists. As we can see, deciding upon the net outcome require consideration of both equilibrium and dynamical aspects.
Factors that could cause dynamical transitions
----------------------------------------------
### Environmental factors {#environmental-factors .unnumbered}
The parameters in our analyses can change due to external factors. One of the most important is temperature [@gillooly_etal-science01]. It is well known for example, that warming can reduce the number of days needed by larvae to complete their development [@bonhomme-ejagr00], making $\gamma$ higher. Keeping everything else equal but $\gamma$, for insects that cannot invade when rare (i.e. displaying Allee effects, $R_{o}<1$), a cooling of the environment will cause the sudden extinction of the insect and a catastrophic collapse of the mutualism, which cannot be simply reverted by warming. For insects that can invade when rare $(R_{o}>1)$, by slowing down larva development, cooling would increase the burden of herbivory over the benefits of pollination making the system more prone to oscillations and less stable (even less under strong herbivory, large $\beta$). Flowering, pollination, herbivory, growth and mortality rates (e.g. $s,a,b,r,m$ and $n$ in equations \[eq:pfla\]) are also temperature-dependent, and they can increase or decrease with warming depending on the thermal impacts on insect and plant metabolisms [@vasseur_mccann-amnat05]. This makes general predictions more difficult. However, we get the general picture that warming or cooling can change the balance between costs and benefits impacting the stability of the plant–insect association.
Dynamical transitions can also be induced by changes in the chemical environment, often as a consequence of human activity. Some pesticides, for example, are hormone retarding agents [@dev-pinsa86]. This means that their release can reduce $\gamma$ altering the balance of the interaction towards more herbivory and less pollination and finally endangering pollination service [@Potts_2010_tree; @Kearns_1998]. In other cases, the chemical changes are initiated by the plants: in response to herbivory, many plants release predator attractants [@allmann_baldwin-science10], which can increase larval mortality $\mu$. If the insect does nothing but harm, this is always an advantage. If the insect is also a very effective pollinator, the abuse of this strategy can cost the plant important pollination services because a dead herbivore today is one less pollinator tomorrow.
Another factor that can increase or decrease larvae maturation rates ($\gamma$), is the level of nutrients present in the plant’s vegetative tissue [@Woods_1999; @Perkins_2004]. On the one hand, the use of fertilizers rich in phosphorus could increase larvae maturation rates [@Perkins_2004]. On the other hand, under low protein consumption *M. sexta* larvae could decrease maturation rate, although *M. sexta* larvae can compensate this lack of proteins by increasing their herbivory levels (i.e. compensatory consumption) [@Woods_1999]. Thus, different external factors related to plant nutrients could indirectly trigger different larvae maturation rates that will potentially modify the interaction dynamics.
### Pollinator’s diet breadth {#pollinators-diet-breadth .unnumbered}
An important factor that can affect the balance between mutualism and herbivory is the diet breadth of pollinators. Alternative food sources for the adults could lead to apparent competition [@holt-tpb77] mediated by pollination, as predicted for the interaction between *D. wrigthii* (Solanacea) and *M. sexta* (Sphingidae) in the presence of *Agave palmieri* (plant) [@bronstein_etal-annbot09]: visitation of *Agave* by *M. sexta* does not affect the pollination benefits received by *D. wrightii*, but it increases oviposition rates on *D*. *wrightii*, increasing herbivory. As discussed before, such an increase in herbivory could explain why oscillations are more widespread when adult insects have alternative food sources $(\phi>0)$ in our PLA model.
Although we did not explore this with our model, the diet breadth of the larva could also have important consequences. In the empirical systems that inspired our model, the larva can have alternative hosts [@alarcon_etal-ecoent08], spreading the costs of herbivory over several species. The local extinction of such hosts could increase herbivory on the remaining ones, promoting unstable dynamics. To explore these issues properly, models like ours must be extended to consider larger community modules or networks, taking into account that there is a positive correlation between the diet breadths of larval and adult stages [@altermatt_pearse-amnat11].
From the perspective of the plant, the lack of alternative pollinators could also lead to increased herbivory and loss of stability. The case of the tobacco plant (*N. attenuata*) and *M. sexta* is illustrative. These moths are nocturnal pollinators, and in response to herbivory by their larvae, the plants can change their phenology by opening flowers during the morning instead. Thus, oviposition and subsequent herbivory can be avoided, whereas pollination can still be performed by hummingbirds [@kessler_etal-cb10]. Although hummingbirds are thought to be less reliable pollinators than moths for several reasons [@irwin-cb10], they are an alternative with negligible costs. Thus, a decline in hummingbird populations will render the herbivore avoidance strategy useless and plants would have no alternative but to be pollinated by insects with herbivorous larvae that promote oscillations.
Conclusions
-----------
Many insect pollinators are herbivores during their larval phases. If pollination and herbivory targets the same plant (e.g. as between tobacco plants and hawkmoths), the overall outcome of the association depends on the balance between costs and benefits for the plant. As predicted by our plant-larva-adult (PLA) model, this balance is affected by changes in insect development: the faster larvae turns into adults the better for the plant, and the more stable the interaction; the slower this development the poorer the outcome for the plant, and the less stable the interaction (oscillations). Under plant–insect oscillations, this balance can be dynamically complex (e.g. periodic alternation between mutualism and antagonism). Since maturation rates play an essential role in long term stability, we predict important qualitative changes in the dynamics due to changes in environmental conditions, such as temperature and chemical compounds (e.g. toxins, hormones, plant nutrients). The stability of these mixed interactions can also be greatly affected by changes in the diet generalism of the pollinators.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Rampal Etienne for the discussions that inspired us to write this article. We thank the comments and suggestions from our colleagues of the Centre for Biodiversity Theory and Modelling in Moulis, France, and the Centre for Australian National Biodiversity and Research at CSIRO in Canberra, Australia. TAR was supported by the TULIP Laboratory of Excellence (ANR-10-LABX-41). FEV was supported by the OCE postdoctoral fellowship at CSIRO.
Appendices {#appendices .unnumbered}
==========
Appendix A: Bifurcations {#appendix-a-bifurcations .unnumbered}
------------------------
Figure \[fig:beta\_vs\_gamma\_phi0\] in the main text shows all outcomes (plant-only, Allee effect, stable coexistence and limit cycles) occurring together in a rectangle at the bottom left corner of the parameter space $\beta$ vs $\gamma$. We enlarged this rectangle in Figure \[fig:app.beta\_vs\_gamma\_phi0\] in order to show the bifurcations of the PLA model as we traverse the parameter space along an elliptical path as indicated.
From Figure \[fig:app.x\_vs\_angle\_phi0\] we can conclude that plant equilibrium biomasses (stable or not) are inversely related with the rate of herbivory $(\beta)$. A similar response occurs regarding oscillations: as long as $\beta$ values are large enough to induce oscillations (the part in the figure marked with circles), such oscillations tend to display lower maxima and minima for larger values of $\beta$, and higher maxima and minima for smaller values instead.
The response of plant biomasses with respect to the insect maturation rate $(\gamma)$ is more complex. For example around the middle part of Figure \[fig:app.x\_vs\_angle\_phi0\] (between the $\pi/2$ and $3\pi/2$ marks), increasing $\gamma$ causes (equilibrium) plant biomass increases if herbivory is high, but decreases if herbivory is low. In contrast, increasing $\gamma$ from very low values causes plant biomass to increase if herbivory is low (between LP and the $3\pi/2$ mark at the right) or decrease when it is high (between BP and the $\pi/2$ mark at the left).
The transitions between stability and limit cycles are typically *super-critical Hopf bifurcations*, in which a stable branch of periodic solutions overlaps a branch of unstable equilibria. The bifurcation diagram (Figure \[fig:app.x\_vs\_angle\_phi0\]) also displays a *sub-critical Hopf bifurcation*, in which an unstable branch of periodic solutions overlaps stable equilibria. In such cases the long term outcome can be stable coexistence or a limit cycle depending on the initial conditions. Given the parameter values in Table \[tab:vars\_and\_pars\], this sub-critical Hopf bifurcation zone was too narrow to be represented in the parameter space (Figure \[fig:app.beta\_vs\_gamma\_phi0\]). Appendix C contains a simulation in which a small change in the initial conditions causes the system to approach an equilibrium or a limit cycle.
The $R_{o}=1$ line in Figure \[fig:beta\_vs\_gamma\_phi0\] can be found analytically. To do this, we need to know when the carrying capacity equilibrium switches between stable and unstable, which depends on the eigenvalues of the jacobian matrix of the PLA model (\[eq:pla\]) evaluated at $(x,y,z)=(1,0,0)$:
$$\left[\begin{array}{ccc}
1-2x+\frac{\sigma\alpha z}{\eta+z}-\beta y & -\beta x & \frac{\sigma\alpha\eta x}{(\eta+z)^{2}}\\
\frac{\epsilon\alpha z}{\eta+z}-\gamma\beta y & -\mu-\gamma\beta x & \frac{\epsilon\alpha\eta x}{(\eta+z)^{2}}+\phi\\
\gamma\beta y & \gamma\beta x & -\nu
\end{array}\right]=\left[\begin{array}{ccc}
-1 & -\beta & \frac{\sigma\alpha}{\eta}\\
0 & -\mu-\gamma\beta & \frac{\epsilon\alpha}{\eta}+\phi\\
0 & \gamma\beta & -\nu
\end{array}\right]\label{eq:jacobian}$$
The eigenvalues of the jacobian are $\lambda_{1}=-1$ and:
$$\lambda_{2}=\frac{-(\mu+\nu+\gamma\beta)\pm\sqrt{(\mu+\nu+\gamma\beta)^{2}-4\left[\nu(\mu+\gamma\beta)-\gamma\beta(\phi+\epsilon\alpha/\eta)\right]}}{2}$$
thus $(x,y,z)=(1,0,0)$ is unstable if at least one of $\lambda_{2}$ have a positive real part. This can only happen when:
$$\frac{(\epsilon\alpha+\phi\eta)\gamma\beta}{\eta\nu(\mu+\gamma\beta)}>1\label{eq:invasion}$$
by which automatically both $\lambda_{2}$ are real (one is negative and the other is positive). The right-hand side of (\[eq:invasion\]) is $R_{o}$ in the main text. Making $R_{o}=1$ and writing $\beta$ as a function of $\gamma$, we obtain a decreasing hyperbolic line with asymptotes $\beta=0$ and $\gamma=0$ as shown in Figures \[fig:beta\_vs\_gamma\_phi0\] and \[fig:beta\_vs\_gamma\_phi1\]. This is yet another reason, a pure technical one this time, that explains why we choose to present our results in the form of a $\beta$ vs $\gamma$ parameter space.
Since the eigenvector of $\lambda_{1}$ is a multiple of $(1,0,0)$, the eigenvectors of $\lambda_{2}$ are orthogonal to $(1,0,0)$, i.e. $v=(0,v_{y},v_{z}),w=(0,w_{y},w_{z})$. This, and the fact that both $\lambda_{2}$ are real if the inequality above holds, means that only perturbations in $y$ and/or $z$, i.e. an insect invasion, would make $(x,y,z)=(1,0,0)$ unstable.
Appendix B: Isocline properties {#appendix-b-isocline-properties .unnumbered}
-------------------------------
Let us assume that the adult phase is very short lived compared with the larval phase and with the dynamics of the plant. In the same way as we did in the case of the flowers, assume that the adults reach a steady-state $dz/dt\approx0$ with respect to the other variables, and that the adult biomass can be approximated by $z\approx\gamma\beta xy/\nu$. Substituting this in the ODE system (\[eq:pla\]), we obtain the two-dimensional system:
$$\begin{aligned}
\dot{x} & =x(1-x)+\sigma\frac{\alpha\gamma\beta x^{2}y}{\eta\nu+\gamma\beta xy}-\beta xy\nonumber \\
\dot{y} & =\epsilon\frac{\alpha\gamma\beta x^{2}y}{\eta\nu+\gamma\beta xy}+\frac{\phi\gamma\beta xy}{\nu}-\gamma\beta xy-\mu y\label{eq:pl}\end{aligned}$$
This system has two trivial isoclines, $x=0$ for the plant and $y=0$ for the insect. The following results only concern the non-trivial isoclines for plants and insects.
### Plant isocline {#plant-isocline .unnumbered}
Making $\dot{x}=0$ in (\[eq:pl\]), the (non-trivial) isocline of the plant can be written as a polynomial in $x$ and $y$:
$$x^{2}y+\beta xy^{2}-(1+\sigma\alpha)xy+\frac{\eta\nu}{\gamma\beta}x+\frac{\eta\nu}{\gamma}y-\frac{\eta\nu}{\gamma\beta}=0\label{eq:pisocline}$$
To characterize the shape of (\[eq:pisocline\]) we start by finding asymptotes. To do this we can rewrite (\[eq:pisocline\]) as a function of $x$:
$$y(x)=\frac{1}{2\beta}\left\{ \frac{-\left(\frac{\eta\nu}{\gamma}-(1+\sigma\alpha)x+x^{2}\right)\pm\sqrt{\left(\frac{\eta\nu}{\gamma}-(1+\sigma\alpha)x+x^{2}\right)^{2}+4\frac{\eta\nu}{\gamma}x(1-x)}}{x}\right\} \label{eq:fisocline(x)}$$
We divide the numerator and the denominator of (\[eq:fisocline(x)\]) by $x$:
$$\begin{aligned}
y(x) & =\frac{1}{2\beta}\left\{ -\frac{\eta\nu}{\gamma x}+(1+\sigma\alpha)-x\pm\sqrt{\frac{1}{x^{2}}\left(\frac{\eta\nu}{\gamma}-(1+\sigma\alpha)x+x^{2}\right)^{2}+\frac{1}{x^{2}}4\frac{\eta\nu}{\gamma}x(1-x)}\right\} \\
& =\frac{1}{2\beta}\left\{ -\frac{\eta\nu}{\gamma x}+(1+\sigma\alpha)-x\pm\sqrt{\left(\frac{\eta\nu}{\gamma x}-(1+\sigma\alpha)+x\right)^{2}+4\frac{\eta\nu}{\gamma}\left(\frac{1}{x}-1\right)}\right\} \end{aligned}$$
and we take the limit when $x$ goes to plus or minus infinity:
$$\begin{aligned}
\lim_{x\to\pm\infty}y(x) & =\frac{1}{2\beta}\lim_{x\to\pm\infty}\left\{ 0+(1+\sigma\alpha)-x\pm\sqrt{(0-(1+\sigma\alpha)+x)^{2}+4\frac{\eta\nu}{\gamma}(0-1)}\right\} \\
& =\frac{1}{2\beta}\lim_{x\to\pm\infty}\left\{ -(x-1-\sigma\alpha)\pm\sqrt{(x-1-\sigma\alpha)^{2}-4\frac{\eta\nu}{\gamma}}\right\} \end{aligned}$$
Note that $|x-1-\sigma\alpha|>\sqrt{(x-1-\sigma\alpha)^{2}-4\frac{\eta\nu}{\gamma}}$. Thus, the square root above can be approximated by $\delta(x)(x-1-\sigma\alpha)$, where $\delta$ is a number between 0 and 1, and $\delta(x)\to1$ as $x\to\pm\infty$. We can continue as follows:
$$\begin{aligned}\lim_{x\to\pm\infty}y(x) & =\frac{1}{2\beta}\lim_{x\to\pm\infty}\{-(x-1-\sigma\alpha)\pm\delta(x)(x-1-\sigma\alpha)\}\\
& =\frac{x-1-\sigma\alpha}{\beta}\lim_{x\to\pm\infty}\frac{\{-1\pm\delta(x)\}}{2}
\end{aligned}
\label{eq:slantlimit}$$
When $x\to\pm\infty$ and $\delta\to1$, the ’+’ branch, $y(x)$ approaches the horizontal asymptote $y=0$. For this ’+’ branch we also have that $-1<\{-1+\delta(x)\}<0$ in (\[eq:slantlimit\]), which means that $y$ is negative when $x\to+\infty$, and positive when $x\to-\infty$. In other words, the horizontal asymptote is approached from below when $x\to+\infty$ and from above when $x\to-\infty$.
When $x\to\pm\infty$ and $\delta\to1$, the ’–’ branch, $y(x)$ approaches the slanted asymptote:
$$y=\frac{1+\sigma\alpha-x}{\beta}\label{eq:pasymptote}$$
which decreases with $x$. For this ’–’ branch we also have that $-1<\{-1-\delta(x)\}/2<-1/2$ in (\[eq:slantlimit\]), which means that when $x\to+\infty$, $y<0$ and $|y|<|(x-1-\sigma\alpha)/\beta|$. In other words, $y$ lies between 0 and the slanted asymptote when $x\to+\infty$.
If we write (\[eq:pisocline\]) as a function of $y$ rather than as a function of $x$, we will find a vertical asymptote $x=0$, and the slanted asymptote (\[eq:pasymptote\]) again. Because (\[eq:pisocline\]) is symmetric regarding the signs of its terms, the properties of the vertical asymptote must consistent with those of the horizontal: $y(x)$ goes towards $+\infty$ when $x=0$ is approached from the left, and towards $-\infty$ when $x=0$ is approached from the right. Also because of symmetry $x$ must lie between 0 and the slanted asymptote when $y\to+\infty$.
The following statements tells us the location of special points of (\[eq:pisocline\]) as well regions in which (\[eq:pisocline\]) cannot be satisfied.
**Lemma 1:** the plant isocline contains the following $(x,y)$ points:
$$\begin{aligned}\mathrm{K} & =(1,0)\\
\mathrm{O} & =(0,\beta^{-1})\\
\mathrm{P} & =(\sigma\alpha-\eta\nu\gamma^{-1},\beta^{-1})\\
\mathrm{Q} & =(1,(\sigma\alpha-\eta\nu\gamma^{-1})\beta^{-1})
\end{aligned}
\label{eq:KOPQ}$$
*Proof:* evaluate (\[eq:pisocline\]) at $x=1$ to get a quadratic equation in $y$ with roots $y=0$ and $y=(\sigma\alpha-\eta\nu/\gamma)/\beta$, this gives points K and Q respectively. Evaluate (\[eq:pisocline\]) at $y=\beta^{-1}$ to get a quadratic equation in $x$ with roots $x=0$ and $x=\sigma\alpha-\eta\nu/\gamma$, this gives points O and P respectively. Points K (the plant’s carrying capacity), and O are always biologically feasible (both have non-negative coordinates).
**Corollary 1:** Simple observation of (\[eq:KOPQ\]) tells us that points P and Q are simultaneously biologically feasible if $\gamma\sigma\alpha>\eta\nu$. Conversely, both are unfeasible if $\gamma\sigma\alpha<\eta\nu$.
**Lemma 2:** Points P and Q lie below the slanted asymptote (\[eq:pasymptote\]).
*Proof:* substitute $y=\beta^{-1}$ in (\[eq:pasymptote\]) to obtain point $(\sigma\alpha,\beta^{-1})$, and substitute $x=1$ in (\[eq:pasymptote\]) to obtain point $(1,\sigma\alpha/\beta)$. Simple inspection shows that point $(\sigma\alpha,\beta^{-1})$ is always to the right of point P, and point $(1,\sigma\alpha/\beta)$ is always above point Q.
**Lemma 3:** the plant isocline crosses the x- and y-axis only at points K and O respectively, and nowhere else.
*Proof:* substituting $y=0$ in (\[eq:pisocline\]) gives only one root $x=1$ (i.e. point K). Substituting $x=0$ in (\[eq:pisocline\]) gives only one root $y=\beta^{-1}$ (i.e. point O).
**Lemma 4:** the plant isocline is not satisfied in the $(-,-)$ quadrant.
*Proof:* let $a,b\geq0$ and substitute $x=-a$ and $y=-b$ in (\[eq:pisocline\]). This leads to:
$$-\left[a^{2}b+\beta ab^{2}+(1+\sigma\alpha)ab+\frac{\eta\nu}{\gamma\beta}a+\frac{\eta\nu}{\gamma}b+\frac{\eta\nu}{\gamma\beta}\right]=0\label{eq:not_in_negneg}$$
since all parameter values are positive, the statement above is false, thus (\[eq:pisocline\]) is not satisfied in the $(-,-)$ quadrant.
Using this information about the asymptotes ($x=0,y=0$ and eq. \[eq:pasymptote\]), and Lemmas 1, 2, 3 and 4 we can conclude that the plant’s isocline must have one of the two forms depicted in figure \[fig:app.piso\]. Corollary 1 explains the form taken in figure \[fig:app.piso\]A, when $\gamma\sigma\alpha<\eta\nu$, and the form in figure \[fig:app.piso\]B, when $\gamma\sigma\alpha>\eta\nu$. These are the two main cases referenced in the main text by (\[eq:pcases\]), where only the positive quadrant is considered. For points between the O–K segment and the axes $\dot{x}>0$, otherwise $\dot{x}<0$.
Figure \[fig:app.pisoshapes\] shows how the positive part of the plant isocline changes as we vary some of the bifurcation parameters. Increasing $\gamma$ or decreasing $\eta$ or $\nu$, causes the isocline to be “compressed” against the asymptote (\[eq:pasymptote\]) and it adopts the shape of a mushroom, the letter $\Omega$ or an anvil. Increasing $\beta$ causes points P and Q to decrease along the vertically axis. It is more difficult to follow the effect of the rest of the parameters, for example increasing $\sigma$ and $\alpha$ cause P and Q to move right and upwards respectively, but they also move the asymptote (\[eq:pasymptote\]) right and upwards, so we cannot tell if this will cause the isocline to adopt a mushroom shape.
![\[fig:app.pisoshapes\]Changes in the shape of the plant’s isocline. (A) As $\gamma$ increases and $\eta,\nu$ decrease, points P and Q move closer to the diagonal asymptote (broken line), and the isocline eventually adopts the form of a mushroom. (B) As $\beta$ increases, O, P, Q and the diagonal asymptote move towards the plant axis and the isocline is compressed vertically.](pisoshapes){width="0.75\paperwidth"}
### Larva isocline {#larva-isocline .unnumbered}
Making $\dot{y}=0$ in (\[eq:pl\]) the larva isocline is:
$$y(x)=\frac{p(x)}{q(x)}\label{eq:lisocline}$$
where the numerator and denominator:
$$\begin{aligned}
p(x) & =\epsilon\alpha\gamma\beta x^{2}-\eta\nu\gamma\beta(1-\phi/\nu)x-\eta\mu\nu\label{eq:polyp}\\
q(x) & =\gamma\beta\left[\gamma\beta(1-\phi/\nu)x+\mu\right]x\label{eq:polyq}\end{aligned}$$
are second order polynomials, i.e. parabolas. By assuming instead $\dot{y}>0$ one obtains (\[eq:lisocline\]) but with a “>” sign, which means that insect biomass grows for points lying below the isocline and conversely decline for points above the isocline.
For function $p(x)$: $p(0)=-\eta\mu\nu<0$ and $\lim_{x\to\pm\infty}p(x)=+\infty$. This means that $p(x)$ has one negative root and one positive root; and also that $p(x)<0$ between the negative and positive roots, and $p(x)>0$ otherwise. Since $p(x)$ is the denominator of (\[eq:lisocline\]), the larva isocline has the same roots as $p(x)$ in the x-axis. The positive root of (\[eq:polyp\]) and (\[eq:lisocline\]) is:
$$x_{0}=\frac{\eta\nu}{2\epsilon\alpha}\left(1-\frac{\phi}{\nu}\right)+\sqrt{\left[\frac{\eta\nu}{2\epsilon\alpha}\left(1-\frac{\phi}{\nu}\right)\right]^{2}+\frac{\eta\mu\nu}{\epsilon\alpha\gamma\beta}}\label{eq:x0}$$
For function $q(x)$: it has one root at $x=0$, a second one at: $$x_{v}=-\frac{\mu}{\gamma\beta(1-\phi/\nu)}\label{eq:xv}$$
and $\lim_{x\to\pm\infty}p(x)=-\infty$. This means that $q(x)>0$ between $0$ and $x_{v}$, and $q(x)<0$ otherwise. Both roots make the denominator of (\[eq:lisocline\]) equal to zero, which means that the larva isocline has two vertical asymptotes, $x=0$ and $x_{v}$.
And finally, the larva isocline has one horizontal asymptote: $$y_{h}=\lim_{x\to\pm\infty}\frac{p(x)}{q(x)}=\frac{\epsilon\alpha}{\gamma\beta(1-\phi/\nu)}\label{eq:yh}$$
Notice that the signs of $x_{v}$ and $y_{h}$ depend on $\phi/\nu$:
$$\begin{cases}
\phi<\nu: & x_{v}<0,y_{h}>0\\
\phi>\nu: & x_{v}>0,y_{h}<0
\end{cases}\label{eq:phi_vs_nu}$$
This information about the parabolas $(p(x),q(x))$, and the signs of the asymptotes $(x_{v},y_{h})$, is enough to sketch the possible shapes of the larva isocline: the isocline crosses the x-axis at the roots of $p(x)$; its jumps to infinity at the roots of $q(x)$; and is positive (negative) whenever $p(x)$ and $q(x)$ have the same (different) signs. According to (\[eq:phi\_vs\_nu\]) we have two main cases:
1. If $\phi<\nu$ the vertical asymptote $x_{v}$ is negative and the horizontal asymptote $y_{h}$ is positive. As we can see, there are two alternatives, depicted by Figure \[fig:app.liso\]A and B. Both are indistinguishable in the positive octant, which is the only part that matters: they both start at the $x_{0}$ in the plant axis and grow up to a plateau $y_{h}$.
2. If $\phi>\nu$ the vertical asymptote $x_{v}$ is positive and the horizontal asymptote $y_{h}$ is negative. In this configuration we also have two alternatives, as depicted in Figures \[fig:app.liso\]C or D. However, we can quickly dismiss alternative D: the insect is meant to grow for points that are below the larva isocline, but since the isocline is decreasing, this automatically means to grow when plant abundance is low rather than high. This is nonsensical because the plant always has a positive effect on insects.
Figure \[fig:app.lisoshapes\] shows how the positive part of the larva isocline responds to some parameter changes. From the equations that define the isocline’s root (\[eq:x0\]) and asymptotes (\[eq:xv\],\[eq:yh\]) we can conclude that increasing $\gamma,\beta$ tends to move the isocline closer to the larva axis.
![\[fig:app.lisoshapes\] (A) For $\phi<\nu$ the larva isocline moves closer to the larva axis and becomes more shallow as $\gamma$ and $\beta$ increase. For $\phi>\nu$ the larva isocline becomes closer to the larva axis.](lisoshapes){width="0.75\paperwidth"}
Appendix C: Additional simulations {#appendix-c-additional-simulations .unnumbered}
----------------------------------
Figure \[fig:app.dynamics\_cicle\_above\_k\] displays limit cycles in the PLA model with plant biomasses entirely above the carrying capacity. The parameters are as in Table \[tab:vars\_and\_pars\] of the main text, but with $\gamma=0.00973,\beta=0.01$. Figure \[fig:app.dynamics\_unscaled\_ucycle\] shows an example where oscillations can damped out or evolve towards a limit cycle depending on the initial conditions. Parameters as in Table \[tab:vars\_and\_pars\] of the main text, but with $\gamma=0.06,\beta=20,\nu=5$. The attraction basins for both outcomes are separated by an unstable orbit, like the one show in the bifurcation plot in Appendix A.
Figure \[fig:app.dynamics\_pfla\_unscaled\] displays the dynamics of plants, flowers, larva and adult insects under the interaction mechanism (\[eq:pfla\]) from which the PLA model is derived in the main text. This simulation uses parameter values from the last column of Table \[tab:vars\_and\_pars\] with $\gamma=0.01,b=0.005$. This figure is comparable to Figure \[fig:dynamics\_muther\] in the main text: the 200 time in units there, become $t=\tau/r=200/0.05=4000$ time units here, and the plant’s carrying capacity there $(x=1)$, becomes $c^{-1}=0.01^{-1}=100$ here.
![\[fig:app.dynamics\_cicle\_above\_k\]Limit cycles in the PLA model, with plants above the carrying capacity (dotted line). Blue:plant, green:larva, red:adult.](dynamics_cycleabovek){width="0.75\paperwidth"}
![\[fig:app.dynamics\_unscaled\_ucycle\]Oscillations in the PLA model started with different initial conditions ([\*]{}). The oscillations can dampen out (blue) or converge to a limit cycle (red).](dynamics_ucycle_3d){width="0.75\paperwidth"}
![\[fig:app.dynamics\_pfla\_unscaled\]Interaction dynamics of plants, larva and adults, with the flowers explicitly considered. Blue:plant, green:larva, red:adult, black:flowers. The dotted line indicates the plant’s carryng capacity.](dynamics_tseries-unscaled){width="0.75\paperwidth"}
|
---
abstract: 'Microwave-field distribution, dissipation, and surface impedance are theoretically investigated for superconductors with laminar grain boundaries (GBs). In the present theory we adopt the two-fluid model for intragrain transport current in the grains, and the Josephson-junction model for intergrain tunneling current across GBs. Results show that the surface resistance $R_s$ nonmonotonically depends on the critical current density $J_{cj}$ at GB junctions, and $R_s$ for superconductors with GBs can be smaller than the surface resistance $R_{s0}$ for ideal homogeneous superconductors without GBs.'
author:
- Yasunori Mawatari
date: 'Dec. 28, 2004'
title: Microwave surface resistance in superconductors with grain boundaries
---
Introduction
============
High-temperature superconductors contain many grain boundaries (GBs), where the order parameter is locally suppressed due to the short coherence length. [@Deutscher87] GBs have attracted much interest for their basic physics as well as for their applications in superconductors, [@Mannhart99; @Larbalestier01; @Hilgenkamp02] and play a crucial role in microwave response and surface resistance $R_s$ of high-temperature superconducting films. [@Hylton88; @Attanasio91; @Halbritter92; @Nguyen93; @Fagerberg94; @Mahel96; @McDonald97; @Gallop97; @Obara01]
Electrodynamics of GB junctions can be described using the Josephson-junction model, and one of the most important parameters that characterize GB junctions is the critical current density $J_{cj}$ for Josephson tunneling current across GBs. [@Barone82; @Tinkham96; @VanDuzer99] The $J_{cj}$ strongly depends on the misorientation angle of GBs. [@Dimos88; @Gurevich98] In $\rm YBa_2Cu_3O_{7-\delta}$ films, $J_{cj}$ can be enhanced [@Hammerl00] and $R_s$ reduced [@Obara01] by Ca doping. The investigation of the relationship between $R_s$ and $J_{cj}$ is needed to understand the behavior of $R_s$ and $J_{cj}$ in Ca doped $\rm YBa_2Cu_3O_{7-\delta}$ films. The $J_{cj}$ dependence of $R_s$, however, has not yet been clarified, and it is not trivial whether GBs enhance the microwave dissipation that is proportional to $R_s$.
In this paper, we present theoretical investigation on the microwave field and dissipation in superconductors with laminar GBs. Theoretical expressions of the surface impedance $Z_s=R_s-iX_s$ of superconductors with GBs are derived as functions of $J_{cj}$ at GB junctions.
Basic Equations
===============
Superconductors with grain boundaries
-------------------------------------
We consider penetration of a microwave field (i.e., magnetic induction $\bm{B}=\mu_0\bm{H}$, electric field $\bm{E}$, and current density $\bm{J}$) into superconductors that occupy a semi-infinite area of $x>0$. We investigate linear response for small microwave power limit, such that the time dependence of the microwave field is expressed by the harmonic factor, $e^{-i\omega t}$, where $\omega/2\pi$ is the microwave frequency that is much smaller than the energy-gap frequency of the superconductors. Magnetic induction $\bm{B}$ is assumed to be less than the lower critical field, such that no vortices are present in the superconductors. (See Ref. for microwave response of vortices.)
The GBs are modelled to have laminar structures as in Ref. ; the laminar GBs that are parallel to the $xz$ plane are situated at $y=ma$, where $a$ is the spacing between grains (i.e., effective grain size) and $m=0,\pm 1,\pm 2,\cdots, \pm\infty$. The thickness of the barrier of GB junctions, $d_j$, is much smaller than both $a$ and the London penetration depth $\lambda$, and therefore, we investigate the thin-barrier limit of $d_j\to 0$, namely, GB barriers situated at $ma-0<y<ma+0$.
Two-fluid model for intragrain current
--------------------------------------
We adopt the standard two-fluid model [@Tinkham96; @VanDuzer99] for current transport in the grain at $ma+0<y<(m+1)a-0$. The intragrain current $\bm{J}=\bm{J}_s+\bm{J}_n$ is given by the sum of the supercurrent $\bm{J}_s=i\sigma_s\bm{E}$ and the normal current $\bm{J}_n=\sigma_n\bm{E}$, where $\sigma_s=1/\omega\mu_0\lambda^2$ and $\sigma_n$ is the normal-fluid conductivity in the grains. The displacement current $\bm{J}_d=-i\omega\epsilon\bm{E}$ with the dielectric constant $\epsilon$ can be neglected for a microwave range of $\omega/2\pi\sim$ GHz. Ampère’s law $\mu_0^{-1}\nabla\times\bm{B}=(\sigma_n+i\sigma_s)\bm{E}$ is thus reduced to $$\bm{E}= -i\omega\Lambda_g^2 \nabla\times\bm{B} ,
\label{E_intra}$$ where $\Lambda_g$ is the intragrain ac field penetration depth defined by $$\Lambda_g^{-2} =\omega\mu_0(\sigma_s -i\sigma_n)
=\lambda^{-2} -i\omega\mu_0\sigma_n .
\label{Lg_intra}$$ Combining Eq. (\[E\_intra\]) with Faraday’s law, $\nabla\times\bm{E} =i\omega\bm{B}$, we obtain the London equation for magnetic induction $\bm{B}=B_z(x,y)\hat{\bm{z}}$ for $y\neq ma$ as $$B_z -\Lambda_g^2\nabla^2 B_z =0 .
\label{B_intra}$$
For ideal homogeneous superconductors without GBs, Eq. (\[B\_intra\]) is valid for $-\infty<y<+\infty$ and the solution is simply given by $B_z(x)= \mu_0 H_0e^{-x/\Lambda_g}$, and the electric field is obtained from Eq. (\[E\_intra\]) as $E_y(x)=-i\omega\mu_0\Lambda_g H_0e^{-x/\Lambda_g}$. The surface impedance $Z_{s0}=R_{s0}-iX_{s0}$ for homogeneous superconductors is given by $Z_{s0}=E_y(x=0)/H_0= -i\omega\mu_0\Lambda_g$. The surface resistance $R_{s0}=\mbox{Re}(Z_{s0})$ and reactance $X_{s0}=-\mbox{Im}(Z_{s0})$ of ideal homogeneous superconductors without GBs are given by [@VanDuzer99] $$\begin{aligned}
R_{s0} &=& \mu_0^2\omega^2\lambda^3\sigma_n/2 ,
\label{Rs0_two-fluid}\\
X_{s0} &=& \mu_0\omega\lambda
\label{Xs0_two-fluid}\end{aligned}$$ for $\sigma_n/\sigma_s\ll 1$ well below the superconducting transition temperature $T_c$.
Josephson-junction model for intergrain current
-----------------------------------------------
We adopt the Josephson-junction model [@Barone82; @Tinkham96; @VanDuzer99] for tunneling current across GBs at $y=ma$. Behavior of the GB junctions is determined by the gauge-invariant phase difference across GBs, $\varphi_j(x)$, and the voltage induced across GB, $V_j(x)$, is given by the Josephson’s relation, $$\int_{ma-0}^{ma+0} E_y dy
= V_j = \frac{\phi_0}{2\pi}(-i\omega\varphi_j) ,
\label{Vj_GB}$$ where $\phi_0$ is the flux quantum. The tunneling current parallel to the $y$ axis is given by the sum of the superconducting tunneling current (i.e., Josephson current) $J_{sj}=J_{cj}\sin\varphi_j$ and the normal tunneling current (i.e., quasiparticle tunneling current) $J_{nj}=\gamma_{nj}V_j$. The critical current density $J_{cj}$ at GB junctions is one of the most important parameters in the present paper, and the resistance-area product of GB junctions corresponds to $1/\gamma_{nj}$. We neglect the displacement current across GBs, $J_{dj}=-i\omega C_jV_j$ where $C_j$ is the capacitance of the GB junctions.
Here we define the Josephson length $\lambda_J$ and the characteristic current density $J_0$ as $$\begin{aligned}
\lambda_J &=& (\phi_0/4\pi\mu_0 J_{cj}\lambda)^{1/2} ,
\label{Lambda-J}\\
J_0 &=& \phi_0/4\pi\mu_0\lambda^3 .
\label{J0}\end{aligned}$$ The ratio $J_{cj}/J_0=(\lambda/\lambda_J)^2$ characterizes the coupling strength of GB junctions. [@Gurevich92] For weakly coupled GBs, namely, $J_{cj}/J_0=(\lambda/\lambda_J)^2\ll 1$ (e.g., high-angle GBs), electrodynamics of the GB junctions can be well described by the weak-link model. [@Barone82; @Tinkham96; @VanDuzer99] For strongly coupled GBs, namely, $J_{cj}/J_0=(\lambda/\lambda_J)^2\agt 1$ (e.g., low-angle GBs), the Josephson-junction model is still valid but requires appropriate boundary condition at GBs, as given in Eq. (4) in Ref. , as pointed out by Gurevich; see also Refs. and .
In the small-microwave-power limit such that $\sin\varphi_j\simeq \varphi_j=2\pi V_j/(-i\omega\phi_0)$ for $|\varphi_j|\ll 1$, the $J_{cj}$ is reduced to $$J_{sj}\simeq J_{cj}\varphi_j =i\gamma_{sj}V_j ,
\label{J-gb_super-linear}$$ where $\gamma_{sj}= 2\pi J_{cj}/\omega\phi_0
=1/2\omega\mu_0\lambda\lambda_J^2$ . The total tunneling current across GB is thus given by $$\left. -\frac{1}{\mu_0} \frac{\partial B_z}{\partial x} \right|_{y=ma}
= J_{sj}+J_{nj} = (i\gamma_{sj} +\gamma_{nj}) V_j .
\label{J-V_inter-sigma}$$ Integration of Faraday’s law, $\partial E_y/\partial x -\partial E_x/\partial y=i\omega B_z$, yields $$\begin{aligned}
\lefteqn{E_x(x,y=ma+0) -E_x(x,y=ma-0)} &&
\nonumber\\
&=& \int_{ma-0}^{ma+0}dy \left[
\frac{\partial E_y(x,y)}{\partial x} -i\omega B_z(x,y) \right]
= \frac{\partial V_j(x)}{\partial x} , \quad
\label{Ex_discontinue}\end{aligned}$$ where we used Eq. (\[Vj\_GB\]). The static version (i.e., $\omega\to 0$) of Eq. (\[Ex\_discontinue\]) corresponds to Eq. (4) in Ref. . Substitution of Eqs. (\[E\_intra\]) and (\[J-V\_inter-sigma\]) into Eq. (\[Ex\_discontinue\]) yields the boundary condition for $B_z$ at $y=ma$, $$\left. -\frac{\partial B_z}{\partial y} \right|_{y=ma+0}
\left. +\frac{\partial B_z}{\partial y} \right|_{y=ma-0}
= \left. \frac{a\Lambda_j^2}{\Lambda_g^2}
\frac{\partial^2 B_z}{\partial x^2} \right|_{y=ma} ,
\label{bc-GB}$$ where $\Lambda_j$ is the characteristic length for ac field penetration into GBs defined by $$\begin{aligned}
\Lambda_j^{-2} &=& \omega\mu_0 a(\gamma_{sj} -i\gamma_{nj})
\nonumber\\
&=& \mu_0 a\left( 2\pi J_{cj}/\phi_0
-i\omega\gamma_{nj}\right) .
\label{Lj_gb}\end{aligned}$$
Surface impedance
=================
Microwave field and surface impedance
-------------------------------------
Equations (\[B\_intra\]) and (\[bc-GB\]) are combined into a single equation for $x>0$ and $-\infty<y<+\infty$ as $$B_z -\Lambda_g^2 \nabla^2B_z
= a\Lambda_j^2 \sum_{m=-\infty}^{+\infty}
\frac{\partial^2 B_z}{\partial x^2} \delta(y-ma) ,
\label{B_all}$$ whose solution is calculated as $$\begin{aligned}
\frac{B_z(x,y)}{\mu_0H_0}
&=& e^{-x/\Lambda_g}
+\frac{2}{\pi}\int_0^{\infty}dk\,
\frac{\cosh[K(y-a/2)]}{\Lambda_g^{\,2}K^2\sinh(Ka/2)}
\nonumber\\
&& \times \frac{k\sin kx}{(2K\Lambda_g^2/a\Lambda_j^2) +k^2\coth(Ka/2)}
\label{Bz_0<y<a}\end{aligned}$$ for $0<y<a$, where $K=(k^2+\Lambda_g^{-2})^{1/2}$. The right-hand side of Eq. (\[B\_all\]) and the second term of the right-hand side of Eq. (\[Bz\_0<y<a\]) reflect the GB effects. See Appendix A for the derivation of Eq. (\[Bz\_0<y<a\]) from Eq. (\[B\_all\]).
Electric field in the grains is obtained from Eq. (\[E\_intra\]) as $E_y= i\omega\Lambda_g^{\,2}\partial B_z/\partial x$, and voltage induced across GB is obtained from Eq. (\[J-V\_inter-sigma\]) as $V_j=\left. i\omega a\Lambda_j^2\partial B_z/\partial x\right|_{y=0}$. The mean electric field $\bar{E}_s$ at the surface of the superconductor is thus calculated as $$\begin{aligned}
\bar{E}_s &\equiv&
\frac{1}{a} \int_{-0}^{a-0}dy\, E_y(x=0,y)
\nonumber\\
&=& \frac{1}{a} \left[ V_j(x=0) +\int_{+0}^{a-0}dy\, E_y(x=0,y) \right]
\nonumber\\
&=& i\omega \left[ \Lambda_j^2
\left. \frac{\partial B_z}{\partial x}\right|_{x=y=0}
+ \frac{\Lambda_g^{\,2}}{a}\int_{+0}^{a-0} dy\,
\left. \frac{\partial B_z}{\partial x}\right|_{x=0} \right] .
\nonumber\\
\label{mean-Es_def}\end{aligned}$$ Substitution of Eq. (\[Bz\_0<y<a\]) into Eq. (\[mean-Es\_def\]) yields the surface impedance $Z_s=R_s-iX_s \equiv\bar{E}_s/H_0$ as $$\begin{aligned}
\frac{Z_s}{-i\omega\mu_0\Lambda_g}
&=& 1+\frac{2}{\pi} \int_0^{\infty} dk \frac{1}{\Lambda_g^{\,3}K^3}
\nonumber\\
&& \times \frac{1}{(K\Lambda_g^2/\Lambda_j^2) +(k^2a/2)\coth(Ka/2)} .
\qquad
\label{Zs_general}\end{aligned}$$ The surface resistance and reactance are given by $R_s= \mbox{Re}(Z_s)$ and $X_s= -\mbox{Im}(Z_s)$, respectively.
Microwave dissipation and surface resistance
--------------------------------------------
The time-averaged electromagnetic energy passing through the surface of a superconductor at $x=0$ and $-0<y<a-0$ is given by the real part of $${\cal E}= \frac{1}{2\mu_0}\int_{-0}^{a-0}dy (E_yB_z^*)_{x=0}
= \frac{a}{2}\bar{E}_sH_0^* ,
\label{Poyntings-vector}$$ where $\bar{E}_s=Z_sH_0$ is defined by Eq. (\[mean-Es\_def\]), and $(B_z)_{x=0}=\mu_0 H_0$. Poynting’s theorem [@Jackson75] states that $\cal E$ is identical to the energy stored and dissipated in the superconductor, $$\begin{aligned}
{\cal E}
&=& \frac{1}{2}
\int_0^{\infty}dx\left[
\int_{+0}^{a-0}dy\, (\sigma_n -i\sigma_s)|\bm{E}|^2 \right.
\nonumber\\
&& \left.\mbox{} +(\gamma_{nj}-i\gamma_{sj})|V_j|^2
-\int_{-0}^{a-0}dy\, \frac{i\omega}{\mu_0}|B_z|^2 \right] .
\label{Poyntings-theorem}\end{aligned}$$ The real parts of Eqs. (\[Poyntings-vector\]) and (\[Poyntings-theorem\]) show that the surface resistance $R_s=\mbox{Re}(\bar{E}_s/H_0)=\mbox{Re}(Z_s)$ is composed of two terms: $$R_s = R_{sg} +R_{sj} .
\label{Rs_Rsg+Rsj}\\$$ The intragrain contribution $R_{sg}$ is from the energy dissipation in the grains, and the intergrain contribution $R_{sj}$ is from the dissipation at GBs: $$\begin{aligned}
R_{sg} &=&
\frac{1}{a|H_0|^2}
\int_0^{\infty}dx\int_{+0}^{a-0}dy\, \sigma_n|\bm{E}|^2 ,
\label{Rsg_dissipation}\\
R_{sj} &=&
\frac{1}{a|H_0|^2} \int_0^{\infty}dx\, \gamma_{nj}|V_j|^2 .
\label{Rsj_dissipation}\end{aligned}$$
Both the intragrain current $|\bm{J}_g|$ around GBs and the intergrain tunneling current $|J_j|$ across GBs are suppressed by the GBs, and are increasing functions of $J_{cj}$. With increasing $J_{cj}$, the intragrain electric field $|\bm{E}|=|\bm{J}_g/(\sigma_n+i\sigma_s)|$ also increases, whereas the intergrain voltage $|V_j|=|J_j/(\gamma_{nj}+i\gamma_{sj})|$ decreases because $\gamma_{sj}\propto J_{cj}$. The dissipation in the grains, $\sigma_n|\bm{E}|^2/2$, and the intragrain contribution to the surface resistance, $R_{sg}$, therefore, tend to [*increase*]{} with increasing $J_{cj}$. The dissipation at GBs, $\gamma_{nj}|V_j|^2/2$, and the intergrain contribution to the surface resistance, $R_{sj}$, on the other hand, [*decrease*]{} with increasing $J_{cj}$.
The surface reactance $X_s=-\mbox{Im}(Z_s)$ is also divided into two contributions, $$X_s = X_{sg} +X_{sj} ,
\label{Xs_Xsg+Xsj}\\$$ where the intragrain contribution $X_{sg}$ and the intergrain contribution $X_{sj}$ are given by $$\begin{aligned}
X_{sg} &=& \frac{1}{a|H_0|^2}
\int_0^{\infty}dx\int_{+0}^{a-0}dy\, \left( \sigma_s|\bm{E}|^2
+\frac{\omega}{\mu_0}|B_z|^2 \right) ,
\nonumber\\
\label{Xsg_energy}\\
X_{sj} &=& \frac{1}{a|H_0|^2} \int_0^{\infty}dx\, \gamma_{sj}|V_j|^2 .
\label{Xsj_energy}\end{aligned}$$ Both $X_{sg}$ and $X_{sj}$ decrease with increasing $J_{cj}$.
Simplified expressions for surface impedance
--------------------------------------------
The following Eqs. (\[Zs\_small-grain\])–(\[Xs\_large-Jc\]) show simplified expressions of the surface impedance $Z_s$, the surface resistance $R_s=\mbox{Re}(Z_s)$, and the surface reactance $X_s=-\mbox{Im}(Z_s)$ for certain restricted cases, assuming $\sigma_n/\sigma_s\ll 1$ and $\gamma_{nj}/\gamma_{sj}\ll 1$ well below the transition temperature.
For small grains of $a\ll\lambda$ such that $\coth(Ka/2)\simeq 2/Ka$, Eq. (\[Zs\_general\]) is reduced to $$Z_s\simeq -i\omega\mu_0 \left(\Lambda_g^2 +\Lambda_j^2\right)^{1/2} .
\label{Zs_small-grain}$$ The right-hand side of Eq. (\[B\_all\]) is reduced to $\Lambda_j^2\partial^2B_z/\partial x^2$ for $a\ll\lambda$, and the effective ac penetration depth is given by $\Lambda_{\rm eff}= (\Lambda_g^2 +\Lambda_j^2)^{1/2}$ as in Ref. , resulting in the surface impedance given by Eq. (\[Zs\_small-grain\]). The $R_s$ and $X_s$ for small grains is obtained as $$\begin{aligned}
\frac{R_s}{R_{s0}} &\simeq&
\left(1+\frac{2\lambda}{a}\frac{J_0}{J_{cj}}\right)^{-1/2}
\left[\,1 +\frac{4\lambda^2\gamma_{nj}}{a\sigma_n}
\left(\frac{J_0}{J_{cj}}\right)^2 \,\right] , \qquad
\label{Rs_small-grain}\\
\frac{X_s}{X_{s0}} &\simeq&
\left(1+\frac{2\lambda}{a}\frac{J_0}{J_{cj}}\right)^{+1/2} ,
\label{Xs_small-grain}\end{aligned}$$ where $R_{s0}$, $X_{s0}$, and $J_0$ are defined by Eqs. (\[Rs0\_two-fluid\]), (\[Xs0\_two-fluid\]) and (\[J0\]), respectively. Equation (\[Rs\_small-grain\]) is decomposed into the intragrain $R_{sg}$ and intergrain $R_{sj}$ contributions, as $R_{sg}/R_{s0}\simeq (1+2\lambda J_0/aJ_{cj})^{-1/2}$ and $R_{sj}/R_{sg}\simeq (4\lambda^2\gamma_{nj}/a\sigma_n) (J_0/J_{cj})^2$, respectively.
Equation (\[Zs\_small-grain\]) is further simplified when $a\ll 2\lambda_J^{\,2}/\lambda$ for small grain and weakly coupled GBs as $$Z_s \simeq -i\omega\mu_0\Lambda_j ,
\label{Zs_small-grain-weak-link}$$ and we have $$\begin{aligned}
\frac{R_s}{R_{s0}} &\simeq&
\frac{2\gamma_{nj}\lambda}{\sigma_n}
\left(\frac{2\lambda}{a}\right)^{1/2}
\left(\frac{J_0}{J_{cj}}\right)^{3/2} ,
\label{Rs_small-grain-weak-link}\\
\frac{X_s}{X_{s0}} &\simeq&
\left(\frac{2\lambda}{a}\right)^{1/2}
\left(\frac{J_0}{J_{cj}}\right)^{1/2} .
\label{Xs_small-grain-weak-link}\end{aligned}$$ Thus, we obtain the dependence of $R_s$ and $X_s$ on the material parameters as $R_s\propto \gamma_{nj}a^{-1/2}J_{cj}^{-3/2}$ and $X_s\propto a^{-1/2}J_{cj}^{-1/2}$, which are independent of $\lambda$. The $R_s$ given by Eq. (\[Rs\_small-grain-weak-link\]) for the small grain and weakly coupled GBs is mostly caused by intergrain dissipation, $R_s\simeq R_{sj}\gg R_{sg}$. For $X_s$ given by Eq. (\[Xs\_small-grain-weak-link\]), on the other hand, both intragrain $X_{sg}$ and intergrain $X_{sj}$ contribute to the total $X_s=X_{sg}+X_{sj}$.
For large $J_{cj}$ (i.e., strong-coupling limit) such that $K\Lambda_g^2/\Lambda_j^2\gg (k^2a/2)\coth(Ka/2)$, Eq. (\[Zs\_general\]) for the surface impedance $Z_s$ is simplified as $$Z_s \simeq -i\omega\mu_0 \left(\Lambda_g +\Lambda_j^2/2\Lambda_g\right) ,
\label{Zs_large-Jcj}$$ and we have $$\begin{aligned}
\frac{R_s}{R_{s0}} &\simeq&
1 -\frac{\lambda}{a}\frac{J_0}{J_{cj}}
+\frac{4\lambda^2\gamma_{nj}}{a\sigma_n}
\left(\frac{J_0}{J_{cj}}\right)^2 ,
\label{Rs_large-Jc}\\
\frac{X_s}{X_{s0}} &\simeq&
1 +\frac{\lambda}{a}\frac{J_0}{J_{cj}} .
\label{Xs_large-Jc}\end{aligned}$$ The first and second terms of the right-hand side of Eq. (\[Rs\_large-Jc\]) correspond to the intragrain contribution, $R_{sg}$, whereas the third term corresponds to the intergrain contribution, $R_{sj}$.
Discussion
==========
![Dependence of surface resistance $R_s=\mbox{Re}(Z_s)$ and surface reactance $X_s=-\mbox{Im}(Z_s)$ \[i.e., Eq. (\[Zs\_general\]) with Eqs. (\[Lg\_intra\]) and (\[Lj\_gb\])\] on critical current density $J_{cj}$ at GB junctions. $R_s$ is normalized to the surface resistance without GB, i.e., $R_{s0}$ given by Eq. (\[Rs0\_two-fluid\]), $X_s$ is normalized to $X_{s0}$ given by Eq. (\[Xs0\_two-fluid\]), and $J_{cj}$ is normalized to $J_0$ defined as Eq. (\[J0\]). Parameters are $\omega/2\pi=10\,$GHz, $\lambda=0.2\,\mu$m, $\sigma_n=10^7\,\Omega^{-1}$m$^{-1}$, and $\gamma_{nj}=10^{13}\,\Omega^{-1}$m$^{-2}$, which yield $R_{s0}=0.25\,$m$\Omega$, $X_{s0}=16\,$m$\Omega$, and $J_0=1.6\times 10^{10}\,$A/m$^2$. (a) Total surface resistance $R_s=R_{sj}+R_{sg}$, intergrain contribution $R_{sj}$ given by Eq. (\[Rsj\_dissipation\]), and intragrain contribution $R_{sg}$ given by Eq. (\[Rsg\_dissipation\]) for $a/\lambda=0.1$. (b) $R_s$ and (c) $X_s$ for $a/\lambda=0.1$, $1$, and $5$. []{data-label="Fig_Rs-Jc"}](fig_RsXs-Jc.eps)
Figure \[Fig\_Rs-Jc\](a) and (b) shows $J_{cj}$ dependence of $R_s$. As shown in Fig. \[Fig\_Rs-Jc\](a), the intergrain contribution $R_{sj}$ is dominant for weakly coupled GBs (i.e., small $J_{cj}/J_0$ regime), whereas the intragrain contribution $R_{sg}$ is dominant for strongly coupled GBs (i.e., large $J_{cj}/J_0$). The $R_{sj}$ [*decreases*]{} with increasing $J_{cj}$ as $R_{sj}\propto J_{cj}^{-1.5}$ \[see Eq. (\[Rs\_small-grain-weak-link\])\], whereas $R_{sg}$ [*increases*]{} with $J_{cj}$. The resulting surface resistance $R_s=R_{sj}+R_{sg}$ [*nonmonotonically*]{} depends on $J_{cj}$ and has a minimum, because $R_s$ is determined by the competition between $R_{sj}$ and $R_{sg}$. As shown in Fig. \[Fig\_Rs-Jc\](c), on the other hand, $X_s$ monotonically decreases with increasing $J_{cj}$ \[i.e., $X_s\propto J_{cj}^{-0.5}$ for weakly coupled GBs as in Eq. (\[Xs\_small-grain-weak-link\])\].
The nonmonotonic dependence of $R_s$ on the grain size $a$ is also seen in Fig. \[Fig\_Rs-Jc\](b). For small $J_{cj}/J_0$ the $R_s$ decreases with increasing $a$ as $R_s\propto a^{-0.5}$ \[see Eq. (\[Rs\_small-grain-weak-link\])\], whereas $R_s$ increases with $a$ for large $J_{cj}/J_0$.
The $R_s$ for strongly coupled GBs can be [*smaller*]{} than $R_{s0}$ for ideal homogeneous superconductors without GBs, namely, $R_s/R_{s0}<1$ for $J_{cj}/J_0\agt 1$. The minimum surface resistance for $\lambda\gamma_{nj}/\sigma_n=0.2$ is $R_s/R_{s0}\approx 0.97$ for $a/\lambda=5$, $R_s/R_{s0}\approx 0.86$ for $a/\lambda=1$, and $R_s/R_{s0}\approx 0.59$ for $a/\lambda=0.1$. The minimum $R_s/R_{s0}$ is further reduced when $\lambda\gamma_{nj}/\sigma_n$ is further reduced.
Theoretical results shown above may possibly be observed by measuring $R_s$, $X_s$, and $J_{cj}$ in Ca doped $\rm YBa_2Cu_3O_{7-\delta}$ films. The enhancement of $J_{cj}$ (Ref. ) and reduction of $R_s$ (Ref. ) by Ca doping are individually observed in $\rm YBa_2Cu_3O_{7-\delta}$, but simultaneous measurements of $J_{cj}$ and $R_s$ are needed to investigate the relationship between $R_s$ and $J_{cj}$. The nonmonotonic $J_{cj}$ dependence of $R_s$ for strongly coupled GBs may be observed in high quality films with small grains $a<\lambda$ and with large $J_{cj}$ on the order of $J_0\sim 10^{10}\,{\rm A/m}^2$ at low temperatures.
Conclusion
==========
We have theoretically investigated the microwave-field distribution in superconductors with laminar GBs. The field calculation is based on the two-fluid model for current transport in the grains and on the Josephson-junction model for tunneling current across GBs. Results show that the microwave dissipation at GBs is dominant for weakly coupled GBs of $J_{cj}\ll J_0$, whereas dissipation in the grains is dominant for strongly coupled GBs of $J_{cj}\gg J_0$. The surface resistance $R_s$ nonmonotonically depends on $J_{cj}$; the $R_s$ decreases with increasing $J_{cj}$ as $R_s\propto J_{cj}^{-1.5}$ for $J_{cj}\ll J_0$, whereas $R_s$ increases with $J_{cj}$ for $J_{cj}\gg J_0$. The intragrain dissipation can be suppressed by GBs, and the surface resistance of superconductors with GBs can be smaller than that of ideal homogeneous superconductors without GBs.
I gratefully acknowledge H. Obara, J.C. Nie, A. Sawa, M. Murugesan, H. Yamasaki, and S. Kosaka for stimulating discussions.
Equation (\[Bz\_0<y<a\]) is derived by solving Eq. (\[B\_all\]) with the boundary condition of $B_z=\mu_0H_0$ at $x=0$, as follows.
We introduce the Fourier transform of $B_z(x,y)$ and $B_z(x,ma)=B_z(x,0)$ as $$\begin{aligned}
\tilde{b}(k,q) &=& \int_0^{\infty}dx \int_{-\infty}^{+\infty}dy\,
B_z(x,y) e^{-iqy}\sin kx ,
\label{bkq_Fourier}\\
\tilde{b}_0(k) &=& \int_0^{\infty}dx\, B_z(x,0) \sin kx
= \int_{-\infty}^{+\infty}\frac{dq}{2\pi}\, \tilde{b}(k,q) ,
\nonumber\\
\label{b0k_Fourier}\end{aligned}$$ respectively. The Fourier transform of Eq. (\[B\_all\]) leads to $$\begin{aligned}
\frac{\tilde{b}(k,q)}{\mu_0H_0} &=& 2\pi\delta(q)\frac{k}{K^2}
\nonumber\\
&& {}+\frac{\alpha k}{K^2+q^2}\sum_m e^{-imqa}
\left[1-\frac{k\tilde{b}_0(k)}{\mu_0H_0}\right] ,
\label{bkq-b0k}\end{aligned}$$ where $K=(k^2+\Lambda_g^{-2})^{1/2}$ and $\alpha= a\Lambda_j^2/\Lambda_g^2$. Substituting Eq. (\[bkq-b0k\]) into Eq. (\[b0k\_Fourier\]), we have $$\begin{aligned}
\frac{\tilde{b}_0(k)}{\mu_0H_0} &=& \frac{k}{K^2}
+\alpha k \left[1-\frac{k\tilde{b}_0(k)}{\mu_0H_0}\right]
\sum_m \int_{-\infty}^{+\infty}\frac{dq}{2\pi}
\frac{e^{-imqa}}{K^2+q^2} ,
\nonumber\\
% &=& \frac{k}{K^2}
% +\alpha k \left[1-\frac{k\tilde{b}_0(k)}{\mu_0H_0}\right]
% \frac{1}{2K}\coth\left(\frac{Ka}{2}\right) ,
\label{b0k-cal}\end{aligned}$$ which is reduced to $$\frac{\tilde{b}_0(k)}{\mu_0H_0} = \frac{1}{k}
-\frac{2}{kK \Lambda_g^2} \frac{1}{2K+\alpha k^2\coth(Ka/2)} .
\label{b0k-sol}$$ $B_z(x,y)$ is calculated from $\tilde{b}(k,q)$ given by Eq. (\[bkq-b0k\]) as $$\begin{aligned}
\frac{B_z(x,y)}{\mu_0H_0} &=& \frac{2}{\pi}\int_0^{\infty}dk
\int_{-\infty}^{+\infty}\frac{dq}{2\pi}\,
\frac{\tilde{b}(k,q)}{\mu_0H_0} e^{iqy}\sin kx
\nonumber\\
&=& e^{-x/\Lambda_g}
+\frac{2\alpha}{\pi} \int_0^{\infty}dk\,
k\sin kx \left[1-\frac{k\tilde{b}_0(k)}{\mu_0H_0}\right]
\nonumber\\
&& {}\times \sum_m \int_{-\infty}^{+\infty}\frac{dq}{2\pi}
\frac{e^{iq(y-ma)}}{K^2+q^2} .
\label{Bz-cal}\end{aligned}$$ Substitution of Eq. (\[b0k-sol\]) into Eq. (\[Bz-cal\]) yields Eq. (\[Bz\_0<y<a\]).
[99]{} G. Deutscher and K.A. Müller, , 1745 (1987). J. Mannhart and H. Hilgenkamp, Physica C [**317-318**]{}, 383 (1999). D. Larbalestier, A. Gurevich, D.M. Feldmann, and A. Polyanskii, Nature [**414**]{}, 368 (2001). H. Hilgenkamp and J. Mannhart, , 485 (2002).
T.L. Hylton, A. Kapitulnik, M.R. Beasley, J.P. Carini, L. Drabeck, and G. Grüner, , 1343 (1988). C. Attanasio, L. Maritato, and R. Vaglio, , 6128 (1991). J. Halbritter, J. Appl. Phys. [**71**]{}, 339 (1992). P.P. Nguyen, D.E. Oates, G. Dresselhaus, and M.S. Dresselhaus, , 6400 (1993). R. Fagerberg, and J.K. Grepstad, J. Appl. Phys. [**75**]{}, 7408 (1994). M. Mahel, Solid State Commun. [**97**]{}, 209 (1996). J. McDonald and J.R. Clem, , 14 723 (1997). J.C. Gallop, A. Cowie, and L.F. Cohen, Physica C [**282-287**]{}, 1577 (1997). H. Obara, A. Sawa, H. Yamasaki, and S. Kosaka, , 646 (2001).
A. Barone and G. Paternò, [*Physics and Applications of the Josephson Effect*]{} (Wiley, New York, 1982). M. Tinkham, [*Introduction to Superconductivity*]{}, 2nd ed. (McGraw-Hill, New York, 1996). T. Van Duzer and C.W. Turner, [*Principles of Superconductive Devices and Circuits*]{}, 2nd ed. (Prentice Hall, New Jersey, 1999).
D. Dimos, P. Chaudhari, J. Mannhart, and F.K. LeGoues, , 219 (1988); D. Dimos, P. Chaudhari, and J. Mannhart, , 4038 (1990). A. Gurevich and E.A. Pashitskii, , 13878 (1998). A. Schmehl, B. Goetz, R. R. Schulz, C. W. Schneider, H. Bielefeldt, H. Hilgenkamp, and J. Mannhart, Europhys. Lett. [**47**]{}, 110 (1999); C.W. Schneider, R.R. Schulz, B. Goetz, A. Schmehl, H. Bielefeldt, H. Hilgenkamp, and J. Mannhart, , 850 (1999); G. Hammerl, A. Schmehl, R.R. Schulz, B. Goetz, H. Bielefeldt, C.W. Schneider, H. Hilgenkamp, and J. Mannhart, Nature (London) [**407**]{}, 162 (2000).
M.W. Coffey and J.R. Clem, , 386 (1991); E.H. Brandt, [*ibid.*]{} [**67**]{}, 2219 (1991).
T.L. Hylton and M.R. Beasley, , 9042 (1989).
A. Gurevich, , R3187 (1992). Yu.M. Ivanchenko and T.K. Soboleva, Phys. Lett. A [**147**]{}, 65 (1990); R.G. Mints and I.B. Snapiro, , 3054 (1995); V.G. Kogan, V.V. Dobrovitski, J.R. Clem, Y. Mawatari, and R.G. Mints, , 144501 (2001).
J.D. Jackson, [*Classical Electrodynamics*]{} (Wiley, New York, 1975).
|
---
abstract: 'Let $\H_n$ be the Iwahori-Hecke algebra of the symmetric group $S_n$, and let $Z(\H_n)$ denote its centre. Let $B=\left\{b_1,b_2,\ldots ,b_t\right\}$ be a basis for $Z(\H_n)$ over $R=\Z[q,q^{-1}]$. Then $B$ is called *multiplicative* if, for every $i$ and $j$, there exists $k$ such that $b_ib_j= b_k$. In this article we prove that no multiplicative bases for $Z(\Z S_n)$ and $Z(\H_n)$ when $n\ge 3$. In addition, we prove that there exist exactly two multiplicative bases for $Z(\Z S_2)$ and none for $Z(\H_2)$.'
address:
- 'School of Computing, Engineering and Mathematics, University of Western Sydney, NSW 2751, Australia'
- 'Department of Mathematics, Shippensburg University, Pennsylvania, USA'
author:
- Andrew Francis
- Lenny Jones
title: 'Multiplicative Bases for the Centres of the Group Algebra and Iwahori-Hecke Algebra of the Symmetric Group'
---
[^1]
[^2]
Introduction
============
Quite a bit is known about the integral linear structure of the centres of Iwahori–Hecke algebras of finite Coxeter groups over the ring $R=\Z [q,q^{-1}]$. For instance, they are known to have a basis consisting of elements that specialize to conjugacy class sums [@GR97]. Our focus here is on the Iwahori–Hecke algebra $\H_n$ of the symmetric group, where there is the suggestion of a multiplicative structure through a connection with symmetric polynomials of Jucys–Murphy elements [@DJ87; @Mat99; @FG:DJconj2006]. This connection has yielded an alternative integral basis involving symmetric polynomials [@FJ:newintbasis], but as yet no multiplicative relations.
The ability to determine exactly how the product of elements of a basis for $Z(\H_n)$, the centre of $\H_n$, decomposes as a linear combination of basis elements would be particularly useful in any homomorphic context. For instance, the Brauer homomorphism plays an important role in the representation theory of the symmetric group, and has been generalized to $\H_n$ [@Jon87]. The fact that the codomain of this generalized Brauer homomorphism can be realized as certain products of elements from $Z(\H_n)$ was instrumental in developing a Green correspondence for $\H_n$ [@Du.GreenCor92]. However, as yet, an explicit determination of the structure constants has not been achieved [@FW-centres-filtered-2008]. Such a description would provide additional insight into the representation theory of $\H_n$. To this end, it is reasonable to search for bases for $Z(\H_n)$ that have nice multiplicative properties.
In this paper we address the question put by Jie Du (private communication) of whether $Z(\H_n)$ might have a basis over $R$ that is *multiplicative* (closed under the multiplication of the algebra). We answer this question in the negative for $Z(\H_n)$ when $n\ge 3$ by showing that no multiplicative bases exist for $Z(\Z S_n)$, the centre of the group algebra. We also prove that there exist exactly two multiplicative bases for $Z(\Z S_2)$ and none for $Z(\H_2)$. We point out that multiplicative bases do exist for $Z(\Q S_n)$ and for $Z(\H_n)$ over $\Q(q)$ for all $n$. For example, nested sums of central primitive orthogonal idempotents do the job.
Definitions and Notation
========================
Let $S_n$ be the symmetric group on $\{1,\dots,n\}$ with generating set of simple reflections $$S:=\{(i\ \ i+1)\mid 1\le i\le n-1\}.$$ Let $\Z S_n$ be the symmetric group algebra over the integers. The centre of $\Z S_n$, which we denote $Z(\Z S_n)$, has dimension equal to the number of partitions of $n$, and a $\Z$-basis consisting of conjugacy class sums. Considered over the rational numbers, $Z(\Q S_n)$ also contains a set of primitive orthogonal idempotents $e_\lambda$ in correspondence with the irreducible characters $\chi_{\lambda}$ for $\lambda\vdash n$. These idempotents form another basis for the centre of $\Q S_n$, and can be expressed in terms of the irreducible characters as follows (see e.g. [@CurtisReiner87v1]): $$e_\lambda=\frac{\chi_\lambda(1)}{n!}\sum_{w\in S_n}\chi_\lambda(w)w.$$ Since the characters are constant on conjugacy classes, this expression can be re-written in terms of class sums $\underline{C}$: $$\label{idempotent:class.elts1}
e_\lambda=\frac{\chi_\lambda(1)}{n!}\sum_{C}\chi_\lambda({w_C})\underline{C}$$ where $w_C$ is a representative of the conjugacy class $C$.
\[Rem: Idempotent Notation\] When the correspondence of a partition to its particular irreducible character, or conjugacy class is not needed, we simply use the notation $\chi_i$, $e_i$ or $C_i$ for an irreducible character, idempotent or conjugacy class, respectively. We also write $w_j$ for an element of the class $C_j$.
Using Remark \[Rem: Idempotent Notation\], we can rewrite (\[idempotent:class.elts1\]) as $$\label{idempotent:class.elts2}
e_i=\frac{\chi_i(1)}{n!}\sum_{j=1}^t\chi_i(w_j)\underline{C_j}.$$
Let $R=\Z[q,q^{-1}]$, where $q$ is an indeterminate. The *Iwahori–Hecke algebra* $\H_n$ of $S_n$ is the unital associative $R$-algebra with generators $\{T_i\mid 1\le i\le n-1\}$, where $T_i$ corresponds to the simple reflection $(i\ \ i+1)\in S$, subject to the relations $$\begin{aligned}
T_{i}T_{j}&=T_{j}T_{i} &\text{if $|i-j|\ge 2$}\ \\
T_{i}T_{i+1}T_{i}&=T_{i+1}T_{i}T_{i+1} &\text{for $1\le i\le n-2$}\ \\
T_{i}^2&=q+(q-1)T_{i} &\text{for $1\le i\le n-1$.}\end{aligned}$$
The Iwahori–Hecke algebra $\H_n$ is a deformation of the symmetric group algebra $\Z S_n$ [@CurtisReiner87v2]. In particular, the specialization of $\H_n$ at $q=1$ is isomorphic to $\Z S_n$. It follows that any basis for $Z(\H_n)$ over $R$ specializes at $q=1$ to a basis for $Z(\Z S_n)$. We denote the specialization of an element $h\in \H_n$, the specialization of every element of a subset $W$ of $\H_n$, or the specialization of every entry in a matrix $M$, at $q=1$ as $h|_{q=1}$, $W|_{q=1}$ or $M|_{q=1}$, respectively. Because of the isomorphism $\H_n|_{q=1}\simeq \Z S_n$, we think of the specialization of any element in $\H_n$ as being an element of $\Z S_n$.
\[Def:mult\] A basis $B=\left\{b_1,b_2,\ldots ,b_t\right\}$ for $Z(\H_n)$ over $R$ is called *multiplicative* if, for every $i$ and $j$, there exists $k$ such that $b_ib_j= b_k$. Specialization at $q=1$ gives analogous definitions for $Z(\Z S_n)$.
Multiplicative Bases
====================
The following preliminaries are needed to establish our main results.
\[Lem:mult form\] Let $\{e_1,e_2,\ldots ,e_t\}$ be a complete set of central primitive idempotents for $Z(\Q S_n)$. Let $B$ be a multiplicative basis for $Z(\Z S_n)$ and let $b\in B$. Then $$b=\sum_{i=1}^t\varepsilon_ie_i,$$ where $\varepsilon_i \in \{-1,0,1\}$.
Consider the sequence $b, b^2, b^3, b^4,\ldots $. Since $B$ is multiplicative and finite, there exist distinct nonnegative integers $k$ and $m$, and $\hat{b}\in B$, such that $$b^{k}=\hat{b}=b^{m}.$$ Since $\left\{e_1,e_2,\ldots ,e_t\right\}$ is a basis for $Z(\Q S_n)$, we can write $b=\sum_{i=1}^tw_ie_i$, where $w_i\in \Q$. Then $$\sum_{i=1}^{t}w_i^{k}e_i=b^k=\hat{b}=b^m=\sum_{i=1}^{t}w_i^{m}e_i,$$ so that $w_i^{k}=w_i^{m}$ for all $i$. Hence, for $w_i\ne 0$, we have that $w_i^{k-m}=1$, which implies that $w_i=\pm 1$ and the lemma is proven.
The following theorem, which we state without proof, is due to W. Burnside [@Isaacs].
\[Thm:Burnside\] Let $G$ be a finite group and let $\chi$ be a nonlinear irreducible character of $G$. Then there exists $g\in G$ such that $\chi(g)=0$.
We are now in a position to prove the following theorem.
There do not exist any multiplicative bases for either $Z(\H_n)$ over $R$ or for $Z(\Z S_n)$ when $n\ge 3$.
Since any basis for $Z(\H_n)$ over $R$ specializes at $q=1$ to a basis for $Z(\Z S_n)$, it is enough to show that no multiplicative basis exists for $Z(\Z S_n)$. Let $\{e_1,e_2,\ldots ,e_t\}$ be a complete set of central primitive idempotents for $Z(\Q S_n)$. By way of contradiction, assume that $B$ is a multiplicative basis for $Z(\Z S_n)$, and let $b\in B$. Considering $b$ as an element of $Z(\Q S_n)$, we have by Lemma \[Lem:mult form\] and (\[idempotent:class.elts2\]) that $$\begin{aligned}
b&=\sum_{i=1}^t\varepsilon_i e_i\\
&=\sum_{i=1}^t\varepsilon_i\left(\frac{\chi_i(1)}{n!}\sum_{j=1}^t\chi_i(w_j)\underline{C_j}\right)\\
&=\sum_{j=1}^t\left(\sum_{i=1}^{t}\varepsilon_i\frac{\chi_i(1)\chi_i(w_j)}{n!}\right)\underline{C_j},
\end{aligned}$$ where $\varepsilon_i\in \{-1,0,1\}$. Since $n\ge 3$, there exists at least one nonlinear irreducible character of $S_n$. Without loss of generality, let $\chi_t$ be nonlinear. Thus, by Theorem \[Thm:Burnside\], there exists at least one conjugacy class, say $C_s$, such that $\chi_t(w_s)=0$. Then the absolute value of the coefficient on $\underline{C_s}$ in $b$ is $$\begin{aligned}
\left| \sum_{i=1}^{t-1}\varepsilon_i\frac{\chi_i(1)\chi_i(w_s)}{n!} \right|
&= \frac{1}{n!}\left| \sum_{i=1}^{t-1}\varepsilon_i\chi_i(1)\chi_i(w_s)\right|\\
& \le \frac{1}{n!}\sum_{i=1}^{t-1}\left|\chi_i(1)\right|\left|\chi_i(w_s)\right|\\
& \le \frac{1}{n!}\sum_{i=1}^{t-1}\chi_i^2(1)\\
&<1.
\end{aligned}$$ Since $b$ is integral, it follows that $\sum_{i=1}^{t-1}\varepsilon_i\frac{\chi_i(1)\chi_i(w_s)}{n!}=0$. But then, the central element $\underline{C_s}$ cannot be written as a linear combination of the elements of $B$, which contradicts the fact that $B$ is a basis for $Z(\Z S_n)$.
The case $n=2$ requires a separate analysis which we give in the following theorem.
There exist exactly two multiplicative bases for $Z(\Z S_2)$ and no multiplicative bases for $Z(\H_2)$ over $R$.
It is easy to see that the two bases $\{(1), (12)\}$ and $\{(1), -(12)\}$ for $Z(\Z S_2)$ are multiplicative. To see that there are no others, let $B=\{b_1,b_2\}$ be a multiplicative basis for $Z(\Z S_2)$, where $b_1=c_1(1)+c_2(12)$ and $b_2=d_1(1)+d_2(12)$, with $c_1,c_2,d_1,d_2\in \Z$. Since $B$ is multiplicative, we have that $b_1^2=b_i$ and $b_2^2=b_j$ for some $i$ and $j$.
Suppose that $b_1^2=b_1$ and $b_2^2=b_1$. Since $b_1^2=b_1$, equating coefficients gives $c_1^2+c_2^2=c_1$ and $2c_2c_2=c_2$, from which we conclude that $c_1=1$ and $c_2=0$. Thus $b_1=(1)$. Then, since $b_2^2=b_1$, we have that either $d_1=\pm 1$ and $d_2=0$ or $d_1=0$ and $d_2=\pm 1$. The first case here implies that $b_2=(1)$, which is impossible since $B$ is a basis and $b_1=(1)$. The second case implies that $b_2=\pm (12)$, and so we get the two bases $\{(1),(12)\}$ and $\{(1),-(12)\}$. The other cases are similar and they yield no new bases.
To prove that no multiplicative basis for $Z(\H_2)$ over $R$ exists is a bit more tedious. Note that $Z(\H_2)=\H_2$ and that $\{1,T_1\}$ is a basis for $\H_2$ over $R$. Suppose that $B=\{b_1,b_2\}$ is a multiplicative basis for $Z(\H_2)$ over $R$. Since $B$ must specialize at $q=1$ to a multiplicative basis for $Z(\H_2)$, we will assume that $b_1|_{q=1}=(1)$ and $b_2|_{q=1}=(12)$. The case that $b_2|_{q=1}=-(12)$ is similar. Then we can write $$b_1=f+gT_1\quad \mbox{and} \quad b_2=r+sT_1,$$ where $f,g,r,s\in R$ with $f|_{q=1}=(1)=s|_{q=1}$ and $g|_{q=1}=0=r|_{q=1}$. Since $B$ is multiplicative, we have that $b_1^2=b_1$ or $b_1^2=b_2$. But $$b_1^2=f^2+g^2q+(2fg+g^2(q-1))T_1,$$ and we see that $b_1^2|_{q=1}=(1)$, which implies that $b_1^2=b_1$. Similarly, $b_2^2=b_1$. Equating coefficients gives: $$\begin{aligned}
f&=f^2+g^2q=r^2+s^2q\label{Eq:Coeff1}\\
g&=2fg+g^2(q-1)=2rs+s^2(q-1).\label{Eq:Coeff2}
\end{aligned}$$ If $g\equiv 0$, then $b_1=fT_1$ and $f^2=f$ since $b_1^2=b_1$. Thus, $f\equiv 1$ and $$\label{Eq:sumofsquares1}
r^2+s^2q=1.$$ Since $s|_{q=1}=1$, we have that $s\not \equiv 0$. Hence, from (\[Eq:Coeff2\]), we get $2r+s(q-1)\equiv 0$, so that $r=-s(q-1)/2$. Substituting into (\[Eq:sumofsquares1\]) gives $s^2=4/(q+1)^2\not \in R$, which is a contradiction. Thus, $g\not \equiv 0$ and from (\[Eq:Coeff2\]) we have that $f=(1-g(q-1))/2$. Then substituting into (\[Eq:Coeff1\]) gives $g^2=1/(q+1)^2\not \in R$, which completes the proof.
Comments and Conclusions
========================
One wonders if some minor adjustment of the idea of a multiplicative basis produces any integral bases for $Z(\H_n)$ or for $Z(\Z S_n)$. Motivated by this speculation, we define the concept of a *quasi-multiplicative* basis.
\[Def:q-mult\] A basis $B=\left\{b_1,b_2,\ldots ,b_t\right\}$ for $Z(\H_n)$ over $R$ is called *quasi-multiplicative* if, for every $i$ and $j$, there exist $k$ and some nonzero element $f_{ij}\in R$, such that $b_ib_j=f_{ij}\cdot b_k$. Specialization at $q=1$ gives an analogous definition for $Z(\Z S_n)$.
We see that if $f_{ij}=1$ for all $i$ and $j$ in Definition \[Def:q-mult\], then $B$ is a multiplicative basis. So Definition \[Def:q-mult\] is a generalization of Definition \[Def:mult\], and every multiplicative basis is a quasi-multiplicative basis. In a forthcoming paper, we investigate the existence of new quasi-multiplicative bases for $Z(\H_n)$ and $Z(\Z S_n)$.
[A]{}
Charles Curtis and Irving Reiner. , volume 1. Wiley, 1987.
Charles Curtis and Irving Reiner. , volume 2. Wiley, 1987.
Richard Dipper and Gordon James. Blocks and idempotents of [H]{}ecke algebras of general linear groups. , 54(1):57–82, 1987.
Jie Du. The [G]{}reen correspondence for the representations of [H]{}ecke algebras of type [$A\sb {r-1}$]{}. , 329(1):273–287, 1992.
Andrew Francis and John J. Graham. . , 306(1):244–267, 2006.
Andrew Francis and Lenny Jones. . , 321(3):866–878, 2009.
Andrew Francis and Weiqiang Wang, *The centers of [I]{}wahori-[H]{}ecke algebras are filtered*, Representation Theory: Proceedings of the 4th International Conference on Representation Theory, Lhasa, Tibet, July 2007 (Jianpan Wang Zongzhu Lin, ed.), Contemporary Mathematics, vol. 478, American Mathematical Society, 2009, pp. 29–37.
Meinolf Geck and Rapha[ë]{}l Rouquier. Centers and simple modules for [I]{}wahori-[H]{}ecke algebras. In [*Finite reductive groups (Luminy, 1994)*]{}, pages 251–272. Birkhäuser Boston, Boston, MA, 1997.
I. Martin Isaacs. . AMS Chelsea Publishing, 2011.
Lenny K. Jones. . PhD thesis, University of Virginia, 1987.
Andrew Mathas. Murphy operators and the centre of [Iwahori]{}-[Hecke]{} algebras of type [$A$]{}. , 9:295–313, 1999.
[^1]: The first author was supported by Australian Research Council Future Fellowship FT100100898.
[^2]: Both authors have benefited greatly from the mentoring of Leonard Scott — in the case of the first author as a postdoc mentor, and in the case of the second author as a PhD advisor. They would like to record their gratitude for his patient and wise tutelage.
|
---
abstract: 'Exponential inequalities are main tools in machine learning theory. To prove exponential inequalities for non i.i.d random variables allows to extend many learning techniques to these variables. Indeed, much work has been done both on inequalities and learning theory for time series, in the past 15 years. However, for the non independent case, almost all the results concern stationary time series. This excludes many important applications: for example any series with a periodic behaviour is nonstationary. In this paper, we extend the basic tools of [@DF15] to nonstationary Markov chains. As an application, we provide a Bernstein-type inequality, and we deduce risk bounds for the prediction of periodic autoregressive processes with an unknown period.'
author:
- 'Pierre Alquier[^1], Paul Doukhan[^2] and Xiequan Fan[^3]'
title: Exponential inequalities for nonstationary Markov Chains
---
Introduction
============
Exponential inequalities are corner stones of machine learning theory. For example, distribution free generalization bounds were proven by Vapnik and Cervonenkis based on Hoeffding’s inequality, see [@Va98]. Model selection bounds in [@M07] also rely on exponential moment inequalities.
To prove such inequalities in the non i.i.d setting is thus crucial to study the generalization ability of machine learning algorithms on time series. As an example, a Bernstein type inequality for $\alpha$-mixing time series is proven in [@MM02]. This result is used by [@SC09] to prove generalization bounds for general learning problems with $\alpha$-mixing observations.
Exponential inequalities and machine learning with non-i.i.d observations actually became an important research direction, a more detailed list of references is given below. However, most of these references assume stationarity. That is, only the independence assumption was removed. The observations are still assumed to be identically distributed, or at least erdogic. This excludes many applications: in addition to trends, data related to a human activity such as in industry or economics has some periodicity (hourly, daily, yearly…) and some regime switching; the same remark applies to data with a physical origin, such as in geology, astrophysics…
In this paper, we generalize the inequalities proven by [@DF15] for time homogeneous Markov chains to non-homogeneous chains. This allows to study a large set of nonstationary processes. We obtain Bernstein and McDiarmid inequalities as well as moments inequalities. As an application, we study periodic autoregressive processes of the form $X_{t} = f_t (X_{t-1}) + \varepsilon_t$ where $f_{t+T}=f_{t}$ for any $t$, for some period $T$. Thanks to our version of Bernstein’s inequality we show that the Empirical Risk Minimizer (ERM) leads to consistent predictions in this setting. We also show that a penalized version of the ERM enjoys the same property even when $T$ is unknown.
The paper is organized as follows. The rest of this introduction is dedicated to a state-of-the-art on exponential inequalities for time series. Section \[section:notations\] introduces the notations and assumptions that will be used in the whole paper. In Section \[section:theorems\], we state an extension of Proposition 2.1 of [@DF15]: this is Lemma \[McD\]. As a proof of concept, we use this lemma to prove a version of Bernstein inequality for nonstationary Markov chains. We also provide Cramer and McDiarmid inequalities based on this lemma. We study periodic autoregressive series in Section \[section:autoregression\]. Finally, Section \[section:proof\] contains the proof of Lemma \[McD\] and of the results in Section \[section:autoregression\].
State of the art
----------------
We refer the reader to [@Bo13] for an overview on exponential and concentration inequalities in the i.i.d case. This book also provides references for applications of these results to machine learning theory.
Exponential inequalities were proven for time under a various range of assumptions. We refer the reader to [@D18] for various approaches on modelling time series.
Inequalities for Markov chains $(X_t)_{t\geq 1}$ are proven in [@Cat03; @A08; @BC10; @JO10; @Win17; @BP18; @P18; @BC19]. Note that most of these inequalities require the chain to be time homogeneous. While this does not imply the chain to be stationary, in some sense the $X_t$’s are asymptotically identically distributed in these papers. For example, consider the powerful renewal technique used in [@BC19] to prove a version of Bernstein inequality. The proof is based on the fact that blocks $(X_{\tau_i},\dots,X_{\tau_{i+1}-1})$ between two [*renewal times*]{} $\tau_i$ and $\tau_{i+1}$ are actually i.i.d. It is thus possible to apply the i.i.d version of Bernstein inequality to these blocks. The spectral technique used in [@P18] still relies on the ergodicity of the Markov chain (we thank the anonymous Referee for pointing out some of these references). Exponential inequalities for hidden Markov chains are given in [@KR08].
It is a well-known fact that Hoeffding’s inequality is not only valid for independent observations, but also for martingales increments (it is sometimes refered to as Hoeffding-Azuma inequality in this case). To decompose a function of the process as a sum of martingales increments is actually one of the most powerful techniques to prove exponential inequalities, see Chapter 3 in [@Bo13]. More exponential inequalities for martingales can be found in [@S+12; @R13b; @BDR15]. This technique is actually used by [@DF15; @Fx2] to prove exponential inequalities for Markov chains.
Markov chains are extremely useful in modelisation and simulations, however, many time series have a very different dependence structure. Mixing coefficients allow to quantify the dependence between observations without giving an explicit structure on this dependence. We refer the reader to [@R17] for a comprehensive introduction. Exponential inequalities for mixing processes are proven in [@Sam00; @RPR09; @R17; @HS17]. Mixing series however exclude many stochastic processes, as discussed in the monograph [@DDp07]. Weak dependance coefficients cover a wider range of processes for which Bernstein type inequalities are proven for example in [@CMS; @DN07; @Win10; @RPR10; @BZ17]. Dynamic systems are examples of processes where only $X_1$ is random, each $X_t$ is then a deterministic fonction of $X_{t-1}$. Based on weak dependence arguments, it is possible to prove exponential inequalities for such processes [@CMS].
Based on such inequalities, it is possible to prove generalization bounds for machine learning algorithms [@SHS09; @SC09; @SK13; @LHTG14; @HS14; @SP15; @KM15; @MSS17; @AB18]. Model selection techniques in the spirit of [@M07] are studied in [@Mei00; @L11; @AW12], and aggregation of estimators in [@ALW13].
In this paper we prove provide tools to prove exponential inequalities for nonstationary, non homogeneous Markov chains. Rather than the renewal or spectral techniques discussed above, we extend the martingale approach of [@DF15] to non-homogeneous chains.
Notations {#section:notations}
=========
From now, all the random variables are defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$. Let $({\mathcal X}, d)$ and $({\mathcal Y}, \delta)$ be two complete separable metric spaces. Let $(\varepsilon_t)_{t \geq 2}$ be a sequence of i.i.d ${\mathcal Y}$-valued random variables. Let $X_1$ be a ${\mathcal X}$-valued random variable independent of $(\varepsilon_t)_{t \geq 2}$. Let $(X_t)_{t \geq 1}$ be the Markov chain given by $$\label{Mchain}
X_t=F_t(X_{t-1}, \varepsilon_t), \quad \text{for $t\geq 2$},$$ where the functions $F_t: {\mathcal X} \times {\mathcal Y} \rightarrow {\mathcal X}$ are such that $$\label{contract}
\sup_{t} {\mathbb E}\big [ d\big(F_t(x, \varepsilon_1), F_t(x', \varepsilon_1)\big) \big] \leq
\rho d(x, x')$$ for some constant $\rho \in [0,1)$, and $$\label{c2}
\sup_t d(F_t(x,y), F_t(x,y')) \leq C \delta(y,y')$$ for some constant $C>0$. In particular, when $F_t \equiv F,$ this is the model studied by [@DF15].
This class of Markov chains, that we call “one-step contracting", contains a lot of pertinent examples. The classical AR(1)-process is given by $X_t = F(X_{t-1},\varepsilon_t)$ where $F(x,y) = ax+y$. Condition is satisfied, and Condition will be satisfied as soon as $|a|<1$. Now, consider a time-varying AR(1) process: $$X_t = a_t X_{t-1} + \varepsilon_t.$$ This process may be non-stationary. Condition is still satisfied, and $
|F_t(x,y) - F_t(x',y)| \leq |a_t| |x-x'|
$ so Condition will be satisfied as soon as $\sup_t |a_t| <1$. This process is studied by [@BD17] under various assumptions: local stationarity, that means a slow variation of $a_t$ as a function of $t$, see [@Da96], and periodicity, that is, for any $t$: $a_{t+T}=a_t$ for some (known) period $T$. If $T$ is unknown, a cross-validation procedure to estimate $T$ is proposed (Remark 2.4) without a consistency result. Below we will propose a penalized procedure with some statistical guarantees.
As a much more general example, consider the following functional auto-regressive model. Let ${\mathcal X}$ be a separable Banach space with norm $|\cdot|$. The functional auto-regressive model is defined by $$X_t=f_t(X_{t-1}) + \varepsilon_t \, ,$$ where $f_t : {\mathcal X} \rightarrow {\mathcal X}$ is such that $$|f_t(x)-f_t(x')|\leq \rho |x-x'| .$$ Clearly (\[Mchain\]) and (\[contract\]) are satisfied once $\rho \in [0, 1)$, see [@DF99] for more examples.
We introduce the natural filtration of the chain ${\mathcal F}_0=\{\emptyset, \Omega \}$ and for $ t \in \mathbb{N}$, $ {\mathcal F}_t = \sigma(X_1, X_2, \ldots, X_t)$.
Consider a separately Lipschitz function $f: {\mathcal X}^n \to {\mathbb R}$ such that $$|f(x_1, \ldots, x_n)-f(x'_1, \ldots, x'_n)| \leq \sum_{t=1}^n d(x_t,x'_t) \, .$$ We define $$\label{Sn}
S_n:=f(X_1, \ldots , X_n) -{\mathbb E}[f(X_1, \ldots , X_n)]\, .$$ The objective of what follows will be to derive inequalities on the tails of $ \mathbb{P}(|S_n|>x) $.
Main results {#section:theorems}
============
[@DF15] proved several exponential and moments inequalities if $F_t \equiv F$, by using a martingale decomposition. We will first extend this martingale decomposition to the general case. As an example, we will use it to prove a Bernstein type inequality. Other inequalities are given in the appendix. Here $(X_t)_{t \geq 1}$ will be a Markov chain satisfying [(\[Mchain\])]{} for some functions $(F_t)_{t \geq 2}$ satisfying [(\[contract\])]{}.
Main lemma: martingale decomposition
------------------------------------
Set $K_t(\rho)=(1-\rho^{t+1})/(1-\rho)$ for $t \ge0,\rho \in [0, 1)$ and $$g_k(X_1, \ldots , X_k)= {\mathbb E}[f(X_1, \ldots, X_n)|{\mathcal F}_k]\, ,$$ and $$d_k=g_k(X_1, \ldots, X_k)-g_{k-1}(X_1, \ldots, X_{k-1})\, .$$ Define $
S_t:=d_1+d_2+\cdots + d_t
$, for $t \in [1, n-1]$, and note that the functional $S_n$ introduced in [(\[Sn\])]{} satisfies indeed $ S_n =d_1+d_2+\cdots + d_n$. Thus $(S_t)$ is a martingale adapted to the filtration ${\mathcal F}_t$, and $(d_t)$ is the martingale difference of $(S_t)$.
Let $P_{X_1}$ denote the distribution of $X_1$ and $P_\varepsilon$ the (common) distribution of the $\varepsilon_t$’s. Let $G_{X_1}$, $G_\varepsilon$ and $H_{t, \varepsilon}$ be defined by $$G_{X_1}(x)=\int d(x, x') P_{X_1}(dx'),$$ $$G_{\varepsilon}(y)= \int C\delta(y,y')P_{\varepsilon}(dy') \text{ and}$$ $$H_{t,\varepsilon}(x,y)= \int d(F_t(x,y),F_t(x,y'))P_{\varepsilon}(dy') \, .$$
We are now in position to state our main lemma.
\[McD\] Assume and , then:
1. The function $g_t$ is separately Lipschitz and $$|g_t(x_1, \ldots, x_t)-g_t(x'_1, \ldots, x'_t)|
\leq \sum_{i=1}^{t-1} d(x_i,x_i')+ K_{n-t}(\rho) d(x_t, x'_t) \, .$$
2. The martingale difference $(d_t)$ is such that $$\begin{aligned}
|d_1| & \leq K_{n-1}(\rho) G_{X_1}(X_1) \, ,
\\
\forall t\in[2,n]\text{, } |d_t| & \leq K_{n-t}(\rho) H_{t,\varepsilon}(X_{t-1}, \varepsilon_t)\, .\end{aligned}$$
3. Assume moreover that the $F_t$’s satisfy [(\[c2\])]{}. Then $H_{t,\varepsilon}(x,y) \leq G_\varepsilon(y)$, and consequently, for $t \in [2,n]$, $$|d_t|\leq K_{n-t}(\rho) G_\varepsilon(\varepsilon_t)\, .$$
The proof of this lemma is given in Section \[section:proof\]. First, we want to show that the inequalities in this lemma can be used to prove exponential inequalities on $S_n$.
Application: Bernstein inequality
---------------------------------
Note that [@V95] and [@D99] obtained some tight Bernstein type inequalities for martingales. Here, we can use the martingale decomposition and apply Lemma \[McD\] to obtain the following result.
\[Bernsteinineq\] Assume that there exist some constants $M>0, V_1\geq 0$ and $V_2\geq 0$ such that, for any integer $k\geq 2$, $$\label{ass:moment}
{\mathbb E} \Big[ G_{X_1}(X_1)^k\Big] \leq \frac {k!}{2} V_1 M^{k-2} \text{, and }
{\mathbb E} \Big[ G_\varepsilon(\varepsilon)^k\Big] \leq
\frac {k!}{2} V_2 M^{k-2} \, .$$ Let $\delta=MK_{n-1}(\rho)$ and $$V_{(n)} = V_1\Big(K_{n-1}(\rho)\Big)^2 + V_2 \sum_{k=2}^n \Big(K_{n-k}(\rho)\Big)^2 .$$ Then, for any $s \in [0, \delta^{-1})$, $$\label{maindfs}
\mathbb{E}\,[e^ {\pm sS_n} ]\leq \exp \left (\frac{s^2 V_{(n)} }{2 (1- s\,\delta )} \right )\, .$$ Consequently, for any $x> 0$, $$\begin{aligned}
{\mathbb P}\big(\pm S_n \geq x\big)
&\leq \exp \left( \frac{-x^2}{V_{(n)}(1+\sqrt{1+2x \delta/V_{(n)}})+x \delta }\right)\, \\
&\leq \exp \left( \frac{-x^2}{2 \,(V_{(n)} +x \delta ) }\right)\, .\end{aligned}$$
The quantity $V_{(n)}$ can be computed explicitely from the definition for each $n$ but note that $$V_1 + (n-1) V_2 \leq V_{(n)} \leq \frac{V_1 + (n-1) V_2}{(1-\rho)^2}.
\label{order_v}$$ *Proof.* For any $s \in [0, \delta^{-1})$, $$\begin{aligned}
\mathbb{E}\,[e^{s d_1 } ]
& \leq 1+ \sum_{i=2}^{\infty} \frac{s^i}{i!}\, \mathbb{E}\,[ | d_1|^i ] \\
&\leq 1+ \sum_{i=2}^{\infty} \frac{s^i}{i!}\, \Big(K_{n-1}(\rho)\Big)^i \mathbb{E}\,\Big[ \Big(G_{X_1}(X_1)\Big)^i \Big] \nonumber\\
&\leq 1+ \sum_{i=2}^{\infty} \frac{s^i}{i!}\, \Big(K_{n-1}(\rho)\Big)^i \frac {i!}{2} V_1 M^{i-2} \\
& = 1+ \frac{s^2V_1 \Big(K_{n-1}(\rho)\Big)^2 }{2 (1 -s\, \delta )}\, \\
&\leq \exp \left (\frac{s^2V_1\Big(K_{n-1}(\rho)\Big)^2}{2 (1 -s\,\delta )} \right ).\end{aligned}$$ We use Lemma \[McD\] for the second inequality, the moment assumption for the third one, and the inequality $1+s \leq e^s,$ for the final inequality. Similarly, for any $k \in [2, n],$ $$\mathbb{E}\,[e^{s d_k }|\mathcal{F}_{k-1}] \leq \exp \left( \frac{s^2V_2\Big(K_{n-k}(\rho)\Big)^2}{2 (1 -s\delta)} \right) \, .$$ By the tower property of conditional expectation, it follows that $$\begin{aligned}
\mathbb{E}\,\big[e^{ sS_n} \big] &=& \mathbb{E}\,\big[ \mathbb{E}\, [e^{ sS_n} |\mathcal{F}_{n-1} ] \big]\\
&=& \mathbb{E}\,\big[ e^ { sS_{ n-1}} \mathbb{E}\, [e^ { sd_n} |\mathcal{F}_{n-1} ] \big] \\
&\leq & \mathbb{E}\,\big[ e^ { sS_{ n-1}} \big] \exp \left( \frac{s^2V_2}{2 (1- s\,\delta)} \right)\\
&\leq & \exp \left(\frac{s^2 V_{(n)}}{2 (1- s\,\delta)} \right),\end{aligned}$$ which gives inequality (\[maindfs\]). Using the exponential Markov inequality, we deduce that, for any $x\geq 0$ $$\begin{aligned}
\mathbb{P}\left( S_{n} \geq x \right)
&\leq& \mathbb{E}\, \big[e^{s\,(S_n -x) } \big ] \nonumber\\
&\leq& \exp \left(-s\,x + \frac{s^2 V_{(n)}}{2 (1- s\,\delta)} \right)\, . \label{fines}\end{aligned}$$ Minimizing the right-hand side with respect to $s$ leads to the result.
McDiarmid and Cramer inequalities
---------------------------------
Here, we state other consequences of Lemma \[McD\]. However, as our applications are based on Bernstein inequality, we postpone the proof of these results to Section \[section:proof\].
When the Laplace transform of the dominating random variables $G_{X_1}(X_1)$ and $G_{\varepsilon}(\varepsilon_k)$ satisfy the Cramér condition, we obtain the following proposition.
\[cram\] Assume that there exist some constants $a>0,$ $K_1\geq 1$ and $K_2\geq 1$ such that $${\mathbb E} \Big[ \exp \Big( a G_{X_1}(X_1)\Big)\Big] \leq K_1$$ and $${\mathbb E} \Big[ \exp \Big( a G_\varepsilon(\varepsilon)\Big)\Big] \leq K_2.$$ Let $$K=\frac{2}{e^{2}} \left( K_1+ K_2\sum_{i=2}^{n} \Big(\frac {K_{n-i}(\rho)}{K_{n-1}(\rho)}\Big)^2 \right)$$ and $ \delta={a}/{K_{n-1}(\rho)} .$ Then, for any $s \in [0, \delta )$, $$\mathbb{E}\,[e^{\pm s S_n}]\leq \exp \left( \frac{s^2 K \delta^{-2} }{1-s \delta^{-1}} \right) \, .$$ Consequently, for any $x> 0$, $$\begin{aligned}
{\mathbb P}\big(\pm S_n \geq x\big) &\leq& \exp \left( \frac{-(x\delta)^2}{2K (1+\sqrt{1+ x \delta / K })+x \delta }\right)\, \\
&\leq& \exp \left( \frac{-(x\delta)^2}{ 4K+ 2 x \delta }\right)\,\, .\end{aligned}$$
Now, consider the case where the increments $d_k$ are bounded. We shall use an improved version of the well known inequality by McDiarmid, stated by [@R13]. For this inequality, we do not assume that [(\[c2\])]{} holds. Thus, Proposition \[classic\] applies to any Markov chain $X_{i}=F_i(X_{i-1}, \varepsilon_i)$ for $F_i$ satisfying [(\[contract\])]{}. Following [@R13], let $$\ell(t)= (t - \ln t -1) +t (e^t-1)^{-1} + \ln(1-e^{-t}) \ \ \ \
\textrm{for all}\ t > 0,$$ and let $$\ell^*(x)=\sup_{t>0}\big(xt- \ell (t) \big) \ \ \ \ \textrm{for all}\ x > 0,$$ be the Young transform of $\ell(t)$. As quoted by [@R13], the following inequality holds $$\begin{aligned}
\ell^*(x) \geq (x^2-2x) \ln(1-x) \ \ \ \ \textrm{for any}\ x \in [0, 1).\end{aligned}$$ Let also $(X_1', (\varepsilon'_i)_{i \geq 2})$ be an independent copy of $(X_1, (\varepsilon_i)_{i \geq 2})$.
\[classic\] Assume that there exist some positive constants $M_k$ such that $$\big \|d(X_1,X_1')
\big \|_\infty \leq M_1$$ and for $k \in [2, n] $, $$\big \|d\big(F_k(X_{k-1}, \varepsilon_k), F_k(X_{k-1}, \varepsilon'_k) \big)
\big \|_\infty \leq M_k .$$ Let $$M^2(n,\rho)= \sum_{k=1}^n \big(K_{n-k}(\rho) M_k \big)^2$$ and $$D (n, \rho)=\sum_{k=1}^n K_{n-k}(\rho) M_k \, .$$ Then, for any $s \geq 0$, $$\begin{aligned}
\label{rio1}
\mathbb{E}[e^{\pm sS_n } ] \ \leq \ \exp \left ( \frac{D^2(n, \rho)}{M^2(n,\rho)}\ \ell \Big( \frac {M^2(n,\rho)\, s} {D(n, \rho)} \Big)\right) \,\end{aligned}$$ and, for any $x \in [0, D(n, \rho)]$, $$\begin{aligned}
\label{rio2}
{\mathbb P}\big(\pm S_n>x\big) \leq \exp \left ( -\frac{D^2(n, \rho)}{M^2(n,\rho)} \ \ell^*\Big( \frac {x} {D(n, \rho)} \Big)\right).\end{aligned}$$ Consequently, for any $x \in [0, D(n, \rho)]$, $$\begin{aligned}
\label{riosbound}
{\mathbb P}\big(\pm S_n>x\big) &\leq& \left ( \frac{D(n, \rho)-x}{D(n, \rho)}\right)^{ \frac{2D(n, \rho)x -x^2}{M^2(n,\rho)}}\!\!\!.\end{aligned}$$
Since $(x^2-2x) \ln(1-x) \geq 2\,x^2$, $\forall x \in [0,1)$, implies the following McDiarmid inequality $$\begin{aligned}
{\mathbb P}\big(\pm S_n>x\big) \ \leq \ \exp \left ( -\frac{2 x^2}{M^2(n,\rho)}
\right) \, .\end{aligned}$$
Taking $ \displaystyle \Delta (n, \rho)=K_{n-1}(\rho)\max_{1\leq k \leq n} M_k$, we obtain, for any $x \in [0, n \Delta(n, \rho)]$, $${\mathbb P}\big(\pm S_n>x\big)
\leq \exp \left ( -n \ell^*\Big( \frac {x} {n \Delta(n, \rho)} \Big)\right)
\leq \exp \left ( -\frac{2 x^2}{n\Delta^2(n,\rho)}
\right)
\, .$$
Application to periodic autoregressive models {#section:autoregression}
=============================================
In this section, we apply Theorem \[Bernsteinineq\] to predict a nonstationary Markov chain. We will use periodic autoregressive predictors. Of course, these predictors will work well when the Markov chain is indeed periodic autoregressive. However, we will state the results in a more general context – when the model is wrong, we simply estimate its best prediction by a periodic autoregression.
Context
-------
Let $(X_t)_{t\geq 1}$ be an $\mathbb{R}^d$-valued process defined by the distribution of $X_1$ and, for $t>0$, $$X_{t} = f_{t}^*(X_{t-1}) + \varepsilon_t ,$$ where the $\varepsilon_t$ are i.i.d and centered, and each $f_t^*$ belong to a fixed family of functions $\mathcal{F}$ with $\forall f\in\mathcal{F}$, $\forall (x,y)\in\mathbb{R}^d$, $\|f(x)-f(y)\| \leq \rho \|x-y\|,\ \rho \in [0, 1).$
We are interested by periodic predictors: $f_{t+T}=f_{t}$, defined by a sequence $(f_1,\dots,f_T)\in\mathcal{F}^T$. Of course, if the series $(X_t)$ actually satisfies $f^*_{t+T}=f^*_{t}$, then this family of predictors can give optimal predictions. But they might also perform well when this equality is not exact (for example, when there is a very small drift).
Prediction is assessed with respect to a non-negative loss function: $\ell(\cdot)$. We assume that $\ell$ is $L$-Lipschitz. Note that this includes the absolute loss, the Huber loss and all the quantile losses. This also includes the quadratic loss if we assume that $X_t$, and hence $\varepsilon_t$, is bounded. Given a sample $X_1,\dots,X_n$ we define the empirical risk, for any $f_{1:T}=(f_1,\dots,f_T)\in \mathcal{F}^T$: $$r_n(f_{1:T}) = \frac{1}{n-1}\sum_{i=2}^n \ell\left(X_i - f_{i[T]} (X_{i-1}) \right) ,$$ where $i[T]\in\{1,\ldots,T\}$ is such that $i-i[T] \in T\cdot\mathbb{Z}.$ We then define $$R_n(f_{1:T}) = \mathbb{E}[r_n(f_{1:T})].$$ Note that when the process has actually $T$-periodic distribution, in the sense that the distribution of the vectors $(X_{kT+1},\dots,X_{(k+1)T})$ are the same for any $k$, then alsmot surely $f_t^* = f^*_{t+T}$ for any $t$ and $$R_n(f_{1:T}) \xrightarrow[n\rightarrow\infty]{} \frac{1}{T}\sum_{t=1}^T \mathbb{E}[ \ell\left(X_t - f_{t} (X_{t-1}) \right)]$$ the prediction averaged over one period, which appears to be equal to $R_{T+1}(f_{1:T})$. We can actually give a more accurate statement.
\[prop-risk\] When the distribution of $(X_{kT+1},\dots,X_{(k+1)T})$ does not depend on $k$, $$| R_{T+1}(f_{1:T}) - R_n(f_{1:T}) | \leq C_0 \frac{T+1}{n-1} ,$$ where $ C_0 = L(1+\rho) \left[ \frac{G_\varepsilon(0)}{1-\rho} + G_{X_1}(0)\right] $.
(All the proofs are postponed to Section \[section:proof\] for the clarity of exposition). The simplest use of Bernstein’s inequality is to control the deviation between $r_n(f_{1:T})$ and $R_n(f_{1:T})$ for a fixed predictor $f_{1:T}$.
\[coro-var\] Assume that the moment assumption in Theorem \[Bernsteinineq\], given by \[ass:moment\], is satisfied. Define $V_{(n)}$ and $\delta$ (depending on $M$, $\rho$, $V_1$ and $V_2$) as in Theorem \[Bernsteinineq\]. Then for any $0\leq s < (n-1)/(L(1+\rho)\delta)$, $$\mathbb{E}\exp\left( \pm s ( r_n(f_{1:T})- R_n(f_{1:T})) \right)
\leq \exp\left( \frac{s^2(1+\rho)^2 L^2 \frac{V_{(n)}}{n-1}}{2(n-1)-2s(1+\rho)\delta L} \right).$$
From above, we know that $$\mathcal{V} := \frac{V_{(n)}}{n-1} \leq \frac{\frac{V_1}{n-1}+V_2}{(1-\rho)^2} \leq \frac{V_1 + V_2}{(1-\rho)^2}$$ that does not depend on $n$.
Estimation with a fixed period
------------------------------
In this subsection we assume that $T$ is known (we will later show how to deal with the case were it is unknown). Thus, we define the estimator $$\hat{f}_{1:T} = (\hat{f}_1,\dots,\hat{f}_T )
= \underset{f_{1:T} = (f_1,\dots,f_T)}{\operatorname{argmin}} r_n(f_{1:T}) .$$
In order to study the statistical performances of $\hat{f}_{1:T}$, a few definitions are in order. For any function $f:\mathbb{R}^d \rightarrow \mathbb{R}^d$ we will use the notation $$\|f\|_{\sup} := \sup_{x\neq 0} \frac{\|f(x)\|}{\|x\|} .$$ When considering linear functions, this actually coincides with the operator norm.
Define the covering number $\mathcal{N}(\mathcal{F},\epsilon)$ as the cardinality of the smallest set $\mathcal{F}_{\epsilon}$ such that $\forall f\in\mathcal{F}$, $\exists f_{\epsilon} \in\mathcal{F}_{\epsilon}$ such that $ \|f-f^\epsilon\|_{\sup}\leq \epsilon $. Define the entropy of $\mathcal{F}$ by $\mathcal{H} (\mathcal{F},\epsilon) = 1\vee \log \mathcal{N}(\mathcal{F},\epsilon)$.
Covering numbers are standard tools to measure the complexity of set of predictors in machine learning.
Consider the class of AR(1) predictors $f(x) = a x$, $|a|\leq \rho$. Define $\mathcal{F}_\epsilon$ as the set of all functions $f(x)=i\varepsilon x$ for $i\in \{0,\pm 1,\dots,\pm \lfloor \rho/\epsilon \rfloor \} $. It is clear that $\mathcal{F}_\epsilon$ satisfies the above definition and that ${\rm card}(\mathcal{F}_\epsilon)\leq 1+2\rho/\epsilon \leq 1+2/\epsilon$. Thus, $ \mathcal{N}(\mathcal{F},\epsilon) \leq 1+2/\epsilon$ and so $\mathcal{H} (\mathcal{F},\epsilon) \leq 1\vee \log\left( 1+2 /\epsilon \right) $. In the VAR(1) case, $f(x) = Ax$ where $\|A\|_{\sup} \leq \rho$. Using the set $\mathcal{F}_\epsilon$ of all matrices with entries in $ (\epsilon/\sqrt{d}) \{0,\pm 1,\dots,\pm \lfloor \rho\sqrt{d}/\epsilon \rfloor \} $, we prove that $ \mathcal{N}(\mathcal{F},\epsilon) \leq (1+2\sqrt{d} /\epsilon)^d$ and thus $\mathcal{H} (\mathcal{F},\epsilon) \leq 1\vee d \log( 1+2\sqrt{d}/\epsilon) $.
We are now in position to state the following result on the convergence of $R_n(\hat{f}_{1:T})$.
\[thm-cvg\] As soon as $ n \geq 1 + 4\delta^2 T \mathcal{H} (\mathcal{F},\frac{1}{Ln}) / \mathcal{V} $ we have, for any $\eta>0$, $$\begin{gathered}
\mathbb{P}\Biggl\{
R_n(\hat{f}_{1:T}) \leq \inf_{f_{1:T}\in\mathcal{F}^T} R_n(f_{1:T})
\\
+
C_1 \sqrt{ \frac{ T\mathcal{H} (\mathcal{F},\frac{1}{Ln})}{n-1} }
+ C_2 \frac{ \log\left(\frac{4}{\eta}\right)}{\sqrt{n-1}}
+ \frac{C_3}{n}
\Biggr\} \geq 1-\eta,\end{gathered}$$ where $C_1 = 4(1+\rho) L\sqrt{\mathcal{V}}$, $C_2 = 2(1+\rho)L\sqrt{\mathcal{V}} + 2\delta$ and $C_3 = 3[G_{\varepsilon}(0)+G_{X_1}(0)]/(1-\rho) + \mathcal{V}/(2\delta)$.
The theorem states that the predictor $\hat{f}_{1:T}$ predict as well as the best possible one up to an estimation error that vanishes at rate $\sqrt{n}$. For example, using (periodic) VAR(1) predictors in dimension $d,$ we get a bound in $$R_n(\hat{f}_{1:T}) \leq \inf_{f_{1:T}\in\mathcal{F}^T} R_n(f_{1:T}) + \mathcal{O}\left( d \sqrt{\frac{T\log(nd)}{n}} \right).$$
When the series is indeed stationary for a known $T$, it is to be noted that $(X_{iT+1},\dots,X_{i(T+1)})_{i\geq 0}$ is a time homogeneous Markov chain. In this case, our technique is not really necessary: it would be possible to apply the inequality from [@DF15]. However, when $T$ is not known, this becomes impossible. In this case, one has to compare the empirical risks of $\hat{f}_{1:T}$ for the various possible $T$’s, and for most of them, $(X_{iT+1},\dots,X_{i(T+1)})_{i\geq 0}$ is not homogeneous. In this case, vectorization cannot help. On the other hand, our inequality can be used for period selection, as detailed in the next subsection.
Period and model selection
--------------------------
We define a penalized estimator in the spirit of [@M07]. Fix a maximal period $T_{\max}$, for example $T_{\max}=\lfloor n/2 \rfloor$. We propose the following penalized estimator for $T$: $$\hat{T} = \arg\min_{1\leq T \leq T_{\max}} \left[ r_n(\hat{f}_{1:T}) +
\frac{C_1}{2} \sqrt{ \frac{ T\mathcal{H} (\mathcal{F},\frac{1}{Ln})}{n-1} }\, \right].$$ Using this estimator, we have the following result.
\[thm-selection\] For any $\eta>0$ we have, $$\begin{gathered}
\mathbb{P}\Biggl\{
R_n(\hat{f}_{1:\hat{T}}) \leq \inf_{1\leq T \leq T_{\max}}\ \inf_{f_{1:T}\in\mathcal{F}^T}
\Biggl[ R_n(f_{1:T})
\\
+
C_1 \sqrt{ \frac{ T\mathcal{H} (\mathcal{F},\frac{1}{Ln})}{n-1} }
+ C_2 \frac{ \log\left(\frac{4 T_{\max}}{\eta}\right)}{\sqrt{n-1}}
+ \frac{C_3}{n} \Biggr]
\Biggr\} \geq 1-\eta,\end{gathered}$$ as soon as $ n \geq 1 + 4\delta^2 T_{\max} \mathcal{H} (\mathcal{F},\frac{1}{Ln}) / \mathcal{V} $.
Note that $\hat{T}$ depends on $C_1=4(1+\rho) L\sqrt{\mathcal{V}} \leq 4(1+\rho) L \sqrt{V_1 + V_2}/(1-\rho) $. While $L$ depends only on the loss that is chosen by the statistician, in many applications $\rho$, $V_1$ and $V_2$ are unknown. We recommend to use an empirical criterion like the slope heuristic, introduced by [@BM06], to calibrate $C_1$. This procedure is as follows:
1. define, for any $c>0$, $ \hat{T}(c) = \arg\min_{1\leq T \leq T_{\max}} [ r_n(\hat{f}_{1:T}) +
c \sqrt{ T} ]$.
2. fix a small step $\epsilon>0$ and define $\hat{c}$ as the maximiser of the jump $J_\epsilon(c)=\sqrt{\hat{T}(c+\epsilon)}-\sqrt{\hat{T}(c)}$.
3. select $\hat{T}(2\hat{c})$.
Many variants, details on fast implementations and references for theoretical results (in the i.i.d case) can be found in see [@BMM12]. A theoretical study of the slope heuristic in the context could be the object of future works.
Simulation study
----------------
As an illustration we simulate $ X_{t+1} = a_t X_t + \varepsilon_t $ for $t=1,\dots,400$, where $a_{t+4}=a_t$, $(a_1,a_2,a_3,a_4) = (0.8,0.5,0.9,-0.7)$ and $\varepsilon_t\sim\mathcal{N}(0,1)$. The data is shown in Figure \[figure-data\] and the autocorrelation function in Figure \[figure-acf\]. It is clear that a statistician trying to estimate an AR(1) model with a fixed coefficient would be puzzled by this situation.
![Simulated data.[]{data-label="figure-data"}](serie.png)
![Autocorrelation function of the data.[]{data-label="figure-acf"}](acf.png)
The dependence of $r_n(\hat{a}_{1:T})$ with respect to $T\in\{1,...,T_{\max}\}$ with $T_{\max}=20$ is shown in Figure \[figure-erm\].
![The empirical risk as a function of $T$.[]{data-label="figure-erm"}](selection.png)
The choice $T=4$ leads to an improvement with respect to $T<4$. On the other hand, we observe a slow linear decrease of $r_n(\hat{a}_{1:4})$, $r_n(\hat{a}_{1:8})$, $r_n(\hat{a}_{1:12})$ …this is a sign of overfitting. And indeed,
1. for $c< 0.008 $, $\hat{T}(c)=20$,
2. for $0.009 < c< 0.239 $, $\hat{T}(c)=4$,
3. for $ 0.240 < c$, $\hat{T}(c)=1$.
Thus, $\hat{c}\simeq 0.0085$ and we choose $\hat{T}(2\hat{c})=\hat{T}(0.017)=4$.
Acknowledgements
----------------
We thank the anonymous Referees for their very constructive comments that helped to improve the clarity of the paper.
Proofs {#section:proof}
======
The first point will be proved by backward induction. The result is obvious for $t=n$, since $g_n=f$. Assume that it is true at step $t$, and let us prove it at step $t-1$. By definition $$\begin{gathered}
g_{t-1}(X_1, \ldots, X_{t-1})
={\mathbb E}[g_t(X_1, \ldots, X_t)|{\mathcal F}_{t-1}] \\
= \int g_t(X_t, \ldots, X_{t-1}, F_t(X_{t-1},y)) P_{\varepsilon}(dy)\, .\end{gathered}$$ It follows that $$\begin{gathered}
\label{triv1}
|g_{t-1}(x_1, \ldots, x_{t-1})-g_{t-1}(x'_1, \ldots, x'_{t-1})|\\ \leq \int |g_t(x_1, \ldots,x_{t-1}, F_t(x_{t-1},y))
\\
-g_t(x'_1, \ldots,x'_{t-1}, F_t(x'_{t-1},y))|
P_{\varepsilon}(dy)\, .\end{gathered}$$ Now, by assumption and condition [(\[contract\])]{}, $$\begin{aligned}
\int |g_t(x_1, \ldots, F_t(x_{t-1},y))
& -g_t(x'_1, \ldots, F_t(x'_{t-1},y))|
P_{\varepsilon}(dy) \nonumber \\
&\leq d(x_1,x'_1)+\cdots + d(x_{t-1}, x'_{t-1})
\\
& + K_{n-t}(\rho) \int d(F_t(x_{t-1},y), F_t(x'_{t-1},y))P_{\varepsilon}(dy)
\nonumber \\
&\leq d(x_1,x'_1)+\cdots + (1+\rho K_{n-t}(\rho)) d(x_{t-1}, x'_{t-1}) \nonumber \\
&\leq d(x_1,x'_1)+\cdots + K_{n-t+1}(\rho) d(x_{t-1}, x'_{t-1}) \, .\end{aligned}$$ Point 1 follows from this last inequality and .
Let us now prove Point 2. First note that $$\begin{aligned}
|d_1|&=&\Big|g_1(X_1)-\int g_1(x)P_{X_1}(dx)\Big| \nonumber \\
&\leq& K_{n-1}(\rho)\int d(X_1,x)P_{X_1}(dx) \nonumber \\
&=& K_{n-1}(\rho) G_{X_1}(X_1) \,\end{aligned}$$ where the inequality comes from the first point of Lemma \[McD\]. In the same way, for $t \geq 2$, $$\begin{aligned}
|d_t| & = \big|g_t(X_1, \cdots, X_t)-{\mathbb E}[g_t(X_1, \cdots, X_t)|{\mathcal F}_{t-1}]\big|\\
&\leq \int \big |g_t(X_1, \cdots, F_t(X_{t-1}, \varepsilon_t))
\\
& \quad \quad -g_t(X_1, \cdots, F_t(X_{t-1}, y))\big| P_{\varepsilon}(dy)\\
&\leq K_{n-t}(\rho)\int d(F_t(X_{t-1},\varepsilon_t),
F_t(X_{t-1}, y))P_{\varepsilon}(dy)\\
& = K_{n-t}(\rho)
H_{t,\varepsilon}(X_{t-1},\varepsilon_t) \, .\end{aligned}$$
Finally, the proof of Point 3 is direct: if [(\[c2\])]{} is true, then $$H_{t,\varepsilon}(x,y)= \int d(F_t(x,y),F_t(x,y'))P_{\varepsilon}(dy')
\leq \int C\delta(y,y')P_{\varepsilon}(dy')= G_\varepsilon(y) \, .$$ The proof of the proposition is now complete.
We state a lemma that will be used in the following proofs.
\[lemma\_moments\] Under the assumptions of Section \[section:autoregression\] we have $$\forall n\in\mathbb{N}\setminus\{0\}\text{, }
\mathbb{E} \|X_n\|
\leq \frac{G_\varepsilon(0)}{1-\rho} + \rho^{n-1} G_{X_1}(0).$$
[*Proof of Lemma \[lemma\_moments\]*]{} By definition of $G_{X_1}$, $ \mathbb{E} \|X_1\| = \int \|x-0\| {\rm d}P_{X_1}(x) = G_{X_1}(0) $. Then, $$\mathbb{E} \|X_n\| = \mathbb{E} \|f_n(X_{n-1}) + \varepsilon \|
\leq \mathbb{E} \|f_n(X_{n-1})\| + \mathbb{E} \|\varepsilon \|
\leq \rho \mathbb{E} \|X_{n-1}\| + G_\varepsilon(0).$$ So, by induction, for $n>1$, $$\begin{aligned}
\mathbb{E} \|X_n\|
& \leq (1+\rho+\dots+\rho^{n-2})G_\varepsilon(0) + \rho^{n-1} G_{X_1}(0) \\
& \leq \frac{G_\varepsilon(0)}{1-\rho} + \rho^{n-1} G_{X_1}(0).\qquad\qquad \qquad\qquad\square
\end{aligned}$$ *Proof of Proposition \[cram\].* Let $\delta=a/K_{n-1}(\rho).$ Since $\mathbb{E}\,[ d_1 ] =0$, it follows that, for any $s \in [0, \delta )$, $$\begin{aligned}
\label{finsa3f}
\mathbb{E}\,[e^{s d_1 } ] &=& 1+ \sum_{j=2}^{\infty} \frac{s^j}{j!}\, \mathbb{E}\,[ (d_1)^j ] \nonumber\\
&\leq& 1+ \sum_{i=2}^{\infty} \Big(\frac{s}{\delta} \Big)^j \, \mathbb{E}\,\Big[\frac{1}{j!} | \delta d_1|^j \Big].\end{aligned}$$ Note that, for $s\geq 0$, $$\begin{aligned}
\label{constantsecond}
\frac{s^j}{j !} e^{-s} &\leq \frac{j^j}{j !} e^{-j} \leq 2 e^{-2}, \quad \text{for any $j\geq 2$,}\end{aligned}$$ where the last inequality follows from the fact that $j^j e^{-j}/j!$ is decreasing in $j$. Notice that the equality in [(\[constantsecond\])]{} is reached at $s=j=2$. By [(\[constantsecond\])]{}, Lemma \[McD\] and the hypothesis of the proposition, we deduce that $$\begin{aligned}
\label{fisa3f}
\mathbb{E}\,\Big[\frac{1}{j!} |\delta d_1|^j \Big] &\leq& \ 2e^{-2} \mathbb{E}\, [e^{\delta |d_1|} ] \nonumber\\
&\leq& \ 2e^{-2} {\mathbb E} \Big[ \exp \Big( a G_{X_1}(X_1)\Big)\Big] \nonumber\\
&\leq& \ 2e^{-2} K_1.\end{aligned}$$ Combining Inequalities (\[finsa3f\]) and (\[fisa3f\]), we get, for any $s \in [0, \delta )$, $$\mathbb{E}\,[e^{s d_1 } ] \ \leq\ 1+ \sum_{j=2}^{\infty} \frac{2}{e^{2}}\Big(\frac{s}{ \delta } \Big)^j K_1
\ =\ 1+ \frac{2}{e^{2}} \frac{s^2 K_1 \delta^{-2} }{1-s \delta^{-1}}
\ \leq\ \exp \left( \frac{2}{e^{2}} \frac{s^2 K_1 \delta^{-2} }{1-s \delta^{-1}} \right).$$ In the same way, for $n\geq i>1$, using Lemma \[McD\], $$\begin{aligned}
\mathbb{E}\,[e^{s d_i }|\mathcal{F}_{i-1} ] &=& 1+ \sum_{j=2}^{\infty} \Big(\frac{s}{\delta} \Big)^j \, \mathbb{E}\,\Big[\frac{1}{j!} | \delta d_i|^j |\mathcal{F}_{i-1} \Big]
\\
&\leq& 1+ \sum_{j=2}^{\infty} \left(\frac{s}{\delta} K_{n-i}(\rho) \right)^j \, \mathbb{E}\,\Big[\frac{1}{j!} | \delta G_{\varepsilon}(\varepsilon_i)|^j |\mathcal{F}_{i-1}\Bigr]
\\
&\leq& 1+ \sum_{j=2}^{\infty} \left(\frac{s}{\delta} \frac{K_{n-i}(\rho)}{K_{n-1}(\rho)} \right)^j \, \mathbb{E}\,\Big[\frac{1}{j!} | a G_{\varepsilon}(\varepsilon_i)|^j\Bigr]
\\
&\leq& 1+ \left(\frac{K_{n-i}(\rho)}{K_{n-1}(\rho)}\right)^{2} \sum_{j=2}^{\infty} \left(\frac{s}{\delta} \right)^j \, 2e^{-2}K_2\end{aligned}$$ where we used again, and the fact that $K_{n-i}(\rho)/K_{n-1}(\rho)\leq 1$ for any $i \in [2, n]$ which implies $[K_{n-i}(\rho)/K_{n-1}(\rho)]^j\leq [K_{n-i}(\rho)/K_{n-1}(\rho)]^2$ for any $j\geq 2$. So, for any $s \in [0, \delta )$, $$\begin{aligned}
\mathbb{E}\,[e^{s d_i }|\mathcal{F}_{i-1}] \ &\leq&
1+ \frac{2}{e^{2}} \frac{s^2 K_2 \delta^{-2} }{1-s \delta^{-1}} \left(\frac{K_{n-i}(\rho)}{K_{n-1}(\rho)}\right)^{2}
\\
&\leq& \ \exp \left(\frac{2}{e^{2}} \frac{s^2 K_2 \delta^{-2} }{1-s \delta^{-1}} \Big(\frac {K_{n-i}(\rho)}{K_{n-1}(\rho)}\Big)^2 \right) \, .\end{aligned}$$ Using the tower property of conditional expectation, we have, for any $s \in [0, \delta )$, $$\begin{aligned}
\mathbb{E}\,\big[e^{ sS_i} \big] &=& \mathbb{E}\,\big[ \,\mathbb{E}\, [e^{ s S_i} |\mathcal{F}_{i-1} ] \big] \nonumber\\
&=& \mathbb{E}\,\big[ e^ { s S_{i-1}} \mathbb{E}\, [e^ { s d_i} |\mathcal{F}_{i-1} ] \big] \nonumber \\
&\leq & \mathbb{E}\,\big[ e^ { s S_{i-1}} \big] \exp \left( \frac{2}{e^{2}} \frac{t^2 K_2 \delta^{-2} }{1-t \delta^{-1}} \Big(\frac {K_{n-i}(\rho)}{K_{n-1}(\rho)}\Big)^2 \right).\end{aligned}$$ By recursion, $$\begin{aligned}
\mathbb{E}\,\big[e^{ sS_i} \big] & \leq
\mathbb{E}\,\big[ e^ { s S_{1}} \big] \exp \left( \frac{2}{e^{2}} \frac{t^2 K_2 \delta^{-2} }{1-t \delta^{-1}} \sum_{i=2}^n \Big(\frac {K_{n-i}(\rho)}{K_{n-1}(\rho)}\Big)^2 \right) \\
& \leq \ \exp \left( \frac{2}{e^{2}} \frac{s^2 K_1 \delta^{-2} }{1-s \delta^{-1}} \right)\exp \left( \frac{2}{e^{2}} \frac{t^2 K_2 \delta^{-2} }{1-t \delta^{-1}} \sum_{i=2}^n \Big(\frac {K_{n-i}(\rho)}{K_{n-1}(\rho)}\Big)^2 \right) \\
& =
\exp \left( \frac{s^2 K \delta^{-2} }{1-s \delta^{-1}} \right),\end{aligned}$$ where $$K=\frac{2}{e^{2}} \bigg( K_1+ K_2\sum_{i=2}^{n} \Big(\frac {K_{n-i}(\rho)}{K_{n-1}(\rho)}\Big)^2 \bigg).$$ Then using the exponential Markov inequality, we deduce that, for any $x\geq 0$ and $s \in [0, \delta )$, $$\begin{aligned}
\mathbb{P}\left( S_{n} \geq x \right)
&\leq& \mathbb{E}\, [e^{s\,(S_n -x) } ] \nonumber\\
&\leq& \exp \left(-s x + \frac{s^2 K \delta^{-2} }{1-s \delta^{-1}} \right)\, .\end{aligned}$$ The minimum is reached at $$s=s(x):= \frac{x\delta^2/K}{x\delta/K + 1 + \sqrt{1+x\delta/K}} .$$ The proposition is proven.
*Proof of Proposition \[classic\].* Denote $$u_{k-1}(x_1, \ldots, x_{k-1})= \text{ess}\inf_{\varepsilon_k} g_k(x_1, \ldots, F_k(x_{k-1}, \varepsilon_k))$$ and $$v_{k-1}(x_1, \ldots, x_{k-1})= \text{ess}\sup_{\varepsilon_k} g_k(x_1, \ldots, F_k(x_{k-1}, \varepsilon_k)).$$ From the proof of Lemma \[McD\], it is easy to see that $$u_{k-1}(X_1, \ldots, X_{k-1}) \leq d_k \leq v_{k-1}(X_1, \ldots, X_{k-1})\, .$$ By Lemma \[McD\] and the hypothesis of the proposition, it follows that $$v_{k-1}(X_1, \ldots, X_{k-1})-u_{k-1}(X_1, \ldots, X_{k-1})
\leq K_{n-k}(\rho) M_k\, .$$ Following exactly the proof of Theorem 3.1 of [@R13] with $\Delta_k = K_{n-k}(\rho) M_k$, we get and . Since $\ell^*(x) \geq (x^2-2x) \ln(1-x)$ for any $x \in [0, 1]$, follows from .
Put $k=\lfloor (n-2)/T \rfloor$, and consider the three possible cases $kT\leq n-2 \leq (k+1/2)T$, $(k+1/2)T< n-2 < (k+1)T-1$ and $n-2 = (k+1)T-1$.
[*First case*]{}. We write: $$\begin{gathered}
\label{split31}
R_n(f_{1:T}) = \mathbb{E}\left[ \frac{1}{n-1}\sum_{i=2}^n \ell\left(X_i - f_{i[T]} (X_{i-1}) \right)\right]
\\
= \frac{ kT R_{T+1}(f_{1:T})}{n-1} + \mathbb{E}\left[ \frac{1}{n-1}\sum_{i=kT+2}^n \ell\left(X_i - f_{i[T]} (X_{i-1}) \right)\right].
\end{gathered}$$ First, $$\begin{aligned}
\left| R_n(f_{1:T})-\frac{ kT R_{T+1}(f_{1:T})}{n-1} \right|
& = \frac{1}{n-1} \sum_{i=kT+2}^{n} \mathbb{E}\left[ | \ell\left(X_i - f_{i[T]} (X_{i-1}) \right)| \right]
\\
& \leq \frac{1}{n-1} \sum_{i=kT+2}^{n} L \mathbb{E} \|X_i\| + \rho L \mathbb{E} \|X_{i-1}\|
\\
& \leq \frac{(n-2)-kT }{n-1} L(1+\rho) \left[ \frac{G_\varepsilon(0)}{1-\rho} + G_{X_1}(0)\right]\end{aligned}$$ where we used Lemma \[lemma\_moments\] and $\rho^{n-1}<1$ for the last inequality. In the same way, $$\begin{aligned}
\left| \frac{ kT R_{T+1}(f_{1:T})}{n-1} - R_{T+1}(f_{1:T})\right| & = \left(1-\frac{kT}{n-1} \right)R_{T+1}(f_{1:T}) \\
& = \frac{(n-1)-kT}{n-1} \frac{1}{T}\sum_{i=2}^{T+1} \mathbb{E}\left[ | \ell\left(X_i - f_{i} (X_{i-1}) \right)| \right]\\
& \leq \frac{(n-2)-kT+1}{n-1} L(1+\rho) \left[ \frac{G_\varepsilon(0)}{1-\rho} + G_{X_1}(0)\right].\end{aligned}$$ Combining both inequalities leads to: $$\begin{aligned}
\left| R_n(f_{1:T})-R_{T+1}(f_{1:T}) \right|
&
\leq \frac{2[(n-2)-kT]+1}{n-1}L(1+\rho) \left[ \frac{G_\varepsilon(0)}{1-\rho} + G_{X_1}(0)\right]
\\
& \leq \frac{T+1}{n-1} C_0\end{aligned}$$ by definition of $C_0$ and as in the first case, $(n-2)-kT \leq T/2$.
[*Case 2*]{}. We write the decomposition of $R_n(f_{1:T})$ in a different way from what we did in : $$\begin{gathered}
R_n(f_{1:T}) = \mathbb{E}\left[ \frac{1}{n-1}\sum_{i=2}^n \ell\left(X_i - f_{i[T]} (X_{i-1}) \right)\right]
\\
= \frac{ (k+1)T R_{T+1}(f_{1:T})}{n-1} - \mathbb{E}\left[ \frac{1}{n-1}\sum_{i=n+1}^{(k+1)T+1} \ell\left(X_i - f_{i[T]} (X_{i-1}) \right)\right].
\end{gathered}$$ Similar derivations lead to $$\left| R_n(f_{1:T})-\frac{ (k+1)T R_{T+1}(f_{1:T})}{n-1} \right|
\leq \frac{(k+1)T+1-n}{n-1} C_0$$ and $$\left| \frac{ (k+1)T R_{T+1}(f_{1:T})}{n-1} - R_{T+1}(f_{1:T})\right|
\leq \frac{(k+1)T+1-n}{n-1}C_0,$$ and combining both inequalities and $(k+1/2)T< n-2$ gives: $$\left| R_n(f_{1:T})- R_{T+1}(f_{1:T}) \right| \leq \frac{T}{n-1} C_0.$$
[*Case 3*]{}. In this case, $$R_n(f_{1:T}) = \frac{(k+1)T R_{T+1}(f_{1:T})}{n-1} = R_{T+1}(f_{1:T})$$ and so $$|R_n(f_{1:T}) - R_{T+1}(f_{1:T}) | = 0 .$$
So, in Cases 1, 2 and 3, the largest bound $$\left| R_n(f_{1:T})-R_{T+1}(f_{1:T}) \right|
\leq \frac{T+1}{n-1}C_0$$ holds[^4].
*Proof of Corollary \[coro-var\].* By definition, we have $ X_t = F_t(X_{t-1},\varepsilon_t) = f_{t}^*(X_{t-1}) + \varepsilon_t$. So $\| F_t(x,\varepsilon_t)-F_t(x',\varepsilon_t) \| = \| f_{t}^*(x) - f_{t}^*(x') \| \leq \rho \|x-x'\| $ and so is satisfied, and $\|F_t(x,y)-F_t(x,y')\|=\|y-y'\|$ so that is satisfied with $C=1$. We consider the random variable $S_n = f(X_1,\dots,X_n)-\mathbb{E}[f(X_1,\dots,X_n)]$, where $$f(x_1,\dots,x_n) = \frac{1}{L(\rho+1)} \sum_{t=2}^n \ell\left(x_t - f_{t[T]}(x_{t-1}) \right).$$ Remark that $$\begin{aligned}
& | f(x_1,\dots,x_n) - f(x_1,\dots,x_t',\dots,x_n) | \\
& \leq \frac{\left| \ell \left(x_{t+1} - f_{(t+1)[T]}(x_{t}) \right) - \ell \left(x_{t+1} - f_{(t+1)[T]}(x_{t}') \right)\right|}{L(\rho+1)}
\\
& \quad \quad + \frac{ \left| \ell \left(x_t - f_{t[T]}(x_{t-1}) \right) - \ell \left(x_t' - f_{t[T]}(x_{t-1}) \right) \right|}{L(\rho +1)}
\\
& \leq \frac{\|f_{(t+1)[T]}(x_{t})-f_{(t+1)[T]}(x_{t}')\| + \|x_t-x_t'\|}{\rho+1} \\
& \leq \|x_t - x_t'\|.\end{aligned}$$ So the assumptions of Theorem \[Bernsteinineq\] are satisfied and $$\mathbb{E}\exp\left( \pm t S_n \right) \leq \exp\left(\frac{t^2 V_{(n)}}{2-2t\delta }\right) .$$ Remind that $S_n = \frac{n-1}{L(1+\rho)}\left[ r_n(f_{1:T})- \mathbb{E}[r_n(f_{1:T})]\right] $, and $ R_n(f_{1:T}) = \mathbb{E}[r_n(f_{1:T})] $, so that by setting $s= {t(n-1)}/{L(1+\rho)}$ we end the proof.
*Proof of Theorem \[thm-cvg\].* Fix $\epsilon > 0$. We have, for any $f_{1:T}\in\mathcal{F}_{\epsilon}$, the deviation inequality from Corollary \[coro-var\]. A union bound on $f_{1:T}\in\mathcal{F}_{\epsilon}$ leads to, for any $s\in\left[0,\frac{n-1}{L(1+\rho)\delta}\right)$, $$\begin{aligned}
& \mathbb{P}\left(\sup_{(f_{1:T})\in\mathcal{F}_{\epsilon}^T}|r_n(f_{1:T})-R_n(f_{1:T})| > x \right) \\
& \leq \sum_{(f_{1:T})\in\mathcal{F}_{\epsilon}} \mathbb{P}\left(|r_n(f_{1:T})-R_n(f_{1:T})| > x \right)
\\
& \leq \sum_{(f_{1:T})\in\mathcal{F}_{\epsilon}^T} \mathbb{E}\exp\left( s |r_n(f_{1:T})-R_n(f_{1:T})| - s x\right)
\\
& \leq 2 \mathcal{N}(\mathcal{F} ,\epsilon)^T \exp\left( \frac{s^2(1+\rho)^2 L^2 \mathcal{V}}{2(n-1)-2s(1+\rho)\delta L} -s x \right).\end{aligned}$$ Now, for any $f_{1:T}\in\mathcal{F}^T$ we construct $f_{1:T}^{\epsilon}=(f_1^\epsilon,\dots,f_T^{\epsilon})$ by chosing, for any $t\in\{1,\dots,T\}$, a function $f_t^\epsilon$ such that $\|f_t-f_t^\epsilon\|_{\sup}\leq \epsilon$, as allowed from the definition of $\mathcal{F}_{\epsilon} $. Obviously $$\begin{gathered}
\left| \ell(X_t-f^\epsilon_{t[T]} (X_{t-1})) - \ell(X_t - f_{t[T]} (X_{t-1}) )\right|
\\
\leq L \|f^\epsilon_{t[T]} (X_{t-1}) - f_{t[T]} (X_{t-1})\|
\leq L\epsilon \|X_{t-1}\|\end{gathered}$$ and as a consequence, $$|r_n(f_{1:T})-r_n(f_{1:T}^\epsilon)| \leq L\epsilon\cdot\frac{\sum_{t=1}^{n-1}\|X_t\|}{n-1},$$ and $$|R_n(f_{1:T})-R_n(f_{1:T}^\epsilon)| \leq \epsilon L\cdot\frac{\sum_{t=1}^{n-1}\mathbb{E}\|X_t\|}{n-1} .$$ Using Theorem \[Bernsteinineq\] with $f(X_1,\dots,X_n)=\sum_{t=1}^{n-1} \|X_t\|$ we have, for any $y>0$, $$\begin{aligned}
& \mathbb{P}\left(\sum_{t=1}^{n-1}\|X_t\| > \sum_{t=1}^{n-1}\mathbb{E}\|X_t\| + y \right) \\
& \leq \mathbb{E}\exp\left[\frac{1}{2\delta}\left(\sum_{t=1}^{n-1}\|X_t\| - \sum_{t=1}^{n-1}\mathbb{E}\|X_t\| - y\right) \right]
\\
& \leq \exp\left(\frac{\left(\frac{1}{2\delta}\right)^2 V_{(n)}}{2\left(1-\frac{1}{2}\right)} - \frac{ y}{2\delta} \right) = \exp\left(\frac{ V_{(n)} }{4\delta^2}-\frac{ y }{2\delta} \right).\end{aligned}$$ Lemma \[lemma\_moments\] leads to $$\begin{aligned}
\sum_{t=1}^{n-1}\mathbb{E} \|X_t\|
& \leq \sum_{t=1}^{n-1} \left[ \frac{G_{\epsilon}(0) }{1-\rho} + \rho^{t-1} G_{X_1}(0) \right] \\
& \leq \frac{ (n-1) G_{\epsilon}(0) + G_{X_1}(0) }{1-\rho} =: z_{\rho,n}\end{aligned}$$ where we introduce the last notation for short. Now let us consider the “favorable” event $$\mathcal{E}= \left\{ \sum_{t=1}^{n-1}\|X_t\| \leq z_{\rho,n} + y \right\}
\bigcap \left\{\sup_{f_{1:T}\in\mathcal{F}_\epsilon}|r_n(f_{1:T})-R_n(f_{1:T})| \leq x \right\} .$$ The previous inequalities show that $$\begin{gathered}
\label{proba-e}
\mathbb{P}\left( \mathcal{E}^c\right)\leq \exp\left(\frac{ V_{(n)}}{4\delta^2}-\frac{ y }{2\delta} \right) \\
+2 \mathcal{N}(\mathcal{F},\epsilon)^T
\exp\left( \frac{s^2(1+\rho)^2 L^2 \mathcal{V}}{2(n-1)-2s(1+\rho)\delta L} -s x \right).\end{gathered}$$ On $\mathcal{E}$, we have: $$\begin{aligned}
R_n(\hat{f}_{1:T})
& \leq R_n(\hat{f}_{1:T}^{\epsilon}) +\epsilon L\frac{z_{\rho,n} }{n-1} \\
& \leq r_n(\hat{f}_{1:T}^{\epsilon}) + x +\epsilon L \frac{z_{\rho,n} }{n-1}
\\
& \leq r_n(\hat{f}_{1:T}) + x + \epsilon L \left[ 2 \frac{z_{\rho,n} }{n-1}+ \frac{y}{n-1}\right]
\\
& = \min_{f_{1:T} \in\mathcal{F}_{\epsilon}^T } r_n(f_{1:T}) + x + \epsilon L \frac{2 z_{\rho,n}+y }{n-1}
\\
& \leq \min_{f_{1:T} \in\mathcal{F}_{\epsilon}^T } R_n(f_{1:T}) + 2x + \epsilon L \frac{2 z_{\rho,n} +y }{n-1}
\\
& \leq \min_{f_{1:T} \in\mathcal{F}^T } R_n(f_{1:T}) + 2x + \epsilon L \frac{3 z_{\rho,n}+y }{n-1} .\end{aligned}$$ In particular, the choice $\epsilon = 1/(Ln)$ ensures: $$\label{almost-done}
R_n(\hat{f}_{1:T}) \leq \min_{f_{1:T} \in\mathcal{F}^T } R_n(f_{1:T}) + 2x + \frac{ 3 z_{\rho,n} + y }{n ( n-1) }.$$ Fix $\eta>0$ and put: $$x = \frac{s(1+\rho)^2 L^2 \mathcal{V}}{2(n-1)-2s(1+\rho)\delta L} + \frac{T\mathcal{H} (\mathcal{F},\frac{1}{Ln}) + \log\left(\frac{4}{\eta}\right)}{s}$$ and $ y = 2\delta\log\left(\frac{2}{\eta}\right) + \frac{ V_{(n)}}{2\delta }.$ Note that, plugged into , these choices ensure $\mathbb{P}(\mathcal{E}^c)\leq \eta/2+\eta/2=\eta$. Put $$s= \frac{1}{(1+\rho)L}\sqrt{ (n-1) T\mathcal{H} (\mathcal{F},\frac{1}{Ln}) \Big/\mathcal{V} }.$$ As soon as $ 2s(1+\rho)\delta L \leq n-1$, that is actually ensured by the condition $ n \geq 1 + 4\delta^2 T \mathcal{H} (\mathcal{F},\frac{1}{Ln}) / \mathcal{V} $, we have: $$\begin{aligned}
x & \leq \frac{s(1+\rho)^2 L^2 \mathcal{V}}{n-1} + \frac{T\mathcal{H} (\mathcal{F},\frac{1}{Ln}) + \log\left(\frac{4}{\eta}\right)}{s}
\\
& = 2(1+\rho) L \sqrt{ \frac{\mathcal{V} T\mathcal{H} (\mathcal{F},\frac{1}{Ln})}{n-1}} \\
& \quad + (1+\rho)L \log\left(\frac{4}{\eta}\right) \sqrt{ \frac{\mathcal{V}}{(n-1)T\mathcal{H} (\mathcal{F},\frac{1}{Ln})}}.\end{aligned}$$ Pluging the expressions of $x$ and $y$ and the definition of $z_{\rho,n}$ into gives: $$\begin{gathered}
R_n(\hat{f}_{1:T}) \leq \min_{f_{1:T} \in\mathcal{F}^T } R_n(f_{1:T}) + 4(1+\rho) L \sqrt{ \frac{\mathcal{V} T\mathcal{H} (\mathcal{F},\frac{1}{Ln})}{n-1} }
\\
+ 2(1+\rho)L \log\left(\frac{4}{\eta}\right) \sqrt{ \frac{\mathcal{V}}{(n-1)T\mathcal{H} (\mathcal{F},\frac{1}{Ln})}}
\\
+ \frac{1}{n} \left[ 3 \frac{G_\epsilon(0) + \frac{G_{X_1}(0)}{n-1}}{1-\rho} + \frac{ 2\delta\log\left(\frac{2}{\eta}\right) + \frac{V_{(n)} }{2\delta }}{n-1}\right]\end{gathered}$$ which ends the proof.
*Proof of Theorem \[thm-selection\].* Fix $\epsilon>0$. For any $1\leq T\leq T_{\max}$ and $f_{1:T}=(f_1,\dots,f_T)\in\mathcal{F}^T$, chose $f_{1:T}^{\epsilon}=(f_1^\epsilon,\dots,f_T^{\epsilon})$ such that for any $i$, $\|f_{i}^{\epsilon}-f_{i}\|_{\sup}\leq \epsilon$. Define the event $$\mathcal{A}= \left\{ \sum_{t=1}^{n-1}\|X_t\| \leq z_{\rho,n} + y \right\}
\bigcap \bigcap_{T=1}^{T_{\max}} \left\{\sup_{f_{1:T}\in\mathcal{F}_\epsilon}|r_n(f_{1:T})-R_n(f_{1:T})| \leq x_{T} \right\}$$ where $z_{\rho,n}$ is defined as in the proof of Theorem \[thm-cvg\] and $x_1,\dots,x_{T_{\max}} > 0 $. We have, for any $s_1,\dots,s_{T_{\max}}< (n-1)/[L\delta(1+\rho)]$, $$\begin{gathered}
\mathbb{P}\left( \mathcal{A}^c\right)\leq \exp\left(\frac{ V_{(n)}}{4\delta^2}-\frac{y }{2\delta } \right)
\\
+2 \sum_{T=1}^{T_{\max}}\mathcal{N}(\mathcal{F},\epsilon)^T \exp\Biggl( \frac{s_T^2(1+\rho)^2 L^2 \mathcal{V}}{2(n-1)-2s_T(1+\rho)\delta L}
-s_T x_T \Biggr) \leq \eta,\end{gathered}$$ the last inequality being ensured by the choice $y=2\delta\log(2/\eta) + V_{(n)}/(2\delta)$ and, for any $T$, $$\begin{aligned}
x_{T} & = \frac{s_T(1+\rho)^2 L^2 \mathcal{V}}{2(n-1)-2s_T(1+\rho)\delta L} \\
& \quad \quad + \frac{T\mathcal{H} (\mathcal{F},\frac{1}{Ln}) + \log\left(4 T_{\max} / \eta \right)}{s_T} ,\\
s_T & = \frac{1}{(1+\rho)L}\sqrt{\frac{(n-1)T\mathcal{H} (\mathcal{F},\frac{1}{Ln})}{\mathcal{V}} }.\end{aligned}$$ Note that this choice also leads to $$x_T \leq \frac{C_1}{2} \sqrt{ \frac{ T\mathcal{H} (\mathcal{F},\frac{1}{Ln})}{n-1}}
+ \frac{C_2}{2} \frac{\log\left(4 T_{\max}/\eta\right)}{\sqrt{n-1}}.$$ On $\mathcal{A}$, we have $ R_n(\hat{f}_{1:\hat{T}}) \leq R_n(\hat{f}_{1:\hat{T}}^{\epsilon}) +\epsilon L z_{\rho,n}/(n-1)$, and $$\begin{aligned}
R_n(\hat{f}_{1:\hat{T}})
& \leq r_n(\hat{f}_{1:\hat{T}}^{\epsilon}) + x_{\hat{T}} +\epsilon L \frac{z_{\rho,n} }{n-1} \\
& \leq r_n(\hat{f}_{1:\hat{T}}) + x_{\hat{T}} + \epsilon L \frac{2 z_{\rho,n}+y}{n-1}
\\
& \leq r_n(\hat{f}_{1:\hat{T}}) + \frac{C_1}{2} \sqrt{ \frac{ \hat{T}\mathcal{H} (\mathcal{F},\frac{1}{Ln})}{n-1}}
+ \frac{C_2}{2} \frac{\log\left(4 T_{\max}/\eta\right)}{\sqrt{n-1}} + \epsilon L \frac{ 2 z_{\rho,n}+y}{n-1}
\\
& = \min_{1\leq T\leq T_{\max}} \Biggr\{ r_n(\hat{f}_{1:T}) + \frac{C_1}{2} \sqrt{ \frac{ T \mathcal{H} (\mathcal{F},\frac{1}{Ln})}{n-1}} \Biggr\}
+ \frac{C_2}{2} \frac{\log\left(4 T_{\max}/\eta\right)}{\sqrt{n-1}}
\\ & \quad
+ \epsilon L \frac{2 z_{\rho,n}+y}{n-1}
\\
& \leq \min_{1\leq T\leq T_{\max}} \min_{f_{1:T} \in\mathcal{F}_{\epsilon}^T } \Biggl\{ r_n(f_{1:T}) + \frac{C_1}{2} \sqrt{ \frac{ T \mathcal{H} (\mathcal{F},\frac{1}{Ln})}{n-1}} \Biggr\}
+ \frac{C_2}{2} \frac{\log\left(4 T_{\max}/\eta\right)}{\sqrt{n-1}}
\\ & \quad
+ \epsilon L \frac{2 z_{\rho,n} + y}{n-1}
\\
& \leq \min_{1\leq T\leq T_{\max}} \min_{f_{1:T} \in\mathcal{F}_{\epsilon}^T } \Biggl\{ R_n(f_{1:T}) +
C_1\sqrt{ \frac{ T \mathcal{H} (\mathcal{F},\frac{1}{Ln})}{n-1}} \Biggr\} +
\frac{C_2}{2} \frac{\log\left(4 T_{\max}/\eta\right)}{\sqrt{n-1}}
\\ & \quad
+ \epsilon L \frac{2 z_{\rho,n} + y}{n-1}
\\
& \leq \min_{1\leq T\leq T_{\max}} \min_{f_{1:T} \in\mathcal{F}^T } \Biggl\{ R_n(f_{1:T}) + C_1\sqrt{ \frac{ T \mathcal{H} (\mathcal{F},\frac{1}{Ln})}{n-1}}\Biggr\} + \frac{C_2}{2} \frac{\log\left(4 T_{\max}/\eta\right)}{\sqrt{n-1}} \\ & \quad
+ \epsilon L \frac{ 3z_{\rho,n}+ y}{n-1}
\\
& \leq \inf_{1\leq T \leq T_{\max}} \inf_{f_{1:T}\in\mathcal{F}^T}
\Biggl[ R_n(f_{1:T}) +
C_1 \sqrt{ \frac{ T\mathcal{H} (\mathcal{F},\frac{1}{Ln}) }{n-1} }
+ C_2 \frac{ \log\left(4 T_{\max}/\eta\right)}{\sqrt{n-1}}
\\ & \quad
+ \frac{C_3}{n} \Biggr].\qquad\qquad\qquad\qquad\qed\end{aligned}$$
Adamczak, R. (2008). A tail inequality for suprema of unbounded empirical processes with applications to [M]{}arkov chains. , 13, 1000–1034.
Alquier, P. and Guedj, B. (2018). Simpler [PAC-B]{}ayesian bounds for hostile data. , 107(5):887–902.
Alquier, P. and Li, X. (2012). Prediction of quantiles by statistical learning and application to [GDP]{} forecasting. In [*International Conference on Discovery Science*]{}, pages 22–36. Springer.
Alquier, P., Li, X., and Wintenberger, O. (2013). Prediction of time series by statistical learning: general losses and fast rates. , 1:65–93.
Alquier, P. and Wintenberger, O. (2012). Model selection for weakly dependent time series forecasting. , 18(3):883–913.
Bardet, J.-M. and Doukhan, P. (2018). Non-parametric estimation of time varying $AR(1)$ processes with local stationarity and periodicity. , 12(2):2323–2354.
Baudry, J.-P., Maugis, C., and Michel, B. (2012). Slope heuristics: overview and implementation. , 22(2):455–470.
Bercu, B., Delyon, B., and Rio, E. (2015). . Springer.
Bertail, P. and Ciolek, G. (2019). . , to appear.
Bertail, P. and Cl[é]{}men[ç]{}on, S. (2010). Sharp bounds for the tails of functionals of [M]{}arkov chains. , 54(3):505–515.
Bertail, P. and Portier, F. (2018). ademacher complexity for [M]{}arkov chains: applications to kernel smoothing and [M]{}etropolis-[H]{}asting. , to appear.
Birgé, L. and Massart, P. (2006). Minimal penalties for gaussian model selection. , 138(1–2):33–73.
Blanchard, G. and Zadorozhnyi, O. (2017). Concentration of weakly dependent [B]{}anach-valued sums and applications to kernel learning methods. .
Boucheron, S., Lugosi, G., and Massart, P. (2013). . Oxford university press.
Catoni, O. (2003). Laplace transform estimates and deviation inequalities. In [*Annales de l’Institut Henri Poincare (B) Probability and Statistics*]{}, 39(1):1–26.
Collet, P., Martinez, S., and Schmitt, B. (2002). Exponential inequalities for dynamical measures of expanding maps of the interval. , 123(3):301–322.
Dahlhaus, R. (1996). On the [Kullback-Leibler]{} information divergence of locally stationary processes. , 62(1):139–168.
de la Pena, V. (1999). A general class of exponential inequalities for martingales and ratios. , 27(1):537–564.
Dedecker, J., Doukhan, P., Lang, G., Rafael, L. R., Louhichi, S., and Prieur, C. (2007). Weak dependence. In [*Weak dependence: With examples and applications*]{}, pages 9–20. Springer.
Dedecker, J. and Fan, X. (2015). Deviation inequalities for separately [L]{}ipschitz functionals of iterated random functions. , 125(1):60–90.
Diaconis, P. and Freedman, D. (1999). Iterated random functions. , 41(1):45–76.
Doukhan, P. (2018). . Springer.
Doukhan, P. and Neumann, M. H. (2007). Probability and moment inequalities for sums of weakly dependent random variables, with applications. , 117:878–903.
Fan, J., Jiang, B., and Sun, Q. (2018). Hoeffding’s lemma for [Markov Chains]{} and its applications to statistical learning. .
Hang, H. and Steinwart, I. (2014). Fast learning from $\alpha$-mixing observations. , 127:184–199.
Hang, H. and Steinwart, I. (2017). A [B]{}ernstein-type inequality for some mixing processes and dynamical systems with an application to learning. , 45(2):708–743.
Joulin, A. and Ollivier, Y. (2010). Curvature, concentration and error estimates for [Markov chain Monte Carlo]{}. , 38(6):2418–2442.
Kontorovich, L. A. and Ramanan, K. (2008). Concentration inequalities for dependent random variables via the martingale method. , 36(6):2126–2158.
Kuznetsov, V. and Mohri, M. (2015). Learning theory and algorithms for forecasting non-stationary time series. In [*Advances in neural information processing systems*]{}, 541–549.
Lerasle, M. (2011). Optimal model selection for density estimation of stationary data under various mixing conditions. , 39(4):1852–1877.
London, B., Huang, B., Taskar, B., and Getoor, L. (2014). collective stability. In [*Artificial Intelligence and Statistics*]{}, 585–594.
Massart, P. (2007). . Lecture Notes in Mathematics 1896, Springer.
McDonald, D. J., Shalizi, C. R., and Schervish, M. (2017). Nonparametric risk bounds for time-series forecasting. , 18(32):1–40.
Meir, R. (2000). Nonparametric time series prediction through adaptive model selection. , 39(1):5–34.
Merlev[è]{}de, F., Peligrad, M., and Rio, E. (2009). ernstein inequality and moderate deviations under strong mixing conditions. In [*High dimensional probability V: the Luminy volume*]{}, pages 273–292. Institute of Mathematical Statistics.
Merlev[è]{}de, F., Peligrad, M., and Rio, E. (2011). A [B]{}ernstein type inequality and moderate deviations for weakly dependent sequences. , 151(3-4):435–474.
Modha, D. S. and Masry, E. (2002). , 42:2133–2145.
Paulin, D. (2018). couplings and spectral methods , 20(79):1–32.
Rio, E. (2013a). Extensions of the [Hoeffding-Azuma]{} inequalities. , 18(54):1–6.
Rio, E. (2013b). On [McDiarmid’s]{} concentration inequality. , 18(44):1–11.
Rio, E. (2017). , Springer, Berlin.
Samson, P.-M. (2000). Concentration of measure inequalities for [M]{}arkov chains and $\phi$-mixing processes. , 28(1):416–461.
Sanchez-Perez, A. (2015). Time series prediction via aggregation: an oracle bound including numerical cost. In [*Modeling and Stochastic Learning for Forecasting in High Dimensions*]{}, Antoniadis, A., Poggi, J.-M. and Brossat, X. (Eds.), Springer, 243–265.
Seldin, Y., Laviolette, F., Cesa-Bianchi, N., Shawe-Taylor, J. and Auer, P. (2012). , 58(12):7086–7093.
Shalizi, C. and Kontorovich, A. (2013). Predictive [PAC]{} learning and process decompositions. In [*Advances in neural information processing systems*]{}, C. Burges, L. Bottou, M. Welling, Z. Ghahramani and K. Weinberger (Eds.), 26:1619–1627.
Steinwart, I. and Christmann, A. (2009). Fast learning from non i.i.d observations. In [*Advances in neural information processing systems*]{}, Y. Bengio, D. Schuurmans, J. Lafferty, C.K.I. Williams and A. Culotta (Eds.), 22:1768–1776.
Steinwart, I., Hush, D., and Scovel, C. (2009). Learning from dependent observations. , 100(1):175–194.
van de Geer, S. (1995). Exponential inequalities for martingales, with application to maximum likelihood estimation for counting processes. , 23(5):1779–1801.
Vapnik, V. (1998). . Wiley, New York.
Wintenberger, O. (2010). Deviation inequalities for sums of weakly dependent time series. , 15:489–503.
Wintenberger, O. (2017). xponential inequalities for unbounded functions of geometrically ergodic [M]{}arkov chains: applications to quantitative error bounds for regenerative [M]{}etropolis algorithms. , 51(1):222–234.
[^1]: CREST, ENSAE, Université Paris Saclay. Pierre Alquier’s work has been supported by GENES and by the French National Research Agency (ANR) under the grant Labex Ecodec (ANR-11-LABEX-0047).
[^2]: AGM UMR8088 University Paris-Seine and CIMFAV, Universidad de Valparaiso, Chile. Paul Doukhan’s work has been developed within the MME-DII center of excellence (ANR-11-LABEX-0023-01) & PAI-CONICYT MEC $N^o 80170072$.
[^3]: CAM, Tianjin University, Tianjin, China. Fan Xiequan has been partially supported by the National Natural Science Foundation of China (Grant $N^o 11601375$.
[^4]: We thank one of the anonymous Referees who suggested improvements in this proof that led to the bound $\frac{T+1}{n-1}C_0$, instead of the bound $\frac{2T+1}{n-1}C_0$ stated in the first version of this paper.
|
---
author:
- 'P. Bonifacio'
- 'E. Caffau'
- 'H.-G. Ludwig'
title: ' Effects of granulation on neutral copper resonance lines in metal-poor stars '
---
-------------------------- --------------- ---------- ---------- ------- ------- ---------- ------- ----------
Star T$_{\rm eff}$ $\log g$ \[Fe/H\] $\xi$ A(Cu) $\sigma$ A(Cu) $\sigma$
K \[cgs\] dex
Cl\* NGC 6752 GVS 4428 6226 4.28 -1.52 0.70 3.23 0.08 2.56 0.16
Cl\* NGC 6752 GVS 200613 6226 4.28 -1.56 0.70 3.01 0.05 2.23 0.07
Cl\* NGC 6397 ALA 1406 6345 4.10 -2.05 1.32 1.33 0.03 0.74 0.05
Cl\* NGC 6397 ALA 228 6274 4.10 -2.05 1.32 1.30 0.03 0.73 0.05
Cl\* NGC 6397 ALA 2111 6207 4.10 -2.01 1.32 1.19 0.02 0.60 0.02
HD 218502 6296 4.13 -1.85 1.00 1.52 0.09 0.95 0.04
-------------------------- --------------- ---------- ---------- ------- ------- ---------- ------- ----------
Introduction
============
In our quest to understand how the Universe evolved from the primordial chemical composition, consisting of H, He, and traces of Li, to the present day complexity, we strive to uncover all the nucleosynthetic channels. This requires to measure the evolution of as many chemical species as possible. For some species this becomes difficult at low metallicities, when all the observable lines become very weak. This is the case for Cu, the measurements of Cu abundances are mainly based on the lines of Mult. 2 at 510.5nm and 578.2nm [@Sneden91; @Mishenina]. However, such lines become very weak at low metallicities, even in cool giants. The strongest line, at 510.5nm, has an equivalent with of a few tenths of picometer for a K giant of metallicity –2.5 and becomes very difficult to measure. This induced several groups [@Bihain; @Cohen08] to push the observations in the near UV, where the resonance lines at 324.7nm and 327.3nm are stronger by a factor of ten and can be measured at the lowest metallicities. These lines are formed in the cool outer layers of the stellar atmosphere. Such layers are formally stable against convection and 1D model atmospheres cannot account for the effects of convective motions (“granulation”) here. The use of 3D hydrodynamical simulations has shown that one effect of this is a steeper temperature gradient in the outer layers than what predicted by 1D models [@A99; @A05]. This effect is often referred to as “overcooling” and is more pronounced for metal-poor stars. In this contribution we wish to investigate the effects of granulation on the formation of the resonance lines in metal-poor stars. To do so we analyse the lines in the turn-off stars of two metal-poor Globular Clusters (NGC 6752 and NGC 6397) and compare the derived abundances, both in 1D and 3D, with those derived for giants, making use of the lines of Mult.2
Observational data and analysis
===============================
Our data consists of spectra acquired with UVES at the ESO Kueyen 8.2m telescope, at a resolution of R$\sim$45000. We have 3 TO stars in the NGC6397, 2 TO stars in NGC6752 and the field TO star HD218502. For each cluster star the total integration time is of the order of 10 hours for each star. The data has already been described in @Pasquini04 and @Pasquini07. The reduced spectra were downloaded from the ESO archive, thanks to the improved strategies for optimal extraction [@ballester], the S/N ratios are greatly improved with respect to what was previously available. We measured the EWs of the lines by fitting a gaussian with the IRAF task [splot]{}. For each star we computed a 1D LTE model atmosphere with the atmospheric parameters given in @Pasquini04 and @Pasquini07 and summarised in Table\[abun\]. We used the ATLAS 9 code [@K93a; @K05] in its Linux version [@SB04; @SB05]. From this we computed for each line a curve of growth with the SYNTHE code [@K93b; @K05; @SB04; @SB05] taking into account the hyperfine structure of the lines. The abundance was determined by interpolation in these curves of growth. For the hydrodynamical models we used the [[CO$^5$BOLD]{}]{} code and used spectrum synthesis on these models to compute curves of growth and “3D corrections”, as defined by @zolfito, with respect to the 1D LHD models. The appropriate 3D correction was found for each star by interpolating in the 3D grid.
Results and conclusions
=======================
From Table \[abun\] it is immediately clear that the 3D corrections are large. The reasons for this can be understood by looking at Fig.1 where the temperature distribution of one of our 3D models is depicted, together with the mean temperature distribution and that of a corresponding LHD model. The overcooling is obvious from the top panel and the contribution functions in the two bottom panels reflect the fact that in these cool outer layers the Cu atoms populate mainly the ground layer, thus contributing for the bulk of the absorption of the line, at variance with what happens in the 1D model (dashed line). The two bottom panels correspond to two different Cu abundances, illustrating how the tendency to prefer the outer layers increases with the increasing number of absorbers. As expected this results in larger 3D corrections for more metal-rich stars. However, one should be always cautious when facing contribution functions like the ones shown in Fig.1. In fact all the computations have been performed in LTE, it is likely that photons coming from the warm streams may produce overionisation in these low density outer layers. If NLTE effects are important a considerable resizing of the outer peak of the contribution function can be expected. A way to check indications of possible NLTE effects is to compare the abundances in the cluster, derived from the resonance lines in TO, with those of giants, derived from the lines of Mult.2. We derived abundance in NGC6397 using the EWs for two giants measured by @gratton82, for NGC6752 we used a UVES spectrum of a giant star (star Cl\* NGC 6752 YGN 30), already studied by @Yong. Neither in 1D nor in 3D giants and dwarfs provide the same abundance. In NGC 6752 the dwarf stars imply a higher abundance, both in 1D and 3D, the reverse is true in NGC 6397. That the problem is with the modelling of the Cu lines is confirmed by the analysis of the giant in NGC6752, for which we were able to measure both the resonance line at 327.3 nm and the 510.5nm line. Both in 1D and 3D the two lines provide abundances which differ by about 0.5dex. We conclude that the resonance lines are not good abundance indicators, a full 3D-NLTE study should be undertaken for these lines. At the same time one may suspect that also the lines of Mult.2 may be affected by deviations from LTE. The Galactic evolution of copper must be placed on solid grounds with a better modelling of the line formation.
We are grateful to L. Pasquini for many useful comments on this work. We acknowledge financial support from EU contract MEXT-CT-2004-014265 (CIFIST). We also acknowledge use of the supercomputing centre CINECA, which has granted us time to compute part of the hydrodynamical models used in this investigation, through the INAF-CINECA agreement 2006,2007.
Asplund, M. 2005, , 43, 481
Asplund, M., Nordlund, [Å]{}., Trampedach, R., & Stein, R. F. 1999, , 346, L17
Ballester, P., et al. 2006, , 6270, 26
Bihain, G., Israelian, G., Rebolo, R., Bonifacio, P., & Molaro, P. 2004, , 423, 777
Caffau, E., & Ludwig, H.-G. 2007, , 467, L11
Cohen, J. G., Christlieb, N., McWilliam, A., Shectman, S., Thompson, I., Melendez, J., Wisotzki, L., & Reimers, D. 2008, , 672, 320
, B., [Steffen]{}, M., & [Dorch]{}, B. 2002, Astronomische Nachrichten, 323, 213
, B., [Steffen]{}, M., [Wedemeyer-B[ö]{}hm]{}, S., & [Ludwig]{}, H.-G. 2003, [CO5BOLD User Manual]{}, [http://www.astro.uu.se/ bf/co5bold\_main.html](http://www.astro.uu.se/~bf/co5bold_main.html)
Gratton, R. G. 1982, , 115, 171
Kurucz, R. 1993a, ATLAS9 Stellar Atmosphere Programs and 2 km/s grid. Kurucz CD-ROM No. 13. Cambridge, Mass.: Smithsonian Astrophysical Observatory, 1993., 13,
, R. 1993b, SYNTHE Spectrum Synthesis Programs and Line Data. Kurucz CD-ROM No. 18. Cambridge, Mass.: Smithsonian Astrophysical Observatory, 1993., 18
, R. L. 2005, Memorie della Società Astronomica Italiana Supplementi, 8, 14
Magain, P. 1986, , 163, 135
Mishenina, T. V., Kovtyukh, V. V., Soubiran, C., Travaglio, C., & Busso, M. 2002, , 396, 189
Pasquini, L., Bonifacio, P., Randich, S., Galli, D., & Gratton, R. G. 2004, , 426, 651
Pasquini, L., Bonifacio, P., Randich, S., Galli, D., Gratton, R. G., & Wolff, B. 2007, , 464, 601
Sneden, C., Gratton, R. G., & Crocker, D. A. 1991, , 246, 354
, L., [Bonifacio]{}, P., [Castelli]{}, F., & [Kurucz]{}, R. L. 2004, Memorie della Società Astronomica Italiana Supplementi, 5, 93
, L. 2005, Memorie della Società Astronomica Italiana Supplementi, 8, 61
, S., [Freytag]{}, B., [Steffen]{}, M., [Ludwig]{}, H.-G., & [Holweger]{}, H. 2004, , 414, 1121
Yong, D., Grundahl, F., Nissen, P. E., Jensen, H. R., & Lambert, D. L. 2005, , 438, 875
|
---
abstract: 'Lam, Gusfield, and Sridhar (2009) showed that a set of three-state characters has a perfect phylogeny if and only if every subset of three characters has a perfect phylogeny. They also gave a complete characterization of the sets of three three-state characters that do not have a perfect phylogeny. However, it is not clear from their characterization how to find a subset of three characters that does not have a perfect phylogeny without testing all triples of characters. In this note, we build upon their result by giving a simple characterization of when a set of three-state characters does not have a perfect phylogeny that can be inferred from testing all pairs of characters.'
author:
- |
Brad Shutters, David Fernández-Baca\
Department of Computer Science, Iowa State University\
[{shutters,fernande}@iastate.edu](mailto:shutters@iastate.edu,fernande@cs.iasate.edu)
bibliography:
- '../Bibs/PhyloBib/PhyloBib.bib'
title: 'A Simple Characterization of the Minimal Obstruction Sets for Three-State Perfect Phylogenies'
---
Introduction
============
The [*$k$-state perfect phylogeny problem*]{} is one of the classic decision problems in computational biology. The input is an $n$ by $m$ matrix $M$ of integers from the set $\{1, \ldots, k\}$. We call a row of $M$ a [*taxon*]{} (plural [*taxa*]{}), a column of $M$ a [*character*]{}, and a value in column $c$ of $M$ a [*state*]{} of character $c$. A [*perfect phylogeny*]{} for $M$ is an undirected tree $t$ with $n$ leaves each labeled by a distinct taxon of $M$ in such a way that, for each character $c$ and each pair $i,j$ of states of $c$, the minimal subtree of $t$ containing all the leaves labeled by a taxon with state $i$ for character $c$ is node-disjoint from the minimal subtree of $t$ containing all the leaves labeled by a taxon with state $j$ for character $c$. The $k$-state perfect phylogeny problem is to decide whether $M$ has a perfect phylogeny. If $M$ has a perfect phylogeny, we say that the characters in $M$ are [*compatible*]{}, otherwise they are [*incompatible*]{}. See [@Baca2001a; @Semple2003a] for more on the perfect phylogeny problem. See Figure \[fig3PPex\] for an example.
[c|c]{} $M$ & a b c\
\
1 & 3 2 1\
2 & 2 1 3\
3 & 2 2 2\
4 & 3 3 2\
5 & 1 1 3\
6 & 2 2 3
(n1) at (-1.3,1) \[circle,inner sep=1.5,fill=black,label=above left:321\] ; (n2) at (-1.3,-1) \[circle,inner sep=1.5,fill=black,label=below left:332\] ; (n3) at (0,0) \[circle,inner sep=1.5,draw=black\] ; (n4) at (1.4,0) \[circle,inner sep=1.5,draw=black\] ; (n5) at (1.4,1.25) \[circle,inner sep=1.5,fill=black,label=above:222\] ; (n6) at (2.8,0) \[circle,inner sep=1.5,draw=black\] ; (n7) at (2.8,-1.25) \[circle,inner sep=1.5,fill=black,label=below:223\] ; (n8) at (4.2,0) \[circle,inner sep=1.5,draw=black\] ; (n9) at (5.5,1) \[circle,inner sep=1.5,fill=black,label=above right:213\] ; (n10) at (5.5,-1) \[circle,inner sep=1.5,fill=black,label=below right:113\] ; (n1) edge\[thick\] (n3); (n2) edge\[thick\] (n3); (n3) edge\[thick\] (n4); (n4) edge\[thick\] (n5); (n4) edge\[thick\] (n6); (n6) edge\[thick\] (n7); (n6) edge\[thick\] (n8); (n8) edge\[thick\] (n9); (n8) edge\[thick\] (n10);
If the number of states of each character is unbounded (so $k$ can grow with $n$), then the perfect phylogeny problem is NP-complete [@Bodlaender1992a; @Steel1992a]. However, if the number of states of each character is fixed, the perfect phylogeny problem is solvable in $O(m^2n)$ (in fact, linear time for $k=2$) [@Gusfield1991a; @Dress1992a; @Kannan1994a; @Agarwala1994a; @Kannan1997a]. Each of these algorithms can also construct a perfect phylogeny for $M$ if one exists. However, since every subset of a compatible set of characters is itself compatible, if no perfect phylogeny exists for $M$, there must be some minimal subset of the characters of $M$ that does not have a perfect phylogeny. We call such a set a [*minimal obstruction set*]{} for $M$. None of the above mentioned algorithms output a minimal obstruction set when there is no perfect phylogeny for $M$.
If the characters in $M$ are two-state characters, then $M$ has a perfect phylogeny if and only if the characters in $M$ are pairwise compatible. Hence, a minimal obstruction set for $k=2$ is of cardinality two [@Buneman1974a; @Meacham1983a; @Steel1992a; @Estabrook1977a]. A recent breakthrough by Lam, Gusfield, and Sridhar [@Lam2009a] shows that if the characters in $M$ are three-state characters, then any minimal obstruction set for $M$ has cardinality at most three. It is conjectured that given an input matrix $M$ of $k$-state characters, there exists a function $f(k)$ such that $M$ has a perfect phylogeny if and only if every subset of $f(k)$ characters of $M$ has a perfect phylogeny [@Fitch1975a; @Johnson1976a; @Fitch1977a; @Meacham1983a; @Gusfield1991a; @Lam2009a; @Habib2011a]. From the discussion above, it follows that $f(2)=2$ and $f(3)=3$. Recent work of Habib and To [@Habib2011a] shows that $f(4) \ge 5$.
If the characters in $M$ are $k$-state characters and the cardinality of a minimal obstruction set for $M$ is bounded above by $f(k)$, then it is preferable to have a test for the existence of such an obstruction set that does not require testing all subsets of $f(k)$ characters in $M$, and ideally one that can be inferred from testing all pairs of the characters in $M$. Since we can decide if $M$ has a perfect phylogeny in $O(m^2n)$ time, and construct a perfect phylogeny in such a case, we should hope to also output a minimal obstruction set in $O(m^2n)$ time when a perfect phylogeny for $M$ does not exist.
Here, we will focus on the three-state perfect phylogeny problem. Hence, we restrict $M$ to be an $n$ by $m$ matrix of integers from the set $\{1,2,3\}$. We build upon the work of Lam, Gusfield, and Sridhar [@Lam2009a] who showed that if $M$ does not have a perfect phylogeny, then $M$ has an obstruction set of cardinality at most three. They also gave a complete characterization of the minimal obstruction sets of cardinality three. However, it is not clear from their characterization how to find such an obstruction set without independently testing all triples of characters in $M$, requiring $O(m^3n)$ time. In this note, we remedy this situation by giving a simple characterization of when a set of three-state characters does not have a perfect phylogeny that can be inferred from testing all pairs of characters in $M$. This leads to a $O(m^2n)$ time algorithm to find an obstruction set when $M$ does not have a perfect phylogeny. If $M$ does admit a perfect phylogeny, then any of the above mentioned algorithms can be used to construct a perfect phylogeny for $M$ in $O(m^2n)$ time.
Preliminaries
=============
Perfect Phylogenies and Partition Intersection Graphs
-----------------------------------------------------
The [*partition intersection graph*]{} of $M$, denoted ${\mathrm{pig}}(M)$, is the graph that has a vertex ${c_i}$ for each character $c$ and each state $i$ of $c$, and an edge between two vertices $c_i$ and $d_j$ precisely if there is a taxon that has both state $i$ for character $c$ and state $j$ for character $d$. Note that there can be no edges between two vertices of the same character. See Figure \[fig3PPpig\] for an example. In this section we give a brief overview of some known results relating three-state perfect phylogenies to partition intersection graphs.
(a1) at (0,0) \[circle,inner sep=1.5,draw=black,label=left:$a_1$\] ; (a2) at (0,-2) \[circle,inner sep=1.5,draw=black,label=left:$a_2$\] ; (a3) at (0,-4) \[circle,inner sep=1.5,draw=black, label=left:$a_3$\] ; (b1) at (2,2) \[circle,inner sep=1.5,draw=black, label=above:$b_1$\] ; (b2) at (4,2) \[circle,inner sep=1.5,draw=black,label=above:$b_2$\] ; (b3) at (6,2) \[circle,inner sep=1.5,draw=black, label=above:$b_3$\] ; (c1) at (8,-2) \[circle,inner sep=1.5,draw=black,label=right:$c_1$\] ; (c2) at (8,0) \[circle,inner sep=1.5,draw=black, label=right:$c_2$\] ; (c3) at (8,-4) \[circle,inner sep=1.5,draw=black,label=right:$c_3$\] ; (a3) edge (b2); (a3) edge (b3); (a2) edge (b1); (a2) edge (b2); (a1) edge (b1); (b1) edge (c3); (b2) edge (c1); (b2) edge (c2); (b2) edge (c3); (b3) edge (c2); (a3) edge (c1); (a3) edge (c2); (a2) edge (c2); (a2) edge (c3); (a1) edge (c3);
A graph $G$ is [*triangulated*]{} if and only if there are no induced chordless cycles of length four or greater. A [*proper triangulation*]{} of ${\mathrm{pig}}(M)$ is a triangulated supergraph of ${\mathrm{pig}}(M)$ such that each edge is between vertices of different characters.
There is a perfect phylogeny for $M$ if and only if ${\mathrm{pig}}(M)$ has a proper triangulation.
For a subset $C=\{c_1,\ldots,c_j\}$ of the characters in $M$, we write $M[c_1,\ldots,c_j]$ to denote $M$ restricted to the columns in $C$. We say that $M$ is [*pairwise compatible*]{} if, for every pair $a,b$ of characters in $M$, there is a perfect phylogeny for $M[a,b]$.
\[thmPairCompatible\] Let $a$ and $b$ be two characters of $M$. Then $M[a,b]$ has a perfect phylogeny if and only if ${\mathrm{pig}}(M[a,b])$ is acyclic.
\[thm3PP\] $M$ has a perfect phylogeny if and only if, for every three characters $a,b,c$ in $M$, $M[a,b,c]$ has a perfect phylogeny.
If three of the characters are incompatible, then either they are not pairwise compatible, or, as the following theorem shows, the edges of their partition intersection graph is a superset (up to renaming of states) of one of a collection of “forbidden” edge sets.
\[thm3StateGametes\] Let $M$ be pairwise compatible. Then, a triple $\{a,b,c\}$ of characters in $M$ is a minimal obstruction set if and only if (under possibly renaming states) ${\mathrm{pig}}(M[a,b,c])$ contains all of the edges of one of graphs of Figure \[figObstructions\].
Solving Three-State Perfect Phylogeny with Two-State Characters
---------------------------------------------------------------
Here we review a result of Dress and Steel [@Dress1992a]. Our exposition closely follows that of [@Gusfield2009a].
Our goal is to derive a matrix of two-state characters $\overline{M}$ from the matrix $M$ of three-state characters. The properties of $\overline{M}$ are such that they enable use to find a perfect phylogeny for $M$. The matrix $\overline{M}$ contains three characters $c(1)$, $c(2)$, $c(3)$ for each character $c$ in $M$, such that all of the taxa that have state $i$ for $c$ in $M$ are given state 1 for character $c(i)$ in $\overline{M}$, and the other taxa are given state 2 for $c(i)$ in $\overline{M}$.
Since every character in $\overline{M}$ has two states, two characters $c(i)$ and $d(j)$ of $\overline{M}$ are incompatible if and only if the two columns corresponding to $c(i)$ and $d(j)$ contain all four of the pairs $(1,1)$, $(1,2)$, $(2,1)$, and $(2,2)$, otherwise they are compatible. This is known as the [*four gametes test*]{} [@Semple2003a].
\[thm3PPto2PP\] There is a perfect phylogeny for $M$ if and only if there is a subset $C$ of the characters of $\overline{M}$ such that
- the characters in $C$ are pairwise compatible, and
- for each character $c$ in $M$, $C$ contains at least two of the characters $c(1)$, $c(2)$, $c(3)$.
Theorem \[thm3PPto2PP\] was used in [@Dress1992a] to give an $O(m^2n)$ time algorithm to decide if there is a perfect phylogeny for $M$. It was also used in [@Gusfield2009a] to reduce the three-state perfect phylogeny problem in polynomial time to the well known 2-SAT problem, which is in $P$.
A Simple Characterization of Minimal Obstruction Sets
=====================================================
In this section, we focus on the case where M is pairwise compatible. Our main result is a characterization of the situation where M does not have a perfect phylogeny that is based on the partition intersection graphs for the pairs of characters in M. Theorem \[thmPairCompatible\] gives a simple characterization of the situation when $M$ is not pairwise compatible. We say that a state $i$ for a character $c$ of $M$ is [*dependent*]{} precisely when there exists a character $d$ of $M$, and two states $j,k$ of $d$, such that $c(i)$ is incompatible with both $d(j)$ and $d(k)$. The character $d$ is a [*witness*]{} that state $i$ of $c$ is dependent.
\[lemDependentState\] Let $c$ be a character of $M$ and let $i$ be a dependent state of $C$. Then no pairwise compatible subset of characters in $\overline{M}$ satisfying Theorem \[thm3PPto2PP\] contains $c(i)$.
Let $I$ be a pairwise compatible subset of the characters in $\overline{M}$ that contains $c(i)$. Since state $i$ of $c$ is dependent, there is a character $b$ in $M$ and two states $j,k$ of $b$, such that $c(i)$ is incompatible with both $b(j)$ and $b(k)$. It follows that $b(j) \not\in I$ and $b(k) \not\in I$. But then $I$ cannot possibly contain two of $b(1)$, $b(2)$, and $b(3)$. Thus, $I$ cannot satisfy the condition required in Theorem \[thm3PPto2PP\].
The next lemma gives a characterization of when a state is dependent using partition intersection graphs. We first introduce some notation: if $p : p_1p_2p_3p_4p_5$ is a path of length four in a graph, then we write $\mathrm{middle}[p]$ to denote $p_3$, the [*middle*]{} vertex of $p$.
\[lemPathDependent\] Let $M$ be pairwise compatible. A state $i$ of a character $c$ of $M$ is a dependent state if and only if there is a character $d$ of $M$ and a path $p$ of length four in ${\mathrm{pig}}(M[c,d])$ with $\mathrm{middle}[p]=c_i$.
W.l.o.g. assume that $i=1$, i.e., $c_i=c_1$.
($\Rightarrow$) Since $1$ is a dependent state of $c$, there exists a character $d$ in $M$ such that $c(1)$ is incompatible with two of $d(1)$, $d(2)$, and $d(3)$. W.l.o.g., assume $c(1)$ is incompatible with both $d(1)$ and $d(2)$. Then, $c_1d_1$ and $c_1d_2$ are edges of ${\mathrm{pig}}(M[c,d])$, and, since $M$ has no cycles, either $d_2c_2$ and $d_1c_3$ or $d_2c_3$ and $d_1c_2$ are edges of $G$. If $d_2c_2$ and $d_1c_3$ are edges of ${\mathrm{pig}}(M[c,d])$, then $c_2d_2c_1d_1c_3$ is the required path of length four. If $d_2c_3$ and $d_1c_2$ are edges of ${\mathrm{pig}}(M[c,d])$, then $c_3d_2c_1d_1c_2$ is the required path of length four.
($\Leftarrow$) Let $d$ be a character of $M$ such that there is a path $p$ of length four in ${\mathrm{pig}}(M[c,d])$ with $\mathrm{middle}[p]=c_1$. Since ${\mathrm{pig}}(M[c,d])$ cannot contain edges between to states of the same character, we can assume w.l.o.g. that $p$ is the path $c_2d_1c_1d_2c_3$. Then, it is easy to verify that $c(1)$ is incompatible with both $d(1)$ and $d(2)$. This is illustrated in Figure \[figPathIncompatible\].
\[lemDependentPP\] If $M$ is pairwise compatible and there is a character $c$ of $M$ that has two dependent states, then no perfect phylogeny exists for $M$.
Let $i$ and $j$ be two dependent states of $c$. Then, by Lemma \[lemDependentState\], no pairwise compatible subset $I$ of the characters of $\overline{M}$ that satisfy the condition required in Theorem \[thm3PPto2PP\] can contain $c(i)$ or $c(j)$. But then $I$ can only contain one of $c(1)$, $c(2)$, or $c(3)$. Hence, no pairwise compatible subset $I$ of the characters of $\overline{M}$ can satisfy the condition required in Theorem \[thm3PPto2PP\]. Hence, by Theorem \[thm3PPto2PP\], there is no perfect phylogeny for $M$.
We now show that the converse of Lemma \[lemDependentPP\] holds.
\[lemPPDependent\] If $M$ is pairwise compatible and has no perfect phylogeny, then there exists a character $c$ of $M$ that has two dependent states.
By Theorem \[thm3StateGametes\], there exists characters $a,b,c$ in $M$ such that $G={\mathrm{pig}}(M[a,b,c])$ (under possibly renaming of states) contains all of the edges of at least one of the subgraphs of Figure \[figObstructions\]. If $G$ contains all of the edges of Figure \[figObstructions\_1\], then $c_3b_1c_1b_2c_2$ is a path witnessing that $c_1$ is dependent and $c_3a_1c_2a_3c_1$ is a path witnessing that $c_2$ is dependent (this is illustrated in Figure \[figPig1Bolded\]). If $G$ contains all of the edges of Figure \[figObstructions\_2\], then $c_3b_1c_1b_2c_2$ is a path witnessing that $c_1$ is dependent and $c_3a_1c_2a_2c_1$ is a path witnessing that $c_2$ is dependent (this is illustrated in Figure \[figPig2Bolded\]). If $G$ contains all of the edges of Figure \[figObstructions\_2\], then $c_3a_2c_1a_1c_2$ is a path witnessing that $c_1$ is dependent and $c_3b_3c_2b_1c_1$ is a path witnessing that $c_2$ is dependent (this is illustrated in Figure \[figPig3Bolded\]). In all three cases, $M$ contains a character that has two dependent states.
Lemmas \[lemDependentPP\] and \[lemPPDependent\] together immediately imply our main theorem.
\[thmMain\] If $M$ is pairwise compatible, then there is a perfect phylogeny for $M$ if and only if there is at most one dependent state of each character $c$ of $M$.
\[obsObstructionSet\] Let $M$ be pairwise compatible and let $c$ be a character of $M$ with two dependent states. Let $a$ be a witness for one dependent state of $c$ and let $b$ be a witness for another dependent state of $c$. Then, the set $\{a,b,c\}$ is an obstruction set for $M$.
This leads to the following $O(m^2n)$ time algorithm to find a minimal obstruction set for $M$, if one exists.\
$M$ is an $n$ by $m$ matrix of integers from the set $\{1,2,3\}$. A minimal obstruction set for $M$ if one exists, otherwise the empty set. $\mathrm{mark}[x_i] \leftarrow \emptyset$; $S \leftarrow \emptyset$; $G \leftarrow {\mathrm{pig}}(M[a,b])$; $\{a,b\}$; $\mathrm{mark}[x_i] \leftarrow \{a,b\} \setminus \{x\}$; $S \leftarrow \{a\} \cup \mathrm{mark}[a_i] \cup \mathrm{mark}[a_j]$; $S \leftarrow \{b\} \cup \mathrm{mark}[b_i] \cup \mathrm{mark}[b_j]$; $S$;
The correctness of the algorithm follows from Theorem \[thmPairCompatible\], Theorem \[thmMain\], Observation \[obsObstructionSet\], and Lemma \[lemPathDependent\]. To see that the algorithm takes $O(m^2n)$ time note that the runtime is dominated by the loop of lines 5-16 which executes once for each of the $O(m^2)$ pairs of characters in $M$. Constructing the partition intersection graph of two three-state characters takes $O(n)$ time. Since the partition intersection graph of two three-state characters is of constant size, each of the other operations performed in the loop take constant time.
We note that if no obstruction set exists for $M$, then a perfect phylogeny for $M$ can be constructed in $O(m^2n)$ time by using one of the existing algorithms for the three-state perfect phylogeny problem [@Dress1992a; @Kannan1994a; @Agarwala1994a; @Kannan1997a; @Gusfield2009a].
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by the National Science Foundation under grants CCF-1017189 and DEB-0829674.
|
---
author:
- 'Giovanni Picogna, Wilhelm Kley'
bibliography:
- 'biblio.bib'
date: 'Received / Accepted '
subtitle: HL Tau system
title: 'How do giant planetary cores shape the dust disk?'
---
[We are observing, thanks to ALMA, the dust distribution in the region of active planet formation around young stars. This is a powerful tool to connect observations with theoretical models and improve our understandings of the processes at play.]{} [We want to test how a multi-planetary system shapes its birth disk and study the influence of the planetary masses and particle sizes on the final dust distribution. Moreover, we apply our model to the HL Tau system in order to obtain some insights on the physical parameters of the planets that are able to create the observed features.]{} [We follow the evolution of a population of dust particles, treated as Lagrangian particles, in two-dimensional, locally isothermal disks where two equal mass planets are present. The planets are kept in fixed orbits and they do not accrete mass.]{} [The outer planet plays a major role removing the dust particles in the co-orbital region of the inner planet and forming a particle ring which promotes the development of vortices respect to the single planetary case. The ring and gaps width depends strongly on the planetary mass and particle stopping times, and for the more massive cases the ring clumps in few stable points that are able to collect a high mass fraction. The features observed in the HL Tau system can be explained through the presence of several massive cores that shape the dust disk, where the inner planet(s) should have a mass on the order of $0.07\,M_\mathrm{Jup}$ and the outer one(s) on the order of $0.35\,M_\mathrm{Jup}$. These values can be significantly lower if the disk mass turns out to be less than previously estimated. Decreasing the disk mass by a factor 10 we obtain similar gap widths for planets with a mass of $10\,M_\oplus$ and $20\,M_\oplus$ respectively. Although the particle gaps are prominent, the expected gaseous gaps would be barely visible.]{}
Introduction {#sec:intro}
============
The planetary cores of giant planets form on a timescale $\sim 1\,\mbox{Myr}$. In this relatively short time-span, a huge number of processes takes place, allowing a swarm of small dust particles to grow several order of magnitudes in size and mass, before the gas disk removal. Until now, the only observational constraints that we had in order to test planet formation models were the gas and dust emissions from protoplanetary nebulae on large scales (on the order of $\sim100\,\mbox{au}$) and the final stage of planet formation through the detection of full-fledged planetary systems. Thanks to the recent advent of a new generation of radio telescopes, like the Atacama Large Millimeter Array (ALMA), we are starting to get some pristine images of the formation process itself, resolving the dust component of protoplanetary disks in the region of active planet formation around young stars.
An outstanding example of this giant leap in the observational data is the young HL Tau system, imaged by ALMA in Bands $3$, $6$, and $7$ (respectively at wavelengths $2.9$, $1.3$ and $0.87\,\mbox{mm}$) with a spatial resolution up to $3.5\,\mbox{au}$. Several features can be seen in the young protoplanetary disk, but the most striking is the presence of several axysimmetric rings in the $\mbox{mm}$ dust disk [@Partnership2014]. Although different mechanisms can be responsible of the observed features [@Flock2015; @Zhang2015], the most straightforward explanation for the ring formation, in the sense that others mechanism require specific initial conditions that reduce their general applicability, is the presence of several planetary cores that grow in their birth disk and shape its dust content. Indeed, in order to have a particle concentration at a particular region, we need a steep pressure gradient in the gaseous disk which can trap particles, ‘sufficiently’ decoupled from the gas, by changing its migration direction. A long-lived high-pressure region can be created even by a small mass planets [@Paardekooper2004], which can effectively carve a deep dust gap and concentrate particles at the gap edges and at corotation in tadpole orbits [@Paardekooper2007; @Fouchet2007].
The aim of this paper is to test how two giant planetary cores shape the dust disk in which they are born, implementing a particle population in the 2D hydro code <span style="font-variant:small-caps;">fargo</span> [@Masset2000], and study the influence of the planetary masses and particle sizes on the final disk distribution. Moreover, we apply our model to the HL Tau system in order to obtain some insights on the physical parameters of the planets creating the observed features.
This paper is organised as follows. In Section \[sec:model\] we discuss under which physical conditions a planet is capable of opening a gap in the dust and gaseous disk in order to define the important physical scales for our model. In Section \[sec:drag\] we define the model adopted for the gas drag. Then, the setup of our simulations is explained in Section \[sec:initsetup\], and the main results are outlined in Section \[sec:res\]. Finally, in Section \[sec:disc\] we discuss our results and their implications and limitation, while the major outcomes are highlighted in Section \[sec:conc\].
Background {#sec:model}
==========
In order to set up our model, we need first to determine what is the minimum mass of a planetary core that is able to open up a gap in the gaseous and dust disk, for a given set of disk parameters, and the relative opening timescale. In particular, we want to understand the influence of the different physical processes modelled on the outcome of the simulation.
Theory of gap formation {#sec:gap}
-----------------------
The theory of gap formation in gaseous disks has been studied extensively in the past, and there are a set of general criteria that a planet must fulfil in order to carve a gap. However, the possibility to open a gap in the dust disk is more complicated, since it depends strongly on the coupling between the dust and the gaseous media, and only recently this problem has been tackled.
### Gaseous gap
The torque exchange between the disk and the planet adds angular momentum to the outer disk regions and removes it from the inner ones. As a result, the disk structure is modified in the regions close to the planet location and, given a minimum core mass and enough time, a gap develops.
The time scale needed to open a gap of half width $x_\mathrm{s}$ can be crudely estimated from the impulse approximation [@Lin1979]. The total torque acting on a planet of mass $M_\mathrm{p}$ and semi-major axis $a$ due to its interaction with the outer disk of surface density $\Sigma$ is [@Lin1979; @Papaloizou2006] $$\label{eq:torqPlan}
\frac{dJ}{dt}=-\frac{8}{27}\frac{G^2M_\mathrm{p}^2 a\Sigma}
{\Omega_\mathrm{p}^2{x_\mathrm{s}}^3}$$ where $\Omega_\mathrm{p}=\sqrt{GM_\star/a^3}$ is the planet orbital frequency around a star of mass $M_\star$. The angular momentum that must be added to the disk in order to remove the gas inside the gap is $$\label{eq:AngMomRem}
\Delta J=2\pi a x_\mathrm{s} \Sigma\frac{dl}{dr}\biggr|_{a}x_\mathrm{s}$$ where $l=\sqrt{GM_\star r}$ is the gas specific angular momentum. Thus, the gap opening time can be estimated as $$\begin{aligned}
\label{eq:topen}
t_\mathrm{open} &= \frac{\Delta J}{|dJ/dt|} =
\frac{27}{8}\pi\frac{1}{q^2\sqrt{GM_\star a}}
\frac{{x_\mathrm{s}}^5}{a^3} \\
&\simeq 33.8\,{\left(\frac{h}{0.05}\right)}^{5}
{\left(\frac{q}{1.25*10^{-4}}\right)}^{-2} P_\mathrm{p}
\end{aligned}$$ where $q=M_\mathrm{p}/M_\star$ is the planet to star mass ratio, and we assumed that the minimum half width of the gap should be $x_\mathrm{s}=H$, where $H$ is the effective disk thickness, and $h=H/r$ is the normalised disk scale height. All values are evaluated at the planet location and the final estimate of the opening time is given in units of the planet orbital time $P_\mathrm{p}$. Although this is a crude estimate, it has been shown with more detailed descriptions that the order of magnitude obtained is correct.
Based on this criterion, given enough time, even a small core can open a gap in an inviscid disk. But, we need to quantify the magnitude of the competing factors that act to prevent or promote its development, in order to obtain a better estimate of the gap opening time scale, and the minimum mass ratio.
#### Thermal condition —
The assumption made for the minimum gap half width in eq. (\[eq:topen\]) is necessary to allow non-linear dissipation of waves generated by the planet [@Korycansky1996] and to avoid dynamical instabilities at the planet location [@Papaloizou1984], which are necessary conditions to clear the regions close to the planet location. This condition, called thermal condition, translates into a first criterion for open up a gap $$\label{eq:gapTherm}
x_\mathrm{s}=1.16a\sqrt{\frac{q}{h}}\geq H = ha$$ for a 2D disk [@Masset2006], which correspond to a minimum planet to star mass ratio of $$\label{eq:gapqTherm}
q_\mathrm{th} \simeq h^3 = 1.25*10^{-4}\,
{\left(\frac{h}{0.05}\right)}^3$$ and a related thermal mass $$\label{eq:gapMTherm}
M_\mathrm{th} \simeq M_\star h^3 = 1.25*10^{-4} M_\star$$
#### Viscous condition —
The viscous diffusion acts to smooth out sharp radial gradients in the disk surface density, preventing the gap clearing mechanism. The time needed by viscous forces to close up a gap of width $x_\mathrm{s}$ is given by the diffusion timescale for a viscous fluid, which can be derived directly from the Navier-Stokes equation in cylindrical polar coordinates [see e.g. @Armitage2010] $$\label{eq:tvisc}
t_\mathrm{visc}=\frac{x_\mathrm{s}^2}{\nu}\simeq 39.8\,
{\left(\frac{\alpha}{0.004}\right)}^{-1} P_\mathrm{p}$$ where $\nu$ is the kinematic viscosity, and $\alpha=\nu\Omega/c_\mathrm{s}^2$ is the Shakura-Sunyaev parameter that measures the efficiency of angular momentum transport due to turbulence.
The minimum mass $q_\mathrm{visc}$ needed to open a gap in a viscous disk is obtain by comparing the opening time due to the torque interaction, eq. (\[eq:topen\]), with the closing time owing to viscous stress eq. (\[eq:tvisc\]) [@Lin1986; @Lin1993] $$\label{eq:gapVisc}
q_\mathrm{visc} \simeq
{\left(\frac{27}{8}\pi\right)}^{1/2}\alpha^{1/2}h^{5/2} \simeq
1.15*10^{-4}{\left(\frac{\alpha}{0.004}\right)}^{1/2}
{\left(\frac{h}{0.05}\right)}^{5/2}$$ Thus, for the parameters chosen, the viscous condition is very similar to the thermal condition.
#### Generalised condition —
A more general semi-analytic criterion, which takes into account the balance between pressure, gravitational, and viscous torques for a planet on a fixed circular orbit has also been derived [@Lin1993; @Crida2006] $$\label{eq:gapCrida}
\frac{3}{4}\frac{H}{r_\mathrm{H}}+\frac{50}{q\ R_\mathrm{e}} < 1$$ where $r_\mathrm{H}=a{(q/3)}^{1/3}$ is the Hill radius and $R_\mathrm{e}$ is the Reynolds number. From the previous relation, and plugging in the parameters used in our analysis we found that the minimum mass ratio is $q\simeq10^{-3}$. However, this criterion was derived for low planetary masses in low viscosity disks, and the behaviour might be considerably different changing those parameters. Moreover, this condition defines a gap as a drop of the mass density to $10\%$ of the unperturbed density at the planet’s location, but even a less dramatic depletion of mass affect planet-disk interaction and could be potentially detected.
### Dust gap
In order to create a gap in the dust disk we need a radial pressure structure induced by the planet. Indeed, also a very shallow gap in the gas will change the drift speed of the dust particles significantly [@Whipple1972; @Weidenschilling1977] favouring the formation of a particle gap. Thus, the minimum mass needed to open up a gap in the dust disk is a fraction of the one needed to clear a gap in the gas. [@Paardekooper2004; @Paardekooper2006] performed extensive 2D simulations, treating the dust as a pressure-less fluid (which is a good approximation for tightly coupled particles), and they found that a planet more massive than $0.05 M_\mathrm{Jup}=0.38\,M_\mathrm{th}$ can open a gap in a mm size disk. Furthermore the dust gap opening time for this lower mass case was $50\,P_\mathrm{p}$, which is about half the timescale to open a gas gap. Previous studies found also a clear dependence of the gap width with the core mass, where larger planets open wider gaps [@Paardekooper2006; @Zhu2014]. Finally, there are some contrasting results regarding the dependence of the gap width respect to the particle size [@Paardekooper2006], but it seems that for less coupled particles the gap is wider for larger particles [@Fouchet2007].
Model {#sec:drag}
=====
Solid particles and gaseous molecules exchange momentum due to drag forces, that depend strongly on the condition of the gas and on shape, size and velocity of the particle. For sake of simplicity we limit ourselves to spherical particles. The drag force acts always in a direction opposite to the relative velocity.
The regime that describes a particular system is defined by any two of three non-dimensional parameters. The Knudsen number $K = \lambda/s$ is the ratio of the two major length scales of the system: the mean free path of the gas molecules $\lambda$ and the particle size $s$. The Mach number $M = v_\mathrm{r}/c_\mathrm{s}$ is the ratio between the relative velocity between particles and gas $\mathbf{v}_\mathrm{r}$, and the gas sound speed $c_\mathrm{s}$. Finally, the Reynolds number $R_\mathrm{e}$ is related to different physical properties of the particle and gaseous media $$R_\mathrm{e} = \frac{2 v_\mathrm{r}s}{\nu_\mathrm{m}}$$ where $\nu_\mathrm{m}$ is the gas molecular viscosity, defined as $$\nu_\mathrm{m} = \frac{1}{3}{\left(\frac{m_0\bar{v}_\mathrm{th}}{\sigma}
\right)}$$ where $m_0$ and $\bar{v}_\mathrm{th}$ are the mass and mean thermal velocity of the gas molecules, and $\sigma$ is their collisional cross section.
There are two main regimes of the drag forces that we are going to study.
Stokes regime
-------------
For small Knudsen number, the particle experience the gas as a fluid. The drag force of a viscous medium with density $\rho_\mathrm{g}
(\mathbf{r}_\mathrm{p})$ acting on a spherical dust particle with radius $s$ can be modelled as [@Landau1959] $$\mathbf{F}_\mathrm{D,S}=-\frac{1}{2}C_\mathrm{D}\pi s^2\rho_\mathrm{g}
(\mathbf{r}_\mathrm{p}) v_\mathrm{r} \mathbf{v}_\mathrm{r}$$ where the drag coefficient $C_\mathrm{D}$ is defined for the various regimes described above as [@Whipple1972; @Weidenschilling1977] $$C_\mathrm{D} \simeq
\begin{cases}
24\,{R_\mathrm{e}}^{-1} & R_\mathrm{e} < 1 \\
24\,{R_\mathrm{e}}^{-0.6} & 1 < R_\mathrm{e} < 800 \\
0.44 & R_\mathrm{e} > 800
\end{cases}$$ for low Mach numbers, however, for our choice of the parameter space high values are not expected in this regime.
Epstein regime
--------------
For low Knudsen numbers, the interaction between particles and single gas molecule becomes important. It can be modelled as [@Epstein1923] $$\mathbf{F}_\mathrm{D,E}= -\frac{4}{3}\pi\rho_\mathrm{g}
(\mathbf{r}_\mathrm{p})s^2\bar{v}_\mathrm{t}
\mathbf{v}_\mathrm{r}$$
General law
-----------
The transition between the Epstein and Stokes regime occurs for a particle of size $s=9\lambda/4$ which in our case is a $m$ size particle in the inner disk, where the mean free path of the gas molecules is defined as [@Haghighipour2003] $$\lambda = \frac{m_0}{\pi a_0^2\rho_\mathrm{g}(r)} =
\frac{4.72*10^{-9}}{\rho_\mathrm{g}} [cm]$$ for a molecular hydrogen particle with $a_0=1.5*10^8\,\mbox{cm}$. In order to model a broad range of particles sizes we adopt a linear combination of Stokes and Epstein regimes [@Supulver2000; @Haghighipour2003] $$\mathbf{F}_\mathrm{D} = (1-f)\mathbf{F}_\mathrm{D,E} +
f\mathbf{F}_\mathrm{D,S}$$ where the factor $f$ is related to the Knudsen number and is defined as: $$f=\frac{s}{s+\lambda}=\frac{1}{1+\mathit{Kn}}$$
Stopping time
-------------
An important parameter to evaluate the strength of the drag force is the stopping time $t_\mathrm{s}$, which can be defined as $$\mathbf{F}_\mathrm{D} = -\frac{m_\mathrm{s}}{t_\mathrm{s}}
\mathbf{v}_\mathrm{r}$$ where $m_\mathrm{s}$ is the mass of the single dust particle of density $\rho_\mathrm{s}$ and, in the Epstein regime, the stopping time can be expressed as $$\label{eq:stop}
t_\mathrm{s}=\frac{s\rho_\mathrm{s}}{\rho_\mathrm{g}\bar{v}_\mathrm{th}}$$ It is useful also to derive an non-dimensional stopping time (or Stokes number) as $$\tau_\mathrm{s}=\frac{s\rho_\mathrm{s}}{\rho_\mathrm{g}
\bar{v}_\mathrm{th}}\Omega_\mathrm{K}(\mathbf{r})$$
Setup {#sec:initsetup}
=====
We used the <span style="font-variant:small-caps;">fargo</span> code [@Masset2000; @Baruteau2008], modified in order to take into account the evolution of partially decoupled particles. An infinitesimally thin disk around a star resembling the observed HL Tau system [@Kwon2011] is modelled. Thus, the vertically integrated versions of the hydrodynamical equations are solved in cylindrical coordinates ($r,\phi,z$), centred on the star where the disk lies in the equatorial plane ($z=0$). The resolution adopted in the main simulations is $256\times512$ with $250\,000$ dust particles for each size, although we tested also a case doubling the resolution in order to test whether our results were resolution dependent.
Gas component {#par:gasdisk}
-------------
The initial disk is axysimmetric and it extends from $0.1$ to $4$ in code units, where the unit of length is $25\,\mbox{au}$. The gas moves with azimuthal velocity given by the Keplerian speed around a central star of mass $1$ ($0.55\,M_\odot$), corrected by the rotating velocity of the coordinate system. We assume no initial radial motion of the gas, since a thin Keplerian disk is radially in equilibrium as gravitational and centrifugal forces approximately balance because pressure effects are small. The initial surface density profile is given by $$\label{eq:surfprof}
\Sigma(r)=\Sigma_0\ r^{-1}$$ where $\Sigma_0$ is the surface density at $r=1$ such as the total disk mass is equal to $0.24$ ($0.13\ M_\odot$) in order to match the value found by @Kwon2011. The disk is modelled with a locally isothermal equation of state, which keeps constant the initial temperature stratification throughout the whole simulation. We assumed a constant aspect ratio $H/r = 0.05$, that corresponds to a temperature profile $$\label{eq:tempprof}
T(r) = T_0\ r^{-1}$$ We introduce a density floor of $\Sigma_\mathrm{floor} = 10^{-9}\ \Sigma_0$ in order to avoid numerical issues. For the inner boundary we applied a zero-gradient outflow condition, while for the outer boundary we adopted a non-reflecting boundary condition. In addition, to maintain the initial disk structure in the outer parts of the disk we implemented a wave killing zone close to the boundary [@deVal-Borro2006], $$\label{eq:dumping}
\frac{d\xi}{dt}=-\frac{\xi-\xi_0}{\tau} {R(r)}^2$$ where $\xi$ represent the radial velocity, angular velocity, and surface density. Those physical quantities are damped towards their initial values on a timescale given by $\tau$ and $R(r)$ is a linear ramp-function decreasing from $1$ to $0$ from $r=3.6$ to the outer radius of the computational domain. The details of the implementation of the boundary conditions can be found in @Muller2012. For the viscosity we adopt a constant $\alpha$ viscosity $\alpha = 0.004$. Furthermore, we discuss the gravitational stability of such initial configuration in Appendix \[sec::stability\].
Dust component {#par:dustdisk}
--------------
The solid fraction of the disk is modelled with $250\,000$ Lagrangian particles for each size considered. We study particles with sizes of $\mbox{mm},\mbox{cm},\mbox{dm},\mbox{m}$, and internal density $\rho_\mathrm{d}=2.6\ \mbox{g/cm}^3$. The initial surface density profile for the dust particles is flat $$\Sigma_\mathrm{s}(r) = \Sigma_\mathrm{s,0}$$ This choice was made in order to have a larger reservoir of particles in the outer disk, since at the beginning of the simulation the planets are slowly growing, thus they are unable to filtrate particles efficiently. The particles were introduced at the beginning of the simulation, and they are evolved with two different integrators, depending on their stopping times. Following the approach by [@Zhu2014], we adopted a semi-implicit Leapfrog-like (Drift-Kick-Drift) integrator in polar coordinates for larger particles, and a fully implicit integrator for particles well coupled to the gas. For the interested reader, we have added in Appendix \[sec::integrators\] the detailed implementation of the two integrators. In this work we do not take into account the back-reaction of the particle on the gaseous disk since we are interested only in the general structure of the dust disk and not on the evolution of dust clumps. Furthermore, for sake of simplicity and to speed up our simulations, we do not consider the effect of the disk self-gravity on the particle evolution. This could be in principle an important factor for the young and massive HL Tau system, although no asymmetric structures related to a gravitationally unstable disk are observed. Finally, we do not consider particle diffusion by disk turbulence, which could be important to prevent strong clumping of particles (Baruteau, private communication), and it will be the subject of a future study. In Figure \[Fig:Ts\] we plotted the stopping times calculated at the beginning of the simulations for the various particle species modelled. The smaller particles (cm, and mm-size) are strongly coupled to the gas in the whole domain, while dm-size particles approach a stopping time of order unity in the outer disk, and m particles in the inner part were we can see also a change in the profile due to the passage from the Epstein to Stokes drag regime.
![Stopping time at the beginning of the simulation for the different particle sizes modelled.\[Fig:Ts\]](Ts){width=".45\textwidth"}
The transition between the Epstein and Stokes regime is clearly visible in Figure \[Fig:Vrad\] where the radial drift velocity at the equilibrium is plotted for the different particles sizes in the whole domain. As the particles approaches a stopping time of order unity their radial velocity grows, so the highest value is associated to the dm particles in the outer disk and the m-size particles in the inner parts. Furthermore, due to the transition between the two drag regimes, the profile of the curves rapidly changes from cm to m-size particles. We point out that when the planet start to clear a gap, the gas surface density inside it drops, and thus the stopping time of particles in horseshoe orbit can increase up to 2 orders of magnitude [@Paardekooper2006]. The transition between the different drag regimes is then expected not only in the inner part of the disk but also near the planet co-orbital regions.
![Radial drift velocity profile at the equilibrium for the different particle sizes modelled.\[Fig:Vrad\]](vr){width=".45\textwidth"}
Planets {#par:planets}
-------
We embed two equal mass planets that orbit their parent star in circular orbits with semi-major axes $a_1=1$ and $a_2=2$. Their mass ranges from $1\,M_\mathrm{th}=0.07\,M_\mathrm{Jup}$ to $10\,M_\mathrm{th}=0.7\,M_\mathrm{Jup}$. Mass accretion is not allowed, and the planets do not feel the disk, so their orbital parameters remain fixed during the whole simulation. The gravitational potential of the planets is modelled through a Plummer type prescription, which takes into account the vertical extent of the disk and avoids the numerical issues related to a point-mass potential. We used a smoothing value of $\epsilon = 0.6\ H$ as this describes the vertically averaged forces very well [@Muller2012]. To prevent strong shock waves in the initial phase of the simulations the planetary core mass is increased slowly over $20$ orbits. Tab. \[tab:sum\] summarises the parameters of the standard model.
Parameter Range
--------------------------------- -------------------------
Planet mass \[$M_\mathrm{th}$\] $1$, $5$, $10$
Dust size \[$\mbox{cm}$\] $0.1$, $1$, $10$, $100$
: Models[]{data-label="tab:sum"}
With these values the gaps are not expected to overlap since, even for the highest mass planet, from eq. (\[eq:gapTherm\]) we have $x_\mathrm{s}\simeq0.18$, thus $$a_1+x_\mathrm{s} < a_2-x_\mathrm{s}$$ The simulations were run for 600 orbital times of the inner planet, which corresponds to $\sim 200$ orbits of the outer planet, and to $\sim 10^5\,\mbox{yr}$, which is a consistent amount of time for a planetary system around a young star like HL Tauri which has only $10^6\,\mbox{yr}$.
Results {#sec:res}
=======
Massive core ($10 M_\mathrm{th}$) {#sec:res10}
---------------------------------
Based on the criteria of gap opening reviewed in Section \[sec:model\], two $10 M_\mathrm{th}$ planets should open up rapidly a gap in the gas and dust disk. We study in detail the disk evolution for these massive cores, focusing on particle concentration which happen mainly at gap edges.
In the following analysis the region of high surface density between the two planets is referred as the ring. It will have an inner and outer edge, which correspond to the outer edge of the inner gap and the inner edge of the outer gap, respectively. Furthermore, there will be the outer edge of the outer gap and the inner edge of the inner gap. The study of the various gap edges, where there is an abrupt change in the surface density profile, is important since those are potentially unstable regions where gas and particles could collect, changing the final surface density distribution of the disk.
### Gas distribution
The inner planet has already opened a clear gap after 100 orbits (Figure \[Fig:Surf10\] - top panel) where, as expected by the general criterion of [@Crida2006], the surface density is an order of magnitude lower than its unperturbed value. Meanwhile, the outer planet is still opening its gap since it has a longer dynamical timescale.
![Gas surface density (top panel) and vorticity (bottom panel) profile after $100$ orbits.\[Fig:Surf10\]](Sigma "fig:"){width=".45\textwidth"} ![Gas surface density (top panel) and vorticity (bottom panel) profile after $100$ orbits.\[Fig:Surf10\]](Vort "fig:"){width=".45\textwidth"}
The steep profile in the surface density close to the ring inner edge can trigger a Rossby wave instability (RWI, [@Li2001]). This instability gives rise to a growing non-axisymmetric perturbation consisting of anticyclonic vortices. A vortex is able to collect a high mass fraction and it can change significantly the final distribution of gas and dust in the disk. An important parameter to study, when considering the evolution of vortices is the gas vorticity, which is defined as $$\omega_\mathrm{z}={(\nabla\times \mathbf{v})}_\mathrm{z}$$ We show its profile in Figure \[Fig:Surf10\] (bottom panel) and its 2D distribution in Figure \[Fig:Surf2D10\] (bottom panel). Comparing the 2D distributions of vorticity and surface density (Figure \[Fig:Surf2D10\]), we see that vorticity peaks where the gap is deeper, and low vorticity regions appear at the centre of spiral arms created by the planets and close to the ring inner edge.
The development of vortices due to the presence of a planet has been studied extensively. However, their evolution in a multi-planet system has not yet been addressed. From Figure \[Fig:Surf2D10\] we see that the outer planet perturbs substantially the co-orbital region of the inner one. There are two competing factors that need to be taken into account in order to estimate the lifetime of a vortex. On the one hand vortex formation is promoted by the enhanced surface density gradient at the ring location due to the combined action of both planets that push away the disk from their location. On the other hand, the periodic close encounters of the outer planet with the vortices enhance the eccentricity of the dust particles trap into it, favouring their escape and thus depleting the solid concentration inside the vortex.
![Gas surface density (top panel) and vorticity distribution (bottom panel) at the end of the simulation for the massive cores case. \[Fig:Surf2D10\]](Surf2D10 "fig:"){width=".45\textwidth"} ![Gas surface density (top panel) and vorticity distribution (bottom panel) at the end of the simulation for the massive cores case. \[Fig:Surf2D10\]](Vort2D10 "fig:"){width=".45\textwidth"}
The capacity of collect particles by a vortex is closely linked to its orbital speed. If a vortex has a Keplerian orbital speed, then dust particles with the same orbital frequency will remain in the vortex for many orbits and slowly drift to its centre due to drag forces. On the other hand, a particle in a vortex orbiting with a non-Keplerian frequency will experience a Coriolis force in the Keplerian reference frame and, if the drag force is unable to counteract it, the particle will leave the vortex location [@Youdin2010].
The evolution of vortices can be also studied from the dynamics of coupled particles, which follow closely the gas dynamics. From the analysis of cm particles near the ring inner edge (Figure \[Fig:Vortevol\]), we can see that after 50 orbits two vortices are already visible and they last for several tens of orbits. The outer planet stretches periodically the vortices, and as a result they slowly shrink in size.
In order to see if the vortices that develop in our simulation are capable of collecting a large fraction of particles, we plot in Figure \[Fig:Vortevol\] the vortices in a co-moving frame with the disk at $r=1.3$. The two vortices follow within a few percent the Keplerian speed, thus they are potentially able to trap a consistent fraction of particles.
![cm-sized particle distribution at different time-steps (50, 60, 70, 80 orbital times) at the ring inner edge for the massive cores case. The evolution of two vortices in a frame co-moving with the disk at $1.3\,\mbox{au}$ is shown. The vortex centre orbits the central star with a velocity close to the background Keplerian speed, which promotes particle trapping inside the vortex. \[Fig:Vortevol\]](VortexEvol){width=".45\textwidth"}
The influence of the outer planet on the development of RWI at the gap edge is studied by running a different model with only one massive planet ($M=10 M_\mathrm{th}$) at $r = 1.0$ (Figure \[Fig:Comp1\]). The particle concentration near the inner ring is weaker in the single core simulation. As a result, the vortices that form are less prominent and, although the perturbation to the ring inner edge is reduced respect to the dual core simulation, their lifetime is shorter.
![Same as Fig. \[Fig:Vortevol\] but for a single massive core at $r=1$. \[Fig:Comp1\]](Comp1pla){width=".45\textwidth"}
### Particle distribution
The evolution of the normalised surface density of the various dust species is shown in Figure \[Fig:Dust10\]. The inner and outer planets carve rapidly (first $50$ orbits) a particle gap, except for the most coupled particles simulated (mm-sized), which are cleared on a longer timescale for the outer planet (see Figure \[Fig:Dust10\] - fourth panel). As stated before, this behaviour is only due to the longer dynamical timescale of the outer planet, and it follows closely the evolution of the gas in the outer gap.
A significant fraction of particles clumps in the co-orbital region with the inner planet for several hundred periods. However, they are finally disrupted by the tidal interaction with the outer planet which excite their eccentricities, causing a close fly-by with the inner planet. The only particles which remain for the all simulated period in the co-orbital region are the m-size ones, although even they are perturbed and a significant mass exchange between the two Lagrangian points (L4 and L5) takes place (see Figure \[Fig:Dust10\] - first panel and Fig. \[Fig:Gap10m\] - bottom right panel). The outer planet is also able to keep a fraction of particles in the co-orbital region for longer timescale, though they are much more dispersed respect to the Lagrangian points. Moreover, in all simulations the L4 Lagrangian point is the most populated.
![Dust normalised surface density distribution for the m (first panel), dm (second panel), cm (third panel), and mm-sized (fourth panel) particles disk and 2 equal mass $10\,M_\mathrm{th}$ cores at $r=1,2$ at 4 different times (50, 100, 300, 600 orbital times) for each case. \[Fig:Dust10\]](1m10M "fig:"){width=".38\textwidth"} ![Dust normalised surface density distribution for the m (first panel), dm (second panel), cm (third panel), and mm-sized (fourth panel) particles disk and 2 equal mass $10\,M_\mathrm{th}$ cores at $r=1,2$ at 4 different times (50, 100, 300, 600 orbital times) for each case. \[Fig:Dust10\]](1dm10M "fig:"){width=".38\textwidth"} ![Dust normalised surface density distribution for the m (first panel), dm (second panel), cm (third panel), and mm-sized (fourth panel) particles disk and 2 equal mass $10\,M_\mathrm{th}$ cores at $r=1,2$ at 4 different times (50, 100, 300, 600 orbital times) for each case. \[Fig:Dust10\]](1cm10M "fig:"){width=".38\textwidth"} ![Dust normalised surface density distribution for the m (first panel), dm (second panel), cm (third panel), and mm-sized (fourth panel) particles disk and 2 equal mass $10\,M_\mathrm{th}$ cores at $r=1,2$ at 4 different times (50, 100, 300, 600 orbital times) for each case. \[Fig:Dust10\]](1mm10M "fig:"){width=".38\textwidth"}
![Particle distribution near the inner massive planetary core location for mm (top left), cm (top right), dm (bottom left) and m (bottom right) size particles at the end of the simulation. The velocity vectors of the particles respect to the planet are shown and the colour scale highlight the radial velocity component. \[Fig:Gap10m\]](gapdust10M){width=".45\textwidth"}
The ring, which forms between the two gaps, gets shortly very narrow and it stabilises in a position close to the 5:3 mean motion resonance (MMR) with the inner planet which seems a stable orbit. As shown in Figure \[Fig:Clumping\] the particles in the ring clump around $5$ symmetric points which gain a high mass.
![dm sized particle distribution at the end of the simulation of the two massive cores. The particle ring between the two planets shrinks with time until it clumps in few stable points that forms a pentagon-like structure with an orbit close to the 5:3 MMR with the inner planet at $r=1.4$. \[Fig:Clumping\]](MMR){width=".45\textwidth"}
There are visible vortices in the particle distribution in the first hundred orbits for the cm size particles at the inner ring edge (third panel). Although these structures are prominent in the particle distribution of cm size dust, they are not visible in the other dust size distributions. The main reason is that for larger particles the ring get shortly very narrow and so there is no time for them to be trapped into the vortex.
The velocity components of the particles shown in Figure \[Fig:Gap10m\] highlight the strong perturbations due to the spiral arm generated by the outer planet which affects mainly the most coupled particles and the gas. The bodies passing close to the planet location, as in the meter case shown in Figure \[Fig:Gap10m\] (bottom right panel), gain a high velocity component that is represented by the long black arrow.
Finally, the particle distributions at the end of the simulation (see Figure \[Fig:Gap10m\]) show the dependence of gap width on particle size. The mm size particles (Fig. \[Fig:Gap10m\] - top left panel) reach a region closer to the planet location, where we overplot the minimum half-gap width as calculated from the eq. (\[eq:gapTherm\]), showing that the dynamics of the smallest particles follow closely that of the gas. The others particles have increasingly larger gaps. The dm particles (Fig. \[Fig:Gap10m\] - bottom left panel) have cleared almost completely the gap region since they have a stopping time closer to one near the planet location and thus their evolution is faster.
Intermediate mass core ($5 M_\mathrm{th}$)
------------------------------------------
The intermediate mass planet should still open up a gap in the gaseous disk, but on a much longer timescale, since from eq. (\[eq:topen\]) the opening time scales as $q^{-2}$.
### Gas distribution
From the density profile of Figure \[Fig:Surf10\] (first panel) we can see that after $100$ orbits there is a visible gap opened by the inner planet, even if it is considerably shallower than the massive cores case, as for the vorticity profile (second panel). The outer planet still perturbs the co-orbital region of the inner one (Figure \[Fig:Surf2D5\]), but its magnitude is reduced and it has has barely modified the unperturbed surface density distribution at its location.
Even though the influence by the outer planet is less dramatic compared to the more massive case, the reduction of the surface density profile’s steepness hinders the development of vortices near the ring inner edge.
![Gas surface density (top panel) and vorticity distribution (bottom panel) at the end of the intermediate mass cores simulation. \[Fig:Surf2D5\]](Surf2D5 "fig:"){width=".45\textwidth"} ![Gas surface density (top panel) and vorticity distribution (bottom panel) at the end of the intermediate mass cores simulation. \[Fig:Surf2D5\]](Vort2D5 "fig:"){width=".45\textwidth"}
### Particle distribution
The intermediate mass inner planet carves a gap in the dust disk after $50$ orbits for all the particles sizes (Figure \[Fig:Dust5m\]). Instead, the outer planet after $50$ orbits is able to clean its gap only for the most decoupled particles (dm and m), while it takes $~300$ orbits to clean a gap in the cm particles (third panel), and it has open only a partial gap in the mm size particles disk at the end of the simulation (fourth panel). The dm particles (second panel) have the highest migration speed in the outer disk, as expected from Figure \[Fig:Vrad\] and after $300$ orbits they are all distributed in narrow regions at gap edges and in co-orbital region with the outer planet. On the other hand, the cm particles (third panel) are more coupled to the gas and the outer planet has not yet carved a gap sufficient to confine them in the outer regions of the disk at the end of the simulation. Thus, we observe after $600$ orbits that they engulf the planet gap (third panel, last screen-shot).
The dependence of the gap width respect to particle size is highlighted in Figure \[Fig:Gap5m\], where we can see a strong difference between the mm particles gap, which follows the gas dynamic, remaining close to the location of the half-gap width ($x_\mathrm{s}$) over-plotted on the first panel, and the gap width of the other particles.
The interaction between the two planet disrupts on the long term the particles clumping around the stable Lagrangian points, as in the more massive case. At the end of the simulation only the m and mm particles (Figure \[Fig:Gap5m\]) are still present in co-orbital region, preferable at the L5 point.
The clumping of particles in few stable points at the ring location is still visible in the intermediate mass case, but only in the m, and dm cases (first and second panels of Figure \[Fig:Dust5m\]).
Furthermore, some vortices are forming in the outer part of the particle disk of cm dust (third panel of Figure \[Fig:Dust5m\]). However, these are numerical artefacts because the disk outer edge is initialized with a sharp density profile cut which, on the long term, favours the development of vortices.
![Dust normalised surface density distribution for the m (first panel), dm (second panel), cm (third panel), and mm-sized (fourth panel) particles disk and 2 equal mass $5\,M_\mathrm{th}$ cores at $r=1,2$ at 4 different times (50, 100, 300, 600 orbital times) for each case. \[Fig:Dust5m\]](1m5M "fig:"){width=".38\textwidth"} ![Dust normalised surface density distribution for the m (first panel), dm (second panel), cm (third panel), and mm-sized (fourth panel) particles disk and 2 equal mass $5\,M_\mathrm{th}$ cores at $r=1,2$ at 4 different times (50, 100, 300, 600 orbital times) for each case. \[Fig:Dust5m\]](1dm5M "fig:"){width=".38\textwidth"} ![Dust normalised surface density distribution for the m (first panel), dm (second panel), cm (third panel), and mm-sized (fourth panel) particles disk and 2 equal mass $5\,M_\mathrm{th}$ cores at $r=1,2$ at 4 different times (50, 100, 300, 600 orbital times) for each case. \[Fig:Dust5m\]](1cm5M "fig:"){width=".38\textwidth"} ![Dust normalised surface density distribution for the m (first panel), dm (second panel), cm (third panel), and mm-sized (fourth panel) particles disk and 2 equal mass $5\,M_\mathrm{th}$ cores at $r=1,2$ at 4 different times (50, 100, 300, 600 orbital times) for each case. \[Fig:Dust5m\]](1mm5M "fig:"){width=".38\textwidth"}
![Particle distribution near the inner intermediate mass planetary core location for mm (top left), cm (top right), dm (bottom left) and m (bottom right) size particles at the end of the simulation. The velocity vectors of the particles respect to the planet are shown and the colour scale shows the relative radial velocity. \[Fig:Gap5m\]](gapdust5M){width=".45\textwidth"}
Low mass core ($1 M_\mathrm{th}$)
---------------------------------
Finally, we explore the low mass core scenario in order to study a case where the particle ring between the two planets does not clump into few stable points and the gaps carved by planets remain narrow, such as in the observed HL Tau system.
### Gas distribution
Figure \[Fig:Surf10\] (top panel) shows that both the inner and outer planet are not massive enough to clear a gap in the gaseous disk within the simulated time. From the 2D distribution of the surface density and vorticity (Figure \[Fig:Surf2D1\]) it is possible to see that the presence of the two planets changes slightly the unperturbed state of the gaseous disk (where the scale has been changed respect to the plots of the more massive cases in order to highlight the small differences).
![Gas surface density (top panel) and vorticity distribution (bottom panel) at the end of the simulation for the two low mass cores. \[Fig:Surf2D1\]](Surf2D1 "fig:"){width=".45\textwidth"} ![Gas surface density (top panel) and vorticity distribution (bottom panel) at the end of the simulation for the two low mass cores. \[Fig:Surf2D1\]](Vort2D1 "fig:"){width=".45\textwidth"}
### Particle distribution
Although the gas profile is not changed considerably due to the small mass of the planets, they are able to open up clear gaps in the dust disk in the first 50 orbits for m and dm size particles (Figure \[Fig:Dust1m\]), while it takes $100$ orbits for the cm size particles and at the end of the simulation it is still clearing the gap for the most coupled particles. As stated before, this process takes longer for the outer planet which is able to open a clear gap only at the end of the simulation for all but the mm size particles where only a partial gap is barely visible.
![Dust normalised surface density distribution for the m (first panel), dm (second panel), cm (third panel), and mm-sized (fourth panel) particles disk and 2 equal mass $1\,M_\mathrm{th}$ cores at $r=1,2$ at 4 different times (50, 100, 300, 600 orbital times) for each case. \[Fig:Dust1m\]](1m1M "fig:"){width=".38\textwidth"} ![Dust normalised surface density distribution for the m (first panel), dm (second panel), cm (third panel), and mm-sized (fourth panel) particles disk and 2 equal mass $1\,M_\mathrm{th}$ cores at $r=1,2$ at 4 different times (50, 100, 300, 600 orbital times) for each case. \[Fig:Dust1m\]](1dm1M "fig:"){width=".38\textwidth"} ![Dust normalised surface density distribution for the m (first panel), dm (second panel), cm (third panel), and mm-sized (fourth panel) particles disk and 2 equal mass $1\,M_\mathrm{th}$ cores at $r=1,2$ at 4 different times (50, 100, 300, 600 orbital times) for each case. \[Fig:Dust1m\]](1cm1M "fig:"){width=".38\textwidth"} ![Dust normalised surface density distribution for the m (first panel), dm (second panel), cm (third panel), and mm-sized (fourth panel) particles disk and 2 equal mass $1\,M_\mathrm{th}$ cores at $r=1,2$ at 4 different times (50, 100, 300, 600 orbital times) for each case. \[Fig:Dust1m\]](1mm1M "fig:"){width=".38\textwidth"}
A significant fraction of particles remains in the co-orbital region with both the inner and outer planet for the all period simulated. As found for the massive core case, their density is higher in the L4 point. Also in this case, the dm size particles are the ones with the fastest dynamical evolution and it is possible to see in Figure \[Fig:Dust1m\] (third panel) that after 300 orbits the outer disk engulfs the outer planet co-orbital region, which is unable to filtrate effectively those particles.
In this case, the ring between the two planets does not clump in a small number of stable points, but it remains wide for the all simulation, though it shrinks with time and its width depends on the particle size (Figure \[Fig:SurfPart1\] -top panel).
From Figure \[Fig:Dust1m\] (first panel) is it possible to see also the formation of ripples just outside the outer planet location. This effect can be recalled from the final eccentricity distribution of the m size particles (Figure \[Fig:SurfPart1\] - bottom panel). This behaviour is due to the eccentricity excitation of particles that passes close to the planet location. For less coupled particles, the interaction with the gas takes several orbital time-steps in order to smooth out the eccentricity, thus these typical structures form. This effect is only visible in the low mass cores simulation since the particle gap is narrower, thus the particles get closer to the planet location and the excitation of their eccentricity is higher.
![Particle surface density (top panel) and eccentricity profile (bottom panel) at the end of the simulation for the different particle species.\[Fig:SurfPart1\]](1MthSigmaPart "fig:"){width=".45\textwidth"} ![Particle surface density (top panel) and eccentricity profile (bottom panel) at the end of the simulation for the different particle species.\[Fig:SurfPart1\]](1MthEccPart "fig:"){width=".45\textwidth"}
In Figure \[Fig:Gap1m\] we focused on the gap structure close to the inner planet location for the different particle sizes at the end of the simulation. We overplot also the minimum gap half-width $x_\mathrm{s}$, in order to test whether this condition is met for the most coupled particles which follow the gas dynamics. From the distribution of mm size particles we can see that the gap is opened exactly at the location of the minimum gap half-width, and there is still a lot of material in the horseshoe region. The gap is considerably wider for the cm size particles
![Particle distribution near the inner low mass planetary core location for mm (top left), cm (top right), dm (bottom left) and m (bottom right) size particles at the end of the simulation. The velocity vectors of the particles respect to the planet are shown and the colour scale shows the relative radial velocity. \[Fig:Gap1m\]](gapdust1M){width=".45\textwidth"}
Finally, in Figure \[Fig:Masstrans\] we highlighted the constant mass transfer that take place through the inner planet location for the most coupled particles which are not effectively filtered by the less massive planet. We plot the radial velocity of the particles with colour a scale in order to emphasise the flow in both directions.
![Mass transfer through the planet position of mm-size particles for the 1 thermal mass planet, which is unable to effectively filtrate them. The particles are plotted together with their velocity vectors and the colorscale indicates their radial velocities.\[Fig:Masstrans\]](Mtrans){width=".45\textwidth"}
Discussion {#sec:disc}
==========
Path to a second generation of planets {#par:plan}
--------------------------------------
The possibility to create the conditions for a second generation of planets by a massive core has already been studied in the past. However, the combined action of a multi-planetary system can achieve the same goal from less massive first generation planets. As we have seen in Sec. \[sec:res10\], two $10\,M_\mathrm{th}=0.7\,M_\mathrm{Jup}$ can trigger vortex formation which are more prominent and live longer compared to the single mass case. Nevertheless, these vortices appear like a transient effect that develop after more than $80$ orbits of the inner planet and last until $120$ orbits, thus it is difficult to relate this behavior to an initial condition effect and it does not depend on the timescale over which the planet are grown in the simulation. Furthermore, vortex formation depends on viscosity and it can persist for longer times in low viscosity disks. Although the main dust collection driver is the ring generated by the combined action of the two planets, vortices can create some characteristic observable features, enhancing locally the dust surface density. Moreover, for both the massive and intermediate mass cores, the particle ring clumps into few symmetric points close to the 5:3 MMR which are stable for several hundred orbits. Taking a dust-to-gas mass ratio of $0.01$ we found that the mass collected in those stable points can reach several Earth masses. However, we point out that this strong mass clumping might be reduced by the introduction of particle diffusion due to disk turbulence.
Comparison with Previous Work {#par:prev}
-----------------------------
This is one of the first studies on the dust evolution and filtration in a multiple planet system thus there are no direct comparison with similar setups. However, recently [@Zhu2014] have performed an extensive analysis of the dust filtration by a single planet in a 2D and 3D disk, from which we have taken some ideas and it is the natural test comparison for the different outcomes of our analysis.
One first interesting comparison between the two results is the possibility to form vortices at the gap edges, taking into account that in their scenario vortex development was favoured by the choice of a non-viscous disk. We found that, even if for single planet the vortices are hindered, the presence of an additional planet can enhance the density at the ring location and promote the development of vortices.
Moreover, we regain the ‘ripple’ formation as in @Zhu2014 for decoupled particles close to the planet location.
[@Zhu2014] found also a direct proportionality between the gap width and the planet mass, where a $9 M_\mathrm{Jup}$ induced a vortex at the gap edge at a distance more than twice the planet semi-major axis. Although we have not modelled such high mass planets we have obtained a similar outcome for our parameter space.
In our simulation we do not find the presence of strong MMR which aid the gap clearing since we do not model particles with stopping times greater than $\sim$ 5, thus the coupling with the gas disrupt the MMR . They became important only when the gas surface density is highly depleted, such as in the ring between planets for the high mass cases, where a 5:3 MMR with the inner planet is found to be a stable location.
Furthermore, as observed in [@Ayliffe2012] there is a strong correlation between particle size and particle gap, where the most coupled particles reach regions closer to the planet location, and are potentially accreted by the planet or they migrate in the inner disk, while less coupled particles are effectively filtrated by the planet.
Comparison with Observations {#par:obs}
----------------------------
Pre-transitional disks are defined observationally as disks with gaps. These features are observed in many cases in the sub-mm dust emission and there is no evidence that the gaseous emission follows the same pattern. The observation of (pre-) transitional disks highlights different physical behaviours that need to be explained. A major problem is the coexistence of a significant accretion rate onto the star (up to the same order as common T Tauri Stars - CTTS) with dust cleared zones, and the absence of near infra-red (NIR) emission. One of the most plausible explanation to solve this issue is the presence of multiple giant planets that can create a common gap and thus enhance the accretion rate across them exchanging torque with the disk, while depleting the dust component, through a filtration mechanism that, together with dust growth, can explain the absence of a strong NIR emission in (pre-) transitional disks compared to full disks [@Zhu2012]. For a full review of the topic see [@Espaillat2014]
In the HL Tau system, although several rings have been observed in its dust emission, it has still a very high accretion rate onto the star $\dot{M} = 2.13*10^{-6}M_\odot/yr$ [@Robitaille2007]. This is a prove that, at least in this system, the rings observed in the dust emission are not related to rings in the gas distribution.
Since we found that a wide ring is observed in our simulations only for the small mass case and a clear gap is visible for the outer planet only in the intermediate and massive core cases, we choose to run a different model with an inner small mass core and an outer intermediate mass core in order to compare the outcome of our simulations with the HL Tau system. We rescaled the system and run a different simulation in order to compare it to the real one, placing the inner planet at $32.3\,\mbox{au}$ corresponding to the D2 gap (see Figure \[Fig:HLTau2\] - top panel) and the outer one at $64.6\,\mbox{au}$, keeping the 2:1 ratio, which is close to the B5 location. Comparing the 2D surface density distribution at the end of the simulation with the deprojected image in the continuum emission and their slices (Figures \[Fig:HLTau2\]-\[Fig:HLTau3\]) we can outline several shared features and differences. The gap created by the inner planet has a very similar configuration as the one observed. On the other hand there are no clear visible features inside its orbit in the observed imaged while a variety of inner structures are visible in the output of the simulation. These features are mainly due to the inner wave generated by the planet. The strong gap that is visible close to the star is instead not physical and it is related to the inner boundary condition. In the outer part of the disk, several differences can be oulined. The major one is the high surface density in the horseshoe region, which is related to our choice of an initial flat profile for the particle distribution. Although this approximation was chosen to extend the simulated time preventing a fast depletion of material from the outer disk, it also favoured the dust trapping by the outer planet. Moreover, the particle ring is more depleted than in the observed image. Thus, we expect that the planetary mass responsible for the observed outer gap should be slightly smaller than the one adopted in this simulation. A final remark is the strong depletion of dust particles just inside the outer planet location due to its dust filtration mechanism, which is clear from the bottom panel of Figure \[Fig:HLTau2\]. Due to this effect, in a multiplanetary system, a planet is not necessary located where the gap is deeper but it could be at the rim of the gap preventing the particles of a certain size to cross its location.
However, a part from these differences, due to our initial choice of the parameter space, the structures obtained from the simulations are similar to what is observed.
![Top panel: deprojected image from the mm continuum of HL Tau. Bottom panel: cross-cuts at PA=$138^\circ$ through the peak of the mm continuum of HL Tau [@Partnership2014]. \[Fig:HLTau2\]](HLTau2){width=".45\textwidth"}
![Top panel: final mm-dust surface density distribution for a inner low mass core and an outer intermediate mass core. Bottom panel: relative surface density distribution. \[Fig:HLTau3\]](pappa2){width=".45\textwidth"}
Dependence on the disk surface density {#par:sta}
--------------------------------------
The dynamical evolution of dust particles is closely linked to their stopping time, which is directly related to the disk surface density through eq. \[eq:stop\], for the Epstein regime. Thus, if we decrease the disk surface density by a factor $10$ in order to stabilize the disk in the isothermal case (see Appendix \[sec::stability\] for a study of the disk stability with a more realistic equation of state), we need to lower by the same factor the particle size to reobtain the same particle dynamics. On the other hand, if we want to keep the particle’s size fixed, in order to compare our results with the ALMA continuum images, we need to decrease the planetary mass, since the gap width depends on the particle stopping time. We tried a different choice of the parameter space to obtain a similar output with a much smaller disk mass and planetary mass cores.
![Top panel: dust distribution for the mm-sized particles disk and 2 planetary cores of $\sim 10$ and $\sim 20\,M_\oplus$ after 250 orbits of the inner planet. Bottom panel: Relative surface density distribution (red solid curve), where the distribution of the higher disk mass case (from Figure \[Fig:HLTau3\] has been overplotted (dashed blue curve). \[Fig:HLTau4\]](pappa1){width=".45\textwidth"}
We report in Fig. \[Fig:HLTau4\] a run with a disk mass of $1/10$ of the test case. The width of the gap created by two planets is similar, as outlined by the bottom panel. However, in this case the planetary mass adopted to open such narrow gaps in the particle disk are much lower: $\sim 10$ and $\sim 20\,M_\oplus$ for the inner and ouer planet, respectively. These lower values of the planetary masses prove that the ability of planets to open gaps in the dust disk is widely applicable, and increase the likelihood of the planetary origin through core accretion in this young system at large radii ($\sim 60\,\mbox{au}$). It will be crucial to define better the disk mass in order to constrain the particle dynamics and the planetary mass growing inside their birth disk.
Conclusion {#sec:conc}
==========
We have implemented a population of dust particles into the 2D hydro code <span style="font-variant:small-caps;">fargo</span> [@Masset2000] in order to study the coupled dynamic of dust and gas. The dust is modelled through Lagrangian particles, which permit us to cover the evolution of both small dust grains and large bodies within the same framework. We have studied in particular the dust filtration in a multi-planetary system to obtain some observable features that can be used to interpret the observations made by modern infrared facilities like ALMA . From the analysis of our simulations we have found that the outer planet
- affects the co-orbital region of the inner one exciting the particles in the Lagrangian points (L4 and L5), which are effectively removed in the majority of the cases,
- increases the surface density in the region between them, creating a particle ring which can clump in a small number of symmetric points, collecting a mass up to several Earth masses,
- promotes the development of vortices at the ring inner edge, increasing the steepness of the surface density profile.
Moreover, when the planets are not massive enough to create a narrow particle ring between the planets, its width depends on the particle size. This could be a potentially observable feature that can link the ring formation with the presence of planets. Furthermore, we confirmed previous results regarding the particle gap, which develops much more quickly than the gaseous one, and is wider for higher mass planets and more decoupled particles.
The features observed in the HL Tau system can be explained through the presence of several massive cores, or lower mass cores depending on the adopted surface density, that shape the dust disk. We have obtained that the inner planet(s) should be on the order of $1\,M_\mathrm{th}=0.07\,M_\mathrm{Jup}$, in order to open a small gap in the dust disk while keeping a wide particle ring. The outer one(s) should have a mass on the order of $5\,M_\mathrm{th}=0.35\,M_\mathrm{Jup}$ in order to open a visible gap. These values are in agreement with those found by [@Kanagawa2015; @DiPierro2015; @Dong2015]. We point out that, decreasing the disk surface density by a factor 10 reduces the required planetary mass to open the observed gaps to a value of $10\,M_\oplus$ and $20\,M_\oplus$ respectively. These reduced values render the planet formation through core accretion more reliable in the young HL Tau system. Although the particle gaps observed are prominent, the expected gaseous gaps would be barely visible.
The limitations of this work are the lack of particle back-reaction on the gas, self-gravity of the disk, and particle diffusion. Furthermore we have not model accretion of particles onto the planet and planet migration. These approximation were chosen in order to study the global evolution of particle distribution with different stopping times and different planet masses, without increasing excessively the computation time. Although the disk is very massive, the asymmetric features typical of a gravitationally unstable disk are not observed in the continuum mm observations that should correctly describe the gas flow, thus we do not expect the real system to be subject to strong perturbations due to its self-gravity. The particle-back reaction plays an important factor when studying the evolution of particle clumps, but it is not expected to change significantly the global dust distribution. However, particle diffusion could have an important role both in reducing the dust migration and preventing strong clumping of particles. We are planning in future works to relax these approximations, running more accurate simulations and testing the contribution of the individual physical process on the final dust filtration and distribution. Moreover, we have limited our analysis to the peculiar case of equal mass planets on a fixed orbit, and changing each one of these conditions can result in a rather different outcome. The possibility to evolve the simulations further in time has been considered since the outer planets where not able to open up a clear gap for the less massive cases. However, in any case they are able to open only a very shallow gap, so the possible influence on the subsequent evolution of the particles close to their position is not expected to be significant. Furthermore, since we have modeled the system for $\sim 10^5\,\mbox{yr}$ around a young star ($\sim 10^6\,\mbox{yr}$), and we need to form the planetary cores in the first place, it does not seem unrealistic to observe a planet at $\sim 60\,\mbox{au}$ still in the gap clearing phase.
We thank an anonymous referee for his useful comments and suggestions. G. Picogna acknowledges the support through the German Research Foundation (DFG) grant KL 650/21 within the collaborative research program ”The first 10 Million Years of the Solar System”. Some simulations were performed on the bwGRiD cluster in Tübingen, which is funded by the Ministry for Education and Research of Germany and the Ministry for Science, Research and Arts of the state Baden-Württemberg, and the cluster of the Forschergruppe FOR 759 ”The Formation of Planets: The Critical First Growth Phase” funded by the DFG.
Disk stability {#sec::stability}
==============
The disk parameters adopted in this work were selected in order to match the observational data [@Kwon2011]. We point out that the disc to star mass ratio is relatively high and, in the locally isothermal approximation, it is gravitationally unstable in the outer regions considering its Toomre parameter $$Q = \frac{h}{r}\frac{M_\star}{M_d}\frac{2(r_{out}-r_{in})}{r} \simeq
\frac{1.6}{r}.$$ However, the isothermal equation of state is usually a poor representation of the temperature distribution in the disk, especially for the inner regions of protoplanetary disks around young stars. In order to validate this model we ran an additional hydro simulation in which we include a more realistic equation of state where the radiative transport and radiative cooling are included. From Fig. \[Fig:sg\] it is clear that the more appropriate equation of state prevent the disk to become even partially unstable.
![Disk surface density after 20 inner planetary orbits for the isothermal case (top panel) and fully radiative case (bottom panel) where the self-gravity of the disk has been considered. \[Fig:sg\]](sgi "fig:"){width=".45\textwidth"} ![Disk surface density after 20 inner planetary orbits for the isothermal case (top panel) and fully radiative case (bottom panel) where the self-gravity of the disk has been considered. \[Fig:sg\]](sgr "fig:"){width=".45\textwidth"}
The different equation of state could in principle affect the gap opening timescale, however the long term evolution of the particle dynamics is not expected to vary significantly. Moreover, the important parameter in our study is the stopping time of the particles, so the results obtained remain valid for different values of the surface density profile, scaling accordingly the dust size, as discussed in Section \[par:sta\].
Integrators {#sec::integrators}
===========
In order to model the dynamics of the particle population in our simulations we tried different integrators.
Semi-implicit integrator in polar coordinates
---------------------------------------------
In order to follow the dynamics of particles well-coupled with the gas, which have a stopping time much smaller than the time step adopted to evolve the gas dynamics, we adopted the semi-implicit Leapfrog (Drift-Kick-Drift) integrator described in @Zhu2014 in polar coordinates. This method guarantees the conservation of the physical quantities for the long term simulations performed in this paper, and at the same time it is faster than an explicit method.
#### Scheme
Half Drift: $$\begin{aligned}
v_{\mathrm{R},n+1} &= v_{\mathrm{R},n} \\
l_{n+1} &= l_n \\
R_{n+1} &= R_n + v_{\mathrm{R},n}\frac{\mathrm{d}t}{2} \\
\phi_{n+1} &= \phi_{n}+\frac{1}{2}\left(
\frac{l_n}{R_n^2}+\frac{l_{n+1}}
{R_{n+1}^2}\right) \frac{\mathrm{d}t}{2}
\end{aligned}$$
Kick: $$\begin{aligned}
R_{n+2} = &\ R_{n+1} \\
\phi_{n+2} = &\ \phi_{n+1} \\
l_{n+2} = &\ l_{n+1} +
\frac{dt}{1+\frac{\mathrm{d}t}{2t_{\mathrm{s},n+1}}}
\left[-{\left(\frac{\partial\Phi}{\partial\phi}\right)}_{n+1} +
\frac{v_{\mathrm{g,\phi},n+1}R_{n+1}-l_{n+1}}{t_{\mathrm{s},n+1}}
\right] \\
v_{\mathrm{R},n+2} = &\ v_{\mathrm{R},n+1} +
\frac{dt}{1+\frac{\mathrm{d}t}{2t_{\mathrm{s},n+1}}}
\Bigg[
\frac{1}{2}\Bigg(
\frac{l_{n+1}^2}{R_{n+1}^3}+
\frac{l_{n+2}^2}{R_{n+2}^3}
\Bigg)-
\Bigg(
\frac{\partial\Phi}{\partial R}
\Bigg)_{n+1} + \\
&\ +\frac{v_{\mathrm{g,R},n+1}-v_{\mathrm{R},n+1}}{t_{\mathrm{s},n+1}}
\Bigg]
\end{aligned}$$
Half Drift: $$\begin{aligned}
v_{\mathrm{R},n+3} &= v_{\mathrm{R},n+2} \\
l_{n+3} &= l_{n+2} \\
R_{n+3} &= R_{n+2} + v_{\mathrm{R},n+3}\frac{\mathrm{d}t}{2} \\
\phi_{n+3} &= \phi_{n+2}+\frac{1}{2}\left(
\frac{l_{n+2}}{R_{n+2}^2}+\frac{l_{n+3}}
{R_{n+3}^2}\right) \frac{\mathrm{d}t}{2}
\end{aligned}$$
where $v_{\mathrm{R}}$ is the radial velocity, $l$ the angular momentum, $R$ the cylindrical radius, and $\phi$ the polar angular coordinate. The index $n$ shows the step at which the various quantities are considered. Further information reguarding to the integrator can be found in @Zhu2014.
Fully-implicit integrator in polar coordinates
----------------------------------------------
For particles with stopping time much smaller than the numerical time step, the drag term can dominate the gravitational force term, causing the numerical instability of the integrator. Thus, it is necessary to adopt a fully implicit integrator following @Bai2010a [@Zhu2014]
#### Scheme
Predictor step: $$\begin{aligned}
R_{n+1} &= R_n + v_{\mathrm{R},n} \mathrm{d}t \\
\phi_{n+1} &= \phi_n+\frac{l_n}{R_n^2}
\mathrm{d}t
\end{aligned}$$
Shift: $$\begin{aligned}
v_{\mathrm{R},n+1} &= v_{\mathrm{R},n} +
\frac{{{\mathop{}\!\mathrm{d}}t}/2}
{1+{\mathop{}\!\mathrm{d}}t{\left(
\frac{1}{2t_{\mathrm{s},n}} +
\frac{1}{2t_{\mathrm{s},n+1}} +
\frac{{\mathop{}\!\mathrm{d}}t}{2t_{\mathrm{s},n}t_{\mathrm{s},n+1}}
\right)}}\cdot \\
& \cdot \Bigg[
-{\left(\frac{\partial\Phi}{\partial R}\right)}_n
-\frac{v_{\mathrm{R},n}-v_{\mathrm{g,R},n}}{t_{\mathrm{s},n}}
+\frac{l_n^2}{R_n^3} +
\Bigg(
-{\left(\frac{\partial\Phi}{\partial R}\right)}_{n+1}+
\nonumber \\
&-\frac{v_{\mathrm{R},n}-
v_{\mathrm{g,R},n+1}}{t_{\mathrm{s},n+1}}
+\frac{l_{n+1}^2}{R_{n+1}^3}
\Bigg)
{\left(1+\frac{{\mathop{}\!\mathrm{d}}t}{t_{\mathrm{s},n}}\right)}
\Bigg] \nonumber \\
l_{n+1} &= l_{n} +
\frac{{{\mathop{}\!\mathrm{d}}t}/2}
{1+{\mathop{}\!\mathrm{d}}t\left(\frac{1}{2t_\mathrm{s,n}}+\frac{1}
{2t_{\mathrm{s},n+1}}+
\frac{{\mathop{}\!\mathrm{d}}t}{2t_{\mathrm{s},n}t_{\mathrm{s},n+1}}\right)}\cdot \\
&\cdot\Bigg[
-{\left(\frac{\partial\Phi}{\partial \phi}\right)}_n
-\frac{l_n - R_n v_{\mathrm{g,\phi},n}}{t_{\mathrm{s},n}}
+\Bigg(
-{\left(\frac{\partial\Phi}{\partial \phi}
\right)}_{n+1} + \nonumber \\
&-\frac{l_n - R_{n+1} v_{\mathrm{g,\phi},n+1}}
{t_{\mathrm{s},n+1}}
\Bigg)
{\left(1+\frac{{\mathop{}\!\mathrm{d}}t}{t_{\mathrm{s},n}}\right)}
\Bigg] \nonumber
\end{aligned}$$
Corrector step: $$\begin{aligned}
R_{n+1} &= R_n + \frac{1}{2}(v_{\mathrm{R},n}+
v_{\mathrm{R},n+1})\mathrm{d}t \\
\phi_{n+1} &= \phi_{n} + \frac{1}{2}\left(
\frac{l_{n}}{R_{n}^2}+\frac{l_{n+1}}
{R_{n+1}^2}\right) \mathrm{d}t
\end{aligned}$$
Particle tests {#sec::partests}
==============
In order to test the numerical integrators described in Sec. \[sec::integrators\], we did an orbital test and a drift test proposed by @Zhu2014.
Orbital tests
-------------
We release one dust particle at $r=1$, $\phi=0$, with $v_\phi=0.7$, and integrate it for $20$ orbits. The time-steps $\Delta t$ are varied between $0.1$ and $0.01$ in units of the orbital time. The results are shown in Figure \[orbital\]. The particle follows an eccentric orbit with $e = 0.51$. The time steps are $\Delta t = 0.1$, compared with the orbital time ($2\pi$). The precession observed is due to the fact that even symplectic integrators cannot simultaneously preserve angulat momentum and energy exactly. The advantage of the semi-implicit scheme is that it does preserve geometric properties of the orbits, while the fully implicit integrator does not. For comparison, the orbit is calculated also with an explicit integrator, but with a much smaller time step $\Delta t=0.01$, showing no visible precession. This behavior is recovered also with the implicit schemes reducing the timestep. Since $\Delta t = 0.01$ is normally the time step used in our planet-disk simulations, our integrators are quite accurate even if we integrate the orbit of particles having moderate eccentricity.
![Orbital evolution of a dust particle released at $r=1$, $\phi=0$, with $v_\phi=0.7$ for the different integrators adopted in the simulations (red and green curves), compared to the solution from an explicit integrator (black curve). \[orbital\]](orbital){width=".45\textwidth"}
2D drift tests
--------------
We model a 2D gaseous disk in hydrostatic equilibrium with $\Sigma\propto r^{-1}$ and release particles with different stopping times from $r=1$. The radial domain is $[0.5, 3]$ with a resolution in the radial direction of $400$ cells. The drift speed at the equilibrium is given by [@Nakagawa1986] $$\label{eq:drifteq}
v_\mathrm{R,d}=\frac{\tau_\mathrm{s}^{-1}v_\mathrm{R,g}-\eta
v_\mathrm{K}}{\tau_\mathrm{s}+\tau_\mathrm{s}^{-1}}$$ where $v_\mathrm{R,g}$ is the gas radial velocity, $\eta$ is the ratio of the gas pressure gradient to the stellar gravity in the radial direction, and we consider $v_\mathrm{R,g}=0$ since we are at the equilibrium.
![Evolution of the particle drift speed in the first 10 orbits for particles with different stopping times. The analytic solution obtained from eq. (\[eq:drifteq\]) is plotted with a black line, while the drift speed obtained from the semi-implicit and fully-implicit integrators are displayed with red and grey lines respectively. \[orbital2\]](Testvrad){width=".45\textwidth"}
Figure \[orbital2\] shows the evolution of the particle radial velocity in the first 10 orbits for the implicit (red) and semi-implicit (grey) integrators together with the analytic solution (black) obtained from eq. (\[eq:drifteq\]). The particle drift speed reaches almost immediately the expected drift speed. The semi-implicit integrator reaches the equilibrium speed on a longer timescale than the fully-implicit one only for the lower stopping time (smaller particles).
|
---
abstract: 'We obtain a double exponential bound in Brauer’s generalisation of van der Waerden’s theorem, which concerns progressions monochromatic with their common difference. Such a result has been obtained independently and in much greater generality by Sanders. Using Gowers’ local inverse theorem, our bound is quintuple exponential in the length of the progression. We refine this bound in the colour aspect for three-term progressions, and combine our arguments with an insight of Lefmann to obtain analogous bounds for the Ramsey numbers of certain nonlinear quadratic equations.'
address: |
School of Mathematics\
University of Manchester\
UK
author:
- Jonathan Chapman
- Sean Prendiville
title: On the Ramsey number of the Brauer Configuration
---
Introduction
============
Schur’s theorem states that in any partition of the positive integers into finitely many pieces, at least one part contains a solution to the equation $x+y = z$. By a theorem of van der Waerden, the same is true for the equation of three-term arithmetic progressions $x+y = 2z$. A common generalisation of these theorems due to Brauer states that, in any finite colouring of the positive integers, there is a monochromatic arithmetic progression of length $k$ which receives the same colour as its common difference.
There is a finitary analogue of these results, asserting that the same holds for colourings of the interval ${\left\{1, 2, \dots, N\right\}}$, provided that $N$ is sufficiently large in terms of the number of colours, as well as the coefficients of the configuration in question. Determining the minimal such number $N$ (the *Ramsey* or *Rado* number of the system) has received much attention for arithmetic progressions, and a celebrated breakthrough of Shelah [@ShelahPrimitive] showed that these numbers (*van der Waerden numbers*) are primitive recursive. One spectacular consequence of Gowers’ work on Szemerédi’s theorem [@GowersSzem] is a bound on van der Waerden numbers which is quintuple exponential in terms of the length of the progression, and double exponential in terms of the number of pieces of the partition.
Using Gowers’ local inverse theorem for the uniformity norms, we obtain a bound for the Ramsey number of the Brauer configuration which is comparable to that obtained for arithmetic progressions.
\[weak brauer theorem\] There exists an absolute constant $C=C(k)$ such that if $r \geqslant 2$ and $N\geqslant \exp\exp(r^{C})$, then any $r$-colouring of $\{1,2,...,N\}$ yields a monochromatic $k$-term progression which is the same colour as its common difference. Moreover, one may ensure that it suffices to assume that $$\label{mainBound}
N\geqslant 2^{2^{r^{2^{2^{k+10}}}}}.$$
A double exponential bound has been obtained independently and in maximal generality by Sanders [@SandersBootstrapping], who bounds the Ramsey number of an *arbitrary* system of linear equations with this colouring property, often termed partition regularity. Our results and those of Sanders are the first quantitatively effective bounds for configurations lacking translation invariance and of ‘true complexity’ [@GowersWolfTrue] greater than one.
Gowers [@GowersSzem] obtains the bound for progressions of length $k+1$, this being the appropriate analogue of the $(k+1)$-point Brauer configuration in Theorem \[weak brauer theorem\]. For a four-point Brauer configuration, we improve the exponent of $r$ on combining our method with an energy-increment argument of Green and Tao [@GT09].
\[improved bound\] There exists an absolute constant $C$ such that if $N\geqslant \exp\exp(Cr\log^2r)$, then in any $r$-colouring of ${\left\{1,2, \dots, N\right\}}$ there exists a monochromatic three-term progression whose common difference receives the same colour.
Due to an insight of Lefmann [@lefmann] we are able to use (a variant of) Theorem \[weak brauer theorem\] to bound the Ramsey number of certain partition regular *nonlinear* equations.
\[intro lefmann\] Let $a_1, \dots, a_s \in {\mathbb{Z}}\setminus{\left\{0\right\}}$ satisfy the following:
1. there exists a non-empty set $I\subset[s]$ such that $\sum_{i\in I}a_{i}=0$;
2. the system $$x_0^{2}\sum_{i\notin I}a_{i}+ \sum_{i\in I}a_{i}x_{i}^{2}=
\sum_{i\in I}a_{i}x_{i}=0.$$ has a rational solution with $x_0 \neq 0$.
Then there exists an absolute constant $C = C(a_1, \dots, a_s)$ such that for $r {\geqslant}2$ and $N {\geqslant}\exp \exp (r^C)$, any $r$-colouring of ${\left\{1, 2, \dots, N\right\}}$ yields a monochromatic solution to the diagonal quadric $$a_1 x_1^2 + \dots + a_s x_s^2 = 0.$$
Previous work {#previous-work .unnumbered}
-------------
Hitherto, little is recorded regarding the Ramsey number of general partition regular systems. Cwalina–Schoen [@CwalinaSchoen] observe that one can use Gowers’ bounds [@GowersSzem] in Szemerédi’s theorem to obtain a bound which is tower in nature, of height proportional to $5r$. Gowers’ methods are well suited to delivering double exponential bounds for so-called translation invariant systems (such as arithmetic progressions), but such systems are far from typical. In Cwalina–Schoen [@CwalinaSchoen], Fourier-analytic arguments are adapted to give an exponential bound on the Ramsey number of a single partition regular equation. The first author [@chapman] has shown how multiplicatively syndetic sets allow one to reduce the tower height to $(1+o(1))r$ for the four-point Brauer configuration $$\label{four point}
x,\ x+d,\ x+2d,\ d.$$
Our method {#our-method .unnumbered}
----------
The approach underlying Theorem \[weak brauer theorem\] generalises that of Roth [@roth] and Gowers [@GowersSzem]. Given a subset $A$ of $[N] := {\left\{1, 2, \dots, N\right\}}$ of density $\delta$, Roth uses a density increment procedure to locate a subprogression $P = a + q\cdot [M]$ of length $M {\geqslant}N^{\exp(-C/\delta)}$ where $A\cap P$ has density at least $\delta$ and is ‘Fourier uniform’, in the sense that all but its trivial Fourier coefficients are small. An application of Fourier analysis (in the form of the circle method) shows that such sets possess of order $\delta^3 M^2$ three-term progressions. This yields a non-trivial three-term progression provided that $N {\geqslant}\exp\exp(C/\delta)$.
The above application of the circle method relies crucially on the translation-dilation invariance of three-term progressions, so that the number of configurations in $P$ is the same as that in $[M]$. Unfortunately, the Brauer configuration is not translation invariant. To overcome the lack of translation-invariance, given a colouring $A_1 \cup \dots \cup A_r = [N]$, we use Gowers’ local inverse theorem for the uniformity norms [@GowersSzem] to run a density increment procedure with respect to the *maximal translate density* $$\sum_{i=1}^r \max_a\frac{|A_i\cap (a + q\cdot [M])|}{M}.$$ This outputs (see Lemma \[thmUniBrau\]) a homogeneous progression $q \cdot [M]$ such that for each colour class $A_i$ there is a translate $a_i + q\cdot [M]$ on which $A_i$ achieves its maximal translate density and on which $A_i$ is suitably uniform. For the four-point Brauer configuration , the correct notion of uniformity is *quadratic uniformity* (as measured by the Gowers $U^3$-norm).
Write $\alpha_i$ for the density of $A_i$ on the maximal translate $a_i + q\cdot[M]$ and $\beta_i$ for its density on the homogeneous progression $q\cdot [M]$. An application of quadratic Fourier analysis shows that the number of four-point Brauer configurations satisfying $${\left\{x, x+d, x+2d\right\}} \subset A_i \cap (a_i + q\cdot [M]) \quad \text{and}\quad d \in A_i \cap q\cdot [M]$$ is of order $\alpha_i^3\beta_i M^2$. By the pigeonhole principle, some colour class has $\beta_i {\geqslant}1/r$, which also implies that $\alpha_i {\geqslant}1/r$, so we deduce that some colour class contains at least $r^{-4} M^2$ Brauer configurations. Unravelling the quantitative dependence in our density increment then yields a double exponential bound on $N$ in terms of $r$.
In §\[secSchurFF\] we give a more detailed exposition of this method for the model problem of Schur’s theorem in the finite vector space ${\mathbb{F}}_2^n$. In §\[general\] we generalise the argument to arbitrarily long Brauer configurations over the integers. We improve our bound for four-point Brauer configurations in §\[four point sec\]. Finally, in §\[lefmann sec\] we show how our methods give comparable bounds for the Ramsey number of certain quadratic equations.
Notation {#notation .unnumbered}
--------
The set of positive integers is denoted by ${\mathbb{N}}$. Given $x\geqslant 1$, we write $[x]:=\{1,2,...,{\left\lfloor x \right\rfloor}\}$. If $f$ and $g$ are functions, and $g$ takes only positive values, then we use the Vinogradov notation $f\ll g$ if there exists an absolute positive constant $C$ such that $|f(x)|\leqslant Cg(x)$ for all $x$. We also write $g\gg f$ or $f=O(g)$ to denote this same property. The letters $C$ and $c$ are used to denote absolute constants, whose values may change from line to line. Typically $C$ denotes a large constant $C>1$, whilst $c$ denotes a small constant $0<c<1$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank Tom Sanders for alerting us to the existence of [@SandersBootstrapping], and for his generosity in synchronising release. The second author thanks Ben Green for suggesting this problem.
Schur in the finite field model {#secSchurFF}
===============================
We illustrate the key ideas of our approach in proving Schur’s theorem over ${\mathbb{F}}_{2}^{n}$. This asserts that, provided the dimension $n$ is sufficiently large relative to the number of colours $r$, any partition ${\mathbb{F}}_{2}^{n}=A_{1}\cup\cdots\cup A_{r}$ possesses a colour class $A_i$ containing vectors $x,y,z$ with $y \neq 0$ and such that $x+y=z$. The goal of this section is to obtain a quantitative bound on the dimension $n$ in terms of $r$.
The argument of this section is purely expository, the resulting bound being slightly worse than that given by a standard application of Ramsey’s theorem (see [@ramseytheory §3.1]) or Schur’s original argument (see [@CwalinaSchoen]). We have since learned that the same ideas are discussed in Shkredov [@Shkredov §5].
\[model schur theorem\] Consider a partition of ${\mathbb{F}}_{2}^{n}$ into $r$ sets $A_{1},...,A_{r}$. If $n$ satisfies $$n> \sqrt{2}\, r^{3}+\log_2(2r),$$ then there exists $i\in[r]$ and $x,y,z\in A_{i}$ with $y\neq 0$ such that $x+y = z$.
One can guarantee that the $x,y,z\in A_{i}$ that are obtained are distinct and non-zero by proceeding as follows. We greedily partition each set $A_{i}\setminus\{0\}$ into two pieces $A^{+}_{i}$ and $A^{-}_{i}$ in such a way that neither $A^{+}_{i}$ nor $A^{-}_{i}$ contain a solution to the equation $x+y=0$. By introducing $A_{0}:=\{0\}$, we obtain a new partition of ${\mathbb{F}}_{2}^{n}$ into $2r+1$ pieces. Applying the above theorem (with $r$ replaced with $2r+1$) to this new partition gives distinct non-zero $x,y,z\in A_{i}$ for some $i$ satisfying $x+y=z$.
Inspired by Cwalina–Schoen [@CwalinaSchoen], we deduce Theorem \[model schur theorem\] from the following dichotomy. This argument is a variant of Sanders’s ‘$99$% Bogolyubov theorem’ [@SandersAdditive], which asserts that the difference set of a dense set contains $99$% of a subspace of bounded codimension.
\[lemSchurDich\] Let $A_1\cup \dots \cup A_r = {\mathbb{F}}_2^n$ be a partition of ${\mathbb{F}}_{2}^{n}$ into $r$ parts. Then there exists a subspace $H\leqslant{\mathbb{F}}_{2}^{n}$ with ${\mathrm{codim}\,}(H) \leqslant \sqrt{2}r^3$ such that for any $i \in [r]$ we have one of the two following possibilities.
- (Sparsity). $$\label{eqnSchurSparse}
|A_i \cap H| < {\tfrac{1}{r}}|H|;$$
- (Expansion). $$\label{eqnSchurExp}
|(A_i - A_i) \cap H| {\geqslant}{\left( 1 - {\tfrac{1}{2r}} \right)} |H|.$$
The idea is that, as one of the colour classes $A_i$ is dense, its difference set $A_i - A_i$ must (by Sanders’ result) contain 99% of a ‘large’ subspace $H$. Were $A_i$ itself to contain more that 1% of this subspace, then we would be done, since then $(A_i - A_i)\cap A_i \neq \emptyset$ and we would obtain the desired Schur triple $x,y,z\in A_{i}$. Unfortunately, this cannot always be guaranteed: consider the case in which $A_i$ is a non-trivial coset of a subspace of co-dimension 1.
To overcome this, we run Sanders’ proof with respect to all of the colour classes simultaneously, constructing a subspace $H$ which is almost covered by $A_i - A_i$ for *all* $i\in[r]$. If such a $H$ were obtainable we would be done as before, since (by the pigeonhole principle) some colour class has large density on $H$. Again, this is slightly too much to hope for, as sets which are ‘hereditarily sparse’ cannot be good candidates for a 99% Bogolyubov theorem. Fortunately, such sets can be accounted for in our argument.
Before we proceed to the proof of Lemma \[lemSchurDich\], let us use this lemma to prove our finite field model of Schur’s theorem.
Let $H$ denote the subspace provided by the dichotomy. By the pigeonhole principle, there exists some $A_i$ satisfying $$|A_i\cap H| \geqslant {\tfrac{1}{r}}|H|.$$ Our assumption on the size of $n$ then implies that $$|A_i \cap H| > {\tfrac{1}{2r}}|H| + 1.$$
Since $A_{i}$ is not sparse on $H$, in the sense of , it must instead satisfy the expansion property . By inclusion–exclusion $$|A_i\cap (A_i-A_i)\cap H| \geqslant |A_i \cap H| + |(A_i-A_i)\cap H| - |H| >1.$$ In particular, the set $A_i\cap (A_i-A_i)$ contains a non-zero element.
A maximal translate increment strategy
--------------------------------------
It remains to prove Lemma \[lemSchurDich\]. Following Sanders [@SandersAdditive], we accomplish this via density increment. We cannot merely increment the density of each individual colour class on translates of different subspaces, since our final dichotomy involves a single subspace $H$ which is uniform for all $A_i$. We therefore have to increment a more subtle notion of density, namely the *maximal translate density* $$\Delta_{H}=\Delta_H(A_1, \dots, A_r) := \sum_{i=1}^r \max_x \frac{|A_i \cap (x+H)|}{|H|}.$$ This is a non-negative quantity bounded above by $r$. It follows that a procedure passing to subspaces $H_0 {\geqslant}H_1 {\geqslant}H_2 {\geqslant}\dots$, which increments $\Delta_{H_i}$ by a constant amount at each iteration, must terminate in a constant number of steps (depending on $r$).
We first observe that to increment $\Delta_{H}$ it suffices to find a subspace where one of the colour classes increases their maximal translate density. To this end, write $$\label{maximal translate density}
\delta_H(A) := \max_x \frac{|A \cap (x+H)|}{|H|}.$$
\[subspace preservation lemma\] Let $H_1 {\geqslant}H_2$ be subspaces of ${\mathbb{F}}_{2}^{n}$. Then for any $A\subset{\mathbb{F}}_{2}^{n}$ we have $$\delta_{H_2}(A) \geqslant \delta_{H_1}(A).$$
As $H_{2}{\leqslant}H_{1}$, we can write $H_{1}$ as a disjoint union of cosets of $H_{2}$. This means that we can find $V\subset H_{1}$ such that $H_1 = \sqcup_{y \in V} (y + H_2)$. Hence $$|A \cap (x + H_1)| = \sum_{y \in V} |A \cap (x+y+H_2)| \leqslant |V| \max_z |A \cap (z+H_2)|.$$ Choosing $x$ so that $A$ has maximal density on $x+H_1$ gives the result.
We now prove Lemma \[lemSchurDich\] using a density increment strategy for the maximal translate density. The argument proceeds by showing that if our claimed dichotomy does not hold, then we may pass to a subspace on which the colour classes have larger maximal translate density.
The process of identifying such a subspace involves the use of Fourier analysis. Given a subspace $H{\leqslant}{\mathbb{F}}_{2}^{n}$ and a function $f:H\to{\mathbb{C}}$, we define the *Fourier transform* $\hat{f}:\hat{H}\to{\mathbb{C}}$ of $f$ by $$\hat{f}(\gamma):=\sum_{x\in H}f(x)\gamma(x).$$ Here $\hat{H}$ denotes the *dual group* of $H$, which is the group of homomorphisms $\gamma:H\to{\mathbb{C}}^\times$. Observe that, for all $x\in H$ and $\gamma\in\hat{H}$, the value $\gamma(x)$ must be $\pm 1$.
We proceed by an iterative procedure, at each stage of which we have a subspace $H = H^{(m)} {\leqslant}{\mathbb{F}}_{2}^n$ of codimension $m$ satisfying $$\Delta_{H^{(m)}} \geqslant \frac{m}{\sqrt{2}r^2}.$$ We initiate this procedure on taking $H^{(0)} := {\mathbb{F}}_{2}^n$. Since $\Delta_{H^{(m)}} \leqslant r$ this procedure must terminate at some $m \leqslant \sqrt{2}r^3$.
Given $H = H^{(m)}$ we define three types of colour class.
- (Sparse colours). $A_i$ is *sparse* if $$\delta_H(A_i) < {\tfrac{1}{r}};$$
- (Dense expanding colours). $A_i$ is *dense expanding* if $\delta_H(A_i) \geqslant {\tfrac{1}{r}}$ and we have the expansion estimate $$\label{schur non expansion}
|(A_i - A_i) \cap H| > {\left( 1 - {\tfrac{1}{2r}} \right)} |H|;$$
- (Dense non-expanding colours). $A_i$ is *dense non-expanding* if it is neither sparse nor dense expanding.
If there are no dense non-expanding colour classes, then the dichotomy claimed in our lemma is satisfied, and we terminate our procedure. Let us show how the existence of a dense non-expanding colour class $A_i$ allows the iteration to continue.
By the definition of maximal translate density, there exists $t$ such that $$|A_i \cap (t + H)| = \delta_H(A_i)|H|.$$ We define dense subsets $A, B\subset H$ by taking $$\label{schur A B defn}
A := (A_i - t)\cap H \quad \text{and} \quad B := H \setminus {\bigl( A_i - A_i \bigr)}.$$ Writing $\alpha$ and $\beta$ for the respective densities of $A$ and $B$ in $H$, our dense non-expanding assumption implies that $\alpha {\geqslant}1/r$ and $ \beta \geqslant 1/(2r)$. Moreover, it follows from our construction that $$\sum_{x -x' = y} 1_A(x) 1_A(x') 1_B(y) = 0.$$ Comparing this to the count $$\sum_{x -x' = y} \alpha 1_H(x) 1_A(x') 1_B(y) = \alpha^2\beta |H|^2,$$ we deduce, on writing $f_{A}:=1_A - \alpha 1_H$, that $${\Bigl| {\mathbb{E}}_{\gamma \in \hat{H}} \overline{\hat{f}_A(\gamma)} \hat{1}_A(\gamma) \hat{1}_B(\gamma) \Bigr|} = {\biggl| \sum_{x -x' = y} f_A(x) 1_A(x') 1_B(y) \biggr|} \geqslant \alpha^2 \beta |H|^2.$$
By Cauchy–Schwarz and Parseval’s identity, there exists $\gamma \neq 1_H$ such that $${\Bigl| \sum_{x \in H} f_A(x) \gamma(x) \Bigr|} \geqslant \frac{\alpha^2\beta}{\sqrt{\alpha\beta}} |H| \geqslant \frac{|H|}{\sqrt{2}r^2}.$$ Partitioning $H$ into level sets of $\gamma$, gives $${\Bigl| \sum_{ \gamma(x) = 1} f_A(x) \Bigr|} + {\Bigl| \sum_{ \gamma(x) = -1} f_A(x) \Bigr|} \geqslant \frac{|H|}{\sqrt{2}r^2}.$$ Since $f_A$ has mean zero, we deduce that the two terms on the left of the above inequality are equal. This implies that there exists $\omega \in{\left\{ \pm 1\right\}}$ such that $$2\sum_{\gamma(x) = \omega} f_A(x) \geqslant \frac{|H|}{\sqrt{2}r^2}.$$
Observe that $H' := {\left\{x \in H : \gamma(x) = 1\right\}}$ is a subspace of $H$ of dimension $|H|-1$. Hence on choosing $y\in H$ with $\gamma(y)=\omega$ we have $$\frac{|A \cap (y+H')|}{|H'|} \geqslant \alpha + \frac{1}{\sqrt{2}r^2}.$$ By combining this with Lemma \[subspace preservation lemma\] and our definition of $A$, we deduce that $$\Delta_{H'} \geqslant \Delta_H + \tfrac{1}{\sqrt{2}r^2} \qquad \text{and} \qquad {\mathrm{codim}\,}(H') = {\mathrm{codim}\,}(H) + 1.$$ We have therefore established that our iteration may continue, completing the proof of the lemma.
Brauer configurations over the integers {#general}
=======================================
In this section we use higher order Fourier analysis to study longer Brauer configurations and prove Theorem \[weak brauer theorem\]. Henceforth, we fix the parameter $k\geqslant 2$ to denote the length of the progression in the Brauer configuration under consideration. We emphasise that this section streamlines substantially if the reader is only interested in a double exponential bound in the colour aspect, as opposed to the more explicit bound .
Given finitely supported $f_{1},f_{2},...,f_{k},g:{\mathbb{Z}}\to{\mathbb{R}}$ we introduce the counting operator $$\label{counting op}
\Lambda(f_{1},f_{2},...,f_{k};g):=\sum_{d,x\in{\mathbb{Z}}}f_{1}(x)f_{2}(x+d)\cdots f_{k}(x+(k-1)d)g(d).$$ For brevity, write $\Lambda(f;g):=\Lambda(f,f,...,f;g)$. For given finite sets $A,B\subset{\mathbb{N}}$, the number of arithmetic progressions of length $k$ in $A$ with common difference in $B$ is given by $\Lambda(1_{A};1_{B})$.
\[propBrauCount\] Let $M\in{\mathbb{N}}$ with $M\geqslant k$. If $B\subset[M/(2k-2)]$, then $$\Lambda(1_{[M]};1_{B})\geqslant{\tfrac{1}{2}}|B|M.$$
Since $B\subset[M/(2k-2)]$, we have $M-(k-1)d\geqslant M/2$ for all $d\in B$. Thus $$\Lambda(1_{[M]};1_{B})=\sum_{d\in B}(M-(k-1)d)\geqslant {\tfrac{1}{2}}|B|M.$$
Gowers norms
------------
Gowers [@GowersNewFour] observed that arithmetic progressions of length four or more are not controlled by ordinary (linear) Fourier analysis. Similarly, four-point Brauer configurations (and longer) require higher order notions of uniformity – they have *true complexity* greater than $1$ (see [@GowersWolfTrue] for further details). To overcome this difficulty, Gowers introduced a sequence of norms which can be used to measure the higher order uniformity of sets and functions.
Let $f:{\mathbb{Z}}\to{\mathbb{R}}$ be a finitely supported function. For each $d\geqslant 2$, the $U^{d}$ *norm* $\lVert f\rVert_{U^{d}}$ of $f$ is defined by $$\label{eqnFinGowNorm}
\lVert f\rVert_{U^{d}} :=\left( \sum_{x\in {\mathbb{Z}}}\sum_{{\mathbf{h}}\in {\mathbb{Z}}^{d}}\,\Delta_{h_1, \dots, h_d}f(x)\right) ^{1/2^{d}},$$ where the *difference operators* $\Delta_{h}$ are defined inductively by $$\Delta_{h}f(x):=f(x)f(x+h)$$ and $$\Delta_{h_1, \dots, h_d}f:=\Delta_{h_{1}}\Delta_{h_{2}}\cdots\Delta_{h_{d}}f.$$
In the literature, and in Gowers’ original paper, it is common to work with functions $f:{\mathbb{Z}}/p{\mathbb{Z}}\to{\mathbb{R}}$, defining $\lVert f\rVert_{U^{d}({\mathbb{Z}}/p{\mathbb{Z}})}$ by summing over $x\in{\mathbb{Z}}/p{\mathbb{Z}}$ and ${\mathbf{h}}\in({\mathbb{Z}}/p{\mathbb{Z}})^{d}$ in . Given a prime $p>N$, one can embed the interval $[N]$ into ${\mathbb{Z}}/p{\mathbb{Z}}$ by reduction modulo $p$. This allows us to identify a function $f:[N]\to{\mathbb{R}}$ with an extension $\tilde{f}:{\mathbb{Z}}/p{\mathbb{Z}}\to{\mathbb{R}}$ on taking $\tilde{f}(x)=0$ for all $x\in({\mathbb{Z}}/p{\mathbb{Z}})\setminus[N]$. One can observe that if $p>2(d+1) N$, then $\lVert f\rVert_{U^{d}}=\lVert \tilde{f}\rVert_{U^{d}({\mathbb{Z}}/p{\mathbb{Z}})}$. This is due to the fact that the interval $[N]\subset{\mathbb{Z}}$ and the embedding of $[N]$ into ${\mathbb{Z}}/p{\mathbb{Z}}$ are *Freiman isomorphic of order $d+1$* (see [@TaoVu §5] for further details).
We note that $$\label{norm bounds}
{\tfrac{1}{2}} N^{\frac{d+1}{2^{d}}} \leqslant {\left\| 1_{[N]}\right\|}_{U^d} \leqslant N^{\frac{d+1}{2^{d}}}.$$ The lower bound follows from inductively applying the Cauchy–Schwarz inequality in the form $${\left\| f\right\|}_{U^{d}} = {\left( \sum_{h_1, \dots, h_{d-1}}{\left| \sum_x \Delta_{h_1, \dots, h_{d-1}} f(x)\right|}^2 \right)}^{1/2^d} {\geqslant}(2N)^{-\frac{d-1}{2^{d}}}{\left\| f\right\|}_{U^{d-1}},$$ the factor of 2 resulting from the observation that if ${\mathrm{supp}}(f) \subset [N]$ then the $h_i$ in contribute only if $h_i \in (-N, N)$. The upper bound is a consequence of the fact that ${\left\| f\right\|}_{U^d}^{2^d}$ counts the number of solutions to a system of $2^d-d-1$ independent linear equations in $2^d$ variables, each weighted by $f$.
\[uniform def\] We say that $A \subset [N]$ is *${\varepsilon}$-uniform of degree $d$* if $$\lVert 1_{A} - {\mathbb{E}}_{[N]}(1_A) 1_{[N]}\rVert_{U^{d+1}}\leqslant {\varepsilon}\lVert 1_{[N]}\rVert_{U^{d+1}},$$ where ${\mathbb{E}}_{[N]}(1_A) := |A\cap [N]|/N$ denotes the density of $A$ on $[N]$. More generally, given $P\subset[N]$, we say that $A$ is *${\varepsilon}$-uniform of degree $d$ on $P$* if $$\lVert 1_{A} - {\mathbb{E}}_{P}(1_A) 1_{P}\rVert_{U^{d+1}}\leqslant {\varepsilon}\lVert 1_{P}\rVert_{U^{d+1}}.$$
Gowers showed that one can study sets which lack arithmetic progressions of length $k$ by considering their uniformity. If a set has density $\alpha$ in $[N]$ and is ${\varepsilon}$-uniform of degree $k$, for some small ${\varepsilon}= {\varepsilon}(k, \alpha)$, then $A$ contains a proportion of $\alpha^k$ of the total progressions of length $k$ in the interval $[N]$. Hence the only way a uniform set can lack $k$-term progressions is if it has few elements.
A similar result holds for Brauer configurations, see for instance [@GreenTaoLinear Appendix C]. In order to avoid the introduction of an (admittedly harmless) absolute constant resulting from the passage to a cyclic group, we give the simple proof.
\[lemGenVon\] Let $f_{1},...,f_{k},g:[N]\to[-1,1]$. Then for each $j\in[k]$ we have $$|\Lambda(f_{1},...,f_{k};g)|\leqslant N^{2}\left(\frac{\lVert f_{j}\rVert_{U^{k}}^{2^{k}}}{N^{k+1}}\right)^{1/2^{k}} \left( \frac{\lVert g\rVert_{U^{k}}^{2^{k}}}{N^{k+1}}\right) ^{1/2^{k}}.$$
We prove the case where $j=k$. The other cases follow on performing a change of variables $x'=x+id$ preceding each application of the Cauchy-Schwarz inequality.
Applying the Cauchy-Schwarz inequality with respect to the $x$ variable shows that $|\Lambda(f_1,...,f_{k};g)|$ is bounded above by $$\begin{aligned}
\left(\sum_{x\in{\mathbb{Z}}}|f_{1}(x)|^{2}\right)^{1/2}\left(\sum_{x,d,d'\in{\mathbb{Z}}}g(d)g(d')\prod_{i=1}^{k-1}f_{i+1}(x+id)f_{i+1}(x+id')\right)^{1/2}.
\end{aligned}$$ Using the fact that $f_{1}$ is a $1$-bounded function supported on $[N]$, and by performing a change of variables $d'=d+h$, we deduce that $$|\Lambda(f_{1},...,f_{k};g)|\leqslant N^{1/2}\left(\sum_{x,d,h\in{\mathbb{Z}}}\Delta_{h}g(d)\prod_{i=1}^{k-1}\Delta_{ih}f_{i+1}(x+id)\right)^{1/2}.$$ By iterating $k-2$ times, we see that $|\Lambda(f_1,...,f_{k};g)|^{2^{k-1}}$ is bounded above by $$N^{2^{k}-k-1}\sum_{{\mathbf{h}}\in{\mathbb{Z}}^{k-1}}\sum_{x\in{\mathbb{Z}}}\Delta_{(k-1)h_1, (k-2)h_2, \dots, h_{k-1}}f_{k}(x) \sum_{d\in{\mathbb{Z}}}\Delta_{h_1, \dots, h_{k-1}}g(d).$$ By applying Cauchy-Schwarz with respect to the ${\mathbf{h}}$ variable, the above sum is at most $ S^{1/2}\lVert g\rVert_{U^{k}}^{2^{k-1}}$, where $S$ is equal to $$\sum_{{\mathbf{h}}\in{\mathbb{Z}}^{k-1}}\left\lvert\sum_{x\in{\mathbb{Z}}}\Delta_{(k-1)h_1, (k-2)h_2, \dots, h_{k-1}}f_{k}(x)\right\rvert^{2}.$$ Since the terms in the sum over $h$ are non-negative, we can extend the summation from ${\mathbb{Z}}^{k-1}$ to $(k-1)^{-1}\cdot {\mathbb{Z}}\times (k-2)^{-1}\cdot {\mathbb{Z}}\times \dots \times {\mathbb{Z}}$, yielding the lemma.
\[corUkCont\] Let $f_1, f_2,g:[N]\to[-1,1]$. Then $$|\Lambda(f_{1};g)-\Lambda(f_{2};g)|\leqslant kN^{2}\frac{\lVert f_{1}-f_{2}\rVert_{U^{k}}}{N^{(k+1)2^{-k}}}.$$
Observe that $\Lambda(f_{1};g)-\Lambda(f_{2};g)$ can be written as the sum of $k$ terms $$\Lambda(f_{1}-f_{2},f_{1},...,f_{1};g)+\Lambda(f_{2},f_{1}-f_{2},f_{1},...,f_{1};g)+\cdots+\Lambda(f_{2},...,f_{2},f_{1}-f_{2};g).$$ Recalling that $g$ is $1$-bounded, the result now follows from the triangle inequality and Lemma \[lemGenVon\].
Lemma \[propBrauCount\] shows us that, for any non-empty $B\subset[N/(2k-2)]$ and $\alpha>0$, we have $$\Lambda(\alpha 1_{[N]};1_{B})\geqslant \tfrac{1}{2}\alpha^{k}|B|N.$$ Combining this with Corollary \[corUkCont\] we see that, if $A\subset[N]$ has density $\alpha>0$ and is ${\varepsilon}$-uniform of degree $k-1$ for some ‘very small’ ${\varepsilon}>0$, then the difference $$|\Lambda(1_{A};1_{B})-\Lambda(\alpha 1_{[N]};1_{B})|$$ is also small. This then implies that $A$ contains an arithmetic progression of length $k$ with common difference in $B$. Hence sets $A$ lacking such arithmetic progression cannot be uniform. A key observation of Gowers is that this lack of uniformity implies that the set $A$ exhibits significant bias towards a long arithmetic progression inside $[N]$.\
**Gowers’ density increment lemma.** *Let $d\geqslant 1$ and $0 < {\varepsilon}{\leqslant}{\tfrac{1}{2}}$. Suppose that $A\subset[N]$ is not ${\varepsilon}$-uniform of degree $d$ as in Definition \[uniform def\]. Then, on setting $$\label{GowersBd}
\eta := {\frac{1}{4}}{\left( \frac{{\varepsilon}}{8(d+2)} \right)}^{2^{d +1+ 2^{d+10}}},$$ there exists an arithmetic progression $P\subset[N]$ such that $$|P|\geqslant \eta N^{\eta} \quad\text{and}\quad
\frac{|A\cap P|}{|P|}\geqslant \frac{|A|}{N}+\eta.$$*
Let $p$ be a prime in the interval $2(d+2) N < p {\leqslant}4(d+2) N$, so that on setting $f := 1_A - {\mathbb{E}}_{[N]}(1_A) 1_{[N]}$ and viewing this as a function on ${\mathbb{Z}}/p{\mathbb{Z}}$ the lower bound in gives $$\sum_{{\mathbf{h}}\in ({\mathbb{Z}}/p{\mathbb{Z}})^{d}} {\left| \sum_{x\in{\mathbb{Z}}/p{\mathbb{Z}}} \Delta_h f(x)\right|}^2 {\geqslant}{\left( \tfrac{{\varepsilon}}{8(d+2)} \right)}^{2^{d+1}} p^{d+2}.$$ Hence, according to Gowers’ [@GowersSzem p.478] definition of $\alpha$-uniformity, $f$ is not $\alpha$-uniform of degree $d$ on ${\mathbb{Z}}/p{\mathbb{Z}}$ with $$\alpha = {\left( \tfrac{{\varepsilon}}{8(d+2)} \right)}^{2^{d+1}}.$$ Applying Gowers’ local inverse theorem for the $U^{d+1}$-norm [@GowersSzem Theorem 18.1] there exists a partition of ${\mathbb{Z}}/p{\mathbb{Z}}$ into (integer) arithmetic progressions $P_1$, …, $P_M$ with $M{\leqslant}p^{1-\alpha^{2^{2^{d+10}}}}$ and such that $$\sum_{j=1}^M {\biggl| \sum_{x\in P_j} f(x) \biggr|} {\geqslant}\alpha^{2^{2^{d+10}}}p.$$ Since $f$ is supported on $[N]$, we may assume that $P_j \subset [N]$ for all $j$. Write $\beta := \alpha^{2^{2^{d+9}}}$. As $f$ has mean zero we may apply (the proof of) [@GowersSzem Lemma 5.15] to obtain a progression $P \subset [N]$ with $|P| {\geqslant}{\tfrac{1}{4}}\beta p^\beta$ which also satisfies $$\sum_{x\in P} f(x) {\geqslant}{\tfrac{1}{4}}\beta |P|.$$
Maximal translate density
-------------------------
As in the previous section, we prove Theorem \[weak brauer theorem\] by a maximal translate density increment argument. For $q, M \in {\mathbb{N}}$ and $A\subset{\mathbb{Z}}$, define the *maximal translate density* $$\delta_{q, M}(A) := \max_{x\in{\mathbb{Z}}} \frac{|A \cap (x+q\cdot[M])|}{M}.$$ Given a collection of non-empty subsets $A_{1},...,A_{r}\subset[N]$, we collate their densities into the quantity $$\Delta(q, M) = \Delta(q, M;\{A_{i}\}_{i=1}^{r}) := \sum_{i=1}^r \delta_{q, M}(A_i).$$ We write $\Delta(q,M)$ when it is clear from the context which collection of sets $\{A_{i}\}_{i=1}^{r}$ we are working with.
In the previous section, where we worked with subspaces of ${\mathbb{F}}_{2}^{n}$, we showed (Lemma \[subspace preservation lemma\]) that the maximal translate density does not decrease when passing to a subspace. This is no longer true when passing to subprogressions in ${\mathbb{Z}}$. However, we can still increment $\Delta(q,M)$ if the subprogression we pass to is not too long.
\[approximately preserving\] Given positive integers $M ,M_1, q, q_1$ and a finite set $A\subset{\mathbb{Z}}$, we have $$\delta_{qq_{1},M_{1}}(A)\geqslant\delta_{q,M}(A)\left( 1-\tfrac{q_{1}M_{1}}{M}\right) .$$
By definition of $\delta_{q,M}$, we can find $t\in{\mathbb{Z}}$ such that $$\delta_{q,M}(A)M=|A\cap (t+q\cdot[M])|.$$ Let $\tilde{A}:=A\cap (t+q\cdot[M])$. Note that $$\delta_{q,M}(\tilde{A})M=|\tilde{A}|=\delta_{q,M}(A)M.$$ Now observe that the collection of translates $\{x+qq_{1}\cdot[M_{1}]:x\in{\mathbb{Z}}\}$ covers ${\mathbb{Z}}$, and each integer $m\in{\mathbb{Z}}$ lies in exactly $M_{1}$ such translates. This gives $$\label{eqnSumTildeC}
\sum_{x\in{\mathbb{Z}}}|\tilde{A}\cap (x+qq_{1}\cdot[M_{1}])|=|\tilde{A}|M_{1}=\delta_{q,M}(A)MM_{1}.$$ Let $\Omega$ be given by $$\Omega:=\{x\in{\mathbb{Z}}: \tilde{A}\cap (x+qq_{1}\cdot[M_{1}])\neq\emptyset\}.$$ Now suppose $x\in\Omega$. Since $\tilde{A}\subset t+q\cdot[M]$, we can find $u\in[M]$ and $u_{1}\in[M_{1}]$ such that $
x-t=q(u-q_{1}u_{1}).
$ From this we see that $$(x-t)\in[q(1-q_{1}M_{1}),q(M-q_{1})]\cap (q\cdot{\mathbb{Z}}).$$ We therefore deduce that $
|\Omega|\leqslant M+q_{1}M_{1}.
$ Applying the pigeonhole principle to (\[eqnSumTildeC\]), we conclude that $$\delta_{qq_{1},M_{1}}(A)\geqslant\delta_{qq_{1},M_{1}}(\tilde{A})\geqslant\delta_{q,M}(A)\tfrac{M}{M+q_{1}M_{1}}.$$ This implies the desired bound.
\[brauer preservation\] Let $M,M_{1},q,q_{1}\in{\mathbb{N}}$, and let $A_{1},...,A_{r}\subset[N]$ be non-empty sets. If $\delta_{qq_1, M_1}(A_i) \geqslant \delta_{q, M}(A_i) + \eta$ for some $i\in[r]$ and some $\eta>0$, then $$\Delta(qq_1, M_1) \geqslant \Delta(q, M) -\tfrac{q_{1}M_{1}}{M}r+ \eta.$$
The following lemma allows us to pass to a subprogression whose common difference and length are sufficiently small to allow for an effective employment of Corollary \[approximately preserving\].
\[preserving density\] Let $q$ and $M$ be positive integers with $M {\leqslant}2{\left\lfloor N/q \right\rfloor}$. For any $A \subset [N]$ there exists an arithmetic progression $P$ of common difference $q$ and length $|P| \in [{\tfrac{1}{2}}M, M]$ such that $$\frac{|A \cap P|}{|P|} {\geqslant}\frac{|A\cap [N]|}{N}.$$
Let us first give the argument for $q = 1$. We partition $[N]$ into the intervals $$(0, {\left\lceil M/2 \right\rceil}] \cup ({\left\lceil M/2 \right\rceil}, 2{\left\lceil M/2 \right\rceil}] \cup \dots \cup (m{\left\lceil M/2 \right\rceil}, N],$$ for some $m$ with $N - m{\left\lceil M/2 \right\rceil} {\leqslant}{\left\lceil M/2 \right\rceil}$. If $N - m{\left\lceil M/2 \right\rceil} = {\left\lceil M/2 \right\rceil}$, then we obtain the result on applying the pigeonhole principle. So we may suppose that $N - m{\left\lceil M/2 \right\rceil} {\leqslant}{\left\lfloor M/2 \right\rfloor}$. The pigeonhole principle again gives the result on partitioning similarly, but with the final interval equal to $((m-1){\left\lceil M/2 \right\rceil}, N]$.
We generalise to $q > 1$ by first partitioning $[N]$ into congruence classes mod $q$. Each such congruence class takes the form $-a + q \cdot [N_a]$ where $0 {\leqslant}a < q$ and $ N_a = {\left\lfloor (N+a)/q \right\rfloor}$. Since $N_a {\geqslant}M/2$ we can use our previous argument to partition $[N_a]$ into intervals, each with length in $[{\tfrac{1}{2}}M, M]$. The result follows once again from the pigeonhole principle.
Uniform translates
------------------
We have shown that a highly uniform set contains many Brauer configurations. In general, one cannot guarantee that one of the colour classes in a finite colouring of $[N]$ is uniform. However, we can use Gowers’ density increment lemma to show that there exists a long arithmetic progression $q \cdot [M]\subset[N]$ such that each colour class is uniform on a translate of $q\cdot [M]$, and on the same translate its density is not diminished.
\[thmUniBrau\] Given $0 < {\varepsilon}{\leqslant}{\tfrac{1}{2}}$ and $d \geqslant 1$, let $\eta = \eta(d, {\varepsilon})$ denote the constant appearing in Gowers’ density increment lemma. Suppose that $$\label{C1bd}
N\geqslant\exp\exp(3r\eta^{-2}).$$ Then for any sets $A_{1},...,A_{r}\subset[N]$ there exists a homogeneous progression $q\cdot[M]\subset[N]$ with $M\geqslant N^{\exp(-3r\eta^{-2})}$ such that the following is true. For each $i\in[r]$, there exists a translate $a_{i}+q\cdot[M]$ on which $A_{i}$ achieves its maximal translate density and on which $A_{i}$ is ${\varepsilon}$-uniform of degree $d$.
We give an iterative procedure, at each stage of which we have positive integers $q_n$ and $M_n$ satisfying $$\label{iteration conclusion}
M_n \geqslant (\eta^2/5r)^{1 + \eta + \dots + \eta^{n-1}} N^{\eta^n}
\quad \text{and}
\quad
\Delta(q_n, M_n) \geqslant n\eta/2.$$ We initiate this on taking $q_0 := 1$ and $M_0 := N$ (the common difference and length of $[N]$). Since $\Delta(q, M) \leqslant r$ this procedure must terminate at some $n \leqslant 2r\eta^{-1}$.
Suppose that we have iterated $n$ times to give $q=q_n$ and $M = M_n$. If, for each $A_{i}$, there is a translate $a_{i}+q\cdot[M]$ on which $A_{i}$ is ${\varepsilon}$-uniform and achieves its maximal translate density, then we terminate our procedure. Suppose then that we can find $A_{j}$ which does not have this property. We now give the iteration step of our algorithm.
By the definition of maximal translate density, there exists $t\in{\mathbb{Z}}$ such that $$|A_j \cap (t + q \cdot [M])| = \delta_{q, M}(A_j)M.$$ Let $A := {\left\{y \in [M] : t+qy \in A_j \right\}}$. Since $A_j$ is not ${\varepsilon}$-uniform of degree $d$ on $t + q\cdot [M]$, we see that $A$ is not ${\varepsilon}$-uniform of degree $d$ on $[M]$. By Gowers’ density increment lemma, we deduce the existence of a progression $P\subset[M]$ of length $|P| \geqslant \eta M^{\eta}$ such that $$|A \cap P | \geqslant \left(\delta_{q, M}(A_j) + \eta \right)|P|.$$
We would like the length and common difference of $P$ to be sufficiently small to allow for the effective employment of Corollary \[brauer preservation\]. Using and one can verify that $\eta M^\eta {\geqslant}2r\eta^{-1}$, so that the integer $ \lfloor \eta |P|/(2r) \rfloor$ is positive. Lemma \[preserving density\] then gives a subprogression $x + q'\cdot [M'] \subset P$, of the same common difference as $P$, such that ${\tfrac{1}{2}}\lfloor \eta |P|/(2r) \rfloor {\leqslant}M' {\leqslant}\lfloor \eta |P|/(2r) \rfloor$ and for some $x$ we have $$\frac{|A\cap (x+q'\cdot[M'])|}{M'}\geqslant\frac{|A\cap P|}{|P|}.$$ Note that, since $P \subset [M]$ has common difference $q'$ we have $q'|P| {\leqslant}M$ and so $q' M' {\leqslant}q' |P| \eta/ (2r) {\leqslant}M \eta/(2r)$. Hence by Corollary \[brauer preservation\] we obtain $$\Delta(q'q, M') \geqslant \Delta(q, M) + {\tfrac{1}{2}}\eta.$$
Again using and one can check that $M' {\geqslant}(\eta^2/5r) M^\eta$, so we obtain with $(q_{n+1}, M_{n+1}) := (q', M')$, and our iteration can continue. Taking this iteration through to completion gives the lemma.
We are now in a position to derive our main theorem.
Let $\eta = \eta(k-1, {\varepsilon})$ be given by in Gowers’ density increment lemma, with ${\varepsilon}$ to be determined, and suppose that $N\geqslant\exp\exp(4r\eta^{-2})$. Let $M,q$ be the positive integers obtained by applying Lemma \[thmUniBrau\] to the partition $[N]=A_{1}\cup\cdots\cup A_{r}$. By the pigeonhole principle, there exists $j\in[r]$ such that $$\label{abBound}
|A_{j}\cap q\cdot[M/(2k-2)]|\geqslant\frac{1}{r}\left\lfloor\frac{M}{2(k-1)}\right\rfloor >\frac{M}{4(k-1)r},$$ the latter following from the fact that $$M {\geqslant}\exp{\left( \exp(-3r\eta^{-2})\log N \right)} {\geqslant}\exp\exp(r\eta^{-2}).$$
Let $t\in{\mathbb{Z}}$ be such that $$|A_{j}\cap (t+q\cdot [M])|=\delta_{q, M}(A_{j})M.$$ We now construct sets $A\subset[M]$ and $B\subset[M/(2k-2)]$ by taking $$A := \{y \in [M] : t+qy \in A_j \} \quad \text{and}\quad
B := \{d \in [M/(2k-2)] : qd \in A_{j}\}.$$ Our goal is to show that $\Lambda(1_{A};1_{B})>0$. If this is the case, then there exists an arithmetic progression of length $k$ in $A$ whose common difference lies in $B$. We can then infer from our construction of $A$ and $B$ the existence of a $(k+1)$-point Brauer configuration in $A_{j}$.
Let $\alpha$ and $\beta$ denote the respective densities of $A$ and $B$ in $[M]$. From the bound it follows that $\alpha,\beta >(4r(k-1))^{-1}$. Combining this with Lemma \[propBrauCount\] gives $$\Lambda(\alpha 1_{[M]};1_{B})\geqslant\tfrac{1}{2}\alpha^{k}\beta M^{2}>\frac{M^{2}}{2^{2k+3}(k-1)^{k+1}r^{k+1}}.$$ Recall that the conclusion of Lemma \[thmUniBrau\] guarantees that $A$ is ${\varepsilon}$-uniform of degree $(k-1)$ (as a subset of $[M]$). Applying Corollary \[corUkCont\] and the upper bound in gives $$|\Lambda(1_{A};1_{B})-\Lambda(\alpha 1_{[M]};1_{B})|\leqslant kM^{2}{\varepsilon}.$$ We therefore obtain $\Lambda(1_{A};1_{B})>0$, and hence the theorem, on taking $$\label{eps choice}
{\varepsilon}^{-1} := k2^{2k+3}(k-1)^{k+1}r^{k+1}.$$ Since we are assuming that $N \geqslant\exp\exp(4r\eta^{-2})$, this choice of ${\varepsilon}$ gives a double exponential bound in $r^{O_k(1)}$.
Finally, we show that the more precise bound suffices. The inequality $\exp\exp(x) {\leqslant}2^{2^{2x}}$ is valid for all $x \geqslant 4$, so that $$\exp\exp(4r \eta^{-2}) {\leqslant}2^{2^{8r\eta^{-2}}}.$$ For our choice of ${\varepsilon}$, we have $$\eta^{-1} {\leqslant}{\left( 16(k+1)k(k-1)^{k+1} 2^{2k+3} r^{k+1} \right)}^{2^{k + 2^{k+9}}}$$ One may check that $$16(k+1)k(k-1)^{k+1} {\leqslant}2^{k^2 + 4},$$ so that, on using $r{\geqslant}2$, we have $$4r\eta^{-2} {\leqslant}4r {\left( 2^{k^2 + 2k+7} r^{k+1} \right)}^{2^{1+k + 2^{k+9}}} {\leqslant}r^{{\left( k+3 \right)}^22^{1+k + 2^{k+9}}} {\leqslant}r^{2^{2^{k+10}}},$$ as required.
An improved bound for four-point configurations {#four point sec}
===============================================
In this section we focus on the four point Brauer configuration , improving our bound on the Ramsey number to $\exp \exp(r^{1+o(1)})$. Instead of finessing the quantitative aspect of Lemma \[thmUniBrau\], we opt to mimic the sparsity–expansion dichotomy of §\[secSchurFF\]. This requires a higher order analogue of the difference set $A - A$, namely $$\mathrm{Step}_3(A) := {\left\{d : A\cap (A-d) \cap (A-2d) \neq \emptyset\right\}}.$$
\[brauer dichotomy lemma\] There exists an absolute constant $C$ such that for $N {\geqslant}\exp \exp (Cr\log^2 r)$ the following holds. For any $r$-colouring $A_1\cup \dots \cup A_r = [N]$ there exist positive integers $q$ and $M$ with $qM\leqslant N$ such that for any $i \in [r]$ we have one of the two following possibilities.
- (Sparsity). $$\label{brauer sparsity}
|A_i \cap q \cdot [3M]| < {\tfrac{1}{r}}M;$$
- (Expansion). $$\label{brauer expansion}
|\mathrm{Step}_3(A_i) \cap q \cdot [M]| > {\left( 1 - {\tfrac{1}{r}} \right)} M.$$
Before proving this, let us first use it to obtain a bound on the Ramsey number of four-point Brauer configurations.
Let $q$ and $M$ denote the numbers provided by the dichotomy. By the pigeonhole principle, there exists a colour class $A_i$ satisfying $$|A_i\cap q \cdot [M]| {\geqslant}{\tfrac{1}{r}}M.$$ So $A_i$ is not sparse in the sense of . It follows that $A_i$ must instead satisfy the expansion property . By inclusion–exclusion $$|A_i\cap \mathrm{Step}_3(A_i)\cap q\cdot [M]| \\
{\geqslant}|A_i \cap q\cdot [M]| + |\mathrm{Step}_3(A_i)\cap q\cdot [M]| - M > 0.$$ In particular $A_i\cap \mathrm{Step}_3(A_i)$ contains a non-zero element.
We may assume that $r {\geqslant}2$, since otherwise the result is simple. Set $q_0 := 1$ and $N_0 := N$. We proceed by an iterative procedure, at each stage of which we have positive integers $q_n$, $N_n$ and $M_i = M_i^{(n)}$ ($1{\leqslant}i {\leqslant}r)$ such that for $n{\geqslant}1$ we have:\
(i) \[length\] $\displaystyle
N_{n} {\geqslant}\exp{\left( -r^{O(1)} \right)}N_{n-1}^{r^{-O(1)}}$;\
(ii) \[order\] $\displaystyle
M_i \in [{\tfrac{1}{2}}N_n, N_n]$ for all $1{\leqslant}i {\leqslant}r$;\
(iii) \[preservation\] $\displaystyle \delta_{q_n,M_i^{(n)}}(A_i) {\geqslant}\delta_{q_{n-1} , M_{i}^{(n-1)}}(A_i )$ for all $1{\leqslant}i {\leqslant}r$;\
(iv) \[density\] $ \displaystyle \delta_{q_n,M_i^{(n)}}(A_i) {\geqslant}{\left( 1+c \right)}\delta_{q_{n-1},M_{i}^{(n-1)}}(A_i)$ for some $ i $ with\
$\delta_{q_{n-1},M_{i}^{(n-1)}}(A_i) {\geqslant}{\tfrac{1}{7r}}$. Here $c = \Omega(1)$ is an absolute constant.\
At stage $n$ of the iteration we classify each colour class $A_i$ according to which of the following hold.
- (Sparse colours). These are the colours $A_i$ for which $$\delta_{q_n, M_i}(A_i) < {\tfrac{1}{7r}}.$$
- (Dense expanding colours). These are the dense colours $\delta_{q_n, M_i}(A_i) {\geqslant}{\tfrac{1}{7r}}$ for which we have the additional expansion estimate $$\label{brauer non expansion}
|\mathrm{Step}_3(A_i) \cap q_n\cdot [N_n/6]| > {\left( 1 - {\tfrac{1}{r}} \right)}{\left\lfloor N_n/6 \right\rfloor}.$$
- (Dense non-expanding colours). The class $A_i$ is *dense non-expanding* if it is neither sparse nor dense expanding.
If there are no dense non-expanding colours, then we terminate our procedure. If $N_n < 100$, then we also terminate our procedure. Let us therefore suppose that $N_n {\geqslant}100$ and there exists a dense non-expanding colour class $A_i$. Our aim is to show how, under these circumstances, the iteration may continue.
By the definition of maximal translate density, there exists $a$ such that $$\label{max density on Ai}
|A_i \cap (a + q_n \cdot [M_i])| M_i^{-1} = \delta_{q_n, M_i}(A_i) {\geqslant}{\tfrac{1}{7r}}.$$ Writing $M := M_i$, we define dense subsets $A, B \subset [M]$ by taking $$\label{brauer A B defn}
A := {\left\{y \in [M] : a+q_ny \in A_i \right\}} \quad \text{and} \quad B := {\left\{d \in [N_n/6] : q_nd \notin \mathrm{Step}_3(A_i)\right\}}.$$ We recall that $M/3 {\geqslant}N_n/6 {\geqslant}M/6$ .
Letting $\alpha$ denote the density of $A$ in $[M]$, we see that $\alpha$ is equal to the left-hand side of . Our assumption that $A_i$ is dense non-expanding and $N_n {\geqslant}100$ together imply that $B$ has size $\gg r^{-1} M$ and that $$\sum_{x, d} 1_A(x) 1_A(x+d)1_A(x+2d) 1_B(d) = 0.$$ Using the notation and (the proof of) Lemma \[propBrauCount\] we have $${\left| \Lambda(1_A; 1_B) -\Lambda(\alpha 1_{[M]}; 1_B)\right|} \gg \alpha^{3} M|B| \gg r^{-4} M^2.$$
From herein, we assume that the reader is familiar with the notation and terminology of Green and Tao [@GT09]. Applying [@GT09 Theorem 5.6] in conjunction with Corollary \[corUkCont\] we obtain a quadratic factor $(\mathcal{B}_1, \mathcal{B}_2)$ of complexity and resolution $\ll r^{O(1)}$ such that the function $f:= {\mathbb{E}}(1_A \mid \mathcal{B}_2)$ satisfies $${\left| \Lambda(f;1_B) -\Lambda(\alpha 1_{[M]}; 1_B)\right|} \gg \alpha^{3} M|B|.$$ Define the $\mathcal{B}_2$-measurable set $$\Omega:= {\left\{x \in [M] : f(x) {\geqslant}(1+c)\alpha\right\}},$$ where $c>0$ is sufficiently small enough to make the following argument valid.
For functions $f_1, f_2, f_3 : [M] \to{\mathbb{R}}$ we have the bound $$\label{L1 bound}
{\left| \Lambda( f_1, f_2, f_3; 1_B)\right|} \leqslant |B|\lVert f_{i}\rVert_{L^{1}} \prod_{j \neq i} {\left\| f_j\right\|}_\infty.$$ Hence a telescoping identity gives ${\left| \Lambda(f;1_B) -\Lambda(f1_{\Omega^c};1_B)\right|} \ll|\Omega||B|$, so that $$|\Omega||B| + {\left| \Lambda(f1_{\Omega^c};1_B) -\Lambda(\alpha1_{[M]};1_B)\right|} \gg \alpha^{3} M|B| .$$ Another telescoping identity in conjunction with gives $$\begin{aligned}
{\left| \Lambda(f1_{\Omega^c};1_B) -\Lambda(\alpha 1_{[M]};1_B)\right|} & \ll \alpha^2|B|
{\left\| f1_{\Omega^c}- \alpha1_{[M]}\right\|}_{L^1}\\
& \ll \alpha^2|B|{\left\| f - \alpha1_{[M]}\right\|}_{L^1} + |B||\Omega|,\end{aligned}$$ so that $$|\Omega|+ \alpha^2 {\left\| f - \alpha1_{[M]}\right\|}_{L^1} \gg \alpha^3 M.$$
Since $f - \alpha1_{[M]}$ has mean zero, its $L^1$-norm is equal to twice that of its positive part. The function ${\left( f - \alpha1_{[M]} \right)}_+$ can only exceed $c \alpha$ on $\Omega$, so taking $c$ small enough gives $$\label{omega lower bound}
|\Omega| \gg \alpha^3M \gg r^{-3} M.$$ As $\mathcal{B}_2$ has complexity and resolution $r^{O(1)}$ it contains at most $\exp(r^{O(1)})$ atoms. By [@GT09 Proposition 6.2] each such atom can be partitioned into a further $$\label{APno}
\exp(r^{O(1)}) M^{1-r^{-O(1)}}$$ disjoint arithmetic progressions. Hence $\Omega$ itself can be partitioned into arithmetic progressions, the number of which is at most . Combining this with and [@GT09 Lemma 6.1], we see that there exists an arithmetic progression $P$ of length at least $$\exp{\left( -r^{O(1)} \right)} M^{1/r^{O(1)}}.$$ on which $A$ has density at least $(1+\tfrac{c}{2})\alpha$. By partitioning $P$ into two pieces and applying the pigeon-hole principle, we may further assume that $|P|q\leqslant M$, where $q$ is the common difference of $P$.
Writing $q_{n+1}$ and $N_{n+1} = M_i^{(n+1)}$ for the common difference and length of $P$, we see that and are satisfied. For all $j \in [r]\setminus{\left\{i\right\}}$ we have $$q_{n+1} N_{n+1} {\leqslant}M = M_{i}^{(n)} {\leqslant}N_n {\leqslant}2M_{j}^{(n)}.$$ Hence we may apply Lemma \[preserving density\] to each colour class $A_j$ with $j \neq i$ to obtain a progression $P_j$ of common difference $q_n$ and length $M_{j}^{(n+1)}$ such that and hold. It follows that our iteration may continue.
Our iterative procedure must terminate at stage $n$ for some $n \ll r^{2}$. To see this, note that the sum of the maximal translate densities $\delta_{q_n ,M_i}(A_i)$ is at most $r$, and this quantity increases by at least $\Omega(1/r)$ at each iteration. Our next task is to improve this upper bound on the number of iterations to $n\ll r \log r$.
Let $A_i$ denote the colour class for which the density increment occurs most often. By the pigeonhole principle this happens on at least $n/r$ occasions. If the density of $A_i$ increments at least $c^{-1}$ times, then its density doubles. After a further ${\tfrac{1}{2}} c^{-1}$ increments the density of $A_i$ quadruples. The density of $A_i$ has therefore increased by a factor of $2^m$ if the number of iterations is at least $$\label{no A_i increments}
{\left\lceil c^{-1} \right\rceil} + {\left\lceil {\tfrac{1}{2}}c^{-1} \right\rceil} + \dots + {\left\lceil {\tfrac{1}{2^{m-1}}}c^{-1} \right\rceil} {\leqslant}m + 2c^{-1}.$$ The first time $A_i$ increments its initial density is at least $1/(7r)$, so if the number of increments experienced by $A_i$ is at least then its final density is at least $2^m /(7r)$. If $n/r > 2c^{-1} + {\left\lceil \log_2(7r) \right\rceil}$ then we obtain a density exceeding 1, a contradiction. It follows that the total number of iterations $n$ satisfies $n = O( r \log_2 r)$.
Having shown that our iteration must terminate in $ n=O(r \log r)$ steps, let us now ensure that termination results from a lack of dense non-expanding colours. This follows if we can ensure that $N_n {\geqslant}100$. Applying the lower bound iteratively we obtain $$N_n {\geqslant}\exp{\left( -r^{O(1)} \right)} N^{r^{-O(n)}}.$$ Using the fact that $n \ll r\log r$, the right-hand side above is at least 100 provided it is not the case that $N {\leqslant}\exp\exp{\left( O(r\log^2 r) \right)}$. Given this assumption, we obtain the conclusion of Lemma \[brauer dichotomy lemma\] on taking $M := {\left\lfloor N_n/6 \right\rfloor}$.
Lefmann quadrics {#lefmann sec}
================
In this section we show how our results can be used to obtain bounds on the Ramsey numbers for quadric equations of the form $$\label{lefquad}
\sum_{i=1}^{s}a_{i}x_{i}^{2}=0.$$ Lefmann [@lefmann Fact 2.8] established the following sufficient condition for equations of the above form to be partition regular.\
**Lefmann’s criterion.**
Lefmann proves this result directly by showing that has a solution in any set of the form $\{x,x+d,...,x+(k-1)d,\lambda d\}$, where $k=1+2\max_{i\in I}|u_{i}|$. We therefore obtain our quantitative version of Lefmann’s result (Theorem \[intro lefmann\]) from the following analogue of Theorem \[weak brauer theorem\].
For positive integers $k,\lambda$ there exists an absolute constant $C = C(k, \lambda)$ such that for any $r {\geqslant}2$ and $N {\geqslant}\exp\exp(r^C)$, if $[N]$ is $r$-coloured then there exists a monochromatic configuration of the form $\{x,x+d,...,x+(k-1)d,\lambda d\}$.
This is essentially the same as the proof of Theorem \[weak brauer theorem\].
[amsalpha]{}
J. Chapman, *Partition regularity and multiplicatively syndetic sets*, preprint (2019), [`arXiv:1902.01149`](https://arxiv.org/abs/1902.01149).
K. Cwalina and T. Schoen, *Tight bounds on additive Ramsey-type numbers*, J. London Math. Soc., **96** (2017) 601–620.
W. T. Gowers. *A new proof of [S]{}zemerédi’s theorem for arithmetic progressions of length four*. Geom. Funct. Anal. **8** (1998), 529–551.
, *A new proof of Szemerédi’s theorem*, Geom. Funct. Anal. **11** (2001), 465–588.
W. T. Gowers and J. Wolf, *The true complexity of a system of linear equations*, Proc. Lond. Math. Soc. **100** (2010), 155–176.
R. Graham, B. Rothschild and J. H. Spencer, *Ramsey theory*. Second edition. Wiley-Interscience Series in Discrete Mathematics and Optimization. 1990.
B. Green and T. Tao, *New bounds for Szemerédi’s theorem, II: A new bound for $r_4(N)$*, Analytic number theory, 180–204, Cambridge Univ. Press, Cambridge, 2009.
, *Linear equations in primes*, Ann. of Math. **171** (2010) 1753–1850.
H. Lefmann, *On partition regular systems of equations*, J. Combin. Theory Ser. A **58** (1991), 35–53.
K. F. Roth, *On certain sets of integers*, J. London Math. Soc. **28** (1953), 104–109. T. Sanders, *Additive structures in sumsets*, Math. Proc. Cambridge Philos. Soc. **144** (2008), 289–316.
, *Bootstrapping partition regularity of linear systems*, forthcoming arXiv preprint.
S. Shelah, *Primitive recursive bounds for van der Waerden numbers*, J. Amer. Math. Soc. **1** (1988), 683–697.
I. D. Shkredov, *Fourier analysis in combinatorial number theory*, Uspekhi Mat. Nauk, **65** (2010) 127–184.
T. Tao and V. Vu, *Additive combinatorics*, volume 105 of Cambridge Studies in Advanced Mathematics. Cambridge University Press (2006).
|
---
abstract: 'We propose a scheme to realize scalable quantum computation in a planar ion crystal confined by a Paul trap. We show that the inevitable in-plane micromotion affects the gate design via three separate effects: renormalization of the equilibrium positions, coupling to the transverse motional modes, and amplitude modulation in the addressing beam. We demonstrate that all of these effects can be taken into account and high-fidelity gates are possible in the presence of micromotion. This proposal opens the prospect to realize large-scale fault-tolerant quantum computation within a single Paul trap.'
author:
- 'S.-T. Wang'
- 'C. Shen'
- 'L.-M. Duan'
title: Quantum Computation under Micromotion in a Planar Ion Crystal
---
Scalable quantum computation constitutes one of the ultimate goals in modern physics [@nielsen2010quantum; @Ladd2010Quantum]. Towards that goal, trapped atomic ions are hailed as one of the most promising platforms for the eventual realization [@Blatt2008Entangled; @Haffner2008Quantum]. The linear Paul trap with an one-dimensional (1D) ion crystal was among the first to perform quantum logic gates [@Cirac1995Quantum; @Monroe1995Demonstration; @Schmidt-Kaler2003Realization] and to generate entangled states [@Turchette1998Deterministic; @Sackett2000Experimental; @Roos2004Science], but in terms of scalability, the 1D geometry limits the number of ions that can be successfully trapped [@Raizen1992Ionic; @Schiffer1993Phase]. Another shortcoming of the 1D architecture is that the error threshold for fault-tolerant quantum computation with short-range gates is exceptionally low and very hard to be met experimentally [@Gottesman2000Fault; @Svore2005Local; @Szkopek2006Threshold].
Generic ion traps, on the other hand, could confine up to millions of ions with a 2D or 3D structure [@Itano1998Bragg; @Drewsen1998Large; @Mortensen2006Observation]. More crucially, large scale fault-tolerant quantum computation can be performed with a high error threshold, in the order of a percent level, with just nearest neighbor (NN) quantum gates [@Raussendorf2007Fault; @Raussendorf2007Topological; @Fowler2009High; @DiVincenzo2009Fault]. This makes 2D or 3D ion crystals especially desirable for scalable quantum computation. Various 2D architectures have been proposed, including microtrap arrays [@Cirac2000Scalable], Penning traps [Porras2006Quantum, Zou2010Implementation, Itano1998Bragg, Mitchell1998Direct]{}, and multizone trap arrays [Kielpinski2002Architecture, Monroe2013Scaling]{}. However, the ion separation distance in microtraps and penning traps is typically too large for fast quantum gates since the effective ion-qubit interaction scales down rapidly with the distance. In addition, fast rotation of the ion crystal in the Penning trap makes the individual addressing of qubits very demanding. Distinct from these challenges, Paul traps provide strong confinement; however, they are hampered by the micromotion problem: fast micromotion caused by the driving radio-frequency (rf) field cannot be laser cooled. It may thus create motion of large amplitudes well beyond the Lamb-Dicke regime [@berkeland1998minimization; @Leibfried2003Quantum], which becomes a serious impediment to high-fidelity quantum gates.
In this paper, we propose a scheme for scalable quantum computation with a 2D ion crystal in a quadrupole Paul trap. We have shown recently that micromotion may not be an obstacle for design of high-fidelity gates for the two-ion case [@Shen2014High]. Here, we extend this idea and show that micromotion can be explicitly taken into account in the design of quantum gates in a large ion crystal. This hence clears the critical hurdle and put Paul traps as a viable architecture to realize scalable quantum computation. In such a trap, DC and AC electrode voltages can be adjusted so that a planar ion crystal is formed with a strong trapping potential in the axial direction. In-plane micromotion is significant, but essentially no transverse micromotion is excited due to negligible displacement from the axial plane. We perform gates mediated by transverse motional modes and show that the in-plane micromotion influences the gate design through three separate ways: (1) It renormalizes the average positions of each ion compared to the static pseudopotential equilibrium positions. (2) It couples to and modifies the transverse motional modes. (3) It causes amplitude modulation in the addressing beam. In contrast to thermal motion, the fluctuation induced by micromotion is coherent and can be taken into account explicitly. Several other works also studied the effect of micromotion on equilibrium ion positions and motional modes [@Landa2012Modes; @Kaufmann2012Precise; @Landa2014Entanglement], or used transverse modes in an oblate Paul trap to minimize the micromotion effect [@Yoshimura2014Creation]. Here, by using multiple-segment laser pulses [@Zhu2006Arbitrary; @Zhu2006Trapped; @Choi2014Optimal], we demonstrate that high-fidelity quantum gates can be achieved even in the presence of significant micromotion and even when many motional modes are excited. Our work therefore shows the feasibility of quadrupole Paul traps in performing large scale quantum computation, which may drive substantial experimental progress.
A generic quadrupole Paul trap can be formed by electrodes with a hyperbolic cross-section. The trap potential can be written as $\Phi (x,\,y,\,z)=\Phi _{\text{DC}}(x,\,y,\,z)+\Phi _{\text{AC}}(x,\,y,\,z)$, where $$\begin{aligned}
\Phi _{\text{DC}}(x,\,y,\,z)& =\frac{U_{0}}{d_{0}^{2}}\left[ (1+\gamma
)x^{2}+(1-\gamma )y^{2}-2z^{2}\right] , \\
\Phi _{\text{AC}}(x,\,y,\,z)& =\frac{V_{0}\cos (\Omega _{T}t)}{d_{0}^{2}}\left( x^{2}+y^{2}-2z^{2}\right) .\end{aligned}$$It contains both a DC and an AC part, with $U_{0}$ being the DC voltage, and $V_{0}$ being the AC voltage forming an electric field oscillating at the radiofrequency $\Omega _{T}$. The parameter $d_{0}$ characterizes the size of the trap and $\gamma $ controls the anisotropy of the potential in the $x$-$y$ plane. We choose $\gamma $ to deviate slightly from zero, so that the crystal cannot rotate freely in the plane, i.e. to remove the gapless rotational mode. The AC part, on the contrary, is chosen to be isotropic in the $x$-$y$ plane. We let $U_{0}<0$ such that the trapping is enhanced along the $z$ direction in order to form a 2D crystal in the $x$-$y$ plane. Disregarding the Coulomb potential first, the equations of motion of ions in such a trap can be written in the standard form of Mathieu equations along each direction: $$\frac{d^{2}r_{\nu }}{d\xi ^{2}}+\left[ a_{\nu }-2q_{\nu }\cos (2\xi )\right]
r_{\nu }=0,$$where $\nu \in \{x,y,z\}$, and the dimensionless parameters are $\xi =\Omega
_{T}t/2$, $a_{x}=8(1+\gamma )eU_{0}/md_{0}^{2}\Omega _{T}^{2}$, $a_{y}=8(1-\gamma )eU_{0}/md_{0}^{2}\Omega _{T}^{2}$, $a_{z}=-16eU_{0}/md_{0}^{2}\Omega _{T}^{2}$, $q_{x}=q_{y}=q=-4eV_{0}/md_{0}^{2}\Omega _{T}^{2}$, $q_{z}=-2q$. Neglecting micromotion, one could approximate the potential as a time-independent harmonic pseudopotential with secular trapping frequencies $\omega _{\nu
}=\beta _{\nu }\Omega _{T}/2$, with $\beta _{\nu }\approx \sqrt{a_{\nu} +q_{\nu}^{2}/2}$ being the characteristic exponents of the Mathieu equations [@mclachlan1951theory; @king1999quantum].\
[**Results**]{}\
**Dynamic ion positions.** Adding Coulomb interactions back, the static equilibrium positions can be found by minimizing the total pseudopotential [@james1998quantum; @Zou2010Implementation], or use molecular dynamics simulation with added dissipation, which imitates the cooling process in experiment [@Zhang2007Molecular; @Schiffer2000Temperature]. In our numerical simulation, we start with $N=127$ ions forming equilateral triangles in a 2D hexagonal structure. We then solve the equations of motion with a small frictional force to find the equilibrium positions $\vec{r}\,^{(0)} =\vec{r}(t\to\infty) =
(x_{1}^{(0)},y_{1}^{(0)}, \cdots, x_{N}^{(0)},y_{N}^{(0)})$, which is the starting point for the expansion of the Coulomb potential. Micromotion is subsequently incorporated by solving the decoupled driven Mathieu equations (see supplementary materials). The average ion positions $\vec{r}\,^{(0)}$ are found self-consistently, which differ slightly from the pseudopotential equilibrium positions (an average of $0.03 \, \mu$m shift). Dynamic ion positions $\vec{r}(t)$ can be expanded successively as $$\vec{r}(t) = \vec{r}\,^{(0)} + \vec{r}\,^{(1)} \cos(\Omega_{T} t) + \vec{r}\,^{(2)} \cos(2\Omega_{T} t) + \cdots. \label{Eq:Positions}$$ Numerically, we found that $\vec{r}\,^{(1)} \approx -\frac{q}{2} \vec{r}\,^{(0)}$ and $\vec{r}\,^{(2)} \approx \frac{q^{2}}{32} \vec{r}\,^{(0)}$, where the expression for $\vec{r}\,^{(1)}$ is consistent with previous results [@Shen2014High; @Landa2012Modes; @Zhang2007Molecular]. Micromotion thus only results in breathing oscillations about the average positions.
Fig. \[Fig:IonPosition\](a) shows the average ion positions $\vec{r}\,^{(0)}$ in the planar crystal. The distribution of NN distance is plotted in figure \[Fig:IonPosition\](b). We choose the voltages $U_{0}$ and $V_{0}
$ such that the ion distance is kept between $6.5\,\mu $m and $10\,\mu $m. This ensures that crosstalk errors due to the Gaussian profile of the addressing beam are negligible, at the same time maintaining strong interaction between the ions. As micromotion yields breathing oscillations, the further away the ion is from the trap center, the larger the amplitude of micromotion becomes. With the furthest ion around $52\,\mu $m from the trap center, the amplitude of micromotion is $-q/2\times 52\approx 1.4\,\mu $m, which is well below the separation distance between the ions but larger than the optical wavelength (see supplementary materials for the distribution of the amplitude of micromotion).\
**Normal modes in the transverse direction.** With the knowledge of ion motion in the $x$-$y$ plane, we proceed to find the normal modes and quantize the motion along the transverse ($z$) direction. As ions are confined in the plane, micromotion along the transverse direction is negligible. The harmonic pseudopotential approximation is therefore legitimate. Expanding the Coulomb potential to second order, we have $\frac{\partial ^{2}}{\partial
z_{i}\partial z_{j}}\Big(\frac{1}{\tilde{r}_{ij}}\Big)\Bigr\rvert_{\vec{r}(t)}=\frac{1}{r_{ij}^{3}}$, where $\tilde{r}_{ij}=\sqrt{(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}+(z_{i}-z_{j})^{2}}$ is the 3D distance and $r_{ij}=\sqrt{(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}}$ is the planar distance between ions $i$ and $j$. To the second order, transverse and in-plane normal modes are decoupled. Note that coupling between the in-plane micromotion and the transverse normal modes has been taken into account in this expansion as the Coulomb potential is expanded around the dynamic ion positions $\vec{r}(t)$. With significant in-plane micromotion, distances between ions are time-dependent, which in turn affects the transverse modes. We can expand the quadratic coefficients in series: $$\dfrac{1}{r_{ij}^{3}}\approx \Big<\dfrac{1}{r_{ij}^{3}}\Big>+M_{ij}\cos
(\Omega _{T}t)+\cdots .$$The time-averaged coefficients $\left\langle 1/r_{ij}^{3}\right\rangle $ can be used to compute the transverse normal modes. The next order containing $\cos (\Omega _{T}t)$ terms can be considered as a time-dependent perturbation to the Hamiltonian. It contributes on the order of $O\left(
q\omega _{k}^{2}/\Omega _{T}^{2}\right) \sim O(qq_{z}^{2})$ in the rotating wave approximation, where $\omega _{k}$ is the transverse mode frequency. The term $\left\langle 1/r_{ij}^{3}\right\rangle
\approx \big(1/r_{ij}^{(0)}\big)^{3}(1-3q^{2}/4)+O(q^{3})$, where $r_{ij}^{(0)}$ is the ion distance computed with $\vec{r}\,^{(0)}$ without considering micromotion (see supplementary materials). Here, the micromotion effect is an overall renormalization in the term $1/r_{ij}^{3}$, so it does not modify the normal mode structure. Instead, it slightly shifts down the transverse mode frequencies (in the order of $O(q^{2})$). Numerically, we found an average reduction of around $0.4\,$kHz in each transverse mode frequency with our chosen parameters. Although mode structure is not altered by this overall renormalization, the discrepancy in equilibrium positions compared to the pseudopotential approximation will modify both the normal mode structure and mode frequencies.
![ **Nearest neighbor quantum gate in a 2D planar crystal.** Two laser beams with a wave vector difference $\Delta k$ aligned in the $z$ direction exert a spin-dependent force on the neighboring ions. Parameters used are: The wave vector difference of addressing beams $\Delta k=8 \, \protect\mu\text{m}^{-1}$; Laser beams are assumed to take a Gaussian profile with a beam waist $w=3 \, \mu$m centered at the average positions of the respective ion; The Lamb-Dicke parameter $\protect\eta_{z} = \Delta k \protect\sqrt{
\hbar/2m\protect\omega_{z} } \approx 0.029$. Other parameters are the same as in Fig. \[Fig:IonPosition\].[]{data-label="Fig:Schematics"}](IonSchematics.jpg){width="47.00000%"}
**High-fidelity quantum gates.** After obtaining the correct transverse normal modes, we now show how to design high-fidelity quantum gates with in-plane micromotion. Since NN gates are sufficient for fault-tolerant quantum computation in a planar crystal, we show as a demonstration that high-fidelity entangling gates can be achieved with a pair of NN ions in the trap center and near the trap edge. One may perform the gate along the transverse direction by shining two laser beams on the two NN ions with wave vector difference $\Delta k\hat{z}$ and frequency difference $\mu $ (see Fig. \[Fig:Schematics\]) [@Leibfried2003Experimental; @Choi2014Optimal]. The laser-ion interaction Hamiltonian is [@Zhu2006Trapped] $H=\sum_{j=1}^{2}\hbar \Omega _{j}\cos (\Delta k \cdot \delta z_{j}+\mu t)\sigma _{j}^{z}$ , where $\Omega _{j}$ is the (real) Raman Rabi frequency for the $j$th ion, $\sigma _{j}^{z}$ is the Pauli-$Z$ matrix acting on the pseudospin space of internal atomic states of the ion $j$, and $\delta z_{j}$ is the ion displacement from the equilibrium position. Quantize the ion motion, $\delta z_{j}=\sum_{k}\sqrt{\hbar /2m\omega _{k}}b_{j}^{k}(a_{k}+a_{k}^{\dag })$, with $b_{j}^{k}$ ($\omega _{k}$) being the mode vector (frequency) for mode $k$ and $a_{k}^{\dagger }$ creates the $k$-th phonon mode. Expanding the cosine term and ignoring the single-bit operation, the Hamiltonian can be written in the interaction picture as $$H_{\text{I}}= - \sum_{j=1}^{2} \sum_{k} \chi_{j} (t) g_{j}^{k} \big(a^{\dag}_{k} e^{i \omega_{k} t}+ a_{k} e^{-i \omega_{k} t} \big) \sigma_{j}^{z},$$ where $\chi_{j} (t) = \hbar \Omega_{j} \sin (\mu t)$, $g_{j}^{k} = \eta_{k} b_{j}^{k}$, and the Lamb-Dicke parameter $\eta_{k}= \Delta k \sqrt{ \hbar/2m\omega_{k} } \ll 1$. The evolution operator corresponding to the Hamiltonian $H_{\text{I}}$ can be written as [@Zhu2006Trapped; @Kim2009Entanglement; @Choi2014Optimal] $$U(\tau) = \exp\Big( i \sum_{j} \phi_{j} (\tau) \sigma_{j}^{z} + i \sum_{j<n} \phi_{jn} (\tau) \sigma_{j}^{z} \sigma_{n}^{z} \Big), \label{Eq:Evolution}$$ where the qubit-motion coupling term $\phi_{j} (\tau) =-i \sum_{k} \alpha_{j}^{k} (\tau) a_{k}^{\dag}- \alpha_{j}^{k*}(\tau) a_{k}$ with $\alpha_{j}^{k}(\tau) = \frac{i}{\hbar} g_{j}^{k} \int_{0}^{\tau} \chi_{j}(t) e^{i \omega_{k}t} dt$ and the two-qubit conditional phase $\phi_{jn}(\tau) = \frac{2}{\hbar^{2}} \sum_{k} g_{j}^{k}g_{n}^{k} \int_{0}^{\tau} \int_{0}^{t_{2}} \chi_{j} (t_{2}) \times \chi_{n}(t_{1}) \sin(\omega_{k}(t_{2}-t_{1}) ) dt_{1} dt_{2}$. To realize a conditional phase flip (CPF) gate between ions $j$ and $n$, we require $\alpha _{j}^{k}\approx 0$ so that the spin and phonons are almost disentangled at the end of the gate, and also $\phi _{jn}(\tau )=\pi /4$. It is worthwhile to note that in deriving Eq. , we dropped single-qubit operations as we are interested in the CPF gate. These fixed single-qubit operations can be explicitly compensated in experiment by subsequent rotations of single spins. (see supplementary materials for more detailed derivation and analysis).
{width="\textwidth"}
As the number of ions increases, transverse phonon modes become very close to each other in frequencies. During typical gate time, many motional modes will be excited. We use multiple-segment pulses to achieve a high-fidelity gate [@Zhu2006Arbitrary; @Zhu2006Trapped]. The total gate time is divided into $m$ equal-time segments, and the Rabi frequency takes the form $\Omega
_{j}(t)=\Omega _{j}^{(i)}\Omega _{j}^{\text{G}}(t)$, with $\Omega _{j}^{(i)}$ being the controllable and constant amplitude for the $i$th segment ($(i-1)\tau /m\leq t<i\tau /m$). Due to the in-plane micromotion, the laser profile $\Omega _{j}^{\text{G}}(t)$ seen by the ion is time-dependent. In our calculation, we assume the Raman beam to take a Gaussian form, with $\Omega _{j}^{\text{G}}(t)=\exp \left\{ -\left[ \big(x_{j}(t)-x_{j}^{(0)}\big)^{2}+\big(y_{j}(t)-y_{j}^{(0)}\big)^{2}\right] /w^{2}\right\} $, where $w$ is the beam waist and $\big(x_{j}^{(0)},y_{j}^{(0)}\big)$ are the average positions for the $j$th ion. Any other beam profile can be similarly incorporated.
To gauge the quality of the gate, we use a typical initial state for the ion spin $|\Phi _{0}\rangle =\left( |0\rangle +|1\rangle \right) \otimes \left(
|0\rangle +|1\rangle \right) /2$ and the thermal state $\rho _{m}$ for the phonon modes at the Doppler temperature. The fidelity is defined as $F=\operatorname{tr}_{m}\left[ \rho _{m}\big|\langle \Psi _{0}|U_{\text{CPF}}^{\dagger }U(\tau
)|\Psi _{0}\rangle \big|^{2}\right] $ tracing over the phonon modes, with the evolution operator $U(\tau )$ and the perfect CPF gate $U_{\text{CPF}}\equiv e^{i\pi \sigma _{1}^{z}\sigma _{2}^{z}}$. For simplicity, we take $\Omega _{j}^{(i)}=\Omega _{n}^{(i)}=\Omega ^{(i)}$ for the ions $j$ and $n$. For any given detuning $\mu $ and gate time $\tau $, we optimize the control parameters $\Omega ^{(i)}$ to get the maximum fidelity $F$. Fig. \[Fig:Gate\] shows the gate infidelity $\delta F=1-F$ and the maximum Rabi frequency $|\Omega |_{\max }=\max_{i}\Omega ^{(i)}$ for the center pair \[(a) and (b)\] and the edge pair \[(c) and (d)\] with $13$ segments and a relatively fast gate $\tau \approx 23\,\mu $s. Detuning $\mu$ can be used as an adjusting parameter in experiment to find the optimal results. All transverse phonon modes are distributed between $0.85 \omega_{z}$ and $\omega_{z}$. We optimize the gate near either end of the spectrum since optimal results typically occur there. Blue solid lines indicate the optimal results with micromotion and red dashed lines show the results for a genuine static harmonic trap, which are almost identical in (a), (b) and (c). It implies that micromotion can almost be completely compensated, but with a stronger laser power for the edge pair. If we apply the optimal result for the static trap to the realistic case with micromotion, the fidelity will be lower as indicated by the black dash-dot lines. This is especially so for the edge pair, where the fidelity is lower than $85\%$ at any detuning. It is therefore critical to properly include the effect of micromotion. With corrected pulse sequences, a fidelity $F>99.99\%$ can be attained with $|\Omega |_{\max }/2\pi \approx 12\,$MHz ($|\Omega |_{\max }/2\pi \approx 22\,
$MHz) for the center (edge) ions. The Rabi frequencies can be further reduced by a slower gate and/or more pulse segments.\
**Noise estimation.** Micromotion of any amplitude does not induce errors to the gates as it has been completely compensated in our gate design. We now estimate various other sources of noise for gate implementation. In considering the effect of in-plane micromotion to the transverse modes, we are accurate to the order of $q^{2}$, so an error of $q^{3}\approx 10^{-4}$ is incurred. The actual error is smaller since the Coulomb potential is an order of magnitude smaller than the trapping potential along the transverse direction. The cross-talk error probability due to beam spillover is $P_{c}=e^{-2(d/w)^{2}}<2\times 10^{-5}$, with the ion distance $d\gtrsim 7\,\mu $m and the beam waist $w=3\,\mu $m. At the Doppler temperature $k_{B}T_{D}/\hbar \approx 2\pi \times 10\,$MHz, thermal spread in positions may degrade the gate fidelity. Similar to micromotion, thermal motion causes the effective Rabi frequency to fluctuate. With $\omega _{x,y}/2\pi \approx 0.2\,$MHz, there is a mean phonon number $\bar{n}_{0}\approx 50$ in the $x$-$y$ plane. It gives rise to thermal motion with average fluctuation in positions, $\delta r\approx 0.23\,\mu $m, which can be estimated as in Ref. . The resultant gate infidelity is $\delta F_{1} \approx (\pi^{2}/4)(\delta r/w)^{4} \approx 10^{-4}$. Lastly, we estimate the infidelity caused by higher-order expansion in the Lamb-Dicke parameter. The infidelity is $\delta F_{2}\approx \pi
^{2}\eta _{z}^{4}(\bar{n}_{z}^{2}+\bar{n}_{z}+1/8)\approx 2\times 10^{-4}$, where $\bar{n}_{z}\approx 5$ is the mean phonon number in the transverse direction [@Zhu2006Trapped]. Other than the effects considered above, micromotion may also lead to rf heating when it is coupled to thermal motion. However, simulation has shown that at low temperature $T<10\,$mK and small $q$ parameters, rf heating is negligible [Zhang2007Molecular,Ryjkov2005Simulations]{}. Heating effect due to rf phase shift and voltage fluctuation should also be negligible when they are well-controlled [@Zhang2007Molecular].\
[**Discussion**]{}\
It is worthwhile to point out that although we have demonstrated the feasibility of our gate design via a single case with $N=127$ ions, the proposed scheme scales for larger crystals. The intuition is that through optimization of the segmented pulses, all phonon modes are nearly disentangled from the quantum qubits at the end of the gate. However, as the number of ions further increases, one would presumably need more and more precise control for all the experimental parameters ($<1\%$ fluctuation in voltage for example). rf heating may also destabilize a much larger crystal [@Buluta2008Investigation], and more careful studies are necessary for larger crystals.
One may also notice that in Ref. , we considered gates mediated by the longitudinal phonon modes, so the effect of micromotion is a phase modulation. Here, we utilize transverse modes so the amplitude of the laser beam is modulated. There are a few advantages in using the transverse modes: first, it is experimentally easier to access the transverse phonon modes in a planar ion crystal; second, in a planar crystal, the transverse direction is tightly trapped, so micromotion along that direction can be neglected; third, the transverse phonon modes do not couple to the in-plane modes and the in-plane micromotion affects the transverse modes via the time-dependence of the equilibrium positions, the effect of which is again suppressed due to tight trapping in the transverse direction.
In summary, we have demonstrated that a planar ion crystal in a quadrupole Paul trap is a promising platform to realize scalable quantum computation when micromotion is taken into account explicitly. We show that the in-plane micromotion comes into play through three separate effects, and each of them can be resolved. This paves a new pathway for large-scale trapped-ion quantum computation.
[10]{} url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
& ** (, ).
*et al.* . ** ****, ().
& . ** ****, ().
, & . ** ****, ().
& . ** ****, ().
, , , & . ** ****, ().
*et al.* . ** ****, ().
*et al.* . ** ****, ().
*et al.* . ** ****, ().
*et al.* . ** ****, ().
, , , & . ** ****, ().
. ** ****, ().
. ** ****, ().
, & . ** ****, ().
*et al.* . ** ****, ().
*et al.* . ** ****, ().
, , , & . ** ****, ().
, , & . ** ****, ().
& . ** ****, ().
, & . ** ****, ().
, & . ** ****, ().
. ** ****, ().
& . ** ****, ().
& . ** ****, ().
, , & ** ****, ().
*et al.* . ** ****, ().
, & . ** ****, ().
& . ** ****, ().
, , , & . ** ****, ().
, , & . ** ****, ().
& . ** ****, ().
, , & . ** ****, ().
*et al.* . ** ****, ().
, , & . ** ****, ().
, , , & . ** (). .
, & . ** ****, ().
, & . ** ****, ().
*et al.* . ** ****, ().
** (, ).
**. Ph.D. thesis, ().
. ** ****, ().
, , , & . ** ****, ().
, , & . ** ****, ().
*et al.* . ** ****, ().
*et al.* . ** ****, ().
*et al.* . ** ****, ().
, & . ** ****, ().
, , & . ** ****, ().
\
We would like to thank T. Choi and Z.-X. Gong for useful discussions. This work was supported by the NBRPC (973 Program) No. 2011CBA00300 (No. 2011CBA00302), the IARPA MUSIQC program, the ARO, and the AFOSR MURI program.\
\
C.S. and L.-M.D. conceived the idea. S.-T.W. and C.S. carried out the calculations. S.-T.W. and L.-M.D. wrote the manuscript. All authors contributed to the discussion of the project and revision of the manuscript.\
\
**Supplementary information** is available.\
**Competing financial interests:** The authors declare no competing financial interests.
Supplementary Information: Quantum Computation under Micromotion in a Planar Ion Crystal
========================================================================================
> In this supplementary information, we provide more details on the iterative method to find dynamic ion positions, and also consider the effect of in-plane micromotion to the transverse normal modes. We also include a more detailed derivation for the Hamiltonian and time-evolution operator for a two-ion entangling gate.
Iterative method to find dynamic ion positions {#App:Iterative}
==============================================
As discussed in the main text, the equations of motion in each direction can be written in the standard form of Mathieu equations (neglecting Coulomb potential): $$\frac{d^{2}r_{\nu}}{d\xi^{2}}+\left[a_{\nu}-2q_{\nu}\cos(2\xi)\right]r_{\nu}=0,$$ where $\nu \in \{x,y,z \}$, $\xi=\Omega_{T} t/2$, and dimensionless parameters $a_{\nu}$ and $q_{\nu}$ are defined in the main text. The characteristic exponents $\beta_{\nu}$ can be computed from $a_{\nu}$ and $q_{\nu}$ iteratively [@mclachlan1951theory]. A pseudopotential can then be obtained with secular frequencies $\omega_{\nu} = \beta_{\nu} \Omega_{T}/2$ and $$e\left(\Phi_{\text{DC}}+\Phi_{\text{AC}}\right)\approx\frac{1}{2}m\omega_{x}^{2}x^{2}+\frac{1}{2}m\omega_{y}^{2}y^{2}+\frac{1}{2}m\omega_{z}^{2}z^{2}.$$ Assuming tight trapping along the $z$ direction, i.e. $\omega_{z}/\omega_{x,y}>10$, a planar crystal is formed in the $x$-$y$ plane. Adding the Coulomb potential $V_{C}$, one acquires a time-independent potential in the plane: $$\begin{aligned}
V_{\text{pseudo}}(x,\, y) =\sum_{i}\left(\frac{1}{2}m\omega_{x}^{2}x_{i}^{2}+\frac{1}{2}m\omega_{y}^{2}y_{i}^{2}\right)+
\sum_{i<j} \frac{e^{2}}{4\pi\epsilon_{0}\sqrt{(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}}}.\end{aligned}$$ $i=1,2,\cdots,N$, where $N$ is the number of ions. Numerically, we start with $N=127$ ions forming equilateral triangles in a 2D hexagonal structure \[Fig. \[FigSup:IonPosition\](a)\], and find the static equilibrium positions $\vec{r}\,^{(0)} = (x_{1}^{(0)},y_{1}^{(0)}, \cdots, x_{N}^{(0)},y_{N}^{(0)})$ under this pseudopotential approximation by solving the classical equations of motion with a frictional force $\left( -\eta(\dot{x}+\dot{y}) \right)$, simulating the cooling process in experiment. This set of static equilibrium positions \[marked by squares in Fig. \[FigSup:IonPosition\](b)\] is the starting point to derive the oscillatory behavior of each ion under micromotion.
In a planar crystal, the ions oscillate slightly around their average positions, so it is appropriate to expand the Coulomb potential around the equilibrium positions $\vec{r}\,^{(0)}$. To the second order, the Coulomb potential can be written in a quadratic form: $$V_{C} \approx \dfrac{1}{2} \vec{r}\,^{T} M_{C} \vec{r} + \vec{g} \,^{T} \vec{r} + \text{constant term}, \label{EqSup:Expansion}$$ where $\vec{r}=(x_{1},y_{1}, \cdots, x_{N}, y_{N})$, $M_{C}$ is a $2N \times 2N$ matrix, and $\vec{g}$ is a $2N$-vector. The trapping potential can also be written in this coordinate basis: $$e\left(\Phi_{\text{DC}}+\Phi_{\text{AC}}\right) = \dfrac{1}{2} \vec{r}\,^{T} M_{DC} \vec{r} + \frac{V_{0}}{d_{0}^{2}} \cos(\Omega_{T} t) \vec{r}\,^{T} I_{2N} \,\vec{r},$$ where $I_{2N}$ is the $2N \times 2N$ identity matrix, and $M_{DC}$ is a diagonal matrix with $2(1+\gamma) eU_{0}/d_{0}^{2}$ in the odd rows (x coordinates), and $2(1-\gamma) eU_{0}/d_{0}^{2}$ in the even rows (y coordinates). Therefore, the total potential energy is $$V = \dfrac{1}{2} \vec{r}\,^{T} (M_{DC}+M_{C}) \vec{r} + \frac{V_{0}}{d_{0}^{2}}\cos(\Omega_{T} t) \vec{r}\,^{T} I_{2N} \,\vec{r} + \vec{g} \,^{T} \vec{r}.$$ Note that the time-dependent part of the potential is isotropic in the coordinates, so it does not couple each Mathieu equations. We can find an orthogonal matrix $Q$ that diagonalizes the first term, i.e. $Q(M_{DC}+M_{C}) Q^{T}= \Lambda$. Using the normal coordinates $ \vec{s} = Q \vec{r}$, the equations of motion form decoupled Mathieu equations: $$\frac{d^{2}s_{i}}{d\xi^{2}}+ (a_{i}-2q_{i} \cos(2\xi)) s_{i}= f_{i},$$ where $a_{i}= 4\Lambda_{ii}/m\Omega_{T}^{2}$, $q_{i} =q= -4 eV_{0}/md_{0}^{2}\Omega_{T}^{2}$, and $f_{i}= -\frac{4}{m\Omega_{T}^{2}} \left(Q\vec{g} \right)_{i}$. The inhomogeneous Mathieu equations can be solved by substituting a special solution in the form of $s_{i} = f_{i} \sum_{n=0}^{\infty} c_{i}^{(n)} \cos(2n\xi)$, and the series coefficients $c_{i}^{(n)}$ can be computed numerically [@Shen2014High]. After that, the ion coordinates can be transformed back to the Cartesian coordinates $\vec{r} = Q^{T} \vec{s}$, where $\vec{r}$ can be expressed successively as $$\vec{r} = \vec{r}\,^{(0)} + \vec{r}\,^{(1)} \cos(2\xi) + \vec{r}\,^{(2)} \cos(4\xi) + \cdots.
\label{EqSup:Positions}$$ $ \vec{r}\,^{(0)}$ now becomes the new average (equilibrium) positions, and can be substituted back to the expansion in equation . The ion positions $\vec{r}$ can be attained self-consistently in this manner. A dynamical expansion of the Coulomb potential around $\vec{r}\,^{(0)} + \vec{r}\,^{(1)} \cos(2\xi)$ may yield a more accurate result for the normal modes in the plane [@Landa2012Modes]. For our purpose, the static expansion is sufficient as we only need accurate ion positions $\vec{r}$ to compute the normal modes along the $z$ direction. Numerically, we found that $\vec{r}\,^{(1)} \approx -\frac{q}{2} \vec{r}\,^{(0)}$ and $\vec{r}\,^{(2)} \approx \frac{q^{2}}{32} \vec{r}\,^{(0)}$, which are consistent with previous results [@Shen2014High; @Landa2012Modes]. Hence, micromotion only results in breathing oscillations about the average positions of each ion. The further the ion is from the center of the trap, the larger the amplitude of micromotion becomes.
Fig. \[FigSup:MicroAmp\] shows the amplitude of micromotion for each ion. The largest amplitude for the edge ion is around $1.35 \, \mu$m, which is well below the ion separation ($7 \sim 10 \, \mu$m), necessary for the formation of a well-defined crystal and for individual addressing.
Normal modes along the transverse direction {#App:Normal}
===========================================
With the knowledge of the motion of ions in the $x$-$y$ plane, we could find the normal modes and quantize the motion along the transverse ($z$) direction. As ions are confined in the plane, micromotion along the transverse direction is negligible. A harmonic pseudopotential is thus valid for the $z$ direction. Expanding the Coulomb potential to second order again, we have $$\begin{aligned}
V_{z} = \dfrac{1}{2}m \omega_{z}^{2} \sum_{i} z_{i}^{2} +
\dfrac{e^{2}}{4\pi \epsilon_{0}} \left[ \sum_{i\neq j} \left(\dfrac{1}{r_{ij}^{3}} \right) z_{i} z_{j} - \sum_{i\neq j} \left(\dfrac{1}{r_{ij}^{3}} \right) z_{i}^{2} \right],\end{aligned}$$ where $r_{ij}= \sqrt{(x_{i}- x_{j})^{2}+ (y_{i}-y_{j})^{2} }$. $x_{i}(t)$ and $y_{i}(t)$ are time-dependent though, due to the in-plane micromotion. From here, we can see explicitly that the transverse modes are decoupled from the planar modes. Expanding the term $1/r_{ij}^{3}(t)$ in series, one has $$\dfrac{1}{r_{ij}^{3}} \approx \Big< \dfrac{1}{r_{ij}^{3}} \Big> + M_{ij} \cos (\Omega_{T} t) + \cdots$$ The matrix element $M_{ij}$ is in the order of $O(q)$ and can be obtained numerically from $\left<\cos(\Omega_{T} t) /r_{ij}^{3} \right>$. To have an intuitive understanding of the effect of micromotion on transverse modes, we take positions $\vec{r}$ in the form of Eq. , obtaining $$\begin{aligned}
\dfrac{1}{r_{ij}^{3}} \approx \left( \dfrac{1}{r_{ij}^{(0)}} \right)^{3} \left(1 -\dfrac{q}{2} \cos(\Omega_{T} t) + \dfrac{q^{2}}{32} \cos(2\Omega_{T} t) \right)^{-3} + O(q^{3}),\end{aligned}$$ where $r_{ij}^{(0)}$ is the zeroth order approximation using the average positions $ \vec{r}\,^{(0)}$ without considering micromotion. Thus, $\left<1 /r_{ij}^{3} \right> \approx \left( 1/r_{ij}^{(0)} \right)^{3}(1-3q^{2}/4) + O(q^{3})$, where we used the fact that $\left< \cos(\Omega_{T} t)\right> =0$ and $\left< \cos^{2}(\Omega_{T} t)\right> =1/2$. From the time-independent term $\left<1 /r_{ij}^{3} \right>$, we diagonalize $V_{z}$ and find the normal modes as well as the eigenenergies in the transverse direction. Subsequently, we quantize the total Hamiltonian (with kinetic energy) and write $H = \sum_{k} \hbar \omega_{k} a_{k}^{\dagger} a_{k}$, where $a_{k}$ is the annihilation operator for the quantized phonon mode, and $\omega_{k}$ is the corresponding eigenfrequency. In the interaction picture, $a_{k} \to a_{k} e^{-i\omega_{k} t}$. The time-dependent term containing $\cos(\Omega_{T} t)$ can then be treated as a perturbation; under the rotating wave approximation, since $\Omega_{T} \gg \omega_{k}$, the term affects the normal modes to the order of $O\left(q\omega_{k}^{2}/ \Omega_{T}^{2} \right) \sim O(qq_{z}^{2})$, which can be safely neglected. Since the first term in $V_{z}$ is diagonal in $z_{i}$ and the second term is reduced by a factor $(1-3q^{2}/4)$ by micromotion, the normal mode structure remains unchanged, and the mode frequencies are reduced slightly.
Two-ion Entangling Gate {#App:Entangle}
=======================
The spin-dependent force on an ion is due to the AC Stark shift on each spin state. A different shift on the two internal spin states of an ion results in a Hamiltonian $$H =\hbar \dfrac{|\Omega_{\text{eg}}|^{2}}{4\delta} \sigma^{z},$$ where $\Omega_{\text{eg}}$ is the Rabi frequency of the laser beam and $\delta$ is the detuning from the excited state. By shining two laser beams at an angle with wave vectors $\mathbf{k}_{1}$, $\mathbf{k}_{2}$ and frequencies $\omega_{1}$, $\omega_{2}$, we have $$\Omega_{\text{eg}}= \Omega_{0} \left( e^{i (\mathbf{k}_{1} \cdot \mathbf{r} + \omega_{1} t+\phi) } + e^{i (\mathbf{k}_{2} \cdot \mathbf{r} + \omega_{2} t )} \right),$$ where $\phi$ is the phase difference between two beams. So we have $$H = \hbar \Omega \left(1+ \cos(\Delta k \cdot z + \mu t + \phi) \right) \sigma^{z},$$ where $\Omega = \Omega_{0}^{2}/2\delta$ is the effective two-photon Rabi frequency, $\Delta k \hat{z}= \mathbf{k}_{1} - \mathbf{k}_{2}$ is aligned along the $z$ direction, and $\mu = \omega_{1} - \omega_{2}$. As we are mostly interested in the two-qubit entangling gate, which is the building block for universal quantum gates, we consider laser beams shining on two ions, and ignore the first term $\hbar \Omega \sigma^{z}$ in the Hamiltonian that only induces single bit operations. We therefore have $$H = \sum_{j=1}^{2} \hbar \Omega_{j} \cos (\Delta k \cdot z_{j} + \mu t + \phi_{j}) \sigma_{j}^{z},$$ The ion position $z_{j} =z_{j0} + \delta z_{j}$, where $z_{j0}$ is the equilibrium position and $\delta z_{j}$ is the small displacement. We dump the term $\Delta k \cdot z_{j0}$ to the phase $\phi_{j}$, and expand the cosine term in the Lamb-Dicke limit $\Delta k \cdot \delta z_{j} \ll 1$, $$\begin{aligned}
H &= \sum_{j=1}^{2} \hbar \Omega_{j} \cos (\Delta k \cdot \delta z_{j} + \mu t + \phi_{j}) \sigma_{j}^{z} \\
&\approx - \sum_{j=1}^{2} \hbar \Omega_{j} \sin(\Delta k \cdot \delta z_{j}) \sin ( \mu t + \phi_{j}) \sigma_{j}^{z} \label{Eq:expansion} \\
& \approx - \sum_{j,k} \hbar \Omega_{j} \sin ( \mu t + \phi_{j}) \Delta k \left[\sqrt{\dfrac{\hbar}{2m\omega_{k}}} b_{j}^{k} a_{k}^{\dag} + \text{H.c.} \right] \sigma_{j}^{z} \notag \\
&= - \sum_{j=1}^{2} \sum_{k} \chi_{j} (t) g_{j}^{k} (a^{\dag}_{k}+ a_{k}) \sigma_{j}^{z}\end{aligned}$$ In step , we drop the cosine-cosine term $\hbar \Omega_{j} \cos(\Delta k \cdot \delta z_{j}) \cos ( \mu t + \phi_{j}) \sigma_{j}^{z} \approx \hbar \Omega_{j} \cos ( \mu t + \phi_{j}) \sigma_{j}^{z} $ since $\Delta k \cdot \delta z_{j} \ll 1$ and it thus does not couple the phonon modes to the spin (in the first-order approximation), resulting in a single-qubit operation. Various terms are defined as $$\delta z_{j} = \sum_{k} \sqrt{\dfrac{\hbar}{2m\omega_{k}}} b_{j}^{k} a_{k}^{\dag} + \text{H.c.}$$ where $b_{j}^{k}$ are the mode vector for mode $k$, $a_{k}^{\dag}$ creates the $k$-th phonon mode (harmonic oscillator mode). The matrix $b_{n}^{k}$ diagonalizes the approximate harmonic potential of the system. $$\begin{aligned}
\chi_{j} (t) &= \hbar \Omega_{j} \sin (\mu t + \phi_{j}) \\
g_{j}^{k} &= \eta_{k} b_{j}^{k}, \quad \text{where} \quad \eta_{k}= \Delta k \sqrt{ \dfrac{\hbar}{2m\omega_{k}} }\end{aligned}$$ $\eta_{k}$ is the Lamb-Dicke parameter, $\eta_{k} \ll 1$ to be valid (for the expansion). For $\Delta k=8 \mu m^{-1}$, $m=171u$ for Ytterbium, and take the transverse mode $\omega_{k} = 2\pi \times 2$MHz. We will have $\eta_{k} \approx 0.03$. Going into the interaction picture and replacing $a_{k} \to a_{k} e^{-i \omega_{k} t}$, we have $$H_{\text{I}}= - \sum_{j=1}^{2} \sum_{k} \chi_{j} (t) g_{j}^{k} (a^{\dag}_{k} e^{i \omega_{k} t}+ a_{k} e^{-i \omega_{k} t}) \sigma_{j}^{z}$$ The evolution operator can be obtained from the Hamiltonian as [@Zhu2006Trapped; @Kim2009Entanglement] $$\begin{aligned}
U(\tau) &= \exp \big( i \sum_{j} \phi_{j} (\tau) \sigma_{j}^{z} + i \sum_{j<n} \phi_{jn} (\tau) \sigma_{j}^{z} \sigma_{n}^{z} \big), \\
\phi_{j} (\tau) &=-i \sum_{k} \alpha_{j}^{k} (\tau) a_{k}^{\dag}- \alpha_{j}^{k*}(\tau) a_{k} \\
\alpha_{j}^{k}(\tau) &= \dfrac{i}{\hbar} g_{j}^{k} \int_{0}^{\tau} \chi_{j}(t) e^{i \omega_{k}t} dt, \label{Eq:Alpha} \\
\phi_{jn}(\tau) &= \dfrac{2}{\hbar^{2}} \sum_{k} g_{j}^{k}g_{n}^{k} \int_{0}^{\tau} \int_{0}^{t_{2}} \chi_{j} (t_{2}) \chi_{n}(t_{1}) \times \sin(\omega_{k}(t_{2}-t_{1})) dt_{1} dt_{2}.\end{aligned}$$ To obtain a two-qubit entangling gate, we need $\alpha_{j}^{k} =0$ so that the spin and phonons are disentangled at the end of the gate, and $\phi_{jn} (\tau) =\pi/4$. This is the starting point to calculate the fidelity of the gate.
|
---
author:
- Andrzej Latka
- Yilong Han
- 'Ahmed M. Alsayed'
- 'Andrew B. Schofield'
- 'A. G. Yodh'
- 'Piotr Habdas[^1]'
title: 'Particle dynamics in colloidal suspensions above and below the glass-liquid re-entrance transition'
---
Introduction
============
Theory, simulation, and experiment have demonstrated that a colloidal system can be driven from a hard-sphere glass to an attractive glass by increasing short-range attractions between colloidal particles [@pham2002mgs; @eckert2002reg; @pham2004ghs; @kaufman2006dir; @simeonova2006doc; @bergenholtz1999ntc; @dawson2000hog; @puertas2002css; @puertas2004dhc]. In colloidal suspensions this effect is typically realized by adding nonadsorbing polymers to the colloidal suspension. Depletion forces [@asakura1954; @illet1995; @yodh2000abh], induced in this way, cause the particles to move closer to one another, and the system exhibits a transition from a hard-sphere glass to an attractive liquid [@pham2002mgs; @eckert2002reg; @pham2004ghs; @kaufman2006dir; @simeonova2006doc]. Increasing the polymer concentration even further causes the system to enter an attractive glass phase [@pham2002mgs; @eckert2002reg; @pham2004ghs; @kaufman2006dir; @simeonova2006doc].
Calculations and molecular dynamics simulations [@bergenholtz1999ntc; @dawson2000hog; @puertas2002css; @puertas2004dhc] suggest that reentrance to the glass phase is due to the existence of two qualitatively different glassy states. In hard-sphere colloidal suspensions the system enters a glass phase through a caging mechanism: as the volume fraction $\phi$ is increased, particles are increasingly trapped by their neighbors, until a critical volume fraction $\phi_{g}\sim0.58$ is reached; then caging becomes effectively permanent, stopping long-range particle motion. In attractive glasses, the attractive part of the potential causes particles to move closer to one another and eventually binds them at contacts. In these glasses, structural arrest is due to bonding. Thus, it is believed that the two types of glasses should have different structural and dynamical properties.
Despite numerous theoretical and experimental studies of the reentrant glass transition in colloidal suspensions, to our knowledge, only a couple of investigations have employed direct microscopic imaging to study the mechanism of this process [@kaufman2006dir; @simeonova2006doc]. Notably, Kaufman and Weitz [@kaufman2006dir] extracted qualitative information from microscopic images about particle motion magnitude and observed that particles in repulsive glasses exhibit cage rattling and escape, while in attractive glasses they exhibit large displacements upon cage escape. In a different vein, Simeonova et. al. [@simeonova2006doc] reported that melting of the hard-sphere glass is accompanied by significant changes in the particle displacement distributions and their moments.
In this Letter, we study this system class as it is brought from a hard-sphere glass into the attractive liquid region using a qualitatively different set of microscopic parameters. Confocal microscopy experiments reveal particles that exhibit motional “events” wherein they move significantly farther than the thermal fluctuations within their cages. We characterize the transition using the properties of these motional events. Interestingly, the average displacement of particles that exhibit motional events increases only slightly as the system is brought from a hard-sphere glass to the attractive liquid. However, the average event duration, expressed in units of Brownian time, decreases by more than an order of magnitude under the same conditions. Thus, effective event motional speed increases with increasing interparticle attractions by almost an order of magnitude. Moreover, as the polymer-concentration-induced attractions increase, the number of particles exhibiting motional events increases by an order of magnitude, and the cluster size of particles exhibiting motional events also increases, i.e. the event motion is correlated over longer length scales.
Experimental
============
The particles used in this study were poly-(methylmethacrylate) (PMMA) spheres, sterically stabilized by a thin layer of poly-12-hydroxystearic acid (radius $a = 1.1$ $\mu m$, polydispersity of $\sim 5 \%$) and dyed with rhodamine. The PMMA particles were suspended in a mixture of cyclohexylbromide/*cis*- and *trans*-decalin which nearly matches the density and the index of refraction of the PMMA particles to the solvent. Coulombic interactions due to surface charges on the colloidal particles were screened by adding 2 mM of tetra-butyl-ammonium chloride [@yethiraj2003cms].
To induce the depletion attraction between PMMA particles, linear polystyrene polymer ($M_w = 7.5\times10^6$ Da; radius of gyration $r_g \approx 106$ nm) was added to the particle suspension. Specifically, a series of samples with polymer concentrations, $c_p$, varying from $0-1.8$ mg/ml was prepared using the following method. For each suspension, a colloidal sample was centrifuged to the random closed-packed volume fraction (RCP). The sample was subsequently diluted by a mixture of the density matched liquid and polymer, yielding a sample particle volume fraction of $\phi=0.60$ (RCP in our samples was measured to be 0.66) and the desired polymer concentration. After 24 hours of homogenization by mixing and tumbling, the colloidal suspension was loaded into a glass microscopy cell along with a small piece of magnetic wire to be used later for reinitiating the sample by stirring. The ratio of the polymer radius of gyration to colloidal particle radius is 0.09.
We used fluorescent and confocal microscopy to capture 2D image slices in 3D samples with a time resolution of 6 s over a time period of 3 hours. Measurements began 10 minutes after stirring, insuring that flows within the sample had time to subside [@courtland2003dva], and measurements were taken at least 35 $\mu m$ away from the cover slip surface to minimize wall effects. The position of each particle within the optical plane was obtained using standard particle tracking techniques [@crocker1996mdv].
A side effect of adding polymer to induce the depletion attraction is to increase the solution viscosity [@pham2004ghs; @kaufman2006dir; @simeonova2006doc]. As a result, at higher polymer concentrations colloidal particles diffuse more slowly. To facilitate comparisons of particle dynamics between samples with different polymer concentration and thus different viscosity, we scale the experimental time for each sample by the time it would take an isolated particle to diffuse its radius in a suspension with the same polymer concentration. Therefore, we analyze particle dynamics in units of Brownian time $t_{B}=\frac{a^{2}}{D_{0}(c_p)}$, where $D_{0}(c_p)$ is the diffusion constant for an isolated particle in the solvent at the polymer concentration $c_p$ [@kaufman2006dir].
Results and Discussion
======================
Figure \[phasediag\] presents a phase diagram of the reentrant glass transition. Solid triangles in Fig. \[phasediag\] correspond to the samples studied, starting from a hard-sphere glass ($c_p = 0$ mg/ml) and ending in the attractive liquid region. The solid lines are schematic, qualitatively indicating the glass transition boundaries. We estimated the lower boundary using sample crystallization (e.g., via the bond orientational order parameter $\Psi_6$). The upper boundary is a conjecture. Samples in the attractive liquid region showed significant crystallization compared to samples in the hard-sphere glass region.
Evidence for the reentrant glass transition has been derived in the past [@pham2002mgs; @eckert2002reg; @pham2004ghs; @kaufman2006dir; @simeonova2006doc]. Here it is apparent in the plot of $D(c_p)/D_0(c_p)$ vs. $c_p$, where $D(c_p)$ is the long time diffusion constant of the particles in suspension at high $\phi$ with polymer concentration $c_p$. Indeed, $D(c_p)/D_0(c_p)$ changes by one to two orders of magnitude as the system evolves from a hard-sphere glass ($c_p = 0$ mg/ml) into the attractive liquid region ($c_p \sim$ 2.0 mg/ml).
Microscopy studies of colloidal suspensions permit determination of particle positions during the entire experiment. Thus, it is possible to quantify the behavior of the most “active” particles in suspension; these “active” particles have been of particular interest recently [@weeks2000tdd; @kegel2000; @weeks2002pcr; @vollmayrlee2001dhb]. We define particles to be active when they move significantly farther than the thermal fluctuations in their cages, following the definition of particle “jumps” in Ref. [@vollmayrlee2001dhb] (see top row Fig. \[exampleevents\]).
For each particle, we calculate running-five-point average position: $\overline{r}(t)=\frac{1}{5}\sum^{t+2}_{i=t-2}r(i)$. Next, we calculate the change in this average particle position, $\Delta \overline{r}$, during the time interval $\Delta t$: $\Delta \overline{r(t)} =
\overline{r(t)}-\overline{r(t-\Delta t)}$. Finally, we compare the average displacement $\Delta \overline{r}$ with average fluctuations ($\sigma$) of the particle during the entire time, $T$, that the particle is tracked: $\sigma^2=\frac{1}{N}\sum^{i=N}_{i=1}(\overline{r(t_i)^2}-\overline{r(t_i)}^2)$ where $N$ is the total number of time steps (see Fig. \[exampleevents\]). If $\Delta \overline{r(t)} > \sqrt{20}\sigma$, then we say that at time $t$ the particle exhibited a *motional event* of duration $\Delta t$ (for further details see Ref.[@vollmayrlee2001dhb]). Typically, $r(t)$ is constant with fluctuations before and after a motional event. To properly identify events, it is important to choose an appropriate $\Delta t$. A short $\Delta t$, e.g. comparable to one Brownian unit, is useful for identifying quick events (jumps), but it is not suitable for identifying more gradual motions; we observe a wide range of motional event durations, including ones that last more than several Brownian time units (bottom row Fig. \[exampleevents\]). Therefore, to identify particles that change their positions over relatively longer times, we perform the above calculation for a range of $\Delta t$’s. Furthermore, we only include particles which exhibit a complete motional event, i.e. an event that began and ended during the time of the experiment.
We first focus attention on the distribution of position displacements, $\Delta \overline{r}$, for motional events. Average particle displacement $\langle \Delta \overline{r} \rangle$ during an event vs. polymer concentration is shown in Figure \[drdt\]a. Interestingly, as the polymer concentration increases, particles exhibiting motional events travel further by only about 0.05 $\mu m$, about one tenth of the particle diameter. This small displacement is on the order of the cage Brownian fluctuations and tracking uncertainty. The observation is somewhat counterintuitive, since one might expect that with increasing polymer concentration, the number of colloidal particles that become stuck to each other increases, thus creating more free space for other particles to jump [@simeonova2006doc]. Our data do not appear to support such a conjecture.
Similarly, we analyzed distributions of event durations. Figure \[drdt\]b presents the average event duration (in Brownian units) vs. polymer concentration. Average event duration decreases from about 170 Brownian units for $c_p = 0$ mg/ml to about 15 Brownian units for $c_p = 0.8$ mg/ml. For polymer concentrations in the attractive liquid region, the average event time saturates at about 15 Brownian units. Thus, as the polymer concentration increases, particles that exhibit motional events do so in a shorter time until the attractive liquid region is reached, wherein all motional events take approximately the same time.
From event displacement and duration information, we calculate particle motional event speed, $\langle \Delta \overline{r}/\Delta t \rangle$, and on Fig. \[eventspeed\] we plot average particle event speed vs. polymer concentration. Particles experiencing motional events move faster with increasing polymer concentration. For the samples in the vicinity of the attractive liquid region, the event speed changes by almost an order of magnitude with respect to the event speed in the hard-sphere sample. Then, for polymer concentrations farther into the attractive liquid region, the event speed saturates.
We next consider the raw number of particle events as a function of polymer concentration as shown in Figure \[eventsall\]a. The number of particle events initially increases from about 3 to almost 100 and then saturates for polymer concentrations in the attractive liquid region. We might expect that as the attractive glass phase is approached, the number of motional events would decrease. However, such a plot (Fig. \[eventsall\]a) does not account for the increase of solvent viscosity with increasing polymer concentration. Thus, we calculate event rate by scaling the number of events by the length of the data in units of Brownian time $t_B$ (Fig. \[eventsall\]b). The “viscosity normalized” event rate increases with polymer concentration by more than an order of magnitude until polymer concentrations of about 1 mg/ml are reached. Therefore, the number of particles that exhibit motional events, and hence are responsible for the relaxation in the samples, increases significantly with polymer concentration as the system fluidizes.
To analyze the collective behaviors of particles exhibiting motional events further, we examine the spatial distributions of particles that exhibit motional events. Figure \[flowhistrogramsandpics\] shows representative microscopy snapshots for three polymer concentrations. White dots are plotted over particles that exhibit motional events with arrows indicating the direction of the motion. As the polymer concentration increases, a particle that exhibited a motional event has, on average, more neighbors that are also moving significantly. Thus, with increasing polymer concentration more particles are moving cooperatively.
To look for spatial correlations of the particles that exhibited a motional event, we analyze the nearest neighbor connectivity and thus identify clusters of connected particles that exhibit a motional event. In Fig. \[clusters\] we plot the frequency distribution of cluster size, $P(N_c)$, vs. number of particles in a cluster, $N_c$, for representative polymer concentrations. For low polymer concentrations, particles have a tendency to move in small clusters. As the polymer concentration increases, the particles move in increasingly bigger clusters. At the highest polymer concentration studied here, $c_p=1.8$ mg/ml, we observed clusters composed of as many as ten particles. Thus, as the attractive liquid region is approached, structural relaxation occurs because of the motion of small numbers of large cooperative clusters of particles that exhibit motional events, rather than mostly solitary particles, as we observe in hard-sphere glass. This effect is also indicated by average cluster size, $N_c$. The average cluster size increases by almost a factor of two with polymer concentration as shown in Figure \[clusters\]b. However, the size distribution of the clusters of particles that exhibited a motional event is likely even broader than presented here since clusters may extend beyond the viewing area of x-y focal plane.
In summary, we have studied a colloidal system with short-range attractive potential in the reentrant region using primarily the properties of “motional events”. We observe the transition from a hard-sphere arrested phase to a liquid-like phase. This transition is characterized by increase in: $D(c_p)/D_0(c_p)$, event speed, and the event rate of moving particles. Interestingly, particles exhibiting a motional event do not move longer distances at higher polymer concentration, but they do move faster (i.e. in Browniant time units). The transition to the reentrant region is also characterized by a growing number of particles that experience motional events. Moreover, the particles experiencing motional events are increasingly spatially correlated with increasing attraction. The particles move in clusters, and the distribution of the cluster size becomes broader and shifts to larger average values with increasing interparticle attraction.
Future microscopy studies should include exploration of re-entrance into the attractive glass region and possibly the influence of particle-to-polymer size ratio on the system dynamics [@ren1993]. Also, since our studies provide only 2-dimensional information, 3-dimensional studies should shed more light on the cluster size and distance traveled by the particles exhibiting motional events. Finally, similar systematic studies of particle dynamics in colloidal suspensions with short-range attractions along the low volume fraction extension of the attractive glass phase line may lead to a unified description of glasses and gels.
We thank Paul J. Angiolillo, Katharina Vollmayr-Lee, Eric R. Weeks, Peter Yunker, and Zexin Zhang for helpful discussions, and Paul J. Angiolillo for comments on the manuscript. PH acknowledges financial support from an award from Research Corporation and Sigma Xi Scientific Research Society, SJU Chapter. AGY acknowledges partial support from the NSF DMR-052002 (MRSEC) and DMR-0804881.
[0]{}
. . . . .
.
. . . .
.
.
. . . . .
. . .
[^1]: E-mail:
|
---
abstract: 'To better understand the inner workings of information spreading, network researchers often use simple models to capture the spreading dynamics. But most models only highlight the effect of local interactions on the global spreading of a single information wave, and ignore the effects of interactions between multiple waves. Here we take into account the effect of multiple interacting waves by using an agent-based model in which the interaction between information waves is based on their novelty. We analyzed the global effects of such interactions and found that information that actually reaches nodes reaches them faster. This effect is caused by selection between information waves: slow waves die out and only fast waves survive. As a result, and in contrast to models with non-interacting information dynamics, the access to information decays with the distance from the source. Moreover, when we analyzed the model on various synthetic and real spatial road networks, we found that the decay rate also depends on the path redundancy and the effective dimension of the system. In general, the decay of the information wave frequency as a function of distance from the source follows a power law distribution with an exponent between -0.2 for a two-dimensional system with high path redundancy and -0.5 for a tree-like system with no path redundancy. We found that the real spatial networks provide an infrastructure for information spreading that lies in between these two extremes. Finally, to better understand the mechanics behind the scaling results, we provide analytical calculations of the scaling for a one-dimensional system.'
author:
- 'A. Mirshahvalad'
- 'A. V. Esquivel'
- 'L. Lizana'
- 'M. Rosvall'
bibliography:
- 'TW\_PRE\_110913.bib'
title: Dynamics of interacting information waves in networks
---
Introduction
============
In today’s society, we are flooded with information. Waves of new ideas, innovations, products, and trends follow each other in quick succession. To better understand the inner workings of the dynamics, researchers often use simple models to capture important spreading mechanisms [@Valente199669; @bikhchandani1998learning; @TheoryOfFads; @Kempe; @goldenberg2001using; @hedetniemi1988survey; @bornholdt2011emergence] on a complex network [@RevModPhys.74.47; @newman2003structure; @boccaletti06; @Sales-Pardo25092007; @ClausetEtAl2008a; @VespignaniNPhys2011; @Jeong2000; @Kleinberg:2000p5066; @Milo2002]. Broadly speaking, there are two classes of such models: threshold models [@granovetter1978threshold; @watts2002simple; @bailey1975mathematical; @hethcote2000mathematics; @karimi2012threshold] and contagion models [@daley1964epidemics; @PhysRevLett.92.218701; @dodds2005generalized; @nekovee2007theory]. Threshold models assume that individuals adopt new information or technology if a certain proportion of their friends have adopted it. This mechanism leads to cascades that, under favorable conditions, can propagate throughout the entire system. Contagion models assume that individuals spread information or rumors much like they spread microbial infections, through interactions. This mechanism can also cause spreading across the entire system, provided that the transmission rate is sufficiently high. Both types of models highlight the effect of local interactions on global spreading, but, in general, they ignore effects of interactions between multiple information waves.
Ideas inherently depend on each other, and waves of new information or technology often interact with one another as they propagate through society. In some systems, information waves integrate or hybridize, while in other systems they compete and replace one another. Here we focus on the latter type of interaction, when waves replace one another entirely, and analyze the global effects of such interactions. For simplicity, we use novelty as a proxy for quality and key trait in the competition between waves [@rosvall2003modeling; @lizana2010time]. Relevant systems include news media reporting of a particular event, release of new software versions, and invention of new technology that makes old technologies obsolete. With interaction between multiple waves, some waves will make it across the system and others will not. Therefore, the wave frequency, or, equivalently, the rate of adoption of individuals, will depend on their position relative to the information source in the system. For example, in ancient times, new methods of metallurgy spread in multiple waves across Europe, and, in modern times, new versions of operating systems spread across the globe. Not everybody upgrades immediately, and our aim in this paper is to analyze how the access to new information depends on the position in a system and the topology of the system. To analyze the effects of interactions between multiple waves, we use a simple agent-based model introduced in ref. [@lizana2011modeling] and further analyzed in ref. [@PhysRevE.85.056116]. In its simplest formulation, a source agent injects multiple waves of new information over time in a given network. At a given rate, neighboring agents adopt the information if it is newer than the information that they already have. We provide analytical results of the wave frequency for a one-dimensional model and use simulations on lattice models between one and two dimensions, as well as on real spatial networks. In this way, we can quantify the effects of interactions between multiple waves and show, for example, that not only the distance from the source, but also the path redundancy, determine the rate of adoption. Moreover, compared to a system with non-interacting waves, new information reaches agents faster, because of selection between waves: slow waves die out and fast waves survive.
We begin by describing the model in detail and then analyze the model in different spatial geometries. In turn, we analyze the model from the perspective of the agents and the information waves, respectively.
Methods
=======
In this section, we first detail the model and then describe the spatial embedding we use to analyze the spreading dynamics.
Model definition {#ModDef}
----------------
The model consists of a number of agents, each of which occupies a node connected to neighboring nodes in a spatial network. The core of the model describes interactions between multiple information waves released at a single source node. At each time $t$, the source node in the center $j=0$ generates a new piece of information tagged by the time when it was generated $a_0(t) = t$. In the same time step, each node $j$ with information of age $a_j(t)$ asks each of its neighbors $k_j$ with probability $\beta$ if $k_j$ has newer information. If $a_{k_{j}}(t) < a_j(t)$, $j$ adopts the new information and sets $a_j(t) = a_{k_{j}}(t)$. Without loss of generality and throughout our analysis, we use $\beta=0.5$. Note that this model formulation is equivalent to one in which agents actively transmit information to each of their neighbors with probability $\beta$, and the neighbors update their information if it is newer than the information they already had. Therefore, if there was only one information wave or if the waves did not interact, the model would describe the standard SI dynamics of susceptible and infected individuals [@anderson1992infectious; @hethcote2000mathematics; @newman2010Intro], and an information wave would always spread across the system. In the presence of multiple interacting waves, however, the information waves will compete with each other as they spread through the system, and only the fast ones will survive and make it across the system.
![[**The spatial and temporal dynamics of the spreading model with multiple interacting waves on a two-dimensional lattice.**]{} **(a)** A snapshot of the dynamics in which each color corresponds to a single information wave. **(b)** The age landscape of the model in which bright colors (light yellow) represent new information and dark colors (dark red) represent old information. []{data-label="Snap"}](Fig1.eps){width="0.995\columnwidth"}
Figure \[Snap\] illustrates the dynamics of the spreading model on a two-dimensional lattice. This figure was produced from the Java applet available in ref. [@Lizana2010Online]. The source of new information is in the center of the lattice. Close to the information source, the diversity of information waves and the competition between them are high. Consequently, agents in this area become updated with high frequency. But far from the source, the wave frequency is lower, because high competition close to the source eliminates some waves. Therefore, nodes in distant areas must wait longer between each update of information. For example, for a line source that is located at one edge of a lattice, the density of wave fronts decays as the square root of distance from the source [@lizana2011modeling]. In this paper, we analytically derive this result for a one-dimensional system and further show that the information wave frequency also depends on the path redundancy, the number of shortest paths between the source and a given node. The path redundancy can be thought of as the effective dimensionality of the system. Higher path redundancy in a system gives nodes better access to new information.
Spatial embedding
-----------------
To analyze the effects of path redundancy, we build synthetic spatial networks with varying degree of path redundancy. The networks range from trees to two-dimensional lattices (Fig. \[schem\]). In the two-dimensional lattice, the number of shortest paths grows exponentially as a function of distance from the source. We construct the networks in two steps:
- We start with a two-dimensional structure with nodes connected in horizontal rows and one vertical column through the source node in the center (Fig. \[schem\](a)).
- We then randomly connect a fraction $\mathcal{R}$ of the remaining disconnected pairs of nodes (Fig. \[schem\](b,c)).
We quantify the path redundancy in terms of $\mathcal{R}$, where $\mathcal{R}=0$ corresponds to a tree and $\mathcal{R}=1$ corresponds to a two-dimensional lattice. Figure \[schem\] schematically shows how, by connecting disconnected pairs in the tree structure in Fig. \[schem\](a), we can increase the path redundancy through intermediate values in Fig. \[schem\](b) to high values in the fully connected two-dimensional lattice in Fig. \[schem\](c).
![[****]{} **Illustration of spatial networks with different degrees of path redundancy.** **(a)** A tree with path redundancy $\mathcal{R}=0$. **(b)** A network with path redundancy $\mathcal{R}=0.33$. **(c)** A two-dimensional lattice with path redundancy $\mathcal{R}=1$[]{data-label="schem"}](Fig2.eps){width="0.95\columnwidth"}
To complement the analysis of synthetic networks, we also analyze two real spatial networks with effective dimensionality between one and two: the road networks of Texas and California [@SNAP]. For all of the described networks, we quantify the wave frequency as a function of distance and path redundancy. For comparison, we compare the results with a null model without interactions between information waves. For the one- and two-dimensional systems, we also quantify the wave-speed distribution as a function of distance and path redundancy.
{width="1.95\columnwidth"}
Results and Discussion
======================
Before we show the results for the speed and wave frequency as a function of distance and path redundancy, we begin by analyzing the dynamics from the information waves’ perspective.
Spatial spreading profile
-------------------------
Unlike non-interacting information waves, many interacting information waves will die out long before they reach the boundaries of the system. For a general idea of how far they spread, we analyzed the spatial spreading profile by measuring the probability distribution of the information waves’ total extension, maximum extension, and maximum penetration distance. The total extension of an information wave over its entire lifetime is the fraction of nodes in the network that, at some point, adopted the corresponding information. This measure captures the aggregated popularity of a piece of information over its entire lifetime. The maximum extension of an information wave at its peak is the maximum fraction of nodes in the network that simultaneously adopted the corresponding information as their latest information. This measure reflects the maximum popularity of a piece of information over its whole lifetime. Finally, the maximum penetration distance of a wave is the longest geodesic distance from the source that the wave ever reached.
Figure \[Cum\_CS\] shows the spatial spreading profiles of interacting waves on three networks with different levels of path redundancy. We used a path redundancy of 0.4 as an intermediate value, because, as we show in the next section, a path redundancy of $\mathcal{R}=0.4$ corresponds to the average path redundancy of the road networks of Texas and California. As Fig. \[Cum\_CS\] shows, the dynamic behavior of this intermediate path redundancy is similar to the maximum path redundancy of the two-dimensional lattice. For all topologies, the competition between waves is most intense close to the source, such that most waves die out small before they have covered much ground (Fig. \[Cum\_CS\](a)). Except for boundary effects, which are significant in some cases, a power law distribution with exponent 1 approximately captures the scaling for all topologies. While the scaling is similar for different degrees of path redundancy, higher path redundancy increases the overall probability of spreading across the system. That is, in a topology with higher path redundancy and more possibilities for a wave to escape from chasing waves, more waves can reach the system boundary. As a result, the fraction of waves that die before reaching the boundary is smaller in a system with higher path redundancy, as seen by the vertically shifted probability densities between high and low path redundancy in Figs. \[Cum\_CS\](b) and (c). With no path redundancy in a tree-like topology, there is only one direction to expand into and chasing waves follow immediately after. Therefore, very few waves occupy many nodes at the same time in a low-dimensional system (Fig. \[Cum\_CS\](b)). In contrast, with higher path redundancy, a fast wave can expand more quickly in multiple directions and reach higher maximum extension size.
Interactions between information waves prevent slow waves from reaching distant parts of the network. For the individuals that propel the spread of the information, this competition affects ($i$) the age of the information that actually reaches them, and ($ii$) the frequency at which new information arrives. In the next section we take the perspective of the waves and, in turn, investigate these two effects in detail.
Information wave speed and frequency
------------------------------------
![\[AgeFig\] [**Probability density of information age.**]{} **(a)** On a tree with 3600 nodes and $\mathcal{R}=0$. **(b)** On a two-dimensional lattice with 3600 nodes and $\mathcal{R}=1$. We quantified the age distribution of arriving waves for nodes that are close to the source ($d=10$) and for nodes that are farther away from the source ($d=28$). Results are obtained by averaging over more than 10,000 different competing waves. ](Fig4.pdf){width="0.8\columnwidth"}
To investigate the extent of novelty for information arriving at a node, we calculated the age distribution of waves that reach a certain area. That is, we measured the average age of hitting waves for nodes at a given geodesic distance $d$ to the source. For comparison, we did the same experiment for multiple non-interacting waves. We ran our experiment on a network with 3600 nodes (ordered in a 60 by 60 square) and quantified the probability distribution of information age for two groups of nodes: those that are close to the source ($d=10$) and those that are far from the source ($d=28$). In Fig. \[AgeFig\], we compare the probability distribution of the information age between interacting and non-interacting waves on two networks: a tree with the lowest possible path redundancy, $\mathcal{R}=0$, and a two-dimensional lattice with the highest possible path redundancy, $\mathcal{R}=1$. In both networks, information packages that reach a node have traveled for a shorter time in systems with interacting waves, because the interaction between waves forms a selection process in which only fast waves survive.
![ [\[Road\] **Wave frequency as a function of distance for interacting and non-interacting information waves on different networks.**]{} The results on the tree and two-dimensional lattice are fitted to a power-law with exponent 0.5 and 0.3, respectively. For the California and Texas road networks, the results are very close to each other and similar to the network with path redundancy $\mathcal{R}=0.4$. All synthetic networks have more than $10^6$ nodes. The results on these networks are achieved by simulating more than 5000 competing waves. The results on the road networks are averaged over more than 25 runs and each run includes more than 30,000 competing waves.](Fig5.pdf){width="0.8\columnwidth"}
Information that reaches a node is newer with than without interaction between waves, because slow waves die out as they move away from the source. That is, nodes far from the source will only be reached by a fraction of all pieces of information that spread from the source. We quantified this effect with the wave frequency, the rate at which new information waves arrive at a node. In Fig. \[Road\], we show how the wave frequency scales as a function of the distance from the source. We quantified the wave frequency as a function of geodesic distance on multiple networks: a tree ($\mathcal{R}=0$) with 1,210,000 nodes (1100 by 1100 square), a two-dimensional lattice ($\mathcal{R}=1.0$) with the same number of nodes, and two synthetic networks with the same number of nodes and path redundancies between the tree and the lattice: one with $\mathcal{R}=0.4$ and one with $\mathcal{R}=0.6$.
For non-interacting waves, the wave frequency is the same for any node at any location and equal to the transition probability $\beta$ (gray line). For interacting waves, the wave frequency decays as a power law of the form $f(d) \sim d^{-\gamma}$ with an exponent $\gamma$ that depends on the path redundancy. In general, for nodes at similar geodesic distance, a topology with higher path redundancy results in higher wave frequency. For example, the wave frequency decays similarly fast for the two road networks and close to the synthetic network with path redundancy $\mathcal{R}=0.4$. Moreover, the exponent $\gamma$ is around 0.2 for the two-dimensional lattice and around 0.5 for the tree structure with no path redundancy.
Historically, the spread of technology and innovation has taken place on spatial networks, such as road and river networks. Trade routes, such as the Silk Road connecting the East and the West, worked as the backbone of the spread of information for centuries. To capture the dynamics on such networks, and on similar networks with intermediate path redundancies, we analyzed the California and Texas road networks. We found that the dynamics on the spatial networks are similar to the synthetic networks with path redundancy $\mathcal{R}=0.4$; the access to new information as a function of distance from the source has a power-law scaling (Fig. \[Road\]). To better understand the origin of this universal power-law behavior, in the next section we derive an analytical calculation for the wave frequency in a one-dimensional system.
Analytical derivation of the wave frequency
-------------------------------------------
In this section, we provide an analytical derivation of the wave-frequency scaling in a one-dimensional system. We derive the wave frequency as a function of distance from the source by working with two quantities: the wave size $s$ and the position $r$ of the outer boundary of the wave, or “position” for short. In one dimension, these two magnitudes are really governed by what happens in the two boundaries in a single time step. Specifically, the following outcomes are possible:
- [ The two boundaries move simultaneously to the right; this happens with probability $\beta^2$. In that case, $s$ remains the same and $r$ increases. This corresponds to the thin horizontal arrows in Fig. \[stateWS\].]{}
- [ The inner boundary remains in the same position, and the outer one moves. This implies that both $r$ and $s$ increase by one, which we represent with thin diagonal arrows in Fig. \[stateWS\]. This happens with probability $\beta (1 - \beta)$. ]{}
- [ The inner boundary moves, and the outer remains in the same position. The size decreases by one and $r$ keeps the previous value. This again happens with probability $\beta (1-\beta)$, and we represent it with thin downward arrows in Fig. \[stateWS\]. ]{}
- [ Both boundaries remain in the same position, in which case neither the wave’s position nor its size change. This happens with probability $(1-\beta)^2$. ]{}
These probabilities sum up to one, but since we are just interested in size and position, and the fourth outcome changes none of these, we remove the fourth outcome and normalize the other probabilities accordingly. Thus the probability for the first outcome becomes $\beta^2/\left( 1 - (1-\beta)^2 \right) = \beta/(2-\beta)$, which we denote by $b$. The second and third case become $\beta (1-\beta) / \left( 1 - (1-\beta)^2 \right) = \beta (\beta-1) / (\beta - 2 )$. The second and third case have the same probability, but we denote them with different letters, $a$ and $c$ respectively, for the sake of clarity. Each of the three kinds of transitions is depicted in figure \[stateWS\] with arrows of different colors.
In the model, the origin is special because a new wave starts there at each time step; no other points share that property. To make the analysis simpler, we will consider an alternative origin one step outwards, where the starting waves have size 1 with probability 1. The state corresponding to this new origin is represented with a green dot in the figure. Effectively, any information we obtain using this new origin is conditioned on the wave actually reaching this first point. We use this fact later to restore the original condition of a new wave at the origin of each time step.
Using the transition probabilities $a$, $b$, and $c$, and guided by Fig. \[stateWS\], we can derive a recursive equation for the probability of having a size-one wave at a given position $r$. We use the fact that, in the state space represented in the diagram, the marginal probability of all the walks starting and ending with the same value for $s$, never going below $s$ and advancing $x$ steps, does not depend on $s$ itself, but only in the number of steps $x$ advanced. In other words, the thick curved arrows in the figure correspond to events with the same probability, even if they use different values for $s$. We call this probability $g(x)$. Note that then $g(r)$ is the probability of finding the wave at position $r$ with size 1.
![ [**State transition diagram of position and size**]{} The position and size of the waves can change by following this diagram’s arrows. The horizontal axis represents the position $r$ of the right boundary, and the vertical axis represents the size $s$ of the wave.[]{data-label="stateWS"}](Fig6.eps){width="0.85\columnwidth"}
We can get an expression for $g(r)$ as follows. If a walk ends at $(r,1)$, it can reach this last state in only two ways: from the left, with a green arrow, or from the state above $(r,2)$, with a gray arrow. If the state is reached from the left, it means that the wave had to go from $(1,1)$ to $(r-1,1)$, and thus we need the value of $g(r-1)$. The case for the wave coming from above is more convoluted and we handle it in the following way. First, because the state above had to go to 2, we can safely assume that there was at least one transition where $s$ changed from 1 to 2. We fix the position of the last of such a transition as $r_1$, such that $(r_1,1)$ and $(r_1+1,2)$ is a segment of the walk. We can see that the walk from $(1,1)$ to $(r_1,1)$ will have probability $g(r_1)$. The segment from $(r_1,1)$ to $(r_1+1,2)$ will have probability $a$. That leaves us with the part of the walk from $(r_1+1,2)$ to $(r,2)$. Because the last transition from $s=1$ to $s=2$ was the one at $r_1$, there is no way that the walk could go to $s<2$ in the section from $(r_1+1,2)$ to $(r,2)$. In other words, this part has probability $g(r-r_1-1)$. With this information, we can write the recurrence for $g(r)$ as $$\label{eqM2}
g(r)=\sum _{r_1=0}^{r-1} \left[ g\left(r_1\right) a\text{ }g\left(r-r_1-1\right) c \right]+ b g(r-1).$$ To solve the above equation, we use a generator function of the form $$\label{eqM3}
G(z)=\sum_{i=0}^{\infty}g(i)\, z^{i}.$$ Since Eq. \[eqM2\] is only valid for $r\geq2$, we first write $$\label{eqGz}
G(z)\text{=}g(0)+z\, g(1)+\sum_{i=2}^{\infty}g(i)\, z^{i},$$ then apply the recursivity to obtain $$\label{eqM4}
\begin{split}
&G(z)=g(0)+z\, g(1)+\sum_{i=2}^{\infty}\, b\, g(i-1)\, z^{i}+
\\
&\sum_{i=2}^{\infty}z^{i}\sum_{i_{1}=0}^{-1+i}ac\, g(i-i_{1}-1)\, g(i_{1}).
\end{split}$$ With some variable substitutions and algebraic manipulation, we can write it as $$\label{eqM5}
\begin{split}
&G(z)=g(0)-b\, z\, g(0)+z\, g(1)+b\, z\, G(z)+
\\
&ac\, z\sum_{j=0}^{\infty}z^{j}\sum_{i_{1}=0}^{j}g\left(j-i_{1}\right)g\left(i_{1}\right)-zac\, g(0)\, g(0).
\end{split}$$ The terms in the sum of the previous expression represent a neat convolution, which can be expressed as the product of generating functions. From there we get the quadratic expression for $G(z)$, $$\label{eqM6}
G(z)=1-b\, z-ac\, z+(b+ac)\, z+b\, z\, G(z)+acz\, G(z)^{2}.$$ From the two solutions, we select $$G(z)=-\frac{-1+b\, z+\sqrt{1-2b\, z-4ac\, z+b^{2}z^{2}}}{2ac\, z}\label{eq:G-definition}.$$
As mentioned in the text, with probability $c g(r)$, the wave is going to die without ever reaching the position $r + 1$. Thus, by summing $c g(x)$ from $x = 1$ to $x = r-1$, we can calculate the fraction of waves alive at a specific position $r$ of any size as $$h(r)=1-c\sum\limits_{x=0}^{r-1} g(x).$$ To calculate the survival probability $h(r)$, we write down the corresponding generating function as $$H(z)=\sum_{i=0}^{\infty}z^{i}-\sum_{r_{0}=0}^{\infty}z^{i}c\sum_{r=0}^{i-1}g(r).$$ The first term in the difference is $\frac{1}{1-z}$ and the second term is again a convolution: $$H(z)\text{=}\frac{1}{1-z}-c\sum_{i=0}^{\infty}z^{i}\sum_{j=0}^{i}g(j)+c\sum_{i=0}^{\infty}z^{i}g(i),$$ such that we get $$H(z)=\frac{c\, z\, G(z)-1}{z-1}.$$ After substituting Eq. \[eqGz\] and doing some simplifications, we obtain $$H(z)=\frac{2}{\sqrt{\left(-4+\beta\left(4+\beta\left(-1+z\right)\right)\right)\left(-1+z\right)}-\beta\,(z-1)}\label{eq:gen-for-H}.$$ The function $H(z)$ has only one principal singularity at $z-1$, and we know that the coefficients $h(r)$ are strictly positive. Therefore, we can apply Corollary 2 in ref. [@flajolet1990singularity] and derive the asymptotic scaling for $h(r)$. By that corollary, we get that, when $r\rightarrow\infty$, $$h(r)\sim\frac{1}{\sqrt{1-\beta}\sqrt{\pi}}\, r^{-1/2}$$ We should remember that $h(r)$ represents the survival probability of the wave once it takes off at the first position inmediatly after the origin, and that happens with probability $\beta$, so the frequency at which new waves are observed at a given point $r$ is $$\label{eq:frequency}
f(r)\sim\frac{\beta}{\sqrt{1-\beta}\sqrt{\pi}}\, r^{-1/2}$$ This expression is valid as long as $\beta<1$, provided that $r$ is sufficiently large. Figure \[SimIn1D\] shows the values of the frequency obtained by simulation and those obtained by the previous equation.
![ [**The wave frequency at different positions $r$.**]{} Dots: the frequency obtained averaging the observation of 2e6 time-steps. Green line: the theoretical prediction according to equation \[eq:frequency\]. []{data-label="SimIn1D"}](Fig7.eps){width="0.95\columnwidth"}
Conclusion
==========
We used a simple agent-based model to capture the observation that waves of new information or technology often interact with one another as they propagate through a system. In the model, we use novelty as a proxy for quality and key trait in the interaction between waves. We showed that information that reaches agents is newer with than without interactions between waves at the cost of lower arrival frequency of information waves. Moreover, high path redundancy has a positive effect on the wave frequency, such that information more easily spreads in a system with multiple routes to targets. In general, the wave frequency decays as a power law of the distance from the source, and analytically we showed that the scaling goes as one over the square root of the distance in a one-dimensional system. Our analysis on road networks of California and Texas showed that these networks provide an infrastructure for information propagation that corresponds to lattice models between one and two dimensions. We conclude that interacting information waves show interesting dynamics that call for further study.
We are grateful to Akhil Kedia for many valuable suggestions. This research was conducted using the resources of High Performance Computing Center North (HPC2N). Martin Rosvall was supported by the Swedish Research Council grant 2012-3729, and Ludvig Lizana wishes to acknowledge financial support from Knut and Alice Wallenberg foundation.
|
---
abstract: 'Thermalization and collective flow of charm ($c$) and bottom ($b$) quarks in ultra-relativistic heavy-ion collisions are evaluated based on elastic parton rescattering in an expanding quark-gluon plasma (QGP). We show that resonant interactions in a strongly interacting QGP (sQGP), as well as parton coalescence, can play an essential role in the interpretation of recent data from the Relativistic Heavy-Ion Collider (RHIC), and thus illuminate the nature of the sQGP and its hadronization. Our main assumption, motivated by recent findings in lattice Quantum Chromodynamics, is the existence of $D$- and $B$-meson states in the sQGP, providing resonant cross sections for heavy quarks. Pertinent drag and diffusion coefficients are implemented into a relativistic Langevin simulation to compute transverse-momentum spectra and azimuthal asymmetries ($v_2$) of $b$- and $c$-quarks in Au-Au collisions at RHIC. After hadronization into $D$- and $B$-mesons using quark coalescence and fragmentation, associated electron-decay spectra and $v_2$ are compared to recent RHIC data. Our results suggest a reevaluation of radiative and elastic quark energy-loss mechanisms in the sQGP.'
author:
- 'Hendrik van Hees$^{1}$, Vincenzo Greco$^{2}$ and Ralf Rapp$^{1}$'
title: 'Heavy-Quark Probes of the Quark-Gluon Plasma at RHIC'
---
*Introduction.* Recent experimental findings at the Relativistic Heavy-Ion Collider (RHIC) have given intriguing evidence for the production of matter at unprecedented (energy-) densities with surprisingly large collectivity and opacity, as reflected by (approximately) hydrodynamic behavior at low transverse momentum ($p_T$) and a substantial suppression of particles with high $p_T$. This has led to the notion of a “strongly interacting Quark-Gluon Plasma” (sQGP), whose microscopic properties, however, remain under intense debate thus far.
Heavy quarks (HQs) are particularly valuable probes of the medium created in heavy-ion reactions, as one expects their production to be restricted to the primordial stages. Recent calculations of radiative gluon energy-loss of charm ($c$) quarks traversing a QGP in central Au-Au collisions at RHIC have found nuclear suppression factors $R_{AA}$$\simeq$0.3-0.4 [@djo04; @arm05], comparable to the observed suppression of light hadrons at high $p_T$, and in line with preliminary (non-photonic) single-electron ($e^\pm$) decay spectra [@phenix-e1; @jac05; @bil05]. The latter also exhibit a surprisingly large azimuthal asymmetry ($v_2$) [@v2-phenix; @v2pre-star; @aki05] in semicentral Au-Au which cannot be reconciled with radiative energy loss, especially if $c$-quarks are hadronized into $D$-mesons via fragmentation. While the underlying transport coefficients [@arm05] exceed their predicted values from perturbative Quantum Chromodynamics (pQCD) by at least a factor of $\sim$5 [@bai02], energy loss due to [*elastic*]{} scattering parametrically dominates toward low $p_T$ (by a factor 1/$\sqrt{\alpha_s}$ [@MT04]). But elastic pQCD cross sections [@Svet88; @MT03] also have to be upscaled substantially to obtain $c$-quark $v_2$ and $R_{AA}$ reminiscent to preliminary $e^\pm$ data, as shown in a recent Langevin simulation for RHIC [@MT04]. In addition, contributions of bottom ($b$) quarks [@Cac05; @djo05] will reduce the effects in the electron-$R_{AA}$ and -$v_2$ at high $p_T$. Quark coalescence approaches suggest that an $e^\pm$-$v_2$ in excess of 10% can only be obtained if [@GKR04] (i) light quarks impart their $v_2$ on $D$-mesons (see also Refs. [@Mol04; @Zhang05]), (ii) the $c$-quark $v_2$ is comparable to that of light quarks.
We are thus confronted with marked discrepancies between pQCD energy-loss calculations and semileptonic heavy-quark (HQ) observables at RHIC. The resolution of this issue is central to the understanding of HQ interactions in the QGP in particular, and to the interpretation of energy loss in general. HQ rescattering also has direct impact on other key observables such as heavy quarkonium production (facilitating regeneration) and dilepton spectra (where $c\bar c$ decays compete with thermal QGP radiation).
In this letter we investigate the HQ energy-loss problem by introducing resonant HQ interactions into a Langevin simulation of an expanding QGP. Our calculations implement a combined coalescence+fragmentation approach for hadronization, as well as bottom contributions, to allow for a quantitative evaluation of pertinent $e^\pm$-spectra ($v_2$ and $R_{AA}$) which is mandatory for a proper interpretation of recent RHIC data. Our main assumption of resonant $D$- and $B$-like states in the sQGP has been shown [@HR05a] to reduce HQ thermalization times by a factor of $\sim$3 compared to pQCD scattering. Theoretical evidence for resonances in the sQGP derives from computations of heavy and light meson correlators within lattice QCD (lQCD) [@AH-prl; @KL03], as well as applications of lQCD-based heavy-quark potentials within effective models [@SZ04; @Wong04; @MR05; @Alberico05]. Except for the mass and width of these states, no further free parameters enter our description, with degeneracies based on chiral and HQ symmetry.
*Heavy-Quark Interactions in the QGP.* Following Ref. [@HR05a] our description of HQ interactions in the QGP focuses on elastic scattering, mediated by resonance excitations on light antiquarks ($\bar q$) as well as (nonresonant) leading order pQCD processes dominated by $t$-channel gluon exchange. The latter correspond to Born diagrams [@com79] regularized by a gluon-screening mass $m_g$=$g T$ with a strong coupling constant, $\alpha_s$=$g^2/(4 \pi)$=0.4. The key assumption [@HR05a] is that a QGP at moderate temperatures $T$$\le$2$T_c$ sustains strong correlations in the lowest-lying color-neutral $D$- and $B$-meson channels. Support for the relevance of such interactions stems from quenched lQCD computations of euclidean mesonic correlation functions, which, after transformation into the timelike regime, exhibit resonance structures for both (heavy) $Q$-$\bar Q$ and (light) $q$-$\bar q$ states [@AH-prl; @KL03]. In addition, applications of lQCD-based $Q$-$\bar Q$ potentials have revealed both bound [@SZ04; @Wong04] and resonance states [@MR05] with dissolution temperatures of $\sim2T_c$, quite compatible with the disappearance of the peak structures in the lQCD spectral functions. Here, we do not attempt a microscopic description of these correlations but cast them into an effective lagrangian with $\bar q$-$Q$-$\Phi$ vertices ($\Phi$=$D$, $B$), at the price of 2 free parameters: the masses of the meson-fields, fixed at $m_{D(B)}$=2(5) GeV, [*i.e.*]{}, 0.5 GeV [*above*]{} the $Q$-$\bar q$ threshold (with quark masses $m_{c(b)}$=1.5(4.5) GeV, $m_{u,d}$=0), and their width, $\Gamma$, obtained from the one-loop $\Phi$ self-energy (which, in turn, dresses the $\Phi$-propagator) with the pertinent coupling constant varied to cover a range suggested by effective quark models [@GK92; @Blasch03], $\Gamma$=0.4-0.75 GeV. The multiplicity of $\Phi$ states follows from chiral and HQ symmetries alone, implying degenerate $J^P$=$0^\pm$ and 1$^\pm$ states. We emphasize again that, besides the mass and width of the $\Phi$ states, no other free parameters (or scale factors) are introduced.
The matrix elements for resonant and pQCD scattering are employed to calculate drag and diffusion coefficients of HQs in a Fokker-Planck approach [@Svet88]. Resonances reduce the thermalization times for both $c$- and $b$-quarks by a factor of $\sim$$3$ compared to pQCD scattering alone [@HR05a].
*Langevin Simulation.* To evaluate thermalization and collective flow of HQs in Au-Au collisions we perform relativistic Langevin simulations [@MT04] embedded into an expanding QGP fireball. In the local rest frame of the bulk matter, the change in position ($\vec x$) and momentum ($\vec p$) of $c$- and $b$-quarks during a time step $\delta t$ is defined by $$\label{langevin}
\delta \vec{x}=\frac{\vec{p}}{E}~\delta t \ , \quad
\delta \vec{p}=-A(t,\vec{p}+\delta \vec{p})~\vec{p}~\delta t + \delta
\vec{W}(t,\vec{p}+\delta \vec{p}) \$$ ($E$: HQ energy), where $\delta \vec{W}$ represents a random force which is distributed according to Gaussian noise [@hae05], $$\label{fluct-force}
P(\delta \vec{W}) \propto \exp \left [-\frac{\hat{B}_{jk} \delta W^j \delta
W^{k}}{4 \delta t} \right] \ .$$ The drag coefficient (inverse relaxation time), $A$, and the inverse of the diffusion-coefficient matrix, $$B_{jk}=B_{0} (\delta^{jk} - \hat{p}^j \hat{p}^k)
+ B_1 \hat{p}^j \hat{p}^k \ ,$$ are given by the microscopic model of Ref. [@HR05a], including $p$- and $T$-dependencies (the latter converts into a time dependence using the fireball model described below). The longitudinal diffusion coefficient is set in accordance with Einstein’s dissipation-fluctuation relation to [@MT04] $B_1$=$TEA$, to ensure the proper thermal equilibrium limit. The latter also requires care in the realization of the stochastic process in Eq. (\[langevin\]); we here use the so-called Hänggi-Klimontovich realization [@hae05] which approaches a relativistic Maxwell distribution if the Einstein relation is satisfied (in the Ito realization, [*e.g.*]{}, an extra term has to be introduced in $A$ [@arnold]). Finally, the HQ momenta are Lorentz-boosted to the laboratory frame with the velocity of the bulk matter at the actual position of the quark, as determined by the fireball flow profile (see below).
The time evolution of Au-Au collisions is modeled by an isentropically expanding, isotropic QGP fireball with a total entropy fixed to reproduce measured particle multiplicities at hadro-chemical freezeout which we assume to coincide with the phase transition at $T_c$=180 MeV [@Ra01]. The temperature at each instant of time is extracted from an ideal QGP equation of state with an effective flavor degeneracy of $N_f$=2.5. Radial and elliptic flow of the bulk matter are parameterized to closely resemble the time-dependence found in hydrodynamical calculations [@kol00], assuming a flow profile rising linearly with the radius. We focus on semicentral collisions at impact parameter $b$=7 fm, with an initial spatial eccentricity of 0.6 and a formation time of $\tau_0$=0.33 fm/c, translating into an initial temperature of $T_0$=340 MeV. The evolution is terminated at the end of the QGP-hadron gas mixed phase (constructed via standard entropy balance [@Ra01]) after about 5 fm/c, at which point the surface flow velocity and momentum anisotropy have reached $v_\perp$=0.5c and $v_2$=5.5% (variations in $\tau_0$ by a factor of two affect the $c$-quark $v_2$ and $R_{AA}$ by 10-20% (less for the $e^\pm$ spectra), while a reduction in the critical temperature to 170 MeV increases (decreases) $v_2^c$ ($R_{AA}$) somewhat less. $D$-meson rescattering in the hadronic phase [@fuchs04] is neglected).
To specify initial HQ $p_T$-distributions, $P_{\text{ini}}(p_{T})$, and especially the relative magnitude of $c$- and $b$-quark spectra (essential for the evaluation of $e^\pm$ spectra), we proceed as follows: we first use modified PYTHIA $c$-quark spectra with $\delta$-function fragmentation to fit $D$ and $D^*$ spectra in d-Au collisions [@star-D]. These spectra are decayed to single-$e^\pm$ which saturate pertinent data from $p$-$p$ and d-Au up to $p_T^e$$\simeq$3.5 GeV [@phenix-pp; @star-dAu]. The missing yield at higher $p_T$ is attributed to contributions from $B$-mesons, resulting in a cross section ratio of $\sigma_{b\bar b}/\sigma_{c\bar
c}$$\simeq$5$\cdot 10^{-3}$ (which is slightly smaller than for pQCD predictions [@Cac05] and implies a crossing of $c$- and $b$-decay electrons at $p_T$$\simeq$5 GeV, as compared to 4 GeV in pQCD).
![Nuclear modification factor (upper panel) and elliptic flow (lower panel) of charm and bottom quarks in $b$=7 fm Au-Au($\sqrt{s}$=200 GeV) collisions based on elastic rescattering in the QGP. Red (green) and blue lines are for $c$- ($b$-) quarks with and without resonance rescattering, respectively, where the bands encompass resonance widths of $\Gamma$=0.4-0.75 GeV.[]{data-label="fig1"}](quark-v2-RAA){width="35.00000%"}
Fig. \[fig1\] summarizes the output of the Langevin simulations for the HQ nuclear modification factor, $R_{AA}$=$P_{\text{fin}}(p_{T})/P_{\text{ini}}(p_{T})$ ($P_{\text{fin}}$: final $p_T$-distributions), and elliptic flow, $v_2$=$\langle {(p_x^2-p_y^2)/(p_x^2+p_y^2)}_{p_T} \rangle$ (evaluated at fixed $p_T$). For $c$-quarks and pQCD scattering only, our results are in fair agreement with those of Ref. [@MT04] (recall that the Debye mass in our calculations is given by $m_g$=$gT$ while in Ref. [@MT04] it was set to $m_g$=1.5$T$ independent of $\alpha_s$). However, both $R_{AA}$ and $v_2$ exhibit substantial sensitivity to the inclusion of resonance contributions, increasing the effects of pQCD scattering by a factor of $\sim$3-5. Also note the development of the plateau in $v_2$($p_T$$>$3 GeV) characteristic for incomplete thermalization of HQs in the bulk matter.
*Hadronization and Single-Electron Spectra.*
Semileptonic single-$e^\pm$ spectra are a valuable tool to investigate heavy-meson spectra in ultrarelativistic heavy-ion collisions, since their decay kinematics largely conserves the spectral properties of the parent particles [@GKR04; @Dong:2004ve]. To compare our results to measured single-$e^\pm$ in Au-Au collisions, the above HQ spectra have to be hadronized. To this end we employ the coalescence approach of Ref. [@GKR04] based on earlier constructed light-quark spectra [@Greco:2003mm]. Quark coalescence has recently enjoyed considerable success in describing, [*e.g.*]{}, the “partonic scaling" of elliptic flow and the large $p$/$\pi$ ratio in Au-Au at RHIC [@Hwa:2002tu; @Greco:2003mm; @Fries:2003kq], as well as flavor asymmetries in $D$-meson production in elementary hadronic collisions [@Rapp:2003wn]. Whereas at low $p_T$ most of the HQs coalesce into $D$- and $B$-mesons, this is no longer the case at higher $p_T$ where the phase space density of light quarks rapidly decreases. Therefore, to conserve HQ number in the $B$- and $D$-meson spectra, the remaining $c$- and $b$-quarks are hadronized using $\delta$-function fragmentation. Finally, single-$e^\pm$ $p_T$- and $v_2$-spectra are computed via $B$- and $D$-meson 3-body decays, and compared to experiment in Fig. \[fig2\]. We find that the effects of resonances are essential in improving the agreement with data, both in terms of lowering the $R_{AA}$ and increasing $v_2$. The $B$-meson contribution reflects itself by limiting $R_{AA}$ and $v_2$ to values above 0.4 and below 10%, respectively, as well as the reduction of $v_2$ above $p_T$$\simeq$3 GeV.
To better illustrate the effects of coalescence we plot in Fig. \[fig3\] calculations where [*all*]{} HQs are fragmented into $D$- and $B$-mesons.
While $R_{AA}$ is significantly reduced, most notably in the $p_T$$\simeq$1-2 GeV region, $v_2$ also decreases reaching at most 6%, which is not favored by current data. It is, however, conceivable that modifications in the fraction of coalescence to fragmentation contributions, as well as improvements in our schematic ($\delta$-function) treatment of fragmentation, will be necessary once more accurate experimental data become available. Additional corrections may also arise from a more precise determination of the $b$/$c$ ratio and nuclear shadowing.
*Conclusion.* We have investigated thermalization and collective flow of $c$- and $b$-quarks within a relativistic Langevin approach employing elastic scattering in an expanding QGP fireball in semicentral Au-Au collisions at RHIC. Underlying drag and diffusion coefficients were evaluated assuming resonant $D$- and $B$-meson correlations in the sQGP, enhancing heavy-quark rescattering. Corresponding $p_T$-spectra and elliptic flow of $c$-quarks exhibit a large sensitivity to the resonance effects, lowering $R_{AA}$ down to 0.2 and raising $v_2$ up to 10%, while the impact on $b$-quarks is small. Heavy-light quark coalescence in subsequent hadronization significantly amplifies the $v_2$ in single-electron decay spectra, but also increases their $R_{AA}$, especially in the $p_T$$\simeq$2 GeV region. Bottom contributions dominate above 3.5 GeV reducing both suppression and elliptic flow. The combined effects of coalescence and resonant heavy-quark interactions are essential in generating a $v_2^e$ of up to 10%, together with $R_{AA}^e$$\simeq$0.5, supplying a viable explanation of current electron data at RHIC without introducing extra scale factors. Our analysis thus suggests that elastic rescattering of heavy quarks in the sQGP is an important component for the understanding of heavy-flavor and single-electron observables in heavy-ion reactions at collider energies. While induced gluon-radiation is expected to be the prevalent interaction with the medium at sufficiently high $p_T$ [@djo04; @arm05; @djo05], it may not be the dominant effect below $p_T$$\simeq$6 GeV or so. A complete picture should clearly include [*both*]{} elastic and inelastic rescattering mechanisms.
*Acknowledgments:* One of us (HvH) thanks the Alexander von Humboldt foundation for support within a Feodor Lynen fellowship. This work was supported in part by a U.S. National Science Foundation CAREER award under grant PHY-0449489.
[99]{} url \#1[\#1]{}urlprefix\[2\]\[\][[\#2](#2)]{}
M. Djordjevic, M. Gyulassy, S. Wicks, Phys. Rev. Lett. [**94**]{}, 112301 (2005).
N. Armesto *et al.*, Phys. Rev. D **71**, 054027 (2005).
S.S. Adler [*et al.*]{} \[PHENIX Collaboration\], Phys. Rev. Lett. [ **94**]{}, 082301 (2005).
B. Jacak [*et al.*]{} \[PHENIX Coll.\], .
J. Bielcik [*et al.*]{} \[STAR Coll.\], . S.S. Adler [*et al.*]{} \[PHENIX Coll.\], Phys. Rev. C [**72**]{}, 024901 (2005).
F. Laue [*et al.*]{} \[STAR Coll.\], J. Phys. G [**31**]{}, S27 (2005).
Y. Akiba [*et al.*]{} \[PHENIX Coll.\], . R. Baier, Nucl. Phys. **A715**, 209 (2003).
G.D. Moore, D. Teaney (2004), .
B. Svetitsky, Phys. Rev. D **37**, 2484 (1988).
M.G. Mustafa, M.H. Thoma, Acta Phys. Hung. A **22**, 93 (2005).
M. Cacciari, P. Nason, R. Vogt, .
M. Djordjevic, M. Gyulassy, R. Vogt, .
V. Greco, C.M. Ko, R. Rapp, Phys. Lett. **B595**, 202 (2004).
D. Molnar, J. Phys. **G31**, S421 (2005).
B. Zhang, L.W. Chen, C.M. Ko, .
H. van Hees, R. Rapp, Phys. Rev. C **71**, 034907 (2005).
M. Asakawa, T. Hatsuda, Phys. Rev. Lett. **92**, 012001 (2004), Nucl. Phys. **A721**, 869 (2003).
F. Karsch, E. Laermann, .
E.V. Shuryak, I. Zahed, Phys. Rev. C **70**, 021901(R) (2004).
C.Y. Wong (2004), .
M. Mannarelli, R. Rapp (2005), .
W.M Alberico *et al.*, .
B.L. Combridge, Nucl. Phys. **B151**, 429 (1979).
F.O. Gottfried, S.P. Klevansky, Phys. Lett. **B286**, 221 (1992).
D. Blaschke *et al.*, Prog. Theor. Phys. Suppl. **149**, 182 (2003).
R. Rapp, Phys. Rev. C **63**, 054907 (2001).
P.F. Kolb, J. Sollfrank, U. Heinz, Phys. Rev. C **62**, 054909 (2000).
C. Fuchs *et al.* (2004), .
J. Adams [*et al.*]{} \[STAR Coll.\], Phys. Rev. Lett. [**94**]{}, 062301 (2005).
S.S. Adler [*et al.*]{} \[PHENIX Coll.\], .
A. Tai [*et al.*]{} \[STAR Coll.\], J. Phys. G [**30**]{}, S809 (2004).
J. Dunkel, P. H[ä]{}nggi, Phys. Rev. E **71**, 016124 (2005).
P. Arnold, Phys. Rev. E **61**, 6099 (2000).
X. Dong *et al.*, Phys. Lett. [**B597**]{}, 328 (2004).
V. Greco, C. M. Ko, P. Levai, Phys. Rev. C [**68**]{}, 034904 (2003).
R.C. Hwa, C.B. Yang, Phys. Rev. C [**67**]{}, 034902 (2003).
R.J. Fries *et al.*, Phys. Rev. C [**68**]{}, 044902 (2003).
R. Rapp, E.V. Shuryak, Phys. Rev. D [**67**]{}, 074036 (2003).
|
---
abstract: 'We study the microstate free entropy $\chi_{\proj}(p_1,\dots,p_n)$ of projections, and establish its basic properties similar to the self-adjoint variable case. Our main contribution is to characterize the pair-block freeness of projections by the additivity of $\chi_{\proj}$ (Theorem \[T-4.1\]), in the proof of which a transportation cost inequality plays an important role. We also briefly discuss the free pressure in relation to $\chi_{\proj}$.'
address:
- 'Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan'
- 'Graduate School of Mathematics, Kyushu University, Fukuoka 810-8560, Japan'
author:
- 'Fumio Hiai$\,^{1,2}$'
- 'Yoshimichi Ueda$\,^{1,3}$'
title: Notes on microstate free entropy of projections
---
[^1] [^2] [^3] [^4]
Introduction {#introduction .unnumbered}
============
The theory of free entropy, initiated and mostly developed by D. Voiculescu in his series of papers [@V1]–[@V6], has become one of the most essential disciplines of free probability theory. For self-adjoint non-commutative random variables, say $X_1,\dots,X_n$, the microstate free entropy $\chi(X_1,\dots,X_n)$ introduced in [@V2] is defined as a certain asymptotic growth rate (as the matrix size $N$ goes to $\infty$) of the Euclidean volume of the set of $N\times N$ self-adjoint matrices $(A_1,\dots,A_n)$ approximating $(X_1,\dots,X_n)$ in moments. It is this microstate theory that settled some long-standing open questions in von Neumann algebras (see the survey [@V-Survey]). On the other hand, the non-microstate free entropy $\chi^*(X_1,\dots,X_n)$ was also introduced in [@V5] based on the non-commutative Hilbert transform and the notion of conjugate variables, without the use of microstates or so-called matrix integrals which are rather hard to handle. Although it is believed that both approaches should be unified and give the same quantity, only the inequality $\chi\le\chi^*$ is known to hold true due to Biane, Capitaine and Guionnet [@BCG] based on an idea of large deviation principle for several random matrices. In his work [@V6] Voiculescu developed another kind of non-microstate approach to the free entropy, the so-called free liberation theory, and introduced the mutual free information $i^*(X_1,\dots,X_n)$ based on it. He suggested there the need to apply the microstate approach to projection random variables because the usual microstate free entropy $\chi$ becomes always zero for projections while $i^*$ does not. Following the suggestion, we here study the microstate free entropy $\chi_\proj(p_1,\dots,p_n)$ of projections $p_1,\dots,p_n$ in the same lines as in [@V2] and [@V4] to provide the basis for future research.
The large deviation principle for random matrices as mentioned above started with the paper of Ben Arous and Guionnet [@BG] and has been almost completed in the single random matrix case (corresponding to the study of $\chi(X)$ for single random variable $X$), see the survey [@Gu-Survey]. We note that such large deviation principle played quite an important role not only for the foundation of free entropy theory but also for getting free analogs of several probability theoretic inequalities (see [@HU1] and the references therein). Recently, one more large deviation was shown in [@HP1] for an independent pair of random projection matrices, including the explicit formula of the free entropy $\chi_\proj(p,q)$ of a projection pair $(p,q)$. This is one of a few large deviation results (indeed the first full large deviation result) in the setting of several random matrices, though the method of the proof is based on the single variable case. Moreover, in [@HU2] we applied it to get a kind of logarithmic Sobolev inequality $\chi_\proj(p,q)\le\ffi^*(p:q)$ between the free entropy $\chi_\proj(p,q)$ and the mutual free Fisher information $\ffi^*(p,q)$ (see [@V6]) for a projection pair. The large deviation result in [@HP1] also plays a crucial role in our study of $\chi_{\proj}$ here.
The paper is organized as follows. After stating the definition and basic properties of $\chi_\proj(p_1,\dots,p_n)$ in §1, we recall in §2 the formula in the case of two variables. In §3 we introduce a certain functional calculus for a projection pair $(p,q)$ and provide a technical tool of separate change of variable formula. This tool is essential in §4 to prove the additivity theorem characterizing the pair-block freeness of projections by the additivity of their free entropy. §5 treats a free analog of transportation cost inequalities for tracial distributions of projections. Its simplest case is needed in the proof of the above additivity theorem while of interest by itself. Finally, along the same lines as in [@Hi], we introduce in §6 the notion of free pressure and compare its Legendre transform with $\chi_\proj(p_1,\dots,p_n)$, thus giving a variational expression of free entropy.
Definition
==========
For $N\in\bN$ let $\U(N)$ be the unitary group of order $N$. For $k\in\{0,1,\dots,N\}$ let $G(N,k)$ denote the set of all $N\times N$ orthogonal projection matrices of rank $k$, that is, $G(N,k)$ is identified with the Grassmannian manifold consisting of $k$-dimensional subspaces in $\bC^N$. With the diagonal matrix $P_N(k)$ of the first $k$ diagonals $1$ and the others $0$, each $P\in G(N,k)$ is diagonalized as $$\label{F-1.1}
P=UP_N(k)U^*,$$ where $U\in\U(N)$ is determined up to the right multiplication of elements in $\U(k)\oplus\U(N-k)$. Hence $G(N,k)$ is identified with the homogeneous space $\U(N)/(\U(k)\oplus\U(N-k))$, and we have a unique probability measure $\gamma_{G(N,k)}$ on $G(N,k)$ invariant under the unitary conjugation $P\mapsto UPU^*$ for $U \in \U(N)$. Via the description as homogeneous space, this corresponds to the measure on $\U(N)/(\U(k)\oplus\U(N-k))$ invarinat under the left multiplication of elements in $\U(N)$ or induced from the Haar probability measure $\gamma_{\U(N)}$ on $\U(N)$. Let $\xi_{N,k}:\U(N)\to G(N,k)$ be the (surjective continuous) map defined by , i.e., $\xi_{N,k}(U):=UP_N(k)U^*$. Then the measure $\gamma_{G(N,k)}$ is more explicitly written as $$\label{F-1.2}
\gamma_{G(N,k)}=\gamma_{\U(N)}\circ\xi_{N,k}^{-1}.$$
Throughout the paper $(\cM,\tau)$ is a tracial $W^*$-probability space. Let $(p_1,\dots,p_n)$ be an $n$-tuple of projections in $(\cM,\tau)$. Following Voiculescu’s proposal in [@V6 14.2] we define the [*free entropy*]{} $\chi_\proj(p_1,\dots,p_n)$ of $(p_1,\dots,p_n)$ as follows. Choose $k_i(N)\in\{0,1,\dots,N\}$ for each $N\in\bN$ and $1\le i\le n$ in such a way that $k_i(N)/N\to\tau(p_i)$ as $N\to\infty$ for $1\le i\le n$. For each $m\in\bN$ and $\eps>0$ set $$\begin{aligned}
\label{F-1.3}
&\Gamma_\proj(p_1,\dots,p_n;k_1(N),\dots,k_n(N);N,m,\eps) \nonumber\\
&\quad:=\biggl\{(P_1,\dots,P_n)\in\prod_{i=1}^nG(N,k_i(N)):
\bigg|{1\over N}\Tr_N(P_{i_1}\cdots P_{i_r})-\tau(p_{i_1}\cdots p_{i_r})\bigg|<\eps
\nonumber\\
&\hskip6cm\mbox{for all $1\le i_1,\dots,i_r\le n$, $1\le r\le m$}\biggr\},\end{aligned}$$ where $\Tr_N$ stands for the usual (non-normalized) trace on the $N\times N$ matrices. We then define $$\begin{aligned}
\label{F-1.4}
&\chi_\proj(p_1,\dots,p_n):=
\lim_{\substack{m\rightarrow\infty \\ \varepsilon \searrow 0}}
\limsup_{N\to\infty} \nonumber\\
&\qquad{1\over N^2}\log\Biggl(\bigotimes_{i=1}^n\gamma_{G(N,k_i(N))}\Biggr)
\bigl(\Gamma(p_1,\dots,p_n;k_1(N),\dots,k_n(N);N,m,\eps)\bigr).\end{aligned}$$ For the justification of the definition of $\chi_{\proj}$, here arises a natural question whether the quantity $\chi_{\proj}(p_1,\dots,p_n)$ depends on the particular choice of $k_i(N)$ or not. The following is the answer to it.
\[P-1.1\] The above definition of $\chi_\proj(p_1,\dots,p_n)$ is independent of the choices of $k_i(N)$ with $k_i(N)/N\to\alpha_i$ for $1\le i\le n$.
For $1\le i\le n$ let $l_i(N)$, $N\in\bN$, be another sequence such that $l_i(N)/N\to\alpha_i$ as $N\to\infty$. For each $N,m\in\bN$ and $\eps>0$, we write $\Gamma(\vec k(N),m,\eps)$ ($\subset\prod_{i=1}^nG(N,k_i(N))$) for the set with respect to $\vec k(N):=(k_1(N),\dots,k_n(N))$, and also $\Gamma(\vec l(N),m,\eps)$ ($\subset\prod_{i=1}^nG(N,l_i(N))$) for the same with respect to $\vec l(N):=(l_1(N),\dots,l_n(N))$. Moreover, we set $\xi_{\vec k(N)}(\vec U):=\bigl(\xi_{N,k_1(N)}(U_1),\dots,\xi_{N,k_n(N)}(U_n)\bigr)$ for $\vec U=(U_1,\dots,U_n)\in\U(N)^n$, and define the subset $\widetilde\Gamma(\vec l(N),m,\eps):=\xi_{\vec l(N)}\circ
\xi_{\vec k(N)}^{-1}\bigl(\Gamma(\vec k(N),m,\eps)\bigr)$ of $\prod_{i=1}^nG(N,l_i(N))$. For every $N\in\bN$ and $U\in\U(N)$, since $$\xi_{N,l_i(N)}(U)-\xi_{N,k_i(N)}(U)=U\bigl(P_N(k_i(N))-P_N(l_i(N))\bigr)U^*,$$ we get $$\big\|\xi_{N,l_i(N)}(U)-\xi_{N,k_i(N)}(U)\big\|_1={|l_i(N)-k_i(N)|\over N},$$ where $\|\cdot\|_1$ denotes the trace-norm with respect to $N^{-1}\Tr_N$. For every $m\in\bN$ and $\eps>0$, there exists an $N_0\in\bN$ such that $N^{-1}|l_i(N)-k_i(N)|<\eps/m$ for all $N\ge N_0$ and $1\le i\le n$. Let us prove that $\widetilde\Gamma(\vec l(N),m,\eps)\subset\Gamma(\vec l(N),m,2\eps)$ whenever $N\ge N_0$. Assume that $N\ge N_0$ and $\vec Q=(Q_1,\dots,Q_n)\in\widetilde\Gamma(\vec l(N),m,\eps)$; then $\vec U=(U_1,\dots,U_n)\in\U(N)^n$ exists so that $\vec Q=\xi_{\vec l(N)}(\vec U)$ and $\vec P=(P_1,\dots,P_n):=\xi_{\vec k(N)}(\vec U)\in\Gamma_{\vec k(N),m,\eps}$. Since $$\|Q_i-P_i\|_1=\big\|\xi_{N,l_i(N)}(U_i)-\xi_{N,k_i(N)}(U_i)\big\|_1
<{\eps\over m},\qquad1\le i\le n,$$ we get for $1\le i_1,\dots,i_r\le n$ and $1\le r\le m$ $$\bigg|{1\over N}\Tr_N(Q_{i_1}\cdots Q_{i_r})
-{1\over N}\Tr_N(P_{i_1}\cdots P_{i_r})\bigg|
\le\sum_{j=1}^r\|Q_{i_j}-P_{i_j}\|_1<\eps$$ so that $$\bigg|{1\over N}\Tr_N(Q_{i_1}\cdots Q_{i_r})-\tau(p_{i_1}\cdots p_{i_r})\bigg|<2\eps,$$ implying $\vec Q\in\Gamma(\vec l(N),m,2\eps)$. Setting $\gamma_{\vec k(N)}:=\bigotimes_{i=1}^n\gamma_{G(N,k_i(N))}$, we now have thanks to $$\begin{aligned}
\gamma_{\vec l(N)}\bigl(\Gamma(\vec l(N),m,2\eps)\bigr)
&\ge\gamma_{\vec k(N)}\bigl(\widetilde\Gamma(\vec l(N),m,\eps)\bigr) \\
&=\bigl(\gamma_{\U(N)}\bigr)^{\otimes n}
\circ\xi_{\vec l(N)}^{-1}\circ\xi_{\vec l(N)}\circ\xi_{\vec k(N)}^{-1}
\bigl(\Gamma(\vec k(N),m,\eps)\bigr) \\
&\ge\bigl(\gamma_{\U(N)}\bigr)^{\otimes n}\circ\xi_{\vec k(N)}^{-1}
\bigl(\Gamma(\vec k(N),m,\eps)\bigr)
=\gamma_{\vec k(N)}\bigl(\Gamma(\vec k(N),m,\eps)\bigr)\end{aligned}$$ whenever $N\ge N_0$. This implies that $$\limsup_{N\to\infty}{1\over N^2}\log
\gamma_{\vec l(N)}\bigl(\Gamma(\vec l(N),m,2\eps)\bigr) \\
\ge\limsup_{N\to\infty}{1\over N^2}\log
\gamma_{\vec k(N)}\bigl(\Gamma(\vec k(N),m,\eps)\bigr),$$ which says that the free entropy given for $\vec k(N)$ is not greater than that for $\vec l(N)$. By symmetry we observe that both free entropies must coincide.
The following are basic properties of $\chi_\proj$. We omit their proofs, all of which are essentially same as in the case of self-adjoint variables in [@V2] or else obvious.
\[P-1.2\] Let $p_1,\dots,p_n$ be projections in $(\cM,\tau)$.
- Negativity[:]{} $\chi_\proj(p_1,\dots,p_n)\le0$.
- Subadditivity[:]{} for every $1\le j<n$, $$\chi_\proj(p_1,\dots,p_n)\le\chi_{\proj}(p_1,\dots,p_j)+\chi_\proj(p_{j+1},\dots,p_n).$$
- Upper semi-continuity[:]{} if a sequence $(p_1^{(m)},\dots,p_n^{(m)})$ of $n$-tuples of projections converges to $(p_1,\dots,p_n)$ in distribution, then $$\chi_\proj(p_1,\dots,p_n)\ge\limsup_{m\to\infty}
\chi_\proj(p_1^{(m)},\dots,p_n^{(m)}).$$
- $\chi_\proj(p_1,\dots,p_n)$ does not change when $p_i$ is replaced by $p_i^\perp:=\1-p_i$ for each $i$.
\[R-1.3\] [We may adopt different ways to introduce the free entropy of an $n$-tuple $(p_1,\dots,p_n)$ of projections in $(\cM,\tau)$. For instance, for each $N\in\bN$ consider two unitarily invariant probability measures $\gamma_{G(N)}^{(1)}$ and $\gamma_{G(N)}^{(2)}$ on $G(N):=\bigsqcup_{k=0}^NG(N,k)$ determined by the weights on $G(N,k)$, $0\le k\le N$, given as $$\gamma_{G(N)}^{(1)}(G(N,k))={1\over N+1},\quad
\gamma_{G(N)}^{(2)}(G(N,k))={1\over2^N}{N\choose k}.$$ For each $m\in\bN$ and $\eps>0$ set $$\begin{aligned}
&\Gamma_\proj(p_1,\dots,p_n;N,m,\eps) \\
&\qquad:=\biggl\{(P_1,\dots,P_n)\in G(N)^n:
\bigg|{1\over N}\Tr_N(P_{i_1}\cdots P_{i_r})-\tau(p_{i_1}\cdots p_{i_r})\bigg|<\eps \\
&\hskip6cm\mbox{for all $1\le i_1,\dots,i_r\le n$, $1\le r\le m$}\biggr\},\end{aligned}$$ and define for $j=1,2$ $$\chi_\proj^{(j)}(p_1,\dots,p_n)
:=\lim_{\substack{m\rightarrow\infty \\ \varepsilon \searrow 0}}
\limsup_{N\to\infty}{1\over N^2}
\log\Bigl(\gamma_{G(N)}^{(j)}\Bigr)^{\otimes n}
\bigl(\Gamma_\proj(p_1,\dots,p_n;N,m,\eps)\bigr).$$ It is fairly easy to see (similarly to the proof of Proposition 1.1) that both $\chi_\proj^{(j)}(p_1,\dots,p_n)$, $j=1,2$, coincide with $\chi_\proj(p_1,\dots,p_n)$ given in . ]{}
Case of two projections
=======================
Let $(p,q)$ be a pair of projections in $(\cM,\tau)$ with $\alpha:=\tau(p)$ and $\beta:=\tau(q)$. Set $$E_{11}:=p\wedge q,\quad E_{10}:=p\wedge q^\perp,\quad
E_{01}:=p^\perp\wedge q,\quad E_{00}:=p^\perp\wedge q^\perp,$$ $$E:=\1-(E_{00}+E_{01}+E_{10}+E_{11}).$$ Then $E$ and $E_{ij}$ are in the center of $\{p,q\}''$ and $(E\{p,q\}''E,\tau|_{E\{p,q\}''E})$ is isomorphic to $L^\infty((0,1),\nu;M_2(\bC))$, where $\nu$ is the measure on $(0,1)$ determined by $$\tau(A) = \frac{1}{2} \int_{(0,1)} \mathrm{Tr}_2(A(x))\,d\nu(x),
\quad A \in L^{\infty}((0,1),\nu;M_2(\mathbb{C})) \cong E\{p,q\}'' E$$ (hence $\nu((0,1))=\tau(E)$). Under this isomorphism, $EpE$ and $EqE$ are represented as $$(EpE)(x)=\bmatrix1&0\\0&0\endbmatrix\ \ \mbox{and}
\ \ (EqE)(x)=\bmatrix x&\sqrt{x(1-x)}\\\sqrt{x(1-x)}&1-x\endbmatrix
\quad\mbox{for $x\in(0,1)$}.$$ In this way, the mixed moments of $(p,q)$ with respect to $\tau$ are determined by $\nu$ and $\{\tau(E_{ij})\}_{i,j=0}^1$. Although $\nu$ is not necessarily a probability measure, we define the free entropy $\Sigma(\nu)$ by $$\Sigma(\nu):=\iint_{(0,1)^2}\log|x-y|\,d\nu(x)\,d\nu(y)$$ in the same way as in [@V1]. Furthermore, we set $$\label{F-2.1}
\rho:=\min\{\alpha,\beta,1-\alpha,1-\beta\},$$ $$\label{F-2.2}
C:=\rho^2B\biggl({|\alpha-\beta|\over\rho},{|\alpha+\beta-1|\over\rho}\biggr)$$ (meant zero if $\rho=0$), where $$\begin{aligned}
B(s,t)&:={(1+s)^2\over2}\log(1+s)-{s^2\over2}\log s
+{(1+t)^2\over2}\log(1+t)-{t^2\over2}\log t \\
&\qquad-{(2+s+t)^2\over2}\log(2+s+t)+{(1+s+t)^2\over2}\log(1+s+t)\end{aligned}$$ for $s,t\geq0$. With these definitions, the following formula of $\chi_\proj(p,q)$ was obtained in [@HP1] as a consequence of the large deviation principle for an independent pair of random projection matrices.
\[P-2.1\][([@HP1 Theorem 3.2, Proposition 3.3])]{}If $\tau(E_{00})\tau(E_{11})=\tau(E_{01})\tau(E_{10})=0$, then $$\begin{aligned}
\chi_\proj(p,q)&={1\over4}\Sigma(\nu)
+{|\alpha-\beta|\over2}\int_{(0,1)}\log x\,d\nu(x) \\
&\qquad\quad+{|\alpha+\beta-1|\over2}\int_{(0,1)}\log(1-x)\,d\nu(x)-C,\end{aligned}$$ and otherwise $\chi_\proj(p,q)=-\infty$. Moreover, $\chi_\proj(p,q)=0$ if and only if $p$ and $q$ are free.
Note that the condition $\tau(E_{00})\tau(E_{11})=\tau(E_{01})\tau(E_{10})=0$ is equivalent to $$\label{F-2.3}
\begin{cases}
\tau(E_{11})=\max\{\alpha+\beta-1,0\}, \\
\tau(E_{00})=\max\{1-\alpha-\beta,0\}, \\
\tau(E_{10})=\max\{\alpha-\beta,0\}, \\
\tau(E_{01})=\max\{\beta-\alpha,0\};
\end{cases}$$ in this case, $\tau(E_{01})+\tau(E_{10})=|\alpha-\beta|$, $\tau(E_{00})+\tau(E_{11})=|\alpha+\beta-1|$ and $\tau(E)=2\rho$.
In the case where $\tau_\proj(p,q)=0$ (equivalently, $p$ and $q$ are free), the measure $\nu$ was computed in [@VDN] as $$\label{F-2.4}
{\sqrt{(x-\xi)(\eta-x)}\over2\pi x(1-x)}\1_{(\xi,\eta)}(x)\,dx$$ with $\xi,\eta:=\alpha+\beta-2\alpha\beta\pm\sqrt{4\alpha\beta(1-\alpha)(1-\beta)}$. It is also worthwhile to note (see [@HP1]) that $\limsup$ in definition can be replaced by $\lim$ in the case of two projections.
In §4 the equivalence between the additivity of $\chi_\proj$ and the freeness of projection variables will be generalized to the case of more than two projections. To do this, we need a kind of separate change of variable formula for $\chi_\proj$ established in the next section.
Separate change of variable formula
===================================
Let $N\in\bN$ and $k,l\in\{0,1,\dots,N\}$. Assume that $0<k\le l$ and $k+l\le N$. Consider a pair $(P,Q)$ of $N\times N$ projection matrices with ${\rm rank}(P)=k$ and ${\rm rank}(Q)=l$, which is distributed under the measure $\gamma_{G(N,k)}\otimes\gamma_{G(N,l)}$ on $G(N,k)\times G(N,l)$. Thanks to the assumptions on $k,l$, for any pair $(P,Q)\in G(N,k)\times G(N,l)$ the so-called sine-cosine decomposition of two projections gives the following representation: $$\begin{aligned}
P&=U\left(\bmatrix I&0\\0&0\endbmatrix\oplus0\oplus0\right)U^*, \label{F-3.1}\\
Q&=U\left(\bmatrix X&\sqrt{X(I-X)}\\\sqrt{X(I-X)}&I-X\endbmatrix
\oplus I\oplus0\right)U^* \label{F-3.2}\end{aligned}$$ in $\bC^N=(\bC^k\otimes\bC^2)\oplus\bC^{l-k}\oplus\bC^{N-k-l}$, where $U$ is an $N\times N$ unitary matrix and $X$ is a $k\times k$ diagonal matrix with the diagonal entries $0\leq x_1\leq x_2\leq \dots\leq x_k\leq 1$. When $x_1,\dots,x_k$ are in $(0,1)$ and mutually distinct, it is easy to see that $U$ is uniquely determined up to the right multiplication of unitary matrices of the form $$\bmatrix T&0\\0&T\endbmatrix\oplus V_1\oplus V_2,
\quad T\in\bT^k,\ V_1\in\U(l-k),\ V_2\in\U(N-k-l).$$ We denote by $V(N,k,l)$ the subgroup of $\U(N)$ consisting of all unitary matrices of the above form so that $\U(N)/V(N,k,l)$ becomes a homogeneous space. Also, let $[0,1]_{\leq}^k$ and $(0,1)_<^k$ denote the sets of $(x_1,\dots,x_k)$ satisfying $0\leq x_1\leq\dots\leq x_N\leq 1$ and $0<x_1<\dots<x_k<1$, respectively. We then consider the continuous map $\Xi_{N,k,l}:
\mathrm{U}(N)/V(k,\ell) \times [0,1]_{\leq}^k \rightarrow G(N,k)\times G(N,l)$ defined by and , that is, $$\begin{aligned}
&\Xi_{N,k,l}([U],X) \\
&\quad :=
\left(
U\left(\begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix} \oplus0\oplus0\right)U^*,
U\left(\begin{bmatrix} X & \sqrt{X(I-X)} \\ \sqrt{X(I-X)} & I-X \end{bmatrix}
\oplus I\oplus0\right)U\right), \end{aligned}$$ where $X$ is regarded as a diagonal matrix. The set $$\left(G(N,k)\times G(N,l)\right)_0:=
\Xi_{N,k,l}\left(\mathrm{U}(N)/V(N,k,l)\times (0,1)^k_<\right)$$ is open and co-negligible with respect to $\gamma_{G(N,k)}\otimes\gamma_{G(N,l)}$ in $G(N,k)\times G(N,l)$ thanks to [@Col Theorem 2.2] (or [@HP1 Lemma 1.1]) and moreover $\Xi_{N,k,l}$ gives a smooth diffeomorphism between $(\U(N)/V(N,k,l))\times(0,1)_<^k$ and $\left(G(N,k)\times G(N,l)\right)_0$. The next lemma will be needed later.
\[L-3.1\] The measure $(\gamma_{G(N,k)}\otimes\gamma_{G(N,l)})\circ\Xi_{N,k,l}$ coincides with $$\gamma_{N,k,l}\otimes\Biggl({1\over Z_{N,k,l}}\prod_{i=1}^kx_i^{l-k}
(1-x_i)^{N-k-l}\prod_{1\le i<j\le k}(x_i-x_j)^2\prod_{i=1}^kdx_i\Biggr),$$ where $\gamma_{N,k,l}$ is the [(]{}unique[)]{} probability measure on $\U(N)/V(N,k,l)$ induced by the Haar probability measure on $\U(N)$ and $Z_{N,k,l}$ is a normalization constant.
Let $\lambda$ be the measure on $\mathrm{U}(N)/V(N,k,l)\times(0,1)^k_<$ transformed from the restriction of $\gamma_{G(N,k)}\otimes\gamma_{G(N,l)}$ to $\left(G(N,k)\times G(N,l)\right)_0$ by the inverse of $\Xi_{N,k,l}$, and $\mu$ be its image measure by the projection map $([U],X) \mapsto X$. The disintegration theorem (see e.g. [@Ma Chapter IV, §6.5]) ensures that there is a $\mu$-a.e. unique Borel map $\lambda_{(\cdot)}$ from $(0,1)^k_<$ to the probability measures on $\U(N)/V(N,k,l)$ such that $\lambda = \int_{(0,1)^k_<} \lambda_X\,d\mu(X)$. Note that $([U],X) \mapsto X$ splits into $\Xi_{N,k,l}$, $(P,Q) \mapsto PQP$ and the map sending $PQP$ to the eigenvalues in increasing order. Hence $\mu$ coincides with the eigenvalue distribution of $PQP$ arranged in increasing order, which is known to be equal to the second component given in the lemma by [@Col Theorem 2.2]. Therefore, it suffices to show that $\lambda_X$ coincides with $\gamma_{N,k,l}$ for $\mu$-a.e. $X \in (0,1)_<^k$. For each $V\in\U(N)$, the unitary conjugation $\mathrm{Ad}\,V\times\mathrm{Ad}\,V : (P,Q)\mapsto(VPV^*,VQV^*)$ on $G(N,k)\times G(N,l)$ and the left-translation $L_V : [U]\mapsto V[U] := [VU]$ on $\mathrm{U}(N)/V(N,k,l)$ satisfy the relation $\Xi_{N,k,l}\circ
(L_V\times\mathrm{id})= (\mathrm{Ad}\,V\times\mathrm{Ad}\,V)\circ\Xi_{N,k,l}$; hence, in particular, $\left(G(N,k)\times G(N,l)\right)_0$ is invariant under the action $\mathrm{Ad}\,V\times\mathrm{Ad}\,V$ for every $V \in \mathrm{U}(N)$. Then, for any bounded Borel function $f$ on $\mathrm{U}(N)/V(N,k,l)\times(0,1)^k)$, one can easily verify that $$\begin{aligned}
&\int_{(0,1)_<^k}\left(\int_{\mathrm{U}(N)/V(N,k,l)} f([U],X)
\,d(\lambda_X\circ L_V)([U])\right)\,d\mu(X) \\
&\qquad\quad=\int_{\mathrm{U}(N)/V(N,k,l)\times(0,1)_<^k} f([U],X)\,d\lambda([U],X),\end{aligned}$$ which means a new disintegration $\lambda = \int_{(0,1)_<^k} \lambda_X\circ L_V\,d\mu(X)$. The uniqueness of the disintegration says that for $\mu$-a.e. $X \in (0,1)_<^k$ one has $\lambda_X = \lambda_X\circ L_V$ for all $V \in \mathrm{U}(N)$. Since $\gamma_{N,k,l}$ is a unique probability measure on $\mathrm{U}(N)/V(N,k,l)$ invariant under the left-translation action of $\mathrm{U}(N)$, it follows that $\lambda_X=\gamma_{N,k,l}$ for $\mu$-a.e. $X \in (0,1)^k_<$ so that $$\lambda=\int_{(0,1)_<^k}\gamma_{N,k,l}\,d\mu(X)=\gamma_{N,k,l}\otimes\mu,$$ as required.
For a pair $(p,q)$ of projections in $(\cM,\tau)$ we introduce a sort of functional calculus via the representation explained in §2 in the following way. Let $\psi$ be a continuous increasing function $\psi$ from $(0,1)$ into itself. With the notations in §2 we define a projection $q(\psi;p)$ in $\{p,q\}''$ by $$q(\psi;p):=Eq(\psi;p)E+E_{00}+E_{01}+E_{10}+E_{11},$$ $$(Eq(\psi;p)E)(x):=\bmatrix\psi(x)&\sqrt{\psi(x)(1-\psi(x))}\\
\sqrt{\psi(x)(1-\psi(x))}&1-\psi(x)\endbmatrix\quad\mbox{for $x\in(0,1)$}.$$ It is obvious that $\tau(q(\psi;p))=\tau(q)$. (The definition itself is possible for general Borel function from $(0,1)$ into $[0,1]$ but the above case is enough for our purpose.) The aim of this section is to prove the following change of variable formula for free entropy of projections.
\[T-3.2\] Let $p_1,q_1,\dots,p_n,q_n,r_{1},\dots,r_{n'}$ be projections in $(\cM,\tau)$ and assume that $\chi_{\rm proj}(p_i,q_i)>-\infty$ for $1\le i\le n$. Let $\psi_1,\dots,\psi_n$ be continuous increasing functions from $(0,1)$ into itself, and $q_i(\psi_i;p_i)$ be the projection defined from $p_i$, $q_i$ and $\psi_i$ as above for $1\le i\le n$. Then we have $$\begin{aligned}
&\chi_{\rm proj}(p_1,q_1(\psi_1;p_1),\dots,p_n,q_n(\psi_n;p_n), r_1,\dots,r_{n'}) \\
&\quad\ge\chi_{\rm proj}(p_1,q_1,\dots,p_n,q_n,r_{1},\dots,r_{n'})
+\sum_{i=1}^n\bigl\{\chi_{\rm proj}(p_i,q_i(\psi_i;p_i))
-\chi_{\rm proj}(p_i,q_i)\bigr\}.\end{aligned}$$ Moreover, if $\psi_1,\dots,\psi_n$ are strictly increasing, then equality holds true in the above inequality.
The proof goes on the essentially same lines as in [@V4] and it is divided into two steps; one is to analyze the case when $\psi_1,\dots,\psi_n$ are all extended to $C^1$-diffeomorphisms from $[0,1]$ onto itself and the other is to approximate, in two stages, the given $\psi_1,\dots,\psi_n$ by $C^{\infty}$-diffeomorphisms from $[0,1]$ onto itself in such a way that the corresponding free entropies converge to those in question. As the first step let us prove the following special case of the theorem.
\[L-3.3\] Let $p_1,q_1,\dots,p_n,q_n,r_{1},\dots,r_{n'}$ be as in Theorem \[T-3.2\]. If $\psi_1,\dots,\psi_n$ are $C^1$-diffeomorphisms from $[0,1]$ onto itself with $\psi_i(0) = 0$, $\psi_i(1) = 1$ and moreover $\psi_i'(x)>0$ for all $x\in[0,1]$, then the equality of Theorem \[T-3.2\] holds true.
Obviously, it suffices to show when $n=1$; hence we assume $n=1$ and write $p=p_1$, $q=q_1$ and $\psi=\psi_1$ for brevity. Let $\nu$ and $\{E_{ij}\}_{i,j=0}^1$ be as in §2 for $(p,q)$. By Propositions \[P-1.2\](iv) and \[P-2.1\] we may assume that $\tau(p) \leq \tau(q) \leq 1/2$ so that $E_{11} = E_{10} = 0$ by . We may further assume that $p$ is non-zero; otherwise there is nothing to do. With the polar decomposition $(1-p)qp = v_{p,q}\sqrt{pqp(p-pqp)}$, we thus represent $p$, $q$ and $q(\psi;p)$ as follows: $$\begin{aligned}
p &= v_{p,q}^* v_{p,q}, \\
q &= pqp + v_{p,q}\sqrt{pqp(p-pqp)}
+ \sqrt{pqp(p-pqp)}v_{p,q}^* + v_{p,q}(p-pqp)v_{p,q}^* \\
&\quad+ \Bigl(q-pqp - (1-p)qp - pq(1-p) - v_{p,q}(p-pqp)v_{p,q}^*\Bigr), \\
q(\psi;p) &= \psi(pqp) + v_{p,q}\sqrt{\psi(pqp)(p-\psi(pqp))} \\
&\quad + \sqrt{\psi(pqp)(p-\psi(pqp))}v_{p,q}^*
+ v_{p,q}(p-\psi(pqp))v_{p,q}^* \\
&\quad + \Bigl(q-pqp - (1-p)qp - pq(1-p) - v_{p,q}(p-pqp)v_{p,q}^*\Bigr), \end{aligned}$$ where $\psi(pqp)$ means the functional calculus of $pqp$. Choose two sequences $k(N)$, $l(N)$ for $N\ge2$ in such a way that $0<k(N) \leq l(N) \leq N/2$ and $k(N)/N \rightarrow \tau(p)$, $l(N)/N \rightarrow \tau(q)$ as $N \rightarrow \infty$. As explained at the beginning of this section, for each $(P,Q) \in (G(N,k(N))\times G(N,l(N)))_0$ there is a unitary $U \in \mathrm{U}(N)$, unique up to $V(N,k(N),l(N))$, for which we have and . Then we can define the map $\Phi_{N,\psi}$ on $(G(N,k(N))\times G(N,l(N)))_0$ by sending $(P,Q)$ to $(P,Q(\psi;P))$ with $$Q(\psi;P) := U\left(\begin{bmatrix}\psi(X) & \sqrt{\psi(X)(I-\psi(X))} \\
\sqrt{\psi(X)(I-\psi(X))} & I-\psi(X) \end{bmatrix} \oplus I\oplus0\right)U^*.$$ With the polar decomposition $(I-P)QP = V_{P,Q}\sqrt{PQP(I-PQP)}$ we have the following expressions: $$\begin{aligned}
Q &= PQP + V_{P,Q}\sqrt{PQP(P-PQP)} \\
&\quad +\sqrt{PQP(P-PQP)}V_{P,Q}^* + V_{P,Q}(P-PQP)V_{P,Q}^* \\
&\quad+ \Bigl(Q-PQP-(I-P)QP-PQ(I-P)-V_{P,Q}(P-PQP)V_{P,Q}^*\Bigr), \\
Q(\psi;P) &= \psi(PQP) + V_{P,Q}\sqrt{\psi(PQP)(P-\psi(PQP))} \\
&\quad +\sqrt{\psi(PQP)(P-\psi(PQP))}V_{P,Q}^* + V_{P,Q}(P-\psi(PQP))V_{P,Q}^* \\
&\quad + \Bigl(Q-PQP-(I-P)QP-PQ(I-P)-V_{P,Q}(P-PQP)V_{P,Q}^*\Bigr). \end{aligned}$$ Upon these expressions, what we now need is to approximate $v_{p,q}$ and $V_{P,Q}$ by polynomials of $p,q$ and $P,Q$, respectively, as stated in the next lemma very similarly to [@HP1 Lemma 2.6] (or [@HP 6.6.4]).
\[L-3.4\] For each $t\geq1$ and $\varepsilon>0$ there exist $N_0, m_0 \in \mathbb{N}$, $\eps_0 >0$ and a real polynomial $G$ such that $\Vert v_{p,q} - (1-p)qp\cdot G(pqp) \Vert_t < \varepsilon$ and such that, for each $N \geq N_0$, if $(P,Q) \in (G(N,k(N))\times G(N,l(N)))_0$ and if $$\label{F-3.3}
\left|\frac{1}{N}\mathrm{Tr}_N((PQP)^m) - \tau((pqp)^m)\right| < \eps_0
\quad \mbox{for $1\leq m \leq m_0$},$$ then $\Vert V_{P,Q} - (1-P)QP\cdot G(PQP)\Vert_t < \varepsilon$. Here, $\|\cdot\|_t$ denotes the Schatten $t$-norm with respect to $\tau$ as well as $N^{-1}\Tr_N$.
We only sketch the proof since it is essentially similar to that of [@HP1 Lemma 2.6]. For each small $\alpha,\beta>0$ we estimate $$\begin{aligned}
\label{F-3.4}
&\Vert v_{p,q} - ((1-p)qp)(\sqrt{pqp(p-pqp)}+\alpha1)^{-1}\Vert_t^t \nonumber\\
&\quad\leq
\frac{1}{2}\left\{\nu((0,\beta)) + \nu((1-\beta,1)) +
\nu([\beta,1-\beta])\left(\frac{\alpha}{\sqrt{\beta(1-\beta)}+\alpha}\right)^t
\right\}\end{aligned}$$ and $$\begin{aligned}
\label{F-3.5}
&\Vert V_{P,Q} - (I-P)QP(\sqrt{PQP(P-PQP)}+\alpha I)^{-1}\Vert_t^t \nonumber\\
&\quad\leq
\frac{1}{N}\#\left\{i : \lambda_i(PQP) < \beta\right\}
+ \frac{1}{N}\#\left\{i : \lambda_i(PQP) > 1-\beta\right\} \nonumber\\
&\quad\qquad + \frac{k(N)}{N}\left(\frac{\alpha}{\sqrt{\beta(1-\beta)}+\alpha}
\right)^t, \end{aligned}$$ where $0 < \lambda_1(PQP) < \cdots < \lambda_{k(N)}(PQP) < 1$ are the eigenvalues of $PQP|_{P\bC^N}$ for $(P,Q) \in (G(N,k(N))\times G(N,l(N)))_0$. For any $\eta>0$ let us choose a $\beta>0$ so that $\nu((0,2\beta)) + \nu((1-2\beta,1)) < \eta^t$. By we get $$\label{F-3.6}
\Vert v_{p,q} - ((1-p)qp)(\sqrt{pqp(p-pqp)}+\alpha 1)^{-1}\Vert_t^t
\leq \frac{\eta^t}{2} + \frac{\tau(E)}{2}\left(
\frac{\alpha}{\sqrt{\beta(1-\beta)}}\right)^t.$$ Note that $\nu$ is non-atomic on $(0,1)$ due to the assumption $\chi_{\mathrm{proj}}(p,q) > -\infty$. Set $\xi_{N,i}:=\min\{x\in[0,1]:\nu((0,x))
=i\tau(E)/k(N)\}$ for $1\le i\le k(N)$; then we get $$\tau((pqp)^m)= \lim_{N\rightarrow\infty}\frac{1}{N}
\sum_{i=1}^{k(N)} \big(\xi_{N,i}\big)^m
\quad\mbox{for all $m \in \mathbb{N}$}.$$ Also choose a constant $C>\sup_{N\ge2}N/k(N)$. By [@HP 4.3.4] there are $m_0 \in \mathbb{N}$ and $\eps_0>0$ such that, for every $N\in\mathbb{N}$ and for every $(\lambda_1,\dots,\lambda_{k(N)}) \in (0,1)_<^{k(N)}$, $$\left|\frac{1}{k(N)}\sum_{i=1}^{k(N)} \lambda_i^m
- \frac{1}{k(N)}\sum_{i=1}^{k(N)}\big(\xi_{N,i}\big)^m\right| < 2C\eps_0
\quad\mbox{for $1\le m\le m_0$}$$ implies $$\label{F-3.7}
\frac{1}{k(N)} \sum_{i=1}^{k(N)} \big|\lambda_i - \xi_{N,i}\big|^m
< \beta \eta^t.$$ Assume . Set $i_0 := \#\{ i : \lambda_i < \beta\}$ and $i_1 := \#\{i : \xi_{N,i}< 2\beta\}$. If $i_1 < i \leq i_0$, then $\big|\lambda_i - \xi_{N,i}\big| = \xi_{N,i} - \lambda_i \geq \beta$ so that we get $i_0 < i_1 + k(N)\eta^t$ by . Since $i_1\tau(E)/k(N) \leq \nu((0,2\beta)) < \eta^t$, we get $i_0 < \tau(E)^{-1}(1+\tau(E))k(N)\eta^t$. If there is no $i_1 < i \leq i_0$, then $i_0 \leq i_1 <\tau(E)^{-1}k(N)\eta^t$. Therefore, $\#\{i:\lambda_i<\beta\}<\tau(E)^{-1}(1+\tau(E))k(N)\eta^t$. Similarly, we have $\#\{i:\lambda_i>1-\beta\}<\tau(E)^{-1}(1+\tau(E))k(N)\eta^t$. Now, choose an $N_0 \in \mathbb{N}$ so that $$\left|\frac{1}{N}\sum_{i=1}^{k(N)} \big(\xi_{N,i}\big)^m
- \tau((pqp)^m)\right| < \eps_0$$ for all $1 \leq m \leq m_0$ and $N \geq N_0$. We then conclude that, for every $N\ge N_0$, if $(P,Q) \in (G(N,k(N))\times G(N,l(N)))_0$ satisfies , then $$\begin{aligned}
\#\{ i : \lambda_i(PQP) < \beta \} &< \frac{1+\tau(E)}{\tau(E)}k(N)\eta^t,
\label{F-3.8}\\
\#\{ i : \lambda_i(PQP) > 1-\beta \} &< \frac{1+\tau(E)}{\tau(E)}k(N)\eta^t.
\label{F-3.9}\end{aligned}$$ Inserting and in we get $$\begin{aligned}
&\Vert V_{P,Q} - (I-P)QP(\sqrt{PQP(P-PQP)}+\alpha I)^{-1}\Vert_t^t \nonumber\\
&\qquad\leq \frac{1+\tau(E)}{\tau(E)}\eta^t + \frac{1}{2}\left(
\frac{\alpha}{\sqrt{\beta(1-\beta)}}\right)^t. \label{F-3.10}\end{aligned}$$ Finally, let $\alpha>0$ be so small as $\alpha/\sqrt{\beta(1-\beta)} < \eta$, and choose a real polynomial $G(x)$ such that $|G(x) - (\sqrt{x(1-x)}+\alpha)^{-1}| < \eta$ for all $x \in [0,1]$. Then by and we obtain $$\Vert v_{p,q} - (1-p)qp\cdot G(pqp)\Vert_t < 2\eta$$ and $$\Vert V_{P,Q} - (I-P)QP\cdot G(PQP)\Vert_t
< \left(\left(\frac{1}{\tau(E)}+{3\over2}\right)^{1/t}+1\right) \eta.$$ The proof is completed if $\eta>0$ was chosen so small as $\left((1/\tau(E)+3/2)^{1/t}+1\right)\eta < \varepsilon$.
[*Proof of Lemma \[L-3.3\].*]{}Choose $k_1(N),\dots,k_{n'}(N)$ so that $k_i(N)/N \rightarrow \tau(r_i)$ as $N\rightarrow\infty$, and set $$\Phi_N := \Phi_{N,\psi}\times\prod_{i=1}^{n'}\mathrm{id}_{G(N,k_i(N))}
\quad\mbox{on}\ (G(N,k(N))\times G(N,l(N)))_0\times\prod_{i=1}^{n'}G(N,k_i(N))$$ and $\gamma_N := \gamma_{G(N,k(N))}\otimes\gamma_{G(N,l(N))}\otimes
\bigotimes_{i=1}^{n'}\gamma_{G(N,k_i(N))}$. Let $m \in \mathbb{N}$ and $\varepsilon>0$ be arbitrary. In the following, for brevity we write $\Gamma_\proj(p,q,r_1,\dots,r_{n'};N,m_0,\eps_0)$ etc. without $k(N),l(N),k_1(N),\dots,k_n(N)$. Thanks to Lemma \[L-3.4\] together with the expressions of $q(\psi;p)$ and $Q(\psi;P)$ above, we can choose $N_0, m_0 \in \mathbb{N}$ and $\varepsilon_0>0$ with $m_0 \geq m$ and $\varepsilon_0 \leq \varepsilon$ such that, for every $N \geq N_0$, if $(P,Q,R_1,\dots,R_{n'}) \in \Gamma_{\mathrm{proj}}(p,q,r_1,\dots,r_{n'};
N,m_0,\varepsilon_0)$ and $(P,Q)\in(G(N,k(N))\times G(N,l(N)))_0$, then $\Phi_N(P,Q,R_1,\allowbreak\dots,R_{n'})$ falls into $\Gamma_{\mathrm{proj}}(p,q(\psi;p),r_1,\dots,r_{n'};\allowbreak N,m,\varepsilon)$. Via $\Xi_{N,k(N),l(N)}$ in the first two coordinates, Lemma \[L-3.1\] enables us to estimate the Radon-Nikodym derivative $d\gamma_N\circ\Phi_N/d\gamma_N$ on a co-negligible subset of $\Gamma_{\mathrm{proj}}(p,q,r_1,\dots,r_{n'};N,m,\varepsilon)$ from below by the infimum value of $$\begin{aligned}
\label{F-3.11}
&\prod_{1\leq i < j \leq k(N)}
\left(\frac{\psi(\lambda_i(PQP))-\psi(\lambda_j(PQP))}
{\lambda_i(PQP)-\lambda_j(PQP)}\right)^2
\,\prod_{i=1}^{k(N)}\psi'(\lambda_i(PQP)) \notag\\
&\qquad\times \prod_{i=1}^{k(N)} \left(\frac{\psi(\lambda_i(PQP))}{\lambda_i(PQP)}
\right)^{l(N)-k(N)}
\,\prod_{i=1}^{k(N)}\left(\frac{1-\psi(\lambda_i(PQP))}{1-\lambda_i(PQP)}
\right)^{N-k(N)-l(N)} \end{aligned}$$ for all $(P,Q) \in (G(N,k(N))\times G(N,l(N)))_0\cap\Gamma_{\mathrm{proj}}
(p,q;N,m_0,\varepsilon_0)$ with the eigenvalue list $\lambda_1(PQP),\dots,\lambda_{k(N)}(PQP)$ in increasing order.
Let $\psi^{[1]}(x,y)$ be the so-called divided quotient of $\psi$, i.e., $$\psi^{[1]}(x,y) :=
\begin{cases}
\frac{\psi(x)-\psi(y)}{x-y} & (x\neq y), \\
\psi'(x) & (x=y).
\end{cases}$$ Then, quantity is rewritten in the coordinate $(P,Q)$ as $$\begin{aligned}
&\mathrm{det}_{k(N)^2\times k(N)^2}\left[ P\otimes P\cdot
\psi^{[1]}(PQP\otimes P,P\otimes PQP)\cdot P\otimes P \right] \\
&\quad\times \left(\mathrm{det}_{k(N)\times k(N)}[P(PQP)^{-1}\psi(PQP)P]
\right)^{l(N)-k(N)} \\
&\quad\times \left(\mathrm{det}_{k(N)\times k(N)}[P(P-PQP)^{-1}(P-\psi(PQP)P]
\right)^{N-k(N)-l(N)} \\
&= \exp\left(\mathrm{Tr}_{k(N)}^{\otimes2}\left(P\otimes P\cdot
\log(\psi^{[1]}(PQP\otimes P, P\otimes PQP))\cdot P\otimes P\right) \right) \\
&\quad\times
\left(\exp\left(\mathrm{Tr}_{k(N)}\left(P\cdot\log\left((PQP)^{-1}\psi(PQP)
\right)\cdot P\right)\right)\right)^{l(N)-k(N)} \\
&\quad\times
\left(\exp\left(\mathrm{Tr}_{k(N)}\left(P\cdot\log\left((P-PQP)^{-1}(P-\psi(PQP))
\right)\cdot P\right)\right)\right)^{N-k(N)-l(N)},\end{aligned}$$ where $\psi^{[1]}(PQP\otimes P,P\otimes PQP)$ is defined on $P\bC^N\otimes P\bC^N$ while $(PQP)^{-1}\psi(PQP)$ and $(P-PQP)^{-1}(P-\psi(PQP))$ are on $P\bC^N$. Let $\delta>0$ be arbitrary. Since $\psi$ is $C^1$, $\log\psi^{[1]}(x,y)$ is continuous on $[0,1]^2$ so that there is a real polynomial $L(x,y)$ on $[0,1]^2$ such that $\Vert \log\psi^{[1]} - L\Vert_{\infty} < \delta$. If $m'\in\mathbb{N}$ is larger than the degree of $L$, then we have, for each $(P,Q) \in \Gamma_{\mathrm{proj}}(p,q;N,m',\varepsilon')$ with an arbitrary $\varepsilon'>0$, $$\begin{aligned}
&\Big|\frac{1}{N^2}\mathrm{Tr}_{N}^{\otimes2}\left(P\otimes P\cdot
\log\psi^{[1]}(PQP\otimes P, P\otimes PQP)\cdot P\otimes P\right) \\
&\phantom{aaaaaaaaaaaaaaa}-
\tau^{\otimes2}(p\otimes p\cdot \log\psi^{[1]}(pqp\otimes p, p\otimes pqp)
\cdot p\otimes p)\Big| \\
&\quad\leq 2\delta +
\Big|\frac{1}{N^2}\mathrm{Tr}_{N}^{\otimes2}(P\otimes P\cdot
L(PQP\otimes P, P\otimes PQP)\cdot P\otimes P) \\
&\phantom{aaaaaaaaaaaaaaaaaaa}-
\tau^{\otimes2}(p\otimes p\cdot L(pqp\otimes p, p\otimes pqp)\cdot
p\otimes p)\Big| \\
&\quad\leq
2\delta+C\varepsilon \end{aligned}$$ with $C>0$ depending only on $L$ (hence on $\delta$). Therefore, for each $\eta>0$ there are $m_1 \in \mathbb{N}$ and $\varepsilon_1>0$ such that $$\begin{aligned}
\exp&\left(\mathrm{Tr}_{k(N)}^{\otimes2}\left(P\otimes P\cdot
\log\left(\psi^{[1]}(PQP\otimes P, P\otimes PQP)\right)\cdot
P\otimes P\right)\right) \notag\\
&\geq\exp\left(N^2\left\{\tau^{\otimes2}(p\otimes p\cdot
\log\psi^{[1]}(pqp\otimes p, p\otimes pqp)\cdot
p\otimes p)-\eta\right\}\right) \label{F-3.12}\end{aligned}$$ for all $(P,Q) \in \Gamma_{\mathrm{proj}}(p,q;N,m',\varepsilon')$ as long as $m' \geq m_1$ and $0<\varepsilon'\leq\varepsilon_1$. Since $x^{-1}\psi(x)$ and $(1-x)^{-1}(1-\psi(x))$ are both bounded below above $0$ on $[0,1]$ due to the assumption on $\psi$, the same argument works for the other two terms $$\begin{gathered}
\exp\left(\mathrm{Tr}_{k(N)}\left(P\cdot\log\left((PQP)^{-1}\psi(PQP)
\right)\cdot P\right)\right), \\
\exp\left(\mathrm{Tr}_{k(N)}\left(P\cdot\log\left((P-PQP)^{-1}(P-\psi(PQP))
\right)\cdot P\right)\right).\end{gathered}$$ Therefore, for each $\eta>0$ there are $m_2 \in \mathbb{N}$ and $\varepsilon_2>0$ such that $$\begin{aligned}
\exp&\left(\mathrm{Tr}_{k(N)}\left(P\cdot\log\left((PQP)^{-1}\psi(PQP)
\right)\cdot P\right)\right) \notag\\
&\geq
\exp\left(N\left\{\tau(p\cdot\log((pqp)^{-1}\psi(pqp))\cdot p)-\eta\right\}\right),
\label{F-3.13}\\
\exp&\left(\mathrm{Tr}_{k(N)}\left(P\cdot\log\left((P-PQP)^{-1}(P-\psi(PQP))
\right)\cdot P\right)\right) \notag\\
&\geq\exp\left(N\left\{\tau(p\cdot\log((p-pqp)^{-1}(p-\psi(pqp)))\cdot p)
-\eta\right\}\right) \label{F-3.14}\end{aligned}$$ for all $(P,Q) \in \Gamma_{\mathrm{proj}}(p,q;N,m',\varepsilon')$ as long as $m'\geq m_2$ and $0<\varepsilon'\leq\varepsilon_2$. Hence, for every $N \geq N_0$, $m'\geq \max\{m_0,m_1,m_2\}$ and $0<\varepsilon'<\min\{\varepsilon_0,\varepsilon_1,\varepsilon_2\}$, we have Take the $\limsup$ as $N\rightarrow\infty$ and the limit as $m\rightarrow\infty$, $\varepsilon\searrow0$ in the above inequality. Since $\eta>0$ is arbitrary, we get $$\begin{aligned}
&\chi_{\mathrm{proj}}(p,q(\psi;p),r_1,\dots,r_{n'}) \\
&\quad\geq \chi_{\mathrm{proj}}(p,q,r_1,\dots,r_{n'})
+ \frac{1}{4}\iint_{(0,1)^2} \log\left|\frac{\psi(x)-\psi(y)}{x-y}\right|
\,d\nu(x)d\nu(y) \\
&\qquad+ {\tau(q)-\tau(p)\over2}
\int_{(0,1)} \log\frac{\psi(x)}{x}\,d\nu(x)
+ {1-\tau(q)-\tau(p)\over2}
\int_{(0,1)} \log\frac{1-\psi(x)}{1-x}\,d\nu(x) \\
&\quad =\chi_\proj(p,q,r_1,\dots,r_{n'})
+\chi_\proj(p,q(\psi;p))-\chi_\proj(p,q) \end{aligned}$$ thanks to Proposition \[P-2.1\]. The reverse inequality can be shown as well if we replace the inequalities – by their reversed versions.
For the second step we present two more technical lemmas. The proof of the next lemma should be compared with that of [@V4 Lemma 4.1].
\[L-3.5\] Let $\mu$ be a measure on $[0,1]$ with no atom at $0,1$, and assume the conditions $$\begin{gathered}
\iint_{(0,1)^2} \log|x-y|\,d\mu(x)\,d\mu(y) > -\infty, \label{F-3.15}\\
\int_{(0,1)} \log x\, d\mu(x) > -\infty, \label{F-3.16}\\
\int_{(0,1)} \log(1-x)\,d\mu(x) > -\infty. \label{F-3.17}\end{gathered}$$ If $\psi$ is a continuous increasing function from $[0,1]$ onto itself with $\psi(0) = 0$, $\psi(1) = 1$, then there exists a sequence of $C^{\infty}$-diffeomorphisms $\psi_j$ from $[0,1]$ onto itself with $\psi_j(0) = 0$, $\psi_j(1) = 1$ such that
- $\psi'_j(x)\geq1/j$ for all $j\in\bN$ and $x \in [0,1]$,
- $\psi_j \longrightarrow \psi$ uniformly on $[0,1]$,
- $\displaystyle{\lim_{j\rightarrow\infty}\iint_{(0,1)^2} \log|x-y|
\,d(\psi_j{}_*\mu)(x)\,d(\psi_j{}_*\mu)(y) = \iint_{(0,1)^2} \log|x-y|
\,d(\psi_*\mu)(x)\,d(\psi_*\mu)(y)}$,
- $\displaystyle{\lim_{j\rightarrow\infty}\int_{(0,1)} \log x
\,d(\psi_j{}_*\mu)(x) = \int_{(0,1)} \log x\,d(\psi_*\mu)(x)}$,
- $\displaystyle{\lim_{j\rightarrow\infty}\int_{(0,1)} \log (1-x)
\,d(\psi_j{}_*\mu)(x) = \int_{(0,1)} \log (1-x)\,d(\psi_*\mu)(x)}$,
where $\psi_*\mu$ is the image measure of $\mu$ by $\psi$. Furthermore, when conditions and/or for $\mu$ are dropped, the conclusion holds without [(iv)]{} and/or [(v)]{} correspondingly.
Extend $\psi$ to a continuous increasing function on the whole $\bR$ periodically, namely, $\psi(x+m) = \psi(x)+m$ for $x \in [0,1]$ and $m\in\mathbb{Z}$. For each $j \in \mathbb{N}$, by – one can choose a $\delta_j \in(0,1/j]$ such that $$\begin{aligned}
\iint_{\{ (x,y) \in (0,1)^2 : |x-y|<\delta_j\}} \log|x-y|\,d\mu(x)\,d\mu(y)
&\geq -1/j, \label{F-3.18}\\
\iint_{\{ (x,y) \in (0,1)^2 : |x-y|<\delta_j\}} d\mu(x)\,d\mu(y)
&\leq 1/(j\log j), \label{F-3.19}\\
\int_{(0,\delta_j)} \log x\,d\mu(x) &\geq -1/j, \label{F-3.20}\\
\mu((0,\delta_j)) &\leq 1/(j\log j), \label{F-3.21}\\
\int_{(1-\delta_j,1)} \log(1-x)\,d\mu(x) &\geq -1/j, \label{F-3.22}\\
\mu((1-\delta_j,1)) &\leq 1/(j\log j). \label{F-3.23} \end{aligned}$$ For each $j$ we choose a $C^{\infty}$-function $\phi_j \geq 0$ supported in $[-1/j,1/j]$ with $\int \phi_j(x)\,dx = 1$ such that $|(\psi*\phi_j)(x) - \psi(x)| \leq \delta_j/2j$ for all $x \in [0,1]$, and define $$\psi_j(x) := {x\over j} + \biggl(1-{1\over j}\biggr)
((\psi*\phi_j)(x) - (\psi*\phi_j)(0))
\quad\mbox{for $x\in[0,1]$}.$$ Then one can immediately see that $\psi_j$ is $C^\infty$, $\psi_j(0)=0$, $\psi_j(1)=1$ and (i), (ii) are satisfied.
For $x,y \in [0,1]$ with $|x-y|\geq\delta_j$ notice that $$\begin{aligned}
&|\psi_j(x) - \psi_j(y)| \\
&\quad={|x-y|\over j}+\biggl(1-\frac{1}{j}\biggr)
|(\psi*\phi_j)(x)-(\psi*\phi_j)(y)| \\
&\quad\geq {|x-y|\over j}+ \biggl(1-\frac{1}{j}\biggr)
\big\{|\psi(x)-\psi(y)| - |(\psi*\phi_j)(x)-\psi(x)|-|\psi(y)-(\psi*\phi_j)(y)|
\big\} \\
&\quad\ge \biggl(1-\frac{1}{j}\biggr)|\psi(x)-\psi(y)|,\end{aligned}$$ and in particular $$\begin{aligned}
\psi_j(x)\ge&\biggl(1-\frac{1}{j}\biggr)\psi(x)
\quad\qquad\ \,\mbox{for $x \in [\delta_j,1)$}, \\
1-\psi_j(x)\ge&\biggl(1-\frac{1}{j}\biggr)(1-\psi(x))
\quad\mbox{for $x \in (0,1-\delta_j]$}.\end{aligned}$$ Hence we have by and $$\begin{aligned}
&\iint_{(0,1)^2} \log|x-y|\,d(\psi_j{}_*\mu)(x)d(\psi_j{}_*\mu)(y) \\
&\qquad\geq
\iint_{\{(x,y) \in (0,1)^2 : |x-y|<\delta_j\}} \log{|x-y|\over j}\,d\mu(x)\,d\mu(y) \\
&\qquad\quad+ \iint_{\{(x,y) \in (0,1)^2 : |x-y|\geq\delta_j\}}\log
\biggl(\biggl(1-\frac{1}{j}\biggr)|\psi(x)-\psi(y)|\biggr)\,d\mu(x)\,d\mu(y) \\
&\qquad\ge -\frac{2}{j} +
\log\biggl(1-\frac{1}{j}\biggr)+\iint_{(0,1)^2}\log|x-y|
\,d(\psi_*\mu)(x)\,d(\psi_*\mu)(y), \\\end{aligned}$$ and also we have by – $$\begin{aligned}
&\int_{(0,1)} \log x\,d(\psi_j{}_*\mu)(x) \\
&\qquad\geq \int_{(0,\delta_j)} \log{x\over j}\,d\mu(x)
+ \int_{[\delta_j,1)} \log\biggl(\biggl(1-\frac{1}{j}\biggr)\psi(x)\biggr)\,d\mu(x) \\
&\qquad\ge -\frac{2}{j} + \log\biggl(1-\frac{1}{j}\biggr)
+ \int_{(0,1)} \log \psi(x)\,d\mu(x),\end{aligned}$$ $$\begin{aligned}
&\int_{(0,1)} \log(1-x)\,d(\psi_j{}_*\mu)(x) \\
&\qquad\geq
\int_{(1-\delta_j,1)} \log{1-x\over j}\,d\mu(x) +
\int_{(0,1-\delta_j]} \log\biggl(\biggl(1-\frac{1}{j}\biggr)(1-\psi(x))\biggr)
\,d\mu(x) \\
&\qquad\ge -\frac{2}{j} + \log\biggl(1-\frac{1}{j}\biggr)
+ \int_{(0,1)} \log(1-\psi(x))\,d\mu(x).\end{aligned}$$ Therefore, $$\begin{aligned}
&\liminf_{j\rightarrow\infty}\iint_{(0,1)^2} \log|x-y|
\,d(\psi_j{}_*\mu)(x)\,d(\psi_j{}_*\mu)(y) \\
&\qquad\geq \iint_{(0,1)^2} \log|x-y|\,d(\psi_*\mu)(x)\,d(\psi_*\mu)(y),\end{aligned}$$ $$\begin{aligned}
\liminf_{j\rightarrow\infty} \int_{(0,1)} \log x\,d(\psi_j{}_*\mu)(x)
&\geq \int_{(0,1)} \log x\, d(\psi_*\mu)(x), \\
\liminf_{j\rightarrow\infty} \int_{(0,1)} \log(1-x)\,d(\psi_j{}_*\mu)(x)
&\geq \int_{(0,1)} \log(1-x)\, d(\psi_*\mu)(x).\end{aligned}$$ On the other hand, Fatou’s lemma says that the reverse inequalities of these three with $\limsup$ in place of $\liminf$ actually hold true. Hence we have (iii)–(v). Finally, the above proof shows the last statement as well.
\[L-3.6\] Let $\mu$ be a measure on $[0,1]$ with no atom at $0,1$, and $\psi$ be a continuous increasing function from $[0,1]$ into itself. Assume that $\mu$ satisfies conditions – in Lemma \[L-3.5\] and also $\psi_*\mu$ does and . Then, there exists a sequence of continuous increasing functions $\psi_m$ from $[0,1]$ onto itself with $\psi_m(0) = 0$, $\psi_m(1) = 1$ such that
- $\int_{(0,1)}|\psi_m(x)-\psi(x)|^2\,d\mu(x) \longrightarrow 0$,
- $\displaystyle{\lim_{m\rightarrow\infty}\iint_{(0,1)^2} \log|x-y|
\,d(\psi_m{}_*\mu)(x)\,d(\psi_m{}_*\mu)(y) = \iint_{(0,1)^2} \log|x-y|
\,d(\psi_*\mu)(x)\,d(\psi_*\mu)(y)}$,
- $\displaystyle{\lim_{m\rightarrow\infty}\int_{(0,1)} \log x
\,d(\psi_m{}_*\mu)(x) = \int_{(0,1)} \log x\,d(\psi_*\mu)(x)}$,
- $\displaystyle{\lim_{m\rightarrow\infty}\int_{(0,1)} \log (1-x)
\,d(\psi_m{}_*\mu)(x) = \int_{(0,1)} \log (1-x)\,d(\psi_*\mu)(x)}$.
Furthermore, when conditions and/or for $\mu$ and $\psi{}_*\mu$ are dropped, the conclusion holds without [(iii)]{} and/or [(iv)]{} correspondingly.
We assume that both $\psi(0)>0$ and $\psi(1)<1$; the other cases can be handled easier. Condition implies $$\begin{aligned}
(-\log m)\nu((0,1/m))^2&\ge\iint_{(0,1/m)^2}\log|x-y|\,d\nu(x)\,d\nu(y)
\longrightarrow0, \label{F-3.24}\\
(-\log m)\nu((1-1/m,1))^2&\ge\iint_{(1-1/m,1)^2}\log|x-y|\,d\nu(x)\,d\nu(y)
\longrightarrow0 \label{F-3.25}\end{aligned}$$ as $m\to\infty$. On the other hand, and imply $$\begin{aligned}
(-\log m)\nu((0,1/m)) &\geq \int_{(0,1/m)} \log x\,d\mu(x) \longrightarrow 0,
\label{F-3.26}\\
(-\log m)\nu((1-1/m,1)) &\geq \int_{(1-1/m,1)}\log(1-x)\,d\mu(x)\longrightarrow 0,
\label{F-3.27}\end{aligned}$$ respectively. For each $m\ge2$ define a function $\psi_m$ on $[0,1]$ by $$\psi_m(x) :=
\begin{cases}
mx\psi(x) & (0\leq x < 1/m), \\
\psi(x) & (1/m \leq x \leq 1-1/m), \\
1-m(1-x)(1-\psi(x)) & (1-1/m < x \leq 1),
\end{cases}$$ which is clearly continuous and increasing with $\psi_m(0)=0$, $\psi_m(1)=1$. Then (i) immediately follows. It is easy to check the following: $$|\psi_m(x) - \psi_m(y)| \geq \begin{cases}
m \psi(0) |x-y| & \text{for $x,y\in(0,1/m)$}, \\
m (1-\psi(1))|x-y| & \text{for $x,y\in(1-1/m,1)$}, \\
|\psi(x)-\psi(y)| & \text{for other $x,y\in(0,1)$}.
\end{cases}$$ Hence we have $$\begin{aligned}
&\iint_{(0,1)^2} \log|x-y|\,d(\psi_m{}_*\mu)(x)\,d(\psi_m{}_*\mu)(y) \\
&\quad\geq
\iint_{(0,1/m)^2} \log (m\psi(0)|x-y|)\,d\mu(x)\,d\mu(y) \\
&\qquad+ \iint_{(1-1/m,1)^2} \log(m(1-\psi(1))|x-y|)\,d\mu(x)\,d\mu(y) \\
&\qquad+ \iint_{(0,1)^2} \log|\psi(x)-\psi(y)|\,d\mu(x)\,d\mu(y) \\
&\quad=(\log m + \log \psi(0))\mu((0,1/m))^2
+ \iint_{(0,1/m)^2} \log|x-y|\,d\mu(x)\,d\mu(y) \\
&\qquad+(\log m + \log(1-\psi(1)))\mu((1-1/m,1))^2
+ \iint_{(1-1/m,1)^2} \log|x-y|\,d\mu(x)\,d\mu(y) \\
&\qquad+ \iint_{(0,1)^2} \log|\psi(x)-\psi(y)|\,d\mu(x)\,d\mu(y) \\
&\quad\longrightarrow \iint_{(0,1)^2} \log|\psi(x)-\psi(y)|\,d\mu(x)\,d\mu(y)\end{aligned}$$ as $m\to\infty$ by , and . Therefore, $$\begin{aligned}
&\liminf_{m\rightarrow\infty}\iint_{(0,1)^2}\log|x-y|
\,d(\psi_m{}_*\mu)(x)\,d(\psi_m{}_*\mu)(y) \\
&\qquad\geq \iint_{(0,1)^2}\log|x-y|\,d(\psi_m{}_*\mu)(x)\,d(\psi_m{}_*\mu)(y). \end{aligned}$$ This together with Fatou’s lemma implies (ii). On the other hand, by for $\mu$ and $\psi{}_*\mu$ we have $$\begin{aligned}
&\int_{(0,1/m)} \log(mx\psi(x))\,d\mu(x) \\
&\qquad=(\log m) \nu((0,1/m)) + \int_{(0,1/m)} \log x\,d\mu(x)
+ \int_{(0,1/m)} \log \psi(x)\,d\mu(x)\longrightarrow 0\end{aligned}$$ thanks to . Furthermore, $$\begin{aligned}
0&\ge\int_{(1-1/m,1)} \log(1-m(1-x)(1-\psi(x)))\,d\mu(x) \\
&\geq \log \psi(1-1/m) \cdot \mu((1-1/m,1)) \longrightarrow 0.\end{aligned}$$ These imply (iii). Similarly, (iv) follows from for $\mu$ and $\psi{}_*\mu$ thanks to .
We are now in the final position to prove Theorem \[T-3.2\] in full generality.
[*Proof of Theorem \[T-3.2\].*]{}As mentioned before we may assume $n=1$, and write $p=p_1$, $q=q_1$ and $\psi=\psi_1$. We may further assume that $\chi_{\mathrm{proj}}(p,q(\psi;p)) > -\infty$ as well as $\chi_\proj(p,q)>-\infty$; otherwise, both sides of the inequality are $-\infty$ thanks to Proposition \[P-1.2\](ii). By Proposition \[P-2.1\] both $\nu$ and $\psi{}_*\nu$ satisfy condition ; moreover they satisfy unless $\tau(p)=\tau(q)$ and also unless $\tau(p)=\tau(\1-q)$. In each case where those equalities of traces occur or not, we choose a sequence $\psi_m$ correspondingly as mentioned in Lemma \[L-3.6\]. Since $$\Vert p\psi_m(pqp)p-p\psi(pqp)p\Vert_2^2
= \int_{(0,1)} |\psi_m(x) - \psi(x)|^2\,d\nu(x) \longrightarrow 0,$$ we get $p\psi_m(pqp)p\rightarrow p\psi(pqp)p$ strongly so that $q(\psi_m;p)\rightarrow q(\psi;p)$ strongly as $m\rightarrow\infty$ due to the definition of $q(\psi;p)$. By Propositions \[P-1.2\](iii) and \[P-2.1\] we see that it suffices to prove the inequality in the case where $\psi(0)=0$ and $\psi(1)=1$. The same argument using Lemma \[L-3.5\] in turn enables us to reduce the proof to Lemma \[L-3.3\], and the proof of the inequality is completed.
To prove the equality of the last statement, let $\psi$ be strictly increasing on $(0,1)$ and define $\tilde \psi$ on $[0,1]$ by $$\tilde \psi(x):=\begin{cases}
0 & \text{($0\le x\le \psi(0+)$)}, \\
\psi^{-1}(x) & \text{($\psi(0+)<x<\psi(1-)$)}, \\
1 & \text{($\psi(1-)\le x\le1$)}.
\end{cases}$$ Furthermore, set $\tilde q:=q(\psi;p)$ and $\tilde\nu:=\psi{}_*\nu$. Then it is clear that $\tilde\nu$ is the measure corresponding to the pair $(p,\tilde q)$ so that $\nu=\tilde \psi{}_*\tilde\nu$ and $q=\tilde q(\tilde \psi;p)$. Hence the inequality established above can be applied to $(p,\tilde q)$ and $\tilde \psi$ too, and we have the reversed inequality as well.
Additivity and freeness
=======================
In this section, we prove the next additivity theorem asserting that the pair-block freeness of projections is characterized by the additivity of their free entropy. For the projection version of free entropy we have no counterpart of the so-called infinitesimal change of variable formula in [@V4 Proposition 1.3], and hence we need to find another route to prove that the additivity implies the freeness.
\[T-4.1\] Let $p_1,q_1,\dots,p_n,q_n,r_1,\dots,r_{n'}$ be projections in $(\cM,\tau)$.
- If $\{p_1,q_1\}$, $\dots$, $\{p_n,q_n\}$, $\{r_1\}$, $\dots$, $\{r_{n'}\}$ are free, then $$\chi_\proj(p_1,q_1,\dots,p_n,q_n,r_1,\dots,r_{n'})
=\chi_\proj(p_1,q_1)+\dots+\chi_\proj(p_n,q_n).$$
- Conversely, if $\chi_\proj(p_i,q_i)>-\infty$ for $1\le i\le n$ and equality holds in [(1)]{}, then $\{p_1,q_1\}$, $\dots$, $\{p_n,q_n\}$, $\{r_1\}$, $\dots$, $\{r_{n'}\}$ are free.
- In particular, $\chi_\proj(p_1,\dots,p_n)=0$ if and only if $p_1,\dots,p_n$ are free.
(1)It suffices to prove the following two assertions:
- If $\{p,q\}$ and $\{p_1,\dots,p_n\}$ are free, then $$\chi_\proj(p,q,p_1,\dots,p_n)=\chi_\proj(p,q)+\chi_\proj(p_1,\dots,p_n).$$
- If $\{p\}$ and $\{p_1,\dots,p_n\}$ are free, then $$\chi_\proj(p,p_1,\dots,p_n)=\chi_\proj(p_1,\dots,p_n).$$
The proofs of these being same, we give only that of (a), which is essentially same as in [@V2; @V-IMRN] (see also [@HP pp. 269–272]).
To prove (a), we may assume that $\chi_\proj(p,q)>-\infty$ and $\chi_\proj(p_1,\dots,p_n)>-\infty$. Choose $k(N),l(N),k_i(N)\in\{0,1,\dots,N\}$ for $N\in\bN$ and $1\le i\le n$ such that $k(N)/N\to\tau(p)$, $l(N)/N\to\tau(q)$ and $k_i(N)/N\to\tau(p_i)$ as $N\to\infty$. For each $m\in\bN$ and $\eps>0$ we set $$\begin{aligned}
\Omega_N(m,\eps)&:=\Gamma_\proj(p,q;k(N),l(N);N,m,\eps) \\
&\qquad\times\Gamma_\proj(p_1,\dots,p_n;k_1(N),\dots,k_n(N);N,m,\eps), \\
\Theta_N(m,\eps)&:=\Gamma_\proj(p,q,p_1,\dots,p_n;k(N),l(N),
k_1(N),\dots,k_n(N);N,m,\eps).\end{aligned}$$ For given $m\in\bN$ and $\eps>0$ one can show as in [@HP 6.4.3] that there exists an $\eps_1>0$ such that $$\lim_{N\to\infty}{\gamma_N(\Omega_N(m,\eps_1)\cap\Theta_N(m,\eps))
\over\gamma_N(\Omega_N(m,\eps_1))}=1,$$ where $\gamma_N:=\gamma_{G(N,k(N))}\otimes\gamma_{G(N,l(N))}\otimes\gamma_{\vec k(N)}$ and $\gamma_{\vec k(N)}:=\bigotimes_{i=1}^n\gamma_{G(N,k_i(N))}$. Hence we have $$\begin{aligned}
&\limsup_{N\to\infty}{1\over N^2}\log\gamma_N(\Theta_N(m,\eps)) \\
&\qquad\ge\limsup_{N\to\infty}{1\over N^2}\log\gamma_N(\Omega_N(m,\eps_1)) \\
&\qquad=\lim_{N\to\infty}{1\over N^2}
\log\bigl(\gamma_{G(N,k(N))}\otimes\gamma_{G(N,l(N))}\bigr)
\bigl(\Gamma_\proj(p,q;k(N),l(N);N,m,\eps_1)\bigr) \\
&\qquad\qquad
+\limsup_{N\to\infty}{1\over N^2}
\log\gamma_{\vec k(N)}
\bigl(\Gamma_\proj(p_1,\dots,p_n;k_1(N),\dots,k_n(N);N,m,\eps_1)\bigr) \\
&\qquad\ge\chi_\proj(p,q)+\chi_\proj(p_1,\dots,p_n).\end{aligned}$$ The above equality is due to [@HP1 Proposition 3.3]. Therefore, $$\chi_\proj(p,q,p_1,\dots,p_n)\ge\chi_\proj(p,q)+\chi_\proj(p_1,\dots,p_n),$$ and the reverse inequality is Proposition 1.2(ii).
\(3) will be proven in Corollary \[C-5.5\] of the next section as a consequence of a transportation cost inequality for projection multi-variables.
(2)We may assume that $p_1,q_1,\dots,p_n,q_n$ are all non-zero. For $1\le i\le n$ let $\nu_i$ be the measure on $(0,1)$ corresponding to the pair $(p_i,q_i)$ (see §2). For each $i$, since $\nu_i$ is non-atomic by the assumption $\chi_\proj(p_i,q_i)>-\infty$, one can choose a continuous increasing function $\psi_i$ from $(0,1)$ into itself such that $\psi_i{}_*\nu_i$ is equal to with $\alpha=\tau(p_i)$, $\beta=\tau(q_i)$. Consider $q_i(\psi_i;p_i)$ constructed from $(p_i,q_i)$ and $\psi_i$ (see §3). Since $\psi_i{}_*\nu_i$ corresponds to the pair $(p_i,q_i(\psi_i;p_i))$, we get $\chi_\proj(p_i,q_i(\psi_i;p_i))=0$. Therefore, by Theorem \[T-3.2\] and the additivity assumption, we have $$\begin{aligned}
&\chi_\proj(p_1,q_1(\psi_1;p_1),\dots,p_n,q_n(\psi_n;p_n),r_1,\dots,r_{n'}) \\
&\qquad\ge\chi_\proj(p_1,q_1,\dots,p_n,q_n,r_1,\dots,r_{n'})
-\sum_{i=1}^n\chi_\proj(p_i,q_i)=0.\end{aligned}$$ This implies by (3) that $p_1,q_1(\psi_1;p_1),\dots,p_n,q_n(\psi_n;p_n),r_1,\dots,r_{n'}$ are free. Since $\nu_i$ and $\psi_i{}_*\nu_i$ are non-atomic, it is plain to see that $\{p_i,q_i\}''=\{p_i,q_i(\psi_i;p_i)\}''$ for $1\le i\le n$. Hence the freeness of $\{p_1,q_1\},\dots,\{p_n,q_n\},\{r_1\},\dots,\{r_{n'}\}$ is obtained.
Asymptotic freeness and free transportation cost inequality
===========================================================
The aim of this section is to prove a transportation inequality for tracial distributions of projection multi-variables. To do so, we first present an asymptotic freeness result for random projection matrices generalizing Voiculescu’s result in [@V0].
Asymptotic freeness for random projection matrices
--------------------------------------------------
Let $\bigl(\{P(s,N),Q(s,N)\}\bigr)_{s\in S}$ be an independent family of pairs of $N\times N$ random projection matrices, and let $k(s,N)$, $l(s,N)$, $n_{11}(s,N)$, $n_{10}(s,N)$, $n_{01}(s,N)$ and $n_{00}(s,N)$ denote the ranks of $P(s,N)$, $Q(s,N)$, $P(s,N)\wedge Q(s,N)$, $P(s,N)\wedge Q(s,N)^\perp$, $P(s,N)^\perp\wedge Q(s,N)$, $P(s,N)^\perp\wedge Q(s,N)^\perp$, respectively. For each $s \in S$ we assume the following:
- $k(s,N)$, $l(s,N)$ and $n_{ij}(s,N)$’s are constant almost surely and $k(s,N)/N$, $l(s,N)/N$ and $n_{ij}(s,N)/N$ converge as $N \rightarrow \infty$.
- The joint distribution of $(P(s,N),Q(s,N))$ is invariant under unitary conjugation $(P,Q) \mapsto (UPU^*,UQU^*)$ for $U\in\U(N)$.
- For each $s\in S$ the distribution measure of $P(s,N)Q(s,N)P(s,N)$ with respect to $N^{-1}\mathrm{Tr}_N$ converges almost surely to a (non-random) measure on $[0,1]$ as $N\rightarrow\infty$.
Let $(R(s',N))_{s'\in S'}$ be an independent family of $N\times N$ random projection matrices, also independent of $\big(\{P(s,N),Q(s,N)\}\big)_{s\in S}$, and assume that each $R(s',N)$ is distributed under the Haar probability measure on $G(N,k(s',N))$ with $0\le k(s',N) \leq N$ such that $k(s',N)/N$ converges. Finally, let $(D(t,N))_{t\in T}$ be a family of $N\times N$ constant matrices such that $\sup_N \Vert D(t,N)\Vert_{\infty} < +\infty$ for each $t \in T$ and $(D(t,N),D(t,N)^*)_{t\in T}$ has the limit distribution. In this setup, we have the following asymptotic freeness result for random projection matrices generalizing [@V0 Theorem 3.11].
\[T-5.1\] With the above notations and assumptions the family $$\Big(\big(\{P(s,N),Q(s,N)\}\big)_{s\in S},\,\big(R(s',N)\big)_{s' \in S'},
\,\bigl\{D(t,N), D(t,N)^* : t \in T\bigr\}\Big)$$ is asymptotically free almost surely as $N\rightarrow\infty$.
Set $n(s,N) := \bigl(N-\sum_{i,j=0}^1 n_{ij}(s,N)\bigr)/2$. By assumption (1), $n(s,N)$ is constant almost surely and $n(s,N)/N$ converges as $N\to\infty$. As before, the sine-cosine decomposition of two projections enables us to represent $$\begin{aligned}
P(s,N) &= U(s,N)\left(\begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix}
\oplus I \oplus I \oplus 0 \oplus 0\right)U(s,N)^*, \\
Q(s,N) &= U(s,N)\left(\begin{bmatrix} X & \sqrt{X(I-X)} \\ \sqrt{X(I-X)} & I-X
\end{bmatrix}\oplus I\oplus0\oplus I\oplus 0\right)U(s,N)^* \end{aligned}$$ in $\mathbb{C}^N = (\mathbb{C}^{n(s,N)}\otimes\mathbb{C}^2)
\oplus\mathbb{C}^{n_{11}(s,N)}\oplus\mathbb{C}^{n_{10}(s,N)}
\oplus\mathbb{C}^{n_{01}(s,N)}\oplus\mathbb{C}^{n_{00}(s,N)}$, where $U(s,N)$ is a random unitary matrix and $X=X(s,N)$ is a diagonal matrix whose diagonal entries are $0\le x_1(s,N)\leq x_2(s,N)\leq\cdots\leq x_{n(s,N)}(s,N)\le1$. Also, we can represent $$R(s',N) = U(s',N)P_{k(s',N)}U(s',N)^*$$ for each $s'\in S'$, where $U(s',N)$ is a unitary random matrix and $P_{k(s',N)}$ the diagonal matrix whose first $k(s',N)$ entries are $1$ and the others $0$. As in the proof of [@HP 4.3.5] we can assume that $(U(s,N))_{s\in S}\sqcup(U(s',N))_{s'\in S'}$ forms an independent family of standard unitary matrices thanks to the independence and assumption (2). We fix $s \in S$ and assume $\lim_{N\rightarrow\infty} n_0(s,N)/N > 0$. (When $n(s,N)/N \to 0$ the discussion below becomes rather trivial.) Write $A(s,N)$ and $B(s,N)$ for the matrices appearing inside $\mathrm{Ad}\,U(s,N)$ in the above representation of $P(s,N)$, $Q(s,N)$, that is, $A(s,N) = U(s,N)^* P(s,N) U(s,N)$ and $B(s,N) = U(s,N)^* Q(s,N) U(s,N)$. By assumption (3) one observes that the empirical distribution $n(s,N)^{-1}\sum_{i=1}^{n(s,N)}\delta_{x_i(s,N)}$ converges to a measure $\rho_s$ on $[0,1]$ weakly in the almost sure sense as $N\rightarrow\infty$. Choose (non-random) $0 \leq \xi_1(s,N) \leq \cdots \leq \xi_{n(s,N)}(s,N) \leq 1$ in such a way that $n(s,N)^{-1}\sum_{i=1}^{n(s,N)}\delta_{\xi_i(s,N)}$ converges to $\rho_s$ weakly as $N\rightarrow\infty$. Let $\Xi(s,N)$ be the diagonal matrix with diagonal entries $\xi_1(s,N),\dots,\xi_{n(s,N)}(s,N)$ and define $$C(s,N) := \begin{bmatrix} \Xi(s,N) & \sqrt{\Xi(s,N)(I-\Xi(s,N))} \\
\sqrt{\Xi(s,N)(I-\Xi(s,N))} & I-\Xi(s,N) \end{bmatrix}
\oplus I \oplus 0 \oplus I \oplus 0.$$ By [@HP 4.3.4] we then have $$\lim_{N\rightarrow\infty}\Vert X(s,N)
- \Xi(s,N)\Vert_{p,n(s,N)^{-1}\mathrm{Tr}_{n(s,N)}} = 0
\ \ \text{almost surely\ \ for all $p\geq1$}$$ so that for any polynomial $F$ $$\lim_{N\rightarrow\infty}\Vert F(B(s,N))
- F(C(s,N))\Vert_{p,N^{-1}\mathrm{Tr}_{N}} = 0
\ \ \text{almost surely\ \ for all $p\geq1$}.$$ Moreover, note that $(A(s,N),C(s,N))_{s\in S}$ has the limit distribution. Under these preparations the proof is completed by the same argument as in [@HP 4.3.5].
Free transportation cost inequality for projections
---------------------------------------------------
Let $\mathcal{A}^{(2n+n')}_{\mathrm{proj}}$ be the universal free product $C^*$-algebra of $2n+n'$ copies of $C^*(\mathbb{Z}_2) = \bC\oplus\bC$, and denote the canonical $2n+n'$ generators of projections by $e_1,f_1,\dots,e_n,f_n,e'_1,\dots,e'_{n'}$. For a given $2n+n'$-tuple $\vec P=(P_1,Q_1,\dots,P_n,Q_n,R_1,\dots,R_{n'})$ of projections in $M_N(\bC)$, there is a unique $*$-homomorphism from $\mathcal{A}^{(2n+n')}_{\mathrm{proj}}$ into $M_N(\bC)$ sending $e_i,f_i,e'_j$ to $P_i,Q_i,R_j$, respectively, which we denote by $h \in \mathcal{A}^{(2n+n')}_{\mathrm{proj}} \mapsto h(\vec P)\in M_N(\bC)$. For $\vec k=(k_1,l_1,\dots,k_n,l_n,k'_1,\dots,k'_{n'})\in\{0,1,\dots,N\}^{2n+n'}$, denote by $G(N,\vec k)$ the product $\prod_{i=1}^n\bigl(G(N,k_i)\times G(N,l_i)\bigr)\times\prod_{j=1}^{n'}G(N,k'_j)$ of Grassmannian manifolds, and by $\mathcal{P}\bigl(G(N,\vec k)\bigr)$ the set of Borel probability measures on $G(N,\vec k)$. Note that each $\lambda \in \mathcal{P}\bigl(G(N,\vec k)\bigr)$ clearly gives rise to the unique tracial state $\hat{\lambda}$ on $\mathcal{A}^{(2n+n')}_{\mathrm{proj}}$ defined by $$\hat{\lambda}(h) :=
\int \frac{1}{N}\mathrm{Tr}_N\bigl(h(\vec P)\bigr)\,d\lambda(\vec P)
\quad\mbox{for $h \in \mathcal{A}^{(2n+n')}_{\mathrm{proj}}$}.$$
Let us denote by $TS\bigl(\mathcal{A}^{(2n+n')}_{\mathrm{proj}}\bigr)$ the set of tracial states on $\mathcal{A}^{(2n+n')}_{\mathrm{proj}}$, and moreover, for each $\vec{\alpha} := (\alpha_1,\beta_1,\dots,\alpha_n,\beta_n,\alpha'_1,\dots,
\alpha'_{n'})\in[0,1]^{2n+n'}$, by $TS_{\vec{\alpha}}(\mathcal{A}^{(2n+n')}_{\proj})$ the set of $\tau \in TS\bigl(\mathcal{A}^{(2n+n')}_{\mathrm{proj}}\bigr)$ such that $\tau(e_i) = \alpha_i$, $\tau(f_i) = \beta_i$ and $\tau(e'_j) = \alpha'_j$. For $\tau_1, \tau_2 \in TS\bigl(\mathcal{A}^{(2n+n')}_{\mathrm{proj}}\bigr)$, the (free probabilistic) Wasserstein distance $W_{2,\mathrm{free}}(\tau_1,\tau_2)$ is defined to be the infimum of $$\sqrt{\tau\Biggl(\sum_{i=1}^n\bigl(
|\sigma_1(e_i) - \sigma_2(e_i)|^2+|\sigma_1(f_i) - \sigma_2(f_i)|^2\bigr)
+\sum_{j=1}^{n'} |\sigma_1(e'_j) - \sigma_2(e'_j)|^2\Biggr)}$$ over all $\tau \in TS\bigl(\mathcal{A}^{(2n+n')}_{\mathrm{proj}}
\bigstar\mathcal{A}^{(2n+n')}_{\mathrm{proj}}\bigr)$ with $\tau\circ\sigma_1=\tau_1$, $\tau\circ\sigma_2=\tau_2$, where $\sigma_1$ and $\sigma_2$ stand for the canonical embedding maps of $\mathcal{A}^{(2n+n')}_{\mathrm{proj}}$ into the left and right copies in $\mathcal{A}^{(2n+n')}_{\mathrm{proj}}
\bigstar\mathcal{A}^{(2n+n')}_{\mathrm{proj}}$, respectively. The next lemma will be one of the keys in proving a free transportation cost inequality.
\[L-5.2\] For each pair $\lambda_1, \lambda_2 \in \mathcal{P}\bigl(G(N,\vec k)\bigr)$ we have $$W_{2,\mathrm{free}}(\hat{\lambda}_1,\hat{\lambda}_2)
\leq \frac{1}{\sqrt{N}} W_{2,HS}(\lambda_1,\lambda_2)
\leq \frac{1}{\sqrt{N}}W_{2,d}(\lambda_1,\lambda_2).$$ Here, $W_{2,HS}$ and $W_{2,d}$ are the usual Wasserstein distances determined by the Hilbert-Schmidt norm $\Vert P-Q\Vert_{HS}$ and the geodesic distance $d(P,Q)$ with respect to the Riemannian metric induced from $\mathrm{Tr}_N$, respectively.
The first inequality is shown in the same way as in [@HU1 Lemma 1.3], while the second immediately follows from the inequality $\Vert P-Q\Vert_{HS} \leq d(P,Q)$ (see e.g. [@He Appendix B]).
Let $\vec{\alpha} \in[0,1]^{2n+n'}$ and $\vec k(N)=(k_1(N),l_1(N),\dots,k_n(N),l_n(N),
k'_1(N),\dots,k'_{n'}(N))\in\{0,1,\dots,N\}^{2n+n'}$ for $N\in\bN$ be given so that $\vec k(N)/N\to\vec\alpha$ as $N\to\infty$. For each $\tau \in TS_{\vec{\alpha}}\bigl(\mathcal{A}^{(2n+n')}_{\mathrm{proj}}\bigr)$ the free entropy $\chi_{\mathrm{proj}}(\tau)$ is defined as follows. We denote by $\Gamma_\proj(\tau;\vec k(N);\allowbreak N,m,\varepsilon)$ the set of all $2n+n'$-tuples $\vec P\in G(N,\vec k(N))$ such that $$\left|\frac{1}{N}\mathrm{Tr}_N\bigl(h(\vec P)\bigr) - \tau(h)\right| < \varepsilon$$ for all monomials $h \in \mathcal{A}^{(2n+n')}_{\proj}$ in $e_1,f_1,\dots,e_n,f_n,r_1,\dots,r_{n'}$ of degree at most $m$. We then define $$\chi_{\mathrm{proj}}(\tau)
:= \lim_{\substack{m\rightarrow\infty \\ \varepsilon \searrow 0}}
\limsup_{N\rightarrow\infty}\,\frac{1}{N^2}\log
\gamma_{\vec k(N)}\bigl(\Gamma_\proj(\tau;\vec k(N);N,m,\varepsilon)\bigr),$$ where $\gamma_{\vec k(N)}:=\bigotimes_{i=1}^n\bigl(\gamma_{G(N,k_i(N))}\otimes
\gamma_{G(N,l_i(N))}\bigr)\otimes\bigotimes_{j=1}^{n'}\gamma_{G(N,k'_j(N))}$ on $G(N,\vec k(N))$. Note that the quantity $\chi_{\proj}(\tau)$ is noting less than $\chi_{\proj}(p_1,q_1,\dots,p_n,q_n,r_1,\dots,r_{n'})$ when $p_i:=\pi_\tau(e_i)$, $q_i:=\pi_\tau(f_i)$ and $r_j:=\pi_\tau(e'_j)$ in the GNS representation of $\mathcal{A}_{\proj}^{(2n+n')}$ associated with $\tau$; hence it is independent of the particular choice of $\vec k(N)$ due to Proposition \[P-1.1\].
In what follows, let $\tau \in TS_{\vec{\alpha}}\bigl(\mathcal{A}^{(2n+n')}_{\mathrm{proj}}\bigr)$ be arbitrarily fixed. Then one can choose a subsequence $N_1 < N_2 < \cdots$ so that $$\label{F-5.1}
\chi_{\proj}(\tau) = \lim_{m\rightarrow\infty}\frac{1}{N_m^2}\log
\gamma_{\vec k(N_m)}\bigl(\Gamma_\proj(\tau;\vec k(N_m);N_m,m,1/m)\bigr).$$ Set $\Gamma_{N_m}:=\Gamma_\proj(\tau;\vec k(N_m);N_m,m,1/m)$ and define $\lambda_{N_m}^{\tau} \in \mathcal{P}\bigl(G(N_m,\vec k(N_m))\bigr)$ by $$d\lambda_{N_m}^{\tau}(\vec P) := \frac{1}{\gamma_{\vec k(N_m)}(\Gamma_{N_m})}
\,\mathbf{1}_{\Gamma_{N_m}}(\vec P)\,d\gamma_{\vec k(N_m)}(\vec P).$$
\[L-5.3\] $\displaystyle \lim_{m\rightarrow\infty} \hat{\lambda}^{\tau}_{N_m} = \tau$ in the weak\* topology.
The proof can be found in [@HU1], even though only the self-adjoint and the unitary cases are treated there.
For $1\le i\le n$ the $C^*$-subalgebra generated by $e_i,f_i$ (obviously identified with $\cA^{(2)}_\proj=C^*(\mathbb{Z}_2\bigstar\mathbb{Z}_2)$) is isomorphic to $$\mathcal{A} := \bigl\{ a(\cdot) = [a_{ij}(\cdot)]_{i,j=1}^2 \in C([0,1];M_2(\bC)) :
\text{$a(0)$, $a(1)$ are diagonals} \bigr\}$$ by the $*$-isomorphism given by $$e_i \mapsto e(t) = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \quad
f_i \mapsto f(t) = \begin{bmatrix} t & \sqrt{t(1-t)} \\ \sqrt{t(1-t)} & 1-t \end{bmatrix},$$ and any tracial state on $\cA$ is of the form $$\tau_{\nu,\{\alpha_{ij}\}}(a) :=
\alpha_{10} a_{11}(0) + \alpha_{01}a_{22}(0) + \alpha_{11} a_{11}(1)
+ \alpha_{00} a_{22}(1) + \int_{(0,1)} \frac{1}{2}\mathrm{Tr}_2(a(t))\,d\nu(t),$$ where $\alpha_{ij}\geq 0$, $\sum_{i,j=0}^1\alpha_{ij} \leq 1$ and $\nu$ is a measure on $(0,1)$ with $\nu((0,1)) = 1-\sum_{i,j=0}^1\alpha_{ij}$. Let $\vec{\psi} = (\psi_1,\dots,\psi_n)$ be an $n$-tuple of continuous functions on $[0,1]$, and for $1\le i\le n$ define the probability distribution $\lambda_N^{\psi_i}$ on $G(N,k_i(N))\times G(N,l_i(N))$ by $$d\lambda_N^{\psi_i}(P,Q) := \frac{1}{Z^{\psi_i}_N}
\exp\bigl(-N\mathrm{Tr}_N(\psi_i(PQP))\bigr)
\,d(\gamma_{G(N,k_i(N))}\otimes\gamma_{G(N,l_i(N))})(P,Q)$$ with the normalization constant $Z^{\psi_i}_N$. For $\tau_{\nu,\{\alpha_{ij}\}} \in
TS_{(\alpha_i,\beta_i)}\bigl(\mathcal{A}_{\proj}^{(2)})$ one has $$\begin{aligned}
&\chi_{\proj}(\tau_{\nu,\{\alpha_{ij}\}})
- \tau_{\nu,\{\alpha_{ij}\}}(\psi_i(efe)) \\
&\qquad=
\frac{1}{4}\Sigma(\nu) + \frac{1}{2}\int_{(0,1)}\bigl((\alpha_{01}+\alpha_{10})\log t
+ (\alpha_{00}+\alpha_{11})\log(1-t) -\psi_i(t)\bigr)\,d\nu(t) - C \end{aligned}$$ with some constant $C$ if $\alpha_{00}\alpha_{11} = \alpha_{01}\alpha_{10}=0$, and otherwise $-\infty$. Thus, a general result on weighted logarithmic energy (see [@ST]) ensures that there is a unique maximizer $\tau_{(\alpha_i,\beta_i)}^{\psi_i} \in
TS_{(\alpha_i,\beta_i)}\bigl(\mathcal{A}_{\proj}^{(2)}\bigr)$ of the functional $\tau \in TS_{(\alpha_i,\beta_i)}\bigl(\mathcal{A}_{\proj}^{(2)}\bigr) \mapsto
\chi_{\proj}(\tau) - \tau(\psi_i(efe))$. Then, we define the tracial state $\tau_{\vec{\alpha}}^{\vec\psi} \in TS\bigl(\mathcal{A}_{\proj}^{(2n+n'}\bigr)$ by $$\tau_{\vec{\alpha}}^{\vec\psi}
:= \Bigl(\bigstar_{i=1}^n \tau_{(\alpha_i,\beta_i)}^{\psi_i}\Bigr) \bigstar
\tau_{(\alpha'_1,\dots,\alpha'_{n'})},
\quad\tau_{(\alpha'_1,\dots,\alpha'_{n'})}
:=\bigstar_{j=1}^{n'}\bigl(\alpha'_j\delta_0 + (1-\alpha'_j)\delta_1\bigr)$$ in the natural identification $\mathcal{A}_{\proj}^{(2n+n')}
= \Bigl(\bigstar_{i=1}^n \mathcal{A}_{\proj}^{(2)}\Bigr)\bigstar
\Bigl(\bigstar_{j=1}^{n'}C^*(\bZ_2)\Bigr)$. Furthermore, we define the joint distribution $$\lambda_N^{\vec\psi} := \Biggl(\bigotimes_{i=1}^n \lambda_N^{\psi_i}\Biggr)\otimes
\Biggl(\bigotimes_{j=1}^{n'} \gamma_{G(N,k'_j(N))}\Biggr)
\quad\mbox{on $G(N,\vec k(N))$}$$ (also considered as a $2n+n'$-tuple of random projection matrices). The next lemma follows from a large deviation result for two projection matrices in [@HP1] and Theorem \[T-5.1\].
\[L-5.4\]
1. $\displaystyle B_{\psi_i} := \lim_{N\rightarrow\infty}\frac{1}{N^2}
\log Z_N^{\psi_i}$ exists for every $1\le i\le n$.
2. $\displaystyle \lim_{N\rightarrow\infty}\hat{\lambda}_N^{\vec\psi}
= \tau_{\vec{\alpha}}^{\vec\psi}$ in the weak\* topology.
When $p_1,q_1,\dots,p_n,q_n$ are absent, we have nothing to do for (1) and moreover (2) immediately follows from Voiculescu’s original result [@V0 Theorem 3.11] rather than Theorem \[T-5.1\] as follows. Let $R_1(N),\dots,R_{n'}(N)$ be an independent family of random projection matrices of ranks $k'_j(N)$ distributed under $\gamma_{G(N,k'_j(N))}$, respectively. Note that $\lambda_N^{\vec\psi}$ in this case is nothing less than $\gamma_N := \bigotimes_{j=1}^{n'}\gamma_{G(N,k'_j(N))}$. For a monomial $h = r_{j_1}\cdots r_{j_m} \in \mathcal{A}_{\proj}^{(n')}$ one has $$\begin{aligned}
\hat\gamma_N(h) = \mathbb{E}\circ\biggl(\frac{1}{N}\mathrm{Tr}_N\biggr)
(R_{j_1}(N)\cdots R_{j_m}(N)), \end{aligned}$$ which converges to $\tau_{(\alpha'_1,\dots,\alpha'_{n'})}(r_{j_1}\cdots r_{j_m})$ thanks to [@V0 Theorem 3.11]. This immediately implies (2) in this special case.
For the general case, i.e., when $p_1,q_1,\dots,p_n,q_n$ really appear, we need to show (1) and $\lim_{N\rightarrow\infty} \hat{\lambda}_N^{\psi_i}
= \tau_{\vec{\alpha}}^{\psi_i}$ weakly\* for each $1\le i\le n$. Both are simple applications of the large deviation result for the empirical eigenvalue distribution of two random projection matrix pair $(P(N),Q(N))$ distributed under $\lambda_N^{\psi_i}$, whose proof is essentially same as in [@HP1 Proposition 2.1] (or in the proof of [@HU2 Theorem 2.1]). Once the latter convergence was established, the above argument would equally work well even in the general setting when [@V0 Theorem 3.11] is replaced by Theorem \[T-5.1\].
With the above lemmas we can now prove the following transportation cost inequality in the essentially same manner as in [@HU1].
\[T-5.4\] Assume that $\psi_i$’s are $C^2$-functions and $\rho :=
\min\bigl\{1 - c_1\Vert \psi'_i\Vert_{\infty} - c_2\Vert\psi''_i\Vert_{\infty}:
1\leq i\leq n\bigr\} > 0$ for some universal constants $c_1, c_2 > 0$ [(]{}for example, one can choose $c_1 = 6$, $c_2 = 9/2$, but these do not seem optimal[)]{}. For every $\tau \in TS_{\vec{\alpha}}\bigl(\mathcal{A}_{\proj}^{(2n+n')}\bigr)$ we have $$\label{F-5.2}
W_{2,\mathrm{free}}\Bigl(\tau,\tau_{\vec{\alpha}}^{\vec\psi}\Bigr) \leq
\sqrt{\frac{2}{\rho}\Biggl(-\chi_{\proj}(\tau)
+ \tau\Biggl(\sum_{i=1}^n \psi_i(pqp)\Biggr) + B_{\vec\psi}\Biggr)}$$ with $B_{\vec\psi} := \sum_{i=1}^n B_{\psi_i}$. In particular, when $p_1,q_1,\dots,p_n,q_n$ are absent, simply becomes $$\label{F-5.3}
W_{2,\mathrm{free}}\bigl(\tau,\tau_{(\alpha'_1,\dots,\alpha'_{n'})}\bigr)
\leq \sqrt{-2\chi_{\mathrm{proj}}(\tau)}.$$
Since $W_{2,\mathrm{free}}$ is lower semi-continuous in the weak\* topology, we have by Lemmas \[L-5.3\] and \[L-5.4\](2) $$W_{2,\mathrm{free}}\Bigl(\tau,\tau_{\vec{\alpha}}^{\vec\psi}\Bigr)
\leq \liminf_{m\rightarrow\infty}
W_{2,\mathrm{free}}\Bigl(\hat{\lambda}^{\tau}_{N_m},\hat{\lambda}^{\vec\psi}_{N_m}\Bigr).$$ By Lemma \[L-5.2\] we also have $$W_{2,\mathrm{free}}\Bigl(\hat{\lambda}^{\tau}_{N_m},
\hat{\lambda}^{\vec\psi}_{N_m}\Bigr)
\leq \frac{1}{\sqrt{N_m}}
W_{2,d}\Bigl(\lambda^{\tau}_{N_m},\lambda^{\vec\psi}_{N_m}\Bigr).$$ We then need to confirm Bakry and Emery’s $\Gamma_2$-criterion [@BE] for $\lambda^{\vec\psi}_N$ with the constant $\rho N$, that is, $$\label{F-5.4}
\mathrm{Ric}\bigl(G(N,\vec k(N))\bigr) + \mathrm{Hess}(\Psi_N) \geq \rho NI_{d(N)},$$ where $\mathrm{Ric}\bigl(G(N,\vec k(N))\bigr)$ is the Ricci curvature tensor of $G(N,\vec k(N))$, $\mathrm{Hess}(\Psi_N)$ is the Hessian of the trace function $$\begin{aligned}
\Psi_N(P_1,Q_1,\dots,P_n,Q_n,R_1,\dots,R_{n'}) :=
N\mathrm{Tr}_N\Biggl(\sum_{i=1}^n \psi_i(P_i Q_i P_i)\Biggr), \end{aligned}$$ and $d(N)$ is the dimension of $G(N,\vec k(N))$, i.e., $$d(N):=2\sum_{i=1}^n\bigl(k_i(N)(N-k_i(N))+l_i(N)(N-l_i(N))\bigr)
+2\sum_{j=1}^{n'}k'_j(N)(N-k'_j(N)).$$ It is known (see [@HU2 Eq.(2.2)]) that $\mathrm{Ric}(G(N,k)) = NI_{2k(N-k)}$ so that we need only to estimate the Hessian $\mathrm{Hess}(\Psi_N^{(i)})$ of the trace function $\Psi_N^{(i)} : (P,Q) \in G(N,k_i(N))\times G(N,l_i(N)) \mapsto
N\mathrm{Tr}_N(\psi_i(PQP))$ from below for each $1\le i\le n$. This can be done by computing $\mathrm{Hess}(\Psi_N^{(i)})$ explicitly, and consequently we can find two universal constants $c_1, c_2 > 0$ so that $$\mathrm{Hess}(\Psi_N^{(i)})
\geq -N(c_1\Vert \psi'_i\Vert_{\infty} + c_2\Vert \psi''_i\Vert_{\infty})
I_{2k_i(N)(N-k_i(N))+2l_i(N)(N-l_i(N))}.$$ Hence is confirmed. See Remark \[R-5.6\] below for more details on this estimate. Thus, by the transportation cost inequality in the Riemannian manifold setting due to Otto and Villani [@OV] we obtain $$\label{F-5.5}
W_{2,d}\Bigl(\lambda^{\tau}_{N_m},\lambda^{\vec\psi}_{N_m}\Bigr)
\leq \sqrt{\frac{2}{\rho N_m}
S\Bigl(\lambda^{\tau}_{N_m},\lambda^{\vec\psi}_{N_m}\Bigr)},$$ where $S\Bigl(\lambda^{\tau}_{N_m},\lambda^{\vec\psi}_{N_m}\Bigr)$ stands for the usual relative entropy. We compute $$\begin{aligned}
S\Bigl(\lambda^{\tau}_{N_m},\lambda^{\vec\psi}_{N_m}\Bigr)
&=\int\log\frac{d\lambda^{\tau}_{N_m}}{d\lambda^{\vec\psi}_{N_m}}
\,\lambda^{\tau}_{N_m} \\
&=-\log\gamma_{\vec k(N_m)}(\Gamma_{N_m})
+ N_m^2 \hat{\lambda}_{N_m}^{\tau}\Biggl(\sum_{i=1}^n \psi_i(e_i f_i e_i)\Biggr)
+ \sum_{i=1}^n\log Z_{N_m}^{\psi_i}. \end{aligned}$$ Consequently, we obtain the desired inequality by taking the limit of as $m\rightarrow\infty$ after divided by $N_m^2$ due to , Lemmas \[L-5.3\] and \[L-5.4\](1). Finally, we should remark that if $p_1,q_1,\dots,p_n,q_n$ disappeared, then the argument would become simpler without estimating the Hessian of $\Psi_N$.
\[R-5.6\]
The computation of $\mathrm{Hess}(\Psi_N^{(i)})$ mentioned in the above proof is outlined here. The tangent space $T_P G(N,k)$ at $P \in G(N,k)$ is identified with the set of $X \in M_N(\mathbb{C})^{sa}$ satisfying $X = PX + XP$, on which our Riemannian metric is given by $\langle X|Y \rangle := \mathrm{Re}\,\mathrm{Tr}_N(YX)$ (this is inherited from that on the Euclidean space $M_N(\mathbb{C})^{sa}$). Moreover, the geodesic curve started at $P$ with tangent vector $X$ is given by $C(t) := \exp(t[X,P])P\exp(-t[X,P])$ for $t \in \mathbb{R}$. (See e.g. [@CPR §2] for a brief summary and references therein.) Since $$\begin{aligned}
&\bigl\langle\mathrm{Hess}(\Psi^{(i)}_N)((C_1(0),C_2(0))(C_1'(0)\oplus C_2'(0)
|C_1'(0)\oplus C_2'(0))\bigr\rangle \\
&\qquad= \frac{d^2}{dt^2}\bigg|_{t=0}N\mathrm{Tr}_N(\psi_i(C_1(t)C_2(t)C_1(t)))\end{aligned}$$ for geodesic curves $C_1(t) \in G(N,k_i(N))$ and $C_2(t) \in G(N,l_i(N))$, it suffices [(]{}for getting the desired inequality in the above proof[)]{} to estimate, at $t=0$, the second derivative of the composition of $\phi(t) := C_1(t)C_2(t)C_1(t) \in M_N(\mathbb{C})^{sa}$ and $X \in M_N(\mathbb{C})^{sa} \mapsto \Phi(X) := N\mathrm{Tr}_N(\psi_i(X))$ with the usual Euclidean structure on $M_N(\mathbb{C})^{sa}$. Passing once to the identification $M_N(\mathbb{C})^{sa} = \mathbb{R}^{N^2}$, we observe that $$(\Phi\circ\phi)''(0)
= \bigl\langle(\nabla^2 \Phi)(\phi(0))\phi'(0)|\phi'(0)\bigr\rangle
+ \bigl\langle(\nabla \Phi)(\phi(0))|\phi''(0)\bigr\rangle$$ thanks to the usual chain rule. By [@HPU Lemma 1.2] we can estimate the operator norms $\Vert (\nabla^2\Phi)(\phi(0)) \Vert_{\infty}$ (for linear operators on $(M_N(\mathbb{C})^{sa},\langle\,\cdot\,|\,\cdot\,\rangle)$) and $\Vert (\nabla\Phi)(\phi(0))\Vert_{\infty}$ (for elements in $M_N(\mathbb{C})^{sa}$) by $N\Vert \psi_i''\Vert_{\infty}$ and $N\Vert \psi_i'\Vert_{\infty}$, respectively, from the above. As mentioned above the tangent vector $C_i'(0) \in M_N(\mathbb{C})^{sa}$ satisfies $C_i'(0) = C_i(0)C_i'(0) + C_i'(0)C_i(0)$ and the geodesic curve $C_i(t)$ must be $$C_i(t) = \exp(t[C_i'(0),C_i(0)])C_i(0)\exp(-t[C_i'(0),C_i(0)]).$$ It follows from these facts that $$\begin{aligned}
\phi'(0)
&=
C'_1(0)C_2(0)C_1(0) + C_1(0)C_2'(0)C_1(0) + C_1(0)C_2(0)C_1'(0), \\
\phi''(0)
&=
[[C_1'(0),C_1(0)],C_1'(0)]C_2(0)C_1(0) \\
&\quad+ C_1(0)[[C_2'(0),C_2(0)],C_2'(0)]C_1(0) \\
&\quad\quad+ C_1(0)C_2(0)[[C_1'(0),C_1(0)],C_1'(0)] \\
&\quad\quad\quad+
2\big\{C_1'(0)C_2'(0)C_1(0) + C_1'(0)C_2(0)C_1'(0) + C_1(0)C_2'(0)C_1'(0)\big\}. \end{aligned}$$ Hence we get the rough estimates $$\begin{aligned}
\Vert\phi'(0)\Vert_{HS}^2
&\leq 6\Vert C_1'(0)\Vert_{HS}^2 + 3\Vert C_2'(0)\Vert_{HS}^2, \\
\Vert\phi''(0)\Vert_{1,\mathrm{Tr}_N}
&\leq 8\Vert C_1'(0)\Vert_{HS}^2 + 4\Vert C_2'(0)\Vert_{HS}^2 \end{aligned}$$ (we used $2C_i'(0)^2 = |[C_i'(0),C_i(0)],C_i'(0)]|$, $i=1,2$, for the latter). Therefore, $$(\Phi\circ\phi)''(0)
\leq N\big\{(8\Vert\psi_i'\Vert_{\infty} + 6\Vert\psi_i''\Vert_{\infty})
\Vert C_1'(0)\Vert_{HS}^2
+ (4\Vert\psi_i'\Vert_{\infty} + 3\Vert\psi_i''\Vert_{\infty})
\Vert C_1'(0)\Vert_{HS}^2\big\}.$$ Since $\Phi\circ\phi(t)$ does not change when $C_1(t), C_2(t)$ are interchanged, one finally finds two universal constants $c_1 = 6>0$, $c_2 = 9/2 > 0$ so that $$|(\Phi\circ\phi)''(0)| \leq N(c_1\Vert\psi_i'\Vert_{\infty}
+c_2\Vert\psi_i''\Vert_{\infty})(\Vert C_1'(0) \Vert_{HS}^2 + \Vert C_2'(0)
\Vert_{HS}^2),$$ which immediately implies the desired inequality.
Finally, it should be pointed out that $(6,9/2)$ can be also chosen for two universal constants $(c_1,c_2)$ in [@HU2 Proposition 3.1].
\[C-5.5\] If $p_1,\dots,p_n$ are projections in $(\cM,\tau)$ and $\chi_{\mathrm{proj}}(p_1,\dots,p_n) = 0$, then $p_1,\dots,\allowbreak p_n$ are free.
This follows from and the fact that $W_{2,\mathrm{free}}$ is a metric on $TS_{\vec\alpha}\bigl(\cA^{(n)}_\proj\bigr)$ where $\vec\alpha:=(\tau(p_1),\dots,\tau(p_n))$.
The corollary was an essential ingredient of the proof of Theorem \[T-4.1\]. In the self-adjoint case, the free transportation cost inequality [@HU1 Theorem 2.2 or Corollary 2.3] provides a new proof of the fact that $X_1,\dots,X_n$ form a free semicircular system if $\chi(X_1,\dots,X_n)$ attains the maximum under the restriction $\tau(X_i^2)=1$, while Voiculescu’s original proof in [@V4] is based on the infinitesimal change of variable formula.
Free pressure
=============
Let $\bigl(\cA^{(n)}_\proj\bigr)^{sa}$ denote the space of self-adjoint elements in the universal $C^*$-algebra $\cA^{(n)}_\proj$ with the canonical projection generators $e_1,\dots,e_n$ as given in the previous section. Elements in $\bigl(\cA^{(n)}_\proj\bigr)^{sa}$ are considered as “free probabilistic hamiltonians on $\bZ_2^{\bigstar n}$." Motivated from the statistical mechanical viewpoint, we introduce the free pressure for those free hamiltonians, and its Legendre transform with respect to the duality between $\bigl(\cA^{(n)}_\proj\bigr)^{sa}$ and $TS\bigl(\cA^{(n)}_\proj\bigr)$ is compared with $\chi_\proj$.
Let $\vec\alpha=(\alpha_1,\dots,\alpha_n)\in[0,1]^n$ and $\vec k(N)=(k_1(N),\dots,k_n(N))\in\{0,1,\dots,N\}$ for $N\in\bN$ be given so that $\vec k(N)/N\to{\vec\alpha}$ as $N\to\infty$. As before we write $G(N,\vec k(N)):=\prod_{i=1}^nG(N,k_i(N))$ and $\gamma_{\vec k(N)}:=\bigotimes_{i=1}^n\gamma_{G(N,k_i(N))}$ for short. For $\vec P=(P_1,\dots,P_n)\in G(N)$ we have the $*$-homomorphism $h\in\cA^{(n)}_\proj\mapsto h(\vec P)\in M_N(\bC)$ sending $e_i$ to $P_i$, $1\le i\le n$. For each $h\in\bigl(\cA^{(n)}_\proj\bigr)^{sa}$ define $$\label{F-6.1}
\pi_{\vec\alpha}(h):=\limsup_{N\to\infty}
{1\over N^2}\log\int_{G(N,\vec k(N))}\exp\Bigl(
-N\Tr_N\bigl(h(\vec P)\bigr)\Bigr)\,d\gamma_{\vec k(N)}(\vec P),$$ which we call the [*free pressure*]{} of $h$ under the trace values $(\alpha_1,\dots,\alpha_n)$.
\[P-6.1\] The above definition of $\pi_{\vec\alpha}(h)$ is independent of the choices of $\vec k(N)$ with $\vec k(N)/N\to\vec\alpha$.
Let $\vec l(N)=(l_1(N),\dots,l_n(N)))$, $N\in\bN$, be another sequence such that $\vec l(N)/N\to\vec\alpha$ as $N\to\infty$. For $h\in\cA^{(n)}_\proj$ and $N\in\bN$ set $$\delta_N(h):=\max_{\vec U\in\U(N)^n}
\bigg|{1\over N}\Tr_N\bigl(h(\xi_{\vec k(N)}(\vec U)\bigr)
-{1\over N}\Tr_N\bigl(h(\xi_{\vec l(N)}(\vec U)\bigr)\bigg|,$$ where $\xi_{\vec k(N)}(\vec U):=(\xi_{N,k_1(N)}(U_1),\dots,\xi_{N,k_n(N)}(U_n))$ for $\vec U=(U_1,\dots,U_n)$ (see §1). For each $h\in\bigl(\cA^{(n)}_\proj\bigr)^{sa}$ we get thanks to $$\begin{aligned}
&\Bigg|{1\over N^2}\log\int_{G(N,\vec k(N))}
\exp\Bigl(-N\Tr_N\bigl(h(\vec P)\bigr)\Bigr)
\,d\gamma_{\vec k(N)}(\vec P) \\
&\qquad\quad-{1\over N^2}\log\int_{G(N,\vec k(N))}
\exp\Bigl(-N\Tr_N\bigl(h(\vec P)\bigr)\Bigr)
\,d\gamma_{\vec k(N)}(\vec P)\Bigg| \\
&\quad=\Bigg|{1\over N^2}\log\int_{\U(N)^n}\exp\Bigl(-N\Tr_N\bigl(
h(\xi_{\vec k(N)}(\vec U)\bigr)\Bigr)
\,d\bigl(\gamma_{\U(N)}\bigr)^{\otimes n}(\vec U) \\
&\qquad\quad-{1\over N^2}\log\int_{\U(N)^n}\exp\Bigl(-N\Tr_N\bigl(
h(\xi_{\vec l(N)}(\vec U)\bigr)\Bigr)
\,d\bigl(\gamma_{\U(N)}\bigr)^{\otimes n}(\vec U)\Bigg| \\
&\quad\le\delta_N(h).\end{aligned}$$ Hence, it suffices to prove that $\delta_N(h)\to0$ as $N\to\infty$ for any $h\in\cA^{(n)}_\proj$. Since $|\delta_N(h_1)-\delta_N(h_2)|\le\|h_1-h_2\|$ for all $h_1,h_2\in\cA^{(n)}_\proj$, we may show that $\delta_N(h)\to0$ for $h=e_{i_1}\cdots e_{i_r}$ with $1\le
i_1,\dots,i_r\le n$. For such $h$, as in the proof of Proposition \[P-1.1\] we have $$\begin{aligned}
\bigg|{1\over N}\Tr_N\bigl(h(\xi_{\vec k(N)}(\vec U)\bigr)
-{1\over N}\Tr_N\bigl(h(\xi_{\vec l(N)}(\vec U)\bigr)\bigg|
&\le\sum_{j=1}^r\|\xi_{N,k_{i_j}(N)}(U_{i_j})-\xi_{N,l_{i_j}(N)}(U_{i_j})\|_1 \\
&\le\sum_{j=1}^r{|k_{i_j}(N)-l_{i_j}(N)|\over N}\longrightarrow0\end{aligned}$$ as $N\to\infty$, and the conclusion follows.
The following are basic properties of $\pi_{\vec\alpha}(h)$; we omit the proofs very similar to those of [@Hi Proposition 2.3] but note that the last assertion of (iv) follows from and Proposition \[P-6.4\](1) below.
\[P-6.2\]
- $\pi_{\vec\alpha}(h)$ is convex on $\bigl(\cA^{(n)}_\proj\bigr)^{sa}$.
- If $h_1,h_2\in\bigl(\cA^{(n)}_\proj\bigr)^{sa}$ and $h_1\le h_2$, then $\pi_{\vec\alpha}(h_1)\ge\pi_{\vec\alpha}(h_2)$.
- $|\pi_{\vec\alpha}(h_1)-\pi_{\vec\alpha}(h_2)|\le\|h_1-h_2\|$ for all $h_1,h_2\in\bigl(\cA^{(n)}_\proj\bigr)^{sa}$.
- If $h_1\in\bigl(\cA^{(j)}_\proj\bigr)^{sa}$ and $h_2\in\bigl(\cA^{(n-j)}_\proj\bigr)^{sa}$ with $1\le j<n$, then $$\pi_{\vec\alpha}(h_1+h_2)\le\pi_{(\alpha_1,\dots,\alpha_j)}(h_1)
+\pi_{(\alpha_{j+1},\dots,\alpha_n)}(h_2)$$ where $h_1+h_2$ is the sum as an element of $\cA^{(n)}_\proj=\cA^{(j)}_\proj\bigstar\cA^{(n-j)}_\proj$. In particular when $j=1$ or $j=2$, equality holds in the above inequality.
\[R-6.3\][Another possible definition of free pressure is to use the probability measure $\gamma_{G(N)}^{(1)}$ or $\gamma_{G(N)}^{(2)}$ on $G(N)$ given in Remark \[R-1.3\]. For $h\in\bigl(\cA^{(n)}_\proj\bigr)^{sa}$ define $$\pi^{(j)}(h):=\limsup_{N\to\infty}{1\over N^2}\log
\int_{G(N)^n}\exp\Bigl(-N\Tr_N\bigl(h(\vec P)\bigr)\Bigr)
\,d\bigl(\gamma_{G(N)}^{(j)}\bigr)^{\otimes n}(\vec P)$$ for $j=1,2$. It is not difficult to show that $$\pi^{(1)}(h)=\pi^{(2)}(h)
=\max\bigl\{\pi_{\vec\alpha}(h):\vec\alpha\in[0,1]^n\bigr\}$$ for every $h\in\bigl(\cA^{(n)}_\proj\bigr)^{sa}$. We simply write $\pi(h)$ for these equal quantities; then $\pi(h)$ has the same properties as in Proposition \[P-6.2\]. However, unlike the free entropy quantities $\chi^{(j)}_\proj$ discussed in Remark \[R-1.3\], $\pi(h)$ does not coincide with $\pi_{\vec\alpha}(h)$; the latter actually depends on $\vec\alpha$. ]{}
In the single projection case, $\cA^{(1)}_\proj=\bC^2$, $\bigl(\cA^{(1)}_\proj\bigr)^{sa}=\bR^2$ and $TS\bigl(\cA^{(1)}_\proj\bigr)=\{\tau_\alpha:0\le\alpha\le1\}$ where $\tau_\alpha(\zeta_1,\zeta_2)=\alpha\zeta_1+(1-\alpha)\zeta_2$ for $(\zeta_1,\zeta_2)\in\bC^2$. Let $0\le\alpha\le1$ and choose $k(N)$ such that $k(N)/N\to\alpha$. For each $h=(h_1,h_2)\in\bR^2$, it is straightforward to check that $$\begin{aligned}
\label{F-6.2}
\pi_\alpha(h)&=\lim_{N\to\infty}{1\over N^2}\log
\int_{G(N,k(N))}\exp\bigl(-N\Tr_N(h(P))\bigr)\,d\gamma_{G(N,k(N))}(P) \nonumber\\
&=-\alpha h_1-(1-\alpha)h_2=-\tau_\alpha(h)\end{aligned}$$ and hence $\chi_\proj(\tau_\alpha)=0=\tau_\alpha(h)+\pi_\alpha(h)$. Moreover, $$\pi(h)=-\min\{h_1,h_2\}=\min\{-\tau_\alpha(h)+\chi(\tau_\alpha):0\le\alpha\le1\}.$$
In the case of two projections, $\cA^{(2)}_\proj=C^*(\bZ_2\bigstar\bZ_2)$ with the canonical projection generators $e,f$. Let $\alpha,\beta\in[0,1]$. The next theorem says that the free entropy $\chi_\proj(\tau)$ for $\tau\in TS_{(\alpha,\beta)}\bigl(\cA^{(2)}_\proj\bigr)$ and the free pressure $\pi_{(\alpha,\beta)}(h)$ for $h\in\bigl(\cA^{(2)}_\proj\bigr)^{sa}$ are the Legendre transforms of each other.
\[P-6.4\]
- In the definition of $\pi_{(\alpha,\beta)}(h)$ in $\limsup$ can be replaced by $\lim$.
- $\pi_{(\alpha,\beta)}(h)=\max\Bigl\{-\tau(h)+\chi_\proj(\tau):
\tau\in TS_{(\alpha,\beta)}\bigl(\cA^{(2)}_\proj\bigr)\Bigr\}$ for every $h\in\bigl(\cA^{(2)}_\proj\bigr)^{sa}$.
- $\chi_\proj(\tau)=\inf\Bigl\{\tau(h)+\pi_{(\alpha,\beta)}(h):
h\in\bigl(\cA^{(2)}_\proj\bigr)^{sa}\Bigr\}$ for every $\tau\in TS_{(\alpha,\beta)}\bigl(\cA^{(2)}_\proj\bigr)$.
- $\pi(h)=\max\Bigl\{-\tau(h)+\chi_\proj(\tau):
\tau\in TS\bigl(\cA^{(2)}_\proj\bigr)\Bigr\}$ for every $h\in\bigl(\cA^{(2)}_\proj\bigr)^{sa}$.
Thanks to the Lipschitz continuity in $h$ of the quantity inside $\limsup$ in as well as both sides of the equality in (2), to prove (1) and (2), we may assume that $h$ is a self-adjoint polynomial of $e,f$ written as $$h=C\1+Ae+Bf+\sum_{k=1}^mA_j(efe)^j+\sum_{j=1}^mB_j(fef)^j
+\sum_{j=1}^mD_j((ef)^j+(fe)^j)$$ with $A,B,C,A_j,B_j,D_j\in\bR$. Set $$h_0:=Ae+Bf+\sum_{k=0}^mC_k(efe)^k$$ with $C_0:=C$, $C_j:=A_j+B_j+D_j$, $1\le j\le m$. We then get $\tau(h)=\tau(h_0)$ and $\Tr_N(h(P,Q))=\Tr_N(h_0(P,Q))$ for $P,Q\in G(N)$ so that $\pi_{(\alpha,\beta)}(h)=\pi_{(\alpha,\beta)}(h_0)$. Hence it is enough to prove (1) and (2) for $h_0$ above. A bit more generally, let $h\in\bigl(\cA^{(2)}_\proj\bigr)^{sa}$ be of the form $$h=Ae+Bf+\psi(efe),$$ where $\psi$ is a real continuous function on $[0,1]$. Choosing $k(N),l(N)$ such that $k(N)/N\to\alpha$ and $l(N)/N\to\beta$, we have $$\begin{aligned}
\label{F-6.3}
&{1\over N^2}\log\int_{G(N,k(N))\times G(N,l(N))}
\exp\bigl(-N\Tr_N(\psi(P,Q))\bigr)
\,d\bigl(\gamma_{G(N,k(N))}\otimes\gamma_{G(N,l(N))}\bigr)(P,Q) \nonumber\\
&\qquad=-A{k(N)\over N}-B{l(N)\over N}+{1\over N^2}\log\int_{[0,1]^n}
\exp\Biggl(-N\sum_{i=1}^N\psi(x_i)\Biggr)\,d\lambda_N(x_1,\dots,x_N),\end{aligned}$$ where $\lambda_N$ is the empirical eigenvalue distribution of $PQP$ when $(P,Q)$ is distributed under $\gamma_{G(N,k(N))}\otimes\gamma_{G(N,l(N))}$. By applying Varadhan’s integral lemma (see [@DZ 4.3.1]) to the large deviation in [@HP1 Theorem 2.2] we have $$\begin{aligned}
\label{F-6.4}
&\lim_{N\to\infty}{1\over N^2}\log\int_{[0,1]^n}
\exp\Biggl(-N\sum_{i=1}^N\psi(x_i)\Biggr)\,d\lambda_N(x_1,\dots,x_N) \nonumber\\
&\quad=\sup_\nu\Biggl\{
-(1-\min\{\alpha,\beta\})\psi(0)-\max\{\alpha+\beta-1,0\}\psi(1)
-{1\over2}\int_{[0,1]}\psi(x)\,d\nu(x) \nonumber\\
&\hskip3cm+{1\over4}\Sigma(\nu)
+{|\alpha-\beta|\over2}\int_{[0,1]}\log x\,d\nu(x) \nonumber\\
&\hskip4cm+{|\alpha+\beta-1|\over2}\int_{[0,1]}\log(1-x)\,d\nu(x)-C\Biggr\},\end{aligned}$$ where $\nu$ runs over all measures on $(0,1)$ with $\nu((0,1))=2\rho$. Here, $\rho$ is in and $C$ in . For $\tau=\tau_{\nu,\{\alpha_{ij}\}}\in
TS_{(\alpha,\beta)}\bigl(\cA^{(2)}_\proj\bigr)$ (see §2 and §5), when $\alpha_{00}\alpha_{11}=\alpha_{01}\alpha_{10}=0$ (this is necessary for $\chi_\proj(\tau)>-\infty$), $\chi_\proj(\tau)$ is given as in Proposition \[P-2.1\] and moreover we get $$\begin{aligned}
\tau(h)&=A\alpha+B\beta+(\alpha_{10}+\alpha_{01}+\alpha_{00})\psi(0)
+\alpha_{11}\psi(1)+{1\over2}\int_{(0,1)}(\psi(x)+\psi(0))\,d\nu(x) \\
&=A\alpha+B\beta+(1-\min\{\alpha,\beta\})\psi(0)
+\max\{\alpha+\beta-1,0\}\psi(1)+{1\over2}\int_{(0,1)}\psi(x)\,d\nu(x)\end{aligned}$$ thanks to . Furthermore, Proposition \[P-2.1\] implies that $\chi_\proj(\tau)$ is concave and weakly\* upper semi-continuous restricted on $TS_{(\alpha,\beta)}\bigl(\cA^{(2)}_\proj\bigr)$. Hence we obtain (1) and (2) by and together with the formulas of $\chi_\proj(\tau)$ and $\tau(h)$. Moreover, (3) follows from (2) due to the duality for conjugate functions (or Legendre transforms). Finally, (4) is obvious from (2) and Remark \[R-6.3\].
Now, we introduce a free entropy-like quantity for tracial states on $\cA^{(n)}_\proj$ (or for $n$-tuples of projections) via the (minus) Legendre transform of free pressure. For each $\vec\alpha\in[0,1]^n$ and $\tau\in TS_{\vec\alpha}\bigl(\cA^{(n)}_\proj\bigr)$ define $$\eta_\proj(\tau):=\inf\Bigl\{\tau(h)+\pi_{\vec\alpha}(h):
h\in\bigl(\cA^{(n)}_\proj\bigr)^{sa}\Bigr\}.$$ Since $\pi_{\vec\alpha}$ is a convex continuous function on $\bigl(\cA^{(n)}_\proj\bigr)^{sa}$ by Proposition \[P-6.2\], the above Legendre transform is reversed so that for every $h\in\bigl(\cA^{(n)}_\proj\bigr)^{sa}$ we have $$\pi_{\vec\alpha}(h)=\sup\Bigl\{-\tau(h)+\eta_\proj(\tau):
\tau\in TS_{\vec\alpha}\bigl(\cA^{(n)}_\proj\bigr)\Bigr\}.$$ For each $h\in\bigl(\cA^{(n)}_\proj\bigr)^{sa}$ there exists a $\tau\in TS_{\vec\alpha}\bigl(\cA^{(n)}_\proj\bigr)$ such that $$\pi_{\vec\alpha}(h)=-\tau(h)+\eta_\proj(\tau).$$ This equality condition is a kind of variational principle and such $\tau$ may be called an equilibrium tracial state associated with $h$ (and $\vec\alpha$).
Moreover, for each $n$-tuple $(p_1,\dots,p_n)$ of projections in $(\cM,\tau)$, we have $\tau_{(p_1,\dots,p_n)}\in TS\bigl(\cA^{(n)}_\proj\bigr)$ defined by $\tau_{(p_1,\dots,p_n)}(h):=\tau(h(p_1,\dots,p_n))$, where $h\in\cA^{(n)}_\proj\mapsto h(p_1,\dots,p_n)\in\cM$ is the $*$-homomorphism sending $e_i$ to $p_i$, $1\le i\le n$. We define $$\eta_\proj(p_1,\dots,p_n):=\eta_\proj(\tau_{(p_1,\dots,p_n)}).$$ It is obvious by definition that the quantity $\eta_\proj(p_1,\dots,p_n)$ has the same properties as $\chi_\proj(p_1,\dots,p_n)$ given in Proposition \[P-1.2\].
\[T-6.5\] Let $p_1,q_1,\dots,p_n,q_n,r_1,\dots,r_{n'}$ be projections in $(\cM,\tau)$.
- $\eta_\proj(p_1,\dots,p_n)\ge\chi_\proj(p_1,\dots,p_n)$.
- If $\{p_1,q_1\}$, $\dots$, $\{p_n,q_n\}$, $\{r_1\}$, $\dots$, $\{r_{n'}\}$ are free, then $$\eta_\proj(p_1,q_1\dots,p_n,q_n,r_1,\dots,r_{n'})
=\chi_\proj(p_1,q_1\dots,p_n,q_n,r_1,\dots,r_{n'}).$$
The proof of (1) is similar to that of [@Hi Theorem 4.5(1)]. By and Proposition \[P-6.4\](3), $\eta_\proj=\chi_\proj$ holds when $n=1$ or $n=2$. Hence (2) is seen from (1) together with the subadditivity of $\eta_\proj$ and the additivity of $\chi_\proj$ in Theorem \[T-4.1\](1).
\[R-6.6\][It is known in a forthcoming paper [@HMU] that if $\chi_\proj(p_1,\dots,p_n)>-\infty$ and $\sum_{i=1}^n\min\{\tau(p_i),\tau(\1-p_i)\}>1$ (this forces $n\ge3$), then $\{p_1,\dots,p_n\}''$ is a non-$\Gamma$ II$_1$ factor. Choose two different $\tau_1,\tau_2\in TS_{\vec\alpha}\bigl(\cA^{(n)}_\proj\bigr)$ such that $\chi_\proj(\tau_1)$ and $\chi_\proj(\tau_2)$ are finite and $\sum_{i=1}^n\min\{\alpha_i,1-\alpha_i\}>1$. For $\tau_0:=(\tau_1+\tau_2)/2$ we get $\eta_\proj(\tau_0)>-\infty$ by the concavity of $\eta_\proj$ on $TS_{\vec\alpha}\bigl(\cA^{(n)}_\proj\bigr)$. But $\chi_\proj(\tau_0)=-\infty$ due to the above mentioned fact. Hence, $\eta_\proj$ and $\chi_\proj$ are not equal in general. ]{}
Finally, we note that the definition is slightly modified in such a way that the modified free pressure $\pi^{(2)}_{\vec\alpha}(g)$ is defined for self-adjoint elements of the minimal $C^*$-tensor product $\cA^{(n)}_\proj\otimes_{\min}\cA^{(n)}_\proj$ and the modified quantity $\tilde\eta_\proj(p_1,\dots,p_n)$ induced from $\pi^{(2)}_{\vec\alpha}$ via Legendre transform is always equal to $\chi_\proj(p_1,\dots,p_n)$. We omit the details concerned with this modification that are essentially same as [@Hi §6].
[99]{}
D. Bakry and M. Emery, Diffusion hypercontractives, in [*Séminaire Probabilités XIX*]{}, Lecture Notes in Math., Vol. 1123, Springer-Verlag, 1985, pp. 177–206.
G. Ben Arous and A. Guionnet, Large deviation for Wigner’s law and Voiculescu’s noncommutative entropy, [*Probab. Theory Related Fields*]{} [**108**]{} (1997), 517–542.
P. Biane, M. Capitaine and A. Guionnet, Large deviation bounds for the law of the trajectories of the Hermitian Brownian motion, [*Invent. Math.*]{} [**152**]{} (2003), 433–459.
B. Collins, Product of random projections, Jacobi ensembles and universality problems arising from free probability, [*Probab. Theory Related Fields*]{} [**133**]{} (2005), 315–344
G. Corach, H. Porta, L. Recht, The geometry of spaces of projections in $C\sp *$-algebras, [*Adv. Math.*]{} [**101**]{} (1993), 59–77.
A. Dembo and O. Zeitouni, [*Large Deviation Techniques and Applications*]{}, Second edition, Springer, New York, 1998.
A. Guionnet, Large deviations and stochastic calculus for large random matrices, [*Probab. Surv.*]{} [**1**]{} (2004), 72–172 (electronic).
O. Henkel, Sphere packing bounds in the Grassmann and Stiefel manifolds, [*IEEE Transactions on Information Theory*]{} [**51**]{} (2005), 3445-3456.
F. Hiai, Free analog of pressure and its Legendre transform, [*Comm. Math. Phys.*]{} [**255**]{} (2005), 229–252.
F. Hiai, T. Miyamoto and Y. Ueda, in preparation.
F. Hiai and D. Petz, [*The Semicircle Law, Free Random Variables and Entropy*]{}, Mathematical Surveys and Monographs, Vol. 77, Amer. Math. Soc., Providence, 2000.
F. Hiai and D. Petz, Large deviations for functions of two random projection matrices, preprint, 2005.
F. Hiai, D. Petz and Y. Ueda, Free transportation cost inequalities via random matrix approximation, [*Probab. Theory Related Fields*]{} [**130**]{} (2004), 199–221.
F. Hiai and Y. Ueda, Free transportation cost inequalities for non-commutative multi-variables, preprint, 2005.
F. Hiai and Y. Ueda, A log-Sobolev type inequality for free entropy of two projections, preprint, 2006.
P. Malliavin, [*Integration and Probability*]{}, Graduate Texts in Mathematics, Vol. 157, Springer-Verlag, New York, 1995.
F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, [*J. Funct. Anal.*]{} [**173**]{} (2000), 361–400.
E.B. Saff and V. Totik, [*Logarithmic Potentials with External Fields*]{} (with Appendix B by Thomas Bloom), Grundlehren der Mathematischen Wissenschaften, Vol. 316. Springer-Verlag, Berlin, 1997.
D. Voiculescu, Limit laws for random matrices and free products, [*Invent. Math.*]{} [**104**]{} (1991), 201-220.
D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory, I, [*Comm. Math. Phys.*]{} [**155**]{} (1993), 71–92.
D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory, II, [*Invent. Math.*]{} [**118**]{} (1994), 411–440.
D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory, IV: Maximum entropy and freeness, in [*Free Probability Theory*]{}, D.V. Voiculescu (ed.), Fields Inst. Commun. [**12**]{}, Amer. Math. Soc., 1997, pp. 293–302.
D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory, V: Noncommutative Hilbert transforms, [*Invent. Math.*]{} [**132**]{} (1998), 189–227.
D. Voiculescu, A strengthened asymptotic freeness result for random matrices with applications to free entropy, [*Internat. Math. Res. Notices*]{} [**1998**]{}, 41–63.
D. Voiculescu, The analogue of entropy and of Fisher’s information measure in free probability theory VI: Liberation and mutual free information, [*Adv. Math.*]{} [**146**]{} (1999), 101–166.
D. Voiculescu, Free entropy, [*Bull. London Math. Soc.*]{} [**34**]{} (2002), 257–278.
D. V. Voiculescu, K. J. Dykema and A. Nica, [*Free Random Variables*]{}, CRM Monograph Ser., Vol. 1, Amer. Math. Soc., 1992.
[^1]: $^1\,$Supported in part by Japan Society for the Promotion of Science, Japan-Hungary Joint Project.
[^2]: $^2\,$Supported in part by Grant-in-Aid for Scientific Research (B)17340043.
[^3]: $^3\,$Supported in part by Grant-in-Aid for Young Scientists (B)17740096.
[^4]: AMS subject classification: Primary: 46L54; secondary: 15A52, 60F10, 94A17.
|
---
abstract: 'We present a new expression for the five-dimensional static Kaluza-Klein black hole solution with squashed $S^3$ horizons and three different charge parameters. This black hole solution belongs to $D = 5$ $N = 2$ supergravity theory, its spacetime is locally asymptotically flat and has a spatial infinity $R \times S^1 \hookrightarrow S^2$. The form of the solution is extraordinary simple and permits us very conveniently to calculate its conserved charges by using the counterterm method. It is further shown that our thermodynamical quantities perfectly obey both the differential and the integral first laws of black hole thermodynamics if the length of the compact extra-dimension can be viewed as a thermodynamical variable.'
address: 'Institute of Theoretical Physics, College of Physics and Electric Information, China West Normal University, Nanchong, Sichuan 637002, People’s Republic of China'
author:
- 'Shuang-Qing Wu[^1], Dan Wen, Qing-Quan Jiang, and Shu-Zheng Yang'
title: 'Thermodynamics of five-dimensional static three-charge STU black holes with squashed horizons'
---
$U(1)^3$ supergravity ,squashed black hole ,thermodynamics 04.20.Jb ,04.65.+e ,04.50.Gh ,04.70.Dy
Introduction
============
Five-dimensional black holes [@TI11] with squashed horizons are of special interest in Kaluza-Klein theory where the fifth dimension is assumed to be compactified into a circle. A simplest example for this in the five-dimensional Einstein-Maxwell theory is the static charged, squashed Kaluza-Klein black hole solution found by Ishihara and Matsuno [@IM06] via applying the so-called squashing transformation to the five-dimensional Reissner-Nordström black hole solution. This black hole has horizon-topology of a $S^3$ sphere that is deformed by the squashing function. At the infinity ($r\to r_\infty$), the spacetime approaches to the direct product of a flat time dimension and the asymptotic structure of the self-dual Taub-NUT instanton with NUT charge $r_\infty/4$. In other words, the black hole spacetime is asymptotically a twisted $S^1$ fibre bundle over $M^{1,3}$, which can be interpreted as a static charged Reissner-Nordström black hole sitting on the self-dual Taub-NUT instanton [@CT10].
Sooner after the work done in Ref. [@IM06], the squashing procedure was then successfully applied to generate new black hole solutions with squashed horizons, due to its simplicity of the method. Subsequently, a large class of new Kaluza-Klein black hole solutions in five dimensions had been constructed so far in vacuum Einstein gravity [@TW06], Einstein-Maxwell-dilaton gravity [@SY06; @NY13], $D = 5$ minimal Einstein-Maxwell-Chern-Simons supergravity [@NIMT08] and $U(1)^3$ supergravity [@ST10]. Later, several black hole solutions with squashed horizons in a five-dimensional Gödel universe had also been presented in Refs. [@MINT08; @TI08; @TIMN09].
In recent years, there are also a lot of attention being paid to various aspects of squashed Kaluza-Klein black holes. For example, thermodynamical properties [@CCO06; @KI07; @KI08; @SSW08; @SN09; @PW10], Hawking radiation [@NY13; @IS07; @CWS08; @WLLR10; @MU11; @HL10; @HL11], perturbation stability [@KMIS08; @IKKMSZ08; @NK10], quasinormal modes [@HWCCL08; @HWC09], gravitational lens [@LCJ10; @CLJ11; @SBV13], geodesic motion [@MI09] and Kerr/CFT correspondence [@PW10] have been investigated for this kind of black holes in recent years. In particular, Cai [*et al.*]{}, [@CCO06] first adopted the boundary counter-term method [@MS06; @AR06] to investigate thermodynamics of the static charged, squashed Kaluza-Klein black holes found in Ref. [@IM06] and showed that it is the counter-term mass, which is equal to the Abott-Deser mass, rather than the Komar mass, that obeys the differential first law when the radius of the compact extra-dimension is considered as a constant. The counter-term method was then frequently applied to calculate the boundary stress-energy tensor and the conserved charges of a large class of Kaluza-Klein black holes [@TW06; @SY06; @CCO06; @KI07; @SSW08; @NY11] with squashed horizons. Moreover, thermodynamics of the five-dimensional squashed Reissner-Nordström black hole was investigated in details in Ref. [@KI08] where the counter-term mass is in agreement with the mass calculated by the background subtraction method if the Kaluza-Klein monopole background is considered as a natural reference spacetime for the squashed Kaluza-Klein black hole.
Our aim of this Letter is mainly concerned with thermodynamics of the three-charge squashed Kaluza-Klein black hole solution to $D = 5$ $N = 2$ ungauged supergravity theory. A rotating solution in this theory was previously presented in Ref. [@ST10] where the author obtained the solution by directly applying the squashing transformation to the three-charge Cvetič-Youm black hole [@CY96] with two equal rotation parameters. Some thermodynamical quantities were given in [@ST10], however they can not consistently fulfil both the differential and the integral first laws of squashed black hole thermodynamics. The reason for this is simply because the ADM mass computed there does not obey the standard first law of thermodynamics for the squashed Kaluza-Klein black holes [@KI07; @KI08]. What is more, the contribution of three dipole charges should but had not been taken into consideration in Ref. [@ST10], when the rotation is included. Therefore, it is clear that thermodynamical properties of this solution had not been correctly studied ever before in the previous literature, unlike the minimal case [@PW10], and it deserves a deeper investigation of its thermodynamics, which consists of the main subject of our work.
However, the rotating version of the three-charge squashed solution presented in Ref. [@ST10] is very complicated after performing the coordinate transformations from ($t, r$) to ($\tau, \rho$) even if one only considers the static case, so the expression for the solution is not suitable for our purpose to study its thermodynamical properties. For the sake of simplicity, in this Letter we shall focus on the nonrotating case only. Our strategy is to seek another new form for the three-charge static squashed black hole solution which is different from the one previously presented in Ref. [@ST10]. To derive the solution, we have applied a triple-repeated lift-boost-reduction procedure [@EE03; @LMP10] to the five-dimensional static squashed Schwarzschild black hole solution which has a normalized Killing time vector at spatial infinity. The derivation is essentially parallel to that of the static three-charge STU black hole solution in Ref. [@HMS96]. The final expressions for the solution are much simpler than those presented in Ref. [@ST10] because three gauge potentials and two dilaton scalar fields associated with it all vanish at the infinity. For this form of the solution, one can very easily calculate its conserved mass and gravitational tension by using the counterterm method, and show that all thermodynamical quantities computed for our three-charge static Kaluza-Klein black hole with squashed $S^3$ horizons perfectly satisfy both the differential and the integral first laws of squashed black hole thermodynamics if the length of the compact extra-dimension can viewed as a thermodynamical variable. When three charges are set to be equal, our results completely reproduce those obtained in Ref. [@CCO06] after some suitable identifications of solution parameters.
The remaining part of our Letter is organized as follows. In Sec. \[Sect2\], we first present a new form of the static three-charge squashed black hole solution. Then, the boundary counterterm method is adopted to calculate the conserved mass and gravitational tension which together with the entropy, horizon temperature, three charges and their corresponding electrostatic potentials completely satisfy both the differential and the integral first laws when we consider the length of the extra-dimension as a thermodynamical variable. Our Letter ends up with a summary of our work and the related future plan.
Static three-charge squashed black hole solution and its thermodynamics {#Sect2}
=======================================================================
In this section, we present a new simple form of the five-dimensional three-charge static squashed black hole solution and investigate its thermodynamics. To generate the solution, we start from the static squashed Schwarzschild solution after performing the appropriate coordinate transformations and use a thrice-repeated sequence [@EE03; @LMP10] of lifting to six dimensions, performing a Lorentz boost, and reducing again to $D = 5$. The final expressions for the metric and three Abelian gauge potentials are concisely given by ds\^2 &=& (h\_1h\_2h\_3)\^[1/3]{} , \[3cSqKK\]\
A\_I &=& d , \[3cgp\] and three scalars are X\_I = , h\_I = 1 +s\_I\^2 , where $c_I = \cosh\delta_I$, and $s_I = \sinh\delta_I$, in which $\delta_I$’s are three charge parameters. In the solution, the coordinate $\rho$ varies from 0 to $\infty$, and ($\theta, \phi, \psi$) are three Eulerian angles, taking the ranges $0 < \theta < \pi$, $0 < \psi < 2\pi$, $0 < \phi < 4\pi$.
At spatial infinity ($\rho \to \infty$), we have $h_I \to 1$ and $X_I \to 1$, so three gauge potentials $A_I$ tend to zero and two dilaton scalar fields ($\varphi_1, \varphi_2$) behave asymptotically like \_1 = +O(\^[-2]{}) , \_2 = +O(\^[-2]{}) , while the metric (\[3cSqKK\]) approaches to ds\^2 = -d\^2 +d\^2 +\^2(d\^2 +\^2d\^2) +\_0(\_0 +\_1)(d+d)\^2 .
Just like the uncharged solution ($\delta_I = 0$), the asymptotic structure of our solution (\[3cSqKK\]) is also a four-dimensional flat Minkowski spacetime with a compact extra-dimension. That is, its spacetime is locally asymptotically flat and has an asymptotic boundary topology $R \times S^1 \hookrightarrow S^2$, and $\p_\phi$ generates the twisted $S^1$ fibre bundle at spatial infinity, with a constant size $2\pi\sqrt{\rho_0(\rho_0+\rho_1)}$. The horizon topology is, however, obviously a squashed $S^3$ sphere.
In the case where all three charges are set to equal ($s_I = s$), we find that by setting $$\tilde{\rho} = \rho +s^2\rho_1 \, , \quad \tilde{\rho}_0 = \rho_0 -s^2\rho_1 \, , \qquad
\rho_+ = c^2\rho_1 \, , \quad \rho_- = s^2\rho_1 \, ,$$ and rescaling the gauge potential by a factor $\sqrt{3}$, the above solution exactly reproduces the static squashed Reissner-Nordström black hole solution presented in Refs. [@IM06; @CCO06].
We have verified that our solution solves the full set of equations of motion derived from the Lagrangian of the $D = 5$, $N = 2$ supergravity theory whose action is &=& \_ d\^5x {\
&& +\^F\^1\_F\^2\_A\^3\_} +\_ K d\^4x , \[act\] where the Chern-Simons term is included for the completeness, but it makes no contribution to the action in the nonrotating case. In the boundary Gibbons-Hawking term, $K$ is the trace of extrinsic curvature $K_{ij} = (n_{i ; j} +n_{j ; i})/2$ for the boundary $\p\mathcal{M}$ with the induced metric $h_{ij}$, $R$ is the bulk scalar curvature, and $F_I = dA_I$ are strengths associated to three $U(1)$’s gauge fields. Two dilaton scalar fields ($\varphi_1, \varphi_2$) are related to three scalars $X_I$ by X\_1 = e\^[-\_1/ -\_2/]{} , X\_2 = e\^[-\_1/ +\_2/]{} , X\_3 = e\^[2\_1/]{} .
It is now the position to investigate thermodynamical properties of our solution given above. An suitable approach for this aim is to make use of the counterterm method [@MS06; @AR06] to compute the conserved charges since our solution is also an asymptotically flat spacetime with boundary topology $R \times S^1\hookrightarrow S^2$. In the method, a counterterm, which is a functional only of the curvature invariants of the induced metric on the boundary, is added to the boundary term at infinity, so that a regular gravitational action is obtained without any modification of the equations of motion.
We consider the following simple counterterm proposed by Mann and Stelea [@MS06] I\_[ct]{} = d\^4x , where $\mathcal{R}$ is the Ricci scalar with respect to the boundary metric $h_{ij}$. Varying the action (\[act\]) with this counterterm leads to the boundary stress-energy tensor T\_[ij]{} = , where $\Psi = \sqrt{2/\mathcal{R}}$, and the covariant derivative is defined with respect to the induced metric $h_{ij}$ on the boundary.
If the boundary geometry has an isometry generated by the Killing vector $\xi$, then $T_{ij}\xi^j$ is divergence free, so the conserved charge associated with it is given by = \_d\^3 S\^i T\_[ij]{}\^j , which represents the conserved mass $M_{ct}$ in the case when $\xi = \p_\tau$, and the gravitational tension $\mathcal{T}$ if $\xi = \p_\phi$.
After some calculation, we find the needed components of the stress tensor as follows T\_ &=& +O(\^[-3]{}) ,\
T\_ &=& +O(\^[-3]{}) ,\
T\_ &=& +O(\^[-4]{}) . Then it is straightforward to calculate the conserved mass and gravitational tension as M\_[ct]{} &=& ,\
&=& + .
On the horizon, the entropy $S = A/4$ and temperature $T = \kappa/(2\pi)$ can be easily obtained as S &=& 4\^2c\_1c\_2c\_3\_1\^[3/2]{}(\_0 +\_1) ,\
T &=& , and three electrostatic potentials are given by \_I = (A\^I\_\^) |\_[=\_1]{} = . Finally, we have three electric charges corresponding to the gauge potentials by completing the integral Q\_I = \_[S\^3]{}X\_I\^[-2]{} F\_I = c\_Is\_I\_1 .
It is not difficult to verify that the above thermodynamical quantities completely satisfy both the different and the integral first laws of black hole thermodynamics dM\_[ct]{} &=& TdS +\_1dQ\_1 +\_2dQ\_2 +\_3dQ\_3 +4d ,\
M\_[ct]{} &=& TS +\_1Q\_1 +\_2Q\_2 +\_3Q\_3 +2 , where $2\pi\mathcal{L}$ is the length of the extra-dimension, and $\mathcal{L} = \sqrt{\rho_0(\rho_0 +\rho_1)}$ can be identified with twice of the NUT charge. In the equal-charge case ($Q_i = Q/\sqrt{3}$), our results exactly recover those obtained in Ref. [@CCO06] where the radius $2\pi\mathcal{L}$ of the extra-dimension was considered as a constant. Here, we view it as a thermodynamical variable for the self-consistence of the integral first law.
Conclusions
===========
In this Letter, we have presented a new form for the static three-charge STU black hole solution with squashed horizons. The expressions for the metric, three gauge potentials and two scalar fields are rather simple and very convenient for us to investigate its thermodynamics. By means of the counterterm method, all thermodynamical quantities of our solution are easily calculated and have been shown to fulfil both differential and integral first laws of the squashed black hole thermodynamics when the length of the extra-dimension is considered as a thermodynamical variable.
It is interesting to extend the present work to the rotating charged case [@ST10] where the form of the squashed black hole solution is definitely inconvenient for studying its thermodynamics. A promising routine to resolve this difficulty is to regenerate this solution from the neutral seed of the rotating Kaluza-Klein squashed black hole solution [@TW06] with a Killing time vector normalized at spatial infinity.
Acknowledgements {#acknowledgements .unnumbered}
================
S.Q. Wu is supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 11275157 and 10975058. Q.Q. Jiang and S.Z. Yang are, respectively, supported by the NSFC under Grant Nos. 11005086 and 11178018.
[99]{}
S. Tomizawa, H. Ishihara, Prog. Theor. Phys. Suppl. **189** (2011) 7. H. Ishihara, K. Matsuno, Prog. Theor. Phys. **116** (2006) 417. Y. Chen, E. Teo, Nucl. Phys. B **838** (2010) 207. T. Wang, Nucl. Phys. B **756** (2006) 86. S.S. Yazadjiev, Phys. Rev. D **74** (2006) 024022. P.G. Nedkova, S.S. Yazadjiev, Eur. Phys. J. C **73** (2013) 2377. T. Nakagawa, H. Ishihara, K. Matsuno, S. Tomizawa, Phys. Rev. D **77** (2008) 044040. S. Tomizawa, arXiv:1009.3568 \[hep-th\].
K. Matsuno, H. Ishihara, T. Nakagawa, S. Tomizawa, Phys. Rev. D **78** (2008) 064016. S. Tomizawa, A. Ishibashi, Class. Quant. Grav. **25** (2008) 245007. S. Tomizawa, H. Ishihara, K. Matsuno, T. Nakagawa, Prog. Theor. Phys. **121** (2009) 823. R.G. Cai, L.M. Cao, N. Ohta, Phys. Lett. B **639** (2006) 354. Y. Kurita, H. Ishihara, Class. Quant. Grav. **24** (2007) 4525. Y. Kurita, H. Ishihara, Class. Quant. Grav. **25** (2008) 085006. C. Stelea, K. Schleich, D. Witt, Phys. Rev. D **78** (2008) 124006. M. Saito, H. Noda, Int. J. Mod. Phys. A **24** (2009) 2357.
J.J. Peng, S.Q. Wu, Nucl. Phys. B **828** (2010) 273. H. Ishihara, J. Soda, Phys. Rev. D **76** (2007) 064022. S.B. Chen, B. Wang, R.K. Su, Phys. Rev. D **77** (2008) 024039. S.W. Wei, R. Li, Y.X. Liu, J.R. Ren, Eur. Phys. J. C **65** (2010) 281. K. Matsuno, K. Umetsu, Phys. Rev. D **83** (2011) 064016. H.L. Li, Europhys. Lett. **92** (2010) 20003.
H.L. Li, Chin. Phys. B **20** (2011) 030402.
M. Kimura, K. Murata, H. Ishihara, J. Soda, Phys. Rev. D **77** (2008) 064015. H. Ishihara, M. Kimura, R.A. Konoplya, K. Murata, J. Soda, A. Zhidenko, Phys. Rev. D **77** (2008) 084019. R. Nishikawa, M. Kimura, Class. Quant. Grav. **27** (2010) 215020. X. He, B. Wang, S.B. Chen, R.G. Cai, C.Y. Lin, Phys. Lett. B **665** (2008) 392. X. He, B. Wang, S.B. Chen, Phys. Rev. D **79** (2009) 084005. Y. Liu, S.B. Chen, J.L. Jing, Phys. Rev. D **81** (2010) 124017. S.B. Chen, Y. Liu, J.L. Jing, Phys. Rev. D **83** (2011) 124019. J. Sadeghi, A. Banijamali, H. Vaez, Astrophys. Space Sci. **343** (2013) 559. K. Matsuno, H. Ishihara, Phys. Rev. D **80** (2009) 104037. R. Mann, C. Stelea, Rhys. Lett. B **634** (2006) 531. D. Astefanesei, E. Radu, Rhys. Rev. D **73** (2006) 044014. P. Nedkova, S. Yazadjiev, Phys. Rev. D **84** (2011) 124040. M. Cvetič, D. Youm, Nucl. Phys. B **476** (1996) 118. H. Elvang, R. Emparan, JHEP **0311** (2003) 035. H. Lü, J.W. Mei, C.N. Pope, Class. Quant. Grav. **27** (2010) 075013. G.T. Horowitz, J.M. Maldacena, A. Strominger, Phys. Lett. B **383** (1996) 151.
[^1]: E-mail: sqwu@phy.ccnu.edu.cn, Corresponding author
|
---
abstract: 'Let $\Lambda \subset \mathbb R^n$ be an algebraic lattice, coming from a projective module over the ring of integers of a number field $K$. Let $\mathcal Z \subset \mathbb R^n$ be the zero locus of a finite collection of polynomials such that $\Lambda \nsubseteq \mathcal Z$ or a finite union of proper full-rank sublattices of $\Lambda$. Let $K_1$ be the number field generated over $K$ by coordinates of vectors in $\Lambda$, and let $L_1,\dots,L_t$ be linear forms in $n$ variables with algebraic coefficients satisfying an appropriate linear independence condition over $K_1$. For each $\varepsilon > 0$ and $\boldsymbol a \in \mathbb R^n$, we prove the existence of a vector $\boldsymbol x \in \Lambda \setminus \mathcal Z$ of explicitly bounded sup-norm such that $$\| L_i(\boldsymbol x) - a_i \| < \varepsilon$$ for each $1 \leq i \leq t$, where $\|\ \|$ stands for the distance to the nearest integer. The bound on sup-norm of $\boldsymbol x$ depends on $\varepsilon$, as well as on $\Lambda$, $K$, $\mathcal Z$ and heights of linear forms. This presents a generalization of Kronecker’s approximation theorem, establishing an effective result on density of the image of $\Lambda \setminus \mathcal Z$ under the linear forms $L_1,\dots,L_t$ in the $t$-torus $\mathbb R^t/\mathbb Z^t$.'
address:
- 'Department of Mathematics, 850 Columbia Avenue, Claremont McKenna College, Claremont, CA 91711'
- 'Steklov Mathematical Institute of the Russian Academy of Sciences, Gubkina 8, 119991, Moscow, Russia'
author:
- Lenny Fukshansky
- Nikolay Moshchevitin
bibliography:
- 'kronecker.bib'
title: 'On an effective variation of Kronecker’s approximation theorem avoiding algebraic sets'
---
[^1]
[^2]
Å[[A]{}]{} Ł[[L]{}]{} Ø[[O]{}]{} [H]{} [c]{}
Introduction {#intro}
============
Let $1,\theta_1,\dots,\theta_t$ be $\que$-linearly independent real numbers. The classical approximation theorem of Kronecker then states that the set of points $$\left\{ (\{ n\theta_1 \}, \dots,\{ n\theta_t \} ) : n \in \zed \right\}$$ is dense in the $t$-torus $\real^t/\zed^t$, where $\{ \cdot \}$ stands for the fractional part of a real number. This result was originally obtained by Kronecker [@kronecker] in 1884, and presents a deep generalization of Dirichlet’s 1842 theorem on Diophantine approximation [@dirichlet]; see, for instance, [@hardy] for a detailed exposition of these classical results.
Kronecker’s theorem can also be viewed as a statement on density of the image of the integer lattice under collection of linear forms in the torus $\real^t/\zed^t$ (compare to the famous Oppenheim conjecture for quadratic forms). Specifically, if $L_1,\dots,L_t$ are linear forms in $n$ variables with real coefficients $b_{ij}$ so that the set of numbers $1$ and $b_{ij}$ are linearly independent over $\que$, then for any $\eps >0$ and $\ba \in \real^t$ there exists $\bx \in \zed^n$ such that $$\label{L-kron}
\| L_i(\bx) - a_i \| < \eps\ \forall\ 1 \leq i \leq t,$$ where $\|\ \|$ stands for the distance to the nearest integer. A nice survey of a wide variety of results related to Kronecker’s theorem is given in [@gonek]. Classical quantitative results in this direction are related to transference theorems for homogeneous and inhomogeneous approximation for the system of linear forms $L_i(\bx)$ (see [@jarnik], Chapter V of [@cass:dioph], [@bl]). In particular, these results give effective bounds for the size of the coordinates of the vector $\bx$ in under the assumption that there are effective lower bounds for $\max_i \| L_i(\bx) \|$ in the homogeneous case. Some additional effective results can also be found in [@malajovich], [@vorselen].
The main goal of this note is somewhat different. We consider linear forms with algebraic coefficients and extend the previously known versions of Kronecker’s theorem in three ways:
1. allow for the approximating vector $\bx$ as in the equation above to come from an algebraic lattice $\Lambda$,
2. exclude vectors from a prescribed union $\Z$ of projective varieties or sublattices not containing this lattice, that is we are interested in approximation vectors $\bx \in \Lambda \setminus \Z$,
3. we obtain effective constants everywhere in our upper bounds.
Effective Diophantine avoidance results, exhibiting solutions to a given problem outside of a prescribed algebraic set can be viewed as statements on distribution of such solutions: not only do small solutions exist, they are also sufficiently well distributed so that it is not possible to “cut them out" by any finite union of varieties. In the recent years, such results were obtained in the general context of Siegel’s lemma (also generalizing Faltings’ version of Siegel’s lemma [@faltings], [@faltings:siegel], [@faltings:siegel_1]) in [@lf:hyper], [@lf:siegel], [@lf:null], [@gaudron], [@gaudron:remond], [@martin], and in the context of Cassels’ theorem on small zeros of quadratic forms and its generalizations in [@lf:smallzeros], [@cfh], [@me_glenn], [@gaudron:remond-1]. We will extend these investigations to Kronecker’s theorem. To obtain effective constants in our bounds we use Liouville-type inequalities (see Remark \[schmidt\] below for stronger non-effective inequalities of similar type, which can be derived from Schmidt’s Subspace Theorem). To give precise statements of our results, we need some notation.
[*1. The lattice.*]{} Let $n \geq 1$ be an integer, and for each vector $\bx \in \real^n$ define the sup-norm $$|\bx| := \max_{1 \leq i \leq n} |x_i|.$$ Let $K$ be a number field of degree $d=r_1+2r_2$ over $\que$, where $r_1$ and $r_2$ are numbers of its real and complex places, respectively, and write $\O_K$ for its ring of integers. Let $1 \leq s \leq w$ be integers, and let $\M \subset K^w$ be an $\O_K$-module such that $\M \otimes_K K \cong K^s$. Write $\D_K(\M)$ for the discriminant of $\M$. Define $\UU_K(\M)$, a fractional $\O_K$-ideal in $K$, to be $$\label{UKM}
\UU_K(\M) = \left\{ \alpha \in K : \alpha\M \subseteq \O_K^w \right\}.$$ We let $\Lambda_K(\M) \subset \real^{wd}$ be the lattice of rank $sd$, which is the image of $\M$ under the standard Minkowski embedding.
[*2. The projective varieties.*]{} Let $m \geq 1$ be an integer. For each $1 \leq i \leq m$, let $\s_i$ be a finite set of homogeneous polynomials in $\real[x_1, \ldots, x_{wd}]$ and $Z(\s_i)$ be its zero set in $\real^{wd}$, that is, $$Z(\s_i) = \{\bx \in \real^{wd} : P(\bx) = 0 \mbox{ for all } P \in \s_i\}.$$ For the collection $\bS: = \{\s_1, \ldots, \s_m\}$ of finite sets of homogeneous polynomials, define $$\label{Z_set}
\Z_{\bS} := \bigcup_{i=1}^m Z(\s_i),$$ and $$\label{def_M}
M_{\bS} := \sum_{i=1}^m \max \{\deg P : P \in \s_i\}.$$ We allow for the possibility that $\Z_{\bS} = \{ \bo \}$, in which case we take instead $M_{\bS} = 1$. Notice that $\Z_{\bS}$ is an algebraic set, which is a union of a finite collection of projective varieties. Assume that the lattice $\Lambda_K(\M)$ is not contained in the set $\Z_{\bS}$.
[*3. The linear forms.*]{} Let $K_1 = K(\Lambda_K(\M))$, i.e. $K_1$ is the number field generated over $K$ by the entries of any basis matrix of the lattice $\Lambda_K(\M)$. Let $B:=(b_{ij})_{1 \leq i \leq t, 1 \leq j \leq wd}$ be a $t \times wd$ matrix with real algebraic entries so that $1,b_{11},\dots,b_{t(wd)}$ are linearly independent over $K_1$, and let $\ell = [E : \que]$ where $E = K_1(b_{11},\dots,b_{t(wd)})$. We will also write $\ell_v = [E_v : \que_v]$ for the local degree of $E$ at every place $v \in M(E)$. Define $t$ linear forms in $wd$ variables $$\label{L_form}
L_i(x_1,\dots,x_{wd}) = \sum_{j=1}^{wd} b_{ij} x_j \in \real[x_1,\dots,x_{wd}]\ \forall\ 1 \leq i \leq t.$$ Our first goal here is to prove the following effective result on density of the image of the set $\Lambda_K(\M) \setminus \Z_{\bS}$ under the linear forms $L_1,\dots,L_t$ in the torus $\real^t / \zed^t$. Let $h$ denote the usual Weil height on algebraic numbers, as well as its extension to vectors with algebraic coordinates; we recall the definition of height along with other necessary notation in Section \[setup\].
\[kron1\] Let $\ba = (a_1,\dots,a_t) \in \real^t$ and $\eps > 0$. There exist $\bx \in \Lambda_K(\M) \setminus \Z_{\bS}$ and $\bp \in \zed^t$ such that $$\left| L_i(\bx) - a_i - p_i \right| < \eps$$ and $$|\bx| \leq \a_K(t,\ell,s) \left( sd M_{\bS} |\D_K(\M)|^{\frac{s}{2}} \right)^{\kk + 1} \left( (wd)^{\frac{3}{2}} h(B) \right)^{\kk} \c_K(\M,\ell,t)\ \eps^{-\ell+1},$$ where the exponent $\kk = \ell^2(t+1)-\ell$ and the constants are $$\a_K(t,\ell,s) = 2^{\ell t (\ell-1) + sr_1 \kk + \frac{sd-1}{2}} (t+1)^{3\ell-1} (t!)^{2\ell}$$ and $$\c_K(\M,\ell,t) = \min \left\{ h(\alpha)^{(\kk+1)sd-1} h(\alpha^{-1})^{\kk} \ :\ \alpha \in \UU_K(\M) \right\}.$$
One special case of Theorem \[kron1\] is when $\Z_{\bS}$ is a union of linear spaces, which means that the point $\bx$ in question is in $\Lambda_K(\M)$ but outside of a union of sublattices of smaller rank than $\Lambda_K(\M)$. What if the rank of such sublattices is equal to the rank of $\Lambda_K(\M)$? The next theorem addresses this situation.
\[kron2\] Let $\ba = (a_1,\dots,a_t) \in \real^t$ and $\eps > 0$. Let $m > 0$ and $\Gamma_1,\dots,\Gamma_m \subset \Lambda_K(\M)$ be proper sublattices of full rank and respective determinants $\D_1,\dots,\D_m$, and let $\D=\D_1 \cdots \D_m$. Then for every $\alpha \in \UU_K(\M)$ there exist $\bx \in \Lambda_K(\M) \setminus \bigcup_{i=1}^m \Gamma_i$ and $\bp \in \zed^t$ such that $$\left| L_i(\bx) - a_i - p_i \right| < \eps$$ and $$|\bx| \leq \left( \b_K(t,\ell,s,w) \left( h(\alpha) h(\alpha^{-1}) h(B) \E_{\alpha} \right)^{\kk} \frac{\D\ \eps^{-\ell+1}}{|\D_K(\M)|^{\frac{sm}{2}}} + 1 \right) \E_{\alpha},$$ where the exponent $\kk = \ell^2(t+1)-\ell$, as in Theorem \[kron1\], the constant $$\b_K(t,\ell,s,w) = 2^{\ell t (\ell-1) + \frac{\kk}{2} + smr_2} (t+1)^{3\ell-1} (t!)^{2\ell} (wd)^{\frac{3\kk}{2}} ,$$ and $\E_{\alpha} =$ $$\label{E-bnd}
\E_{\alpha}(\M,\Gamma_1,\dots,\Gamma_m) := 2^{\frac{sr_1-1}{2}} h(\alpha)^{sd-1} |\D_K(\M)|^{\frac{s}{2}} \left( \sum_{i=1}^m \frac{\D}{\D_i} - m + 1 \right) + \D^{\frac{1}{sd}}.$$
\[simpler\] The bounds of Theorems \[kron1\] and \[kron2\] can be recorded in a slightly weaker simplified form as $$|\bx| \ll \left( \det \Lambda_K(\M) \right)^{\kk + 1} h(B)^{\kk} \c_K(\M,\ell,t)\ \eps^{-\ell+1}$$ and $$|\bx| \ll \left( \sum_{i=1}^m \frac{\D}{\D_i} \right)^{\kk+2} ( \det \Lambda_K(\M) )^{\kk-m+1} h(B)^{\kk} \c_K(\M,\ell,t)\ \eps^{-\ell+1},$$ respectively, where the constants in the $\ll$ Vinogradov notation depend on the number field $K$ and the integer parameters $t, \ell, s, m$. The expression $\c_K(\M,\ell,t)$ can be viewed as a certain measure of arithmetic complexity of $\M$; in particular, if $\M \subseteq \O_K^w$, then $\c_K(\M,\ell,t) = 1$.
Here is a sketch of the proofs of Theorems \[kron1\] and \[kron2\]. We first construct a point $\bwy \in \Lambda_K(\M)$ of controlled sup-norm, which is outside of $\Z_{\bS}$ or $\bigcup_{i=1}^m \Gamma_i$, respectively: in the first case, we use the classical Minkowski’s Successive Minima Theorem and a version of Alon’s Combinatorial Nullstellensatz [@alon] (we use the convenient formulation developed in [@null]), while in the second we employ a recent result of Henk and Thiel [@martin] on points of small norm in a lattice outside of a union of full-rank sublattices. We use $\bwy$ to construct an infinite sequence of points $n \bwy$ satisfying the above conditions, and use an effective version of Kronecker’s original theorem to obtain a value of the index $n$ (depending on $\eps > 0$) for which the required inequalities on values of linear forms are satisfied. In other words, our avoidance strategy is to follow the line $n\bwy$ until a necessary point is found. One may wish to use a similar strategy, but following a higher dimensional subspace of the ambient space in the hope of a better bound, however it is difficult to guarantee avoiding our fixed algebraic set with such strategy. A convenient effective version of Kronecker’s theorem that we use is worked out in Section \[kron\]. It should be remarked that the most important feature of approximation results such as our Theorems \[kron1\] and \[kron2\] is the exponent on $\eps$ in the bounds for $|\bx|$. As we show, this exponent is the same as in the corresponding bound of the effective version of Kronecker’s theorem that we use.
In Section \[setup\] we introduce the necessary notation and provide all the details of our setup. We derive an effective version of Kronecker’s theorem in Section \[kron\]. We then prove Theorem \[kron1\] in Section \[proof-kron\] and Theorem \[kron2\] in Section \[sublattices\].
Notation and setup {#setup}
==================
Let the notation be as in Section \[intro\]. Here we introduce some additional notation needed for our algebraic setup. Let the number field $K$ have discriminant $\D_K$, $r_1$ real embeddings $\sigma_1,\dots,\sigma_{r_1}$ of $K$, and $r_2$ conjugate pairs of complex embeddings $\tau_1,\taubar_1,\dots,\tau_{r_2},\taubar_{r_2}$, then $d=r_1+2r_2$. For each $\tau_k$, write $\Re(\tau_k)$ for its real part and $\Im(\tau_k)$ for its imaginary part. Let us write $M(K)$ for the set of all places of $K$, then the archimedean places of $K$ are in correspondence with the embeddings of $K$, and we choose the absolute values $|\ |_{v_1},\dots,|\ |_{v_{r_1+r_2}}$ so that for each $a \in K$ $$|a|_{v_k} = |\sigma_k(a)|\ \forall\ 1 \leq k \leq r_1$$ and $$|a|_{v_{r_1+k}} = |\tau_k(a)| = \sqrt{ \Re(\tau_k(a))^2 + \Im(\tau_k(a))^2}\ \forall\ 1 \leq k \leq r_2,$$ where $|\ |$ stands for the usual absolute value on $\real$ or $\cee$, respectively. For each $v \in M(K)$, we write $K_v$ for the completion of $K$ at $v$, and for each $n \geq 1$ we define a local norm $|\ |_v : K_v^n \to \real$ by $$|\ba|_v := \max_{1 \leq j \leq n} |a_j|_v,$$ for each $\ba = (a_1,\dots,a_n) \in K_v^n$. Then the extended Weil height on $K^n$ is given by $$h(\ba) = \prod_{v \in M(K)} \max \{ 1, |\ba|_v \}^{d_v/d},$$ where $d_v = [K_v : \que_v]$ is the local degree of $K$ at $v$, so that $\sum_{v \mid u} d_v = d$ for each $u \in M(\que)$.
For each integer $n \geq 1$, define the standard Minkowski embedding $\rho^n_K : K^n \to \real^{nd}$ by $$\rho^n_K(\ba) := \left( \sigma^n_1(\ba),\dots,\sigma^n_{r_1}(\ba),\Re(\tau^n_1(\ba)), \Im(\tau^n_1(\ba)),\dots,\Re(\tau^n_{r_2}(\ba)),\Im(\tau^n_{r_2}(\ba)) \right).$$ We will now use Minkowski embedding to construct lattices from $\O_K$-modules and outline some of their main properties; see [@me_glenn] for further details. Let $1 \leq s \leq w$ be integers, and let $\M \subset K^w$ be an $\O_K$-module such that $\M \otimes_K K \cong K^s$. By the structure theorem for finitely generated projective modules over Dedekind domains (see, for instance [@lang]), $$\M = \left\{ \sum_{j=1}^s \beta_j \bwy_j : \bwy_j \in \O_K^w,\ \beta_j \in \I_j \right\}$$ for some $\O_K$-fractional ideals $\I_1,\dots,\I_s$ in $K$. By Proposition 13 on p.66 of [@lang], the discriminant of $\M$ is then $$\label{module_disc}
\D_K(\M) := \D_K \prod_{j=1}^s \Nn(\I_j)^2,$$ where $\Nn(\I_j)$ is the norm of the fractional ideal $\I_j$.
Let $\Lambda_K(\M) := \rho^w_K(\M)$ be an algebraic lattice of rank $sd$ in $\real^{wd}$, then a direct adaptation of Lemma 2 on p.115 of [@lang] implies that the determinant of $\Lambda_K(\M)$ is $$\label{det_lkm}
\det(\Lambda_K(\M)) = 2^{-sr_2} |\D_K(\M)|^{\frac{s}{2}} = 2^{-sr_2} |\D_K|^{\frac{s}{2}} \prod_{j=1}^s \Nn(\I_j),$$ where the last identity follows by above. Let $\bx \in \Lambda_K(\M)$, then $\bx = \rho^w_K(\ba)$ for some $\ba \in \M$ and $$\label{min_bnd}
|\bx| \geq \frac{1}{\sqrt{2}} h(\alpha)^{-1},$$ for any $\alpha \in \UU_K(\M)$ by inequality (54) of [@me_glenn]. Let $v \in M(K)$ be an archimedean place, and assume first that it corresponds to a real embedding $\sigma_j$ for some $1 \leq j \leq r_1$, then $|\ba|_v = |\bx|$. On the other hand, if $v$ corresponds to a complex embedding $\tau_j$ for some $1 \leq j \leq r_2$, then $|\ba|_v \leq \left( \sum_{j=1}^{wd} x_j^2 \right)^{1/2} \leq \sqrt{wd}\ |\bx|$. Hence for each $v \mid \infty$, $$\label{max_bnd}
|\bx| \leq |\ba|_v \leq \sqrt{wd}\ |\bx|.$$
Let $L_1,\dots,L_t$ be the linear forms defined in . For each $1 \leq i \leq t$, we define $$|L_i|_v = \max_{1 \leq j \leq wd} |b_{ij}|_v,$$ for each place $v \in M(E)$, and define the height of $L_i$ to be $$h(L_i) = h(b_{i1},\dots,b_{i(wd)}) = \prod_{v \in M(E)} \max \{ 1, |L_i|_v \}^{\ell_v/\ell}.$$ We similarly define the height of the matrix $B$ to be $$h(B) = h(b_{11},\dots,b_{t(wd)}),$$ then $h(L_i) \leq h(B)$ for all $1 \leq i \leq t$. We are now ready to proceed.
An effective version of Kronecker’s theorem {#kron}
===========================================
In this section we derive an effective version of Kronecker’s theorem, which we then use to prove Theorems \[kron1\] and \[kron2\]. Similar to the setup in the beginning of Section \[intro\], let $1,\theta_1,\dots,\theta_t$ be $\que$-linearly independent real algebraic numbers. For each $1 \leq j \leq t$, let $f_j(x) \in \zed[x]$ be the minimal polynomial of $\theta_j$ of degree $d_j$, $|f_j|$ be the maximum of absolute values of the coefficients of $f_j$, and $A_j$ be the leading coefficient of $f_j$, so $A_j \leq |f_j|$. By Lemma 3.11 of [@waldschmidt], $$\frac{1}{2^{d_j}}\ |f_j| \leq h(\theta_j)^{d_j} \leq \sqrt{d_j + 1}\ |f_j|,$$ for every $1 \leq j \leq t$. Define $A$ to be the least common multiple of $A_1,\dots,A_t$, so $$\label{A_bnd}
A \leq \prod_{j=1}^t |f_j| \leq \prod_{j=1}^t (2 h(\theta_j))^{d_j}.$$
Let $F=\que(\theta_1,\dots,\theta_t)$ be a number field of degree $e \geq t+1$, then $e \leq \prod_{j=1}^t d_j$. Let $\theta_{t+1},\dots,\theta_{e-1} \in F$ be such that $$1 = \theta_0,\theta_1,\dots,\theta_t,\theta_{t+1},\dots,\theta_{e-1}$$ form a $\que$-basis for $F$. Let $\sigma_1,\dots,\sigma_e$ be the embeddings of $F$ into $\cee$. We recall Liouville inequality. For any $\bm = (m_0,\dots,m_t,0,\dots,0) \in \zed^e$, $$\label{L1}
A^e \prod_{i=1}^e \left| \sum_{j=0}^{e-1} \sigma_i(\theta_j) m_j \right| \geq 1,$$ and so $$\label{L2}
A^e \left( (t+1) \max_{1 \leq i \leq e, 0 \leq j \leq t} |\sigma_i(\theta_j)| \right)^{e-1} |\bm|^{e-1} \| m_1 \theta_1 + \dots + m_t \theta_t \| \geq 1.$$ Now observe that $$\max_{1 \leq i \leq e, 0 \leq j \leq t} |\sigma_i(\theta_j)| \leq \max_{1 \leq j \leq t} h(\theta_j)^{d_j},$$ and so define $$\label{const-c1}
\C_1 = \C_1(\theta_1,\dots,\theta_t) := \left( (t+1) \max_{1 \leq j \leq t} h(\theta_j)^{d_j} \right)^{e-1} \prod_{j=1}^t (2 h(\theta_j))^{ed_j}.$$ Then for any $\bo \neq \bm \in \zed^t$, $$\label{mth}
\| m_1 \theta_1 + \dots + m_t \theta_t \| \geq \C_1^{-1} |\bm|^{-e+1}.$$ We will now apply a transference homogeneous-inhomogeneous argument. A transference principle of this sort was first described in Chapter V, §4 of [@cass:dioph]; the particular stronger result we are applying here is obtained in [@bl]. Let us write $$M(\bwy) = \sum_{i=1}^t \theta_i y_i$$ for $\bwy = (y_1,\dots,y_t) \in \zed^t$, and let $$L_j(x) = \theta_j x,\ 1 \leq j \leq t$$ for $x \in \zed$. Then guarantees that for any $\bo \neq \bwy \in \zed^t$ with $|\bwy| \leq Y$, $$\| M(\bwy) \| \geq \C_1^{-1} Y^{-(e-1)}.$$ Now applying the transference Lemma 3 of [@bl] to these linear forms, we have that for every $\ba = (a_1,\dots,a_t) \in \real^t$ there exists $x \in \zed$ such that $|x| \leq 2^{-t} ((t+1)!)^2 \C_1 Y^{e-1}$ and $$\max_{1 \leq j \leq t} \| L_j(x) - a_j \| \leq 2^{-t} ((t+1)!)^2 Y^{-1}.$$ Letting $Q = \left( 2^{t} ((t+1)!)^{-2} Y \right)^{e-1}$, we obtain that $$\max_{1 \leq j \leq t} \| L_j(x) - \alpha_j \| \leq Q^{-\frac{1}{e-1}}$$ for some $0 \neq x \in \zed$ with $|x| \leq 2^{-et} ((t+1)!)^{2e} \C_1 Q$. Taking $\eps = Q^{-\frac{1}{e-1}}$ immediately yields the following effective version of Kronecker’s theorem.
\[KR\] Let $1,\theta_1,\dots,\theta_t$ be $\que$-linearly independent real algebraic numbers, and let $e = [\que(\theta_1,\dots,\theta_t) : \que]$. Let $\C_1$ be given by above, and let $\eps > 0$. Then for any $(a_1,\dots,a_t) \in \real^t$ there exists $q \in \zed \setminus \{0\}$ such that $$\label{qtheta}
\|q \theta_j - a_j \| \leq \eps,\ 1 \leq j \leq t$$ and $$|q| \leq 2^{-et} ((t+1)!)^{2e} \C_1 \eps^{-e+1}.$$ In particular, if $h(\theta_j) \leq H$ for all $1 \leq j \leq t$ and $\max \{ e, d_1,\dots,d_t \} \leq \ell$, then $$|q| \leq \left( 2^{\ell t (\ell-1)} (t+1)^{3\ell-1} (t!)^{2\ell} H^{\ell^2(t+1)-\ell} \right) \eps^{-\ell+1}.$$
\[schmidt\] Stronger non-effective results can be derived as corollaries of Schmidt’s Subspace Theorem. For instance, results discussed in Chapter 6, §2 of [@schmidt1980] together with the transference principles of Chapter V, §4 of [@cass:dioph] and [@bl] imply, for any $\eps > 0$ and $\ba \in \real^t$ under the assumptions of Theorem \[KR\], the existence of $q \in \zed$ satisfying \[qtheta\] such that $$|q| \leq \C'(\delta) \eps^{-t-\delta},$$ for any $\delta > 0$, where the constant $\C'(\delta)$ is non-effective. This would result in the same exponent on $\eps$ in the bounds for $|q|$ in Theorems \[kron1\] and \[kron2\], but with non-effective constants.
Proof of Theorem \[kron1\] {#proof-kron}
==========================
Here we present the proof of our first result. Since $\Lambda_K(\M) \nsubseteq \Z_{\bS}$, $\Lambda_K(\M) \nsubseteq Z(\s_i)$ for all $1 \leq i \leq m$, and so for each $i$ at least one polynomial $P_i$ in $\s_i$ is not identically zero on $\Lambda_K(\M)$. Clearly for each $1 \leq i \leq m$, $$Z(\s_i) \subseteq Z(P_i) := \left\{ \bx \in \real^{wd} : P_i(\bx) = 0 \right\}.$$ Define $$P(\bx) = \prod_{i=1}^m P_{i}(\bx),$$ so that $\Z_{\bS} \subseteq Z(P)$ and $\deg(P) \leq M_{\bS}$, while $\Lambda_K(\M) \nsubseteq Z(P)$. Indeed, $Z(P)$ is the union of hypersurfaces $Z(P_1),\dots,Z(P_m)$, and a lattice cannot be covered by a finite union of hypersurfaces unless it is contained in one of them. We will next construct a point $\bwy \in \Lambda_K(\M)$ of controlled sup-norm such that $P(\bwy) \neq 0$.
Let $V = \spn_{\real} \Lambda_K(\M)$ be the $sd$-dimensional subspace of $\real^{wd}$ spanned by the lattice $\Lambda_K(\M)$. For a positive real number $\mu$, let us write $$C_V(\mu) := \left\{ \bx \in V : |\bx| \leq \mu \right\}$$ for the $sd$-dimensional cube with side-length $2\mu$ centered at the origin in $V$, so $C_V(\mu) = \mu C_V(1)$. Let $0 < \lambda_1 \leq \lambda_2 \leq \dots \leq \lambda_{sd}$ be the successive minima of $\Lambda_K(\M)$ with respect to the cube $C_V(1)$. In other words, for each $1 \leq i \leq sd$, $$\lambda_i := \min \left\{ \mu \in \real_{>0} : \dim_{\real} \spn_{\real} \left( \Lambda_K(\M) \cap C_V(\mu) \right) \geq i \right\}.$$ Let $\bv_1,\dots,\bv_{sd}$ be a collection of linearly independent vectors in $\Lambda_K(\M)$ corresponding to these successive minima, then $|\bv_i| = \lambda_i$. Since the volume of $sd$-dimensional cube $C_V(1)$ is $2^{sd}$, Minkowski’s Successive Minima Theorem (see, for instance, [@cass:geom] or [@gruber_lek]) implies that $$\frac{\det(\Lambda_K(\M))}{(sd)!} \leq \prod_{i=1}^{sd} |\bv_i| \leq \det(\Lambda_K(\M)),$$ where $\frac{1}{\sqrt{2}} h(\alpha)^{-1} \leq |\bv_1| \leq \dots \leq |\bv_{sd}|$, by . This means that $$\label{vi_bnd}
|\bv_1| \leq \dots \leq |\bv_{sd}| \leq \left( \sqrt{2} h(\alpha) \right)^{sd-1} \det(\Lambda_K(\M)).$$ Let $I(M_{\bS}) = \{ 0,1,2,\dots,M_{\bS} \}$ be the set of the first $M_{\bS}+1$ non-negative integers. For each $\bxi \in I(M_{\bS})^{sd}$, define $$\bv(\bxi) = \sum_{i=1}^{sd} \xi_i \bv_i,$$ then $$\label{v_xi_bnd}
|\bv(\bxi)| = \max_{1 \leq j \leq wd} \left| \sum_{i=1}^{sd} \xi_i v_{ij} \right| \leq sd |\bxi| |\bv_{sd}| \leq sd M_{\bS} \left( \sqrt{2} h(\alpha) \right)^{sd-1} \det(\Lambda_K(\M)),$$ by . Assume that $P(\bv(\bxi)) = 0$ for each $\bxi \in I(M_{\bS})^{sd}$. Then Theorem 4.2 of [@null] implies that $P(\bx)$ must be identically zero on $V$, which would contradict the fact that $P$ does not vanish identically on $\Lambda_K(\M)$. Hence there must exist some $\bxi \in I(M_{\bS})^{sd}$ such that $P$ does not vanish at the corresponding $\bwy := \bv(\bxi)$, and $|\bwy| \leq sd M_{\bS} \left( \sqrt{2} h(\alpha) \right)^{sd-1} \det(\Lambda_K(\M))$ by . Since $P(\bx)$ is a homogeneous polynomial, it must be true that $P(n \bwy) \neq 0$ for every $n \in \zed_{>0}$. On the other hand, by our construction $$n \bwy = n \sum_{i=1}^{sd} \xi_i \bv_i \in \spn_{\zed} \left\{ \bv_1,\dots,\bv_{sd} \right\} \subseteq \Lambda_K(\M),$$ and so $\left\{ n \bwy \right\}_{n \in \zed_{>0}}$ gives an infinite sequence of points in $\Lambda_K(\M)$ outside of $\Z_{\bS}$. For each such point, we have $$L_i(n \bwy) = n L_i(\bwy),\ \forall\ 1 \leq i \leq t.$$ Let us define, for each $1 \leq i \leq t$, $$\label{theta-i}
\theta_i := L_i(\bwy) = \sum_{j=1}^{wd} b_{ij} y_j \neq 0,$$ since $y_j \in K_1$, not all zero, and $b_{ij}$ are $K_1$-linearly independent. Notice that $\theta_1,\dots,\theta_t \in E$, and hence all of them are algebraic numbers of degree $\leq \ell$.
Let $\alpha \in \UU_K(\M)$. Then, by , for each archimedean $v \in M(E)$, $$\begin{aligned}
\label{th-1}
\max \{1, |\theta_i|_v \} & \leq & \max \{ 1, (wd)^{\frac{3}{2}} |L_i|_v |\bwy| \} \leq (wd)^{\frac{3}{2}} \max \{ 1, |\bwy| \} \max \{ 1, |L_i|_v \} \nonumber \\
& \leq & \sqrt{2}\ (wd)^{\frac{3}{2}} h(\alpha) |\bwy| \max \{ 1, |L_i|_v \},\end{aligned}$$ by . By , $|\bwy| \leq sd M_{\bS} \left( \sqrt{2} h(\alpha) \right)^{sd-1} \det(\Lambda_K(\M))$, and hence $$\label{theta_bnd}
\max \{1, |\theta_i|_v \} \leq sd (wd)^{\frac{3}{2}} M_{\bS} \left( \sqrt{2} h(\alpha) \right)^{sd} \det(\Lambda_K(\M)) \max \{ 1, |L_i|_v \}.$$ Now suppose $v \in M(E)$ is non-archimedean. Then $\alpha y_j$ is an algebraic integer for each $1 \leq j \leq wd$, and hence $|\alpha y_j|_v = |\alpha|_v |y_j|_v \leq 1$, meaning that $$\max \{ 1, |y_1|_v,\dots,|y_{wd}|_v \} \leq \max \{ 1, |\alpha|^{-1}_v \}.$$ Then $$\begin{aligned}
\label{th-2}
\max \{1, |\theta_i|_v \} & \leq & \max \{1, |L_i|_v \} \max \{ 1, |y_1|_v,\dots,|y_{wd}|_v \} \nonumber \\
& \leq & \max \{ 1, |\alpha^{-1}|_v \} \max \{1, |L_i|_v \},\end{aligned}$$ for each non-archimedean $v \in M(E)$. Taking a product over all places of $E$, we obtain: $$\begin{aligned}
h(\theta_i) & = & \prod_{v \in M(E)} \max \{1, |\theta_i|_v \}^{\frac{\ell_v}{\ell}} = \left( \prod_{v \mid \infty} \max \{1, |\theta_i|_v \}^{\ell_v} \times \prod_{v \nmid \infty} \max \{1, |\theta_i|_v \}^{\ell_v} \right)^{\frac{1}{\ell}} \\
& \leq & sd (wd)^{\frac{3}{2}} M_{\bS} \left( \sqrt{2} h(\alpha) \right)^{sd} \det(\Lambda_K(\M)) h(L_i) \prod_{v \nmid \infty} \max \{ 1, |\alpha^{-1}|_v \}^{\frac{\ell_v}{\ell}} \\
& \leq & sd (wd)^{\frac{3}{2}} M_{\bS} \left( \sqrt{2} h(\alpha) \right)^{sd} h(\alpha^{-1}) \det(\Lambda_K(\M)) h(L_i).\end{aligned}$$ Recalling that $h(L_i) \leq h(B)$ for all $1 \leq i \leq t$, we obtain $$\label{hti}
h(\theta_i) \leq 2^{\frac{sd}{2}} sd (wd)^{\frac{3}{2}} M_{\bS} h(\alpha)^{sd} h(\alpha^{-1}) \det(\Lambda_K(\M)) h(B),$$ for each $1 \leq i \leq t$, where the choice of $\alpha \in \UU_K(\M)$ is arbitrary.
We will now show that $1,\theta_1,\dots,\theta_t$ are $\que$-linearly independent. Suppose not, then there exist $c_0,c_1,\dots,c_t \in \que$, not all zero, such that $$c_0 = \sum_{i=1}^t c_i \theta_i = \sum_{i=1}^t \sum_{j=1}^{wd} c_i y_j b_{ij},$$ where not all $c_i y_j$ are equal to zero. Recall that $\bwy \in \Lambda_K(\M)$, meaning that coordinates of $\bwy$ are in $K_1$, hence all $c_i y_j$ are in $K_1$. This contradicts the assumption that $1,b_{11},\dots,b_{1(wd)}$ are linearly independent over $K_1$. Hence $1,\theta_1,\dots,\theta_t$ must be linearly independent over $\que$.
Now let $\ba = (a_1,\dots,a_t) \in \real^t$ and $\eps > 0$, as in the statement of our theorem. Then, by and Theorem \[KR\], there exists $q \in \zed$ and $\bp \in \zed^t$ such that $$\begin{aligned}
\label{q_bnd}
|q| & \leq & 2^{\ell t (\ell-1)} (t+1)^{3\ell-1} (t!)^{2\ell} \times \nonumber \\
& & \times \left( 2^{\frac{sd}{2}} sd (wd)^{\frac{3}{2}} M_{\bS} h(\alpha)^{sd} h(\alpha^{-1}) \det(\Lambda_K(\M)) h(B) \right)^{\ell^2(t+1)-\ell} \eps^{-\ell+1}\end{aligned}$$ and $$\left| q \theta_i - a_i - p_i \right| < \eps\ \forall\ 1 \leq i \leq t.$$ Letting $\bx = q\bwy$, we see that $q\theta_i = L_i(\bx)$ for each $1 \leq i \leq t$ and $|\bx| = |q| |\bwy|$. Combining these observations with , and and taking a minimum over all $\alpha \in \UU_K(\M)$ finishes the proof of the theorem.
Proof of Theorem \[kron2\] {#sublattices}
==========================
Let $\Gamma_1,\dots,\Gamma_m$ be full-rank sublattices of $\Lambda_K(\M)$ of respective determinants $\D_1,\dots,\D_m$. Let $\Omega = \cap_{i=1}^m \Gamma_i$, then $\Omega$ also has full rank and $$\D := \D_1 \cdots \D_m \geq \det \Omega.$$ We write $\lambda_i$ for the successive minima of $\Lambda_K(\M)$ and $\lambda_i(\Omega)$ for the successive minima of $\Omega$. Theorem 1.2 of [@martin] implies that there exists $\bwy \in \Lambda_K(\M) \setminus \bigcup_{i=1}^m \Gamma_i$ such that $$|\bwy| < \frac{\det \Lambda_K(\M)}{\lambda_1(\Omega)^{sd-1}} \left( \sum_{i=1}^m \frac{\D}{\D_i} - m + 1 \right) + \lambda_1(\Omega).$$ Our first goal is to make this bound more explicit in terms of the parameters of $\M$. First notice that by Minkowski’s Successive Minima Theorem, $$\lambda_1(\Omega) \leq \left( \prod_{i=1}^{sd} \lambda_i(\Omega) \right)^{1/sd} \leq \left( \det \Omega \right)^{1/sd} \leq \D^{1/sd}.$$ We also need a lower bound on $\lambda_1(\Omega)$. Observe that $\lambda_1(\Omega) \geq \lambda_1$, while $\lambda_1 \geq \frac{1}{\sqrt{2}} h(\alpha)^{-1}$ for any $\alpha \in \UU_K(\M)$, by above. Putting these estimates together, we see that $$\label{y_O_bnd}
|\bwy| < \left( \sqrt{2} h(\alpha) \right)^{sd-1} \det \Lambda_K(\M) \left( \sum_{i=1}^m \frac{\D}{\D_i} - m + 1 \right) + \D^{1/sd}$$ for any $\alpha \in \UU_K(\M)$.
Since $\bwy \in \Lambda_K(\M)$ and $|\Lambda_K(\M) : \Gamma_i| = \D_i/\det \Lambda_K(\M)$ for each $1 \leq i \leq m$, it follows that $$\left( g |\Lambda_K(\M) : \Gamma_i| \right) \bwy = \frac{g \D_i}{\det \Lambda_K(\M)} \bwy \in \Gamma_i,$$ for every $g \in \zed$, and hence $$\left( \frac{g \D_1 \cdots \D_m}{\left( \det \Lambda_K(\M) \right)^m} \right) \bwy = \left( \frac{g \D}{\left( \det \Lambda_K(\M) \right)^m} \right) \bwy \in \Omega,$$ for every $g \in \zed$. Therefore, it must be true that $$\left( \frac{g \D}{\left( \det \Lambda_K(\M) \right)^m} + 1 \right) \bwy \in \Lambda_K(\M) \setminus \bigcup_{i=1}^m \Gamma_i,$$ for every $g \in \zed$. For brevity, let us write $\D' = \frac{\D}{\left( \det \Lambda_K(\M) \right)^m}$.
From here on, the argument is largely similar to the proof of Theorem \[kron1\] above, but with some notable changes. For each $1 \leq i \leq t$, let $\theta_i$ be as in for our choice of $\bwy \in \Lambda_K(\M) \setminus \bigcup_{i=1}^m \Gamma_i$ satisfying as above, then $$L_i((g\D'+1) \bwy) = (g\D'+1) \theta_i\ \forall\ 1 \leq i \leq t.$$ Using with instead of , we obtain that $\max \{1, |\theta_i|_v \} \leq$ $$(wd)^{\frac{3}{2}} \left( \left( \sqrt{2} h(\alpha) \right)^{sd} \det \Lambda_K(\M) \left( \sum_{i=1}^m \frac{\D}{\D_i} - m + 1 \right) + \D^{\frac{1}{sd}} \sqrt{2} h(\alpha) \right) \max \{ 1, |L_i|_v \}$$ for all archimedean $v \in M(E)$, while for the non-archimedean $v \in M(E)$, $$\max \{1, |\theta_i|_v \} \leq \max \{ 1, |\alpha^{-1}|_v \} \max \{1, |L_i|_v \},$$ as in . Taking the product over all places of $E$, we have for every $1 \leq i \leq t$: $$\begin{aligned}
\label{th-4}
h(\theta_i) & \leq & (wd)^{\frac{3}{2}} \sqrt{2} h(\alpha) h(\alpha^{-1}) h(B) \times \nonumber \\
& & \times \left( \left( \sqrt{2} h(\alpha) \right)^{sd-1} \det \Lambda_K(\M) \left( \sum_{i=1}^m \frac{\D}{\D_i} - m + 1 \right) + \D^{\frac{1}{sd}} \right),\end{aligned}$$ and $1,\theta_1,\dots,\theta_t$ (and hence $1, \D' \theta_1,\dots, \D' \theta_t$) are $\que$-linearly independent by the same reasoning as in the proof of Theorem \[kron1\].
Now let $\ba = (a_1,\dots,a_t) \in \real^t$ and $\eps > 0$, as in the statement of our theorem. Notice that for each $1 \leq i \leq t$, $$\left| (g\D'+1) \theta_i - a_i - p_i \right| = \left| g (\D' \theta_i) + (\theta_i - a_i) - p_i \right|,$$ for any integers $p_1,\dots,p_t$. Then, applying Theorem \[KR\] to approximate the vector $(\theta_1 - a_1,\dots,\theta_t - a_t)$ by the fractional parts of the integer multiples of the vector $(\D' \theta_1,\dots,\D' \theta_t)$, we conclude that there exists $g \in \zed$ and $\bp \in \zed^t$ such that $$\begin{aligned}
\label{g_bnd}
|g| & \leq & 2^{\ell t (\ell-1)} (t+1)^{3\ell-1} (t!)^{2\ell} \times \nonumber \\
& & \times \left( (wd)^{\frac{3}{2}} \sqrt{2} h(\alpha) h(\alpha^{-1}) h(B) \E_{\alpha}(\M,\Gamma_1,\dots,\Gamma_m) \right)^{\ell^2(t+1)-\ell}\ \eps^{-\ell+1},\end{aligned}$$ where $\E_{\alpha}(\M,\Gamma_1,\dots,\Gamma_m)$ is as in , and $$\left| g (\D' \theta_i) + (\theta_i - a_i) - p_i \right| < \eps\ \forall\ 1 \leq i \leq t.$$ Letting $\bx = (g\D'+1)\bwy$, we see that $(g\D'+1) \theta_i = L_i(\bx)$ for each $1 \leq i \leq t$ and $|\bx| = |g\D'+1| |\bwy|$. Combining these observations with , and finishes the proof of the theorem.
[**Acknowledgement.**]{} We are grateful for the wonderful hospitality of the Oberwolfach Research Institute for Mathematics: an important part of this work has been done during our Research in Pairs stay at the Institute. We would also like to thank the referee for the helpful remarks.
[^1]: Fukshansky was supported by the NSA grant H98230-1510051 and Simons Foundation grant \#519058.
[^2]: Moshchevitin was supported by RNF Grant No. 14-11-00433.
|
---
abstract: 'We consider an inverse problem for Schrödinger operators on connected equilateral graphs with standard matching conditions. We calculate the spectral determinant and prove that the asymptotic distribution of a subset of its zeros can be described by the roots of a polynomial. We verify that one of the roots is equal to the mean value of the potential and apply it to prove an Ambarzumian type result, i.e., if a specific part of the spectrum is the same as in the case of zero potential, then the potential has to be zero.'
address: 'Department of Differential Equations Institute of Mathematics Budapest University of Technology and Economics H 1111 Budapest, Műegyetem rkp. 3-9.'
author:
- Márton Kiss
title: Spectral determinants and an Ambarzumian type theorem on graphs
---
[Introduction]{}
Quantum graphs arise naturally as simplified models in mathematics, physics, chemistry, and engineering [@BerkolaikoKuchment2013book]. Ambarzumian’s theorem in inverse spectral theory refers to a setting when a differential operator can be reconstructed from at most one spectrum due to the presence of a constant eigenfunction. The original theorem from 1929 states for $q\in C[0,\pi]$ that if the eigenvalues of $$\begin{aligned}
\left.
\begin{array}{cc}
-y''+q(x)y=\lambda y\\
y'(0)=y'(1)=0
\end{array}
\right\}\end{aligned}$$ are $\lambda_n=n^2\pi^2$ ($n\ge 0$), then $q=0$ [@A]. Eigenvalues other than zero are used only through eigenvalue asymptotics to get $\int_0^{1}q=0$; hence a subsequence $\lambda_r=r^2\pi^2+o(1)$ of them is sufficient to reach the same conclusion even if $q\in L^1(0,\pi)$. On finite intervals inverse eigenvalue problems have a vast literature. In general we mention [@Horvath2005Annals] and the basic paper [@B1] as well as the works referenced by and referencing these. For Ambarzumian’s theorem a recent stability result is found in [@Horvath2015]. Let us turn to the list of extensions to graphs. On a tree with edges of equal length knowing the smallest eigenvalue $0$ exactly and a specific part of the spectrum approximately is enough for recover to the zero potential [@CarlsonPivovarchik2007ambarzumian]. On a tree with different edge lengths this is still true (see Lemma 4.4 of [@LawYanagida2012]), however, the required set of eigenvalues is given by an existence proof. On the other hand, having information about the entire spectrum allows applying a trace formula, even if a detailed description of the structure of the spectrum is not available. The paper [@Davies2013] uses heat extension in an abstract framework; this allows arbitrary graphs with arbitrary edge lengths at the expense of requiring information from the entire spectrum. For a summary on differential operators on graphs, see [@PokornyiBorovskikh2004; @Kuchment2008]. In this paper we consider a connected graph $G(V,E)$ with edges of equal length. The graph can contain loops and multiple edges. We parametrize each edge with $x\in(0,1)$. This gives an orientation on $G$. We consider a Schrödinger operator with potential $q_j(x)\in L^1(0,1)$ on the edge $e_j$ and with Neumann (or Kirchhoff) boundary conditions (sometimes called standard matching conditions), i.e., solutions are required to be continuous at the vertices and, in the local coordinate pointing outward, the sum of derivatives is zero. More formally, consider the eigenvalue problem $$\begin{aligned}
\label{sch}
-y''+q_j(x)y=\lambda y\end{aligned}$$ on $e_j$ for all $j$ with the conditions $$\begin{aligned}
\label{continuity}
y_j(\kappa_j)=y_k(\kappa_k)\end{aligned}$$ if $e_j$ and $e_k$ are incident edges attached to a vertex $v$ where $\kappa=0$ for outgoing edges, $\kappa=1$ for incoming edges (and can be both $0$ or $1$ for loops); and in every vertex $v$ $$\begin{aligned}
\label{Kirchhoff}
\sum_{e_j\textrm{ leaves }v} y_j'(0)=\sum_{e_j\textrm{ enters }v} y_j'(1)\end{aligned}$$ (loops are counted on both sides).
The *spectral determinant* or alternatively *functional determinant* or *characteristic function* of the problem (\[sch\])-(\[Kirchhoff\]) is a meromorphic function whose zeros coincide with its spectrum. Spectral determinants have been subject to continuous attention in the theoretical physics literature for the last twenty years [@CurrieWatson2005; @KacPivovarchik2011; @CarlsonPivovarchik2008SpectralAsymptotics; @Pankrashkin2006; @AkkermansComtetDesboisMontambauxTexier2000; @Desbois2000; @Friedlander2006; @HarrisonKirstenTexier2012; @Texier2010]. As a tool for proving our Ambarzumian type results, we give a formula for the spectral determinant of Schrödinger operators (see (\[detM\])) and for the asymptotic distribution of some of its zeros. It is already known that for finite connected graphs, the main term of the eigenvalue asymptotics can be obtained from Weyl’s law and the next terms depend on complicated combinations of $\int_0^1q_j$, $j=1,\ldots,|E|$ ([@MollerPivovarchik2015book], p. 213). We express some of these combinations as the roots of a polynomial (see (\[eigpol\])) and prove that for *any* connected, equilateral graph there is a root equal to the mean value of the potential. For the convenience of the reader, we emphasize this result in the context of spectral determinants:
[1.1’]{}\[spectraldet\] Consider a connected graph $G(V,E)$ with edges of equal length. The spectral determinant of Schrödinger operators on $G$ with standard matching conditions has a sequence of roots which asymptotically differ by the mean value of the potential from the corresponding sequence of roots of the spectral determinant of the free Schrödinger operator. Precisely, the problem (\[sch\])-(\[Kirchhoff\]) has a sequence of eigenvalues $\lambda_k=(2k\pi)^2+\frac{1}{|E|}\sum_{j}\int_0^{1}q_j+o(1)$ ($k\in\mathbb{Z}^+$). Moreover, if $G$ is a bipartite graph, the problem (\[sch\])-(\[Kirchhoff\]) has a sequence of eigenvalues $\lambda_k=(k\pi)^2+\frac{1}{|E|}\sum_{j}\int_0^{1}q_j+o(1)$.
[[**Remark.** ]{}]{}It would be interesting to prove Theorem \[spectraldet\] in the case of possibly different edge lengths and to describe the asymptotic distribution of the eigenvalues in question.
For our Ambarzumian type result, we need an improved version of this theorem:
\[equallengthdirect\] Consider the eigenvalue problem (\[sch\])-(\[Kirchhoff\]). There are exactly $|E|-|V|+2$ eigenvalues (counting multiplicities) such that $\lambda=(2k\pi)^2+O(1)$ as $k\to\infty$ ($k\in\mathbb{Z}^+$). Among these eigenvalues at least one has the asymptotics $\lambda=(2k\pi)^2+\frac{1}{|E|}\sum_{j}\int_0^{1}q_j+o(1)$. Moreover, if $G$ is a bipartite graph, the same is true for $k$ instead of $2k$, i.e., there are exactly $|E|-|V|+2$ eigenvalues (counting multiplicities) such that $\lambda=(k\pi)^2+O(1)$ and at least one has the asymptotics $\lambda=(k\pi)^2+\frac{1}{|E|}\sum_{j}\int_0^{1}q_j+o(1)$.
[[**Remark.** ]{}]{}Although $G$ is a directed graph, if we reverse an edge $e_j$ and change the potential to $q_j(1-x)$ on it, we get an eigenvalue problem with the same eigenvalues and eigenfunctions (which are reversed with respect to the new direction on $e_j$).
[[**Remark.** ]{}]{}For $q=0$ and $\lambda=2k^2\pi^2$ ($k\in\mathbb{Z}^+$) one eigenfunction is $\cos{2k\pi x}$ and in each circle there is a Dirichlet eigenfunction $\pm\sin{2k\pi x}$ on its edges (depending on the direction) and $0$ everywhere else. In the bipartite case we can assume that $V$ is a disjoint union of $V_1$ and $V_2$, and that every edge points from $V_1$ to $V_2$. Then for $k$ odd, we can take $\cos{k\pi x}$ on all edges; besides, there are Dirichlet eigenfunctions, $\pm\sin{k\pi x}$ alternately on the edges of a circle and $0$ elsewhere. Hence in both cases there are $|E|-|V|+2$ independent eigenfunctions. We shall prove that the multiplicity is not greater.
\[equallengthinverse\] Consider the eigenvalue problem (\[sch\])-(\[Kirchhoff\]). If $\lambda=0$ is the smallest eigenvalue and for infinitely many $k\in\mathbb{Z}^+$ there are $|E|-|V|+2$ eigenvalues (counting multiplicities) such that $\lambda=(2k\pi)^2+o(1)$, then $q=0$ a.e. on $G$. Moreover, if $G$ is a bipartite graph, the same is true for $k$ instead of $2k$, i.e., if $\lambda=0$ is the smallest eigenvalue and for infinitely many $k\in\mathbb{Z}^+$ there are $|E|-|V|+2$ eigenvalues (counting multiplicities) such that $\lambda=(k\pi)^2+o(1)$, then $q=0$ a.e. on $G$.
[[**Remark.** ]{}]{}For a tree $|E|-|V|+2=1$, and a tree is bipartite, hence Theorem \[equallengthinverse\] is a generalization of Theorem 1.2 in [@CarlsonPivovarchik2007ambarzumian].
If the graph represents an electrical circuit in which each edge has a unit resistance, the *effective resistance* of an edge can be computed (or in mathematics, defined) by terms of the graph Laplacian. A result of Kirchhoff [@Kirchhoff1847] is that the effective resistance of an edge $e$ can be expressed as the number of spanning trees containing $e$ divided by the number of all spanning trees.
\[equallengthequalresistances\] Consider the eigenvalue problem (\[sch\])-(\[Kirchhoff\]). If every non-loop edge of $G$ has the same effective resistance $r<1$, the multiplicities required by Theorem \[equallengthinverse\] can be weakened to $|E|-|V|+1$.
Interesting examples are the complete graph (if $|V|>2$) or a graph with one point and a loop, which corresponds to the case of periodic boundary conditions treated in [@ChengWangWu2010].
[The proof]{}
Denote by $c_j(x,\lambda)$ the solution of (\[sch\]) which satisfies the conditions $c_j(0,\lambda)-1 =c_j'(0,\lambda) = 0$ and by $s_j(x,\lambda)$ the solution of (\[sch\]) which satisfies the conditions $s_j(0,\lambda) =s_j'(0,\lambda)-1 = 0$. Each $y_j(x,\lambda)$ may be written as a linear combination $$\begin{aligned}
y_j(x,\lambda)=A_j(\lambda)c_j(x,\lambda)+\tilde B_j(\lambda)s_j(x,\lambda).\end{aligned}$$ Then $y_j(0,\lambda)=A_j(\lambda)$ is the same on each outgoing edge; hence we choose to index the functions $A(\lambda)$ by vertices, and then $$\begin{aligned}
\label{eigfun}
y_j(x,\lambda)=A_v(\lambda)c_j(x,\lambda)+\tilde B_j(\lambda)s_j(x,\lambda),\end{aligned}$$ if $e_j$ starts from $v$. If the eigefunctions are normalized, i.e., $\sum_j\|y_j(x,\lambda)\|_2^2=1$, then $A_v(\lambda)=O(1)$, $\tilde B_j(\lambda)=O(\sqrt{\lambda})$ ([@CarlsonPivovarchik2007ambarzumian]). For later calculations it is more convenient to work with the $O(1)$-variables $A_v(\lambda)$ and $B_j(\lambda)=\frac{\tilde B_j(\lambda)}{\sqrt{\lambda}}$.
The coefficients $A_v$ and $B_j$ form a $(|V|+|E|)$-vector, which satisfies $|V|$ Kirchhoff conditions at the vertices and $|E|$ continuity conditions at the incoming ends of edges, namely, for all $v\in V(G)$, $$\begin{aligned}
\sum_{e_j:\ldots\to v} \frac{1}{\sqrt{\lambda}}A_{v_j}(\lambda)c'_j(1,\lambda)+B_j(\lambda)s_j'(1,\lambda)-\sum_{e_j:v\to\ldots}B_j(\lambda)=0,\end{aligned}$$ where in the first sum $v_j$ denotes the starting point of $e_j$; and for all $e_j\in E(G)$, $$\begin{aligned}
A_u(\lambda)c_j(1,\lambda)+\sqrt{\lambda}B_j(\lambda)s_j(1,\lambda)-A_v(\lambda)&=0,\end{aligned}$$ if $e_j$ points from $u$ to $v$.
The matrix of this homogeneous linear system of equations has the form $M=\left[\begin{array}{cc}A&B\\C&D\end{array}\right]$, where
- $A$ is like an adjacency matrix; $a_{vu}=\frac{1}{\sqrt{\lambda}}\sum c_j'(1,\lambda)$, the sum is taken on edges pointing from $u$ to $v$;
- $B$ and $C$ are like incidence matrices;\
$b_{vj}=\left\{\begin{array}{ccc}s_j'(1,\lambda)&\textrm{ if $e_j$ ends in $v$}\\-1&\textrm{ if $e_j$ starts from $v$}\\s_j'(1,\lambda)-1&\textrm{ if $e_j$ is a loop in $v$}\\0&\textrm{otherwise}\end{array}\right.$\
$c_{jv}=\left\{\begin{array}{ccc}-1&\textrm{ if $e_j$ ends in $v$}\\c_j(1,\lambda)&\textrm{ if $e_j$ starts from $v$}\\-1+c_j(1,\lambda)&\textrm{ if $e_j$ is a loop in $v$}\\0&\textrm{otherwise}\end{array}\right.$
- $D$ is a diagonal matrix, $d_{jj}=\sqrt{\lambda}s_j(1,\lambda)$.
The determinant of the matrix $M$ is the so-called spectral determinant of the problem (\[sch\])-(\[Kirchhoff\]).
[**Example 1.** ]{} Consider a single vertex with a loop. Then $$M=M_1=\left[\begin{array}{c|c}
\frac{1}{\sqrt{\lambda}}c'(1,\lambda)&s'(1,\lambda)-1\\\hline
-1+c(1,\lambda)&\sqrt{\lambda}s(1,\lambda)
\end{array}\right].
$$
[**Example 2.** ]{} Consider a star graph with root $r$ and vertices $u$, $v$ and $w$. Let $e_1$, $e_2$ and $e_3$ point from $u$, $v$ and $w$ to $r$, respectively. We choose to index rows and columns by $r$, $u$, $v$, $w$, $e_1$, $e_2$, $e_3$, in that order. Then the matrix $M=M_2$ is: $$\left[\begin{array}{cccc|ccc}
0&\mkern-12mu\frac{1}{\sqrt{\lambda}}c_1'(1,\lambda)&\mkern-12mu\frac{1}{\sqrt{\lambda}}c_2'(1,\lambda)&\mkern-12mu\frac{1}{\sqrt{\lambda}}c_3'(1,\lambda)&s_1'(1,\lambda)&s_2'(1,\lambda)&s_3'(1,\lambda)\\
0&0&0&0&-1&0&0\\
0&0&0&0&0&-1&0\\
0&0&0&0&0&0&-1\\\hline
\mkern-6mu-1&c_1(1,\lambda)&0&0&\mkern-6mu\sqrt{\lambda}s_1(1,\lambda)&0&0\\
\mkern-6mu-1&0&c_2(1,\lambda)&0&0&\mkern-18mu\sqrt{\lambda}s_2(1,\lambda)&0\\
\mkern-6mu-1&0&0&c_3(1,\lambda)&0&0&\mkern-18mu\sqrt{\lambda}s_3(1,\lambda)\mkern-6mu
\end{array}\right]\mkern-6mu,$$ with determinant $\frac{1}{\sqrt{\lambda}}\sum_{j=1}^{3}c_j'(1,\lambda)\prod_{p\ne j}c_p(1,\lambda)$ (corresponding to formula (5) of [@Pivovarchik2005ambarzumian]).
The elements of $M$ have the following asymptotics for $\lambda=k^2\pi^2+d+o(1)$ (see [@CarlsonPivovarchik2007ambarzumian] eq. (2.3) or [@LawYanagida2012] Lemma 3.1): $$\begin{aligned}
\frac{1}{\sqrt{\lambda}}c_j'(1,\lambda)&=(-1)^k\frac{1}{2\sqrt{\lambda}}(\int_0^{1}q_j-d)+o(\frac{1}{\sqrt{\lambda}}),\label{asym1}\\
s_j'(1,\lambda)&=(-1)^k+o(\frac{1}{\sqrt{\lambda}}),\label{asym2}\\
c_j(1,\lambda)&=(-1)^k+o(\frac{1}{\sqrt{\lambda}}),\label{asym3}\\
\sqrt{\lambda}s_j(1,\lambda)&=(-1)^k\frac{1}{2\sqrt{\lambda}}(d-\int_0^{1}q_j)+o(\frac{1}{\sqrt{\lambda}}).\label{asym4}\end{aligned}$$
[[**Remark.** ]{}]{}Using these asymptotics, we get $\det M_1=c'(1,\lambda)s(1,\lambda)+o(\frac{1}{\lambda})$ for $k$ even (using the Wronskyan would yield only $o(\frac{1}{\sqrt{\lambda}})$), and $\det M_2=\frac{1}{\sqrt{\lambda}}\sum_{j=1}^{3}c_j'(1,\lambda)+o(\frac{1}{\sqrt{\lambda}})$, for all $k$. These are special cases of (\[detM\]) below.
\[incidence\] If $\lambda=(2k)^2\pi^2+O(1)$, or $\lambda=k^2\pi^2+O(1)$ and $G$ is bipartite, then every $|V|\times |V|$ submatrix of $C$ (and of $B$) has determinant at most $o(\frac{1}{\sqrt{\lambda}})$.
Leaving out the $o(\frac{1}{\sqrt{\lambda}})$ terms from the submatrix we make only $o(\frac{1}{\sqrt{\lambda}})$ error in its determinant. What we get is an incidence matrix of a graph with $|V|$ vertices and $|V|$ edges. This must contain a circle, hence the corresponding rows are dependent. The proof for $B$ is similar.
The determinant of $M$ is $O(\lambda^{-\frac{1}{2}(|E|-|V|+2)})$.
Look at the terms in the Laplace expansion. Taking $(|V|-1)$ factors from $B$ (and consequently from $C$) we have to take $(|E|-|V|+2)$ factors of magnitude $O(\frac{1}{\sqrt{\lambda}})$ from $A$ and $D$; otherwise, we get smaller terms, using the previous lemma.
The next statement is a variant of the Matrix Tree Theorem ([@Kirchhoff1847]; see also [@Lovasz1993Combinatorial], p. 252, [@Bollobas1998Modern], Theorem II.12, [@Tutte2001book], Theorem VI.29).
\[direct\] If $\lambda=k^2\pi^2+O(1)$ and $k$ is even or $G$ is bipartite, then $$\begin{aligned}
\label{detM}
\det M=(-1)^{k|V|}\sum_{\tau}\!\left(\!\frac{1}{\sqrt{\lambda}}\sum_{e_j\in G} c_j'(1,\lambda)\!\right)\!\!\prod_{e_j\notin\tau}\sqrt{\lambda}s_j(1,\lambda)+o(\lambda^{-\frac{|E|-|V|+2}{2}}).\end{aligned}$$ where the sum is taken for all spanning trees $\tau$ of $G$.
The main terms in the Laplace expansion are those which contain exactly $(|E|-|V|+1)$ elements from $D$. The product of a fixed set of $(|E|-|V|+1)$ elements in $D$ is weighted by the determinant of the respective minor, with all other elements of $D$ substituted by zero. The remaining rows in $C$ and columns in $B$ look like an ordered (or unordered) incidence matrix of the graph $\tau$ spanned by the remaining $(|V|-1)$ edges for $k$ even (or odd, respectively). If $\tau$ contains a circle, then the determinant of the minor is $o(\frac{1}{\sqrt{\lambda}})$. Otherwise $\tau$ is a spanning tree of $G$; then the determinant is the sum of the elements in $A$ (plus $o(\frac{1}{\sqrt{\lambda}})$), as it follows from the next two lemmas.
Let $\tau$ be a spanning tree of $G$ and $R$ the ordered vertex-edge incidence matrix for $\tau$. Consider the matrix $M_1=\left[\begin{array}{cc}X&R\\-R^T&Y\end{array}\right]$, where $Y$ is a $(|V|-1)\times(|V|-1)$ zero matrix and $X$ has only one nonzero element $s$. Then the determinant of $M_1$ is $s$, independently of the position of the nonzero element in $X$.
In the Laplace expansion every nonzero term (in fact there is only one) contains an element from $X$, hence it is enough to prove this for $s=1$. Let the indices of the nonzero element in $X$ be $uv$. First we prove that the determinant of $M_1$ is independent of $v$. Indeed, there is a path in the tree between two arbitrary vertices, hence we can add to (or subtract from) row $u$ the rows corresponding to the vertices of that path. Similarly, the determinant does not depend on $u$. Reversing an edge in $G$ does not change the determinant; using this and by adding rows (and corresponding columns) we can assume that $\tau$ is a path. Then taking $u=|V|$, $v=1$ and expanding the determinant from left to the right we get $((-1)^{|V|)})^{|V|-1}=1$.
Let $G$ be a bipartite graph, $\tau$ a spanning tree of $G$ and $R$ the unordered incidence matrix for $\tau$. Consider the matrix $M_1=\left[\begin{array}{cc}X&-R\\-R^T&Y\end{array}\right]$, where $Y$ is a $(|V|-1)\times(|V|-1)$ zero matrix and $X$ has only one nonzero element, $x_{uv}=s$, such that $(uv)\in E(G)$. Then the determinant of $M_1$ is $(-1)^{|V|}s$, independently of $u$ and $v$.
There is only one nonzero term in the Laplace expansion of the determinant, which contains only $\pm1$’s besides $s$, hence $\det M_1=\pm s$. If $V$ is a disjoint union of $V_1$ and $V_2$ such that all edges connect $V_1$ to $V_2$, then let us multiply by $(-1)$ the rows in $R$ corresponding to $V_1$ and the columns in $R^T$ corresponding to $V_2$, respectively. This multiplies the determinant by $(-1)^{|V|}$ leaving the nonzero element of $X$ unchanged for $(uv)\in E(G)$. Hence the statement follows from the previous lemma.
Substituting the asymptotics (\[asym1\])-(\[asym4\]) we get
If $\lambda=k^2\pi^2+d+o(1)$ and $k$ is even or $G$ is bipartite, then $$\begin{aligned}
\det M=(-1)^{k|E|}\left(\frac{1}{2\sqrt{\lambda}}\right)^{|E|-|V|+2}p(d)+o(\lambda^{-\frac{1}{2}(|E|-|V|+2)}),\end{aligned}$$ where $$\begin{aligned}
\label{eigpol}
p(d)=\sum_{\tau}\left(\sum_{e_j\in G}\int_0^{1}q_j-|E|d\right)\prod_{e_j\notin\tau}(d-\int_0^{1}q_j),\end{aligned}$$ the outer sum is taken for all spanning trees $\tau$ of $G$.
[**Proof of Theorem \[equallengthdirect\].**]{}
$\lambda$ is an eigenvalue of the eigenvalue problem (\[sch\])-(\[Kirchhoff\]) if and only if $\det M(\lambda)=0$. Let the distinct roots of $p(d)$ be $d_1,\ldots,d_l$. By the previous corollary for $\lambda=k^2\pi^2+O(1)$ (if $k$ is even or $G$ is bipartite) the distinct roots of $\det M(\lambda)$ are exactly of the form $\lambda=k^2\pi^2+d_j+o(1)$ $(1\le j\le l)$. As it is seen from (\[eigpol\]), one of them is $\lambda=k^2\pi^2+\frac{1}{|E|}\sum_{e_j\in G}\int_0^{1}q_j+o(1)$.
It remains to prove that the total multiplicities of the eigenvalues $\lambda=k^2\pi^2+O(1)$ are exactly $|E|-|V|+2$. A direct calculation shows that this is true for $q=0$. Indeed, then $\det M$ is a polynomial of $\cos{\sqrt{\lambda}}$ and $\sin{\sqrt{\lambda}}$, hence its zeros are $2\pi$-periodic (in the bipartite case $\pi$-periodic) in $\sqrt{\lambda}$. Hence $\lambda=k^2\pi^2+O(1)\implies\sqrt{\lambda}=k\pi+o(1)$ implies $\sqrt{\lambda}=k\pi$ with finitely many exceptions. For $\lambda=k^2\pi^2$ $A$ and $D$ are zero matrices, thus the rank of $M$ is $2(|V|-1)$, and its nullspace is exactly $(|E|-|V|+2)$-dimensional. Then consider the eigenvalue problem $-y''+tq_j(x)y=\lambda y$ with the boundary conditions (\[continuity\])-(\[Kirchhoff\]) for $t\in[0,1]$. The corresponding operator $T(t)$ forms a self-adjoint holomorphic family and its eigenvalues $\lambda_n(t)$ and normalized eigenfunctions $g_n(t)$ can be represented by holomorphic functions of $t$ (see Example VII-3.5 and Theorem VII-3.9 in [@K]). $$\begin{aligned}
\lambda_n'(t)=\langle g_n(t),T'(t)g_n(t)\rangle=\int_Gg_n^2(t)q \end{aligned}$$ is bounded by $\|g_n\|_{\infty}^2\|q\|_{1}=O({\|q\|_{1}})$ (see (\[eigfun\]) and the paragraph below it). Hence $|\lambda_n(1)-\lambda_n(0)|=O(1)$ and the total multiplicity of eigenvalues $\lambda=k^2\pi^2+O(1)$ is the same for all $q\in L^1$.
[**Proof of Theorem \[equallengthinverse\].**]{}
According to Theorem \[equallengthdirect\], $$\begin{aligned}
\int_Gq=\sum_{e_j\in G}\int_0^{1}q_j=0. \end{aligned}$$ Let us denote the operator of the eigenvalue problem (\[sch\])-(\[Kirchhoff\]) by $L$. $\langle \varphi,L\varphi\rangle\ge\lambda_0=0$ and equality holds if and only if $\varphi$ is an eigenfunction of $L$. It follows that the constant $1$ must be an eigenfunction corresponding to the eigenvalue $0$. Substituting this to (\[sch\]) gives $q(x)=0$.
[**Proof of Theorem \[equallengthequalresistances\].**]{}
The equality of the effective resistances $r<1$ implies that the number of spanning trees not containing a fixed edge is the same nonzero integer for every non-loop edge. Suppose that $\sum_{e_j\in G}\int_0^{1}q_j\ne0$. Then by (\[eigpol\]) $$\begin{aligned}
\prod_{e_j\textrm{is a loop}}(d-\int_0^{1}q_j)\sum_{\tau}\prod_{\substack{e_j\textrm{is not a loop}\\e_j\notin\tau}}(d-\int_0^{1}q_j)=\sum_{\tau}d^{|E|-|V|+1},\end{aligned}$$ as loops are not in spanning trees. Hence if $e_j$ is a loop then $\int_0^{1}q_j=0$, and the sum of the roots of the second factor must also be zero. This is $(1-r)\sum_{e_j\textrm{is not a loop}}\int_0^{1}q_j$ multiplied by the number of spanning trees. Thus $\sum_{e_j\in G}\int_0^{1}q_j=0$ and we can proceed as in the proof of Theorem \[equallengthinverse\].
[10]{}
Eric Akkermans, Alain Comtet, Jean Desbois, Gilles Montambaux, and Christophe Texier, *Spectral determinant on quantum graphs*, Annals of Physics **284** (2000), no. 1, 10–51.
V. Ambarzumian, *[Über eine Frage der Eigenwerttheorie]{}*, Zeitschrift für Physik **53** (1929), 690–695.
G Berkolaiko and P Kuchment, *Introduction to quantum graphs*, Mathematical Surveys and Monographs, AMS, 2013.
Béla Bollobás, *Modern graph theory*, Springer, New York, 1998.
G. Borg, *[Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe]{}*, Acta Math. **78** (1946), 1–96.
Robert Carlson and Vyacheslav Pivovarchik, *Ambarzumian’s theorem for trees*, Electronic Journal of Differential Equations **2007** (2007), no. 142, 1–9.
[to3em]{}, *Spectral asymptotics for quantum graphs with equal edge lengths*, Journal of Physics A: Mathematical and Theoretical **41** (2008), no. 14, 145202.
YH Cheng, Tui-En Wang, and Chun-Jen Wu, *[A note on eigenvalue asymptotics for Hill’s equation]{}*, Applied Mathematics Letters **23** (2010), no. 9, 1013–1015.
Sonja Currie and Bruce A Watson, *Eigenvalue asymptotics for differential operators on graphs*, Journal of computational and applied mathematics **182** (2005), no. 1, 13–31.
E.B Davies, *An inverse spectral theorem*, Journal of Operator Theory **69** (2013), no. 1, 195–208.
Jean Desbois, *[Spectral determinant of Schrödinger operators on graphs]{}*, Journal of Physics A: Mathematical and General **33** (2000), no. 7, L63.
Leonid Friedlander, *[Determinant of the Schrödinger Operator on a Metric Graph]{}*, Contemporary Mathematics **415** (2006), 151–160.
J M Harrison, K Kirsten, and C Texier, *[Spectral determinants and zeta functions of Schrödinger operators on metric graphs]{}*, Journal of Physics A: Mathematical and Theoretical **45** (2012), no. 12, 125206.
M. Horváth, *[Inverse spectral problems and closed exponential systems]{}*, Ann. of Math. **162** (2005), no. 2, 885–918.
[to3em]{}, *[On the stability in Ambarzumian theorems]{}*, Inverse Problems **31** (2015), no. 2, 025008.
I Kac and V Pivovarchik, *On multiplicity of a quantum graph spectrum*, Journal of Physics A: Mathematical and Theoretical **44** (2011), no. 10, 105301.
T. Kato, *[Perturbation theory for linear operators]{}*, Springer-Verlag, Berlin Heidelberg New York, 1980.
Gustav Kirchhoff, *[[Ü]{}ber die Aufl[ö]{}sung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Str[ö]{}me gef[ü]{}hrt wird]{}*, Annalen der Physik **148** (1847), no. 12, 497–508, (English transl. IRE Trans. Circuit Theory **CT-5** (1958), 4–7.).
Pavel Kuchment, *Quantum graphs: an introduction and a brief survey*, in: Analysis on Graphs and Its Applications (2008), 291–314.
Chun-Kong Law and Eiji Yanagida, *[A solution to an Ambarzumyan problem on trees]{}*, Kodai Mathematical Journal **35** (2012), no. 2, 358–373.
László Lovász, *Combinatorial problems and exercises*, North-Holland Pub. Co, Amsterdam London, 1993.
Manfred M[ö]{}ller and Vyacheslav Pivovarchik, *[Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and their Applications]{}*, Operator Theory: Advances and Applications, Springer, 2015.
Konstantin Pankrashkin, *[Spectra of Schr[ö]{}dinger operators on equilateral quantum graphs]{}*, Letters in Mathematical Physics **77** (2006), no. 2, 139–154.
VN Pivovarchik, *[Ambarzumian’s Theorem for a Sturm-Liouville boundary value problem on a star-shaped graph]{}*, Functional Analysis and Its Applications **39** (2005), no. 2, 148–151.
Yu V Pokornyi and AV Borovskikh, *Differential equations on networks (geometric graphs)*, Journal of Mathematical Sciences **119** (2004), no. 6, 691–718.
Christophe Texier, *$\zeta$-regularized spectral determinants on metric graphs*, Journal of Physics A: Mathematical and Theoretical **43** (2010), no. 42, 425203.
William Thomas Tutte, *Graph theory*, volume 21 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2001.
|
---
abstract: 'We review the status of String Gas Cosmology after the 2015 Planck data release. String gas cosmology predicts an almost scale-invariant spectrum of cosmological perturbations with a slight red tilt, like the simplest inflationary models. It also predicts a scale-invariant spectrum of gravitational waves with a slight blue tilt, unlike inflationary models which predict a red tilt of the gravitational wave spectrum. String gas cosmology yields two consistency relations which determine the tensor to scalar ratio and the slope of the gravitational wave spectrum given the amplitude and tilt of the scalar spectrum. We show that these consistency relations are in good agreement with the Planck data. We discuss future observations which will be able to differentiate between the predictions of inflation and those of string gas cosmology.'
address: 'Physics Department, McGill University, 3600 University St., Montreal, QC, H3A 2T8, Canada'
author:
- 'Robert H. Brandenberger'
title: String Gas Cosmology after Planck
---
April 2015
Introduction
============
The recently released Planck results [@Planck] have further confirmed the predictions of the six parameter $\Lambda$CDM cosmological model. Even though these six parameters describe properties of the current universe, their values are quite mysterious without going back to processes which happened in the very early universe. In particular, the origin of the almost scale-invariant spectrum of almost adiabatic curvature fluctuations with a slight red tilt which has now been confirmed by Planck and many other experiments has no explanation in the context of late time cosmology.
Inflationary cosmology [@Guth] is the current paradigm for explaining the overall homogeneity and spatial flatness of space and the origin of the observed spectrum of fluctuations [@Mukh]. The inflationary scenario is based on the potential energy of some scalar field yielding a period of (almost) exponential expansion of space. In simple models of inflation, however, the energy scale at which inflation takes place is of the order of $10^{16} {\rm GeV}$, much closer to the Planck scale than to scales which have been explored in particle physics experiments, and in fact even closer to the string scale [@GSW]. Hence, in order to truly understand the mechanism of inflation, it appears to be necessary to tackle superstring cosmology.
Another reason for turning to string theory as a framework for inflationary cosmology is the “$\eta$ problem” (see e.g. Section 27.2 in [@etaproblem]). In order not to produce a too large amplitude of fluctuations, the scalar field potential must be sufficiently flat. For simple scalar field potentials, the slow-roll inflationary dynamics takes place at field values larger than the Planck mass $m_{pl}$. This is also a field range for which the slow-roll inflationary trajectory is a local attractor in initial condition space [@initial], even including linear metric fluctuations [@Hume] [^1]. In order to understand physics in this field range, it seems necessary to embed inflation into an ultraviolet complete theory of quantum gravity such as superstring theory.
In most current approaches to superstring cosmology, the theory is treated in an effective field theory limit in which scalar fields motivated by string physics are coupled to Einstein gravity or dilaton gravity (see e.g. [@reviews] for recent reviews on inflation in this framework). However, this approach misses one of the key symmetries of string theory, namely the “T-duality” symmetry [@Pol]. To illustrate one manifestation of this symmetry, assume that space is toroidal. In this case, the perturbative spectrum of string states constains both string momentum and winding modes. The energy of the string momentum modes scales as $1/R$, where $R$ is the radius of the torus, whereas the energy in the winding modes increases linearly in $R$. Thus, the spectrum of string states is unchanged under the transformation $R \rightarrow 1/R$ (in string units). The string vertex operators are consistent with this symmetry, and if we postulate that the symmetry extends to the non-perturbative level, we predict the existence of D-branes [@Pol; @Boehm].
Another feature of superstring theory which is missed in the effective field theory description is the existence of an infinite tower of string oscillatory modes which leads to a maximal temperature for a gas of closed strings in thermal equilibrium, the [*Hagedorn temperature*]{} [@Hagedorn]. It is self-evident that both the T-duality symmetry of string theory and the existence of a limiting temperature should play an important role in superstring cosmology. The [*string gas cosmology*]{} (SGC) [@BV] scenario which will be reviewed here is an approach to superstring cosmology which is based on the T-duality symmetry of string theory and string thermodynamics [@Nivedita]. As was realized much later [@NBV], string gas cosmology yields an alternative to inflation for explaining the origin of the inhomogeneities and anisotropies which are now mapped out by cosmological experiments such as the Planck satelllite mission.
Some readers may be under the impression that observations have proven that there was a period of inflation in the early universe. However, all that the observations show is that there is some mechanism which produces an almost scale-invariant spectrum of nearly adiabatic fluctuations with a small red tilt on scales which were larger than the Standard Big Bang horizon at early times (times comparable to the time $t_{rec}$ of last scattering). That such a spectrum yields acoustic oscillations in the angular power spectrum of the cosmic microwave background (CMB) and baryon acoustic oscillations in the galaxy power spectrum was realized [@Sunyaev] more than a decade before the development on the inflationary scenario. It is true that cosmological inflation is the first model where such a spectrum emerges from first principles, but it is not the only one (see e.g. [@RHBrev1] for a recent review of alternatives).
Alternatives include the [*Pre-Big-Bang*]{} and [*Ekpyrotic*]{} scenarios [@Ekp] (in which a light entropic mode obtains a scale-invariant spectrum during the contracting phase [@NewEkp]), the [*conformal universe*]{} [@Rubakov] or [*pseudo-conformal universe*]{} [@Khoury] (in which a scale-invariant spectrum is generated from an emergent phase by a moving Galileon field which induces squeezing of the curvature fluctuations), the [*matter bounce*]{} scenario [@Wands; @Fabio] in which a scale-invariant spectrum of curvature perturbations is generated from initial vacuum fluctuations on scales which exit the Hubble radius during the matter-dominated phase of contraction, and [*string gas cosmology*]{}, the topic of this review. In the same way that inflation may find an embedding within superstring theory, it is also possible that one of the alternatives to inflation ends up being realized in string theory. For example, the Ekpyrotic scenario was initially motivated from a stringy construction, heterotic M-theory [@Horava], and there is a recent realization [@Costas] of the matter-bounce scenario in which an S-brane originating from extra states becoming massless at a T-dual point mediates the transition from contraction to expansion. String gas cosmology is, obviously, from the outset firmly rooted in superstring theory.
In the following we first present a review of string gas cosmology (see e.g [@SGCreviews] for more detailed reviews). We then confront the predictions of string gas cosmology with recent observations, and we close this article with a discussion of some major challenges facing the scenario.
String Gas Cosmology
====================
Standard Big Bang Cosmology is based on coupling a gas of point particles to a background space-time. “String Gas Cosmology” [@BV] (see also [@Perlt]) is a modest extension of this setup which maintains the key new symmetries (T-duality) and new degrees of freedom (string oscillatory and winding modes) which distinguish string theory from point particle theories. This is achieved by replacing the gas of point particles by a gas of strings. To be specific, we consider a theory of closed superstrings, and we assume that the background space is toroidal (extensions to toroidal orbifolds are considered in [@BG1]). We also assume that the string coupling constant is small such that the strings are the light degrees of freedom (compared to branes - for an analysis of the role of branes in string gas cosmology see [@Stephon]).
Following what is done in Standard Big Bang Cosmology, we assume that matter is in thermal equilibrium. The infinite tower of string oscillatory modes then leads immediately to a crucial difference between point particle cosmology and string cosmology: the temperature of the gas of strings has a maximal temperature $T_H$, the “Hagedorn temperature” [@Hagedorn] which is given by the string scale. Let us start with a large dilute box of strings at low temperatures. In this case almost all of the energy is in the momentum modes whose energy increases as the box contracts. This leads to a rising temperature, as in particle cosmology. However, once the temperature approaches $T_H$, the energy density is high enough to excite the string oscillatory modes. Further contraction of the box will lead to a growing tower of oscillatory modes being excited at approximately constant temperature. Once the box size decreases below the string scale, the energy of the gas of strings will drift into the winding modes which become less energetic as the universe contracts, leading to a deceasing temperature. Figure 1 [@BV] shows how the temperature of a gas of heterotic superstrings in a box of radius $R$ varies as the radius changes. The vertical axis is temperature, the horizontal axis the radius of the box in units of the string length. The two different curves corresponding to different total amounts of entropy - the larger the entropy the more extended the [*Hagedorn phase*]{}, the phase where $T$ is close to $T_H$.
To obtain a cosmological scenario we need not only kinematics of string gas cosmology, but also dynamics. At the present time we do not have a first principles dynamics which comes from string theory. Neither Einstein gravity nor dilaton gravity can be applied in the Hagedorn phase since these frameworks are not consistent with the symmetries of string theory which are expected to be present in the Hagedorn phase. There are two possible dynamical scenarios. In the first, the universe starts in a quasi-static Hagedorn phase with all spatial dimensions wrapped by strings. The decay of string winding modes into loops triggers the dynamical breaking of the T-duality symmetry of the string gas and the transition to the radiation stage of Standard Big Bang cosmology. Figure 2 is a sketch of the time evolution of the cosmological scale factor $a(t)$ according to this scenario [@BV].
An alternative scenario is that the universe undergoes a cosmological bounce in which $R$ starts out at the far left of Figure 1, i.e. with $R \ll l_s$, where $l_s$ is the string length, passes through the Hagedorn phase, and then enters the usual radiation phase of Standard Cosmology. In terms of the light variables, the phase with $R \ll l_s$ and $R$ increasing corresponds to a contracting phase. Thus, this second scenario corresponds to a cosmological bounce mediated by a gas of strings [@Biswas]. However, in the following we will explore the first possibility, namely that the universe starts in a long quasi-static Hagedorn phase.
String theory is mathematically well-defined in ten space-time dimensions. The idea of string gas cosmology is to start in a thermal state in which all spatial dimensions are equivalent, and to then explain why only three spatial dimensions effectively de-compactify [@BV], as opposed to the usual approach in string motivated field theory cosmology where one assumes from the outset that our three spatial dimensions are special and one then requires some ad-hoc compactification mechanism. Thus, we start with a thermal gas of strings in which the momentum and winding modes about all spatial dimensions are excited. The presence of winding modes prevents spatial sections from expanding. The only way that a spatial dimension can become large is if the winding modes about that dimension can decay. Decay of winding modes, however, requires winding mode interactions. Since such interactions require the string world sheets to intersect, the interaction probability is negligible in more than three spatial dimensions, as long as there are no long-range forces between the strings [@BV] (see also [@Mairi] for a numerical study). The three dimensions in which winding modes can annihilate may not all start to expand at exactly the same time, but, as long as some string winding modes are still present there is an isotropization mechanism which is at work [@Scott1].
In the string gas cosmology setup the six spatial dimensions in which the winding modes were not able to annihilate are confined by the gas of winding and momentum modes to remain at the string scale [@Scott2; @Subodh1]. Thus, the size moduli of string theory are naturally stabilized in string gas cosmology. This corresponds to moduli stabilization at enhanced symmetry points [@Scott3; @Alex]. In the case of heterotic superstring theory it can be shown explicitly [@Subodh2] that this moduli trapping mechanism is consistent with late time cosmology. The presence of winding modes can also trap shape moduli of the internal dimensions, as was shown in [@Edna]. The one modulus which is not stabilized by intrinsic stringy effects is the dilaton. The dilaton, however, can be stabilized by invoking gaugino condensation [@Frey], without destabilizing the size moduli. Gaugino condensation then leads to (typically high scale) supersymmetry breaking [@Wei]. The bottom line is that moduli stabilization, the Achilles heel of many other approaches to string cosmology, appears to be in good controle in the context of string gas cosmology.
String Gas Cosmology, Structure Formation and the Planck Results
================================================================
In inflationary cosmology and some of its alternatives, the source of cosmological perturbations is quantum vacuum fluctuations [@Mukh]. A justification of this idea is as follows: the exponential expansion of space dilutes the density of any excitations which may have been present at the beginning of the inflationary phase, leaving behind a vacuum state of matter. In contrast, in string gas cosmology the initial state is a hot gas of strings in thermal equilibrium at a temperature close to $T_H$, the Hagedorn temperature. Hence, in string gas cosmology the source of inhomogeneities is thermal fluctuations of a gas of strings. As realized in [@NBV], this leads to an almost scale-invariant spectrum of curvature perturbations at late times, and similarly [@BNPV2] to an almost scale-invariant spectrum of gravitational waves. The spectrum of cosmological perturbations has a small red tilt, like in the case of inflation. However, the spectrum of gravitational waves has a small blue tilt [@BNPV2], unlike in the case of inflation where a red tilt is inevitable (provided matter is used which obeys the usual energy conditions).
Cosmological fluctuations should be viewed as a superposition of small amplitude plane wave inhomogenities. If we expand the full equations of motion for space-time and matter to linear order in these fluctuations, each wave will evolve independently. In inflationary cosmology, each wave corresponds to a harmonic oscillator which begins in its vacuum state on sub-Hubble scales and is stretched by the accelerated expansion of space to super-Hubble lengths where the wave function is squeezed and classicalizes via the intrinsic nonlinearities of the system [@KPS; @Martineau]. We now compare this setup to what happens in string gas cosmology.
A space-time diagram of string gas cosmology is shown in Fig. 3. The vertical axis is time, the horizontal axis indicates physical spatial length. The Hagedorn phase corresponds to times earlier than $t_R$. During the Hagedorn phase space is static and hence the Hubble radius is infinite. After the decay of the string winding modes our three dimensional space starts to expand according to the usual laws of Standard Big Bang cosmology. The Hubble radius drops to a microscopic value at $t_R$ and then expands linearly as shown in the blue curve. The physical wavelength of fluctuation modes in constant in the Hagedorn phase and then increases in proportion to the scale factor $a(t)$ after $t_R$.
We first compare the ways in which inflation and string gas cosmology, respectively, solve the horizon problem of Standard Big Bang cosmology and lead to the possibility of a causal structure formation mechanism. In inflationary cosmology it is the accelerated expansion of space which renders the horizon much larger than the Hubble radius and ensures that the past light cone of our current observer fits into the horizon at $t_{rec}$, the time of recombination. In string gas cosmology a long Hagedorn phase will similarly allow the horizon to become much larger than the Hubble radius at $t_R$. In inflationary cosmology, it is again the accelerated expansion of space which allows fluctuation modes which are currently observed on large scales to be pushed far outside of the Hubble radius. In string gas cosmology, the wavelengths of the perturbation modes is constant during the Hagedorn phase, but the Hubble radius decreases dramatically such that modes become super-Hubble at the end of the Hagedorn phase. In both cases, fluctuations not only have a wavelength smaller than the horizon, but also smaller than the Hubble radius. This enables a causal generation mechanism.
Assuming that the string scale is close to the scale of particle physics Grand Unification, which is the preferred value for heterotic superstring particle phenomenology [@GSW], the physical wavelength of fluctuation modes which are observed today is of the order $1 {\rm mm}$. While this scale seems microscopic from the point of view of cosmology, it is very large compared to the string scale or the Planck scale. In inflationary cosmology, the wavelength of these fluctuations is exponentially smaller than this scale at the beginning of the period of inflation, thus leading to the “trans-Planckian problem” for fluctuations [@Jerome]: it is not justified to use Einstein gravity and low energy effective classical matter physics to study the origin and early evolution of fluctuations. In contrast, in string gas cosmology the fluctuation modes of interest to us are safely in the far infrared for all times, and thus safe from the trans-Planckian problem.
In contrast to the case of inflationary cosmology, in string gas cosmology the inhomogeneities are not vacuum fluctuations, but rather thermal fluctuations. Importantly, they are not thermal fluctuations of a gas of point particles, but of a gas of fundamental strings. Hence, the thermal fluctuations are described by string thermodynamics (see e.g. [@Nivedita]). The computation of the spectrum of cosmological fluctuations and gravitational waves now proceeds as follows [@NBV; @BNPV2]: we first compute the matter fluctuations in the Hagedorn phase, using relations of string thermodynamics [@Nivedita]. In a second step, we use the Einstein constraint equations to relate the matter fluctuations to metric fluctuations. This is done mode by mode at the time when the wavelength crosses the Hubble radius. Finally, we evolve the metric perturbations on super-Hubble scales until they re-enter the Hubble radius at late times using the equations of the theory of cosmological perturbations (see e.g. [@MFB; @RHBfluctrev] for reviews).
Our method relies on three key assumptions: firstly the existence of an initial quasi-static phase containing a thermal gas of strings. Secondly, we posit a fast transition from the Hagedorn phase to the radiation phase of Standard cosmology. Thirdly, we assume the validity of Einstein gravity in the far infrared, an assumption which we use both in converting matter fluctuations to metric ones, and in evolving the perturbations to late times.
We begin with the ansatz for a space-time metric containing both linear cosmological fluctuations (also called “scalar metric fluctuations”) and gravitational waves (“tensor metric fluctuations”): $$d s^2 \, = \, a^2(\eta) \bigl( (1 + 2 \Phi(x, \eta) )d\eta^2 -
[ (1 - 2 \Phi)\delta_{ij} + h_{ij} ]d x^i d x^j\bigr) \,$$ where $\eta$ is conformal time, $a(\eta)$ is the scale factor describing the background cosmology, $\Phi(x, \eta)$ are the cosmological perturbations which depend on the spatial coordinates $x$ and on time, and the transverse traceless tensor $h_{ij}(x, \eta)$ describes the gravitational waves (see [@MFB; @RHBfluctrev]). We have chosen a gauge (coordinate system) in which the metric corresponding to the cosmological perturbations is diagonal, and assumed that there is no anisotropic stress (which leads to the fact that there is only one non-trivial function $\Phi$ characterizing these fluctuations. Note that Latin indices represent spatial coordinates.
The Einstein constraint equations determine the scalar and tensor metric fluctuations in terms of the fluctuations of the energy-momentum tensor. Specifically, $$\label{scalarexp}
\langle |\Phi(k)|^2\rangle \, = \, 16 \pi^2 G^2
k^{-4} \langle\delta T^0{}_0(k) \delta T^0{}_0(k)\rangle \, ,$$ and $$\label{tensorexp}
\langle |h(k)|^2\rangle \, = \, 16 \pi^2 G^2
k^{-4} \langle\delta T^i{}_j(k) \delta T^i{}_j(k)\rangle \, ,$$ where $G$ is Newton’s gravitational constant, where in the last equation $h$ is the amplitude of each of the two polarization modes of gravitational waves, and on the right hand side an average of the off-diagonal spatial matrix elements (i.e. $i \neq j$) is implicit. The expectation values on the right hand side of the above equations indicate thermal expectation values.
Since the wavelengths of the modes we are interested in are always in the far infrared compared to the string scale, we can safely use the perturbed Einstein equations to study the evolution of the fluctuations. From the theory of cosmological fluctuations we know that, since the equation of state parameter $1 + w$ (where $w$ is the ratio of pressure $p$ to energy density $\rho$) does not change by more than a factor of order one during the transition from the Hagedorn phase to the Standard Cosmology phase (in contrast to what happens in inflationary cosmology during the transition between the inflationary phase and the post-inflation period), both $\Phi$ and $h$ remain constant while on super-Hubble scales. Hence, it will be the values of $\Phi$ and $h$ computed at Hubble radius crossing at the end of the Hagedorn phase which are relevant for current observations.
Let us now turn to the determination of the initial fluctuations. In general, for thermal fluctuations the energy density perturbations are determined by the specific heat capacity $C_V$ (in a volume $V$), and by the temperature $T$. If $C_V$ is the specific heat capacity in a volume of radius $R$, the resulting energy density variations are given by $$\label{drho}
\langle \delta\rho^2 \rangle \, = \, \frac{T^2}{R^6} C_V \, .$$ What is special about string thermodynamics is that for strings in a compact space of radius $R$, the specific heat capacity has holographic scaling with $R$ [@Nivedita; @Ali] $$\label{specheat}
C_V \, \approx \, 2 \frac{R^2/l_s^3}{T \left(1 - T/T_H\right)}\, .,$$ where $l_s$ is the string length.
By combining these equations we can compute the power spectrum of cosmological fluctuations which is defined as \[power2\] P\_(k) [1 ]{} k\^3 |(k)|\^2 . Inserting the value of $\Phi(k)$ at Hubble radius crossing in terms of the density fluctuations from (\[scalarexp\]), making use its representation (\[drho\]) in terms of the specific heat capacity, and then making the replacement (\[specheat\]) we obtain the following expression for the power spectrum in terms of the temperature $T(k)$ when the mode $k$ exits the Hubble radius: \[sresult\] P\_(k) = ()\^4[T(k) ]{} [1 ]{} , , where $l_{pl}$ is the Planck length (determined by $G$).
As can be seen from Figure 1, to first approximation the temperature $T(k)$ is independent of $k$. To next order, however, we notice that $T(k)$ in a slightly decreasing function of $k$ as $k$ increases, since the temperature starts to decrease as we near the exit from the Hagedorn phase. Thus, string gas cosmology, like inflation, predicts a roughly scale-invariant spectrum of cosmological fluctuations with a slight red tilt [@NBV]. The spectral tilt $n_s - 1$ is given by [@BNP] \[scalartilt\] n\_s-1 = (1- )\^[-1]{} k , which is negative since $T(k)$ is a decreasing function of $k$,
The above computation contains two parameters which from our point of view are free, the first being the ratio of the string length to the Planck length, and the second the deviation of the temperature from $T_H$ during the Hagedorn phase. The latter depends on the total entropy of the system [@BV], the former is set by the string model. Making use of the value of the string length preferred for particle physics reasons in [@GSW], and taking the ratio of temperatures on the right hand side of (\[sresult\]) to be of the order one, we obtain an amplitude of the spectrum which is consistent with observations. However, from the effective theory analysis which we have presented, we must take $l_s$ and $1 - T(k_0)/T_H$ as free parameters (where $k_0$ is the pivot scale). These parameters can be fixed by demanding agreement with the current CMB data. Once these parameters are fixed, however, both the amplitude and spectral tilt of the gravitational wave spectrum are determined.
As indicated in (\[tensorexp\]), the gravitational wave spectrum is given by the off-diagonal pressure fluctuations. String thermodynamics allows the computation of all stree-energy tensor correlation functions. The result for the off-diagonal stress correlation function is [@Ali] $$<|T_{ij}(R)|^2> \, \sim \, {T \over {l_s^3 R^4}} (1 - T/T_H)
\ln^2{\left[\frac{1}{l_s^3 T R^{-2}}(1 - T(k)/T_H)\right]}\, .$$ Note that the factor $(1 - T/T_H)$ is in the numerator instead of in the denominator as in the case of the energy density correlation function. Inserting this expression into (\[tensorexp\]) yields the following result for the power spectrum of gravitational waves \[tresult\] P\_h(k) \~ ()\^4 (1 - T(k)/T\_H)\^2 . This corresponds to a scale-invariant spectrum, but this time with a blue tilt. The tensor tilt $n_t$ is \[cons1\] n\_t = k = -(n\_s-1)(2 - 1) . To first approximation, the magnitude of the blue tilt of the tensor spectrum equals the magnitude of the red tilt of the scalar spectrum.
The difference in the sign of the tilt of the gravitational wave spectrum is the key and most rebust criterion which differentiates between cosmological inflation and string gas cosmology [@BNPV2; @BNP]. The reason why the tilt in inflation is negative (i.e. a red spectrum) is that the amplitude of the gravitational wave spectrum is set by the Hubble constant $H$, and that during inflation $H(t)$ is a decreasing function of time (as long as matter satisfies the “null energy condition”). Thus, long wavelength fluctuations exit the Hubble radius at a larger value of $H$ and thus with a larger amplitude of gravitational waves. On the other hand, in string gas cosmology the gravitational wave spectrum is determined in terms of the off-diagonal pressure fluctuations which are proportional to the pressure. Earlier in the Hagedorn phase the pressure is closer to zero, and hence large-scale modes exit the Hubble radius with a smaller amplitude of the pressure perturbations than small wavelength modes, and a blue spectrum results. Since the temperature $T(k)$ in the Hagedorn phase is close to $T_H$, the consistency relations approximately reads n\_t - (n\_s - 1) . Based on the Planck data [@Planck], string gas cosmology thus predicts n\_t = 0.03 0.01 .
The tensor to scalar ratio $r$ is also a prediction of string gas cosmology. By combining (\[sresult\]) and (\[tresult\]) we obtain \[cons2\] r(k) = (1 - )\^2 \^2 . Note that, as in the case of inflationary cosmology, the value of $r$ depends on the scale $k$.
At this stage, the background of string gas cosmology is not sufficiently developed to be able to make a specific prediction for the tensor to scalar ratio. From (\[sresult\]) and (\[scalartilt\]) we see that the amplitude and tilt of the scalar spectrum depend on the ratio $l_{pl} / l_s$, the factor $1 - T/T_H$ and on the $dT(k) / dk$. The last factor is not known since we do not have an analytical description of the exit from the Hagedorn phase. Just considering the factors $(1 - T/T_H)$ we would expect r \~ (1 - T/T\_H)\^2 which is expected to be significantly smaller than $1$. It would be interesting to model the exit from the Hagedorn phase in order to obtain an actual prediction for $r$.
The Planck and joint Planck/BICEP2 results for $r$ yield a bound of $r < 0.1$. Thus, the results are completely consistent with the predictions of string gas cosmology. To differentiate inflation and string gas cosmology it will be crucial to determine the tensor tilt. The original BICEP2 results [@BICEP2] favored a blue tilt of the tensor spectrum [@YiWang], and hence favored string gas cosmology over inflation as was stressed in [@BNP].
The experimental prospects for measuring the tensor tilt depend on the amplitude $r$. For a value $r = 0.05$, a careful analysis of B-mode polarization data including de-lensing will allow an identification of a tensor tilt variance of $\sigma(n_t) = 0.04$ [@Simard], very close to the prediction of string gas cosmology. The current upper bound on the tensor tilt are n\_t < 0.15 (making use of constraints from pulsar timing, direct detection experiments and nucleosynthesis [@Stewart]).
Since the fluctuations in string gas cosmology are of thermal origin, thermal non-Gaussianities will be produced. However, these non-Gaussianities are Poisson suppressed on scales larger than the characteristic scale of the thermal fluctuations, the inverse temperature. Since the temperature in the Hagedorn phase is close to the string scale, the non-Gaussianities on observable scales will be highly suppressed [@YiWang2]. Hence, observing non-Gaussianities on cosmological scales would be a serious challenge for string gas cosmology. On the other hand, the Planck satellite has not seen any non-Gaussianities, and hence also in this respect the predictions of string gas cosmology are consistent with current observations.
There is one type of non-Gaussianities which could be present in string gas cosmology: if we are dealing with a string theory in which cosmic superstrings [@Witten] are stable (see [@Myers] for a discussion of the criteria for this to be the case), then string gas cosmology would leave behind a network of cosmic strings in our three dimensional space. As studied in detail in the case of cosmic strings (see e.g. [@CSreviews] for reviews on cosmic strings and cosmology), the network of cosmic superstrings would take on a “scaling solution” in which the network of strings looks identical at all times when all lengths are scaled to the Hubble radius. The scaling solution corresponds to a fixed number $N$ of infinite string segments crossing each Hubble volume, and a distribution of string loops with radius smaller than $t$ which are produced by the interactions between the infinite strings. As a consequence of the energy which is trapped in the strings, cosmic superstrings (like cosmic strings) leave behind clear signatures in cosmological observations. The power spectrum of the string-induced fluctuations is approximately scale-invariant (see e.g. [@Turok]). The non-Gaussianities are prominent in position space maps: line discontinuities in CMB temperature maps [@KS], rectangles in the sky with direct B-mode polarization [@Holder1], and thin wedges in 21cm redshift maps (extended in the sky over degree scale but thin in redshift direction) [@Holder2]. The current bound on the string tension $\mu$ from not detecting any string-specific signatures is G < 2 10\^[-7]{} . This bound comes from combining Planck data with that of smaller angular scale telescopes [@Dvorkin; @PlanckTD]. With a dedicated position space search using Planck data, a reduction of this limit might be possible. A study of the potential of the South Pole Telescope to contrain $G \mu$ indicated [@Danos] that an improvement of the bound by one order of magnitude should be possible (for a recent discussion of signals of cosmic strings in new observational windows the reader is referred to [@RHBCSrev]).
Discussion and Conclusions
==========================
Assuming the cosmological background given in Fig. 1, string gas cosmology naturally solves the horizon problem of the Standard Big Bang model. In contrast to inflationary cosmology, the string gas cosmology does not provide a mechanism to produce spatial flatness, and it also assumes a large initial size and entropy of space. As shown above, string gas cosmology leads to a structure formation scenario which yields predictions which are in good agreement with all current observations, and makes predictions for future observations with which the model can be distinguished from cosmological inflation.
The Achilles heel of string gas cosmology is the fact that at the current time we do not have a mathematical desciption of the Hagedorn phase. In this phase, non-perturbative string theory will be crucial. Einstein and dilaton gravity are not applicable since these background actions are not consistent with the key symmetries which distinguish string theory from point particle theory. A possible framework to study the dynamics of the Hagedorn phase is “double field theory” [@Hull], a setup in which the number of dimensions is doubled, the two copies being related by T-duality.
References {#references .unnumbered}
==========
[99]{}
P. A. R. Ade [*et al.*]{} \[Planck Collaboration\], “Planck 2015 results. XIII. Cosmological parameters,” arXiv:1502.01589 \[astro-ph.CO\];\
P. A. R. Ade [*et al.*]{} \[BICEP2 and Planck Collaborations\], “Joint Analysis of BICEP2/$Keck Array$ and $Planck$ Data,” Phys. Rev. Lett. [**114**]{}, no. 10, 101301 (2015) \[arXiv:1502.00612 \[astro-ph.CO\]\]. A. Guth, “The Inflationary Universe: A Possible Solution To The Horizon And Flatness Problems,” Phys. Rev. D [**23**]{}, 347 (1981);\
K. Sato, “First Order Phase Transition Of A Vacuum And Expansion Of The Universe,” Mon. Not. Roy. Astron. Soc. [**195**]{}, 467 (1981);\
A. A. Starobinsky, “A New Type of Isotropic Cosmological Models Without Singularity,” Phys. Lett. B [**91**]{}, 99 (1980);\
R. Brout, F. Englert and E. Gunzig, “The Creation Of The Universe As A Quantum Phenomenon,” Annals Phys. [**115**]{}, 78 (1978). V. Mukhanov and G. Chibisov, “Quantum Fluctuation And Nonsingular Universe. (In Russian),” JETP Lett. [**33**]{}, 532 (1981) \[Pisma Zh. Eksp. Teor. Fiz. [**33**]{}, 549 (1981)\]. M. B. Green, J. H. Schwarz and E. Witten, “Superstring Theory. Vols. 1 & 2” (Cambridge Univ. Press, Cambridge, 1987).
D. Baumann, “TASI Lectures on Inflation,” arXiv:0907.5424 \[hep-th\]. R. H. Brandenberger and J. H. Kung, “Chaotic Inflation as an Attractor in Initial Condition Space,” Phys. Rev. D [**42**]{}, 1008 (1990). H. A. Feldman and R. H. Brandenberger, “Chaotic Inflation With Metric and Matter Perturbations,” Phys. Lett. B [**227**]{}, 359 (1989). T. Vachaspati and M. Trodden, “Causality and cosmic inflation,” Phys. Rev. D [**61**]{}, 023502 (1999) \[gr-qc/9811037\]. A. Ijjas, P. J. Steinhardt and A. Loeb, “Inflationary paradigm in trouble after Planck2013,” Phys. Lett. B [**723**]{}, 261 (2013) \[arXiv:1304.2785 \[astro-ph.CO\]\]. L. Berezhiani and M. Trodden, “How Likely are Constituent Quanta to Initiate Inflation?,” arXiv:1504.01730 \[hep-th\]. D. S. Goldwirth and T. Piran, “Initial conditions for inflation,” Phys. Rept. [**214**]{}, 223 (1992). D. Baumann and L. McAllister, “Inflation and String Theory,” arXiv:1404.2601 \[hep-th\];\
C. P. Burgess, M. Cicoli and F. Quevedo, “String Inflation After Planck 2013,” JCAP [**1311**]{}, 003 (2013) \[arXiv:1306.3512 \[hep-th\]\];\
D. Baumann and L. McAllister, “Advances in Inflation in String Theory,” Ann. Rev. Nucl. Part. Sci. [**59**]{}, 67 (2009) \[arXiv:0901.0265 \[hep-th\]\]. J. Polchinski, “String theory. Vols. 1 & 2” (Cambridge Univ. Press, Cambridge, 1998)
T. Boehm and R. Brandenberger, “On T duality in brane gas cosmology,” JCAP [**0306**]{}, 008 (2003) \[hep-th/0208188\]. R. Hagedorn, “Statistical Thermodynamics Of Strong Interactions At High-Energies,” Nuovo Cim. Suppl. [**3**]{}, 147 (1965). R. H. Brandenberger and C. Vafa, “Superstrings In The Early Universe,” Nucl. Phys. B [**316**]{}, 391 (1989). N. Deo, S. Jain, O. Narayan and C. I. Tan, “The Effect of topology on the thermodynamic limit for a string gas,” Phys. Rev. D [**45**]{}, 3641 (1992). A. Nayeri, R. H. Brandenberger and C. Vafa, “Producing a scale-invariant spectrum of perturbations in a Hagedorn phase of string cosmology,” Phys. Rev. Lett. [**97**]{}, 021302 (2006) \[arXiv:hep-th/0511140\];\
R. H. Brandenberger, A. Nayeri, S. P. Patil and C. Vafa, “String gas cosmology and structure formation,” Int. J. Mod. Phys. A [**22**]{}, 3621 (2007) \[hep-th/0608121\]. R. A. Sunyaev, Y. .B. Zeldovich, “Small scale fluctuations of relic radiation,” Astrophys. Space Sci. [**7**]{}, 3-19 (1970);\
P. J. E. Peebles, J. T. Yu, “Primeval adiabatic perturbation in an expanding universe,” Astrophys. J. [**162**]{}, 815-836 (1970).
R. H. Brandenberger, “Cosmology of the Very Early Universe,” AIP Conf. Proc. [**1268**]{}, 3 (2010) \[arXiv:1003.1745 \[hep-th\]\];\
R. H. Brandenberger, “Unconventional Cosmology,” Lect. Notes Phys. [**863**]{}, 333 (2013) \[arXiv:1203.6698 \[astro-ph.CO\]\]. M. Gasperini and G. Veneziano, “Pre - big bang in string cosmology,” Astropart. Phys. [**1**]{}, 317 (1993) \[hep-th/9211021\];\
M. Gasperini and G. Veneziano, “The Pre - big bang scenario in string cosmology,” Phys. Rept. [**373**]{}, 1 (2003) \[hep-th/0207130\]. J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok, “The Ekpyrotic universe: Colliding branes and the origin of the hot big bang,” Phys. Rev. D [**64**]{}, 123522 (2001) \[hep-th/0103239\]. A. Notari and A. Riotto, “Isocurvature perturbations in the ekpyrotic universe,” Nucl. Phys. B [**644**]{}, 371 (2002) \[hep-th/0205019\];\
J. L. Lehners, P. McFadden, N. Turok and P. J. Steinhardt, “Generating ekpyrotic curvature perturbations before the big bang,” Phys. Rev. D [**76**]{}, 103501 (2007) \[hep-th/0702153 \[HEP-TH\]\];\
P. Creminelli and L. Senatore, “A Smooth bouncing cosmology with scale invariant spectrum,” JCAP [**0711**]{}, 010 (2007) \[hep-th/0702165\];\
E. I. Buchbinder, J. Khoury and B. A. Ovrut, “New Ekpyrotic cosmology,” Phys. Rev. D [**76**]{}, 123503 (2007) \[hep-th/0702154\]. V. A. Rubakov, “Harrison-Zeldovich spectrum from conformal invariance,” JCAP [**0909**]{}, 030 (2009) \[arXiv:0906.3693 \[hep-th\]\]. K. Hinterbichler and J. Khoury, “The Pseudo-Conformal Universe: Scale Invariance from Spontaneous Breaking of Conformal Symmetry,” JCAP [**1204**]{}, 023 (2012) \[arXiv:1106.1428 \[hep-th\]\]. D. Wands, “Duality invariance of cosmological perturbation spectra,” Phys. Rev. D [**60**]{}, 023507 (1999) \[gr-qc/9809062\]. F. Finelli and R. Brandenberger, “On the generation of a scale invariant spectrum of adiabatic fluctuations in cosmological models with a contracting phase,” Phys. Rev. D [**65**]{}, 103522 (2002) \[hep-th/0112249\]. P. Horava and E. Witten, “Eleven-dimensional supergravity on a manifold with boundary,” Nucl. Phys. B [**475**]{}, 94 (1996) \[hep-th/9603142\]. C. Kounnas, H. Partouche and N. Toumbas, “Thermal duality and non-singular cosmology in d-dimensional superstrings,” Nucl. Phys. B [**855**]{}, 280 (2012) \[arXiv:1106.0946 \[hep-th\]\]; R. H. Brandenberger, C. Kounnas, H. Partouche, S. P. Patil and N. Toumbas, “Cosmological Perturbations Across an S-brane,” JCAP [**1403**]{}, 015 (2014) \[arXiv:1312.2524 \[hep-th\]\]. R. H. Brandenberger, A. Nayeri, S. P. Patil and C. Vafa, “String gas cosmology and structure formation,” Int. J. Mod. Phys. A [**22**]{}, 3621 (2007) \[hep-th/0608121\];\
T. Battefeld and S. Watson, “String gas cosmology,” Rev. Mod. Phys. [**78**]{}, 435 (2006) \[hep-th/0510022\];\
R. H. Brandenberger, “String Gas Cosmology,” arXiv:0808.0746 \[hep-th\]\
R. H. Brandenberger, “String Gas Cosmology: Progress and Problems,” Class. Quant. Grav. [**28**]{}, 204005 (2011) \[arXiv:1105.3247 \[hep-th\]\]. J. Kripfganz and H. Perlt, “Cosmological Impact of Winding Strings,” Class. Quant. Grav. [**5**]{}, 453 (1988). R. Easther, B. R. Greene and M. G. Jackson, “Cosmological string gas on orbifolds,” Phys. Rev. D [**66**]{}, 023502 (2002) \[hep-th/0204099\]. S. Alexander, R. H. Brandenberger and D. A. Easson, “Brane gases in the early universe,” Phys. Rev. D [**62**]{}, 103509 (2000) \[hep-th/0005212\]. T. Biswas, R. Brandenberger, A. Mazumdar and W. Siegel, “Non-perturbative Gravity, Hagedorn Bounce & CMB,” JCAP [**0712**]{}, 011 (2007) \[hep-th/0610274\]. M. Sakellariadou, “Numerical experiments on string cosmology,” Nucl. Phys. B [**468**]{}, 319 (1996) \[hep-th/9511075\]. S. Watson and R. H. Brandenberger, “Isotropization in brane gas cosmology,” Phys. Rev. D [**67**]{}, 043510 (2003) \[hep-th/0207168\]. S. Watson and R. Brandenberger, “Stabilization of extra dimensions at tree level,” JCAP [**0311**]{}, 008 (2003) \[hep-th/0307044\]. S. P. Patil and R. Brandenberger, “Radion stabilization by stringy effects in general relativity and dilaton gravity,” Phys. Rev. D [**71**]{}, 103522 (2005) \[arXiv:hep-th/0401037\]; S. Watson, “Moduli stabilization with the string Higgs effect,” Phys. Rev. D [**70**]{}, 066005 (2004) \[arXiv:hep-th/0404177\]; L. Kofman, A. Linde, X. Liu, A. Maloney, L. McAllister and E. Silverstein, “Beauty is attractive: Moduli trapping at enhanced symmetry points,” JHEP [**0405**]{}, 030 (2004) \[arXiv:hep-th/0403001\]; S. P. Patil and R. H. Brandenberger, “The cosmology of massless string modes,” JCAP [**0601**]{}, 005 (2006) \[arXiv:hep-th/0502069\]; R. Brandenberger, Y. K. Cheung and S. Watson, “Moduli stabilization with string gases and fluxes,” JHEP [**0605**]{}, 025 (2006) \[arXiv:hep-th/0501032\]; R. J. Danos, A. R. Frey and R. H. Brandenberger, “Stabilizing moduli with thermal matter and nonperturbative effects,” Phys. Rev. D [**77**]{}, 126009 (2008) \[arXiv:0802.1557 \[hep-th\]\]; S. Mishra, W. Xue, R. Brandenberger and U. Yajnik, “Supersymmetry Breaking and Dilaton Stabilization in String Gas Cosmology,” JCAP [**1209**]{}, 015 (2012) \[arXiv:1103.1389 \[hep-th\]\]. R. H. Brandenberger, A. Nayeri, S. P. Patil and C. Vafa, “Tensor Modes from a Primordial Hagedorn Phase of String Cosmology,” Phys. Rev. Lett. [**98**]{}, 231302 (2007) \[hep-th/0604126\]. C. Kiefer, I. Lohmar, D. Polarski and A. A. Starobinsky, “Pointer states for primordial fluctuations in inflationary cosmology,” Class. Quant. Grav. [**24**]{}, 1699 (2007) \[astro-ph/0610700\]. P. Martineau, “On the decoherence of primordial fluctuations during inflation,” Class. Quant. Grav. [**24**]{}, 5817 (2007) \[astro-ph/0601134\]. J. Martin and R. H. Brandenberger, “The TransPlanckian problem of inflationary cosmology,” Phys. Rev. D [**63**]{}, 123501 (2001) \[hep-th/0005209\]. V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, “Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions,” Phys. Rept. [**215**]{}, 203 (1992). R. H. Brandenberger, “Lectures on the theory of cosmological perturbations,” Lect. Notes Phys. [**646**]{}, 127 (2004) \[arXiv:hep-th/0306071\]. A. Nayeri, “Inflation free, stringy generation of scale-invariant cosmological fluctuations in D = 3 + 1 dimensions,” arXiv:hep-th/0607073. R. H. Brandenberger, A. Nayeri and S. P. Patil, “Closed String Thermodynamics and a Blue Tensor Spectrum,” Phys. Rev. D [**90**]{}, no. 6, 067301 (2014) \[arXiv:1403.4927 \[astro-ph.CO\]\]. P. A. R. Ade [*et al.*]{} \[BICEP2 Collaboration\], “Detection of $B$-Mode Polarization at Degree Angular Scales by BICEP2,” Phys. Rev. Lett. [**112**]{}, no. 24, 241101 (2014) \[arXiv:1403.3985 \[astro-ph.CO\]\]. K. M. Smith, C. Dvorkin, L. Boyle, N. Turok, M. Halpern, G. Hinshaw and B. Gold, “Quantifying the BICEP2-Planck Tension over Gravitational Waves,” Phys. Rev. Lett. [**113**]{}, no. 3, 031301 (2014) \[arXiv:1404.0373 \[astro-ph.CO\]\];\
Y. Wang and W. Xue, “Inflation and Alternatives with Blue Tensor Spectra,” JCAP [**1410**]{}, no. 10, 075 (2014) \[arXiv:1403.5817 \[astro-ph.CO\]\]. G. Simard, D. Hanson and G. Holder, “Prospects for Delensing the Cosmic Microwave Background for Studying Inflation,” arXiv:1410.0691 \[astro-ph.CO\];\
L. Boyle, K. M. Smith, C. Dvorkin and N. Turok, “On testing and extending the inflationary consistency relation for tensor modes,” arXiv:1408.3129 \[astro-ph.CO\]. A. Stewart and R. Brandenberger, “Observational Constraints on Theories with a Blue Spectrum of Tensor Modes,” JCAP [**0808**]{}, 012 (2008) \[arXiv:0711.4602 \[astro-ph\]\]. B. Chen, Y. Wang, W. Xue and R. Brandenberger, “String Gas Cosmology and Non-Gaussianities,” arXiv:0712.2477 \[hep-th\]. E. Witten, “Cosmic Superstrings,” Phys. Lett. B [**153**]{}, 243 (1985). E. J. Copeland, R. C. Myers and J. Polchinski, “Cosmic F and D strings,” JHEP [**0406**]{}, 013 (2004) \[hep-th/0312067\]. A. Vilenkin and E.P.S. Shellard, *Cosmic Strings and other Topological Defects* (Cambridge Univ. Press, Cambridge, 1994);\
M. B. Hindmarsh and T. W. B. Kibble, “Cosmic strings,” Rept. Prog. Phys. [**58**]{}, 477 (1995) \[arXiv:hep-ph/9411342\];\
R. H. Brandenberger, “Topological defects and structure formation,” Int. J. Mod. Phys. A [**9**]{}, 2117 (1994) \[arXiv:astro-ph/9310041\]. N. Turok and R. H. Brandenberger, “Cosmic Strings And The Formation Of Galaxies And Clusters Of Galaxies,” Phys. Rev. D [**33**]{}, 2175 (1986). N. Kaiser and A. Stebbins, “Microwave Anisotropy Due To Cosmic Strings,” Nature [**310**]{}, 391 (1984). R. J. Danos, R. H. Brandenberger and G. Holder, “A Signature of Cosmic Strings Wakes in the CMB Polarization,” Phys. Rev. D [**82**]{}, 023513 (2010) \[arXiv:1003.0905 \[astro-ph.CO\]\]. R. H. Brandenberger, R. J. Danos, O. F. Hernandez and G. P. Holder, “The 21 cm Signature of Cosmic String Wakes,” JCAP [**1012**]{}, 028 (2010) \[arXiv:1006.2514 \[astro-ph.CO\]\]. C. Dvorkin, M. Wyman and W. Hu, “Cosmic String constraints from WMAP and the South Pole Telescope,” Phys. Rev. D [**84**]{}, 123519 (2011) \[arXiv:1109.4947 \[astro-ph.CO\]\]. P. A. R. Ade [*et al.*]{} \[Planck Collaboration\], “Planck 2013 results. XXV. Searches for cosmic strings and other topological defects,” Astron. Astrophys. [**571**]{}, A25 (2014) \[arXiv:1303.5085 \[astro-ph.CO\]\]. R. J. Danos and R. H. Brandenberger, “Canny Algorithm, Cosmic Strings and the Cosmic Microwave Background,” Int. J. Mod. Phys. D [**19**]{}, 183 (2010) \[arXiv:0811.2004 \[astro-ph\]\];\
S. Amsel, J. Berger and R. H. Brandenberger, “Detecting Cosmic Strings in the CMB with the Canny Algorithm,” JCAP [**0804**]{}, 015 (2008) \[arXiv:0709.0982 \[astro-ph\]\];\
A. Stewart and R. Brandenberger, “Edge Detection, Cosmic Strings and the South Pole Telescope,” JCAP [**0902**]{}, 009 (2009) \[arXiv:0809.0865 \[astro-ph\]\]. R. H. Brandenberger, “Searching for Cosmic Strings in New Observational Windows,” Nucl. Phys. Proc. Suppl. [**246-247**]{}, 45 (2014) \[arXiv:1301.2856 \[astro-ph.CO\]\]. C. Hull and B. Zwiebach, “Double Field Theory,” JHEP [**0909**]{}, 099 (2009) \[arXiv:0904.4664 \[hep-th\]\].
[^1]: However, beyond linear order the story may be very different [@Tanmay; @Paul; @Mark]. Also, in the case of small field inflation there is an initial condition problem [@Piran].
|
---
abstract: 'Large $H$–selfadjoint random matrices are considered. The matrix $H$ is assumed to have one negative eigenvalue, hence the matrix in question has precisely one eigenvalue of nonpositive type. It is showed that this eigenvalue converges in probability to a deterministic limit. The weak limit of distribution of the real eigenvalues is investigated as well.'
address: |
Faculty of Mathematics and Computer Science\
Jagiellonian University\
Łojasiewicza 6\
30-348 Kraków\
Poland
author:
- Michał Wojtylak
title: 'On a class of $H$–selfadjont random matrices with one eigenvalue of nonpositive type.'
---
[^1]
Introduction {#introduction .unnumbered}
============
The main object of this survey are random matrices that are not symmetric, but are selfadjoint with respect to an indefinite inner product. Spectrum and numerical range of some classes of such matrices were considered recently in [@NCR1; @NCR2], although, in the present paper we consider different instances. The inefinite linear algebra motivation of the present research is the follwoing. Consider an invertible, hermitian–symmetric matrix $H\in{\mathbb{C}}^{n\times n}$. We say that $X\in{\mathbb{C}}^{n\times n}$ is $H$–selfadjoint if $X^*H=HX$. This is the same as to say that $A$ is selfadjoint with respect to an inner product $$[x,y]_H:=y^*Hx,\quad x,y\in{\mathbb{C}}^n.$$ Note that this inner product is not positive definite if $H$ has negative eigenvalues. In the literature the space ${\mathbb{C}}^n$ with the inner product $[\cdot,\cdot]_H$ is also called $\Pi_\kappa$–space (where $\kappa$ is the number of negative eigenvalues of $H$) or Pontryagin space, the infinite dimensional case is considered as well [@bognar; @langerio]. In the present paper the case when $$\label{H}
H =\left[
\begin{array}{cc}
-1 & 0 \\
0 & I_{N}
\end{array}
\right],$$ is considered. It is easy to check that for such $H$ each $H$–selfadjoint matrix has the form $$\label{A}
X=\left[
\begin{array}{cc}
a & -b^*\\
b & C \\
\end{array}
\right],$$ with $x\in{\mathbb{R}}$, $b\in{\mathbb{C}}^N$ and a hermitian–symmetric matrix $C\in{\mathbb{C}}^{N\times N}$. Due to the famous theorem of Pontryagin [@pontr] the matrix $X$ has precisely one eigenvalue $\beta$, for which the corresponding eigenvector $x$ satisfies $[x,x]_H\leq 0$. The problem of tracking the nonpositive eigenvalue was considered for example in [@DHS3; @SWW]. In those papers the setting was non–random and $X$ was in the family of one dimensional extensions of a fixed operator in an infinite dimensional $\Pi_1$–space. The aim of the present work is to investigate the behavior of $\beta$ when $X$ is a large random matrix. We show that the main method of [@DHS3; @SWW] – the use of Nevanlinna functions with one negative square – can be adapted to the random setting as well.
A classical result of Wigner [@wigner] says that if the random variables $y_{ij}$, $0\leq i\leq j<+\infty$ are real, i.i.d with mean zero and variance equal one, then the distribution of eigenvalues of a matrix $$Y_N=\frac{1}{\sqrt{N}} [y_{ij}]_{ij=0}^N,$$ where $y_{ji}=y_{ij}$ for $j>i$, converge weakly in probability to the Wigner semicircle measure. Note that by multiplying the first row of $Y_N$ by -1 we obtain a $H$–selfadjoint matrix $X_N$. A result of a preliminary numerical experiment with gaussian $y_{ij}$ is plotted in Figure \[fig\].
![Eigenvalues of a random matrix $X_{100}$ computed with R [@R] []{data-label="fig"}](eig1.pdf){width="200pt"}
Note that the spectrum of $X_N$ is real, except two eigenvalues, lying symmetrically with respect to the real line. Although we pay a special attention to the above case, we study the behavior of the eigenvalue of nonpositive type in a more general setting. Namely, we assume that the random matrix $X_N$ in ${\mathbb{C}}^{N\times N}$ is of a form $$X=\left[
\begin{array}{cc}
a_N & -b_N^*\\
b_N & C_N \\
\end{array},\right],$$ with $a_N$, $b_N$ and $C_N$ being independent. Furthermore, the vector $b_N$ is a column of a Wigner matrix and $a_N$ converges weakly to zero. The only assumption on $C_N$ is that the limit distribution of its eigenvalues converge weakly in probability, see (R0)–(R3) for details. In Theorem \[main\] we prove that under these assumptions the non–real eigenvlaues converge in probability to a deterministic limit that can be computed knowing the limit distribution of eigenvalues of $C_N$. In the case when $C_N$ is a Wiegner matrix the nonreal eigenvalues converge to $\pm\operatorname{i}\sqrt{2}/2$, cf. Theorem \[WiegnerTh\]. Furthermore, under a technical assumption of continuity of the entries of $X_N$, we show in Theorem \[realev\] that the limit distribution of the real eigenvalues of $X_N$ coincides with the limit distribution of eigenvalues for the matrices $C_N$. Again, in the case when $C_N$ is a Wiegner matrix we obtain a more precise result. Namely, in Theorem \[WiegnerTh\] we show that the real eigenvalues $\zeta_2^N{,\dots,}\zeta_N^N$ of $X_N$ and the eigenvalues $\lambda_1^N{,\dots,}\lambda_N^N$ of $C_N$ satisfy the following inequalities: $$\lambda_1^N < \zeta_2^N < \lambda_2^N < \cdots <\lambda_{N-1}^N < \zeta_{N}^N < \lambda_N^N.$$ It shows that the nonreal eigenvalue of $X_N$ plays an analogue role as the largest eigenvalue in one–dimensional, symmetric perturbations of Wiegner matrices. This fact relates the present paper to the current work on finite dimensional perturbations of random matrices, see [@BG1; @BG2; @BG3; @cap1; @capitaine; @FP; @knowles] and references therein. Also note that $X_N$ is a product of a random and deterministic matrix, such products were already considered in the literature, see e.g. [@vershynin].
The author is indebted to Anna Szczepanek for preliminary numerical simulations and Maxim Derevyagin for his valuable comments. Special thanks to Anna Kula, Patryk Pagacz and Janusz Wysoczański.
Functions of class $\mathcal{N}_1$ {#sN1}
==================================
The Nevanlinna functions with negative squares play a similar role for the class of $H$–selfadjoint matrices as the class of ordinary Nevanlinna function plays for hermitian–symmetric matrices. This phenomenon has its roots in in operator theory, we refer the reader to [@DHS1; @DHS3; @KL71; @KL77; @KL81] and papers quoted therein for a precise description of a relation between $\mathcal{N}_\kappa$–functions and selfadjoint operators in Krein and Pontryagin spaces. We begin with a very general definition of the class $\mathcal{N}_\kappa$, but we immediately restrict ourselves to certain subclasses of those functions.
We say that $Q$ is *a generalized Nevanlinna function of class $\mathcal N_{\kappa}$* [@KL71; @L] if it is meromorphic in the upper half–plane ${\mathbb{C}}^+$ and the kernel $$N(z,w)=\frac{Q(z)-\overline{Q(w)}}{z-\bar w}$$ has precisely $\kappa$ negative squares, that is for any finite sequence $z_1{,\dots,}z_k\in{\mathbb{C}}^+$ the hermitian–symmetric matrix $$[N(z_i,z_j)]_{ij=1}^k$$ has not more then $\kappa$ nonpositive eigenvalues and for some choice of $z_1{,\dots,}z_k$ it has precisely $\kappa$ nonpositive eigenvalues. In the present paper we use this definition with $\kappa=0,1$.
The class $\mathcal{N}_0$ is the class of ordinary Nevanlinna functions, i.e. the functions that are holomorphic in ${\mathbb{C}}^+$ with nonnegative imaginary part. By $M^+_b({\mathbb{R}})$ we denote the set of positive, bounded Borel measures on ${\mathbb{R}}$. For $\mu\in M^+_b({\mathbb{R}})$ we define the Stieltjes transform as $$\hat\mu(z)=\int_{\mathbb{R}}\frac 1{t-z}dt ,\quad z\in{\mathbb{C}}\setminus\operatorname{supp}\mu.$$ Clearly, $\hat\mu$ belongs to the class $\mathcal{N}_0$ and the values of $\hat\mu$ in the upper half–plane determine the measure uniquelly by the Stieltjes inversion formula. Although not every function of class $\mathcal{N}_0$ is a Stieltjes transform of a Borel measure (cf. [@donoghue]), this subclass of $\mathcal{N}_0$ functions will be sufficient for present reasonings. Also, we will be interested in a special subclass of $\mathcal{N}_1$ functions, namely in the functions of the form below. We refer the reader to the literature [@DHS1; @DLLSh] for representation theorems for $\mathcal{N}_\kappa$ functions.
\[N1\] If $\mu\in M_b^+({\mathbb{R}})$, $a\in{\mathbb{R}}$ then $$\label{Qform}
Q(z)=\hat\mu(z) + a - z$$ is a holomorphic function in ${\mathbb{C}}^+$ and belongs to the class $\mathcal{N}_1$. Furthermore, there exists precisely one $z_0\in{\mathbb{C}}$ such that either $z_0\in{\mathbb{C}}^+$ and $$\label{x_}
Q(z_0)=0,$$ or $z_0\in{\mathbb{R}}$ and $$\label{x_0}
\lim_{z \hat\to z_0}\frac{Q(z)}{z-z_0}\in (-\infty,0].$$
The symbol $\hat\to$ above denotes the non-tangential limit: $$z\in{\mathbb{C}}^+,\quad z\to z_0,\quad \pi/2-\theta \leq \arg(z-z_0)\leq \pi/2+\theta,$$ with some $\theta\in(0,\pi/2)$. We call $z_0\in{\mathbb{C}}^+\cup{\mathbb{R}}$ the *generalized zero of nonpositive type* (*GZNT*) of $Q(z)$. The first part of the Proposition can be found e.g. in [@KL77], while for the proof of the ’Furthermore’ part in the general context[^2] we refer the reader to [@L Theorem 3.1, Theorem 3.1’]. In view of the above proposition we can define a function $$G:M_b^+({\mathbb{R}})\times{\mathbb{R}}\to {\mathbb{C}}^+$$ by saying that $G(\mu,a)$ is the GZNT of the function $\hat\mu(z) + a - z$. The following proposition plays a crucial role in our arguments.
\[cont\] The function $G$ is jointly continuous with respect to the weak topology on $M_b^+({\mathbb{R}})$ and the standard topology on ${\mathbb{R}}$.
Assume that $(\mu_n)_n{\subset}M^+_b({\mathbb{R}})$ converges weakly to $\mu\in M^+_b({\mathbb{R}})$ and $a_n\in{\mathbb{R}}$ converges to $a\in{\mathbb{R}}$ with $n\to\infty$. Take a compact $K$ in the open upper half–plane, with nonempty interior. Then $\hat\mu_n$ converges uniformly to $\hat\mu$ on the set $K$. Indeed, if $r=\sup_{t\in{\mathbb{R}},\ z\in K}1/|t-z|$ then $$\sup_{z\in K} |\hat\mu_n(z) - \hat\mu_0 (z)| \leq r |\mu_n - \mu_0|({\mathbb{R}}),$$ the latter clearly converging to zero with $n\to\infty$. In consequence, $\hat\mu_n(z) + a_n - z$ converges to $\hat\mu(z) + a - z$ uniformly on $K$ with $n\to\infty$. By [@LaLuMa] the GZNT of $\hat\mu_n(z) + a_n - z$ converges to the GZNT of $\hat\mu(z) + a - z$, which finishes the proof.
$H$–selfadjoint matrices
========================
In this section we review basic properties of selfadjoint matrices in indefinite inner product spaces introducing the concept of a canonical form and showing its relation with $\mathcal{N}_1$–functions. Let $H\in{\mathbb{C}}^{(n+1)\times( n+1)}$ ($n\in{\mathbb{N}}\setminus{\left\{0\right\}}$) be an invertible, Hermitian–symmetric matrix. We say that $X\in{\mathbb{C}}^{(n+1)\times( n+1)}$ is *$H$–selfadjoint* if $X^*H=HX$. Our main interest will lie in the matrix $$\label{H}
H =\left[
\begin{array}{cc}
-1 & 0 \\
0 & I_{n}
\end{array}
\right],$$ where $I_n$ denotes the identity matrix of size $n\times n$. As it was already mentioned, each $H$–selfadjoint matrix has the form $$\label{A}
X =\left[
\begin{array}{cc}
a & -b^*\\
b & C \\
\end{array}
\right],$$ with $a\in{\mathbb{R}}$, $b\in{\mathbb{C}}^n$ and hermitian–symmetric $C\in{\mathbb{C}}^{n\times n}$. Due to [@GLR] there exists an invertible matrix $S$ and a pair of matrices $H',S'\in{\mathbb{C}}^{(n+1)\times(n+1)}$ such that $X=S^{-1}X'S$ $H=S^* H' S$ and $X',H'$ are of one of the following forms:
- $$X'={\begin{bmatrix} \beta & 0\\ 0 & \bar \beta \end{bmatrix}} \oplus \operatorname{diag}(\zeta_2{,\dots,}\zeta_{n}),\quad H'= {\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}} \oplus I_{n-1},$$ with $\beta\in{\mathbb{C}}^+$, $\zeta_2{,\dots,}\zeta_{n}\in{\mathbb{R}}$.
- $$X'= [\beta]\oplus\operatorname{diag}( \zeta_1{,\dots,}\zeta_{n}),\quad H'= [-1] \oplus I_n,$$ with $\beta\in{\mathbb{R}}$, $\zeta_1{,\dots,}\zeta_{n}\in{\mathbb{R}}$.
- $$X'={\begin{bmatrix} \beta & 1\\ 0 & \beta \end{bmatrix}} \oplus \operatorname{diag}(\zeta_2{,\dots,}\zeta_{n}), \quad H'=\gamma {\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}} \oplus I_{n-1},$$ with $\beta\in{\mathbb{R}}$, $\zeta_2{,\dots,}\zeta_{n}\in{\mathbb{R}}$, $\gamma\in{\left\{-1,1\right\}}$.
- $$X'={\begin{bmatrix} \beta & 1 & 0 \\ 0 & \beta & 1\\ 0 & 0 & \beta \end{bmatrix}} \oplus \operatorname{diag}(\zeta_3{,\dots,}\zeta_{n}), \quad H'= {\begin{bmatrix} 0 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 0 \end{bmatrix}} \oplus I_{n-2},$$ with $\beta\in{\mathbb{R}}$, $\zeta_3{,\dots,}\zeta_{n}\in{\mathbb{R}}$.
It is easy to verify that in each case $X'$ is $H'$-symmetric. The pair $(X',H')$ is called *the canonical form* of $(X,H)$. We refer the reader to [@GLR] for the proof and for canonical forms for general H–symmetric matrices and to [@bognar; @langerio] for the infinite–dimensional counterpart of the theory. At this point is enough to mention that the canonical form is uniquely determined (up to permutations of the numbers $\zeta_i$) for each pair $(X,H)$, where $X$ is $H$–selfadjoint. Note that in each of the cases $\beta$ is an eigenvalue of $X$ and there exists a corresponding eigenvector $x\in{\mathbb{C}}^{n+1}$ satisfying $[x,x]_H\leq0$, furthermore, $\beta$ is the only eigenvalue in ${\mathbb{C}}^+\cup{\mathbb{R}}$ having this property. Therefore, we will call $\beta$ the *eigenvalue of nonpositive type of $X$*.
Observe that the function $$\label{QX}
Q(z)=a-z+b^*(C-z)^{-1}b$$ is an $\mathcal{N}_1$–function. Indeed, if $D=UCU^*=\operatorname{diag}(\lambda_1{,\dots,}\lambda_n)$ is a diagonalization of the hermitian–symmetric matrix $C$ and $d=Ub$ then $$Q(z)=a-z +\sum_{j=1}^n \frac{|d_j|^2}{\lambda_j-z}=a-z+\hat\mu(z),\quad\text{where}\quad \mu=\sum_{j=1}^n|d_j|^2 \delta_{\lambda_j},$$ and we may apply Proposition \[N1\]. The following lemma is a standard in the indefinite linear algebra theory. We present the proof for the reader’s convenience.
\[ZZ\] Let $X$ and $Q$ be defined by and , respectively. A point $\beta\in{\mathbb{C}}^+\cup{\mathbb{R}}$ is the eigenvalue of nonpositive type of $X$ if and only if it is the GZNT of $Q(z)$. Furthermore, the algebraic multiplicity of $\beta$ as an eigenvalue of $X$ equals the order of $\beta$ as a zero of $Q(z)$.
First note that, due to the Shur complement formula[^3], $$-\frac{1}{Q(z)}={e^*H(X-z)^{-1}e},$$ where $e$ denotes the first vector of the canonical basis of ${\mathbb{C}}^{n+1}$. Let $(X',H')$ be the canonical form of $(X,H)$ and let $S$ be the appropriate transformation. Consequently, $$\label{-1/Q}
-\frac{1}{Q(z)}= {(Se_1)^* H' (X'-z)^{-1} Se_1}.$$ Below we evaluate this expression in each of the Cases 1–4. Let $f=[f_0{,\dots,}f_n]^\top=Se$. Note that $$\label{ef}
f_1^* H' f_1= e_1^*He_1=-1,$$ independently on the Case.
Case 1. Observe that $f_0\bar f_1\neq 0$, otherwise $f^*H'f \geq 0$, which contradicts . Due to one has $$-\frac{1}{Q(z)}= \frac{f_0\bar f_1}{\beta-z}+\frac{f_1 \bar f_0}{\bar\beta-z} +\sum_{j=2}^{n} \frac{ |f_{j}|^2}{\zeta_j-z}.$$ Hence, $\beta\in{\mathbb{C}}^+$ is a simple pole of $-1/Q$ and consequently it is the GZNT of $Q$ and a simple zero of $Q$.
Case 2. Observe that $|f_0|^2>\sum_{j=1}^n|f_j|^2$, otherwise $f^*H'f \geq 0$, which contradicts . Due to one has $$-\frac{1}{Q(z)}= \frac{-|f_0|^2}{\beta-z}+\sum_{j=1}^{n} \frac{ |f_{j}|^2}{\zeta_j-z}.$$ Hence, the residue of $-1/Q$ in $\beta$ is less then zero. Consequently $Q(\beta)=0$, $Q'(\beta)<0$ and $\beta$ is the GZNT of $Q$.
Case 3. Observe that $|f_1|^2>0$, otherwise $f^*H'f \geq 0$, which contradicts . Due to one has $$-\frac{1}{Q(z)}= \frac{2\gamma\operatorname{Re}f_0\bar f_1}{\beta-z}+ \frac{-\gamma|f_1|^2}{(\beta-z)^2}+ \sum_{j=2}^{n} \frac{ |f_{j}|^2}{\zeta_j-z}.$$ Hence, $\beta$ is pole of $-1/Q$ of order 2. Consequently, $Q(\beta)=Q'(\beta)=0$, $Q''(\beta)\neq 0$ and $\beta$ is the GZNT.
Case 4. Observe that $|f_2|^2>0$, otherwise $f^*H'f\geq 0$, which contradicts . Due to one has $$-\frac{1}{Q(z)}= \frac{2\operatorname{Re}f_0\bar f_2+|f_1|^2}{\beta-z}+ \frac{-2\operatorname{Re}f_1\bar f_2}{(\beta-z)^2}+\frac{|f_2|^2}{(\beta-z)^3}+ \sum_{j=3}^{n} \frac{ |f_{j}|^2}{\zeta_j-z},$$ Hence, $\beta$ is pole of $-1/Q$ of order 3. Consequently, $Q(\beta)=Q'(\beta)=Q''(\beta)=0$, $Q'''(\beta)\neq0$ and $\beta$ is the GZNT of $Q$.
Random $H$–selfadjoint matrices
===============================
By $X_N$, $H_N$ we understand the following pair of a random and deterministic matrix in ${\mathbb{C}}^{(N+1)\times(N+1)}$ $$\label{XH}
X_N=\left[
\begin{array}{cc}
a_N & -b_N^*\\
b_N & C_N \\
\end{array}
\right], \quad
H_N={\begin{bmatrix} -1 & 0 \\ 0 & I_{N} \end{bmatrix}},$$ where $a_N$ is a real–valued random variable, $b_N$ is a random vector in ${\mathbb{C}}^N$, and $C_N$ is a hermitian–symmetric random matrix in ${\mathbb{C}}^{N\times N}$. Note that $X_N$ is $H_N$–symmetric. By $\lambda_1^N\leq\cdots \leq\lambda_N^N$ we denote the eigenvalues of $C_N $ and by $\nu_N$ we denote the random measure on ${\mathbb{R}}$ $$\nu_N=\frac 1N \sum_{j=1}^N \delta_{\lambda_j^N}.$$ Recall that $$\label{RStj}
\hat \nu_N(z) = \frac { \operatorname{tr}(C_N-z)^{-1}}N.$$ The assumptions on $X_N$ are as follows:
- The random variable $a_N$ is independent on the entries of the vector $b_N$ and on the entries of the matrix $C_N$ for each $N>0$, futhermore $a_N$ converges with $N\to \infty$ to zero in probability.
- The random vector $b_N$ is of the form $$b_N:= \frac1{\sqrt{N}}[x_{j0}]_{j=1{,\dots,}N},$$ where $[x_{j0}]_{j>0 }$ are i.i.d. random variables, independent on the entries of $C_N$ for $N>0$, of zero mean with $E|x_{j0}|^2={s}^2$ for $ j>0 $.
- The random measure $\nu_N$ converges with $N\to \infty$ to some non–random measure $\mu_0$ weakly in probability
All the results below hold also in the case when all variables $x_{j0}$ ($j>0$) are real, in this situation $b_N^*$ is just the transpose of $b_N$. The entries of $C_N$ might be as well real or complex. In Section \[sW\] we will consider two instances of the matrix $C_N$: a Wiegner matrix and a diagonal matrix. In the case when $C_N$ is a Wiegner matrix the proposition below is a consequence of the isotropic semicircle law [@Erdos; @knowles]. We present below a simple proof of the general case, based on the ideas in [@MP69].
\[BQ\] Assume that [(R1)]{} and [(R2)]{} are satisfied. Then for each $z\in{\mathbb{C}}^+$ $$b_N^*(C_N -z)^{-1} b_N \to {s}^2\ \hat\mu_0(z) \quad (N\to\infty)$$ in probability.
By ${\left\Verty\right\Vert}$ we denote the euclidean norm of $y\in{\mathbb{C}}^n$.
In the light of Chebyshev’s inequality, and assumption (R2) it is enough to show that $$\label{rr}
\lim_{N\to\infty}E\left|b_N^*(C_N-z)^{-1}b_N - {s}^2\frac{\operatorname{tr}(C_N-z)^{-1}}N\right|^2 =0.$$ Observe that $$\label{summand1}
E \left|b_N^*(C_N-z)^{-1} b_N -{s}^2\frac{\operatorname{tr}(C_N-z)^{-1}}N\right|^2=$$ $$\left(E\left|b_N^*(C_N-z)^{-1} b_N\right|^2 - {s}^4E\left|\frac{\operatorname{tr}(C_N-z)^{-1}}N\right|^2 \right)-$$ $$\label{summand2}
2\operatorname{Re}E\left({s}^2\overline{\frac{\operatorname{tr}(C_N-z)^{-1}}N}\left(b_N^*(C_N-z)^{-1} b_N-{s}^2 \frac{\operatorname{tr}(C_N-z)^{-1}}N\right)\right).$$ First we prove that the summand equals zero. Indeed, conditioning on the $\sigma$–algebra generated by the entries of the matrix $C_N$ and setting $$[c_{ij}]_{ij=1}^N=(C_N-z)^{-1}$$ one obtains $$E\left({s}^2\overline{\frac{\operatorname{tr}(C_N-z)^{-1}}N}\left(b_N^*(C_N-z)^{-1} b_N- {s}^2\frac{\operatorname{tr}(C_N-z)^{-1}}N\right)\right) =$$ $$E\left( {s}^2 \sum_{i=1}^N \frac{\overline{ c_{ii}}}N \left( \sum_{jk=1}^N c_{jk}\frac{x_{0j}\overline{x_{0k}}}{N} - {s}^2 \sum_{j=1}^N \frac{c_{jj}}N\right) \right)=$$ $$E\left( {s}^2 \sum_{i=1}^N \frac{\overline{ c_{ii}}}N \left( \sum_{j=1}^N c_{jj}\frac{s^2}{N} - {s}^2 \sum_{j=1}^N \frac{c_{jj}}N\right) \right)=0.$$
Next, observe that $$E|b_N^*(C_N-z)^{-1} b_N|^2 = E \sum_{ijkl=1}^N c_{ij} \overline{c_{kl}}\frac{ x_{0i} \overline{ x_{0j}} x_{0k} \overline{x_{0l}}}{N^2} =$$ $${s}^4 \sum_{ij=1}^N \frac{ E( c_{ii} \overline{ c_{jj}} )}{N^2} + {s}^4 \sum_{ij=1}^N \frac{E(c_{ij}\overline{c_{ij}})}{N^2}=
{s}^{4}E\left|\frac{ \operatorname{tr}(C_N-z)^{-1}}{N}\right|^2 +{s}^{4}E \sum_{ij=1}^N \frac{c_{ij}\overline{c_{ij}}}{N^2}.$$ This allows us to estimate by $$$$ [s]{}\^4 | E \_[ij=1]{}\^N | \^4 E=N, $$$$ $$$$ which finishes the proof of .
Let $U_N$ be a unitary matrix, such that $U_NC_NU^*_N$ is diagonal and let $d_N=[d^N_1{,\dots,}d^N_N]^\top=U_Nb_N$. Denote by $\mu_N$ the measure defined by $$\mu_N = \sum_{j=1}^N | d_{j}^N |^2 \delta_{\lambda^N_j},$$ and observe that $\hat\mu_N(z)=b_N^*(C_N-z)^{-1}b_N$.
\[conv\] Assume that [(R1)]{} and [(R2)]{} are satisfied. Then the sequence of random measures $\mu_N$ converge weakly with $N\to\infty$ to $\mu_0$ in probability.
First note that almost surely $\mu_N({\mathbb{R}}) \to {s}^2\mu_0({\mathbb{R}})$ with $N\to \infty$. Indeed, $$\mu_N({\mathbb{R}})= \sum_{j=1}^N |d_j^N|^2 = {\left\Vertd_N\right\Vert}^2 = {\left\Vertb_N\right\Vert}^2=\frac1N\sum_{j=1}^N |x_{0j}|^2,$$ which converges almost surely to ${s}^2$ by the strong law of large numbers. Furthermore, Proposition \[BQ\] shows that $\hat\mu_N(z)$ converges in probability to $\hat\mu_0(z)$ for every $z\in{\mathbb{C}}^+$. Repeating the proof of Theorem 2.4.4 of [@AGZ] we get the weak convegence of $\mu_N$ in probability.
Main results
============
\[main\] If [(R0) – (R2)]{} are satisfied then the eigenvalue of nonpositive type $\beta_N$ of $X_N$ converges in probability to the GZNT $\beta_0$ of the $\mathcal{N}_1$–function $$Q_0(z)=-z+{s}^2\hat\mu_0(z).$$
Consider a sequence of $\mathcal{N}_1$–functions $$\label{QN}
Q_N(z)=a_N-z+\hat\mu_N(z).$$ Recall that each of those functions has precisely one GZNT which, by definition of $\mu_N$ and Lemma \[ZZ\], is the eigenvalue of nonpositive type $\beta_N$ of $X_N$. Recall that $a_N$ converges to zero in probability by (R0) and $\mu_N$ converges to $\mu_0$ in probability by Proposition \[conv\]. Let $d$ be any metric that metrizises the topology of weak convergence on $M_b^+({\mathbb{R}})$. Since $\beta_N$ is a continuous function of $\mu_N$ and $a_N$ (Proposition \[cont\]), for each ${\varepsilon}>0$ one can find $\delta >0$ such that for each $N>0$ the event ${\left\{|a_N|<\delta,\ d(\mu_N,\mu_0)<\delta\right\}}$ is contained in ${\left\{ |\beta_N - \beta_0|<{\varepsilon}\right\}}$. Using the assumed in (R0) independence of $\mu_N$ and $a_N$ one obtains $$P(|\beta_0-\beta_N|\geq{\varepsilon}) \leq P(|a_N|\geq\delta) \cdot P( d(\mu_N,\mu_0)\geq\delta ).$$ Hence, $\beta_N$ converges to $\beta_0$ in probability.
As it was explained in Section \[N1\], each matrix $X_N$ has, besides the eigenvalue $\beta_N$ of nonpositive type, a set of real eigenvaules $\zeta_{k_N}^N{,\dots,}\zeta_N^N$, where $k_N=1$ in Case 1 and 3, $k_N=2$ in Case 2 and $k_N=3$ in Case 3. By $\tau_N$ we denote the empirical measure connected with these eigenvalues: $$\tau_N=\frac1N\sum_{j=k_N} ^N \delta_{\zeta_j}^N.$$
\[realev\] If [(R0)–(R2)]{} are saisfied and the random variables ${\left\{x_{0j}: j>0\right\}}$ are continuous, then the measure $\tau_N$ converges weakly in probability to $\mu_0$.
We use the notations $U_N,d_N$ and $\mu_N$ from the previous section, let also $Q_N$ be given by . Note that the set ${\left\{y\in{\mathbb{C}}^N:(U_N y)_{j}=0\right\}}$ is of Lebesgue measure zero. Hence, with probability one $d_j^N\neq0$ for $j=1{,\dots,}N$, $N>0$. Therefore, $\hat\mu_N(z)$ is a rational function almost surely with poles of order one in $\lambda_1^N{,\dots,}\lambda_N^N$. Furthermore, $$Q_N(z) = \frac{(a_N-z)\ \prod_{j=1}^N(\lambda^N_j-z)+\sum_{i=1}^N|d^N_i|^2\prod_{j\neq i} (\lambda^N_j-z) }{\prod_{j=1}^N(\lambda^N_j-z)}.$$ In consequence, $Q_N$ has exactly $N+1$ zeros counting multiplicities, all of them different from $\lambda_1^N{,\dots,}\lambda_N^N$. Due to the Schur complement argument, each of those zeros is an eigenvalue of the matrix $X_N\in{\mathbb{C}}^{(N+1)\times(N+1)}$. Furthermore, due to Lemma \[ZZ\] the algebraic multiplicity of $\beta_N$ as eigenvalue of $X_N$ equals the order of $\beta_N$ as a zero of $Q_N$. In consequence, the spectrum of $X_N$ coincides with the zeros of $Q_N$ and $\beta_N$ is the only zero of order possibly greater then one[^4].
On the other hand, the function $\hat\mu_N$ is increasing on the real line with simple poles in $\lambda_1^N{,\dots,}\lambda_N^N$. Hence, in each of the intervals $(\lambda_j^N,\lambda_{j+1}^N)$ $(j=1{,\dots,}N-1)$ there is an odd number of zeros of $Q_N$, counting multiplicities. Consequently, in each of the intervals $(\lambda_j^N,\lambda_{j+1}^N)$ $(j=1{,\dots,}N-1)$ there is precisely one zero of $Q_N$, except possibly one interval that contains three zeros of $Q_N$. Out of these three zeros of $Q_N$ either one or two of them belong to the set ${\left\{\zeta_{k_N}^N{,\dots,}\zeta_{N}^N\right\}}$, accordingly to the canonical form of $X_N$. Hence, in each of the intervals $(\lambda_j^N,\lambda_{j+1}^N)$ $(j=1{,\dots,}N-1)$ there is precisely one of the eigenvalues $\zeta_{k_N}^N{,\dots,}\zeta_{N}^N$, except possibly one interval that contains two of the eigenvalues $\zeta_{k_N}^N{,\dots,}\zeta_{N}^N$. Consequently, the weak limit of $\tau_N$ in probability equals the weak limit of $\nu_N$.
Two instances {#sW}
=============
In the present section we consider two instances of $C_N$: the Wigner matrix and the diagonal matrix. These both cases appear naturally as applications of main results. We refer the reader to [@P72] for a scheme joining both examples.
Consider an $H$–selfadjoint real Wigner matrix $$\label{XW}
X_N:= \frac1{\sqrt{N}}\ H_N [x_{ij}]_{ij=0}^N,$$ with $x_{ij}$ real, $x_{ij}= x_{ji}$ $(0\leq i < j <\infty)$, i.i.d., of zero mean and variance equal to ${s}^2$, and let $H_N$ be defined as in . Clearly $X_N$ is $H_N$–selfadjoint and satisfies (R0)–(R2) with $\mu_0$ equal to the Wiegner semicircle measure $\sigma$. The Stieltjes transform of the $\sigma$ equals $$\hat\sigma(z) = \frac{-z+\sqrt{z^2 - 4{s}^2}}{2{s}^2}.$$ It is easy to check that $\beta_0=\frac{ \sqrt{2}}2 {s}\operatorname{i}$ is a zero of $Q_0(z)=-z+{s}^2\hat\sigma(z)$. Hence, $\beta_0$ is the GZNT of $Q_0$ and we have proved the first part of the theorem below.
\[WiegnerTh\] Let $X_N$ be defined by . Then
- $\beta_N$ converges in probability to $
\beta_0=\frac{ \sqrt{2}}2 {s}\operatorname{i};
$
- if, additionally, the random variables $b_{ij}$ $(0\leq i<j<\infty)$ are continuous, then the probability of an event that there are precisely $N-1$ real eigenvalues $\zeta^N_2<\cdots <\zeta^N_N$ of $X_N$ and the inequalities $$\label{lambdazeta}
\lambda_1^N < \zeta_2^N < \lambda_2^N < \cdots <\lambda_{N-1}^N < \zeta_{N}^N < \lambda_N^N.$$ are satisfied, converges with $N$ to 1.
\(ii) Assume that $$\label{beta}
|\beta_N-\beta_0|\leq \frac{\sqrt2}4s.$$ Then the canonical form of $(X_N,H_N)$ is as in Case 1. In consequence, there are exactly $N-1$ real eigenvalues $\zeta^N_2{,\dots,}\zeta^N_N$ of $X_N$. Let us recall now the arguments from proof of Theorem \[realev\]. The function $\hat\mu_N$ is increasing on the real line with simple poles in $\lambda_1^N{,\dots,}\lambda_N^N$. In each of the intervals $(\lambda_j^N,\lambda_{j+1}^N)$ $(j=1{,\dots,}N-1)$ there at least one of the eigenvalues $\zeta_{2}^N{,\dots,}\zeta_{N}^N$. Consequently, each of the intervals $(\lambda_j^N,\lambda_{j+1}^N)$ $(j=1{,\dots,}N-1)$ contains precisely one of the eigenvalues $\zeta_{2}^N{,\dots,}\zeta_{N}^N$. To finish the proof it is enough to note that by point (i) for every ${\varepsilon}>0$ there exists $N_0>0$ such that for $N>N_0$ the probability of is greater then $1-{\varepsilon}$.
![The real and imaginary part of $\beta_N$, with real, gaussian entries of $X_N$ and $s^2=1$, computed with R [@R].[]{data-label="figreim"}](Re.pdf "fig:"){width="200pt"}\
-0.9cm ![The real and imaginary part of $\beta_N$, with real, gaussian entries of $X_N$ and $s^2=1$, computed with R [@R].[]{data-label="figreim"}](Im.pdf "fig:"){width="200pt"}
The numerical simulations of values of $\operatorname{Re}\beta_N$ and $\operatorname{Im}\beta_N$ can be found in Figure \[figreim\]. Note that $\beta_0$ lies in open upper half–plane and is satisfied. We provide now an example when $\beta_0\in{\mathbb{R}}$ and show that each number in $[0,\infty)$ can be the limit in . Let $a_N=0$, ${x_{i0}}$ ($i=1,2,{,\dots,}$) be independent real variables of zero mean and variance ${s}^2$ and let $C_N=\operatorname{diag}(c_1{,\dots,}c_N)$, where the random variables ${\left\{c_j:j=1,\dots\right\}}$ are i.i.d. and independent on ${x_{i0}}$ ($i=1,2,{,\dots,}$). Furthermore, let the law of $c_j$ (which is simultaneously the limit measure $\mu_0$) be given by a density $$\phi(t)=\begin{cases}\frac{ 3 t^2}2 &: t\in [-1,1] \\ 0 &: t\in{\mathbb{R}}\setminus[-1,1]\end{cases} .$$ An easy calculation shows that $$\lim_{z\hat\to 0}\frac{\hat\mu_0(z)}{z}= 3.$$ Hence, $$\lim_{z\hat\to 0}\frac{-z+{s}^2\hat\mu_0(z)}{z}=-1+ 3{s}^2$$ and the function $$Q_0(z)=-z+\hat\mu_0(z)=-z+\int_{{\mathbb{R}}}\frac{\phi(t) }{t-z} dt$$ has a GZNT at $z=0$ if ${s}^2\leq1/3$. Note that $\beta_0=0$ lies in the support of $\mu_0$. The case ${s}^2=1/3$ is plotted in Figure \[F3\]. Only the imaginary part is displayed, since the numerical computation of the real part of $\beta_N$ might be not reliable in case $\beta_N\in{\mathbb{R}}$. One may observe that the convergence of $\beta_N$ is worse in Figure \[figreim\]. Also, the canonical form of $(X_N,H_N)$ changes with $N$, contrary to the case when $H_NX_N$ is a Wigner matrix. In the case ${s}^2<1/3$ in numerical simulations point $\beta_N$ is real for all $N$.
![The imaginary part of $\beta_N$.[]{data-label="F3"}](Im3.pdf){width="200pt"}
[99]{}
G.W. Anderson, A. Guionnet, O. Zeitouni, *An Introduction to Random Matrices*, Cambridge University Press 2010.
F. Benaych-Georges, A. Guionnet, and M. Maïda, Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices, arXiv:1009.0145.
F. Benaych-Georges, A. Guionnet, and M. Maïda, Large deviations of the extreme eigenvalues of random deformations of matrices, *Prob. Theor. Rel. Fields* (2010), 1–49.
F. Benaych-Georges and R.R. Nadakuditi, The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices, *Adv. Math.* 227 (2011), 494–521.
J. Bognár, [*Indefinite Inner Product Spaces*]{}, Springer–Verlag, New York–Heidelberg, 1974.
M. Capitaine, C. Donati-Martin, and D. Féral, The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations, *Ann. Prob.* 37 (2009), 1–47.
M. Capitaine, C. Donati-Martin, D. Féral and M. Février, Free convolution with a semicircular distribution and eigenvalues of spiked deformations of Wigner matrices, *Electronic Journal of Probability*, 16 (2011), 1750–1792.
V. Derkach, S. Hassi, and H.S.V. de Snoo, Operator models associated with Kac subclasses of generalized Nevanlinna functions, *Methods of Functional Analysis and Topology*, 5 (1999), 65–87.
V.A. Derkach, S. Hassi, and H.S.V. de Snoo, Rank one perturbations in a Pontryagin space with one negative square, *J. Functional Analysis*, 188 (2002), 317–349.
A. Dijksma, H. Langer, A. Luger, and Yu. Shondin, A factorization result for generalized Nevanlinna functions of the class ${\mathbf{N}}_\kappa$, *Integral Equations Operator Theory*, 36 (2000), 121–125.
W.F. Donoghue, [*Monotone matrix functions and analytic continuation*]{}, Springer–Verlag, New York-Heidelberg, 1974.
L. Erdös, Rigidity of eigenvalues of generalized Wigner matrices, *Adv. Math. Preprint* 229 (2012), 1435–1515.
D. Féral and S. Péché, The largest eigenvalue of rank one deformation of large Wigner matrices, *Comm. Math. Phys.* 272 (2007), 185–228.
I. Gohberg, P. Lancaster and L. Rodman: *Indefinite Linear Algebra and Applications*. Birkhäuser–Verlag, 2005.
A. Knowles, J. Yin, The Isotropic Semicircle Law and Deformation of Wigner Matrices, arXiv:1110.6449v2.
M.G. Kreĭn and H. Langer, The defect subspaces and generalized resolvents of a Hermitian operator in the space $\Pi _{\kappa
}$, *Funkcional. Anal. i Prilozen.* 5 (1971), 54–69 \[Russian\].
M.G. Kreĭn and H. Langer, Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher operatoren im Raume $\Pi _\kappa $ zusammenhangen. I. Einige Funktionenklassen und ihre Dahrstellungen, *Math. Nachr.*, 77 (1977), 187–236.
M.G. Kreĭn and H. Langer, Some propositions on analytic matrix functions related to the theory of operators in the space $\Pi _\kappa $, *Acta Sci. Math.* (Szeged), 43 (1981), 181–205.
H. Langer, A characterization of generalized zeros of negative type of functions of the class $\mathbf{N}_{\kappa}$”, *Oper. Theory Adv. Appl.*, 17 (1986), 201–212.
H. Langer, A. Luger, V. Matsaev, Convergence of generalized Nevanlinna functions, *Acta Sci. Math.* (Szeged), 77 (2011), 425–437.
M. A. Marchenko, L. A. Pastur, Distribution of eigenvalues in some ensembles of random matrices, *Mat. Sb.*, 72(114) (1967), 507–536 \[Russian\].
S. N. Chandler-Wilde, R. Chonchaiya, M. Lindner, Eigenvalue problem meets Sierpinski triangle: computing the spectrum of a non-self-adjoint random operator, *Oper. Matrices* 5 (2011), 633ñ648.
S. N. Chandler-Wilde, R. Chonchaiya, M. Lindner, On the Spectra and Pseudospectra of a Class of Non-Self-Adjoint Random Matrices and Operators, arXiv:1107.0177.
L. A. Pastur, On the spectrum of random matrices, *Theoretical and Mathematical Physics*, 10 (1972), 102–112, \[Russian\].
A. Pizzo, D. Renfrew, and A. Soshnikov, On finite rank deformations of Wigner matrices, to appear in Ann. Inst. Henri Poincaré (B), arXiv:1103.3731.
L.S. Pontryagin, Hermitian operators in spaces with indefinite metric, *Izv. Nauk. Akad. SSSR*, Ser. Math. 8 (1944), 243–280 \[Russian\].
R Development Core Team (2011). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org/.
H.S.V. de Snoo, H. Winkler, M. Wojtylak, Zeros and poles of nonpositive type of Nevanlinna functions with one negative square, *J. Math. Ann. Appl.*, 382 (2011), 399-417.
R. Vershynin, Spectral norm of products of random and deterministic matrices, *Probab. Theory Related Fields* 150 (2011), 471–509.
E. P. Wigner. Characteristic vectors of bordered matrices with infinite dimensions. *Annals Math.*, 62(1955), 548–564.
[^1]: The research was supported by Polish Ministry of Science and Higher Education with a Iuventus Plus grant.
[^2]: For arbitrary $\mathcal{N}_1$ function $z_0=\infty$ can be also the GZNT, in that case $\lim_{z \hat\to \infty} zQ(z) \in [0,\infty)$. However, this is clearly not possible for $Q$ of the form .
[^3]: It is well known [@KL77] that $-1/Q$ belongs to $\mathcal{N}_1$ provided that $Q$ belongs to $\mathcal{N}_1$, however, this information is not essential for the proof.
[^4]: In other words: $e$ is almost surely a cyclic vector of $X_N$.
|
---
author:
- |
Yoshikazu [Fujiwara]{}, Hidekatsu [Nemura]{}$^{*}$, Yasuyuki [Suzuki]{}$^{**}$\
Kazuya [Miyagawa]{}$^{***}$ and Michio [Kohno]{},$^{****}$
title: 'Three-Cluster Equation Using Two-Cluster RGM Kernel'
---
\#1 \#1[[\#1 ]{}]{} \#1[[\#1 \^2]{}]{} 3[N (3/2)]{} 1[N (1/2)]{} 1[\^3S\_1]{} 1[\^3D\_1]{} å[[a\^a]{}]{} ł \#1[Eq. (\[\#1\])]{} \#1[[\#1 ]{}]{} \#1[[\#1 \^2]{}]{}
Introduction
============
All the present-day quark-model descriptions of the nucleon-nucleon ($NN$) and hyperon-nucleon ($YN$) interactions incorporate important roles of the quark-gluon degrees of freedom in the the short-range region and the meson-exchange processes dominated in the medium- and long-range parts of the interaction. [@OY84] For example, we have introduced one-gluon exchange Fermi-Breit interaction and effective meson-exchange potentials acting between quarks, and have achieved very accurate descriptions of the $NN$ and $YN$ interactions with limited number of parameters. [@FU96a; @FU96b; @FU01b] We hope that the derived interaction in these models can be used for a realistic calculation of few-baryon systems like the hypertriton and various types of baryonic matter. This program, however, involves a non-trivial problem of how to extract the effective two-baryon interaction from the microscopic quark-exchange kernel. The basic baryon-baryon interaction is formulated as a composite-particle interaction in the framework of the resonating-group method (RGM). If we rewrite the RGM equation in the form of the Schr[" o]{}dinger-type equation, the interaction term becomes non-local and energy dependent. Furthermore, the RGM equation sometimes involves redundant components due to the effect of the antisymmetrization, which is related to the existence of the Pauli-forbidden states. In such a case, the full off-shell $T$-matrix is not well defined in the standard procedure which usually assumes simple energy-independent local potentials. [@GRGM] Since these features are related to the characteristic description of the short-range part in the quark model, it would be desirable if one can use the quark-exchange kernel directly in the application to many-baryon systems.
In this paper, we propose a new type of three-cluster equation which uses two-cluster RGM kernel for the inter-cluster interaction. We assume, for simplicity, three identical clusters having only one Pauli-forbidden state for the inter-cluster relative motion, but the extension to general systems is rather straightforward. We first consider a two-cluster RGM equation and the structure of $T$-matrix constructed from the two-cluster RGM kernel (RGM $T$-matrix). Next, we formulate a three-cluster equation which employs the two-cluster RGM kernel and a projection operator on the pairwise Pauli-allowed space. This equation is converted to the Faddeev equation which uses a non-singular part of the RGM $T$-matrix. Finally, we show some examples of the present formulation with respect to the $0^+$ ground states of three di-neutron ($3d^\prime$) and $3\alpha$ systems. The calculation is performed in the variational method, using the translationally-invariant harmonic oscillator basis. For the $3d^\prime$ system, the result of the Faddeev calculation is also presented. Detailed comparison is made with respect to the more desirable three-cluster RGM calculation, and to some other approximations like “renormalized RGM” and the well-known orthogonality condition model (OCM). [@SA68; @SA77; @HO74]
$T$-matrix of the two-cluster RGM kernel
========================================
We use the same notation as used in Ref. and write a two-cluster RGM equation as $$\begin{aligned}
\left[\,\varepsilon-H_0-V^{\rm RGM}(\varepsilon)\,\right] \chi=0\ ,
\label{form1}\end{aligned}$$ where $\varepsilon$ is the relative energy, $\varepsilon=E-E^{\rm int}$, between the two clusters, $H_0$ is the relative kinetic-energy operator, and $$\begin{aligned}
\VRGM=V_{\rm D}+G+\varepsilon K\ ,
\label{form2}\end{aligned}$$ is the RGM kernel composed of the direct potential $V_{\rm D}$, the sum of the exchange kinetic-energy and interaction kernels, $G=G^{\rm K}+G^{\rm V}$, and the exchange normalization kernel $K$. We assume that there exists only one Pauli-forbidden state $|u\rangle$, which satisfies the eigen-value equation $K |u\rangle=\gamma |u\rangle$ with the eigen-value $\gamma=1$. The projection operator on the Pauli-allowed space for the relative motion is denoted by $\Lambda=1-|u\rangle \langle u|$. Using the basic property of the Pauli-forbidden state $|u\rangle$, $(H_0+V_{\rm D}+G)|u\rangle=\langle u|(H_0+V_{\rm D}+G)=0$, we can separate $\VRGM$ into two distinct parts: $$\begin{aligned}
\VRGM=V(\varepsilon)+v(\varepsilon)\ ,
\label{form3}\end{aligned}$$ where $$\begin{aligned}
& & V(\varepsilon)=(\varepsilon-H_0)-\Lambda (\varepsilon-H_0) \Lambda
=\varepsilon |u\rangle \langle u| +\Lambda H_0 \Lambda-H_0\ ,\nonumber \\
& & v(\varepsilon)=\Lambda \VRGM \Lambda
=\Lambda \left( V_{\rm D}+G+\varepsilon K\right) \Lambda\ .
\label{form4}\end{aligned}$$ Note that $\Lambda V(\varepsilon) \Lambda=0$ and $\Lambda v(\varepsilon) \Lambda=v(\varepsilon)$; namely, $V(\varepsilon)$ may be considered as an operator acting in the Pauli-forbidden space, while $v(\varepsilon)$ an operator acting in the Pauli-allowed space. Using these properties, we can express as $$\begin{aligned}
\Lambda \left[\,\varepsilon-H_0-v(\varepsilon)\,\right] \Lambda \chi=0\ .
\label{form5}\end{aligned}$$ The separation of $\VRGM$ in enables us to deal with the energy dependence of the exchange RGM kernel in the Pauli-forbidden space and that in the allowed space, separately. Let us generalize to $$\begin{aligned}
\CVwe=V(\omega)+v(\varepsilon)\ ,
\label{form6}\end{aligned}$$ which we use in the following three-cluster formulation. We will see that the energy dependence involved $V(\omega)$ can be eliminated by the orthogonality condition to the Pauli-forbidden state.
Since the direct application of the $T$-matrix formalism to leads to a singular off-shell $T$-matrix, [@GRGM] we first consider the subsidiary equation $$\begin{aligned}
\left[\,\omega -H_0-\VRGM\,\right] \chi=0\ ,
\label{form7}\end{aligned}$$ with $\omega \neq \varepsilon$, and extract a singularity-free off-shell $T$-matrix starting from the standard $T$-matrix formulation for the “potential” $\VRGM$. A formal solution of the $T$-matrix equation $$\begin{aligned}
T(\omega,\varepsilon)=\VRGM+\VRGM G^{(+)}_0(\omega)
T(\omega,\varepsilon)
\label{form8}\end{aligned}$$ with $G^{(+)}_0(\omega)=1/(\omega-H_0+i0)$ is given by $$\begin{aligned}
T(\omega,\varepsilon) & = & \widetilde{T}(\omega, \varepsilon)
+(\omega-H_0)|u\rangle {1 \over \omega-\varepsilon}
\langle u|(\omega-H_0) \ ,\nonumber \\
\widetilde{T}(\omega, \varepsilon) & = & T_v(\omega, \varepsilon)
-\left(1+T_v(\omega, \varepsilon) G^{(+)}_0(\omega)\right)|u \rangle
{1 \over \langle u|G^{(+)}_v(\omega, \varepsilon)|u \rangle} \nonumber \\
& & \times \langle u| \left(1+G^{(+)}_0(\omega)
T_v(\omega, \varepsilon)\right) \ ,
\label{form9}\end{aligned}$$ where $T_v(\omega, \varepsilon)$ is defined by $$\begin{aligned}
T_v(\omega, \varepsilon)=v(\varepsilon)+v(\varepsilon) G^{(+)}_0(\omega)
T_v(\omega, \varepsilon) \ .
\label{form10}\end{aligned}$$ This result is obtained through the expression for the full Green function given by $$\begin{aligned}
G^{(+)}(\omega,\varepsilon)={1 \over \omega-H_0-\VRGM+i0}
=G^{(+)}_{\Lambda}(\omega, \varepsilon)
+|u\rangle {1 \over \omega-\varepsilon}
\langle u|\ ,\qquad
\label{form11}\end{aligned}$$ where $$\begin{aligned}
G^{(+)}_{\Lambda}(\omega, \varepsilon)
& = & G^{(+)}_v(\omega, \varepsilon)-G^{(+)}_v(\omega, \varepsilon)|u\rangle
{1 \over \langle u|G^{(+)}_v(\omega, \varepsilon)|u \rangle} \nonumber \\
& & \times \langle u|G^{(+)}_v(\omega, \varepsilon)\ ,
\label{form12}\end{aligned}$$ and $G^{(+)}_v(\omega, \varepsilon)
=1/(\omega-H_0-v(\varepsilon)+i0)$ is the solution of $$\begin{aligned}
G^{(+)}_v(\omega, \varepsilon)=G^{(+)}_0(\omega)+G^{(+)}_0(\omega)
\,v(\varepsilon) G^{(+)}_v(\omega, \varepsilon)\ .
\label{form13}\end{aligned}$$ In fact, the simple relationship $$\begin{aligned}
V(\varepsilon)=V(\omega)-(\omega-\varepsilon)|u\rangle \langle u|
\label{form13-1}\end{aligned}$$ yields $$\begin{aligned}
\omega-H_0-\VRGM=\Lambda \left(\omega-H_0-v(\varepsilon)\right)\Lambda
+(\omega-\varepsilon)|u\rangle \langle u|\ .
\label{form13-2}\end{aligned}$$ Then, if one uses the property $$\begin{aligned}
& & \Lambda \left[\,\omega-H_0-v(\varepsilon)\,\right]\Lambda
\cdot G^{(+)}_{\Lambda}(\omega, \varepsilon) \nonumber \\
& & =G^{(+)}_{\Lambda}(\omega, \varepsilon) \cdot
\Lambda \left[\,\omega-H_0-v(\varepsilon)\,\right]\Lambda=\Lambda\ ,
\label{form13-3}\end{aligned}$$ it is easily found that $$\begin{aligned}
& & \left[\,\omega-H_0-\VRGM\,\right]
\ G^{(+)}(\omega, \varepsilon) \nonumber \\
& & =\left\{\,\Lambda \left[\,\omega-H_0-v(\varepsilon)\,\right]\Lambda
+(\omega-\varepsilon)|u\rangle \langle u|\,\right\}
\left\{\,G^{(+)}_{\Lambda}(\omega, \varepsilon)
+|u\rangle {1 \over \omega-\varepsilon}
\langle u|\,\right\} \nonumber \\
& & =\Lambda+|u\rangle \langle u|=1\ .
\label{form13-4}\end{aligned}$$ The expression of $T(\omega,\varepsilon)$ in is most easily obtained from $$\begin{aligned}
G^{(+)}(\omega, \varepsilon)=G^{(+)}_0(\omega)
+G^{(+)}_0(\omega)\,T(\omega,\varepsilon)\,G^{(+)}_0(\omega)\,
\label{form13-5}\end{aligned}$$ or $$\begin{aligned}
T(\omega,\varepsilon)=\left(\omega-H_0\right)
\,G^{(+)}(\omega, \varepsilon)\,\left(\omega-H_0\right)
-\left(\omega-H_0\right)\ .
\label{form13-6}\end{aligned}$$
The basic relationship which will be used in the following is $$\begin{aligned}
G^{(+)}_0(\omega) T(\omega,\varepsilon)
& = & G^{(+)}(\omega, \varepsilon ) \VRGM \nonumber \\
& = & G^{(+)}_\Lambda(\omega, \varepsilon) \VRGM
+|u\rangle {1 \over \omega-\varepsilon}
\langle u|\VRGM \nonumber \\
& = & G^{(+)}_\Lambda(\omega, \varepsilon) \CVwe
-|u\rangle \langle u|
+|u\rangle {1 \over \omega-\varepsilon}
\langle u|(\omega-H_0) \nonumber \\
& = &G^{(+)}_0(\omega) \widetilde{T}(\omega, \varepsilon)
+|u\rangle {1 \over \omega-\varepsilon}
\langle u|(\omega -H_0)\ ,
\label{form14}\end{aligned}$$ where $\widetilde{T}(\omega, \varepsilon)$ satisfies $$\begin{aligned}
G^{(+)}_0(\omega) \widetilde{T}(\omega, \varepsilon)
=G^{(+)}_\Lambda(\omega, \varepsilon) \CVwe
-|u\rangle \langle u| \ .
\label{form15}\end{aligned}$$ These can be shown by using Eqs.(\[form11\]) and (\[form13-1\]). The full $T$-matrix, $T(\omega, \varepsilon)$, in is singular at $\varepsilon=\omega$, while $\widetilde{T}(\omega, \varepsilon)$ does not have such singularity. For $\varepsilon \neq \omega$, $T(\omega, \varepsilon)$ satisfies the relationship $$\begin{aligned}
\langle u|G^{(+)}_0(\omega) T(\omega,\varepsilon)|\phi \rangle
=\langle \phi | T(\omega,\varepsilon) G^{(+)}_0(\omega)
|u \rangle=0
\label{form16}\end{aligned}$$ for the plane wave solution $|\phi \rangle$ with the energy $\varepsilon$, i.e., $(\varepsilon-H_0)|\phi \rangle=0$. This relationship is a direct result of more general relationship $$\begin{aligned}
\langle u| \left[\,1+G^{(+)}_0(\omega) \widetilde{T}(\omega, \varepsilon)
\,\right]=\left[\, 1+\widetilde{T}(\omega, \varepsilon)
G^{(+)}_0(\omega)\,\right] |u \rangle=0 \ ,
\label{form17}\end{aligned}$$ which is simply seen from .
We note that $\widetilde{T}(\omega, \varepsilon)$ satisfies $$\begin{aligned}
& & \widetilde{T}(\omega, \varepsilon)=\CVwe
-|u\rangle \langle u |(\omega-H_0)+\CVwe G^{(+)}_0(\omega)
\widetilde{T}(\omega, \varepsilon)\ , \nonumber \\
& & \widetilde{T}(\omega, \varepsilon)=\CVwe
-(\omega-H_0)|u\rangle \langle u |+\widetilde{T}(\omega, \varepsilon)
G^{(+)}_0(\omega) \CVwe \ .
\label{form18}\end{aligned}$$ However, these asymmetric forms of the $T$-matrix equations do not determine the solution $\widetilde{T}(\omega, \varepsilon)$ uniquely, since the resolvent kernel $\left[\,1-\CVwe G^{(+)}_0(\omega)\,\right]^{-1}$ has a singularity related to the existence of the trivial solution $|u\rangle$: $$\begin{aligned}
\langle u|\left[\,1-\CVwe G^{(+)}_0(\omega)\,\right]=0\ .
\label{form19}\end{aligned}$$ The driving term, $\CVwe-|u\rangle \langle u |(\omega-H_0)$, etc., guarantees the existence of the solution, except for an arbitrary admixture of $(\omega-H_0)|u\rangle$ component. In order to eliminate this ambiguity and to make $\widetilde{T}(\omega, \varepsilon)$ symmetric, one has to impose some orthogonality conditions, which will be discussed in a separate paper.
Three-cluster equation
======================
Let us consider a system composed of three identical spinless particles, interacting via the two-cluster RGM kernel $\VRGM$. The energy dependence involved in $\VRGM$ should be treated properly by calculating the expectation value of the two-cluster subsystem, at least for $v(\varepsilon)$. On the other hand, the energy dependence involved in $V(\varepsilon)$ is of kinematical origin, and could be modified so as to be best suited to the three-cluster equation. The three-body equation we propose is expressed as $$\begin{aligned}
P \left[\,E-H_0-\VRGMA-\VRGMB-\VRGMC\,\right] P \Psi=0\ ,
\label{three1}\end{aligned}$$ where $H_0$ is the free three-body kinetic-energy operator and $\VRGMA$ stands for the RGM kernel for the $\alpha$-pair, etc. The two-cluster relative energy $\varepsilon_\alpha$ in the three-cluster system is self-consistently determined through $$\begin{aligned}
\varepsilon_\alpha=\langle P \Psi|
\,h_\alpha+\VRGMA\,|P \Psi \rangle\ ,
\label{three1-2}\end{aligned}$$ using the normalized three-cluster wave function $P \Psi$ with $\langle P \Psi| P \Psi \rangle=1$. Here $h_\alpha$ is the free kinetic-energy operator for the $\alpha$-pair. Also, $P$ is the projection operator on the \[3\] symmetric Pauli-allowed space as defined below. [@HO74] We solve the eigen-value problem $$\begin{aligned}
\sum_\alpha|u_\alpha \rangle \langle u_\alpha | \Psi_\lambda \rangle
=\lambda~| \Psi_\lambda \rangle
\label{three2}\end{aligned}$$ in the \[3\] symmetric model space, $|\Psi_\lambda \rangle \in [3]$, and define $P$ as a projection on the space spanned by eigen-vectors with the eigen-value $\lambda=0$: $$\begin{aligned}
P=\sum_{\lambda=0}|\Psi_\lambda \rangle \langle \Psi_\lambda |\ .
\label{three3}\end{aligned}$$ It is easy to prove that $P$ has the following properties: $$\begin{aligned}
&({\rm i}) &~\Lambda_\alpha P=P \Lambda_\alpha=P \quad \hbox{for} \quad
\hbox{}^\forall \alpha\ ,\nonumber \\
&({\rm ii}) &~\hbox{when}~\Psi \in [3],
~\hbox{}^\forall~\langle u_\alpha|\Psi \rangle=0
\longleftrightarrow P \Psi=\Psi\ ,\nonumber \\
&({\rm iii}) &~\hbox{when}~\Psi \in [3],~P \Psi=0
\longleftrightarrow \hbox{}^\exists |f\rangle\ ,\nonumber \\
& & ~\hbox{such~that}
~\Psi=|u_\alpha\rangle |f_\alpha\rangle+|u_\beta\rangle |f_\beta\rangle
+|u_\gamma\rangle |f_\gamma\rangle\ .
\label{three4}\end{aligned}$$ Note that all these relations are considered in the \[3\] symmetric model space, and $P$ and $Q \equiv 1-P$ are both \[3\] symmetric three-body operators. Using the property (i), we can simplify as $$\begin{aligned}
P \left[\,E-H_0-\va-\vb-\vc\,\right] P \Psi=0\ .
\label{three5}\end{aligned}$$
In order to derive the Faddeev equation corresponding to , it is convenient to rewrite or (\[three5\]) as $$\begin{aligned}
P \left[\,E-H_0-\CVA-\CVB-\CVC\,\right] P \Psi=0\ ,
\label{three6}\end{aligned}$$ where $\CVA$ is the three-body operator defined by $\CVwe$ in through $$\begin{aligned}
\CVA & = & {\cal V}_\alpha (E-h_{\bar \alpha}, \varepsilon_\alpha)
\nonumber \\
& = & (E-H_0)-\Lambda_\alpha (E-H_0) \Lambda_\alpha
+v_\alpha(\varepsilon_\alpha)\ .
\label{three7}\end{aligned}$$ Here $h_{\bar \alpha}$ is the kinetic-energy operator between the $\alpha$-pair and the third particle. The last equation of is derived by using $h_\alpha+h_{\bar \alpha}=H_0$. The validity of is easily seen from, for example, $P\CVB P=P \Lambda_\beta \CVB \Lambda_\beta P
=P v_\beta(\varepsilon_\beta) P$, which uses the property (i) of . The expression behind the first $P$ in the left-hand side of is symmetric with respect to the exchange of the three particles. [^1] Thus, by applying the property (iii) of , we find $$\begin{aligned}
& & \left[\,E-H_0-\CVA-\CVB-\CVC\,\right] P \Psi \nonumber \\
& & =|u_\alpha\rangle |f_\alpha\rangle+|u_\beta\rangle |f_\beta\rangle
+|u_\gamma\rangle |f_\gamma\rangle\ ,
\label{three8}\end{aligned}$$ where $|f\rangle$ is an unknown function, and $|f_\beta\rangle$ and $|f_\gamma\rangle$ are simply obtained from $|f_\alpha\rangle$ by the cyclic permutations. Here we invoke the standard ansatz to set $$\begin{aligned}
P \Psi=\psi_\alpha+\psi_\beta+\psi_\gamma\ ,
\label{three9}\end{aligned}$$ and define $\psi_\alpha$ as the solution of $$\begin{aligned}
\left(\,E-H_0\,\right) \psi_\alpha
=\CVA P \Psi+|u_\alpha\rangle |f_\alpha\rangle\ .
\label{three10}\end{aligned}$$ This equation can be written as $$\begin{aligned}
\left[\,E-H_0-\CVA\,\right] \psi_\alpha
=\CVA (\psi_\beta+\psi_\gamma)+|u_\alpha\rangle |f_\alpha\rangle\ ,
\label{three11}\end{aligned}$$ or by using as $$\begin{aligned}
\Lambda_\alpha \left[\,E-H_0-\va\,\right] \Lambda_\alpha \psi_\alpha
=\CVA (\psi_\beta+\psi_\gamma)+|u_\alpha\rangle |f_\alpha\rangle\ .
\label{three12}\end{aligned}$$ The unknown function $|f_\alpha\rangle$ is determined if we multiply this equation by $\langle u_\alpha|$ from the left and use $\langle u_\alpha|\CVA=\langle u_\alpha|(E-H_0)$: $$\begin{aligned}
|f_\alpha\rangle=-\langle u_\alpha|\,E-H_0\,| \psi_\beta+\psi_\gamma
\rangle\ .
\label{three13}\end{aligned}$$ Thus we obtain $$\begin{aligned}
\Lambda_\alpha \left[\,E-H_0-\va\,\right] \Lambda_\alpha \psi_\alpha
& = & \CVA (\psi_\beta+\psi_\gamma) \nonumber \\
& & -|u_\alpha\rangle \langle u_\alpha|\,E-H_0\,| \psi_\beta+\psi_\gamma
\rangle\ .
\label{three14}\end{aligned}$$ By employing the two-cluster relation in the three-cluster model space, $$\begin{aligned}
G^{(+)}_{\Lambda_\alpha}(E, \varepsilon_\alpha) \CVA
=G^{(+)}_0(E) \TTE + |u_\alpha\rangle \langle u_\alpha| \ ,
\label{three15}\end{aligned}$$ and the relationship, $$\begin{aligned}
& & G^{(+)}_{\Lambda_\alpha}(E, \varepsilon_\alpha)
\,\Lambda_\alpha \left[\,E-H_0-\va\,\right] \Lambda_\alpha
\nonumber \\
& & =\Lambda_\alpha \left[\,E-H_0-\va\,\right] \Lambda_\alpha
\,G^{(+)}_{\Lambda_\alpha}(E, \varepsilon_\alpha)=\Lambda_\alpha\ ,
\label{three16}\end{aligned}$$ yields $$\begin{aligned}
\Lambda_\alpha \psi_\alpha
=G^{(+)}_0(E) \TTE (\psi_\beta+\psi_\gamma)
+|u_\alpha\rangle \langle u_\alpha| \psi_\beta+\psi_\gamma \rangle\ .\qquad
\label{three17}\end{aligned}$$ Since $\langle u_\alpha| \psi_\beta+\psi_\gamma\rangle=
-\langle u_\alpha| \psi_\alpha\rangle$ from , we finally obtain $$\begin{aligned}
\psi_\alpha=G^{(+)}_0(E) \TTE (\psi_\beta+\psi_\gamma)\ .
\label{three18}\end{aligned}$$ Note that $\TTE$ is essentially the non-singular part of the two-cluster RGM $T$-matrix : $$\begin{aligned}
\TTE=\widetilde{T}_\alpha(E-h_{\bar \alpha}, \varepsilon_\alpha)\ ,
\label{three19}\end{aligned}$$ and that the solution of automatically satisfies $\langle u_\alpha| \psi_\alpha+\psi_\beta+\psi_\gamma\rangle=0$ due to . Since $\psi_\alpha+\psi_\beta+\psi_\gamma \in [3]$, the property (ii) of yields $\Psi=P \Psi$ if we set $\Psi=\psi_\alpha+\psi_\beta+\psi_\gamma$. We can also start from and derive by using the properties (i) and (ii) of , thus establish the equivalence between and .
Three di-neutron system
=======================
As a simplest non-trivial example, we first consider three di-neutron ($d^\prime$) system, where the internal wave function of the $d^\prime$ is assumed to be $(0s)$ harmonic oscillator (h.o.) wave function. The normalization kernel $K$ for the $2d^\prime$ system is given by $K=\Lambda=1-|u\rangle \langle u|$ and the $\Lambda
(\varepsilon K) \Lambda$ term in $v(\varepsilon)$ disappears. Here $|u\rangle$ is the $(0s)$ h.o. wave function given by $u(\br)=\langle \br|u\rangle
=(2\nu/\pi)^{3/4} e^{-\nu \br^2}$. We assume a very simple two-nucleon interaction of the Serber type $$\begin{aligned}
v_{ij}=-v_0~e^{-\kappa r^2}~{1 \over 2}\left(1+P_r\right) \ ,
\label{deut1}\end{aligned}$$ according to Ref.. This paper deals with a schematic model of the almost forbidden state with $v_0=90$ MeV, but this strength is too weak to give a bound state for the $3d^\prime$ system. We use the following parameter set in the present calculation: $\nu=0.12~\hbox{fm}^{-2}$, $\kappa=0.46~\hbox{fm}^{-2}$ and $v_0=153~\hbox{MeV}$. With this value of $v_0$, the $2d^\prime$ system is slightly bound.[^2]
In order to solve the three-cluster equation (\[three1\]), we first prepare \[3\] symmetric translationally-invariant h.o. basis according to the Moshinsky’s method [@MO96].[^3] The \[3\] symmetric Pauli-allowed states, which we denote by $\varphi^{[3](\lambda \mu)}_{a, n}(\bfrho, \bflambda)$, are explicitly constructed by the diagonalization procedure in . Here, $\bfrho=(\bX_1-\bX_2)/\sqrt{2}$ and $\bflambda
=(\bX_1+\bX_2-2\bX_3)/\sqrt{6}$ are the Jacobi coordinates for the center-of-mass coordinates $\bX_i$ ($i=1$ - 3) of the three $d^\prime$ clusters. These eigen-states are specified by the $SU_3$ quantum number $(\lambda \mu)$ and a set of the other quantum numbers $n$, which includes the total h.o. quanta $N$. We then perform the variational calculation using these basis states. Namely, we first expand $P \Psi$ as $$\begin{aligned}
P \Psi=\sum_{(\lambda, \mu), n} c^{(\lambda, \mu)}_n
\varphi^{[3](\lambda \mu)}_{a, n}(\bfrho, \bflambda)\ .
\label{deut2}\end{aligned}$$ In the following, we use a simplified notation $n$ to represent the set of $(\lambda \mu)$ and $n$ (and also the possible $K$ quantum number if the total orbital angular momentum $L \neq 0$). Since $\Psi$ is \[3\] symmetric, the three interaction terms in give the same contribution. This leads to the eigen-value equation $$\begin{aligned}
& & \sum_{n^\prime}\left(E~\delta_{nn^\prime}-H_{nn^\prime}\right)
c_{n^\prime}=0\ , \nonumber \\
& & H_{nn^\prime}=(H_0)_{nn^\prime}
+3 \left[\,(V_{\rm D})_{nn^\prime}+G_{nn^\prime}+\varepsilon K_{nn^\prime}
\,\right]\ .
\label{deut3}\end{aligned}$$ Here $K_{nn^\prime}$ term in the $3d^\prime$ problem is trivially zero since the \[3\] symmetric allowed basis does not contain the $(0s)$ component from the very beginning. This implies that our $d^\prime d^\prime$ interaction is energy independent and the self-consistency for $\varepsilon$ is not necessary. On the other hand, the $3\alpha$ case which will be discussed in the next section requires to determine $\varepsilon$ through $$\begin{aligned}
\varepsilon={\sum_{n,n^\prime}\left[h_{nn^\prime}
+(V_{\rm D})_{nn^\prime}+G_{nn^\prime}\right] c_n c_{n^\prime} \over
1-\sum_{n,n^\prime}K_{nn^\prime} c_n c_{n^\prime}}\ .
\label{deut4}\end{aligned}$$ The two-body matrix elements in the three-body space, $\CO_{nn^\prime}=\langle \varphi^{[3]}_n |\CO|
\varphi^{[3]}_{n^\prime}\rangle$, can be calculated by using the power series expansion of the complex GCM kernel and $SU_3$ Clebsch-Gordan (C-G) coefficients of the type $\langle (N_1 0) \ell (N_2 0) \ell || (\lambda \mu)
0 0 \rangle$. Fortunately, a concise expression is given by Suzuki and Hecht [@SU83] for this particular type of $SU_3$ C-G coefficients with $L=0$.
Table I shows the lowest eigen-values of with an increasing number of total h.o. quanta $N$ included in the calculation. The number of basis states rapidly increases, as the maximum $N$ becomes larger. The listing is terminated when the number of basis states $n_{\rm Max}$ is over 1000, which is reached around $N\sim 60$. The convergence of the $3d^\prime$ system is rather slow, since the bound-state energy is especially small in this particular system. The best value obtained in the variational calculation is $E_{3d^\prime}=-0.4323$ MeV, using 2,927 basis states with $N \leq 88$. We have also solved the Faddeev equation (\[three18\]), and obtained $E_{3d^\prime}=-0.4375$ MeV and $-0.4378$ MeV, when the partial waves up to $\ell=4$ and 6 are included in the calculation, respectively. The final value $E_{3d^\prime}=-0.438$ MeV can also be compared with $E^{\rm RGM}_{3d^\prime}=-1.188$ MeV, which is obtained by the stochastic variational method [@SU98] for the $3d^\prime$ RGM. Our result by the three-cluster equation gives 0.75 MeV less bound, compared with the full microscopic $3d^\prime$ RGM calculation.
$3\alpha$ system
================
In this system, the structure of the $2\alpha$ normalization kernel $K$ is more involved. In the relative $S$-wave we have two Pauli-forbidden states, $(0s)$ and $(1s)$, while in the $D$-wave only one $(0d)$ h.o. state is forbidden. The relative motion between the two $\alpha$ clusters starts from $N=4$ h.o. quanta The eigen-value $\gamma_N$ for $K$ is given by $\gamma_N=2^{2-N}-3\delta_{N, 0}$, which is 1 ($N=0$ or 2), 1/4 ($N=4$), 1/16 ($N=6$), $\cdots$ . The rather large value $\gamma_4=1/4$ makes the self-consistent procedure through very important. For the two-body effective interaction, we use the Volkov No.2 force with $m=0.59$, following the $3\alpha$ RGM calculation by Fukushima and Kamimura [@FU78]. The h.o. constant for the $\alpha$ cluster is $\nu=0.275~\hbox{fm}^{-2}$, which gives the $\alpha$ cluster internal energy $E_\alpha=-27.3$ MeV for the $(0s)^4$ configuration, if the Coulomb interaction is included. (Cf. $E^{\rm exp't}_\alpha=-28.3$ MeV.) We have carried out the $2\alpha$ RGM calculation by using this parameter set, and found that the present $2\alpha$ system is bound with the binding energy $E_{2\alpha}=-0.245$ MeV. (Cf. $E^{\rm exp't}_{2\alpha}=92$ keV.)
Table I lists the convergence of the lowest $3\alpha$ eigen-values with respect to the maximum total h.o. quanta $N$. We find that the final values of $E_{3\alpha}$ and $\varepsilon$ are $E_{3\alpha}=-5.97$ MeV and $\varepsilon=9.50$ MeV. If we compare this with $E^{\rm RGM}_{3\alpha}=-7.5$ MeV (Cf. $E^{\rm exp't}_{3\alpha}=-7.3$ MeV) by the $3\alpha$ RGM calculation [@FU78], we find that our result is 1.5 MeV less bound. The amplitude of the lowest shell-model component with the $SU_3$ (04) representation is $c_{(04)}=0.790$. We think that the underbinding compared to the three-cluster RGM calculation is reasonable, since our three-cluster equation misses some attractive effect due to the genuine three-cluster exchange kernel. Oryu et al. carried out $3\alpha$ Faddeev calculation using $2\alpha$ RGM kernel. [@OR94] They obtained very large binding energy, $E_{3\alpha}=-28.2$ MeV with the Coulomb force turned off. Since the effect of the Coulomb force is at most 6 MeV,[^4] this value is too deep. This is because they did not treat the $\varepsilon K$ term in the RGM kernel properly and the effect of $P$ in is not fully taken into account in their Faddeev formalism. In order to see the importance of the $\varepsilon K$ term in , it is useful to see the contribution of this term in the lowest h.o. (04) configuration. The decomposition of $E^{(04)}_{3\alpha}=12.634$ MeV with $N=8$ in Table I is $$\begin{aligned}
H & = & H_0+3\,(V_{\rm D}+G^{\rm K}+G^{\rm V}+{V_{\rm D}}^{\rm CL}
+G^{\rm CL}+\varepsilon K) \nonumber \\
& = & 125.4+3(-36.54-15.68+6.54+2.58-0.78+25.12/4) \nonumber \\
& = & 125.4-3 \times 37.6=12.6~\hbox{MeV}\ .
\label{alpha1}\end{aligned}$$ This example shows very clearly that the self-consistent procedure for the energy-dependence of the RGM kernel in the allowed model space is sometimes very important.
Since the present calculation employs the h.o. basis, it is very easy to examine another approximation which eliminate the explicit energy dependence involved in $v(\varepsilon)$. This approximation is related to the proper normalization of the two-cluster relative wave function $\chi$ in through $\psi=\sqrt{1-K}\chi$, and we call this approximation the renormalized RGM. In this formulation, the interaction generated from the RGM kernel is expressed as $$\begin{aligned}
v=\left({1 \over \sqrt{1-K}}\right)^\prime\,(h_0+V_{\rm D}+G)
\,\left({1 \over \sqrt{1-K}}\right)^\prime - \Lambda h_0 \Lambda\ ,
\label{alpha2}\end{aligned}$$ where the prime in $(1/\sqrt{1-K})^\prime$ implies the inversion of $\sqrt{1-K}$ in the allowed model space. (See Ref. and the discussion in Ref..) The column $E^{\rm RN}_{3\alpha}$ in Table I shows the result of this approximation. We find $E^{\rm RN}_{3\alpha}=-4.99$ MeV, which is 0.98 MeV less bound in comparison with our result. This may result from rather inflexible choice of the $3\alpha$ Hamiltonian, caused by the lack of the self-consistency. Table I also shows the result by $3\alpha$ OCM ($E^{\rm OCM}_{3\alpha}$), whose procedure is to use $v=\Lambda (V_{\rm D}+V^{\rm CL}_{\rm D}) \Lambda$. We find $E^{\rm OCM}_{3\alpha}=-4.68$ MeV, which is further 0.31 MeV less bound. In this case, $2\alpha$ OCM gives a larger binding energy, $E^{\rm OCM}_{2\alpha} \leq -0.4$ MeV, than the $2\alpha$ RGM. If we readjust the potential parameter to fit the $2\alpha$ binding energy, we would apparently obtain an even worse result. It was realized a long time ago that a simple choice of the direct potential $V_{\rm D}$ for the effective interaction $V^{\rm eff}$ in OCM gives a poor result. [@TA81]
Summary
=======
The main purpose of this study is to find an optimum equation for three-cluster systems interacting via pairwise two-cluster RGM kernel. This is a necessary first step to apply the quark-model baryon-baryon interactions to few-baryon systems. We have found that the orthogonality condition to the pairwise Pauli-forbidden states is a compulsory condition to assure a reasonable result. The inherent energy dependence of the two-cluster exchange RGM kernel should be treated self-consistently, if the eigen-values of the exchange normalization kernel $K$ are non-negligible in the allowed space. The proposed three-cluster equation has a nice feature that the equivalent three-cluster Faddeev equation is straightforwardly formulated using the non-singular part of the full $T$-matrix derived from the RGM kernel. We have applied this equation to simple systems composed of the three di-neutrons and three $\alpha$ clusters. The equivalent Faddeev equation is also solved for the three di-neutron system. The Faddeev calculation for the $3\alpha$ system will be reported in a separate paper. [@FU02] For the ground state of the three $\alpha$ system, the obtained binding energy is 1.5 MeV less bound, in comparison with the full microscopic three-cluster RGM calculation. We think this satisfactory, since three-cluster RGM calculation of the few-baryon systems using quark-model baryon-baryon interaction is still beyond the scope of feasibility. The application to the hypertriton using our quark-model nucleon-nucleon and hyperon-nucleon interactions [@FU96b; @FU01b] is now under way.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture (No. 12640265).
[99]{} M. Oka and K. Yazaki, in [*Quarks and Nuclei*]{}, ed. W. Weise (World Scientific, Singapore, 1984), p. 489. Y. Fujiwara, C. Nakamoto and Y. Suzuki, Phys. Rev. Lett. [**76**]{} (1996), 2242. Y. Fujiwara, C. Nakamoto and Y. Suzuki, Phys. Rev. [**C54**]{} (1996), 2180. Y. Fujiwara, T. Fujita, M. Kohno, C. Nakamoto and Y. Suzuki, Phys. Rev. [**C65**]{} (2002), 014002. Y. Fujiwara, M. Kohno, C. Nakamoto and Y. Suzuki, Prog. Theor. Phys. [**104**]{} (2000), 1025. S. Saito, Prog. Theor. Phys. [**40**]{} (1968), 893; [**41**]{} (1969), 705. S. Saito, Prog. Theor. Phys. Suppl. No. 62 (1977), 11. H. Horiuchi, Prog. Theor. Phys. [**51**]{} (1974), 1266; [**53**]{} (1975), 447. S. Saito, S. Okai, R. Tamagaki and M. Yasuno, Prog. Theor. Phys. [**50**]{} (1973), 1561. See, for example, M. Moshinski and Y. F. Smirnov, [*The Harmonic Oscillator in Modern Physics*]{}, Contemporary Concepts in Physics, Vol. 9 (Harwood Academic Publishers, Amsterdam, 1996), Chapter III.17, p. 58. Y. Fujiwara and H. Horiuchi, Memoirs of the Faculty of Science, Kyoto University, Series A of Physics, Astrophysics, Geophysics and Chemistry, Vol. XXXVI, No. 2, Article 1, (1983), 197. Y. Suzuki and K. T. Hecht, J. Math. Phys. [**24**]{} (1983), 785. Y. Suzuki and K. Varga, [*Stochastic Variational Approach to Quantum-Mechanical Few-Body Problems*]{}, Lecture Note in Physics, Vol. m54 (Springer-Verlag, Berlin, Heidelberg, 1998); K. Varga, Y. Suzuki and R. G. Lovas, Nucl. Phys. [**A571**]{} (1994), 447-466; K. Varga and Y. Suzuki, Phys. Rev. [**C52**]{} (1995), 2885-2905. Y. Fukushima and M. Kamimura, Proc. Int. Conf. Nuclear Structure, Tokyo, 1977, J. Phys. Soc. Japan [**44**]{} (1978) Suppl. p. 225. S. Oryu, K. Samata, T. Suzuki, S. Nakamura and H. Kamata, Few-Body Systems [**17**]{} (1994), 185. See, for example, Y. C. Tang, [*Microscopic Description of the Nuclear Cluster Theory*]{}, Lecture Notes in Physics [**145**]{}, Topics in Nuclear Physics II, edited by T. T. S. Kuo and S. S. M. Wong (Springer-Verlag, Berlin, 1981), 571. Y. Fujiwara, Y. Suzuki, K. Miyagawa, M. Kohno, H. Nemura, submitted to Prog. Theor. Phys. (2002).
----- --------------- ----------------- --------------- --------------- --------------- ------------------------ -------------------------
$N$ $n_{\rm Max}$ $E_{3d^\prime}$ $n_{\rm Max}$ $\varepsilon$ $E_{3\alpha}$ $E^{\rm RN}_{3\alpha}$ $E^{\rm OCM}_{3\alpha}$
4 1 3.256 $-$ $-$ $-$ $-$ $-$
6 3 2.828 $-$ $-$ $-$ $-$ $-$
8 6 0.7373 1 25.120 12.634 12.634 23.570
10 10 0.5585 3 18.023 3.615 4.575 11.422
12 16 0.1169 7 14.857 $-0.343$ 0.874 5.322
14 23 0.0523 12 13.046 $-2.454$ $-1.194$ 1.827
16 32 $-0.0868$ 19 11.920 $-3.678$ $-2.449$ $-0.323$
18 43 $-0.1351$ 28 11.182 $-4.439$ $-3.255$ $-1.703$
20 56 $-0.1972$ 39 10.682 $-4.929$ $-3.788$ $-2.614$
22 71 $-0.2313$ 52 10.339 $-5.252$ $-4.148$ $-3.227$
24 89 $-0.2660$ 68 10.099 $-5.470$ $-4.394$ $-3.647$
26 109 $-0.2899$ 86 9.931 $-5.619$ $-4.565$ $-3.939$
28 132 $-0.3117$ 107 9.812 $-5.721$ $-4.685$ $-4.143$
30 158 $-0.3284$ 131 9.727 $-5.792$ $-4.769$ $-4.289$
32 187 $-0.3431$ 158 9.666 $-5.842$ $-4.829$ $-4.394$
34 219 $-0.3550$ 188 9.623 $-5.878$ $-4.872$ $-4.469$
36 255 $-0.3653$ 222 9.591 $-5.903$ $-4.904$ $-4.525$
38 294 $-0.3740$ 259 9.568 $-5.921$ $-4.926$ $-4.565$
40 337 $-0.3815$ 300 9.551 $-5.934$ $-4.943$ $-4.595$
42 384 $-0.3879$ 345 9.539 $-5.943$ $-4.954$ $-4.618$
44 435 $-0.3934$ 394 9.530 $-5.950$ $-4.964$ $-4.635$
46 490 $-0.3982$ 447 9.524 $-5.955$ $-4.971$ $-4.647$
48 550 $-0.4025$ 505 9.519 $-5.959$ $-4.976$ $-4.657$
50 614 $-0.4061$ 567 9.515 $-5.962$ $-4.979$ $-4.664$
52 683 $-0.4094$ 634 9.512 $-5.964$ $-4.982$ $-4.669$
54 757 $-0.4122$ 706 9.510 $-5.965$ $-4.984$ $-4.674$
56 836 $-0.4147$ 783 9.508 $-5.966$ $-4.986$ $-4.677$
58 920 $-0.4169$ 865 9.507 $-5.967$ $-4.987$ $-4.679$
60 1010 $-0.4189$ 953 9.506 $-5.968$ $-4.988$ $-4.681$
----- --------------- ----------------- --------------- --------------- --------------- ------------------------ -------------------------
: The lowest eigen-values for $3d^\prime$ and $3\alpha$ systems, obtained by diagonalization using \[3\] symmetric translationally-invariant h.o. basis. The orthogonality condition by the projection operator $P$ in is imposed. $N$ stands for the maximum total h.o. quanta included in the calculation, and $n_{\rm Max}$ the number of the basis states with the orbital angular momentum $L=0$. The three-cluster equation is used for $E_{3d^\prime}$ and $E_{3\alpha}$, while the energy-independent renormalized interaction and $v=\Lambda
(V_{\rm D}+V^{\rm CL}_{\rm D}) \Lambda$ are used in for $E^{\rm RN}_{3\alpha}$ and $E^{\rm OCM}_{3\alpha}$, respectively. []{data-label="table"}
[^1]: The two-cluster relative energies, $\varepsilon_\alpha$, $\varepsilon_\beta$ and $\varepsilon_\gamma$, are actually all equal, since we are dealing with the three identical particles.
[^2]: The $S$-wave phase shift for the $2d^\prime$ scattering shows that the $2d^\prime$ system gets bound between $v_0=151$ MeV and 152 MeV.
[^3]: This process is most easily formulated using the theory of Double Gel’fand polynomials. [@KI83].
[^4]: In our calculation, $E_{3\alpha}=-11.42$ MeV when the Coulomb interaction is turned off, which implies that the effect of the Coulomb interaction is 5.45 MeV.
|
---
abstract: 'Surface waves on a liquid air interface excited by a vertical vibration of a fluid layer (Faraday waves) are employed to investigate the phase relaxation of ideally ordered patterns. By means of a combined frequency-amplitude modulation of the excitation signal a periodic expansion and dilatation of a square wave pattern is generated, the dynamics of which is well described by a Debye relaxator. By comparison with the results of a linear theory it is shown that this practice allows a precise measurement of the phase diffusion constant.'
address: |
$^1$Fakultät für Physik und Elektrotechnik, Universität des Saarlandes, 66041 Saarbrücken, Germany\
$^2$Laboratoire de Physique Statistique, Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, France\
$^3$Max-Planck-Institut für Polymerforschung, Ackermannweg 10, 55128 Mainz, Germany\
$^{4}$Czestochowa Technical University, Electrical Eng. Dep., Al. Armii Krajowej 17, PL-42200 Czestochowa, Poland
author:
- 'A.V. Kityk$^{1,4}$, C. Wagner$^2$, K. Knorr$^1$, H. W. Müller$^{3,1}$'
title: Phase relaxation of Faraday surface waves
---
Our understanding of spatio-temporal pattern formation in non-equilibrium fluid systems has greatly benefitted [@cross93] from recent quantitative experiments in combination with the development of new theoretical concepts. One of them is the so-called [*amplitude equation*]{} approach [@newell69], which is based on the linear instability of a homogeneous state and leads naturally to a classification of patterns in terms of characteristic wave numbers and frequencies. A different but equally universal description, the [*phase dynamics*]{} [@pomeau79], applies to situations where a periodic spatial pattern experiences long wavelength phase modulations. This approach, originally introduced in the context of thermal convection, has proven to be useful to understand the stability and the relaxation of periodic patterns, wave number selection, and defect dynamics. In many paradigmatic pattern forming systems such as thermal convection in a fluid layer heated from below (Rayleigh Bénard convection, RBC) or the formation of azimuthal vortices in the gap between two rotating cylinders (Taylor Couette flow, TCF) the dominating wave number is dictated by the geometry and thus inconvenient to be changed in a given experimental setup (for instance by a mechanical ramp of the layer thickness [@kramer82]).
Faraday waves are surface waves on the interface between two immiscible fluids, excited by a vertical vibration of the container. Beyond a sufficiently large excitation amplitude the plane interface undergoes an instability (Faraday instability) and standing surface waves appear, oscillating with a frequency one half of the drive. This type of parametric wave instability is attractive as the wavelength of the pattern is [*dispersion*]{} rather than [*geometry*]{} controlled. Just by varying the drive frequency the wave number can be tuned in a wide range. In that sense the Faraday setup is well suited for the study of phase dynamics.
Nevertheless recent research activity in this field was mainly dedicated to the exploration of the processes underlying the selection of patterns with a fixed wavelength. Faraday [@faraday31] was the first to provide a quantitative study of this system, revealing that a sinusoidal vibration may induce a periodic array of squares. Later on, more complicated patterns with up to a 12-fold rotational symmetry (quasi-periodic structures) have been observed [@douady90; @muller98]. Here the amplitude equation technique contributed considerably to unfold the governing spatio-temporal resonance mechanisms. Applied to a set of modes ${\bf k}_i$ with different orientations but fixed wavelength, $|{\bf k}_i|=k$ the resulting set of Landau equations lead to a semi-quantitative understanding [@milner91] of pattern selection in this system. Motivated by these advances the idea came up to apply more complicated drive signals composed of two or more commensurable frequencies [@edwards93]. That way the simultaneous excitation of distinct wavelengths gave rise to novel surface patterns in form of superlattices [@edwards93; @kudrolli98; @arbell00]. Only recently the phase information carried by the participating modes was found to have a crucial influence on the visual appearance of the convection structures [@wagner01].
In comparison to other classical pattern forming systems such as RBC or TCF, the phase dynamics in the Faraday system is much less explored. In usual Faraday experiments the drive frequency (or frequency composition including relative amplitudes) is held fixed while the overall drive amplitude is ramped in order to record the bifurcation sequence of appearing structures. To our knowledge none of the previous investigations used the excitation frequency $\omega$ as the primary control parameter rather than the drive amplitude $a$. That way it is particularly simple to impose phase perturbations on ordered patterns and to study their relaxation dynamics. Moreover, doing phase dynamics on the Faraday system has the additional advantage of rather quick relaxation times, which in typical setups are one and two order of magnitude faster than for instance in RBC.
The present paper reports a systematic investigation of phase relaxation on Faraday surface waves. Our study is focused on the relaxational dynamics of an ideal surface pattern with a square tesselation. By evaluating the relaxation time of the pattern in response to small changes of the frequency, the phase diffusion coefficient has been measured. The experimental results are found to be in good agreement with the findings of the linear theory evaluated for a system of infinite lateral extension.
The experimental setup consists of a black cylindrical container built out of anodized aluminium, and filled to a height $h$ of $4.2$ mm with a silicone oil (kinematic viscosity $\nu=21.4
\times 10^{-6}$ m$^2$/s, density $\rho=949$ kg/m$^3$, surface tension $\sigma=17.3 \times 10^{-3}$ N/m). In order to study finite size effects, we used three different containers, with inner diameters $L_1=265$ mm, $L_2=185$ mm, and $L_3=125$ mm. A glass plate covering the container was used to prevent evaporation, pollution and temperature fluctuations of the liquid. Furthermore, to avoid uncontrolled changes of the viscosity, density, and surface tension of the liquid, all the measurements have been performed at a constant temperature of $30\pm
0.1^\circ$C. The Faraday waves were excited by an electromagnetic shaker vibrating vertically with an acceleration allowing for simultaneous amplitude [*and*]{} frequency modulations in the form $a(t)\cos{\omega(t)}$. The corresponding input signal was produced by a waveform generator via a D/A-converter. The instantaneous acceleration was measured by a piezoelectric sensor. In a preparatory experiment undertaken with a sinusoidal (i.e. unmodulated) drive $a \cos{\omega t}$ the critical acceleration amplitude $a_c(\omega)$ for the onset of the Faraday instability was determined by visual inspection of the interface while quasi-statically ramping $a$ at fixed $\omega=2 \pi f$ (see Fig. \[fig1\]). Throughout the investigated frequency interval $70$ Hz$<f<110$ Hz the surface patterns, which appear at a supercritical drive of less than about $1.1 \times a_c$, always consisted of an ordered square wave pattern, which – after some healing time – was free of defects (Fig. \[fig1\]b). In order to study the dynamics of phase-perturbed patterns we have carried out measurements of the average wave number $k(t)$ of the Faraday pattern in response to small changes of the drive frequency $\omega(t)$ around a mean value $\omega_0$. The $\omega$-modulation has been accomplished in two different ways: (i) By discontinuous jumps (back and forth) between frequencies $\omega_0-\Delta \omega/2$ and $\omega_0+\Delta \omega/2$ with a repetition period $T$ between $100$-$300$s, sufficiently large for the pattern to relax. (ii) By a sinusoidal modulation of the drive frequency according to $\omega(t)=\omega_0 + \Delta \omega \sin{(\Omega t)}$, with $T=1/F=2 \pi/\Omega$ between $2$ and $1000$s. (within the frequency range of our study, the response time of the shaker to small changes of the drive frequency is less than $10$ ms and thus negligible). In both cases the vibration amplitude $a(t)$ was co-modulated in such a way that the instantaneous supercritical drive $\varepsilon=a(t)/a_c[\omega(t)]-1$ remained constant (see Fig. \[fig1\]a). The bandwidth $\Delta \omega$ of the modulation needed to be confined to a few Hz in order to avoid the occurrence of dislocation-type defects. Under those conditions the square pattern remained practically ideal without perturbations (see Fig. \[fig1\]b), just expanding and contracting (breathing) in a homogeneous manner with the modulation period $T$. To obtain the temporal wave number dependence a full frame CCD camera surrounded by a set of 4 incandescent lamps was mounted some distance above the container. About $100$ pictures of the light reflected from the surface were taken at consecutive instances of maximum surface excursion from which the spatially averaged wave number $k(t)$ was extracted by evaluating the position of the principal peaks in a 2-dimensional FFT.
The wave number $k(t)$ followed the modulation in a relaxational manner. For the discontinuous modulation (i) this is directly apparent from Fig. \[fig2\]. Here the relaxation time $\tau$ has been derived by fitting the exponential decay of the data. In the type (ii) experiment $k(t)$ oscillates around a mean value $k_0$ with an amplitude $\Delta k_m$ and a temporal phase lag $\delta$ (see Fig. \[fig3\]). Introducing the complex wave number increment $\Delta k^\star={\rm Re} [\Delta k^\star] + i \,
{\rm Im}[\Delta k^\star]=\Delta k_m \, e^{i \delta}$, its real and imaginary parts are plotted in Fig. \[fig4\] as a function of the modulation frequency $F$. The solid curves of this figure are fits of a linear Debye relaxator, where $\Delta k^*=\Delta
k(\Omega =0)/[1+i\Omega \tau]$. Fig. \[fig5\] shows the relaxation time $\tau$, as obtained from both types of experiments, as a function of the mean wave number $k_0$, the container diameter $L$ and the reduced drive strength $\varepsilon$. A dependence on the (small) modulation amplitude $\Delta \omega$ could not be detected. The experimental data reveal a linear increase with the wave number $k_0$ and a proportionality to the square of the container size $L$.
The dependence of the relaxation time $\tau$ on, respectively, $k_0$, $L$, and $\varepsilon$ can be understood in terms of the phase diffusion approach [@pomeau79]. Here one takes advantage of the fact that local disturbances of the elevation amplitude die out rapidly, while long-wavelength phase perturbation survive on a much longer (diffusive) time scale. To streamline the arguments and to work out the basic physics we consider a 1-dimensional surface elevation profile in form of stripes as given by $\zeta(x,t) \propto [e^{i k_0 x +
\varphi(x,t)} + c.c.]\cos{(\omega t/2)} $. Here $\partial_x
\varphi=\Delta k$ describes the spatio-temporal variation of the local wave number around the underlying base pattern with the wave number $k_0$. Following the phase diffusion approach the phase $\varphi$ obeys a diffusion equation of the form $\partial_t \varphi = D_\parallel
\partial_{xx} \varphi $ with the diffusion constant (valid close above onset) of the form $$\label{diffusion}
D_\parallel=\frac{\xi_0^2}{\tau_0} \, \frac{\varepsilon - 3
\xi_0^2 (k_0-k_c)^2}{\varepsilon - \xi_0^2 (k_0-k_c)^2}.$$ The coefficients $\tau_0^{-1}=\partial \lambda /
\partial \varepsilon|_{k=k_c,\varepsilon=0}$ and $\xi_0^2=-\tau_0/2 (\partial^2 \lambda /
\partial k^2)|_{k=k_c,\varepsilon=0}$ are given in terms of the linear growth rate $\lambda=\lambda(\varepsilon,k)$ at which plane wave perturbations grow out of the plane undeformed interface, when the Faraday instability sets in. Here $k_c$ is the wave number at onset of the Faraday instability. For weakly damped ($\nu
k_0^2/\omega \ll1$) capillary waves \[$k_0 \simeq
(\rho/\sigma)^{1/3}\, (\omega/2)^{2/3}$\] on a deep ($kh\gg 1$) fluid layer (reasonable approximations in our experiment) one obtains approximately $\tau_0^{-1} \simeq 2 \nu k_0^2$ and $\xi_0^2=(1/2) [9 \sigma/(16 \nu^2 \rho)] k_0^{-3}$. By decomposition of the phase perturbations into a set of discrete Fourier modes compatible with the finite container dimension, $\varphi = \sum_{n=1}^\infty a_n \sin{(n \pi/L)}$, the mode $n=1$ has the slowest decay time $$\label{tau}
\tau=\frac{L^2}{\pi^2} D_\parallel^{-1}= \frac{\nu \rho}{\sigma}
\, k_0 \, \frac{16 L^2}{9 \pi^2 } \, \frac{\varepsilon -
\xi_0^2(k_0-k_c)^2}{\varepsilon - 3 \xi_0^2 (k_0-k_c)^2}$$ and thus determines the relaxation time of the wave number of the experiment. The dashed line in Fig. \[fig5\] is the prediction according to Eq. (\[tau\]). For a quantitatively more reliable check we also evaluated the coefficients $\tau_0$ and $\xi_0^2$ numerically from a linear analysis, which takes into account the finite filling level as well as gravitational contributions to the wave dispersion. The respective result is shown by the solid line in Fig. \[fig5\]a. Furthermore, the predicted quadratic dependence of $\tau$ on the container dimension $L$ is verified by Fig. \[fig5\]b. The theoretical prediction Eq. (\[tau\]) also implies a slight dependence of $\tau$ on the drive amplitude $\varepsilon$. However, checking for this feature requires to account for the fact that also the mean wave number $k_0$ is affected by the drive strength. We deduced the empiric dependence $k_0/k_c=1-\beta \varepsilon$ with $\beta=0.264$ from a control run where $\varepsilon$ was slowly ramped at fixed $f$. Inserting this result into the last term on the right hand side of Eq. (\[tau\]) leads to the following expression $$\label{correction}
\frac{\tau(\varepsilon)}{\tau(\varepsilon=0)}=\frac{1- \xi_0^2
k_c^2 \beta^2 \varepsilon}{1- 3 \xi_0^2 k_c^2 \beta^2
\varepsilon}.$$ Although this relation (see solid line in Fig. \[fig5\]c) gives a reasonable estimate for the $\varepsilon$-dependence of $\tau$ it does not correctly reflect the empiric dependence. Apparently this is a finite size effect, which is expected to become significant at small values of $\varepsilon$. Roth et al. [@roth94] recently demonstrated that the effective phase diffusion constant, measurable in RBC and TCF relaxation experiments, depends sensitively on the aspect ratio $\alpha=\sqrt{\varepsilon}L/\xi_0$, defined by the quotient between the container dimension and the linear correlation length. Taking TCF as an example, a decrease of $\alpha$ from $50$ down to $15$ (which in our experiment corresponds to a reduction of $\varepsilon$ form $9\%$ to $2 \%$) implies a decay of the relaxation time by $20-30\%$. This is of the same order of magnitude as the value observed in our measurements (see Fig. \[fig5\]c).
[*Acknowledgements*]{} — We thank M. Lücke for helpful comments and J. Albers for his support. This work is financially supported by the Deutsche Forschungsgemeinschaft.
M. C. Cross, P. C. Hohenberg, Rev. Mod. Phys. [**65**]{}, 851 (1993).
A. C. Newell, J. A.Whitehead, J. Fluid Mech. [**38**]{}, 279 (1969); L. A. Segel, J. Fluid Mech [**38**]{}, 203 (1969).
Y. Pomeau, P. Manneville, J. Phys. Lett. [**40**]{}, L-609 (1979).
L. Kramer, E. Ben-Jacob, H. Brand, M. C. Cross, Phys. Rev. Lett. [**49**]{}, 1891 (1982); Y. Pomeau, S. Zaleski, J. Phys. Lett. [**44**]{}, L135 (1983); I. Rehberg, E. Bodenschatz, B. Winkler, and F. H. Busse Phys. Rev. Lett. 59, 282-284 (1987); L. Ning, G. Ahlers, and D. S. Cannell Phys. Rev. Lett. 64, 1235-1238 (1990).
M. Faraday, Philos. Trans. R. Soc. London [**52**]{}, 319 (1831).
S. Douady, J. Fluid Mech. [**221**]{}, 383 (1990); B. Christiansen, P. Alstrom, M. T. Levinsen, Phys. Rev. Lett. [**68**]{}, 2157 (1992); J. Fluid Mech. [**291**]{}, 3231 (1998); K. Kumar, K.M. S. Bajaj, Phys. Rev. E[**52**]{}, R4606 (1995); A. Kudrolli, J. P. Gollub, Physica D [**97**]{}, 133 (1996); D. Binks, W. van der Water, Phys. Rev. Lett. [**78**]{}, 4043 (1997).
for a recent review see also: H. W. Müller, R. Friedrich, and D. Papathanassiou, [*Theoretical and experimental studies of the Faraday instability*]{}, in: [*Lecture notes in Physics*]{}, ed. by F. Busse and S. C. Müller, Springer (1998).
S. T. Milner, J. Fluid Mech. [**225**]{}, 81 (1991); W. Zhang, J. Vinals, Phys. Rev E [**53**]{}, R4283 (1996); J. Fluid Mech. [**336**]{}, 301 (1997); [**341**]{}, 225 (1997);P. Chen, J. Vinals, Phys. Rev. Lett. [**79**]{}, 2670 (1997); Phys. Rev. E [**60**]{}, 559 (1999).
W. S. Edwards, S. Fauve, Phys. Rev E [**47**]{}, R788 (1993); J. Fluid Mech. [**278**]{}, 123 (1994).
A. Kudrolli, B. Pier, J. P. Gollub, Physica D [**123**]{}, 99 (1998).
H. Arbell, J. Fineberg, Phys. Rev. Lett. [**85**]{}, 756 (2000); [**84**]{}, 654 (2000).
C. Wagner, H. W. Müller, K. Knorr, [*Pattern formation at the bi-critical point of the Faraday instability*]{} submitted, pattsol: http://arxiv.org/ps/nlin.PS/0107018
D. Roth, M. Lücke, M. Kamps, R. Schmitz, Phys. Rev. E [**50**]{}, 2756 (1994).
|
---
abstract: 'Extended dynamical mean-field theory (EDMFT) is insufficient to describe non-local effects in strongly correlated systems, since corrections to the mean-field solution are generally large. We present an efficient scheme for the construction of diagrammatic extensions of EDMFT that avoids usual double counting problem by using an exact change of variables (the Dual Boson formalism) to distinguish the correlations included in the dynamical mean-field solution and those beyond. With a computational efficiency comparable to EDMFT$+$GW approach, our scheme significantly improves on the charge order transition phase boundary in the extended Hubbard model.'
author:
- 'E. A. Stepanov'
- 'A. Huber'
- 'E. G. C. P. van Loon'
- 'A. I. Lichtenstein'
- 'M. I. Katsnelson'
bibliography:
- 'DB\_GW.bib'
title: ' From local to non-local correlations: the Dual Boson perspective '
---
Introduction
============
The description of strongly correlated electronic systems is still one of the most challenging problems in condensed matter physics, despite a lot of efforts and plenty of suggested theories. One of the most popular approaches is the dynamical mean-field theory (DMFT) [@PhysRevLett.62.324; @RevModPhys.68.13; @RevModPhys.78.865; @WernerCasula], which provides an approximate solution of the (multiband) Hubbard model by mapping it to a local impurity problem. Although DMFT neglects non-local correlation effects, it is able to capture important properties of the system such as the formation of Hubbard bands [@Hubbard238; @Hubbard401] and the Mott transition [@mott1974metal; @RevModPhys.70.1039]. Later, an extended dynamical mean-field theory (EDMFT) [@PhysRevLett.77.3391; @PhysRevB.61.5184; @PhysRevLett.84.3678; @PhysRevB.63.115110; @PhysRevB.66.085120] was introduced to include collective (bosonic) degrees of freedom, such as charge or spin fluctuations, into DMFT. Unfortunately, these collective excitations have a strongly non-local nature, so a dynamical mean-field approach is insufficient and it was necessary to develop some extensions, we will call them EDMFT++, to treat non-local correlations.
The quantities of physical interest in EDMFT++ are the electronic self-energy $\Sigma_{{\mathbf{k}}\nu}$ and polarization operator $\Pi_{{\mathbf{q}}\omega}$. The main idea of the dynamical mean-field approach is that all local correlations are already accounted for in the effective local impurity problem which results in the self-consistency conditions on the local part of lattice Green’s function and susceptibility. The mean-field ideology implies that in the EDMFT approach, the local self-energy and polarization operator are given by those of the impurity model. To go beyond, one needs to determine the corrections $\bar{\Sigma}_{{\mathbf{k}}\nu}$ and $\bar{\Pi}_{{\mathbf{q}}\omega}$ to the electronic self-energy and polarization operator that describe non-local excitations.
However, as soon as one goes beyond the dynamical mean-field level, the *non-local* corrections also change the *local* parts of $\Sigma_{{\mathbf{k}}\nu}$ and $\Pi_{{\mathbf{q}}\omega}$. Since the local self-energy has contributions both from the dynamical mean-field solution and from the non-local corrections, great care should be taken to avoid double counting of correlation effects when merging EDMFT with a diagrammatic approach.
The EDMFT$+$GW approach [@PhysRevLett.90.086402; @PhysRevLett.109.226401; @PhysRevB.87.125149; @PhysRevB.90.195114] combines GW diagrams [@PhysRev.139.A796; @GW1; @GW2] for the self-energy and polarization operator with EDMFT. In an attempt to avoid double counting, all local contributions of the GW diagrams are subtracted and only the *purely non-local* part of $\bar{\Sigma}_{{\mathbf{k}}\nu}$ and $\bar{\Pi}_{{\mathbf{q}}\omega}$ is used to describe non-local correlations. Excluding double counting is necessary for a correct construction of the theory, however the EDMFT$+$GW procedure is not unique and is the subject of hot discussions [@PhysRevB.91.235114].
More complicated approaches to treat non-local effects with the impurity problem as a starting point are D$\Gamma$A [@PhysRevB.75.045118], 1PI [@PhysRevB.88.115112] and DMF$^2$RG [@PhysRevLett.112.196402]. These extensions of DMFT include two-particle vertex corrections in their diagrams. However, these methods cannot describe the collective degrees of freedom arising from non-local interactions that are of interest here. On the other hand, the recent TRILEX [@PhysRevB.92.115109; @2015arXiv151206719A] approach was introduced to treat diagrammatically both fermionic and bosonic excitations. In this method the exact Hedin form [@PhysRev.139.A796] of the lattice self-energy and polarization operator are approximated by including the full impurity fermion-boson vertex in the diagrams.
Instead of trying to construct the proper dynamical mean-field extension in terms of lattice Green’s functions, one can take a different route and introduce so-called dual fermions (DF) [@PhysRevB.77.033101] and dual bosons (DB) [@Rubtsov20121320; @PhysRevB.90.235135; @PhysRevB.93.045107] and then deal with new *dual* degrees of freedom. In these methods the local impurity model still serves as the starting point of a perturbation expansion, so (E)DMFT is reproduced as the non-interacting dual problem. It is important to point out that the self-energy and polarization operator in DF and DB are free from double counting problems by construction: there is no overlap between the impurity contribution to the self-energy and polarization operator and local parts of dual diagrams since the impurity model deals with *purely local* Green’s functions only and the dual theory is constructed from *purely non-local* building blocks. The impurity contribution has been excluded already on the level of the bare dual Green’s function and interaction. Contrary to the existing methods, the DB approach does allow to describe strongly non-local collective excitations such as plasmons [@PhysRevLett.113.246407].
The self-energy and polarization operator in self-consistent DB are built up as a ladder consisting of full fermion-fermion and fermion-boson vertices obtained from the local impurity problem. For computational applications, particularly those aimed at realistic multi-orbital systems, it can be convenient to use simpler approximations that do not require the computational complexity of the full two-particle vertex. To that end, we construct EDMFT++ schemes that do not require the full two-particle vertex, that exclude double counting using the dual theory, and that contain the most essential parts of the non-local physics. We illustrate this by means of the charge-order transition in the extended Hubbard model.
EDMFT++ theories
================
The extended Hubbard model serves as the canonical example of a strongly correlated system where non-local effects play a crucial role. In momentum space, its action is given by the following relation $$\begin{aligned}
S=-\sum_{{\bf k}\nu\sigma} c^{*}_{{\bf k}\nu\sigma}[i\nu+\mu-\varepsilon^{\phantom{*}}_{\bf k}]c^{\phantom{*}}_{{\bf k}\nu\sigma}+\frac{1}{2}\sum_{{\bf q}\omega}U^{\phantom{*}}_{\bf q}\rho^{*}_{{\mathbf{q}}\omega}\rho^{\phantom{*}}_{{\mathbf{q}}\omega}.
\label{eq:action}\end{aligned}$$ Here we are interested only in the charge fluctuations, so in the following we suppress the spin labels on Grassmann variables $c^{*}_{{\mathbf{q}}\nu}$ ($c^{\phantom{*}}_{{\mathbf{q}}\nu}$) corresponding to creation (annihilation) of an electron with momentum ${\mathbf{k}}$ and fermionic Matsubara frequency $\nu$. The interaction $U_{\bf q}=U+V_{\bf q}$ consists of the on-site and nearest-neighbour interactions respectively. The charge fluctuations are given by the complex bosonic variable $\rho_{\omega}=~n_\omega-{\ensuremath{\left\langle n \right\rangle}}\delta_{\omega}$, where $n^{\phantom{*}}_\omega = \sum_{\nu\sigma}c^{*}_{\nu}c^{\phantom{*}}_{\nu+\omega}$ counts the number of electrons and $\omega$ is a bosonic Matsubara frequency. The chemical potential $\mu$ is chosen in such a way that the average number of electrons per site is one (half-filling). Finally, $\varepsilon_{{\mathbf{k}}}$ is the Fourier transform of the hopping integral $t$ between neighboring sites.
First of all, since we are interested in the EDMFT++ theories, let us briefly remind the main statements of the extended dynamical mean-field theory. In EDMFT, the lattice action is split up into a set of single-site local impurity actions $S_{\text{imp}}$ and a non-local remaining part $S_{\rm rem}$ $$\begin{aligned}
S =& \sum_{j} S^{(j)}_{\rm imp} + S^{\phantom{(j)}}_{\rm \hspace{-0.1cm}\phantom{i}rem},
\label{eq:actionlatt}\end{aligned}$$ which are given by the following explicit relations $$\begin{aligned}
S_{\text{imp}}=&-\sum_{\nu} c^{*}_{\nu}[i\nu+\mu-\Delta^{\phantom{*}}_{\nu}]c^{\phantom{*}}_{\nu} \notag\\
+&\, \frac{1}{2}\,\sum_{\omega}\,{\cal U}^{\phantom{*}}_{\omega}\, \rho^{*}_{\omega} \rho^{\phantom{*}}_{\omega}\label{eq:imp_action},\\
S_{\text{rem}} =&-\sum_{{\bf k}\nu} c^{*}_{{\bf k}\nu}[\Delta^{\phantom{*}}_{\nu}-\varepsilon^{\phantom{*}}_{\bf k}]c^{\phantom{*}}_{{\bf k}\nu} \notag\\
+&\, \frac{1}{2}\,\sum_{{\bf q}\omega}\,(U^{\phantom{*}}_{\bf q}-{\cal U}^{\phantom{*}}_{\omega})\, \rho^{*}_{{\bf q}\omega} \rho^{\phantom{*}}_{{\bf q}\omega}. \label{eq:rem_action}\end{aligned}$$ Since the impurity model is solved exactly, our goal is to move most of the correlation effects into the impurity, so that the remainder is only weakly correlated. For this reason, two hybridization functions $\Delta_{\nu}$ and $\Lambda_{\omega}$ are introduced to describe the interplay between the impurity and external fermionic and bosonic baths respectively. These functions are determined self-consistently for an optimal description of local correlation effects. The local bare interaction of the impurity model is then equal to ${\cal U}_{\omega} = U + \Lambda_{\omega}$. The impurity problem can be solved using, e.g., continuous-time quantum Monte Carlo solvers [@RevModPhys.83.349; @PhysRevLett.104.146401], and one can obtain the local impurity Green’s function $g_{\nu}$, susceptibility $\chi_{\omega}$ and renormalized interaction ${\cal W}_{\omega}$ as $$\begin{aligned}
g_{\nu} &= -{\ensuremath{\left\langle c^{\phantom{*}}_{\nu}c^{*}_{\nu} \right\rangle}}_\text{imp}, \\
\chi_{\omega} &= -{\ensuremath{\left\langle \rho^{\phantom{*}}_{\omega}\rho^{*}_{\omega} \right\rangle}}_\text{imp}, \\
{\cal W}_{\omega} &= {\cal U}_{\omega} + {\cal U}_{\omega}\chi_{\omega}{\cal U}_{\omega},\end{aligned}$$ where the average is taken with respect to the impurity action . One can also introduce the local impurity self-energy $\Sigma_{\rm imp}$ and polarization operator $\Pi_{\rm imp}$ $$\begin{aligned}
\Sigma^{\phantom{*}}_{\rm imp} &= i\nu + \mu - \Delta^{\phantom{*}}_{\nu} - g^{-1}_{\nu}, \\
\Pi^{-1}_{\rm imp} &= \chi^{-1}_{\omega} + {\cal U}^{\phantom{*}}_{\omega},\end{aligned}$$ that are used as the basis for the EDMFT Green’s function $G_{\rm E}$ and renormalized interaction $W_{\rm E}$ defined as $$\begin{aligned}
G_{\rm E}^{-1} &= \hspace{0.1cm}G_{0}^{-1}-\Sigma^{\phantom{1}}_{\rm imp}
= \hspace{0.1cm}g^{-1}_{\nu}-(\varepsilon^{\phantom{*}}_{{\mathbf{k}}} - \Delta^{\phantom{*}}_{\nu}), \\
W_{\rm E}^{-1} &= W_{0}^{-1}-\Pi^{\phantom{1}}_{\rm imp}
= U_{{\mathbf{q}}}^{-1}-(\chi^{-1}_{\omega}+{\cal U}^{\phantom{*}}_{\omega})^{-1}.\end{aligned}$$ Here $G_{0}=(i\nu+\mu-\varepsilon_{{\mathbf{k}}})^{-1}$ is the bare lattice Green’s function and $W_{0}$ is the bare interaction, which is equal to $U_{{\mathbf{q}}}$, or $V_{{\mathbf{q}}}$ in the case of $UV$–, or $V$– decoupling respectively [@PhysRevLett.109.226401; @PhysRevB.87.125149].
Importantly, a solution of every EDMFT++ theory can be exactly written in terms of EDMFT Green’s functions and renormalized interactions as follows $$\begin{aligned}
G_{{\mathbf{k}}\nu}^{-1} &= \hspace{0.1cm}G_{0}^{-1}-\Sigma^{\phantom{1}}_{{\mathbf{k}}\nu} \hspace{0.05cm}
= \hspace{0.1cm}G_{\rm E}^{-1}-\tilde{\Sigma}^{\phantom{1}}_{{\mathbf{k}}\nu},
\label{eq:nonlocal_Sigma} \\
W_{{\mathbf{q}}\omega}^{-1} &= W_{0}^{-1}-\Pi^{\phantom{1}}_{{\mathbf{q}}\omega}
= W_{\rm E}^{-1}-\tilde{\Pi}^{\phantom{1}}_{{\mathbf{q}}\omega},
\label{eq:nonlocal_Pi}\end{aligned}$$ where $\Sigma_{{\mathbf{k}}\nu}$ and $\Pi_{{\mathbf{q}}\omega}$ are the exact, unknown in general, self-energy and polarization operator of the model respectively, and $\bar{\Sigma}_{{\mathbf{k}}\nu}=~\Sigma_{{\mathbf{k}}\nu}-~\Sigma_{\rm imp}$ and $\bar{\Pi}_{{\mathbf{q}}\omega}=~\Pi_{{\mathbf{q}}\omega}-\Pi_{\rm imp}$ are the corrections to the dynamical mean-field solution. With EDMFT as a starting point, the goal of EDMFT++ theories is to approximate these corrections. As pointed out above, $\bar{\Sigma}_{{\mathbf{k}}\nu}$ and $\bar{\Pi}_{{\mathbf{q}}\omega}$ should be introduced without double counting with an effective local impurity problem, but still can give a local contributions to the lattice self-energy and polarization operator.
Attempt to construct a simple exact numerical method to obtain non-local self-energy was presented in the Ref. . This, so called bold diagrammatic Monte Carlo method, is numerically exact, but very expensive for the realistic calculations. In this paper we will be focused on the less expensive diagrammatic methods.
(E)DMFT$+$GW approach
---------------------
Historically, the EDMFT$+$GW approach [@PhysRevLett.109.226401; @PhysRevB.87.125149; @PhysRevB.90.195114] introduced the first approximations for $\bar{\Sigma}_{{\mathbf{k}}\nu}$ and $\bar{\Pi}_{{\mathbf{q}}\omega}$. Here, the self-energy and polarization operator diagrams from the GW approximation [@PhysRev.139.A796; @GW1; @GW2] are added to the dynamical mean-field solution to treat non-local correlations, $$\begin{aligned}
\Sigma^{\rm GW}_{{\mathbf{k}}\nu} &=-\sum\limits_{{\mathbf{q}},\omega}G_{{\mathbf{k}}+{\mathbf{q}},\nu+\omega}W_{{\mathbf{q}}\omega},\label{eq:Sigma_Pi_GW}\\
\Pi^{\rm GW}_{{\mathbf{q}}\omega} &= \,2\,\sum\limits_{{\mathbf{k}},\nu}\,G_{{\mathbf{k}}+{\mathbf{q}},\nu+\omega}G_{{\mathbf{k}}\nu},
\label{eq:Sigma_Pi_GW1}\end{aligned}$$ where the coefficient “2” in Eq. accounts for the spin degeneracy. To avoid double counting between the impurity correlations and the GW correlations, only the non-local part of Eqs. - is used, i.e., $\bar{\Sigma}^{\rm GW}_{{\mathbf{k}}\nu}=~\Sigma^{\rm GW}_{{\mathbf{k}}\nu} -~\Sigma^{\rm GW}_{\rm loc}$ and $\bar{\Pi}^{\rm GW}_{{\mathbf{q}}\omega} = \Pi^{\rm GW}_{{\mathbf{q}}\nu} - \Pi^{\rm GW}_{\rm loc}$. Since the local interaction $U$ has already been accounted for the impurity problem, the bare non-local interaction in Eq. can be taken in the form of $V$– decoupling ($W_0 = V_{{\mathbf{q}}}$), which leads to simple separation of local and non-local contributions to the self-energy $\bar{\Sigma}_{{\mathbf{k}}\nu}$. Unfortunately, this form of renormalized interaction leads to overestimation of non-local correlation effects [@PhysRevLett.109.226401; @PhysRevB.87.125149]. On the other hand, the form of $UV$– decoupling ($W_0 = U_{{\mathbf{q}}}$) is more consistent with standard perturbation theory for the full Coulomb interaction, but leads to the problems with separation of local and non-local parts of the diagrams as shown in Appendix \[app:GW\]. Therefore, the form of the renormalized interaction and the way to avoid the double counting in general is a topic of hot discussions nowadays [@PhysRevB.91.235114].
Note that hereinafter the name $V$– or $UV$– decoupling in the EDMFT++ theories implies only the form of interaction $W_{0}$ used in the self-energy diagrams beyond the dynamical-field level. Since the aim of the paper is to compare the existing schemes of exclusion of the double counting, the form of the self-energy diagrams in these both cases remains the same. Our notations can differ from those introduced in the previous works on EDMFT++ theories by the presence of additional diagrams in the different versions of decoupling schemes (see for example Ref. )
It should be noted, that there is another clear way to avoid the double counting problem, namely simply ignoring non-local interactions in the dynamical mean-field part of the action and including them in the non-local corrections only. The impurity model then corresponds to DMFT, i.e., ${\cal U}_\omega = U$. Then, the non-local renormalized interaction in Eq. be can taken in the form of $V$– decoupling as $W_0 = V_{{\mathbf{q}}}$, and the local part of this self-energy diagram is automatically zero. Although, the DMFT$+$GW approach is free from double counting by construction, it is less advanced than EDMFT+GW, since it ignores screening of the local interaction by non-local processes.
Local vertex corrections beyond the EDMFT {#sec:LocVertCor}
-----------------------------------------
The exact self-energy and polarization operator of the lattice problem are given by the following relations [@PhysRev.139.A796] $$\begin{aligned}
\Sigma_{{\mathbf{k}}\nu} &= -\sum\limits_{{\mathbf{q}}\omega}G^{\phantom{{\mathbf{k}}}}_{{\mathbf{k}}+{\mathbf{q}},\nu+\omega}W^{\phantom{{\mathbf{k}}}}_{{\mathbf{q}}\omega}\Gamma^{{\mathbf{k}}{\mathbf{q}}}_{\nu\omega} =
\includegraphics[width=0.16\linewidth]{Hedin_Sigma1.pdf}~, \label{eq:HedinSigma}\\
\Pi_{{\mathbf{q}}\omega} &= \,2\,\sum\limits_{{\mathbf{k}}\nu}G^{\phantom{{\mathbf{k}}}}_{{\mathbf{k}}+{\mathbf{q}},\nu+\omega}G^{\phantom{{\mathbf{k}}}}_{{\mathbf{k}}\nu}\,\Gamma^{{\mathbf{k}}{\mathbf{q}}}_{\nu\omega}=
\includegraphics[width=0.16\linewidth]{Hedin_Pi1.pdf}~, \label{eq:HedinPi}\end{aligned}$$ where $\Gamma^{{\mathbf{k}}{\mathbf{q}}}_{\nu\omega}$ is the exact three-point Hedin vertex. Unfortunately, the full three-point vertex of the considered problem is unknown, and the self-energy and polarization operator can be found only approximately. The most important correlation effects beyond EDMFT and the GW-diagrams are expected in the frequency-dependence of the fermion-boson vertex [@PhysRevB.90.235135; @PhysRevB.92.115109]. For this reason, the recent TRILEX [@PhysRevB.92.115109; @2015arXiv151206719A] approach with application to the Hubbard model was introduced. In this approach the exact Hedin vertex is approximated by the full local three-point vertex of impurity problem, which results in $$\begin{aligned}
\Sigma^{\rm TRILEX}_{{\mathbf{k}}\nu} &= -\sum\limits_{{\mathbf{q}}\omega}G^{\phantom{{\mathbf{k}}}}_{{\mathbf{k}}+{\mathbf{q}},\nu+\omega}W^{\phantom{{\mathbf{k}}}}_{{\mathbf{q}}\omega}\gamma^{\phantom{1}}_{\nu\omega},\\
\Pi^{\rm TRILEX}_{{\mathbf{q}}\omega} &= \,2\,\sum\limits_{{\mathbf{k}}\nu}G^{\phantom{{\mathbf{k}}}}_{{\mathbf{k}}+{\mathbf{q}},\nu+\omega}G^{\phantom{{\mathbf{k}}}}_{{\mathbf{k}}\nu}\,\gamma^{\phantom{1}}_{\nu\omega},
\label{eq:TRILEX_diagr}\end{aligned}$$ where $\gamma_{\nu\omega}$ is the full three-point vertex of impurity problem determined below (see Eq. ). Thus, the local parts of the self-energy and polarization operator are identically equal to the local impurity quantities $\Sigma_{\rm imp}$ and $\Pi_{\rm imp}$ respectively. Moreover, it is possible to split $\Sigma^{\rm TRILEX}_{{\mathbf{k}}\nu}$ and $\Pi^{\rm TRILEX}_{{\mathbf{q}}\omega}$ into the local impurity part and non-local contribution as it was shown in Ref. $$\begin{aligned}
\Sigma^{\rm TRILEX}_{{\mathbf{k}}\nu} &= \Sigma_{\rm imp} + \bar{\Sigma}^{\rm TRILEX}_{{\mathbf{k}}\nu},\\
\Pi^{\rm TRILEX}_{{\mathbf{q}}\omega} &= \Pi_{\rm imp} + \bar{\Pi}^{\rm TRILEX}_{{\mathbf{q}}\omega},\end{aligned}$$ where $$\begin{aligned}
\bar{\Sigma}^{\rm TRILEX}_{{\mathbf{k}}\nu} &= -\sum\limits_{{\mathbf{q}}\omega}\bar{G}^{\rm TRILEX}_{{\mathbf{k}}+{\mathbf{q}},\nu+\omega}\bar{W}^{\rm TRILEX}_{{\mathbf{q}}\omega}\gamma^{\phantom{1}}_{\nu\omega},\\
\bar{\Pi}^{\rm TRILEX}_{{\mathbf{q}}\omega} &= \,2\,\sum\limits_{{\mathbf{k}}\nu}\bar{G}^{\rm TRILEX}_{{\mathbf{k}}+{\mathbf{q}},\nu+\omega}\,\bar{G}^{\rm TRILEX}_{{\mathbf{k}}\nu}\gamma^{\phantom{1}}_{\nu\omega},\end{aligned}$$ and $\bar{G}^{\rm TRILEX}_{{\mathbf{k}}\nu} = G_{{\mathbf{k}}\nu}^{\phantom{1}}-g_{\nu}$, $\bar{W}^{\rm TRILEX}_{{\mathbf{q}}\omega}=~W^{\phantom{1}}_{{\mathbf{q}}\omega} - {\cal W}^{\phantom{1}}_{\omega}$ are non-local parts of the full lattice Green’s function and renormalized interaction respectively. Therefore, TRILEX approach is nothing more then an (E)DMFT+GW approximation with the same exclusion of double counting, where the GW diagrams are additionally dressed with the local three-point vertex from one side. In this case, the lattice Green’s function and renormalized interaction are given by the same Dyson Eqs. - with $\bar{\Sigma}^{\rm TRILEX}_{{\mathbf{k}}\nu}$ and $\bar{\Pi}^{\rm TRILEX}_{{\mathbf{q}}\omega}$ introduced beyond the dynamical mean-field level.
The main advantage of the TRILEX approach compared to existing diagrammatic methods is a computational efficiency due to the use of only the three-point vertex $\gamma_{\nu\omega}$ to threat non-local correlations. Nevertheless, even with this vertex function one can approximate the exact Hedin form of the self-energy and polarization function in a better way.
It is of course true, that if the self-energy and polarization operator in the exact form of Eqs. - do not contain any non-local propagators, then these quantities are given by the impurity $\Sigma_{\rm imp}$ and $\Pi_{\rm imp}$ respectively. Therefore, the improvements concern only the contributions $\bar{\Sigma}^{\rm TRILEX}_{{\mathbf{k}}\nu}$ and $\bar{\Pi}^{\rm TRILEX}_{{\mathbf{q}}\omega}$, written in terms of non-local propagators and local impurity vertex functions. As it was mentioned above, the self-consistency condition on the local parts of the Green’s function and renormalized interaction cannot also fix the local parts of $\Sigma_{{\mathbf{k}}\nu}$ and $\Pi_{{\mathbf{q}}\omega}$ at the same time. Therefore, additional local contributions to the self-energy and polarization operator, hidden in the non-local structure of the exact three-point vertex, can appear from the diagrams introduced beyond the dynamical mean-field level. For example, the Hedin vertex with the same lattice indices at the all three external points can contain non-local parts $$\begin{aligned}
\includegraphics[width=0.55\linewidth]{Vertex1.pdf}.
\label{eq:Vertex}\end{aligned}$$ Therefore, these contributions are not provided by the local impurity problem and should be taken into account.
It is worth mentioning, that the Hedin form of the self-energy and polarization operator is exact for the theories with only one type of propagators. As soon as one includes the vertex functions of impurity problem in the diagrams, all propagators become effectively separated into the two different types. Now, since the correction to the dynamical mean-field level is introduced in terms of only one (non-local) type of lines and all local lines are gathered in the local vertices, the Hedin form does not provide the exact result for the self-energy and polarization as shown in Refs. .
, .
In order to discuss this more details, let us take a closer look on the Hedin diagram for the self-energy. Above we discussed the case of only local propagators. Now let us assume, that Hedin vertex contains at least one non-local Green’s function $\tilde{G}_{{\mathbf{k}}\nu}$ and renormalized interaction $\tilde{W}_{{\mathbf{q}}\omega}$. Then, the self-energy diagram can be reduced to the form of two renormalized three-point vertices with the non-local propagators in between as shown in Fig. \[fig:Hedin\] a). It may also happen, that one particular contribution to the lattice self-energy does not contain the non-local renormalized interaction at all. This case is shown in Fig. \[fig:Hedin\] b). The last case without the non-local Green’s functions is not considered here due to appearance of the higher-order vertex functions of impurity problem in the diagrams. The same procedure can be done for the polarization operator. Then, if we restrict ourselves only to the lowest order vertex function $\gamma_{\nu\omega}$, the self-energy and polarization operator introduced beyond the dynamical mean-field level are $$\begin{aligned}
\includegraphics[width=0.3\linewidth]{SigmaPi_a1.pdf}~,\label{eq:Sigma2}\\
\includegraphics[width=0.3\linewidth]{SigmaPi_a2.pdf}~,\label{eq:Pi2}\end{aligned}$$ where, according to the above discussions, the three-point vertices appear at the both sides of GW diagrams.
,\
.
The illustration of the importance to have the three-point vertex functions on the both sides is also shown in Fig. \[fig:Two\]. Top row corresponds to a theory constructed from only one type of Green’s functions. Then, the fermion-boson vertices are composed of the same propagators as the remainder of the diagram, and it is always possible to “move” all vertex correction to the right side of diagram and obtain the Hedin form for the self-energy [@PhysRev.139.A796]. On the other hand, if the vertex functions are constructed from a different type of propagators (for example $g_{\nu}$ and ${\cal W}_{\omega}$ obtained from impurity problem) then the Green’s function $G$ and renormalized interaction $W$, it is no longer possible to obtain the Hedin form for this diagram. More clearly, the Hedin form is hidden inside of the impurity vertices. “Moving” the right part of the diagram to the right, as in the bottom row of Fig. \[fig:Two\], gives a diagram with the same Hedin structure, but with different propagators.
So, if one prefers to work with the bare lattice propagators and to use the Hedin form of self-energy, then it would be consistent to approximate the exact Hedin vertex using the same bare lattice quantities without inclusion of any other types of propagators. If, instead, a combination of Green’s functions and impurity vertices coming from different models is used, the renormalized vertices should be included at both ends of the GW diagram for the self-energy and polarization operator.
In order to take the above corrections into account and to compare the double counting exclusion schemes, one can introduce the EDMFT$+$GW$\gamma$ approach in the same way as EDMFT+GW by including the local impurity vertex $\gamma_{\nu\omega}$ in the GW diagrams as $$\begin{aligned}
\Sigma^{{\rm GW}\gamma}_{{\mathbf{k}}\nu} &= -\sum\limits_{{\mathbf{q}},\omega}\gamma_{\nu\omega}G_{{\mathbf{k}}+{\mathbf{q}},\nu+\omega}W_{{\mathbf{q}}\omega}\gamma_{\nu+\omega,-\omega},
\label{eq:Sigma_Pi_GWd}\\
\Pi^{{\rm GW}\gamma}_{{\mathbf{q}}\omega} &= \,2\,\sum\limits_{{\mathbf{k}},\nu}\gamma_{\nu\omega}G_{{\mathbf{k}}+{\mathbf{q}},\nu+\omega}\,G_{{\mathbf{k}}\nu}\gamma_{\nu+\omega,-\omega}.
\label{eq:Sigma_Pi_GWd1}\end{aligned}$$ Similarly to the EDMFT+GW case, only the non-local parts $\bar{\Sigma}^{{\rm GW}\gamma}_{{\mathbf{k}}\nu}$ and $\bar{\Pi}^{{\rm GW}\gamma}_{{\mathbf{q}}\omega}$ of the self-energy and the polarization operator are used beyond the EDMFT. Then, the lattice quantities are given by the same equations -.
Dual Boson approach
===================
A different way of accounting for non-local correlations beyond EDMFT is given by the Dual Boson approach [@Rubtsov20121320; @PhysRevB.93.045107]. The non-local part $S_{\rm rem}$ of the lattice action can be rewritten in terms of new *dual* variables $f^{*},f,\phi$ by performing a Hubbard–Stratonovich transformation, which leads to the dual action $$\begin{aligned}
\tilde{S} &=
- \sum_{{\mathbf{k}}\nu}f^{*}_{{\mathbf{k}}\nu} \tilde{G}_{0}^{-1} f^{\phantom{*}}_{{\mathbf{k}}\nu} - \frac{1}{2}\sum_{{\mathbf{q}}\omega} \phi^{*}_{{\mathbf{q}}\omega}\tilde{W}_{0}^{-1}\phi^{\phantom{*}}_{{\mathbf{q}}\omega}+ \tilde{V}
\label{eq:dual_action}\end{aligned}$$ with the bare dual propagators $$\begin{aligned}
\tilde{G}_{0} &= \,G_{\rm E} - g_{\nu},
\label{eq:barefermionpropagator}\\
\tilde{W}_{0} &= W_{\rm E}-{\cal W}_{\omega},
\label{eq:barebosonpropagator}\end{aligned}$$ and the full dual interaction $\tilde{V}$ that includes the impurity vertex functions $\gamma^{n,m}$ with $n$ fermion and $m$ boson lines to all orders in $n$ and $m$, as detailed in Appendix \[ap:dualtr\]. The first two terms in $\tilde{V}$ are given by the following relation $$\begin{aligned}
\tilde{V}
&=\gamma^{2,1}_{\nu,\omega}\,f^{*}_{\nu}f^{\phantom{*}}_{\nu+\omega}\phi^{*}_{\omega}
+ \frac14\,\gamma^{4,0}_{\nu,\nu',\omega}\,f^{*}_{\nu}f^{*}_{\nu'}f^{\phantom{*}}_{\nu+\omega}
f^{\phantom{*}}_{\nu'-\omega}.\end{aligned}$$ We define the three-point vertex $\gamma^{2,1}_{\nu\omega}$ in the same way as it is done in the TRILEX [@PhysRevB.92.115109; @2015arXiv151206719A] approach: $$\begin{aligned}
\gamma^{2,1}_{\nu\omega}&=
g^{-1}_{\nu}g^{-1}_{\nu+\omega}\alpha^{-1}_{\omega}
{\ensuremath{\left\langle c^{\phantom{*}}_{\nu}c^{*}_{\nu+\omega}\rho^{\phantom{*}}_{\omega} \right\rangle}},
\label{eq:3pvertex}\end{aligned}$$ where $\alpha_{\omega}={\cal W}_{\omega}/{\cal U}_{\omega}=(1+{\cal U}_{\omega}\chi_{\omega})$ is the local renormalization factor. In order to shorten notations, hereinafter we call the three-point vertex as $\gamma_{\nu\omega}$. The four-point vertex function $\gamma^{4,0}_{\nu\nu'\omega}$ can be determined similarly to the previous papers on the Dual Boson formalism [@Rubtsov20121320; @PhysRevB.93.045107] $$\begin{aligned}
\gamma^{4,0}_{\nu\nu'\omega} = g^{-1}_{\nu}g^{-1}_{\nu'}g^{-1}_{\nu'-\omega}g^{-1}_{\nu+\omega} \Big[&{\ensuremath{\left\langle c^{\phantom{*}}_{\nu}c^{\phantom{*}}_{\nu'}c^{*}_{\nu'-\omega}
c^{*}_{\nu+\omega} \right\rangle}}- \notag\\
&\,\,g_{\nu}g_{\nu'}(\delta_{\omega}-\delta_{\nu',\nu+\omega})\Big].\end{aligned}$$ Then, the dual Green’s function $\tilde{G}_{{\mathbf{k}}\nu} =~ -{\ensuremath{\left\langle f^{\phantom{*}}_{{\mathbf{k}}\nu}f^{*}_{{\mathbf{k}}\nu} \right\rangle}}$ and renormalized dual interaction $\tilde{W}_{{\mathbf{q}}\omega} =~ -{\ensuremath{\left\langle \phi^{\phantom{*}}_{{\mathbf{q}}\omega}\phi^{*}_{{\mathbf{q}}\omega} \right\rangle}}$, as well as dual self-energy $\tilde{\Sigma}_{{\mathbf{k}}\nu}$ and polarization operator $\tilde{\Pi}_{{\mathbf{q}}\omega}$, can be obtained diagrammatically [@Rubtsov20121320; @PhysRevB.90.235135; @PhysRevB.93.045107]. These dual quantities have usual connection $$\begin{aligned}
\tilde{G}_{{\mathbf{k}}\nu}^{-1} &= \,\tilde{G}^{-1}_{0}\, - \tilde{\Sigma}^{\phantom{1}}_{{\mathbf{k}}\nu},\\
\tilde{W}_{{\mathbf{q}}\omega}^{-1} &= \tilde{W}^{-1}_{0} - \tilde{\Pi}^{\phantom{1}}_{{\mathbf{q}}\omega}.
\label{eq:X-Pi}\end{aligned}$$
To close the circle, the Green’s function $G_{{\mathbf{k}}\nu}$ and renormalized interaction $W_{{\mathbf{q}}\omega}$ of the original model can be exactly expressed in terms of dual quantities via the similar Dyson Eqs. - as follows $$\begin{aligned}
G_{{\mathbf{k}}\nu}^{-1} &= \,G^{-1}_{\rm E}\, - \tilde{\Sigma}^{'}_{{\mathbf{k}}\nu},\label{eq:dual_G-Sigma}\\
W_{{\mathbf{q}}\omega}^{-1} &= W^{-1}_{\rm E} - \tilde{\Pi}^{'}_{{\mathbf{q}}\omega},
\label{eq:dual_X-Pi}\end{aligned}$$ where the self-energy and polarization operator introduced beyond EDMFT are $$\begin{aligned}
\tilde\Sigma^{'}_{{\mathbf{k}}\nu} &= \frac{\tilde\Sigma_{{\mathbf{k}}\nu}}{1+g_{\nu}\tilde\Sigma_{{\mathbf{k}}\nu}},\label{eq:Sigmap}\\
\tilde\Pi^{'}_{{\mathbf{q}}\omega} &= \frac{\tilde\Pi_{{\mathbf{q}}\omega}}{1+{\cal W}_{\omega}\tilde\Pi_{{\mathbf{q}}\omega}}\label{eq:Pip}.\end{aligned}$$
It should be noted, that the bare dual Green’s function and renormalized interaction are strongly non-local due to the EDMFT self-consistency conditions $$\begin{aligned}
\sum_{{\mathbf{k}}}G_{\rm E}\, &= g_{\nu},\\
\sum_{{\mathbf{q}}}W_{\rm E} &= {\cal W}_{\omega}.\end{aligned}$$ Therefore, the dual theory is free from the double-counting problem by construction, and the local impurity contribution is excluded from the diagrams on the level of the bare propagators -. The DB relations up to this point are exact and derived without any approximations.
It is worth mentioning, that the non-interacting dual theory ($\tilde{V}=0$) is equivalent to EDMFT. However, even in the weakly-interacting limit of the original model, $U\to~0$, the fermion-boson vertex $\gamma^{2,1}$ is non-zero and equal to unity, as shown in Appendix \[ap:dualtr\] and previous works on the DB approach. Thus, the Dual Boson formalism explicitly shows, that corrections to EDMFT are not negligible. Therefore, the dynamical mean-field level is insufficient for describing non-local bosonic excitations, because the interactions between the non-local fermionic and bosonic degrees of freedom are always relevant.
Dual diagrams for the self-energy and polarization operator
-----------------------------------------------------------
The impurity vertices $\gamma^{n,m}$ are computationally expensive to calculate for large $n$ and $m$. Practical DB calculations are usually restricted to $\gamma^{4,0}$ and $\gamma^{2,1}$, since that is sufficient to satisfy conservation laws and since processes involving higher-order vertices can be suppressed with the appropriate self-consistency condition [@PhysRevB.93.045107].
As it was shown above, the dual theory can be rewritten in terms of lattice quantities (see Eqs. -), where the dual diagrams are constructed in terms of only one type of bare propagators, i.e. the non-local part of EDMFT Green’s function and renormalized interaction given by Eqs. -. Local parts of the bare EDMFT propagators, namely $g_{\nu}$ and ${\cal W}_{\omega}$, are of the second type and hidden in the full local vertex functions of impurity problem, which serve as the bare interaction vertices in dual space. Then, with the same logic presented in Section \[sec:LocVertCor\], the DB self-energy and polarization operator in the ladder approximation are given by $$\begin{aligned}
&\includegraphics[width=0.5\linewidth]{SigmaDual.pdf}~,\\
&\includegraphics[width=0.5\linewidth]{PiDual.pdf}\hspace{-1.9cm},\end{aligned}$$ where the renormalized vertex functions are taken in the ladder approximation (see Fig. \[fig:PiPi’\] top line). Note, that here the splitting of propagators into the two parts is nominal and matters only for the dual theory when all diagrams are written in terms of only one non-local type of bare propagators. In general, the initial lattice theory works only with one type of Green’s function and renormalized interaction, namely the bare EDMFT propagators, that for the local case we call as impurity $g_{\nu}$ or ${\cal W}_{\omega}$ and for non-local as dual $\tilde{G}_0$ or $\tilde{W}_0$. Since the dual theory gives the correction to the lattice quantities, the dual contributions $\tilde{\Sigma}^{'}_{{\mathbf{k}}\nu}$ and $\tilde{\Pi}^{'}_{{\mathbf{q}}\omega}$ introduced beyond EDMFT should be irreducible with respect to the both impurity and dual propagators.
![Structure of the vertex corrections in different theories in case of one (top line) and two (bottom line) types of propagators. Solid straight and wave lines correspond to the Green’s function and renormalized interaction of one type and the dashed lines to those of the second type respectively.[]{data-label="fig:PiPi’"}](PiLadder_a.pdf "fig:"){width="0.95\linewidth"}\
![Structure of the vertex corrections in different theories in case of one (top line) and two (bottom line) types of propagators. Solid straight and wave lines correspond to the Green’s function and renormalized interaction of one type and the dashed lines to those of the second type respectively.[]{data-label="fig:PiPi’"}](PiLadder_b.pdf "fig:"){width="0.95\linewidth"}\
![Structure of the vertex corrections in different theories in case of one (top line) and two (bottom line) types of propagators. Solid straight and wave lines correspond to the Green’s function and renormalized interaction of one type and the dashed lines to those of the second type respectively.[]{data-label="fig:PiPi’"}](PiLadder_c.pdf "fig:"){width="0.95\linewidth"}
Let us turn to the more detailed explanation. As it was shown in Eqs. - the lattice self-energy and polarization operator introduced beyond EDMFT are not given by the dual $\tilde{\Sigma}_{{\mathbf{k}}\nu}$ and $\tilde{\Pi}_{{\mathbf{q}}\omega}$ and have the form of Eqs. -. Note, that denominators in the expressions for $\tilde{\Sigma}^{'}_{{\mathbf{k}}\nu}$ and $\tilde{\Pi}^{'}_{{\mathbf{q}}\omega}$ have very important physical meaning. The DB theory works with the [*full*]{} vertex functions of impurity problem, that obviously contain one-particle reducible contributions. Therefore, the denominators in Eqs. - exclude these one-particle reducible contributions from the diagrams for the self-energy and polarization operator in order to avoid the double counting in the Dyson Eqs. -. Similar discussions were presented in [@PhysRevB.88.115112] with application to the DF approach.
![Diagrammatic representation of the second and the third order contribution to the renormalized interaction.[]{data-label="fig:W"}](WLadder_a.pdf "fig:"){width="0.95\linewidth"}\
![Diagrammatic representation of the second and the third order contribution to the renormalized interaction.[]{data-label="fig:W"}](WLadder_b.pdf "fig:"){width="0.95\linewidth"}\
![Diagrammatic representation of the second and the third order contribution to the renormalized interaction.[]{data-label="fig:W"}](WLadder_c.pdf "fig:"){width="0.95\linewidth"}
To show this more explicitly, let us consider the following example. The dual polarization operator in the form of full two-particle ladder can be written in a matrix form as (see the second line of Fig. \[fig:PiPi’\] for the diagrammatic representation) $$\begin{aligned}
\tilde{\Pi}_{{\mathbf{k}}\omega} = \frac{\gamma{}\tilde{G}\tilde{G}\gamma}{1+\left[\gamma{}\right]^{-1}\gamma^{4,0}\tilde{G}\tilde{G}\gamma},\end{aligned}$$ where $\gamma^{4,0}$ is the full local four-point vertex of impurity problem. Using these relations, equation can be rewritten as (see the third line of Fig. \[fig:PiPi’\]) $$\begin{aligned}
\tilde\Pi^{'}_{{\mathbf{q}}\omega} &= \frac{\gamma{}\tilde{G}\tilde{G}\gamma{}}{1+\left[\gamma{}\right]^{-1}\left(\gamma^{4,0}+\gamma{}{\cal W}_{\omega}\gamma\right)\tilde{G}\tilde{G}\gamma}.\end{aligned}$$ Here $$\begin{aligned}
&\gamma^{4,0}_{\rm irr}=\gamma^{4,0}+\gamma{\cal W}_{\omega}\gamma,\notag\\
&\includegraphics[width=3.1cm]{Gir.pdf}
\label{eq:red-irr-vertex}\end{aligned}$$ is identically the irreducible part $\gamma^{4,0}_{\rm red}$ of full four-fermionic vertex function of impurity problem with respect to the renormalized interaction ${\cal W}_{\omega}$. Then the polarization operator $\tilde{\Pi}^{'}$ introduced beyond EDMFT is nothing more then normal dual polarization operator $\tilde{\Pi}$ taken in the form of full dual ladder, but with irreducible four-point vertices $\gamma^{4,0}_{\rm irr}$ instead of full vertices $\gamma^{4,0}$ of impurity problem. Therefore, the exact relation automatically corrects the structure of polarization operator, which is irreducible with respect to the dual renormalized interaction, to be also irreducible with respect to the impurity interaction ${\cal W}_{\omega}$.
Let us then compare the second and the third order term of diagrammatic expansion of Eq. shown in Fig. \[fig:W\] $$\begin{aligned}
W^{(2)}_{{\mathbf{q}}\omega} &= W^{\rm E}_{{\mathbf{q}}\omega}\tilde{\Pi}^{'}_{{\mathbf{q}}\omega}W^{\rm E}_{{\mathbf{q}}\omega}, \label{eq:W2}\\
W^{(3)}_{{\mathbf{q}}\omega} &= W^{\rm E}_{{\mathbf{q}}\omega}\tilde{\Pi}^{'}_{{\mathbf{q}}\omega}W^{\rm E}_{{\mathbf{q}}\omega}\tilde{\Pi}^{'}_{{\mathbf{q}}\omega}W^{\rm E}_{{\mathbf{q}}\omega}. \label{eq:W3}\end{aligned}$$ After the substitution of the the second term of $\tilde{\Pi}^{'}$ to Eq. and of the first term of $\tilde{\Pi}^{'}$ to Eq. we get $$\begin{aligned}
W^{(2)}_{{\mathbf{q}}\omega}=&-W_{\rm E}\gamma{}GG\gamma^{4,0}_{\rm irr}GG\gamma{}W_{\rm E}, \label{eq:W2n}\\
W^{(3)}_{{\mathbf{q}}\omega}=&\,\,\,\,\,\,\,W_{\rm E}\gamma{}GG\gamma{}{\cal W}_{\omega}\gamma{}GG\gamma{}W_{\rm E} \label{eq:W3n}\\
&+W_{\rm E} \gamma{}GG\gamma{}(W_{\rm E}-{\cal W}_{\omega})\gamma{}GG\gamma{}W^{\rm E}_{{\mathbf{q}}\omega}, \notag\\
W^{(2)+(3)}_{{\mathbf{q}}\omega} =&-W_{\rm E}\gamma{}GG\gamma^{4,0}_{\rm irr}GG\gamma{} W_{\rm E} \label{eq:W23n}\\
&+W_{\rm E} \gamma{}GG\gamma{}W_{\rm E}\gamma{}GG\gamma{}W^{\rm E}_{{\mathbf{q}}\omega}.\notag\end{aligned}$$ Then one can see, that the first term in Eq. exactly gives the reducible contribution to full four-point vertex function that was excluded from Eq. by the denominator of $\tilde{\Pi}^{'}$. If one neglects this denominator, it will immediately lead to the double counting in Dyson Eq. .
The same holds for the self-energy, where all contributions, coming from the denominator, give corrections to the six-point vertices $\gamma^{6,0}$ and $\gamma^{2,2}$ and neglect the reducible contributions with respect to the local impurity Green’s function $g_{\nu}$. Previous DB studies did not account for the six- and higher-point vertices, because they are negligibly small in the both large and small $U$ limits [@PhysRevB.93.045107]. Therefore, from the one point of view, if the ladder approximation for the dual self-energy does not contain these six-point vertices, then the denominator in Eq. should be neglected, because otherwise it will cancel the reducible terms in Dyson Eq. with respect to the impurity $g_{\nu}$. In the other hand, the one of advantages of the DB formalism is the fact, that all dual diagrams are written in terms of full impurity vertices instead of irreducible ones. Therefore, in the strong interaction limit, where the formal diagrammatic expansion cannot be performed, the full high-order vertices are small, which is not the case for the irreducible ones. Thus, writing the dual diagrams in terms of full vertices, it allows us to exclude the terms with the six-point vertices from the self-energy. Then, the presence of denominator in Eq. helps to include irreducible contributions of the high-order vertices when their full contributions are negligibly small.
DB$-$GW approach
----------------
With the four-fermion vertex $\gamma^{4,0}$, the Dual Boson approach can obviously include more correlation effects than EDMFT$+$GW, at a significant computational cost. However, it is also possible to construct a EDMFT++ approach from DB that does not require the full two-particle vertex. Taking $\gamma^{4,0}=0$, the fermion-boson vertex $\gamma_{\nu\omega}$ can be approximated as unity, as was discussed above, and the expressions for the dual $\tilde{\Sigma}_{{\mathbf{k}}\nu}$ and $\tilde{\Pi}_{{\mathbf{q}}\omega}$ operator are $$\begin{aligned}
\tilde{\Sigma}^{\text{DB$-$GW}}_{{\mathbf{k}}\nu}
&= -\sum\limits_{{\mathbf{q}}\omega}\tilde{G}_{{\mathbf{k}}+{\mathbf{q}},\nu+\omega}\tilde{W}_{{\mathbf{q}}\omega},\\
\tilde{\Pi}^{\text{DB$-$GW}}_{{\mathbf{q}}\omega} &= \,2\, \sum\limits_{{\mathbf{k}}\nu}\tilde{G}_{{\mathbf{k}}+{\mathbf{q}},\nu+\omega}\tilde{G}_{{\mathbf{k}}\nu}.
\label{eq:dual_GW_limit}\end{aligned}$$ We call this the DB$-$GW approximation. According to the above discussions, in this simplest case the denominator in Eqs. - should be excluded, since we are interested in the contribution of only lower-order vertex function, so we can take $\tilde{\Sigma}^{'}_{{\mathbf{k}}\nu}=\tilde{\Sigma}^{\rm DB-GW}_{{\mathbf{k}}\nu}$ and $\tilde{\Pi}^{'}_{{\mathbf{q}}\omega}=\tilde{\Pi}^{\rm DB-GW}_{{\mathbf{q}}\omega}$. Thus we see, that the EDMFT$+$GW and DB$-$GW approaches start with the same form of the self-energy and polarization operator diagrams and with similar propagators based on the same EDMFT quantities $G_{\rm E}$ and $W_{\rm E}$. The difference between the two approaches lies in the way double counting is excluded from these diagrams, which for DB$-$GW case is shown in Eqs. -. This results in different self-energies $\tilde\Sigma_{{\mathbf{k}}\nu}$, and polarization operators $\tilde\Pi_{{\mathbf{q}}\omega}$ that are used to treat non-local effects beyond the EDMFT in these two different cases. Since the local and non-local correlation effects are intertwined in a complicated way, it is more efficient to exclude double counting already on the level of bare EDMFT Green’s function and bare interaction in the dual formalism, rather than to remove the local contribution of the full diagram. This happens naturally in the exact dual Hubbard–Stratonovich transformation.
It is worth mentioning, that the dual renormalized interaction $\tilde{W}_{{\mathbf{q}}\omega}$ does not depend on the form of decoupling. As it is shown in Eq. , both $UV$– and $V$– decoupling forms lead to the same result $U_{{\mathbf{q}}}-{\cal U}_{\omega}=V_{{\mathbf{q}}}-\Lambda_{\omega}$ for the interaction accounted beyond the dynamical mean-field level in the DB theory. It is then easy to see, that DMFT+GW theory in a $V$– decoupling form excludes the impurity interaction in a proper way, since the dual renormalized interaction in case of $\Lambda_{\omega}=0$ has exactly the form of $V$– decoupling. Due to the problems arising in EDMFT+GW approach in $UV$– decoupling form mentioned in Appendix \[app:GW\] we take renormalized interaction for the EDMFT$+$GW($\gamma$) theories in the form of $V$– decoupling for the later comparison with DB results.
Local vertex corrections in DB method
-------------------------------------
To add vertex corrections to the DB$-$GW approach, one can take the second order diagrams for the dual self-energy $\tilde{\Sigma}^{(2)}_{{\mathbf{k}}\nu}$ and polarization operator $\tilde{\Pi}^{(2)}_{{\mathbf{q}}\omega}$ , which are dressed with the full local impurity fermion-boson vertices $\gamma_{\nu\omega}$ as $$\begin{aligned}
\tilde{\Sigma}^{{\rm GW}\gamma}_{{\mathbf{k}}\nu} &= -\sum\limits_{{\mathbf{q}}\omega}\gamma^{\phantom{1}}_{\nu\omega}\tilde{G}_{{\mathbf{k}}+{\mathbf{q}},\nu+\omega}\tilde{W}_{{\mathbf{q}}\omega}\gamma^{\phantom{1}}_{\nu+\omega,-\omega}, \\
\tilde{\Pi}^{{\rm GW}\gamma}_{{\mathbf{q}}\omega} &= \,2\, \sum\limits_{{\mathbf{k}}\nu}\gamma^{\phantom{1}}_{\nu\omega}\tilde{G}_{{\mathbf{k}}+{\mathbf{q}},\nu+\omega}\tilde{G}_{{\mathbf{k}}\nu}\gamma^{\phantom{1}}_{\nu+\omega,-\omega}.
\label{eq:dual_GWd_limit}\end{aligned}$$ Similarly to the DB$-$GW approach we neglect the denominator in Eqs. - and repeat all calculations in the same way.
The four approaches are summarized in Fig. \[fig:4methods\], showing the self-energy and polarization operator diagram, where square brackets $[\ldots]_{{\rm nloc}}$ denote the exclusion of the local part. The computational recipes for the all the EDMFT++ theories is shown in Fig. \[fig:recipe\].
![Self-energy and polarization operator for EDMFT++ approaches. The square brackets $[\ldots]_{{\rm nloc}}$ denote exclusion of the local part. DMFT$+$GW is not listed here, it has the same diagrams as EDMFT$+$GW and only differs in their choice of ${\cal U}_\omega$.[]{data-label="fig:4methods"}](4methods.pdf){width="0.9\linewidth"}
Numerical results
=================
To test the EDMFT++ schemes, we study the charge-order transition in the square lattice Hubbard model, a popular testing ground for extensions of EDMFT [@PhysRevB.87.125149; @PhysRevB.90.195114; @PhysRevB.90.235135]. Here we show calculations where first $\Delta_\nu$ and $\Lambda_\omega$ are determined self-consistently on the EMDFT level for all schemes. Then, the non-local correlation effects are included. Having the same impurity problem as the starting point for all approaches allows us to compare clearly the the effect from the extensions only. We use $t=1/4$, $\beta=50$ and a $32\times{}32$ lattice. The resulting phase boundary between the charge-ordered phase (CO) and the Fermi liquid (FL), determined in the same way as in [@PhysRevB.90.235135], is shown in Fig. \[fig:phase\]. The checkerboard CO phase is characterized by a divergent charge susceptibility at the wave vector ${\mathbf{q}}= (\pi, \pi)$. The phase boundary may therefore be located by looking for zeros of the inversed susceptibility $X^{-1}_{\omega=0, {\mathbf{q}}=(\pi, \pi)}$. Note that the renormalized interaction $W_{{\mathbf{q}}\omega}$ in DMFT$+$GW, EDMFT$+$GW and EDMFT$+$GW$\gamma$ approaches is taken in form of $V$– decoupling due to the above discussions.
![The recipe to construct an EDMFT++ theory. DMFT$+$GW is obtained by taking ${\cal U}_\omega = U$ instead of determining it self-consistently. []{data-label="fig:recipe"}](recipe.pdf){width="0.9\linewidth"}
Since ordering is unfavorable for the interaction energy for $V<U/4$, the true phase boundary is expected to be above the $V=U/4$ line. Indeed, the full DB result is above this line [@PhysRevB.90.235135]. In all other EDMFT++ approximations with fewer correlation effects, the phase transitions occurs at smaller $V$. The DB$-$GW$\gamma$ approximation performs best in this respect, and is close to the DB phase boundary for all values of $U$. The two approximations that include local vertex corrections via $\gamma_{\nu\omega}$ perform better than their counterparts without, and both DB based approaches outperform their EDMFT$+$GW counterpart.
At $U=0$, we expect the Random Phase Approximation (RPA) to give a reasonable prediction for the phase boundary. RPA limit is recovered by all shown EDMFT++ approaches, but already at relatively small $U=0.5$, strong differences between the methods becomes clear.
In the opposite limit of large $U$, EDMFT itself starts to give an accurate phase boundary, since it accounts for all local effects and those are most important at large $U$. Both DB-based approaches converge to EDMFT at $U=2.5$, whereas both EDMFT$+$GW approaches give a phase boundary at the same, slightly smaller $V$.
We even observe that DMFT$+$GW performs better than EDMFT$+$GW, although it is simpler. Although DMFT$+$GW contains fewer correlation effects than EDMFT$+$GW, it is free from double counting by construction. This clearly shows the huge role that double counting can play. On the other hand, comparison of DMFT$+$GW and DB$-$GW schemes confirms the fact, that inclusion of bosonic correlations already on the impurity level is also very important and provides the better starting point for extension of dynamical mean-field theory.
In Fig. \[SigmaNL\], we show the polarization operator corrections $\bar{\Pi}_{{\mathbf{q}}\omega}$ at high-symmetry ${\mathbf{q}}$-points, according to the EDMFT$+$GW($\gamma$) and DB$-$GW($\gamma$) approaches. The behaviour of all methods is different, especially in the approaches that contain the frequency dependent vertex function $\gamma_{\nu\omega}$ in the diagrams. This shows that the main difference in the approaches lies in their description of the collective excitations.
The fermion-boson vertex exhibits less structure as the metallicity of the system is increased and becomes mostly flat as the phase boundary to the CO phase is approached [@PhysRevB.90.235135]. The influence of non-local interaction $V$ on the three-point vertex function $\gamma_{\nu\omega}$ is shown in Fig. \[fig:vertex\]. The effects of the three-leg vertex are also visible in the non-local part of polarization operator in the difference between DB$-$GW and DB$-$GW$\gamma$ (or between EDMFT$+$GW and EDMFT$+$GW$\gamma$) approaches (see Fig. \[SigmaNL\]).
![$U-V$ phase diagram in EDMFT, DB and EDMFT++ theories at inverse temperature $\beta=50$. The dashed line shows $V=U/4$, the dot at $U=0$ shows the starting point of RPA data. CO and FL denote charge-ordered and Fermi-liquid metallic phases, respectively. The EDMFT and DB data are taken from [@PhysRevB.90.235135], EDMFT$+$GW data practically coincides with results shown in [@PhysRevLett.109.226401; @PhysRevB.87.125149] papers.[]{data-label="fig:phase"}](Phase-boundary.pdf){width="0.9\linewidth"}
Conclusions
===========
We have presented a recipe to create approximations beyond EDMFT that take into account non-local correlation effects while simultaneously avoiding double counting issues. By properly including non-locality we see an improvement in the phase boundary between the charge-ordered phase and the Fermi liquid. Even in weakly and moderately interacting systems, the phase boundary is shifted significantly upwards compared to traditional EDMFT$+$GW. In fact, EDMFT+GW is even improved upon by DMFT$+$GW, which neglects the effect of non-local interactions on the impurity model but does avoid double containing. This allows us to study the physics in a larger part of parameter space, where EDMFT$+$GW has undergone a spurious transition. This is important for accurately determining the charge-ordering transition in real materials and in surface systems.
The approaches presented here work without requiring the computationally expensive full two-particle vertex. The frequency dependence of the much simpler fermion-boson vertex already contains most of the relevant physics, and including it via DB$-$GW$\gamma$ gives a phase boundary close to the full DB result. Without access to the fermion-boson vertex, deviations are bigger. In both cases, however, properly avoiding double counting of correlation effects greatly improves the results.
$\begin{array}{cc}
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip, width=4.1cm]{pi_nonloc_u23_v02_g.pdf}&
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip, width=4.2cm]{pi_nonloc_u23_v02_m.pdf}
\end{array}$
The authors would like to thank Alexey Rubtsov for fruitful discussions and Lewin Boehnke for useful comments. E.A.S., E.G.C.P.v.L. and M.I.K. acknowledge support from ERC Advanced Grant 338957 FEMTO/NANO and from NWO via Spinoza Prize, A.I.L. from the DFG Research Unit FOR 1346, A.H. from the DFG via SPP 1459 and computer time at the North-German Supercomputing Alliance (HLRN).
Dual transformations {#ap:dualtr}
====================
The dual transformations of the non-local part of the action $S_{\rm rem}$ can be made in the same way as in previous works on DB approach. In order to define the three-point vertex in the TRILEX way, here we introduce a different rescaling of the dual bosonic fields. The partition function of our problem is given by $$\begin{aligned}
Z=\int D[c^{*},c] \, e^{-S}\end{aligned}$$ where the action $S$ is given by . Performing the Hubbard–Stratonovich transformations one can introduce the new ${\it dual}$ variables $f^{*},f,\phi$ $$\begin{aligned}
&e^{\,\sum\limits_{{\bf k}\nu\sigma} c^{*}_{{\bf k}\nu\sigma}[\Delta_{\nu\sigma}-\varepsilon_{\bf k}]c^{\phantom{*}}_{{\bf k}\nu\sigma}} = D_{f}\times\notag\\
&\int D[f^{*},f]\,e^{-\sum\limits_{{\bf k}\nu\sigma}\left\{ f^{*}_{{\bf k}\nu\sigma}[\Delta_{\nu\sigma}-\varepsilon_{\bf k}]^{-1}f^{\phantom{*}}_{{\bf k}\nu\sigma} + c^{*}_{\nu\sigma}f^{\phantom{*}}_{\nu\sigma} + f^{*}_{\nu\sigma}c^{\phantom{*}}_{\nu\sigma}\right\}},\notag\\
&e^{\,\frac12\sum\limits_{{\bf q}\omega} \rho^{*}_{{\bf q}\omega}[\Lambda_{\omega}-V_{\bf q}]\rho^{\phantom{*}}_{{\bf q}\omega}} \,\,\,= D_{\,b}\times\notag\\
&\int D[\phi]\,e^{-\frac12\sum\limits_{{\bf q}\omega}\left\{ \phi^{*}_{{\bf q}\omega}[\Lambda_{\omega}-V_{\bf q}]^{-1}\phi^{\phantom{*}}_{{\bf q}\omega} + \rho^{*}_{\omega}\phi^{\phantom{*}}_{\omega} + \phi^{*}_{\omega}\rho^{\phantom{*}}_{\omega}\right\}}.\end{aligned}$$ Terms $D_{f} = {\rm det}[\Delta_{\nu\sigma}-\varepsilon_{\bf k}]$ and $D^{-1}_{\,b} = \sqrt{{\rm det}[\Lambda_{\omega}-V_{\bf q}]}$ can be neglected, because they does not contribute to expectation values. Rescaling the fermionic fields $f_{{\bf k}\nu\sigma}$ as $f^{\phantom{1}}_{{\bf k}\nu\sigma}g^{-1}_{\nu\sigma}$, the bosonic fields $\phi_{{\mathbf{q}}\omega}$ as $\phi_{{\mathbf{q}}\omega}\alpha^{-1}_{\omega}$, where $\alpha_{\omega}=~(1+{\cal U}_{\omega}\chi_{\omega})$, and integrating out the original degrees of freedom $c^*$ and $c$ we arrive at the dual action $$\begin{aligned}
\tilde{S} &=
- \sum_{{\mathbf{k}}\nu}f^{*}_{{\mathbf{k}}\nu} \tilde{G}_{0}^{-1} f^{\phantom{*}}_{{\mathbf{k}}\nu} - \frac{1}{2}\sum_{{\mathbf{q}}\omega} \phi^{*}_{{\mathbf{q}}\omega}\tilde{W}_{0}^{-1}\phi^{\phantom{*}}_{{\mathbf{q}}\omega}+ \tilde{V}.\end{aligned}$$ with the bare dual propagators $$\begin{aligned}
\tilde{G}_{0} &= [g^{-1}_{\nu}+\Delta_{\nu}-\varepsilon_{\bf k}]^{-1} - g_{\nu} = G_{\rm E} - g_{\nu} ,
\\
\tilde{W}_{0} &= \alpha_{\omega}^{-1}\left[[U_{\bf q}-{\cal U}_{\omega}]^{-1} - \chi_{\omega}\right]^{-1}\alpha_{\omega}^{-1}
= W_{\rm E}-{\cal W}_{\omega},\end{aligned}$$ and the dual interaction term $\tilde{V}$. The explicit form of the dual interaction can be obtained by expand the $c^{*}$ and $c$ dependent part of partition function in an infinite row and integrating out these degrees of freedom as follows $$\begin{aligned}
&\int
e^{-\sum\limits_{\nu\omega}\left\{c^{*}_{\nu}g^{-1}_{\nu}f^{\phantom{*}}_{\nu} + f^{*}_{\nu}g^{-1}_{\nu}c^{\phantom{*}}_{\nu} + \rho^{*}_{\omega}\alpha_{\omega}^{-1}\phi^{\phantom{*}}_{\omega} + \phi^{*}_{\omega}\alpha_{\omega}^{-1}\rho^{\phantom{*}}_{\omega}\right\}}\notag\\
&\,\,\,\,\,\,\,\,e^{-S_{\rm imp}[c^*,c]}\,D[c^{*},c] =
f^{*}_{\nu_1}f^{\phantom{*}}_{\nu_2}{\ensuremath{\left\langle c^{\phantom{*}}_{\nu_1}c^{*}_{\nu_2} \right\rangle}}
g^{-1}_{\nu_1}g^{-1}_{\nu_2} \notag\\
&+\frac12\phi^{*}_{\omega_1}\phi^{\phantom{*}}_{\omega_2}
{\ensuremath{\left\langle \rho^{\phantom{*}}_{\omega_1}\rho^{*}_{\omega_2} \right\rangle}}\alpha_{\omega_1}^{-1}\alpha_{\omega_2}^{-1} \notag\\
&-f^{*}_{\nu_1}f^{\phantom{*}}_{\nu_2}\phi^{*}_{\omega_3}
{\ensuremath{\left\langle c^{\phantom{*}}_{\nu_1}c^{*}_{\nu_2}\rho^{\phantom{*}}_{\omega_3} \right\rangle}}g^{-1}_{\nu_1}g^{-1}_{\nu_2}\alpha_{\omega_3}^{-1}\notag\\
&+\frac14\,f^{*}_{\nu_1}f^{*}_{\nu_2}f^{\phantom{*}}_{\nu_3}
f^{\phantom{*}}_{\nu_4}{\ensuremath{\left\langle c^{\phantom{*}}_{\nu_1}c^{\phantom{*}}_{\nu_2}
c^{*}_{\nu_3}c^{*}_{\nu_4} \right\rangle}}g^{-1}_{\nu_1}g^{-1}_{\nu_2}
g^{-1}_{\nu_3}g^{-1}_{\nu_4} + \ldots \notag\\
&=-f^{*}_{\nu}g^{-1}_{\nu}f^{\phantom{*}}_{\nu} - \frac12\phi^{*}_{\omega}\alpha_{\omega}^{-1}\chi^{\phantom{*}}_{\omega}\alpha_{\omega}^{-1}\phi^{\phantom{*}}_{\omega}\notag\\ &-f^{*}_{\nu_1}f^{\phantom{*}}_{\nu_2}\phi^{*}_{\omega_3}
{\ensuremath{\left\langle c^{\phantom{*}}_{\nu_1}c^{*}_{\nu_2}\rho^{\phantom{*}}_{\omega_3} \right\rangle}}g^{-1}_{\nu_1}g^{-1}_{\nu_2}\alpha_{\omega_3}^{-1}\notag\\
&+\frac14\,f^{*}_{\nu_1}f^{*}_{\nu_2}f^{\phantom{*}}_{\nu_3}
f^{\phantom{*}}_{\nu_4}{\ensuremath{\left\langle c^{\phantom{*}}_{\nu_1}c^{\phantom{*}}_{\nu_2}
c^{*}_{\nu_3}c^{*}_{\nu_4} \right\rangle}}g^{-1}_{\nu_1}g^{-1}_{\nu_2}
g^{-1}_{\nu_3}g^{-1}_{\nu_4} + \ldots \notag\\
&=e^{-\left\{ f^{*}_{\nu}g^{-1}_{\nu}f^{\phantom{*}}_{\nu} + \frac12\phi^{*}_{\omega}\alpha_{\omega}^{-1}\chi^{\phantom{*}}_{\omega}\alpha_{\omega}^{-1}\phi^{\phantom{*}}_{\omega} +\tilde{V}\right\}}.\end{aligned}$$ Therefore dual interaction has the form of infinite expansion on the full vertices of the local impurity problem $$\begin{aligned}
\tilde{V} = &\,\,f^{*}_{\nu_1}f^{\phantom{*}}_{\nu_2}\phi^{*}_{\omega_3}
{\ensuremath{\left\langle c^{\phantom{*}}_{\nu_1}c^{*}_{\nu_2}\rho^{\phantom{*}}_{\omega_3} \right\rangle}}g^{-1}_{\nu_1}g^{-1}_{\nu_2}\alpha_{\omega_3}^{-1}-\notag\\
&\frac14\,f^{*}_{\nu_1}f^{*}_{\nu_2}f^{\phantom{*}}_{\nu_3}
f^{\phantom{*}}_{\nu_4}g^{-1}_{\nu_1}g^{-1}_{\nu_2}
g^{-1}_{\nu_3}g^{-1}_{\nu_4}
\big\{{\ensuremath{\left\langle c^{\phantom{*}}_{\nu_1}c^{\phantom{*}}_{\nu_2}
c^{*}_{\nu_3}c^{*}_{\nu_4} \right\rangle}}- \notag\\
& {\ensuremath{\left\langle c^{\phantom{*}}_{\nu_1}c^{*}_{\nu_4} \right\rangle}}
{\ensuremath{\left\langle c^{\phantom{*}}_{\nu_2}c^{*}_{\nu_3} \right\rangle}} + {\ensuremath{\left\langle c^{\phantom{*}}_{\nu_1}c^{*}_{\nu_3} \right\rangle}}
{\ensuremath{\left\langle c^{\phantom{*}}_{\nu_2}c^{*}_{\nu_4} \right\rangle}} \big\}+\ldots.\end{aligned}$$ Here we define the three- and four-point vertex functions as ($\gamma_{\nu\omega}$ is the shorthand notation for the $\gamma^{2,1}_{\nu\omega}$) $$\begin{aligned}
\gamma^{\phantom{1}}_{\nu\omega}&=
g^{-1}_{\nu}g^{-1}_{\nu+\omega}\alpha^{-1}_{\omega}
{\ensuremath{\left\langle c^{\phantom{*}}_{\nu}c^{*}_{\nu+\omega}\rho^{\phantom{*}}_{\omega} \right\rangle}},\\
\gamma^{4,0}_{\nu\nu'\omega} &= g^{-1}_{\nu}g^{-1}_{\nu'}g^{-1}_{\nu'-\omega}g^{-1}_{\nu+\omega} \Big[{\ensuremath{\left\langle c^{\phantom{*}}_{\nu}c^{\phantom{*}}_{\nu'}c^{*}_{\nu'-\omega}
c^{*}_{\nu+\omega} \right\rangle}}- \notag\\
&\hspace{3.5cm}g_{\nu}g_{\nu'}(\delta_{\omega}-\delta_{\nu',\nu+\omega})\Big],\end{aligned}$$ with the simple connection between them $$\begin{aligned}
\gamma^{\phantom{1}}_{\nu\omega} = \alpha_{\omega}^{-1}\sum_{\nu'}\big[1 - \gamma^{4,0}_{\nu\nu'\omega}g_{\nu'}g_{\nu'-\omega}\big].
\label{eq:ConGamma}\end{aligned}$$ In the weakly-interacting limit, namely $U\to0$, the renormalization factor $\alpha_{\omega}$ goes to unity and the four-point vertex $\gamma^{4,0}$ is zero, as detailed in the previous works [@Rubtsov20121320; @PhysRevB.90.235135; @PhysRevB.93.045107] on the DB approach. Then, the three-point vertex can be reduced to its bare form $\gamma_0=1$. Frequency dependence of the full local three-point vertex function $\gamma_{\nu\omega}$ and the influence of non-local interaction $V$ is shown in Fig. \[fig:vertex\].
$\begin{array}{cc}
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip, width=4.1cm]{vert_u15_m0.pdf}&
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip, width=4.1cm]{vert_u15_m6.pdf}\\
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip, width=4.1cm]{vert_u23_m0.pdf}&
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip, width=4.1cm]{vert_u23_m6.pdf}
\end{array}$
Then, the two first terms in $\tilde{V}$ are given by $$\begin{aligned}
\tilde{V}
&=\gamma^{\phantom{1}}_{\nu\omega}\,f^{*}_{\nu}f^{\phantom{*}}_{\nu+\omega}\phi^{*}_{\omega}
+ \frac14\,\gamma^{4,0}_{\nu\nu'\omega}\,f^{*}_{\nu}f^{*}_{\nu'}f^{\phantom{*}}_{\nu+\omega}
f^{\phantom{*}}_{\nu'-\omega}.\end{aligned}$$ The dual Green’s function $\tilde{G}_{{\mathbf{k}}\nu} =~ -{\ensuremath{\left\langle f^{\phantom{*}}_{{\mathbf{k}}\nu}f^{*}_{{\mathbf{k}}\nu} \right\rangle}}$ and renormalized dual interaction $\tilde{W}_{{\mathbf{q}}\omega} =~ -{\ensuremath{\left\langle \phi^{\phantom{*}}_{{\mathbf{q}}\omega}\phi^{*}_{{\mathbf{q}}\omega} \right\rangle}}$, as well as dual self-energy $\tilde{\Sigma}_{{\mathbf{k}}\nu}$ and polarization operator $\tilde{\Pi}_{{\mathbf{q}}\omega}$, can be obtained diagrammatically [@Rubtsov20121320; @PhysRevB.90.235135; @PhysRevB.93.045107]. These dual quantities have usual connection $$\begin{aligned}
\tilde{G}_{{\mathbf{k}}\nu}^{-1} &= \,\tilde{G}^{-1}_{0}\, - \tilde{\Sigma}^{\phantom{1}}_{{\mathbf{k}}\nu},\\
\tilde{W}_{{\mathbf{q}}\omega}^{-1} &= \tilde{W}^{-1}_{0} - \tilde{\Pi}^{\phantom{1}}_{{\mathbf{q}}\omega}.\end{aligned}$$
Finally, lattice Green’s function $G_{{\mathbf{k}}\nu}$ and susceptibility $X_{{\mathbf{q}}\omega}$ can be expressed in terms of dual propagators via exact relations $$\begin{aligned}
G_{{\mathbf{k}}\nu} = \,&-[\varepsilon_{\bf k}-\Delta_{\nu}]^{-1} \notag\\
&+[\varepsilon_{\bf k}-\Delta_{\nu}]^{-1}\, g^{-1}_{\nu}\,\tilde{G}^{\phantom{1}}_{{\bf k}\nu}\,g^{-1}_{\nu}[\,\varepsilon_{\bf k}-\Delta_{\nu}]^{-1},\\
X_{{\mathbf{q}}\omega} = \, &- [U_{\bf q}-{\cal U}_{\omega}]^{-1} \notag\\
&+[U_{\bf q}-{\cal U}_{\omega}]^{-1}\alpha_{\omega}^{-1}\tilde{W}^{\phantom{1}}_{{\bf q}\omega}\alpha_{\omega}^{-1}[U_{\bf q}-{\cal U}_{\omega}]^{-1}.\end{aligned}$$ One can also rewrite the last relation and obtain the relation for the full dual renormalized interaction $$\begin{aligned}
\alpha^{-1}_{\omega}\tilde{W}^{\phantom{1}}_{{\bf q}\omega}\alpha^{-1}_{\omega} &= [U_{\bf q}-{\cal U}_{\omega}] +
[U_{\bf q}-{\cal U}_{\omega}]X_{{\mathbf{q}}\omega}[U_{\bf q}-{\cal U}_{\omega}],
\label{eq:tildeW}\end{aligned}$$ to show that the dual propagator $\tilde{W}_{{\bf q}\omega}$ is evidently a renormalized interaction in the non-local part of the action, where the impurity interaction is excluded on the level of the bare interaction. It is worth mentioning, that for the case of $\Lambda_{\omega}=0$, which corresponds to the DMFT theory as a basis, the renormalized interaction is exactly that of the usual $V$– decoupling.
Comparison of the different decoupling schemes with the DB approach {#app:GW}
===================================================================
As a consequence of the exact dual transformations presented in Appendix \[ap:dualtr\], the renormalized interaction introduced beyond the DMFT when bosonic hybridization function $\Lambda_{\omega}$ is equal to zero (i.e. $U_{\omega}=U$) should be taken in the form of $V$– decoupling . Contrary to DMFT, impurity model in the EDMFT approach contains non-zero bosonic retarded interaction $\Lambda_{\omega}$, thus the renormalized interaction in EDMFT++ theories has neither $UV$–, nor $V$– decoupling form. In this case the bare non-local interaction $U_{{\mathbf{q}}}-{\cal U}_{\omega}$ for small $\Lambda_{\omega}$ (i.e. $U_{\omega}\simeq{}U$) is closer to $V_{{\mathbf{q}}}$ then to $U_{{\mathbf{q}}}=U+V_{{\mathbf{q}}}$, and therefore in this paper we take $W_{{\mathbf{q}}}$ in the form of $V$– decoupling for all EDMFT++ theories.
The one more argument to avoid treating the renormalized interaction in the $UV$– decoupling form is the fact, that in this case EDMFT+GW reproduces the results of GW approach in the region close to the phase boundary. Indeed, the self-energy and polarization operator for the GW approach are given by Eqs. - respectively. EDMFT$+$GW approach uses only non-local parts of these diagrams beyond the dynamical mean-field solution. They can be written as follows $$\begin{aligned}
\bar{\Sigma}^{\rm E+GW}_{{\mathbf{k}}\nu} &= -\sum\limits_{{\mathbf{q}}\omega}\bar{G}^{\rm E+GW}_{{\mathbf{k}}+{\mathbf{q}},\nu+\omega}\bar{W}^{\rm E+GW}_{{\mathbf{q}}\omega}, \label{eq:SigmaUVd}\\
\bar{\Pi}^{\rm E+GW}_{{\mathbf{q}}\omega} &= \,2\,\sum\limits_{{\mathbf{k}}\nu}\bar{G}^{\rm E+GW}_{{\mathbf{k}}+{\mathbf{q}},\nu+\omega}\,\bar{G}^{\rm E+GW}_{{\mathbf{k}}\nu},\end{aligned}$$ where $\bar{G}^{\rm E+GW}_{{\mathbf{k}}\nu} = G_{{\mathbf{k}}\nu}^{\phantom{1}}-g_{\nu}$, $\bar{W}^{\rm E+GW}_{{\mathbf{q}}\omega}=~W^{\phantom{1}}_{{\mathbf{q}}\omega} - {\cal W}^{\phantom{1}}_{\omega}$ are non-local parts of the full lattice Green’s function and renormalized interaction respectively. Then, the full self-energy and polarization operator of the lattice problem can be written as $$\begin{aligned}
\Sigma_{{\mathbf{k}}\nu}^{\phantom{1}} &= \Sigma_{\rm imp}^{\phantom{1}} + \bar{\Sigma}^{\rm E+GW}_{{\mathbf{k}}\nu},\\
\Pi^{\phantom{1}}_{{\mathbf{q}}\omega} &= \Pi_{\rm imp}^{\phantom{1}} + \bar{\Pi}^{\rm E+GW}_{{\mathbf{q}}\omega},\end{aligned}$$ where $$\begin{aligned}
\Sigma_{\rm imp} &= -\sum\limits_{\omega}g_{\nu+\omega}{\cal W}_{\omega}\gamma^{\phantom{1}}_{\nu\omega},\\
\Pi_{\rm imp} &= \,2\,\sum\limits_{\nu}g_{\nu+\omega}\,\,g_{\nu}\,\gamma^{\phantom{1}}_{\nu\omega},\label{eq:HedinPiImp}\end{aligned}$$ are the exact self-energy and polarization operator of impurity problem written in the Hedin form. Then, one can rewrite the full lattice self-energy and polarization operator as $$\begin{aligned}
\Sigma_{{\mathbf{k}}\nu} &= -\sum\limits_{{\mathbf{q}}\omega}G_{{\mathbf{k}}+{\mathbf{q}},\nu+\omega}W_{{\mathbf{q}}\omega} -\sum\limits_{\omega}g_{\nu+\omega}{\cal W}_{\omega}\left(\gamma^{\phantom{1}}_{\nu\omega}-1\right) \notag\\
&= \, \Sigma^{\rm GW}_{{\mathbf{k}}\nu}-\sum\limits_{\omega}g_{\nu+\omega}{\cal W}_{\omega}\left(\gamma^{\phantom{1}}_{\nu\omega}-1\right), \label{eq:gwSigma}\\
\Pi_{{\mathbf{q}}\omega} &= \,2\,\sum\limits_{{\mathbf{k}}\nu}G_{{\mathbf{k}}+{\mathbf{q}},\nu+\omega}\,G_{{\mathbf{k}}\nu} + \,2\,\sum\limits_{\nu}g_{\nu+\omega}g_{\nu}\left(\gamma^{\phantom{1}}_{\nu\omega}-1\right) \notag\\
&= \, \Pi^{\rm GW}_{{\mathbf{q}}\omega} + \,2\,\sum\limits_{\nu}g_{\nu+\omega}g_{\nu}\left(\gamma^{\phantom{1}}_{\nu\omega}-1\right).
\label{eq:gwPi}\end{aligned}$$
Therefore, in the region, where the value of the three-point vertex $\gamma_{\nu\omega}$ is close to the value of the bare three-point vertex $\gamma_{0}=1$, EDMFT$+$GW approach reproduces the result of GW method. Thus, contribution of the exactly solvable impurity model in this region is lost. It happens, because one cancels very big local contribution from the GW diagrams in order to avoid the double-counting problem, and then this local contribution suppresses the contribution of the local impurity model. It turns out, that EDMFT$+$GW approach cancels too much from the diagrams introduced beyond the dynamical mean-field level and treating of the double-counting problem can be done in a better way.
To see this, one can compare dual way of exclusion of the double counting with $UV$– decoupling scheme. Since the inner self-consistency for the diagrams beyond the dynamic mean-field level is used, it is hard to compare the resulting diagrams of these two approaches. Nevertheless, let us consider polarization operator in the first iteration, when the only bare EDMFT Green’s functions enter the diagrams. It is sufficient due to the fact, that non-local self-energy $\tilde{\Sigma}_{{\mathbf{k}}\nu}$ is small in our region of interest. Then, one can see, that polarization operator for EDMFT+GW and DB$-$GW has the same form $$\begin{aligned}
\tilde{\Pi}^{0}_{{\mathbf{q}}\omega} &= \,2\,\sum\limits_{{\mathbf{k}}\nu}\tilde{G}_{0}\,\tilde{G}_{0},\end{aligned}$$ where $\tilde{G}_{0} = G_{\rm E}-g_{\nu}$. Then, one can obtain for the difference between the renormalized interactions used in EDMFT$+$GW and DB$-$GW the following relation $$\begin{aligned}
&\left[W_{{\mathbf{q}}\omega} - {\cal W}_{\omega}\right] - \tilde{W}_{{\mathbf{q}}\omega}
= \frac{W_{\rm E}}{1-\tilde{\Pi}^{0}_{{\mathbf{q}}\omega}W_{\rm E}} - \frac{\tilde{W}_{0}}{1-\tilde{\Pi}^{0}_{{\mathbf{q}}\omega}\tilde{W}_{0}} - {\cal W}_{\omega} \notag\\
&= {\cal W}_{\omega}[1-\tilde{\Pi}^{0}_{{\mathbf{q}}\omega}\tilde{W}_{0}]^{-1}[1-\tilde{\Pi}^{0}_{{\mathbf{q}}\omega}W_{\rm E}]^{-1}-{\cal W}_{\omega} \notag\\
&= {\cal W}_{\omega}\tilde{\Pi}^{0}_{{\mathbf{q}}\omega}\left[ \tilde{W}_{{\mathbf{q}}\omega}+W_{{\mathbf{q}}\omega}+\tilde{W}_{{\mathbf{q}}\omega}\tilde{\Pi}^{0}_{{\mathbf{q}}\omega}W_{{\mathbf{q}}\omega}
\right].\label{eq:UVdecoupling}\end{aligned}$$ Therefore, the self-energy in the form of $UV$– decoupling additionally to the non-local dual contribution accounts for some diagrams that have local renormalized interaction ${\cal W}_{\omega}$ in their structure. In the Dual Boson formalism all local propagators are gathered in the local vertex functions of impurity problem, including the local renormalized interaction ${\cal W}_{\omega}$, which is a part of the local four-point vertex $\gamma^{4,0}$. For example, the first term in the right hand side of Eq. gives the following contribution to the self-energy $$\begin{aligned}
\includegraphics[width=0.6\linewidth]{UVdecoupling1.pdf}\,\,,
\label{eq:dc1}\end{aligned}$$ which is a part of the dual diagram for the self-energy shown in Fig. \[fig:Hedin\]a). The second term in the right hand side of Eq. , when one takes only the local part of the EDMFT renormalized interaction in Eq. , namely $W_{{\mathbf{q}}\omega}=\frac{W_{\rm E}}{1-\tilde{\Pi}^{0}_{{\mathbf{q}}\omega}W_{\rm E}}\sim \frac{{\cal W}_{\omega}}{1-\tilde{\Pi}^{0}_{{\mathbf{q}}\omega}{\cal W}_{\omega}}$, is then equal to $$\begin{aligned}
\includegraphics[width=0.6\linewidth]{UVdecoupling2.pdf},
\label{eq:dc2}\end{aligned}$$ which is again a part of the dual diagram for the self-energy shown in Fig. \[fig:Hedin\]b). This fact leads to the two important problems in the EDMFT$+$GW approach. First of all, these additional self-energy diagrams in case of $UV$– decoupling presented above are very selective and account for the only local renormalized interaction ${\cal W}_{\omega}$ instead of the full local four-point vertex functions $\gamma^{4,0}$, as the DB approach does. This selective choice is not well-controlled and may result in over- or under-estimation of interaction effects. Also, the existence of the local propagators as a part of non-local interaction shows, that EDMFT$+$GW approach in the form of $UV$– decoupling is not able to separate local and non-local degrees of freedom in a proper way. This leads, in particular, to the double-counting problem in the next-order non-local diagrams introduced beyond EDMFT. Indeed, if one does not restrict himself to the simplest GW diagram accounted beyond the dynamical mean-field level and additionally includes the four-point vertex functions $\gamma^{4,0}$ in the diagrams for the self-energy (for example the diagrams shown in Fig. \[fig:Hedin\]b), then, as it was shown in Eqs. -, the GW diagram for the self-energy would have contributions with the local ${\cal W}_{\omega}$, that would already be accounted for these additional diagrams with the local four-point vertices.
Let us study what happens in the region, where the impurity renormalized interaction ${\cal W}_{\omega}$ gives the main contribution in the full local four-point vertex $\gamma^{4,0}$. In this region EDMFT$+$GW solution should be close to the Dual Boson ladder approximation with the self-energy and polarization operator diagrams shown in Fig. \[fig:Hedin\] a), b). Substituting “$-{\cal W}_{\omega}$” for the four-point vertex $\gamma^{4,0}$ in Eq. and using Eq. and relation $\alpha^{-1} =~1-~\Pi_{\rm imp}\,{\cal U}_{\omega}$ one can get trivial solution $\sum_{\nu}g_{\nu+\omega}g_{\nu}(\gamma_{\nu\omega}-1)=0$. Therefore, as it was shown in Eqs. -, in this region EDMFT$+$GW in the $UV$– decoupling form reproduces result of the GW approach. In the other regions, where the bare vertex $\gamma_{0}=1$ does not give the main contribution to the full three-point vertex $\gamma_{\nu\omega}$, EDMFT$+$GW shows a different result then GW approach, but unfortunately, it is not correct to approximate full local vertex $\gamma^{4,0}$ by the local ${\cal W}_{\omega}$ there. As it was pointed out above, the one of advantages of the DB formalism is the fact, that the full impurity vertices, in particular full four-point vertex $\gamma^{4,0}$, are used in the dual diagrams for the self-energy and polarization operator. This full vertex $\gamma^{4,0}$ is small and consists of the two large contributions: reducible ($\gamma^{4,0}_{\rm red}$) and irreducible ($-\gamma{\cal W}_{\omega}\gamma$) with respect to renormalized interaction ${\cal W}_{\omega}$. These two contributions compensate each other as shown in Eq. \[eq:red-irr-vertex\]. If one accounts for only one large irreducible contribution to the vertex function, it leads to incorrect description of the collective excitations and problems mentioned above.
Finally, one can rewrite Eq. as follows $$\begin{aligned}
\tilde{W}_{{\mathbf{q}}\omega}
= W_{{\mathbf{q}}\omega} - {\cal W}_{\omega}\Big[1 + \tilde{\Pi}^{0}_{{\mathbf{q}}\omega}\tilde{W}_{{\mathbf{q}}\omega} +
\tilde{\Pi}^{0}_{{\mathbf{q}}\omega}&W_{{\mathbf{q}}\omega} \notag\\
+\,\,\tilde{\Pi}^{0}_{{\mathbf{q}}\omega}\tilde{W}_{{\mathbf{q}}\omega}
\tilde{\Pi}^{0}_{{\mathbf{q}}\omega}&W_{{\mathbf{q}}\omega} \Big],\end{aligned}$$ and see, that DB excludes not the full local renormalized interaction ${\cal W}_{\omega}$ of impurity model from the full lattice interaction $W_{{\mathbf{q}}\omega}$, but the local interaction, that is renormalized by non-local polarization and non-local interactions $W_{{\mathbf{q}}\omega}$ and $\tilde{W}_{{\mathbf{q}}\omega}$. Therefore, the DB approach, which is free from the double-counting problem by construction, excludes less contribution from the full lattice renormalized interaction then the EDMFT$+$GW approach, and effects of the impurity model are not suppressed in our calculations.
|
---
abstract: 'We construct exact non-perturbative massive solutions in the gravitational Higgs mechanism. They confirm the conclusions of arXiv:1102.4991, which are based on non-perturbative Hamiltonian analysis for the relevant metric degrees of freedom, that while perturbatively unitarity may not be evident, no negative norm state is present in the full nonlinear theory. The non-perturbative massive solutions do not appear to exhibit instabilities and describe vacuum configurations which are periodic in time, including purely longitudinal solutions with isotropic periodically expanding and contracting spatial dimensions, “cosmological strings" with only one periodically expanding and contracting spatial dimension, and also purely non-longitudinal (“traceless") periodically expanding and contracting solutions with constant spatial volume. As an aside we also discuss massive solutions in New Massive Gravity. While such solutions are present in the linearized theory, we argue that already at the next-to-linear (quadratic) order in the equations of motion (and, more generally, for weak-field configurations) there are no massive solutions.'
---
**Non-perturbative Massive Solutions in**
**Gravitational Higgs Mechanism**
Zurab Kakushadze$^\S$$^\dag$[^1]
*$^\S$ Quantigic$^\circledR$ Solutions LLC*
*1127 High Ridge Road \#135, Stamford, CT 06905[^2]*
*$^\dag$ Department of Physics, University of Connecticut*
*1 University Place, Stamford, CT 06901*
(May 7, 2013)
Introduction and Summary
========================
The gravitational Higgs mechanism gives a non-perturbative and fully covariant definition of massive gravity [@KL; @thooft; @ZK; @Oda; @ZK1; @ZK2; @Oda1; @Demir; @Cham1; @Oda2; @JK; @dS; @Cham2; @Ber2; @Cham3; @Unitarity]. The graviton acquires mass via spontaneous breaking of the underlying general coordinate reparametrization invariance by scalar vacuum expectation values.[^3]
Within the perturbative framework, unitarity requires that, in order not to propagate a negative norm state at the quadratic level in the action, the graviton mass term be of the Fierz-Pauli form [@FP]. Furthermore, higher order terms should be such that they do not introduce additional degrees of freedom that would destabilize the background [@BD].[^4]
In [@Unitarity], following [@dS], it was argued that perturbation theory appears to be inadequate, among other things, for the purposes of addressing the issue of unitarity.[^5] In [@Unitarity] a non-perturbative Hamiltonian analysis for the relevant metric degrees of freedom was performed and it was argued that the full nonlinear theory is free of ghosts. The main result of [@Unitarity] is that, in the gravitational Higgs mechanism in the Minkowski background, [*non-perturbatively*]{} the Hamiltonian is bounded from below.[^6] In fact, the results of [@Unitarity] are in complete agreement with the [*full*]{} Hamiltonian analysis performed in [@JK] for the original model proposed by ’t Hooft [@thooft] (see below), in which the [*full*]{} (gauge-fixed) Hamiltonian is explicitly positive-definite and coincides with the Hamiltonian of [@Unitarity] for the relevant metric degrees of freedom.[^7]
In fact, this holds irrespective of whether the perturbative mass term is of the Fierz-Pauli form, including in the simplest case with no higher-derivative couplings in the scalar sector, first discussed in [@thooft]. Even though perturbatively the trace of the metric fluctuations is a propagating ghost, non-perturbatively the theory “resums” and the non-perturbative Hamiltonian is bounded from below.[^8]
In this paper we further verify the results of [@Unitarity] by constructing exact non-perturbative massive solutions in the gravitational Higgs mechanism. These non-perturbative massive solutions do not appear to exhibit instabilities and describe vacuum configurations which are periodic in time. For example, in $D=3$ in the model of [@thooft], we have purely longitudinal solutions with isotropic periodically expanding and contracting spatial dimensions with the metric ($0\leq \rho < 1$) $$\label{iso}
ds^2 = -d\tau^2 + \left[1 - \rho~\cos(M\tau)\right]~\delta_{ij}dx^i dx^j~,$$ where $M$ is the naive perturbative graviton mass.[^9] This is an exact solution. When the amplitude of the oscillations $\rho$ is large, the perturbative expansion breaks down as is evident upon examining the full nonlinear Hamiltonian ${\cal H}$ for the longitudinal mode: $$\label{H-intro}
{{\cal H} \over 2\kappa M} = {M^2\over 2}~\sqrt{{2e^{-2q} - e^{-4q}\over{M^2 + 4(\partial_t q)^2}}} = {M\over 2} - {{(\partial_t q)^2 + M^2 q^2}\over M} +\dots~,$$ which non-perturbatively is positive-definite (the first equality in (\[H-intro\])), but perturbatively, when expanded in the weak-field approximation to the second order in $q$ and $\partial_t q$, has a [*fake*]{} ghost of mass $M$ (the second equality in (\[H-intro\])).[^10]
Also in the same model of [@thooft], in any dimension $D$, we find exact solutions we refer to as “cosmological strings", of the form: $$ds^2 = e^{\omega(t)}~\left[-dt^2 + (dx^1)^2\right] + \sum_{i=2}^{D-1} (dx^i)^2~.$$ These solutions are 2-dimensional cosmological defects in a $D$-dimensional space-time with only one periodically expanding and contracting spatial dimension. The periodic function $\omega(t)$ has a period $T$, which depends on the oscillation amplitude. For small amplitudes $T\approx 2\pi/M$, where $M$ is the naive perturbative graviton mass.[^11]
Once higher-derivative terms are added in the scalar sector,[^12] there are other exact solutions, including, [*e.g.*]{}, $D=3$ purely non-longitudinal (“traceless") periodically expanding and contracting solutions with constant spatial volume with the metric of the form $$\begin{aligned}
&&ds^2 = -d\tau^2 + \left[1 + \rho~\cos(M_1\tau)\right](dx^1)^2 + \left[1 - \rho~\cos(M_1\tau)\right](dx^2)^2 + \nonumber\\
&&\,\,\,2\rho~\sin(M_1\tau)~(dx^1)(dx^2)~,\end{aligned}$$ where the amplitude of oscillations $\rho$ and the mass parameter $M_1$ are determined by the higher-derivative couplings in the scalar sector (see Subsection 7.3 for details).
Upon gauging away the scalar degrees of freedom, the simplest non-perturbative massive gravity action (in the Minkowski background) can be written as $$S_{MG} = M_P^{D-2}\int d^Dx \sqrt{-G}\left[ R + \mu^2\left(D - 2 - G^{MN}\eta_{MN}\right)\right]~,$$ which corresponds to the model of [@thooft] without higher-derivative couplings in the scalar sector.[^13] This action, albeit nonlinear, allows to obtain various exact solutions, as we have done in this paper.[^14] Since non-perturbatively there is no ghost [@Unitarity], it might be possible to utilize this model in pursuing one of the original motivations of [@thooft], namely, string theory description of QCD, where massless spin-2 modes somehow must acquire mass.[^15] Another motivation for massive gravity – its large-scale modification – is cosmology, including in the context of the currently observed accelerated expansion of the Universe [@Nova; @Nova1] and the cosmological constant.
As an aside we also discuss massive solutions in New Massive Gravity [@BHT]. While such solutions are present in the linearized theory, we argue that already at the next-to-linear (quadratic) order in the equations of motion (and, more generally, for weak-field configurations) there are no massive solutions.
The rest of the paper is organized as follows. In Sections 2 and 3 we discuss the gravitational Higgs mechanism in a general background, which results in linearized massive gravity with the Fierz-Pauli mass term for the appropriately tuned cosmological constant. In Section 4 we discuss non-perturbative massive gravity via the gravitational Higgs mechanism and review the non-perturbative Hamiltonian for the relevant metric modes and its positive-definiteness. In Section 5 we discuss restrictions on higher curvature terms that we use in our discussion of New Massive Gravity in Section 8. In Section 6 we discuss non-perturbative massive solutions corresponding to the longitudinal mode. In Section 7 we discuss non-perturbative non-longitudinal massive solutions. In Section 8 we discuss massive solutions in New Massive gravity.
Gravitational Higgs Mechanism
=============================
The goal of this section is to obtain massive gravity in a general background via the gravitational Higgs mechanism. This is a generalization of the setup of [@dS] for the gravitational Higgs mechanism in the de Sitter space to a general background. Thus, let $$\label{action.0}
S = S_G + {1\over 2} \int d^Dx ~G_{MN} T^{MN} ~,$$ where $$S_G \equiv M_P^{D-2}\int d^Dx \sqrt{-G}\left[ R - {\widetilde \Lambda} + {\cal O}\left(R^2\right)\right]$$ is an arbitrary generally covariant action constructed from the metric $G_{MN}$ and its derivatives, ${\widetilde \Lambda}$ is the cosmological constant, ${\cal O}\left(R^2\right)$ stands for higher curvature terms constructed from $R$, $R_{MN}$ and $R_{MNST}$, and we have included a coupling to the conserved energy momentum tensor $T^{MN}$: $$\nabla_N T^{NM} = 0~.$$ Let ${\widetilde G}_{MN}$ be a background solution to the equations of motion corresponding to (\[action.0\]): $$R_{MN} - {1\over 2} G_{MN} \left[R - {\widetilde \Lambda}\right] + {\cal O}\left(R^2\right) = {M_P^{2-D}\over 2\sqrt{-G}} T_{MN}~,$$ The background metric ${\widetilde G}_{MN}$ generally is a function of the coordinates $x^S$: ${\widetilde G}_{MN} = {\widetilde G}_{MN}(x^0,\dots, x^{D - 1})$.
Let us now introduce $D$ scalars $\phi^A$. Let us normalize them such that they have dimension of length. Let us define a metric $Z_{AB}$ for the scalars as follows: $$Z_{AB}(\phi^0,\dots,\phi^{D-1}) \equiv {\delta_A}^M {\delta_B}^N {\widetilde G}_{MN}(\phi^0,\dots, \phi^{D - 1})~,$$ [*i.e.*]{}, we substitute the space-time indices $M$ and $N$ in ${\widetilde G}_{MN}$ with the global scalar indices $A$ and $B$, and substitute $x^0\rightarrow \phi^0,\dots ,x^{D-1}\rightarrow \phi^{D-1}$ in the functional form of ${\widetilde G}_{MN}$.
Next, consider the induced metric for the scalar sector: $$Y_{MN} = Z_{AB} \nabla_M\phi^A \nabla_N\phi^B~.$$ Also, let $$\label{Y}
Y\equiv Y_{MN}G^{MN}~.$$ The following action, albeit not most general,[^16] will suffice for our purposes here: $$S_Y = M_P^{D-2}\int d^Dx \sqrt{-G}\left[ R - \mu^2 V(Y) + {\cal O}\left(R^2\right)\right] + {1\over 2} \int d^Dx ~G^{MN} T_{MN} ~,
\label{actionphiY}$$ where [*a priori*]{} the “potential" $V(Y)$ is a generic function of $Y$, and $\mu$ is a mass parameter, while the cosmological constant ${\widetilde \Lambda}$ is subsumed in the definition of $V(Y)$ (see below).
The equations of motion read: $$\begin{aligned}
\label{phiY}
&& \nabla^M\left(V^\prime(Y)Z_{AB} \nabla_M \phi^B\right) = {1\over 2} {\partial Z_{BC}\over \partial\phi^A} G^{MN} \nabla_M \phi^B~\nabla_N\phi^C ~V^\prime(Y)~,\\
\label{einsteinY}
&& R_{MN} - {1\over 2}G_{MN} R + {\cal O}\left(R^2\right) - \mu^2\left[ V^\prime(Y) Y_{MN}
-{1\over 2}G_{MN} V(Y)\right] = \nonumber\\
&& = {M_P^{2-D}\over 2\sqrt{-G}} T_{MN}~,\end{aligned}$$ where prime denotes a derivative w.r.t. $Y$. Multiplying (\[phiY\]) by $\nabla_S\phi^A$ and contracting indices, we can rewrite the scalar equations of motion as follows: $$\label{phiY.1}
\partial_M\left[\sqrt{-G} V^\prime(Y) G^{MN}Y_{NS}\right] - {1\over 2}\sqrt{-G} V^\prime(Y) G^{MN}\partial_S Y_{MN} = 0~.$$ Since the theory possesses full diffeomorphism symmetry, (\[phiY.1\]) and (\[einsteinY\]) are not all independent but linearly related due to Bianchi identities. Thus, multiplying (\[einsteinY\]) by $\sqrt{-G}$, differentiating w.r.t. $\nabla^N$ and contracting indices we arrive at (\[phiY.1\]).
We are interested in finding solutions of the form: $$\begin{aligned}
\label{solphiY}
&&\phi^A = {\delta^A}_M~x^M~,\\
\label{solGY}
&&G_{MN} = {\widetilde G}_{MN}~.\end{aligned}$$ Since on this solution $Y_{MN} = {\widetilde G}_{MN}$ and $Y = D$, (\[phiY.1\]) is automatically satisfied. Furthermore, (\[einsteinY\]) implies that $$R_{MN} - {1\over 2} G_{MN} \left[R - {\widetilde \Lambda}\right] + {\cal O}\left(R^2\right) = {M_P^{2-D}\over 2\sqrt{-G}} T_{MN}~,$$ provided that $$\label{cosm.const}
V(D) - 2 V^\prime(D) = {\widetilde \Lambda} / \mu^2~.$$ For [*non-generic*]{} potentials this condition can be constraining. For example, if $V(Y) = a + Y$ and ${\widetilde \Lambda} = 0$, we have $a = -(D-2)$ implying that the vacuum energy density $\Lambda \equiv \mu^2 V(0) = a \mu^2$ in the unbroken phase ($\phi^A \equiv 0$) must be negative. However, since $\mu$ is an arbitrary mass scale, for [*generic*]{} potentials $V(Y)$ the above condition is not constraining. In particular, generically there is no restriction on the vacuum energy density $\Lambda = \mu^2 V(0)$ in the unbroken phase, which can be positive, negative or zero, even if one requires that the mass term for the graviton in the linearized theory is of the Fierz-Pauli form (see below).
Linearized Massive Gravity
==========================
In this section we discuss linearized fluctuations in the background given by (\[solphiY\]) and (\[solGY\]). Since diffeomorphisms are broken spontaneously, the equations of motion are invariant under the full diffeomorphism invariance. The scalar fluctuations $\varphi^A$ can therefore be gauged away using the diffeomorphisms: $$\label{diffphiY}
\delta\varphi^A =\nabla_M \phi^A \xi^M = {\delta^A}_M ~\xi^M~.$$ However, once we gauge away the scalars, diffeomorphisms can no longer be used to gauge away any of the graviton components $h_{MN}$ defined as: $$G_{MN} = {\widetilde G}_{MN} + h_{MN}~,$$ where ${\widetilde G}_{MN}$ is the background metric defined in the previous section. We will use the notation $h \equiv {\widetilde G}^{MN} h_{MN}$.
After setting $\varphi^A = 0$, we have $$\begin{aligned}
&& Y_{MN} = {\widetilde G}_{MN}~,\\
&& Y = Y_{MN} G^{MN} = D - h + {\cal O}\left(h^2\right) ~,\end{aligned}$$ Due to diffeomorphism invariance, the scalar equations of motion (\[phiY\]) are related to (\[einsteinY\]) via Bianchi identities. We will therefore focus on (\[einsteinY\]). Linearizing the terms containing $V$, we obtain: $$\begin{aligned}
&&R_{MN} - {1\over 2}G_{MN} \left[R - {\widetilde \Lambda}\right] + {\cal O}\left(R^2\right)- \nonumber\\
&& -{M^2\over 2} \left[{\widetilde G}_{MN} h - \zeta h_{MN}\right] + {\cal O}\left(h^2\right) = \nonumber\\
&& = {M_P^{2-D}\over 2\sqrt{-G}} T_{MN}~,\label{lin.eom}\end{aligned}$$ where $$\begin{aligned}
&&M^2 \equiv \mu^2 \left[V^\prime(D) - 2 V^{\prime\prime}(D)\right]~,\label{M}\\
&&\zeta M^2 \equiv 2\mu^2 V^\prime(D)~.\label{zeta}\end{aligned}$$ This corresponds to adding a graviton mass term of the form $$-{M^2\over 4} \left[\zeta h_{MN}h^{MN} - h^2\right]$$ to the gravity action $S_G$, and the Fierz-Pauli combination corresponds to taking $\zeta = 1$. This occurs for a special class of potentials with $$\label{tune-V}
V^\prime(D) = -2V^{\prime\prime}(D)~.$$ Thus, as we see, we can obtain the Fierz-Pauli combination of the mass term for the graviton if we tune [*one*]{} combination of couplings. In fact, this tuning is nothing but the tuning of the vacuum energy density $\Lambda = \mu^2 V(0)$ in the unbroken phase – indeed, (\[tune-V\]) relates the vacuum energy density in the unbroken phase to higher derivative couplings.
Linear Potential
----------------
The simplest potential is given by [@thooft]: $$V = a + Y$$ From (\[cosm.const\]) we have $$\label{a.lin}
a = {{\widetilde \Lambda}\over \mu^2} - (D - 2)$$ The vacuum energy density in the unbroken phase $\Lambda = \mu^2 V(0)$ is negative for Minkowski solutions. Also, for this potential (\[tune-V\]) cannot be satisfied, [*i.e.*]{}, we cannot have the Fierz-Pauli mass term, which requires $\zeta = 1$ in (\[zeta\]) and instead we have $\zeta = 2$. Perturbatively, the trace $h \equiv {\widetilde G}^{MN} h_{MN}$ is a propagating ghost degree of freedom. However, as was argued in [@Unitarity], non-perturbatively there is no ghost and the Hamiltonian for the relevant degrees of freedom (see below) is bounded from below. To achieve the Fierz-Pauli mass term, higher derivative terms for scalars are needed [@ZK1].
Quadratic Potential
-------------------
Thus, consider a simple example: $$\label{quad.pot}
V = a + Y + \lambda Y^2~.$$ The first term corresponds to the vacuum energy density $\Lambda = \mu^2 V(0)$ in the unbroken phase, the second term is the kinetic term for the scalars (which can always be normalized such that the corresponding coefficient is 1 by adjusting $\mu$), and the third term is a four-derivative term. From (\[tune-V\]) we then have: $$\lambda = -{1\over{2(D+2)}}~,$$ and the graviton mass is given by: $$M^2 = {4\mu^2\over{D+2}}~.$$ Moreover, from (\[cosm.const\]) we have: $$a = {{\widetilde \Lambda}\over \mu^2} - {{D^2 + 4D - 8}\over {2(D+2)}}~,$$ which is nothing but tuning of the vacuum energy density $\Lambda = \mu^2 V(0)$ in the unbroken phase against the higher derivative coupling $\mu^2 \lambda$.
Here the following remark is in order. In the above example with the quadratic potential (\[quad.pot\]) the vacuum energy density $\Lambda = \mu^2 V(0)$ in the unbroken phase must be negative in the case of the Minkowski background (${\widetilde \Lambda} = 0$). However, generically, there is no restriction on $\Lambda$, even in the case of the Minkowski background, if we allow cubic and/or higher order terms in $V(Y)$, or consider non-polynomial $Y(Y)$.
Exponential Potential
---------------------
Thus, consider another simple example: $$\label{exp.pot}
V = a + b~e^{-\lambda Y}~.$$ Then from (\[tune-V\]) we then have: $$\label{FP.exp}
\lambda = {1\over 2}~,$$ and from (\[cosm.const\]) we have: $$a = {{\widetilde \Lambda}\over \mu^2} - 2\, b\, e^{-D/2}~.$$ In this case the vacuum energy density in the unbroken phase $\Lambda = \mu^2 V(0) = a + b$ can therefore be positive, negative or zero depending on the value of the parameter $b$ irrespective of the value of ${\widetilde \Lambda}$.
Square-root Potential
---------------------
Finally, consider the following non-polynomial potential: $$\label{sq.rt}
V = a + \sqrt{Y + b}~.$$ From (\[tune-V\]) we have: $$b = -(D-1)~,$$ and from (\[cosm.const\]) we further have $$a = {{\widetilde \Lambda}\over \mu^2}~.$$ An interesting feature of this square-root potential is that there is no unbroken phase as we must have $Y\geq (D-1)$.
Non-perturbative Massive Gravity
================================
The gravitational Higgs mechanism provides a non-perturbative definition of massive gravity in a general background. Thus, if we use diffeomorphisms to gauge away the scalars by fixing them to their background values (\[solphiY\]), the full non-perturbative action is then given by $$S_{MG} = M_P^{D-2}\int d^Dx \sqrt{-G}\left[ R - \mu^2 V(G^{MN}{\widetilde G}_{MN}) + {\cal O}\left(R^2\right)\right] + {1\over 2} \int d^Dx ~G^{MN} T_{MN} ~,
\label{massivegravity}$$ which describes massive gravity in the background metric ${\widetilde G}_{MN}$. The equation of motion reads $$\begin{aligned}
\label{R.eom}
&& R_{MN} - {1\over 2}G_{MN} R + {\cal O}\left(R^2\right) - \mu^2\left[ V^\prime(G^{ST}{\widetilde G}_{ST}) {\widetilde G}_{MN}
-{1\over 2}G_{MN} V(G^{ST}{\widetilde G}_{ST})\right] = \nonumber\\
&& = {M_P^{2-D}\over 2\sqrt{-G}} T_{MN}~,\end{aligned}$$ with the Bianchi identity $$\label{phi.eom}
\partial_M\left[\sqrt{-G} V^\prime(G^{RT}{\widetilde G}_{RT}) G^{MN}{\widetilde G}_{NS}\right] - {1\over 2}\sqrt{-G} V^\prime(G^{RT}{\widetilde G}_{RT}) G^{MN}\partial_S {\widetilde G}_{MN} = 0~,$$ which is equivalent to the gauge-fixed equations of motion for the scalars.
An Example: Schwarzschild Background
------------------------------------
This construction allows to define massive gravity in nontrivial curved backgrounds. The de Sitter background was discussed in [@dS]. Another example is the Schwarzschild background, for which the background metric ${\widetilde G}_{MN}$ is given by: $$ds^2 = {\widetilde G}_{MN} dx^A dx^B = -A^2 dt^2 + B^2 dr^2 + C^2 \gamma_{ab} dx^a dx^b~,$$ where $$\begin{aligned}
&&A = B^{-1} = \sqrt{1 - {r_* / r}}~,\\
&&C = r~,\end{aligned}$$ and $\gamma_{ab}$ is a metric on the unit sphere $S^{d-1}$, $d = D - 1$. This, provides an explicit construction of massive gravity in the Schwarzschild background (not to be confused with a construction of Schwarzschild-like solutions in massive gravity).
Unitarity: Minkowski Background
-------------------------------
Following [@dS] and [@Unitarity], in this subsection we discuss unitarity of massive gravity via the gravitational Higgs mechanism in the Minkowski background by studying the full non-perturbative Hamiltonian for conformal and helicity-0 modes. This suffices to deduce whether there are any additional degrees of freedom that destabilize the background, assuming that transverse-traceless graviton components are positive-definite.[^17]
To identify the relevant modes in the full nonlinear theory, let us note that in the linearized theory the potentially “troublesome” mode is the longitudinal helicity-0 mode $\rho$. However, we must also include the conformal mode $\omega$ as there is kinetic mixing between $\rho$ and $\omega$. In fact, $\rho$ and $\omega$ are not independent but are related via Bianchi identities. Therefore, in the linearized language one must look at the modes of the form[^18] $$\label{param.lin}
h_{MN} = \eta_{MN}~\omega + \nabla_M\nabla_N \rho~.$$ Furthermore, based on symmetry considerations, namely, the $SO(D-1)$ invariance in the spatial directions,[^19] we can focus on field configurations independent of spatial coordinates [@dS; @Unitarity]. Indeed, for our purposes here we can compactify the spatial coordinates on a torus $T^{D-1}$ and disregard the Kaluza-Klein modes. This way we reduce the $D$-dimensional theory to a classical mechanical system, which suffices for our purposes here. Indeed, with proper care (see [@dS]), if there is a negative norm state in the uncompactified theory, it will be visible in its compactified version, and vice-versa.
Let us therefore consider field configurations of the form: $$\label{param.full}
G^{MN} = {\rm diag}(g(t)~\eta^{00}, f(t)~\eta^{ii})~,$$ where $g(t)$ and $f(t)$ are functions of time $t$ only. The action (\[massivegravity\]) then reduces as follows:[^20] $$\begin{aligned}
\label{compact}
S_{MG} = -\kappa \int dt ~g^{-{1\over 2}} f^{-{{D-1}\over 2}}
\left\{Q(gU^2) + \mu^2~V(g + \Omega)\right\}~,\end{aligned}$$ where $$\begin{aligned}
&&\kappa\equiv {M_P^{D-2} W_{D-1}}~,\\
&&Q(x)\equiv \sum_{k=1}^{\infty} c_k x^k~,\label{sumk}\\
&&c_1 \equiv (D-1)(D-2)~,\\
&&U\equiv {1\over 2}\partial_t\ln(f)~,\\
&&\Omega\equiv (D-1)f~,\end{aligned}$$ and $W_{D-1}$ is the volume in the spatial dimensions ([*i.e.*]{}, the volume of $T^{D-1}$). Also, the $k=1$ term in (\[sumk\]) corresponds to the Einstein-Hilbert term, while the $k>1$ terms (the sum over $k$, [*a priori*]{}, can be finite or infinite) correspond to the ${\cal O}\left(R^2\right)$ terms and we are assuming that the ${\cal O}\left(R^2\right)$ are such that only first time-derivatives of $f$ appear, which imposes certain conditions on the higher curvature terms (see below). Also, note that $g$ is a Lagrange multiplier. We can integrate out $g$ and obtain the corresponding action for $f$. It is then this action that we need to test for the presence of a negative norm state.
However, there is a simple way to see if the Hamiltonian is positive-definite. The equation of motion for $g$ reads: $$\label{geq}
\mu^2\left[V(g+\Omega) - 2gV^\prime(g +\Omega)\right] = 2gU^2 Q^\prime(gU^2) - Q(gU^2)~.$$ For our purposes here it is more convenient to work with the canonical variable $q$, where $$\begin{aligned}
&&q\equiv {1\over 2} \ln(f)~,\\
&&\Omega = (D-1)e^{2q}~,\\
&&U = \partial_t q~,\end{aligned}$$ and the action reads: $$\begin{aligned}
\label{compact1}
S_{MG} = \int dt~L = -\kappa \int {dt} ~g^{-{1\over 2}} e^{-(D-1)q}
\left\{Q(g U^2) + \mu^2~V(g + \Omega)\right\}~,\end{aligned}$$ where $L$ is the Lagrangian. This action corresponds to a classical mechanical system with a lagrange multiplier $g$.
Next, the conjugate momentum is given by $$p = {{\partial L} \over {\partial(\partial_t q)}} = -2\kappa e^{-(D-1)q} g^{{1\over 2}} U Q^\prime(gU^2)~,
\label{momentum}$$ where we have used (\[geq\]) to eliminate terms containing $${\hat g} \equiv {{\partial g} \over {\partial(\partial_\tau q)}}~,$$ and the Hamiltonian is given by $$\label{Hamiltonian}
{\cal H} = p~\partial_t q - L = 2 \kappa\mu^2 g^{{1\over 2}} e^{-(D-1)q}V^\prime(g + \Omega)~,$$ where we again used (\[geq\]).
Actually, this Hamiltonian can be obtained up to a normalization constant from (\[phiY.1\]), which for the field configurations (\[param.full\]) reduces to $$\partial_t\left[g^{{1\over 2}} e^{-(D-1)q}V^\prime(g + \Omega)\right]=0~.$$ This is nothing but the condition that the Hamiltonian is constant.
### Linear Potential
The Hamiltonian is evidently positive-definite for the linear potential $V = a + Y$ of [@thooft],[^21] in complete agreement with the full Hamiltonian analysis of [@JK] (see [@Unitarity] for details). As was pointed out in [@Unitarity], the absence of ghosts in the full non-perturbative theory despite the fact that for the linear potential we do not have the Fierz-Pauli mass-term, is yet another illustrative example of pitfalls of linearization.
### Quadratic Potential {#quad.subsection}
The proof that the Hamiltonian is positive-definite for the quadratic potential (\[quad.pot\]) is given in [@Unitarity] for the Einstein-Hilbert case ($c_{k>1} = 0$ in (\[sumk\])). The proof in the case of general $Q(gU^2)$ is essentially the same with the substitution $c_1 gU^2 \rightarrow Q(gU^2)$ assuming that $c_{k>1} \geq 0$.
### Exponential Potential
The Hamiltonian is evidently positive-definite for the exponential potential (\[exp.pot\]) with $b\lambda < 0$. In fact, ghosts are absent even without requiring (\[FP.exp\]), which is the requirement that the linearized mass term be of the Fierz-Pauli form.
### Square-root Potential {#sq.rt.subsection}
The Hamiltonian is evidently positive-definite for the square-root potential (\[sq.rt\]). This is consistent with the full Hamiltonian analysis of [@JK] for this potential.[^22]
Unitarity: de Sitter Background
-------------------------------
The unitarity argument for the de Sitter background closely parallels that for the Minkowski background. In the de Sitter case we consider field configurations of the form: $$\label{param.full.dS}
G^{MN} = {\rm diag}(g(t)~{\widetilde G}^{00}, f(t)~{\widetilde G}^{ii})~,$$ where $g(t)$ and $f(t)$ are functions of time $t$ only, and ${\widetilde G}_{MN}$ is the de Sitter metric, which we can choose as follows: $${\widetilde G}_{MN} = {\rm diag}(\eta_{00}, e^{-2Ht}\eta_{ii})~,$$ where $H$ is the Hubble parameter. The above unitarity argument for the Minkowski background is essentially unchanged if we define $$q\equiv {1\over 2}\ln(f) + Ht$$ as the canonical variable.
Restrictions on Higher Curvature Terms
======================================
In the previous section we assumed that (\[sumk\]) does not contain higher derivative terms. This imposes restrictions on higher curvature terms. Here we discuss these restrictions in the case of terms quadratic in curvature. The most general quadratic curvature contribution to the action is given by: $$M_P^{D-2}\int d^Dx \sqrt{-G}\left[\alpha R^2 + \beta R^{MN} R_{MN} + \gamma\left(R^{MNST}R_{MNST} - 4R^{MN}R_{MN} + R^2\right)\right]~.$$ The Gauss-Bonnet combination does not introduce higher derivatives. The other two terms do not contribute higher derivative terms provided that $$\label{alphabeta}
4(D-1)\alpha + D\beta = 0~,$$ and in this case we have ($c_{k>2} = 0$): $$c_2 = {(D-1) (D-2) (D-4)\over 12}~\left[4(D-3)\gamma - (D-2)\beta\right]~.$$ Note that $c_2$ vanishes for $D=4$. The same is the case for $D>4$ and $$\beta={4(D-3)\gamma\over(D-2)}~,$$ when the quadratic curvature contribution is of the form: $$M_P^{D-2}\int d^Dx \sqrt{-G} ~C^{MNST}C_{MNST}~,$$ where $C_{MNST}$ is the Weyl tensor and the theory is at a critical point discussed in [@crit] with a unique vacuum.
Here we should point out that our discussion of higher-curvature terms in this section is motivated primarily by our discussion of New Massive Gravity in Section 8 (as opposed to in the context of the gravitational Higgs mechanism).
Non-perturbative Longitudinal Solutions
=======================================
In this section we discuss non-perturbative solutions to the equations of motion in the Minkowski background with the aim to further demonstrate that, despite apparent issues with perturbative unitarity, [*e.g.*]{}, in the case of the linear potential where perturbatively there is a propagating ghost, non-perturbatively no negative-norm states appear and the background appears to be stable. In most cases solving highly nonlinear equations of motion analytically is challenging. However, there are two cases of interest where the equations of motion are tractable. In this section we will assume that no higher-curvature terms are present.
Linear Potential
----------------
Let us start with the linear potential. From (\[geq\]) we have $$g = {{\Omega - (D-2)}\over {1 + c_1~{U^2\over\mu^2}}}~,$$ where we have used (\[a.lin\]) together with ${\widetilde \Lambda} = 0$. Furthermore, from the condition that the Hamiltonian is constant, we have: $$\label{g.lin}
g~e^{-2(D-1)q} = \nu^2~,$$ where $$\label{nu}
\nu \equiv {{\cal H}\over{2\kappa\mu^2}} > 0~.$$ We therefore have the following first-order equation for $q$: $$\label{q.lin}
(D-1)(D-2)~{\nu^2\over\mu^2}~\left(\partial_t q\right)^2 = (1 - \nu^2) - (e^{-2q} - 1)^2~\sum_{k=0}^{D-3} (k+1)~e^{-2kq}~,$$ where we have used the following identity $$n~x^{n-1} - (n-1)~x^n = 1 - (x-1)^2~\sum_{k=0}^{n-2}(k+1)~x^k$$ along with the definitions of $\Omega$ and $c_1$. Note that from (\[q.lin\]) we have $\nu\leq 1$.
The equation of motion (\[q.lin\]) can be solved in the $D=3$ case, which suffices for our purposes here: $$e^{2q} = {1\over \nu^2}\left[1+\sqrt{1-\nu^2}~\cos\left(\sqrt{2}\mu(t-t_0)\right)\right]~,$$ where $t_0$ is an integration constant. Note that, due to (\[g.lin\]), we have $$g^{1\over 2} = \nu~e^{2q}~.$$ The metric is given by $$ds^2 = -g^{-1}~dt^2 + e^{-2q}~\delta_{ij}dx^i dx^j~.$$ Let us redefine the time coordinate as follows: $$\label{tau-t}
d\tau \equiv g^{-{1\over 2}}~dt~.$$ Then we have $$\tau - \tau_0 = {1\over{\sqrt{2}\mu}}~\arccos\left({{\sqrt{1-\nu^2} + \cos\left(\sqrt{2}\mu(t-t_0)\right)}\over{1 + \sqrt{1-\nu^2}~\cos\left(\sqrt{2}\mu(t-t_0)\right)}}\right)~,$$ where $\tau_0$ is an integration constant. In terms of this new time coordinate $\tau$ we have: $$ds^2 = -d\tau^2 + e^{-2q}~\delta_{ij}dx^i dx^j~,$$ where $$e^{-2q} = 1 - \sqrt{1-\nu^2}~\cos\left(\sqrt{2}\mu(\tau-\tau_0)\right)~.$$ The non-perturbative Hamiltonian for the linear potential in general dimension $D$ is given by: $${\cal H}_{\mbox{\small{non-pert}}} = 2\kappa\mu^3~\sqrt{{(D-1)e^{-2(D-2)q} - (D-2)e^{-2(D-1)q}\over{\mu^2 + (D-1)(D-2)(\partial_t q)^2}}}~.$$ If we naively expand the Hamiltonian in the weak-field approximation to the second order in $q$ and $\partial_t q$, we obtain: $${\cal H}_{\mbox{\small{pert}}} = 2\kappa\left\{\mu^2 - (D-1)(D-2)\left[{1\over 2}\left(\partial_t q\right)^2 + \mu^2~q^2\right] + \dots\right\}~,$$ where the ellipses stand for higher order terms in $q$ and $\partial_t q$. This naive expansion produces a ghost of mass $M_* = \sqrt{2}\mu$, which is precisely the mass of the perturbative would-be ghost – the trace $h$ – in (\[lin.eom\]) when $\zeta = 2$.[^23] However, non-perturbatively there is no ghost as the Hamiltonian is positive. Instead, we have non-perturbative oscillations with the same mass parameter $M_*$, but with the amplitude controlled by $\nu$. This amplitude is small for $\nu = 1 - \epsilon$, where $\epsilon\ll 1$. However, if $\nu$ is not close to 1, then the perturbative expansion breaks down. Therefore, the presence of a ghost in the perturbative expansion is merely an artifact of linearization.
For the sake of completeness, let us note that the equation of motion for general $D$ in terms of the time coordinate $\tau$ reads: $$(D-1)(D-2)~{e^{-2(D-1)q}\over\mu^2}~\left(\partial_\tau q\right)^2 = (1 - \nu^2) - (e^{-2q} - 1)^2~\sum_{k=0}^{D-3} (k+1)~e^{-2kq}~,$$ with $$g^{1\over 2} = \nu~e^{(D-1)q}~,$$ and $\tau$ is related to $t$ via (\[tau-t\]).
Square-root Potential {#square-root-potential-1}
---------------------
As was argued in [@Unitarity], in the case of the quadratic potential the Fierz-Pauli point is not special in any way as far as non-perturbative unitarity is concerned, in fact, non-perturbatively the Hamiltonian is bounded from below for a continuous range of values of the four-derivative coupling $\lambda$ in (\[quad.pot\]) smoothly interpolating between the linear potential ($\zeta = 2$ in (\[lin.eom\])) and the Fierz-Pauli point ($\zeta=1$ in (\[lin.eom\])). The aim of this subsection is to understand if there is anything “special" happening at the Fierz-Pauli point, if not from the non-perturbative unitarity standpoint, at least at the level of the non-perturbative equations of motion. The latter are cumbersome to analyze for the quadratic potential. However, they are tractable for the square-root potential (\[sq.rt\]) in the Minkowski background at the Fierz-Pauli point, [*i.e.*]{}, $a=0$ and $b=-(D-1)$.
From (\[geq\]) we have $${(D-1)(f-1)\over\sqrt{g + (D-1)(f-1)}}= c_1~{gU^2\over\mu^2}~.$$ Furthermore, from the condition that the Hamiltonian is constant, we have: $$\label{g.sq}
{g^{1\over 2}~f^{-{{D-1}\over 2}} \over 2 \sqrt{g + (D-1)(f-1)}} = \nu~,$$ where $\nu$ is defined in (\[nu\]). Let us define a new time coordinate ${\widetilde t}$ via $$d{\widetilde t} = g^{-3/4} dt~,$$ We then have the following equation for $f$: $$(f-1)f^{{D+3}\over 2} = {{D - 2}\over 8\nu\mu^2} ~\left(\partial_{\widetilde t} f\right)^2~.$$ Let us define $${\widetilde f} \equiv f^{-1}~.$$ We have $$\label{tilde.f}
(1-{\widetilde f}){\widetilde f}^{{3-D}\over 2} = {{D - 2}\over 8\nu\mu^2} ~\left(\partial_{\widetilde t} {\widetilde f}\right)^2~.$$ One solution is ${\widetilde f} \equiv 1$, [*i.e.*]{}, $f\equiv 1$, which implies $\nu = 1/2$. Note that $g$ is arbitrary in this case and can be absorbed into the definition of the time coordinate.
To see if there are any other solutions, let us consider the $D=3$ case, which suffices for our purposes here. In $D=3$ a nontrivial solution to (\[tilde.f\]) is given by $$\label{sol.tilde.f}
{\widetilde f} = 1 - 2\nu\mu^2~\left({\widetilde t} - {\widetilde t}_0\right)^2~,$$ where ${\widetilde t}_0$ is an integration constant. This implies that ${\widetilde f} \leq 1$ and $f \geq 1$. Note that the metric is given by $$ds^2 = -g^{1\over 2} d{\widetilde t}^2 + {\widetilde f}~\delta_{ij} dx^i dx^j~,$$ and from (\[g.sq\]) we have $$g = {{8\nu^2 f^2 (f - 1)}\over{1 - 4\nu^2 f^2}}~,$$ so we must have $f < 1/(2\nu)$ and ${\widetilde f} > 2\nu$, [*i.e.*]{}, $\nu < 1/2$.
On the other hand, from (\[R.eom\]) (with the higher curvature terms and the source term set to zero), we have for the scalar curvature $$R = \mu^2~{{D-1}\over {D-2}}~{{g + (D-1)f - D}\over\sqrt{g + (D-1)(f-1)}}~.$$ So, in the above solution (\[sol.tilde.f\]), when $f$ approaches $1/(2\nu)$ from below, which occurs at $${\widetilde t}_{\pm} = {\widetilde t}_0 \pm {1\over \mu}~\sqrt{{1\over 2\nu} - 1}~,$$ the scalar curvature $R$ diverges ($R\rightarrow +\infty$) as $g$ diverges and we have singularities at ${\widetilde t}_\pm$. Furthermore, as $f$ approaches 1 from above, which occurs when $$\left|{\widetilde t} -{\widetilde t}_0\right| \rightarrow 0~,$$ the scalar curvature $R$ also diverges ($R\rightarrow -\infty$) as $g$ goes to zero.
As in the previous subsection, let us introduce the time coordinate $\tau$ defined via $$d\tau = g^{-{1\over 2}}dt = g^{1\over 4}d{\widetilde t}~.$$ Then the metric reads $$ds^2 = -d\tau^2 + {\widetilde f}~\delta_{ij} dx^i dx^j~.$$ Since $$g^{1\over 4} \sim \left|{\widetilde t} - {\widetilde t}_\pm\right|^{-{1\over 4}}$$ as ${\widetilde t}$ approaches ${\widetilde t}_\pm$, it takes finite time $\tau$ to reach the singularities at ${\widetilde t} = {\widetilde t}_\pm$, which are therefore true (not coordinate) singularities. The singularity at ${\widetilde t} = {\widetilde t}_0$ is also a true singularity. Therefore, the solution (\[sol.tilde.f\]) with $\nu < 1/2$ is a singular solution, which appears to be non-physical, and should be discarded.[^24] This leaves us with the sole non-singular solution $f\equiv 1$ with $\nu = 1/2$.
Thus, what is “special" at the Fierz-Pauli point is that non-perturbatively there appears to be no dynamics associated with the longitudinal mode. In contrast, as we saw in the previous subsection for the linear potential, at a non-Fierz-Pauli point we have oscillating non-perturbative solutions with the mass scale $M_* = \sqrt{2}\mu$, same as the mass of the would-be perturbative ghost; however, non-perturbatively there is no ghost and the Hamiltonian is bounded from below.
Full Equations of Motion
------------------------
In this subsection we check that the solutions we found using the above Hamiltonian approach satisfy the full equations of motion. We are looking for the solutions of the form: $$\label{ua}
G_{MN} = {\rm diag}(u(t)~\eta_{00}, e^{2A(t)}~\eta_{ii})~.$$ Note the relation to our notations above: $u = g^{-1}$ and $A = -q$. We have: $$\begin{aligned}
&&R_{ij} = \left[{A^{\prime\prime}\over u} - {A^\prime u^\prime \over 2u^2} + {(D-1)\over u} (A^\prime)^2\right]~e^{2A}~\eta_{ij}~,\\
&&R_{00} = -(D-1)\left[A^{\prime\prime} + (A^\prime)^2 - {A^\prime u^\prime \over 2u}\right]~,\\
&&R_{i0} = 0~,\end{aligned}$$ where a prime on $A$ and $u$ denotes a time derivative (not to be confused with a prime on the potential $V$ defied above). The equations of motion (\[R.eom\]) and (\[phi.eom\]) read (assuming the Minkowski background metric ${\widetilde G}_{MN} = \eta_{MN}$ and no source or higher curvature terms): $$\begin{aligned}
&&R_{MN} = \mu^2 \left[V^\prime(X)~\eta_{MN} + {{V(X) - X~V^\prime(X)}\over {D-2}}~G_{MN}\right]~,\\
&&\partial_M\left[\sqrt{-G}V^\prime(X) G^{MN}\right] = 0~,\label{Bianchi}\end{aligned}$$ where $$X\equiv u^{-1} + (D-1)e^{-2A}~.$$ Therefore, for the metric (\[ua\]) we have: $$\begin{aligned}
&&A^{\prime\prime} - {A^\prime u^\prime \over 2u} + (D-1)(A^\prime)^2 = \mu^2 u \left[V^\prime(X)e^{-2A} + {{V(X) - X~V^\prime(X)}\over {D-2}}\right],\\
&&(D-1)\left[A^{\prime\prime} + (A^\prime)^2 - {A^\prime u^\prime \over 2u}\right] = \mu^2 \left[V^\prime(X) + {{V(X) - X~V^\prime(X)}\over {D-2}}~u\right],\\
&&u = {1\over \nu^2} e^{2(D-1)A}~\left[V^\prime(X)\right]^2~,\end{aligned}$$ where $\nu$ is an integration constant and the last equation for $u$ follows from the Bianchi identity (\[Bianchi\]). Combining these equations, we have: $$\begin{aligned}
&&(D-1)(D-2){\nu^2\over\mu^2}(A^\prime)^2 = \nonumber\\
&&\,\,\, (D-1)V^\prime(X) e^{2(D-2)A} + \left[V(X) - X~V^\prime(X)\right] e^{2(D-1)A} - \nu^2 V^\prime(X)~.\end{aligned}$$ For the linear potential $V(X) = X - (D-2)$ we recover (\[q.lin\]). Furthermore, the second-order equation for $A$ reads $$A^{\prime\prime} = {\mu^2\over\nu^2}~\left[e^{2(D-2)A} - e^{2(D-1)A}\right]~,$$ which follows from the first-order equation (\[q.lin\]).
Non-perturbative Non-longitudinal Solutions
===========================================
In the previous section we saw that non-perturbatively in the case of the linear potential (non-Fierz-Pauli point) we have oscillating solutions corresponding to the longitudinal mode with the mass parameter $M_* = \sqrt{2}\mu$, same as the mass of the would-be perturbative ghost; however, non-perturbatively there is no ghost, the Hamiltonian is bounded from below, and the aforementioned non-perturbative oscillating solutions appear to be well-behaved. In the case of the square-root potential (Fierz-Pauli point) there appears to be no dynamics associated with the longitudinal mode. In this section we study full non-perturbative equations including non-longitudinal modes, [*i.e.,*]{} the modes other than $f$ (and $g$). Our goal is to study non-perturbative massive solutions. For massive solutions we can focus on field configurations with no spatial dependence. This can be thought about in two ways. In the rest frame of a massive object the spatial momenta vanish, so a solution to the (nonlinear) “wave" equation depends only on time.[^25] Alternatively, as in Subsection 4.2, we can compactify the spatial coordinates on a torus $T^{D-1}$ and disregard the Kaluza-Klein modes. This is a valid approximation for slow-moving massive objects.
We will therefore look for solutions of the following form, which depend only on time $t$: $$\begin{aligned}
&&G_{00} \equiv -u(t)~,\\
&&G_{ij} \equiv G_{ij}(t)~,\\
&&G_{i0} \equiv 0~.\end{aligned}$$ While $u(t) > 0$ in the metric can be absorbed into the definition of time via $d\tau = u^{1/2} dt$, we must keep it for the purpose of solving the equations of motion because diffeomorphisms are broken by the potential $V$ and $u$ contributes into $V$.
For the above field configurations we have: $$\begin{aligned}
\label{Rij}
&& R_{ij} = {1\over 2u}~G_{ij}^{\prime\prime} + {1\over 4u} \left(\phi - {u^\prime\over u}\right)~G_{ij}^\prime - {1\over 2u}~f_{ij}~,\\
&& R_{00} = -{1\over 2}~\phi^\prime + {u^\prime\over 4u}~\phi - {1\over 4}~G^{ij}~f_{ij}~,\\
&& R_{i0} = 0~,\end{aligned}$$ where $$\begin{aligned}
&&\phi \equiv G^{ij}~G_{ij}^\prime~,\\
&&f_{ij} \equiv G_{ik}^\prime~G_{jl}^\prime~G^{kl}~.\end{aligned}$$ The equations of motions read: $$\begin{aligned}
\label{R-V}
&&R_{MN} = \mu^2 \left[V^\prime(X)~\eta_{MN} + {{V(X) - X~V^\prime(X)}\over {D-2}}~G_{MN}\right]~,\\
&&\partial_M\left[\sqrt{-G}V^\prime(X) G^{MN}\right] = 0~,\label{Bianchi1}\end{aligned}$$ where $$X\equiv u^{-1} + G^{ij}\eta_{ij}~.$$ When $u\equiv 1$ and $G_{ij} = \eta_{ij}$ (the Minkowski background), we have $X = D$.
From the Bianchi identity (\[Bianchi1\]) we have $$\label{u}
u = {1\over \nu^2}\det(G_{ij})\left[V^\prime(X)\right]^2~,$$ where $\nu$ is an integration constant.
In the general case solving the above nonlinear equations analytically is challenging (albeit not impossible – see Subsection 7.3 below). For the linear potential $V(X) = X - (D-2)$ we have substantial simplifications. First, from (\[u\]) we have $$\begin{aligned}
&& u = {1\over \nu^2}\det(G_{ij})~,\\
&& u^\prime = \phi~u~,\end{aligned}$$ and (\[Rij\]) simplifies as follows: $$R_{ij} = {1\over 2 u}\left[G_{ij}^{\prime\prime} - f_{ij}\right]~.$$ So the equations of motion read: $$\begin{aligned}
&&G_{ij}^{\prime\prime} - f_{ij} = 2\mu^2 u\left[\eta_{ij} - G_{ij}\right]~,\\
&&\phi^\prime - {1\over 2}~\phi^2 + {1\over 2}~G^{ij} f_{ij} = 2\mu^2 [1 - u]~.\end{aligned}$$ From the $G_{ij}$ equation we also have $$\label{phiprime}
\phi^\prime = 2\mu^2 u\left[G^{ij}\eta_{ij} - (D - 1)\right]~.$$ We will now analyze the above equations.
Three Dimensions
----------------
In $D = 3$ we have the following simplification: $$\begin{aligned}
&&f_{ij} = \phi~G_{ij}^\prime - \xi~G_{ij}~,\\
&&\xi \equiv {\det(G_{ij}^\prime)\over\det(G_{ij})}~,\label{xi}\end{aligned}$$ so we have $$\begin{aligned}
\label{Gij}
&&G_{ij}^{\prime\prime} - \phi~G_{ij}^\prime + \xi~G_{ij} = 2\mu^2 u\left[\eta_{ij} - G_{ij}\right]~,\\
&&\phi^\prime - \xi = 2\mu^2 [1 - u]~.\end{aligned}$$ Using (\[phiprime\]) we also have $$\xi = 2\mu^2 \left[u(G^{ij}\eta_{ij} - 1) - 1\right]~.$$ Let $$\chi \equiv \eta^{ij}G_{ij}~.$$ If we assume that $\chi = 2$, then from (\[Gij\]) it follows that $\xi = 0$ and $\det(G_{ij}^\prime) = 0$. However, in $D = 3$ this would imply that $G_{ij} = \eta_{ij}$. Indeed, if $\chi = 2$, then we have $G_{ij} = \eta_{ij} + a~(\sigma_3)_{ij} + b~(\sigma_1)_{ij}$, where $\sigma_1$ and $\sigma_3$ are the Pauli matrices, and $\det(G_{ij}^\prime) = -(a^\prime)^2 - (b^\prime)^2$, so $\xi = 0$ implies that $a$ and $b$ are constant (and can be set to 1 by rescaling the spatial coordinates). This means that for the linear potential in $D=3$ nontrivial solutions always involve a mixture of the longitudinal and non-longitudinal modes.
To summarize, our equations of motion read: $$\begin{aligned}
&& G_{ij}^{\prime\prime} - \phi~G_{ij}^\prime + \xi~G_{ij} = 2\mu^2 u\left[\eta_{ij} - G_{ij}\right]~,\\
&& \xi = 2\mu^2 \left[u(G^{ij}\eta_{ij} - 1) - 1\right]~,\\
&& u = {1\over \nu^2}\det(G_{ij})~,\\
&& \phi = {u^\prime \over u}~.\end{aligned}$$ We have the following solution: $$\begin{aligned}
&&G_{11} = u~,\\
&&G_{22} = 1~,\\
&&G_{12} = 0~,\\
&&\nu = 1~,\end{aligned}$$ and $u$ is a solution to the following equation: $$\left({u^\prime \over u}\right)^\prime = 2\mu^2 [1 - u]~.$$ Let $$u \equiv e^\omega~.$$ Then we have $$\omega^{\prime\prime} = 2 \mu^2\left[1 - e^\omega\right]~,$$ and $$\label{omega}
(\omega^\prime)^2 + \Phi(\omega) = 4\mu^2\eta~,$$ where $\eta$ is an integration constant, $$\Phi(\omega)\equiv 4\mu^2\left[e^\omega - 1 - \omega\right]~,$$ and we have $\Phi(\omega)\geq 0$ and $\eta\geq 0$. For $\eta = 0$ we have a trivial solution $\omega \equiv 0$ (and $u \equiv 1$). For $\eta > 0$ we have solutions oscillating between $\omega_-$ and $\omega_+$, where $\omega_-$ and $\omega_+$ are the negative and positive roots of the equation $$\label{omegapm}
e^{\omega_\pm} - 1 - \omega_\pm = \eta~,$$ and the period of these oscillations is given by: $$\label{T}
T = {1\over \mu}~F(\eta)~,$$ where $$F(\eta) \equiv \int_{\omega_-}^{\omega_+} {d\omega\over\sqrt{\eta + 1 + \omega - e^\omega}}~.$$ For $\eta \ll 1$ we have $T\approx 2\pi/(\sqrt{2}\mu)$, which is consistent with fact that in this case perturbatively the mass of both the longitudinal (see Subsection 6.1) and of the non-longitudinal (see (\[zeta\])) modes is $\sqrt{2}\mu$. However, if $\eta$ is not small, then the oscillation period depends on the amplitude, which is controlled by $\eta$. Furthermore, non-perturbatively we have no ghost.
The above oscillating solutions are interesting because only one of the spatial directions expands and contracts, while the other does not. Such unisotropically oscillating solutions may have interesting applications in the context of cosmology.
Finally, let us note that the above solution can be rotated, and the following is also a solution to the equations of motion: $$\begin{aligned}
&&G_{11} = u~\cos^2(\alpha) + \sin^2(\alpha)~,\\
&&G_{22} = u~\sin^2(\alpha) + \cos^2(\alpha),\\
&&G_{12} = (u - 1)\cos(\alpha)\sin(\alpha)~,\\
&&\nu = 1~,\end{aligned}$$ where $u$ is the same as above and $\alpha$ is a constant.
“Cosmological Strings"
----------------------
Looking at the equations of motion for the linear potential for general $D$, it is evident that solutions with only one oscillating spatial direction exist in any $D$. Up to $SO(D-1)$ rotations, these solutions are given by: $$\begin{aligned}
&&G_{11} = u \equiv e^\omega~,\\
&&G_{ii} = 1,~~~i > 1\\
&&G_{ij} = 0~,~~~i\not=j\\
&&\nu = 1~,\end{aligned}$$ and the metric is given by $$\label{string}
ds^2 = e^{\omega(t)}~\left[-dt^2 + (dx^1)^2\right] + \sum_{i=2}^{D-1} (dx^i)^2~,$$ where $\omega$ is an oscillating solution to the equation (\[omega\]) with the period $T$ given by (\[T\]). In terms of the time coordinate $\tau$ defined via $$d\tau \equiv e^{\omega/2}~dt~,$$ we have the metric $$ds^2 = -d\tau^2 + e^\omega(dx^1)^2 + \sum_{i=2}^{D-1} (dx^i)^2~.$$ The oscillation period in terms of the time coordinate $\tau$ is given by $${\widetilde T} = {1\over \mu}~{\widetilde F}(\eta)~,$$ where $${\widetilde F}(\eta) \equiv \int_{\omega_-}^{\omega_+} {e^{\omega/2}d\omega\over\sqrt{\eta + 1 + \omega - e^\omega}}~,$$ and $\omega_\pm$ are defined in (\[omegapm\]).
These solutions are 2-dimensional cosmological defects – “cosmological strings" – in a $D$-dimensional space-time. Unlike static cosmic strings, cosmological strings are not static objects. The cosmological sting solutions we found here have the length scale (“warp" factor) in one spatial dimension oscillating in time, while the other $(D-2)$ spatial dimensions remain static and flat.
Cosmological strings arise in the gravitational Higgs mechanism because diffeomorphisms are spontaneously broken. Indeed, for the metric of the form (\[string\]) and, more generally, for any metric of the form $$ds^2 = \gamma_{\mu\nu}dx^\mu dx^\nu + \sum_{i=2}^{D-1} (dx^i)^2~,$$ where the metric $\gamma_{\mu\nu}$, $\mu,\nu = 0,1$ is independent of $x^i$, $i>1$, we have $$\begin{aligned}
&&R_{\mu\nu} = {1\over 2}\gamma_{\mu\nu} R~,\\
&&R_{ij} = 0~,~~~i,j>1~,\\
&&R_{\mu i} = 0,~~~i>1~,\end{aligned}$$ and the Einstein tensor $E_{MN}\equiv R_{MN} - {1\over 2}G_{MN} R$ is given by: $$\begin{aligned}
&&E_{\mu\nu} = 0~,\\
&&E_{ij} = -{1\over 2}\eta_{ij} R~,~~~i,j>1~,\\
&&E_{\mu i} = 0~,~~~i>1.\end{aligned}$$ So, to have such solutions, we must have non-vanishing $E_{ij}$ in the transverse directions $i,j>1$. This is precisely what transpires in the gravitational Higgs mechanism with the linear potential $V(X) = X - (D-2)$. In fact, it is not difficult to see that such solutions exist for no other potential $V(X)$ (albeit [*a priori*]{} this does not exclude more general cases – see footnote \[foot\] hereof).
Constant-volume Solutions
-------------------------
Above we discussed non-perturbative solutions for the linear potential. In this subsection we discuss purely non-longitudinal solutions, which we will also refer to as “traceless" solutions, for which $\eta^{ij}G_{ij} = D-1$, [*i.e.*]{}, $h_{ij}\equiv G_{ij} - \eta_{ij}$ is traceless: $h\equiv \eta^{ij} h_{ij} = 0$. For the general potential $V(X)$ we can achieve a simplification by working in $D=3$ and considering constant-volume solutions for which $$\label{detG}
\left[\det(G_{ij})\right]^\prime = 0~.$$ In this case the equations of motion simplify as follows: $$\begin{aligned}
&&G^{\prime\prime}_{ij} + \xi~G_{ij} = 2\mu^2~u~\left(V^\prime(X)~\eta_{ij} + \left[V(X) - X~V^\prime(X)\right]~G_{ij}\right)~,\\
&&\xi = -2\mu^2~\left(V^\prime(X) + \left[V(X) - X~V^\prime(X)\right]~u\right)~,\label{xi-eom}\end{aligned}$$ where the first equation follows from the $ij$ component of the equations of motion (\[R-V\]), while the second equation follows from the $00$ component, and $$\begin{aligned}
&&\xi\equiv {{\det(G^\prime_{ij})}\over{\det(G_{ij})}}~,\\
&&X\equiv u^{-1} + G^{ij}\eta_{ij}~.\end{aligned}$$ The $00$ component of the metric, $u\equiv -G_{00}$, is determined via (\[u\]), which follows from the Bianchi identity (\[Bianchi1\]): $$\label{u1}
u = {1\over \nu^2}\det(G_{ij})\left[V^\prime(X)\right]^2~,$$ where $\nu$ is an integration constant. Note that in $D=3$, when $\eta^{ij}h_{ij} = 0$, we have $$G^{ij} = \left[\eta^{ij} - h^{ij}\right]~\left[\det(G_{ij})\right]^{-1}~,$$ where $h^{ij}\equiv\eta^{ik}\eta^{jl}h_{kl}$ and $\eta_{ij}h^{ij} = 0$. This implies that $G^{ij}\eta_{ij} = 2\left[\det(G_{ij})\right]^{-1}$ is constant as we have (\[detG\]), so both $X$ and $u$ are also constant. Then from (\[xi-eom\]) it follows that $\xi$ is also constant. We therefore have: $$\begin{aligned}
&&h_{ij}^{\prime\prime} = - {\widetilde M}^2~h_{ij}~,\\
&&(1+u)~V^\prime(X) + 2u~\left[V(X) - X~V^\prime(X)\right] = 0~,\label{u-X}\\
&&\xi=-\mu^2(1-u)V^\prime(X)~,\\
&&X = u^{-1} + 2~\left[\det(G_{ij})\right]^{-1}~,\\
&&{\widetilde M}^2 \equiv 2\mu^2 u V^\prime(X)~.\end{aligned}$$ A solution is given by $$\begin{aligned}
&&h_{11} = -h_{22} = \rho~\cos({\widetilde M}(t-t_0))~,\\
&&h_{12} = \rho~\sin({\widetilde M}(t-t_0))~,\\
&&u = {{1-\rho^2}\over{1+\rho^2}}~,\end{aligned}$$ with $$\begin{aligned}
&& X = {{3+\rho^2}\over{1-\rho^2}}~,\\
&&\det(G_{ij}) = 1-\rho^2~,\\
&&\xi = -{\rho^2~\over{1-\rho^2}}~{\widetilde M}^2,\end{aligned}$$ and $\rho$ is subject to (\[u-X\]). Using $\rho^2 = (X-3)/(X+1)$ and $u=2/(X-1)$, (\[u-X\]) reduces to $$\label{X}
4V(X) - (3X - 1)V^\prime(X) = 0~.$$ Note that due to (\[cosm.const\]) this equation always has at least one solution, namely, $X=3$ (so $u=1$ and $\rho=0$), which is simply the Minkowski background. Also note that, in agreement with our results in Subsection 7.1, for the linear potential $V(X) = X - 1$, this is the only solution. However, for nonlinear potentials we can have other nontrivial solutions to (\[X\]). Such a solution $X = X_*$ must satisfy two conditions: $X_* > 3$ (so that $\rho^2$ is positive – in this case $u$ is also positive) and $V^\prime(X_*) > 0$ (so that ${\widetilde M}^2$ is positive).
We have such a solution already for a quadratic potential: $$\label{VX*}
V(X) = \lambda\left[X^2 + 2(X_* + 2)X - (2X_* + 1)\right]~.$$ For this potential (\[X\]) has two solutions: $X=3$ and $X=X_*$. We can choose $X_* > 3$. We also have $V^\prime(X_*) = 4\lambda(X_* + 1)$, which is positive for $\lambda > 0$. If we compute perturbative mass squared and $\zeta$ via (\[M\]) and (\[zeta\]), we get $M^2 = 2\mu^2\lambda (X_* + 3) > 0$ and $\zeta = 2(X_* + 5)/(X_* + 3)$, so perturbatively the traceless components have positive mass squared, and so does the trace component (as $1 < \zeta < 3$ – see footnote \[foot2\] hereof), albeit perturbatively the trace component is a propagating ghost (this is a non-Fierz-Pauli point), while non-perturbatively there is no ghost as the Hamiltonian (\[Hamiltonian\]) is evidently positive-definite in this case.
Let us rescale the time coordinate via $$d\tau^2 \equiv u~dt^2$$ Then the metric reads $$\begin{aligned}
&&ds^2 = -d\tau^2 + \left[1 + \rho~\cos(M_1\tau)\right](dx^1)^2 + \left[1 - \rho~\cos(M_1\tau)\right](dx^2)^2 + \nonumber\\
&&\,\,\,2\rho~\sin(M_1\tau)~(dx^1)(dx^2)~,\end{aligned}$$ where $M_1^2 \equiv {\widetilde M}^2 / u = 2\mu^2 V^\prime(X_*)$.
Finally, let us mention that in the above constant-volume traceless solutions the amplitude $\rho$ of the oscillations is determined by the higher-derivative couplings in the scalar sector. Indeed, $\rho$ is fixed by $X_*$, which in turn is fixed via the higher-derivative coupling once we normalize the kinetic term, [*i.e.*]{}, the term linear in $X$ in (\[VX\*\]), by setting $2\lambda(X_*+2) = 1$. Also note that $\rho$ is small when $X_* - 3$ is small.
An Aside: New Massive Gravity
=============================
In this section, as an illustrative aside, we discuss a simple application of the methods discussed in the previous section to New Massive Gravity [@BHT], which appears to confirm the results of [@dRGPTY] obtained via a different framework. For computational convenience, our discussion here will be for general dimension $D$ and a general higher derivative gravity action (without any scalars or breaking of the diffeomorphism invariance) subject to the restrictions on higher curvature terms that (\[sumk\]) does not contain higher derivative terms (see Section 5). We will then apply it to the particular case of New Massive Gravity.
For the field configurations of the form $$G^{MN} = {\rm diag}(g(t)~\eta^{00}, f(t)~\eta^{ii})~,$$ the dimensionally reduced action reads $$\begin{aligned}
\label{NMG}
S = -\kappa \int dt ~g^{-{1\over 2}} e^{-(D-1)q}~Q(gU^2)~,\end{aligned}$$ where $$U\equiv \partial_t q \equiv {1\over 2}\partial_t \ln(f)$$ and $Q(gU^2)$ is given by (\[sumk\]). The $g$ equation of motion reads: $$\label{g.eom.nmg}
2gU^2Q^\prime(gU^2) = Q(gU^2)~.$$ The Hamiltonian ${\cal H} = 0$, which is due to the fact that diffeomorphisms are unbroken. Due to (\[g.eom.nmg\]), the $q$ equation of motion reduces to $$\partial_t\left[g^{1\over 2}UQ^\prime(gU^2)\right] = 0~.$$ Let us introduce the time coordinate $\tau$ via $$d\tau = g^{-{1\over 2}} dt~.$$ The metric reads $$ds^2 = -d\tau^2 + e^{-2q}~\eta_{ij}dx^idx^j~.$$ We then have $$\left(\partial_\tau q\right)^2 = x_*~,$$ where $x_*$ is a solution to the equation $$\label{vacua}
2x_*Q(x_*) = Q(x_*)~,$$ and $$Q(x) \equiv \sum_{k=1}^\infty c_k x^k~.$$ So, we have a Minkowski solution as $x_* = 0$ is a solution to (\[vacua\]). Depending on $Q$, there might also exist other solutions. Consider the case of quadratic $Q$: $$Q(x) = c_1 x + c_2 x^2~.$$ Then we have an additional solution $$x_* = -{c_1\over 3c_2}~.$$ For the action $$S = M_P^{D-2}\int d^Dx \sqrt{-G}\left[\sigma R + \alpha R^2 + \beta R^{MN} R_{MN}\right]~,$$ where $$\begin{aligned}
&&\beta \equiv {1\over m^2}~,\\
&&\sigma = \pm 1~,\end{aligned}$$ we have (see Section 5) $$\begin{aligned}
&&\alpha = -{D\over 4(D-1)m^2}~,\\
&&c_1 = \sigma(D-1)(D-2)~,\\
&&c_2 = {(D-1)(D-2)^2(4-D)\over 12 m^2}~,\\
&&x_* = -{4\sigma m^2\over (4-D)(D-2)}~,\end{aligned}$$ where we are assuming $D\not =4$.[^26] For $\sigma=-1$ we have de Sitter background $q = H\tau$ with the Hubble parameter $H$ and the cosmological constant ${\widetilde \Lambda}$: $$\begin{aligned}
&&H^2 = {4m^2\over(4-D)(D-2)}~,\\
&&{\widetilde \Lambda} = (D-1)(D-2)H^2 = 4(D-1)m^2/(4-D)~.\end{aligned}$$ In $D=3$ this reproduces the known results in New Massive Gravity. Our point here is that, with the restrictions of Section 5, only first time-derivatives appear in action (\[NMG\]) and no trouble is expected with the longitudinal mode. This is evident in the language of $g$ and $f$, and appears to be much more nontrivial in the usual perturbative parametrization (see [@dRGPTY]). In particular, non-perturbatively there is no unusual dynamics associated with the longitudinal mode, which simply determines the choice of the background (Minkowski or de Sitter). Therefore, there appears to be no issue with unitarity in New Massive Gravity. What is less evident is whether there exist non-perturbative massive solutions for the non-longitudinal modes as they do in the gravitational Higgs mechanism (see Section 7). In this regard, it would be interesting to apply the methods discussed in Section 7 to New Massive Gravity and see if there exist non-perturbative massive solutions to the (highly nonlinear) equations of motions. We will do this in the remainder of this section.
No Cosmological Strings in New Massive Gravity
----------------------------------------------
In the gravitational Higgs mechanism, in the previous section, we found cosmological string solutions. Here we show that such solutions are absent in New Massive Gravity. As before, for computational convenience we will work in the general dimension $D$. We start with the action (\[NMG\]). The equations of motion read: $$\label{EK}
\sigma E_{MN} + K_{MN} = 0~,$$ where $$\begin{aligned}
&& E_{MN} \equiv R_{MN} - {1\over 2} G_{MN} R~,\label{E}\\
&&K_{MN} \equiv (2\alpha +\beta)(G_{MN}\nabla^2 - \nabla_M\nabla_N)R + \beta\nabla^2 E_{MN} + \nonumber\\
&&\,\,\,2\alpha R \left(R_{MN} - {1\over 4}G_{MN} R\right) + 2\beta\left(R_{MSNT} - {1\over 4}G_{MN}R_{ST}\right)R^{ST}~.\label{K}\end{aligned}$$ For any metric of the form $$ds^2 = \gamma_{\mu\nu}dx^\mu dx^\nu + \sum_{i=2}^{D-1} (dx^i)^2~,$$ where the metric $\gamma_{\mu\nu}$, $\mu,\nu = 0,1$ is independent of $x^i$, $i>1$, we have $$\begin{aligned}
&& E_{\mu\nu} = 0~,\\
&& E_{ij} = -{1\over 2}\eta_{ij}R~,~~~i,j > 1\\
&& K_{\mu\nu} = (2\alpha+\beta)\left[(\gamma_{\mu\nu}\nabla^2 - \nabla_\mu\nabla_\nu)R + {1\over 4}\gamma_{\mu\nu}R^2\right]~,\\
&& K_{ij} = \left[\left(2\alpha + {1\over 2}\beta\right)\nabla^2 R - {1\over 4}(2\alpha + \beta)R^2\right]\eta_{ij}~, ~~~i,j>1.\end{aligned}$$ The equations of motion (\[EK\]) then imply that $$\begin{aligned}
&&\nabla_\mu\nabla_\nu R = -{1\over 4}\gamma_{\mu\nu} R^2~,\\
&&(3\alpha + \beta)R^2 + \sigma R = 0~,\end{aligned}$$ which only have a trivial solution $R = 0$ assuming $\sigma \not=0$. For $\sigma = 0$ (no Einstein-Hilbert term) we can satisfy the second equation if the condition $3\alpha + \beta = 0$ is satisfied.[^27] However, these cannot be massive solutions. This can be seen by transforming the 2-dimensional metric $\gamma_{\mu\nu}$ into a conformally flat form. Alternatively, if we look for solutions that depend only on time $t$ (see the discussion at the beginning of Section 7), then we invariably have $R=0$. So, there are no cosmological strings in New Massive Gravity.[^28]
Analysis of Massive Solutions
-----------------------------
In this subsection we study full non-perturbative equations in New Massive Gravity in the context of non-perturbative massive solutions. The theory has full diffeomorphism invariance. Therefore, we can always set $G_{00} = 1$ and $G_{i0} = 0$. For the remaining metric components we look for solutions that depend only on time $t$. We then have $$\begin{aligned}
&&R_{00} = -{1\over 2} R + z~,\\
&&G^{ij}~R_{ij} = {1\over 2} R + z~,\label{RG}\\
&&R_{i0} = 0~.\end{aligned}$$ Using our formulas in Section 7, in $D=3$, which we focus on here, we have $$z = {\xi\over 4}~,$$ where $\xi$ is defined in (\[xi\]).
In $D=3$ we have the following identity: $$\label{RGR}
R_{ik}R_{jl}G^{kl} = R_{ij}\left(R_{kl}G^{kl}\right) - G_{ij}~{\det(R_{kl})\over\det(G_{kl})}~.$$ We are assuming that $\det(G_{kl})>0$. It then follows that $$\begin{aligned}
\label{RMNsq}
R_{MN}R^{MN} = {1\over 2}~R^2 + 2~z^2 - 2~{\det(R_{kl})\over\det(G_{kl})}~.\end{aligned}$$ On the other hand, the equation of motion (\[EK\]) in $D=3$ implies that $$\label{traceK}
R_{MN}R^{MN} - {3\over 8}~R^2 = -m^2~R~,$$ where we have set $\beta = 1/m^2$ and $\sigma=-1$. Furthermore, $$\begin{aligned}
\label{EOMEK}
&&E_{MN} = K_{MN}~,\\
&&m^2~K_{MN} = {1\over 4}\left(G_{MN}\nabla^2 - \nabla_M\nabla_N\right)R + \nabla^2 E_{MN} + \nonumber\\
&&\,\,\, {9\over 4}R_{MN}R - 4R_{MS}{R^S}_N -{3m^2\over 2}G_{MN} R - {1\over 4}G_{MN}R^2~.\end{aligned}$$ From (\[RMNsq\]) and (\[traceK\]) it follows that $$\label{R1}
R ={2\over m^2}\left[{\det(R_{kl})\over\det(G_{kl})} - z^2 - {1\over 16}~R^2\right]~.$$ Furthermore, from the $00$ component of the equation of motion (\[EOMEK\]), we also have: $$\label{R2}
R = {2\over 3m^2} \left[z^{\prime\prime} + \left(m^2 +{7\over 4}~R\right)~z - 4~z^2 - {1\over 8}~R^2\right]~,$$ where we have taken into account that $E_{00} = R_{00} + R/2 = z$ and that $\nabla_0^2 E_{00} = E_{00}^{\prime\prime}$. Finally, the $ij$ components of the equation of motion (\[EOMEK\]) read $$\label{Rij1}
-\nabla_0^2 R_{ij} + \left({1\over 4}R - 4z - m^2\right)R_{ij} + \left(m^2 R + 4z^2 + {1\over 4}R^{\prime\prime}\right)G_{ij} = 0~,$$ where we have used (\[RGR\]), (\[RG\]) and (\[R1\]). Let $${\overline R}_{ij} \equiv R_{ij} - {1\over 2}G_{ij}\left(G^{kl}R_{kl}\right) = R_{ij} - G_{ij}\left({R\over 4} + {z\over 2}\right)~.$$ Note that $G^{ij}{\overline R}_{ij} = 0$. Using (\[RG\]) and the fact that $\nabla_0^2 z = z^{\prime\prime}$, from (\[Rij1\]) we get $$\label{overlineR}
-\nabla_0^2 {\overline R}_{ij} + \left({1\over 4}R - 4z - m^2\right){\overline R}_{ij} = 0~.$$ Also, $$\det(R_{ij}) = \det({\overline R}_{ij}) + \left({R\over 4} + {z\over 2}\right)^2~\det(G_{ij})~,$$ so (\[R1\]) reads $$R = {2\over m^2}\left[{\det({\overline R}_{kl})\over\det(G_{kl})} - {3\over 4}~z^2 - {1\over 4}~R~z\right]~.$$ Note that $G_{ij} = e^{-2Ht}\eta_{ij}$ is a solution to (\[R1\]), (\[R2\]) and (\[Rij1\]) for $H^2=4m^2$. Indeed, in this case we have $R_{ij} = 2H^2 G_{ij}$, $R_{00} = -2H^2$, $z = H^2$ and $R=6H^2$. This is the de Sitter solution discussed above. However, what we are interested in here is finding massive solutions. Just as in the case of gravitational Higgs mechanism, we expect that if such solutions exist, they should exist for weak-field configurations.
### Linearized Approximation
In the linearized approximation $$G_{ij} \equiv \eta_{ij} + h_{ij}~,$$ and one keeps only the terms linear in $h_{ij}$ in the equations of motion. Thus, we have $$\begin{aligned}
&&R_{ij}^{(1)} = {1\over 2}~h_{ij}^{\prime\prime}~,\\
&&R_{00}^{(1)} = -{1\over 2}~h^{\prime\prime}~,\\
&&R_{i0}=0~,\\
&&R^{(1)} = h^{\prime\prime}~,\label{h}\end{aligned}$$ where $h\equiv \eta^{ij}h_{ij}$. Note that the leading term in $z$ is quadratic in $h_{ij}$, so $z^{(1)} = 0$. Both (\[R1\]) and (\[R2\]) then require that $$R^{(1)} = 0~,$$ and at this order the trace component vanishes: $h^{(1)} = 0$. We have $$\begin{aligned}
&&K_{MN}^{(1)} = -{1\over m^2}~\partial_t^2 E^{(1)}_{MN}~,\\
&&E^{(1)}_{MN} = R^{(1)}_{MN}~,\end{aligned}$$ so the equations of motion (\[EK\]) give $$\left[\partial_t^2 + m^2\right]~\partial_t^2 h_{ij}^{(1)} = 0~,$$ which are solved by solutions to the equations $$\label{hij1}
\left[\partial_t^2 + m^2\right]~h_{ij}^{(1)} = 0~,$$ which are massive oscillating solutions.
### Next-to-linear (Quadratic) Order
Let $h_{ij}^{(1)}$ be an arbitrary nontrivial solution of (\[hij1\]). We have $h^{(1)} = 0$ and $$\label{z1}
z^{(2)} = {1\over 4}~\det(\partial_t h^{(1)}_{ij})~.$$ Using (\[hij1\]) we then have $$\begin{aligned}
\label{z2}
&& z^{(2)} = C - {m^2\over 4}~\det(h_{ij}^{(1)})~,\\
&& \partial_t^2 z^{(2)} = {m^4\over 2}~\det(h_{ij}^{(1)}) - 2m^2~z^{(2)} = m^4~\det(h_{ij}^{(1)}) - 2m^2~C~.\end{aligned}$$ Here $C$ is an integration constant ($C^\prime \equiv 0$) – (\[z2\]) can be obtained by differentiating (\[z1\]) w.r.t. $t$ and using (\[hij1\]). Note that, unless $h^{(1)}_{ij} \equiv 0$, we have $\det(\partial_t h^{(1)}_{ij}) < 0$ and $\det(h^{(1)}_{ij}) < 0$ as $h^{(1)}=0$, which implies that $C < 0$. Conversely, if $C=0$, then $h^{(1)}_{ij} \equiv 0$.
Next, we have $$\det(R^{(1)}_{ij}) = {m^4\over 4}~\det(h^{(1)}_{ij})~.$$ At the quadratic order in $h_{ij}$ and its derivatives, (\[R1\]) and (\[R2\]) then give, respectively: $$\begin{aligned}
\label{R(2)}
&& R^{(2)} = {m^2\over 2}~\det(h^{(1)}_{ij})~,\\
&& R^{(2)} = {m^2\over 2}~\det(h^{(1)}_{ij}) - {2C\over 3}~,\end{aligned}$$ which implies that $C = 0$ and $h^{(1)}_{ij} \equiv 0$. Thus, as we see, the quadratic order equations of motion have no massive oscillating solutions around the Minkowski background.
### Non-perturbative Argument
The fact that perturbatively there are no massive solutions at the next-to-linear order indicates that either massive solutions do not exist or they cannot be treated perturbatively. Here we wish to see this without doing perturbative expansion. If there are massive solutions, we expect that they should be traceless, [*i.e.*]{}, in the decomposition $G_{ij} = \eta_{ij} + h_{ij}$, without assuming that $h_{ij}$ are small, we have $h\equiv\eta^{ij}h_{ij} = 0$. For such field configurations we have $\eta^{ij}G_{ij} = 2$ and $$\eta^{ij}{\overline R}_{ij} = 3z -{1\over 2}R~.$$ From (\[overlineR\]) we then have $$\label{R-second}
R^{\prime\prime} = 8m^2 R - 16Rz + R^2 + 48 z^2~,$$ where we have used (\[R2\]). Note that this equation is exact for all traceless field configurations with $\eta^{ij}G_{ij} = 2$. It is now clear that there are no massive solutions corresponding to weak-field configurations. Indeed, for such field configurations $|R|$ and $|z|$ are small compared with $m^2$, so the last three terms on the r.h.s. of (\[R-second\]) are small and can be neglected. However, the resulting equation $R^{\prime\prime} = 8m^2 R$ does not correspond to oscillating massive solutions; instead, it has exponentially growing solutions, which implies that the weak-field assumption is incorrect to begin with.[^29]
Acknowledgments {#acknowledgments .unnumbered}
---------------
I would like to thank Alberto Iglesias for reading the manuscript and valuable comments.
[99]{}
Z. Kakushadze and P. Langfelder, “Gravitational Higgs Mechanism”, Mod. Phys. Lett. A15 (2000) 2265, arXiv:hep-th/0011245.
G. ’t Hooft, “Unitarity in the Brout-Englert-Higgs Mechanism for Gravity”, arXiv:0708.3184.
Z. Kakushadze, “Gravitational Higgs Mechanism and Massive Gravity”, Int. J. Mod. Phys. A23 (2008) 1581, arXiv:0709.1673.
I. Oda, “Gravitational Higgs Mechanism with a Topological Term”, arXiv:0709.2419.
Z. Kakushadze, “Massive Gravity in Minkowski Space via Gravitational Higgs Mechanism”, Phys. Rev. D77 (2008) 024001, arXiv:0709.1673.
Z. Kakushadze, “Massless Limit of Gravitational Higgs Mechanism”, Int. J. Geom. Meth. Mod. Phys. 05 (2008) 157, arXiv:0711.0386.
M. Maeno and I. Oda, “Classical Solutions of Ghost Condensation Models", Mod. Phys. Lett. B22 (2009) 3025, arXiv:0801.0827; “Massive Gravity in Curved Cosmological Backgrounds”, Int. J. Mod. Phys. A24 (2009) 81, arXiv:0808.1394.
D.A. Demir and N.K. Pak, “General Tensor Lagrangian from Gravitational Higgs Mechanism”, Class. Quantum. Grav. 26 (2009) 105018, arXiv:0904.0089.
A.H. Chamseddine and V. Mukhanov, “Higgs for Graviton: Simple and Elegant Solution”, JHEP 1008 (2010) 011, arXiv:1002.3877.
I. Oda, “Higgs Mechanism for Gravitons”, arXiv:1003.1437; “Remarks on Higgs Mechanism for Gravitons”, arXiv:1004.3078.
J. Kluson, “Hamiltonian Analysis of the Higgs Mechanism for Graviton”, Class. Quant. Grav. 28 (2011) 155014, arXiv:1005.5458.
A. Iglesias and Z. Kakushadze, “Massive Gravity in de Sitter Space via Gravitational Higgs Mechanism”, Phys. Rev. D82 (2010) 124001, arXiv:1007.2385.
L. Alberte, A.H. Chamseddine and V. Mukhanov, “Massive Gravity: Resolving the Puzzles”, JHEP 1012 (2010) 023 , arXiv:1008.5132.
L. Berezhiani and M. Mirbabayi, “Unitarity Check in Gravitational Higgs Mechanism”, Phys. Rev. D83 (2011) 067701, arXiv:1010.3288.
L. Alberte, A.H. Chamseddine and V. Mukhanov, “Massive Gravity: Exorcising the Ghost”, JHEP 1104 (2011) 004, arXiv:1011.0183.
A. Iglesias and Z. Kakushadze, “Non-perturbative Unitarity of Gravitational Higgs Mechanism", Phys. Rev. D84 (2011) 084005, arXiv:1102.4991.
M.J. Duff, “Dynamical Breaking of General Covariance and Massive Spin-2 Mesons”, Phys. Rev. D12 (1975) 3969.
C. Omero and R. Percacci, “Generalized Nonlinear Sigma Models In Curved Space And Spontaneous Compactification”, Nucl. Phys. B165 (1980) 351.
M. Gell-Mann and B. Zwiebach, “Space-Time Compactification Due To Scalars”, Phys. Lett. B141 (1984) 333.
R. Percacci, “The Higgs Phenomenon in Quantum Gravity”, Nucl. Phys. B353 (1991) 271.
M.B. Green and C.B. Thorn, “Continuing between Closed and Open Strings”, Nucl. Phys. B367 (1991) 462.
W. Siegel, “Hidden Gravity in Open-String Field Theory”, Phys. Rev. D49 (1994) 4144, arXiv:hep-th/9312117.
M. Porrati, “Higgs Phenomenon for 4-D Gravity in Anti de Sitter Space”, JHEP 0204 (2002) 058, arXiv:hep-th/0112166.
N. Arkani-Hamed, H. Georgi and M.D. Schwartz, “Effective Field Theory for Massive Gravitons and Gravity in Theory Space”, Annals Phys. 305 (2003) 96, arXiv:hep-th/0210184.
A.H. Chamseddine, “Spontaneous Symmetry Breaking for Massive Spin-2 Interacting with Gravity”, Phys. Lett. B557 (2003) 247, arXiv:hep-th/0301014.
I.A. Bandos, J.A. de Azcarraga, J.M. Izquierdo and J. Lukierski, “Gravity, p-branes and a Space-time Counterpart of the Higgs Effect”, Phys. Rev. D68 (2003) 046004, arXiv:hep-th/0301255.
N. Arkani-Hamed, H.-C. Cheng, M.A. Luty and S. Mukohyama, “Ghost Condensation and a Consistent Infrared Modification of Gravity”, JHEP 0405 (2004) 074, arXiv:hep-th/0312099.
P. Creminelli, A. Nicolis, M. Papucci and E. Trincherini, “Ghosts in Massive Gravity”, JHEP 0509 (2005), arXiv:hep-th/0505147.
N. Arkani-Hamed, H.-C. Cheng, M.A. Luty, S. Mukohyama and T. Wiseman, “Dynamics of Gravity in a Higgs Phase”, JHEP 0701 (2007) 036, arXiv:hep-ph/0507120.
M. Leclerc, “The Higgs Sector of Gravitational Gauge Theories”, Annals Phys. 321 (2006) 708, arXiv:gr-qc/0502005.
I. Kirsch, “A Higgs Mechanism for Gravity”, Phys. Rev. D72 (2005) 024001, arXiv:hep-th/0503024.
E. Kiritsis, “Product CFTs, Gravitational Cloning, Massive Gravitons and the Space of Gravitational Duals”, JHEP 0611 (2006) 049, arXiv:hep-th/0608088.
Z. Berezhiani, D. Comelli, F. Nesti and L. Pilo, “Spontaneous Lorentz Breaking and Massive Gravity”, Phys. Rev. Lett. 131101 (2007) 99, arXiv:hep-th/0703264.
M.V. Bebronne and P.G. Tinyakov, “Massive Gravity and Structure Formation”, Phys. Rev. D76 (2007) 084011, arXiv:0705.1301.
R. Jackiw, “Lorentz Violation in Diffeomorphism Invariant Theory”, arXiv:0709.2348.
M.V. Bebronne and P.G. Tinyakov, “Black hole solutions in massive gravity”, JHEP 0904 (2009) 100, arXiv:0902.3899.
C. de Rham and G. Gabadadze, “Generalization of the Fierz-Pauli Action”, Phys. Rev. D82 (2010) 044020, arXiv:1007.0443.
S.F. Hassan, S. Hofmann and M. von Strauss, “Brane Induced Gravity, its Ghost and the Cosmological Constant Problem”, JCAP 1101 (2011) 020, arXiv:1007.1263.
C. de Rham, G. Gabadadze and A.J. Tolley, “Resummation of Massive Gravity”, Phys. Rev. Lett. 106 (2011) 231101, arXiv:1011.1232; “Comments on (super)luminality", arXiv:1107.0710; “Ghost free Massive Gravity in the Stückelberg language", arXiv:1107.3820; “Helicity Decomposition of Ghost-free Massive Gravity", arXiv:1108.4521.
S.F. Hassan and R.A. Rosen, “On Non-Linear Actions for Massive Gravity," JHEP 1107 (2011) 009, arXiv:1103.6055; “Resolving the Ghost Problem in non-Linear Massive Gravity," Phys. Rev. Lett. 108 (2012) 041101, arXiv:1106.3344.
A. Gruzinov, “All Fierz-Paulian massive gravity theories have ghosts or superluminal modes", arXiv:1106.3972.
A.H. Chamseddine and V. Mukhanov, “Massive Gravity Simplified: A Quadratic Action", JHEP 1108 (2011) 091, arXiv:1106.5868; “Hidden Ghost in Massive gravity", arXiv:1302.4367.
A.H. Chamseddine and M.S. Volkov, “Cosmological solutions with massive gravitons", Phys. Lett. B704 (2011) 652, arXiv:1107.5504.
G. D’Amico, C. de Rham, S. Dubovsky, G. Gabadadze, D. Pirtskhalava and A.J. Tolley, “Massive Cosmologies”, arXiv:1108.5231.
L. Berezhiani, G. Chkareuli, C. de Rham, G. Gabadadze and A. J. Tolley, “On Black Holes in Massive Gravity", arXiv:1111.3613.
A.E. Gumrukcuoglu, C. Lin and S. Mukohyama, “Cosmological perturbations of self-accelerating universe in nonlinear massive gravity", JCAP 03 (2012) 006, arXiv:1111.4107.
A. De Felice, A.E. Gumrukcuoglu and S. Mukohyama, “Massive gravity: nonlinear instability of the homogeneous and isotropic universe", Phys. Rev. Lett. 109 (2012) 171101, arXiv:1206.2080.
S. Deser and A. Waldron, “Acausality of Massive Gravity", Phys. Rev. Lett. 110 (2013) 111101, arXiv:1212.5835.
L. Berezhiani, G. Chkareuli and G. Gabadadze, “Restricted Galileons", arXiv:1302.0549.
K. Zhang, P. Wu and H. Yu, Phys. Rev. D87 (2013) 063513, arXiv:1302.6407.
A. De Felice, A.E. Gumrukcuoglu, C. Lin and S. Mukohyama, “Nonlinear stability of cosmological solutions in massive gravity", JCAP 1305 (2013) 0351, arXiv:1303.4154.
L. Berezhiani, G. Chkareuli, C. de Rham, G. Gabadadze and A.J. Tolley, “Mixed Galileons and Spherically Symmetric Solutions", arXiv:1305.0271.
S. Deser, K. Izumi, Y.C. Ong and A.Waldron, “Massive Gravity Acausality Redux", arXiv:1306.5457.
M. Fierz and W. Pauli, “On Relativistic Wave Equations for Particles of Arbitrary Spin in an Electromagnetic Field”, Proc. Roy. Soc. Lond. A173 (1939) 211.
D.G. Boulware and S. Deser, “Can gravitation have a finite range?", Phys. Rev. D6 (1972) 3368.
S. Deser and R. I. Nepomechie, “Gauge Invarinace versus Masslessness in de Sitter Spaces", Annals Phys. 154 (1984) 396.
A. Higuchi, “Forbidden Mass Range for Spin-2 Field Theory in de Sitter Spacetimes", Nucl. Phys. B282 (1987) 397.
S. Deser and A. Waldron, “Gauge Invariances and Phases of Massive Higher Spins in (A)dS", Phys. Rev. Lett. 87 (2001) 031601, arXiv:hep-th/0102166.
G. Gabadadze and A. Iglesias, “Special Massive Spin-2 on de Sitter Space", JCAP 0802 (2008) 014, arXiv:0801.2165.
G. Gabadadze, A. Iglesias and Y. Shang, “General Massive Spin-2 on de Sitter Background", arXiv:0809.2996.
G. Dvali and M. Shifman, “Surviving on the Slope: Supersymmetric Vacuum in the Theories Where It Is Not Supposed to Be”, Phys. Lett. B454 (1999) 277, arXiv:hep-th/9901111.
Z. Kakushadze, “Localized (Super)Gravity and Cosmological Constant”, Nucl. Phys. B589 (2000) 75, arXiv:hep-th/0005217.
A.G. Riess et al., “Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant", Astron. J. 116 (1998) 1009, arXiv:astro-ph/9805201.
S. Perlmutter et al., “Measurements of Omega and Lambda from 42 High-Redshift Supernovae", Astrophys. J. 517 (1999) 565, arXiv:astro-ph/9812133.
E.A. Bergshoeff, O. Hohm and P.K. Townsend, “Massive Gravity in Three Dimensions", Phys. Rev. Lett. 102 (2009) 201301, arXiv:0901.1766; “More on Massive 3D Gravity", Phys. Rev. D79 (2009) 124042, arXiv:0905.1259.
S. Deser, H. Liu, H. Lu, C.N. Pope, T. Cagri Sisman, B. Tekin, “Critical Points of D-Dimensional Extended Gravities", Phys. Rev. D83 (2011) 061502, arXiv:1101.4009.
C. de Rham, G. Gabadadze, D. Pirtskhalava, A.J. Tolley and I. Yavin, “Nonlinear Dynamics of 3D Massive Gravity", arXiv:1103.1351.
[^1]: Email: zura@quantigic.com
[^2]: DISCLAIMER: This address is used by the corresponding author for no purpose other than to indicate his professional affiliation as is customary in scientific publications. In particular, the contents of this paper are limited to Theoretical Physics, have no commercial or other such value, are not intended as an investment, legal, tax or any other such advice, and in no way represent views of Quantigic$^\circledR$ Solutions LLC, the website or any of their other affiliates.
[^3]: For earlier and subsequent related works, see, [*e.g.*]{}, [@Duff; @OP; @GMZ; @Perc; @GT; @Siegel; @Por; @AGS; @Ch; @Ban; @AH1; @CNPT; @AH2; @Lec; @Kir; @Kiritsis; @Ber; @Tin; @Jackiw; @Tin1; @RG; @HHS; @RG1; @Gruzinov; @HR; @Cham4; @Cham5; @MC; @BH; @Muk1; @Muk2; @DW1; @Gal1; @ZWY; @Muk3; @Gal2; @DW2], and references therein.
[^4]: For recent related works, see, [*e.g.*]{}, [@AGS; @AH1; @CNPT; @AH2; @RG; @RG1; @HR; @Cham4; @Gal1; @Gal2], and references therein.
[^5]: In [@Gal2] it is argued that in the massive gravity models of [@RG; @RG1] perturbation theory breaks down already at a very low scale.
[^6]: The same conclusion was reached in [@dS] in the de Sitter case. For prior works on massive gravity in the de Sitter space, see, [*e.g.*]{}, [@DN; @Higuchi; @DW; @GI; @GIS], and references therein.
[^7]: In [@JK] the full Hamiltonian analysis also was performed for another model with non-polynomial “potential" – see subsection \[sq.rt.subsection\] hereof for details. The full Hamiltonian analysis is technically challenging in general case. However, the Hamiltonian analysis of [@Unitarity] for the relevant metric degrees of freedom is tractable in other cases with nonlinear potentials, [*e.g.*]{}, the quadratic potential – see subsection \[quad.subsection\] hereof for details.
[^8]: However, neither [@Unitarity] nor this paper attempt to address the question of whether there is any superluminal propagation of signals or the related issue of (a)causality. In this regard, we emphasize that the results of [@Gruzinov; @DW1; @DW2] are [*intrinsically perturbative*]{} and do not appear to apply to the full non-perturbative definition of the gravitational Higgs mechanism. To see if there is any superluminal propagation of signals in the full non-perturbative theory, it appears that one might have to develop some new non-perturbative methods, which is clearly beyond the scope of this paper (and off the cuff it is not even evident what such non-perturbative methods would entail). The non-Fierz-Pauli model of [@thooft] is the “least non-perturbative" and might provide a fruitful testing ground in this context. We also note that the issue of superluminal propagation does not appear to be relevant in the context of the application of the gravitational Higgs mechanism to a string theory description of QCD – which was one of the primary motivations in ’t Hooft’s paper [@thooft] – if QCD is to be described by string theory, all known consistent versions of which contain massless gravity, then the graviton should presumably somehow acquire mass, and the gravitational Higgs mechanism is one way of approaching this problem.
[^9]: Perturbatively, the traceless components of the graviton are positive-definite with mass $M$, while the trace component is a ghost with the same mass. Non-perturbatively, there is no ghost.
[^10]: In (\[H-intro\]) $q$ is a canonical variable for the longitudinal mode, $\kappa$ is a constant, $t$ is the time coordinate (related to but not the same as $\tau$ in (\[iso\])) w.r.t. which the conjugate momentum for $q$ is defined, and the ellipses stand for higher order terms in $q$ and $\partial_t q$ (see $\S\S$ 4.2 and 6.1 for details).
[^11]: The explicit form of $\omega$ is given in Subsections 7.1 and 7.2.
[^12]: As was first discussed in [@ZK1], higher-derivative terms are required to obtain the Fierz-Pauli term in the perturbative expansion.
[^13]: See Section 2 for the discussion of more general actions that allow for the Fierz-Pauli mass term in the linearized gravity.
[^14]: It would also be interesting to construct black hole solutions in this model.
[^15]: Because of the tachyon problem, it is difficult to speculate about embedding the gravitational Higgs mechanism into bosonic string theory. However, such embedding appears to be both straightforward and natural in the context of supersymmetric string theory, where typically there is an abundance of scalars with flat directions (and no non-derivative couplings). (In this context, the role of the time-like scalar is played by a pseudoscalar dual to a $p$-form with $p=D-2$.) In supersymmetric string theory there appears to exist no evident obstruction to the aforesaid scalars acquiring coordinate-dependent vacuum expectation values thereby spontaneously breaking the diffeomorphism invariance and resulting in the gravitational Higgs mechanism. Such a string background would, however, appear to be non-perturbative (strongly coupled, and, in fact, time-dependent), which bodes well with the apparent non-perturbative nature of the gravitational Higgs mechanism. In this regard, let us note a difference between the gravitational Higgs mechanism and its gauge theory counterpart. In the latter scalar vacuum expectation values are constant, while in the former they depend linearly on space-time coordinates. In fact, the background is not even static. It would take infinite energy to destabilize such a background, which should therefore be stable. This is reminiscent to infinite-tension domain walls discussed in [@DS; @ZK3].
[^16]: \[foot\]One can consider a more general setup with the scalar action constructed not just from $Y$, but from $Y_{MN}$, $G_{MN}$ and $\epsilon_{M_0\dots M_{D-1}}$, see, [*e.g.*]{}, [@ZK1; @Demir; @Cham1; @Oda2]. However, a simple action containing a scalar function $V(Y)$ suffices to capture all qualitative features of the gravitational Higgs mechanism.
[^17]: In some cases with higher curvature terms transverse-traceless graviton components are not necessarily positive-definite. In such cases our discussion here assumes that no higher-curvature terms are present.
[^18]: However, as was discussed in detail in [@dS; @Unitarity], this naive intrinsically perturbative parametrization is inadequate, among other things, for addressing the issue of unitarity.
[^19]: Indeed, negative norm states cannot arise from purely space-like components or spatial derivatives, and are due to time-like components and/or time derivatives.
[^20]: Here we omit the source term as it does not affect the unitarity analysis.
[^21]: In [@thooft] it was observed that perturbatively the theory was non-unitary.
[^22]: The Hamiltonian analysis of [@JK] for this potential apparently assumes $b>0$. However, it appears that it can also be performed for $b<0$.
[^23]: \[foot2\]For general $\zeta\not=1$, perturbatively the mass of the propagating trace component $h$ is given by $M_*^2 = \zeta(D-\zeta)\mu^2/(D-2)(\zeta - 1)$ – see, [*e.g.*]{}, [@ZK1] for details.
[^24]: Perhaps inclusion of higher-curvature terms could smooth out the singularities. Note that this solution passes through the epoch of vanishing (small) curvature.
[^25]: [*E.g.*]{}, in the case of massive Klein-Gordon equation $\left(\partial^M\partial_M - m^2\right)\phi = 0$, in the rest frame the solution is given by $\phi = a~\cos(Et) + b~\sin(Et)$, where the energy $E = m$.
[^26]: For $D=4$ we have $c_2 = 0$ and there is only $x_* = 0$ solution.
[^27]: This condition is not satisfied in New Massive Gravity in $D=3$. This condition is the same as (\[alphabeta\]) in $D=4$, in which case we simply have Weyl gravity up to the Gauss-Bonnet combination, which is a total derivative in $D=4$.
[^28]: In the gravitational Higgs mechanism we found cosmological string solutions for the linear potential.
[^29]: We can show directly that there are no constant-volume traceless solutions. Constant-volume solutions are defined as those with $\left[\det(G_{ij})\right]^\prime \equiv 0$. For such solutions we have $R = -2z$. Then (\[R2\]) and (\[R-second\]) imply that $z = 0$ or $z = 6m^2/25$. However, for traceless configurations $z < 0$ unless $h_{ij}\equiv 0$. This shows that there are no constant-volume traceless solutions.
|
---
abstract: 'We propose and study an approach to realize quantum switch for single-photon transport in a coupled superconducting transmission line resonator (TLR) array with one controllable hopping interaction. We find that the single-photon with arbitrary wavevector can transport in a controllable way in this system. We also study how to realize controllable hopping interaction between two TLRs via a Cooper pair box (CPB). When the frequency of the CPB is largely detuned from those of the two TLRs, the variables of the CPB can be adiabatically eliminated and thus a controllable interaction between two TLRs can be obtained.'
author:
- 'Jie-Qiao Liao'
- 'Jin-Feng Huang'
- 'Yu-xi Liu'
- 'Le-Man Kuang'
- 'C. P. Sun'
title: 'Quantum switch for single-photon transport in a coupled superconducting transmission line resonator array'
---
Coupled cavity arrays (CCAs) [@CCAreview] have recently attracted considerable attentions of both theorists and experimentalists. The CCAs have been proposed to implement quantum simulators for many-body physics, such as discovering new matter phases of photons [@HBP06; @GTCH06; @RF07] and providing a new platform to study spin systems [@ASB07; @HBP07a]. The CCAs are also suggested to manipulate photons for optical quantum information processing [@HLSS08; @BAB07; @ASYE07]. Moreover, photon transport in the CCAs has been investigated [@ZLS07; @ZGLSN08; @HZSS07; @ZGSS08; @GIZS08]. There are several possible ways to construct the CCAs, for example: (i) coupled defect cavities in photonic crystals [@Vuckovic]; (ii) coupled toroidal microresonators [@AKSV03]; and (iii) coupled superconducting transmission line resonators (TLRs) [@ZGLSN08; @HZSS07].
In CCAs, there have been many proposals to realize quantum switch [@sun; @Switch], which is used to control single-photon transport [@ZGLSN08; @Lukin-np; @Fanpaper1; @Fanpaper2]. For example, the reflection and transmission of photons in a coupled resonator waveguide can be controlled by a tunable two-level quantum system [@ZGLSN08; @Switch], acting as a controller.
Here, we study another approach to control the single-photon transport in a CCA, which consists of a chain of TLRs [@Wal04; @Blais04]. In our proposal, the controllable transport is realized by a tunable coupling. As we know, how to control coupling between two solid devices is a major challenge in scalable quantum computing circuits [@you; @you1; @liu; @miro; @yingdan; @yingdan2; @Hu]. To obtain a tunable coupling, we propose that a Cooper pair box (CPB) acts as a coupler. When the frequency of the coupler is largely detuned from those of the two resonators, the variables of the coupler can be adiabatically eliminated and thus a controllable interaction can be induced. Compared with previous approach [@ZGLSN08], this approach has following advantage: dynamical variables of the coupler are adiabatically eliminated, therefore the coupler is a passive controlling element, which makes robust to prevent from the environment of the coupler.
As shown in Fig. \[configuration\], one-dimensional CCA is a chain of $N$ cavities, each is only coupled to its nearest-neighbor ones, Fig. \[configuration\](a) and (b) are the site lattice model and the schematic diagram of coupled TLR array, respectively. The TLRs are assumed to have the same frequency. We also assume that the coupling strength between two nearest-neighbor TLRs is the same, except one between the $l$-th and $(l+1)$-th TLRs. The Hamiltonian of the system is $$\begin{aligned}
\label{Hamiltonian}H&=&\omega\sum_{n}a_{n}^{\dag}a_{n}-t\sum_{n}(a_{n}^{\dag}a_{n+1}+a_{n+1}^{\dag}a_{n})\nonumber\\
&&-\lambda t(a_{l}^{\dag}a_{l+1}+a_{l+1}^{\dag}a_{l}),\end{aligned}$$ hereafter we take $\hbar=1$. Here, we assume that all TLRs have the same frequency $\omega$. $a^{\dag}_{n}$ and $a_{n}$ are the creation and annihilation operators of $n$-th TLR; $t$ is the coupling strength between the $n$-th ($n\neq l$) and $(n+1)$-th TLRs; $\lambda=(t'-t)/t$ is introduced to denote the relation between $t$ and $t'$, where $t'$ is the coupling strength between the $l$- and $(l+1)$- TLRs. Obviously, $-1<\lambda<0$ corresponds to $0<t'<t$, while $\lambda\geq 0$ implies $t'\geq t$. Below we will first study how to control the single-photon transport by changing coupling strength $t'$, and then answer question how to realize controllable coupling $t'$.
In the case of $t'=t$, the Hamiltonian in Eq. (\[Hamiltonian\]) is reduced to the usual bosonic tight binding model $H_{\textrm{btb}}=\omega\sum_{n}a_{n}^{\dag}a_{n}-t\sum_{n}(a_{n}^{\dag}a_{n+1}+a_{n+1}^{\dag}a_{n})$ as shown in Ref. [@Data], which describes an $N$-site lattice model with nearest-neighbor coupling. It is well known that, under the periodic boundary condition, the bosonic tight binding Hamiltonian can be diagonalized as $H_{\textrm{btb}}=\sum_{k}\Omega_{k}a_{k}^{\dag}a_{k}$ by using the Fourier transformation $a_{k}=\sum_{n}\exp(iknd_{0})a_{n}/\sqrt{N}$, where $d_{0}$ is the site distance. Below, $d_{0}$ is taken as units. We choose the wavevectors $k=2\pi m/N$ within the first Brillouin zone, i.e., $-N/2<m\leq N/2$. The corresponding dispersion relation is $\Omega_{k}=\omega-2t\cos k$, which is an energy band structure. For $t>0$, the wavevectors $k=\pm\pi/2$ correspond to the energy band center, while the wavevectors $k=0$ and $k=\pm\pi$ correspond to the bottom and top of the energy band, respectively.
![(Color online) Schematic configuration for controllable transport of single photon: (a) one-dimensional site lattice model for the coupled cavity array; (b) schematic diagram of coupled superconducting transmission line resonator array.[]{data-label="configuration"}](configuration.eps){width="8"}
Let us now define a total excitation number operator $\hat{N}=\sum_{n}a_{n}^{\dag}a_{n}$. It is straightforward to show that $\hat{N}$ commutes with the model Hamiltonian (\[Hamiltonian\]), i.e., $[\hat{N},H]=0$, which implies that the total excitation number $\hat{N}$ is a conserved observable. We now restrict our discussion to the single excitation subspace since we only consider the single-photon transport. In this case, a general state can be written as $|\Omega\rangle=\sum_{n}A_{n}|1_{n}\rangle$, where we have introduced the basis state $|1_{n}\rangle=|0\rangle\otimes...\otimes|1\rangle_{n}\otimes...\otimes|0\rangle$, which represents the state that the $n$- TLR has one photon while other TLRs have no photon. $A_{n}$ is the probability amplitude of the state $|1_{n}\rangle$. Using the discrete scattering method proposed in Ref. [@ZGLSN08] and according to the eigenequation $H|\Omega\rangle=\Omega|\Omega\rangle$, we have
$$\begin{aligned}
\label{eq:2a}-t(A_{n+1}+A_{n-1})&=(\Omega-\omega)A_{n},\hspace{0.3 cm} n\neq \{l,l+1\},\\
\label{eq:2b}-t'A_{l+1}-tA_{l-1}&=(\Omega-\omega)A_{l},\\
\label{eq:2c}-A_{l+2}-t'A_{l}&=(\Omega-\omega)A_{l+1}.\end{aligned}$$
For the coherent transport of a single-photon with the energy $\Omega=\omega-2t\cos k$, we can assume the following forms for the probability amplitudes
$$\begin{aligned}
\label{eq:3a}A_{n}&=e^{ikn}+re^{-ikn},\hspace{0.3 cm}(n\leq l),\\
\label{eq:3b}A_{n}&=se^{ikn},\hspace{0.3 cm}(n\geq l+1).\end{aligned}$$
Here $r$ and $s$ are the reflection and transmission amplitudes, respectively. Obviously, Eqs. (\[eq:3a\]) and (\[eq:3b\]) are the solutions of Eq. (\[eq:2a\]). Substituting Eqs. (\[eq:3a\]) and (\[eq:3b\]) into Eqs. (\[eq:2b\]) and (\[eq:2c\]), we can obtain the transmission coefficient $$\begin{aligned}
\label{Tcoefficient}T(\lambda,k)=\frac{4(\lambda+1)^{2}\sin^{2}k}{\lambda^{2}(\lambda+2)^{2}+4(\lambda+1)^{2}\sin^{2}k},\end{aligned}$$ and the reflection coefficient $R(\lambda,k)=|s|^{2}=1-T(\lambda,k)$. Eq. (\[Tcoefficient\]) shows that the reflection and transmission coefficients $R(\lambda,k)$ and $T(\lambda,k)$ are function of the parameter $\lambda$ and the wavevector $k$ of the incident photon, and they are independent of other variables, e.g., the site position parameter $l$, the cavity frequency $\omega$, and the coupling constant $t$.
![(Color online) The transmission coefficient $T$ versus the parameter $\lambda$ for different wavevectors k=0.01, $\pi/8$, $\pi/4$, and $\pi/2$ is plotted.[]{data-label="Transmission"}](Transmission.eps){width="7" height="5"}
Equation (\[Tcoefficient\]) shows two symmetry relations $T(\lambda,k)=T(\lambda,-k)$ and $T(\lambda,\pi/2-k)=T(\lambda,\pi/2+k)$. Therefore we need only to analyze the transmission coefficient within the region $0\leq
k\leq\pi/2$. In this region, there are four special cases: (1) $T(\lambda\neq0,0)=0$, when the wavevector $k=0$, for $\lambda\neq0$, the input single photon is reflected completely; (2) $T(\lambda=-1,k)=0$, when $\lambda=-1$, the coupling between the $l$- and $(l+1)$- cavities is switched off, so for any value of the wavevector $k$, the transmission coefficient is zero; (3) $T(\lambda\rightarrow\infty,k)=0$, when $\lambda\rightarrow\infty$, namely, $t'\gg t$, the transmission coefficient is zero for any $k$. Physically, when $t'\gg t$, the Hamiltonian (\[Hamiltonian\]) is approximated to $H(t'\gg
t)\approx -t'(a^{\dagger}_{l}a_{l+1}+a^{\dagger}_{l+1}a_{l})$. The input photon will stay in the $l$- and $(l+1)$- cavities once it arrives the $l$- cavity; (4) $T(\lambda=0,k)=1$, $\lambda=0$ implies $t'=t$, the present model reduces to the usual bosonic tight binding model, so the photon with any wavevector can be perfectly transported.
To observe the effect on the transmission coefficient $T$ for general wavevector $k$ and parameter $\lambda$, in Fig. \[Transmission\], the transmission coefficient $T$ is plotted as a function of the parameter $\lambda$ for wavevectors $k=0.01$, $\pi/8$, $\pi/4$, and $\pi/2$. Fig. \[Transmission\] indicates that there are two regions, $-1\leq\lambda\leq0$ and $0\leq\lambda$, in which controllable transport of single photon can be achieved. The transmission coefficient $T$ can be tuned from $0$ to $1$ by changing the coupling strength $t'$, namely $\lambda$. When $t'=0$, the transmission coefficient $T=0$. With the increase of the coupling strength $t'\rightarrow t$, the transmission coefficient $T$ gradually approaches to 1. For $t'\geq t$, the transmission coefficient $T$ approaches to $0$ with the increase of the coupling $t'\rightarrow\infty$. In this region, the larger wavevector $k$ corresponds to the larger parameter range of $\lambda$. In both regions, the controllable transport of single-photon with arbitrary wavevector $k$ can be realized. Therefore, our approach for single-photon transport can cover complete bandwidth.
Let us now focus the problem on how to realize controllable coupling between two TLRs [@Switch; @miro]. The system we considered is shown in Fig. \[TwoTLRs\]. Two TLRs are coupled to a CPB through capacitors $C_{l}$ and $C_{r}$, respectively. We assume that the two TLRs are identical, that is, they have the same length $d$ and capacitance $C_{0}$ (inductance $L_{0}$) per unit length. We consider only single-modes of the two TLRs in near resonant with the CPB. The free Hamiltonian of the two TLRs is $$\begin{aligned}
\label{TLRhamiltonian}H_{\textrm{TLR}}=\omega
a_{l}^{\dag}a_{l}+\omega a_{r}^{\dag}a_{r},\end{aligned}$$ where $a^{\dagger}_{l}$ ($a^{\dagger}_{r}$) and $a_{l}$ ($a_{r}$) are the creation and annihilation operators of the resonant modes with frequency $\omega$ for the left (right) TLR, respectively.
![(Color online) Schematic diagram for two TLRs (the left and the right ones), which are coupled to a CPB through two capacitors $C_{l}$ and $C_{r}$, respectively. The CPB is biased by a magnetic flux $\Phi_{x}$[]{data-label="TwoTLRs"}](TwoTLRs.eps){width="7"}
The CPB is a superconducting loop interrupted by two identical Josephson junctions with the capacitance $C_{J}$ and the Josephson energy $E^{(0)}_{J}$. To obtain a tunable Josephson coupling energy, an external magnetic flux $\Phi_{x}$ is applied through the superconducting loop. The Hamiltonian of the CPB is $$\begin{aligned}
H'_{\textrm{CPB}}&=&E_{C}\,n^{2}-E_{J}(\Phi_{x})\cos\varphi,\label{CPBhamiltonian}\end{aligned}$$ where $n$ is the number operator of Cooper-pair charges on the island connected to the CPB, and $\varphi$ is the superconducting phase difference across the Josephson junction. The charging energy $E_{C}$ and effective Josephson energy $E_{J}(\Phi_{x})$ of the CPB are $E_{C}=2e^{2}/(C_{l}+C_{r}+2C_{J})$ and $E_{J}(\Phi_{x})=2E^{(0)}_{J}\cos(\pi\Phi_{x}/\Phi_{0})$, respectively. Here, we assume that the charging energy and the effective Josephson energy satisfy the condition $E_{J}(\Phi_{x})\gg
E_{C}$. Under this condition, the spectrum of the lowest energy levels of the CPB can be described approximately by a harmonic oscillator [@yingdan2]. That is, we expand $E_{J}(\Phi_{x})\cos\varphi$ around $\varphi=0$ up to $\mathcal{O}(\varphi^{2})$, and then Eq. (\[CPBhamiltonian\]) becomes $$\begin{aligned}
\label{e6} H_{\textrm{CPB}}=\omega_{b}b^{\dag}b, \hspace{0.3 cm}
\omega_{b}=\sqrt{2E_{C}E_{J}(\Phi_{x})}. \label{eq:6}\end{aligned}$$ The annihilation and creation operators $b$ and $b^{\dag}$ in Eq. (\[eq:6\]) are defined in terms of $\varphi=\sqrt[4]{E_{C}/(2E_{J}(\Phi_{x}))}(b+b^{\dag})$ and $n=-i\sqrt[4]{E_{J}(\Phi_{x})/(8E_{C})}(b-b^{\dag})$.
We assume that the linear dimension of the CPB is much smaller than wavelengths of the TLRs, and choose the position of the CPB at the origin of the axis. Then the quantized voltages at the left and right TLRs are $$\begin{aligned}
V_{j}(0)=-i\sqrt{\frac{\omega}{dC_{0}}}(a_{j}-a^{\dag}_{j}),
\hspace{0.3 cm} j=l,r.\end{aligned}$$ According to circuit theory, we know that the voltage at the island is $\Phi_{0}\dot{\varphi}/(2\pi)$. Therefore, the Coulomb interaction induced by the two capacitors $C_{l}$ and $C_{r}$ is $$\begin{aligned}
\label{Coupling}
H_{I}=\sum_{j=l,r}\frac{C_{j}}{2}\left(V_{j}(0)-\frac{\Phi_{0}}{2\pi}\dot{\varphi}\right)^{2}.\end{aligned}$$ In fact, capacitors $C_{l}$ and $C_{r}$ induce a direct Coulomb interaction between the two TLRs with the strength $\propto
C_{l}C_{r}$. However, this direct interaction is much smaller than the interaction between the two TLRs and the CPB given by Eq. (\[Coupling\]) with strengths $\propto C_{\Sigma l}C_{r}$ and $\propto C_{\Sigma r}C_{l}$ under the condition $\{C_{\Sigma
l},C_{\Sigma r}\}>>\{C_{l},C_{r}\}$, where $C_{\Sigma
l}=C_{0}d/2+C_{l}$ and $C_{\Sigma r}=C_{0}d/2+C_{r}$ are the sum capacitors connected to the left and right TLRs, respectively [@Li]. For instance, using current experimental parameters [@Frunzio] $C_{0}d/2\sim 1.6$ pF and $C_{l}=C_{r}=6$ fF, we find that the interaction between the TLRs and the CPB is larger than the direct interaction between two TLRs by three orders of magnitude.
Using Eqs. (\[TLRhamiltonian\]-\[Coupling\]), the total Hamiltonian of the system described in Fig. \[TwoTLRs\] is $$\begin{aligned}
\label{e11} H&=&\omega_{l} a_{l}^{\dag}a_{l}+\omega_{r}
a_{r}^{\dag}a_{r}+\omega'_{b}b^{\dag}b\nonumber\\
&&+g_{l}(a_{l}b^{\dag}+ba_{l}^{\dag})+g_{r}(a_{r}b^{\dag}+ba_{r}^{\dag}),\end{aligned}$$ where we have introduced the renormalized frequencies
$$\begin{aligned}
\label{e12a}\omega_{j}&=\omega\left(1+\frac{C_{j}}{dC_{0}}\right),\hspace{1 cm} j=l,r,\\
\label{e12b}\omega'_{b}&=\omega_{b}+(C_{l}+C_{r})\omega_{b}^{2}\left(\frac{\Phi_{0}}{2\pi}\right)^{2}
\left(\frac{E_{C}}{2E_{J}(\Phi_{x})}\right)^{\frac{1}{2}},\end{aligned}$$
and the coupling strengths $$\begin{aligned}
\label{e13}
g_{j}&=-C_{j}\omega_{b}\frac{\Phi_{0}}{2\pi}\sqrt{\frac{\omega}{dC_{0}}}
\left(\frac{E_{C}}{2E_{J}(\Phi_{x})}\right)^{\frac{1}{4}},\hspace{0.3cm}
j=l,r.\end{aligned}$$ It should be noted that we have made the rotation wave approximation when Eq. (\[e11\]) is derived.
Equation (\[e11\]) describes that two TLRs are coupled to the CPB, which serves as a coupler. To obtain controllable coupling between the two TLRs, we restrict the system in the large detuning regime, where the frequency differences between the two TLRs and the CPB are much larger than their coupling constants, i.e., $\Delta_{l}\gg
g_{l}$ and $\Delta_{r}\gg g_{r}$. Here, $\Delta_{j}=\omega'_{b}-\omega_{j}$ for $j=l,r$ are the dutuning between the frequencies of the TLRs and that of the CPB. By adiabatically eliminating the degree of freedom of the CPB, we obtain an effective interaction between the two TLRs. That is, we perform a unitary transform $U=\exp[g_{l}(a_{l}b^{\dag}-ba_{l}^{\dag})/\Delta_{l}
+g_{r}(a_{r}b^{\dag}-ba_{r}^{\dag})/\Delta_{r}]$ for the Hamiltonian in Eq. (\[e11\]) and use the Hausdorff expansion up to the first order in the small parameter $g_{j}/\Delta_{j}$ with $j=l,\,r$, then we obtain an effective Hamiltonian $$\begin{aligned}
\label{e15} H_{\textrm{eff}}&=&\omega'_{l}
a_{l}^{\dag}a_{l}+\omega'_{r}
a_{r}^{\dag}a_{r}+g(a_{r}a_{l}^{\dag}+a_{l}a_{r}^{\dag}),\label{eq:15}\end{aligned}$$ where we have defined the Stark-shifted frequencies $\omega'_{j}=\omega_{j}+g^{2}_{j}/\Delta_{j}$ for $j=l,r$, and the effective coupling strength $$\begin{aligned}
\label{e17}
g=\frac{g_{l}g_{r}(\Delta_{l}+\Delta_{r})}{2\Delta_{l}\Delta_{r}}.\end{aligned}$$ Note that the effective Hamiltonian of the CPB $H_{\textrm{CPB}}=\omega''_{b}b^{\dag}b$ with $\omega''_{b}=\omega'_{b}-g^{2}_{l}/\Delta_{l}-g^{2}_{r}/\Delta_{r}$ has been neglected in Eq. (\[eq:15\]). It is obvious that the Hamiltonian (\[e15\]) describes an effective interaction between the two TLRs. According to Eqs. (\[eq:6\]) and (\[e12b\]), the frequency of the CPB can be tuned by the external magnetic flux $\Phi_{x}$. Correspondingly, the detunings $\Delta_{l}$ and $\Delta_{r}$ between the TLRs and the CPB can be tuned, thus the coupling constant $g$ can be tuned. When the detunings are very larger than the coupling constants between the TLRs and the CPB, the effective coupling constant $g$ between the two TLRs are negligibly small, and then the interaction between the two TLRs is switched off. For example, if we assume that the two transmission line resonators are identical, i.e., $\Delta_{l}=\Delta_{r}=\Delta$ and $g_{l}=g_{r}=g'$, and we take the parameters [@Frunzio]: $\omega=2\pi\times3$ GHz, $C_{l}=C_{r}=6$ fF, $C_{0}d=1.6$ pF, $E_{C}=2\pi\times0.35$ GHz, $E^{(0)}_{J}\sim 10^{3}E_{C}$, then we calculate $g\approx1.1\sim23$ MHz corresponding to $\cos(\pi\Phi_{x}/\Phi_{0})\approx0.02\sim1$. In this region, the conditions $E_{J}(\Phi_{x})\gg E_{C}$ and $\Delta\gg g'$ are satisfied.
In conclusion, we have studied a quantum switch for single-photon transport in a coupled TLR array with one controllable hopping interaction. We have found that the controllable single-photon transport, for an arbitrary wavevector of photons, in the coupled TLR array can be realized by tuning one of the coupling constants. How to realize the controllable coupling between two TLRs is also studied. We have proposed that a CPB serves as a coupler to connect the two TLRs. In the regime of $E_{J}(\Phi_{x})\gg E_{C}$, the CPB is approximately described as a harmonic oscillator. Under the large detuning condition, we have obtained an effective interaction between the TLRs by adiabatically eliminating the variables of the CPB. This induced effective coupling can be controlled by the external magnetic flux $\Phi_{x}$ through the CPB.
This work is supported by the NSFC with Grant Nos. 10474104, 60433050, 10704023, and 10775048; NFRPC Nos. 2006CB921205, 2005CB724508, and 2007CB925204.
[99]{} M. J. Hartmann, F. G. S. L. Brand$\tilde{a}$o, and M. B. Plenio, Laser and Photon Rev. **2**, No. 6, 527 (2008) and references therein.
M. J. Hartmann, F. G. S. L. Brand$\tilde{a}$o, and M. B. Plenio, Nat. Phys. **2**, 849 (2006).
A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, Nat. Phys. **2**, 856 (2006).
D. Rossini and R. Fazio, Phys. Rev. Lett. **99**, 186401 (2007).
D. G. Angelakis, M. F. Santos, and S. Bose, Phys. Rev. A **76**, R031805 (2007).
M. J. Hartmann, F. G. S. L. Brand$\tilde{a}$o, and M. B. Plenio, Phys. Rev. Lett. **99**, 160501 (2007).
M. X. Huo, Y. Li, Z. Song, and C. P. Sun, Phys. Rev. A [**77**]{}, 022103 (2008).
S. Bose, D. G. Angelakis, and D. Burgarth, J. Mod. Opt. [**54**]{}, 2307 (2007).
D. G. Angelakis, M. F. Santos, V. Yannopapas, and A. Ekert, Phys. Lett. A. [**362**]{}, 377 (2007).
L. Zhou, J. Lu, and C. P. Sun, Phys. Rev. A [**76**]{}, 012313 (2007).
L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, Phys. Rev. Lett. [**101**]{}, 100501 (2008).
F. M. Hu, L. Zhou, T. Shi, and C. P. Sun, Phys. Rev. A [**76**]{}, 013819 (2007).
L. Zhou, Y. B. Gao, Z. Song, and C. P. Sun, Phys. Rev. A [**77**]{}, 013831 (2008).
Z. R. Gong, H. Ian, L. Zhou, and C. P. Sun, Phys. Rev. A [**78**]{}, 053806 (2008).
H. Altug and J. Vuckovic, Appl. Phys. Lett. **84**, 161 (2004).
D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, Nature [**421**]{}, 925 (2003).
C. P. Sun, L. F. Wei, Y. X. Liu, and F. Nori, Phys. Rev. A **73**, 022318 (2006).
M. Mariantoni, F. Deppe, A. Marx, R. Gross, F. K. Wilhelm, and E. Solano, Phys. Rev. B **78**, 104508 (2008).
D. E. Chang, A. S. S[ø]{}rensen, E. A. Demler, and M. D. Lukin, Nat. Phys. **3**, 807 (2007).
J. T. Shen and S. Fan, Phys. Rev. Lett. **95**, 213001 (2005).
J. T. Shen and S. Fan, Phys. Rev. Lett. **98**, 153003 (2007).
A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature **431**, 162 (2004).
A. Blais, R. S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A **69**, 062320 (2004).
J. Q. You and Franco Nori, Phys. Rev. B **68**, 064509 (2003).
J. Q. You, J. S. Tsai, and F. Nori, Phys. Rev. Lett. **89**, 197902 (2002).
Y. X. Liu, L. F. Wei, J. S. Tsai, and F. Nori, Phys. Rev. Lett. **96**, 067003 (2006).
M. Grajcar, Y. X. Liu, F. Nori, and A. M. Zagoskin, Phys. Rev. B **74**, 172505 (2006).
Y. D. Wang, Z. D. Wang, and C. P. Sun, Phys. Rev. B **72**, 172507 (2005).
Y. D. Wang, P. Zhang, D. L. Zhou, and C. P. Sun, Phys. Rev. B **70**, 224515 (2004).
Y. Hu, Y. F. Xiao, Z. W. Zhou, and G. C. Guo, Phys. Rev. A [**75**]{}, 012314 (2007).
S. Data, *Quantum Transport: Atom to Transistor* (Cambridge Univ. Press, Cambridge, 2005).
J. Li, K. Chalapat, and G.S. Paraoanu, Phys. Rev. B **78**, 064503 (2008).
L. Frunzio, A. Wallraff, D. I. Schuster, J. Majer, and R. J. Schoelkopf IEEE Trans. on App. Superconductivity **15**, 860 (2005).
|
---
abstract: 'The longitudinal resistivity $\rho_{xx}$ of two-dimensional electron gases formed in wells with two subbands displays ringlike structures when plotted in a density–magnetic-field diagram, due to the crossings of spin-split Landau levels (LLs) from distinct subbands. Using spin density functional theory and linear response, we investigate the shape and spin polarization of these structures as a function of temperature and magnetic-field tilt angle. We find that (i) some of the rings “break” at sufficiently low temperatures due to a quantum Hall ferromagnetic phase transition, thus exhibiting a high degree of spin polarization ($\sim 50 $%) within, consistent with the NMR data of Zhang *et al.* \[Phys. Rev. Lett. [**98**]{}, 246802 (2007)\], and (ii) for increasing tilting angles the interplay between the anticrossings due to inter-LL couplings and the exchange-correlation (XC) effects leads to a collapse of the rings at some critical angle $\theta_c$, in agreement with the data of Guo *et al.* \[Phys. Rev. B [**78**]{}, 233305 (2008)\].'
author:
- 'Gerson J. Ferreira'
- 'Henrique J. P. Freire'
- 'J. Carlos Egues'
title: 'Many-body effects on the $\rho_{xx}$ ringlike structures in two-subband wells'
---
The fascinating quantum Hall regime hosts a number of fundamental physical phenomena, being also relevant for metrology (standard for resistance) and as an alternate means to precisely determine the fine structure constant. The spectrum of two-dimensional electron gases (2DEGs) in the quantum Hall regime is quantized into highly degenerate Landau levels (LLs) [@Chakraboty95]. At opposite-spin LL crossings near the Fermi level, a ferromagnetic instability of the 2DEG may arise thus leading to a quantum Hall ferromagnetic phase. This spontaneous spin polarization of the electrons lowers the repulsive Coulomb energy of the Fermi sea, because electrons with parallel spins avoid each other due the Pauli exclusion principle. Even for vanishingly small Zeeman splittings the exchange-energy gain can stabilize quantum Hall ferromagnetism at low enough temperatures [@Quinn85; @GirvinBook1997; @Jungwirth00].
Quantum Hall ferromagnetism has been extensively studied in the quantum Hall regime via magnetotransport measurements [@qhf-exp]. Near LL crossings in tilted magnetic fields $B$, the longitudinal resistivity $\rho_{xx}$ vs $B$ of wells with a singly occupied subband exhibits ubiquitous hysteretic spikes, which signals quantum Hall ferromagnetism [@spikes; @jungwirthprl; @Freire07PRL]. In two-subband wells, spin-split LLs from distinct subbands cross even without a tilted $B$ field and can form closed loops \[ABCD loop, Fig. \[fig1\](a)\]. Quite generally, $\rho_{xx}$ is directly related to the energy spectrum near the Fermi level (linear response) and hence the topology in Fig. 1(a) translates into ringlike structures [@Muraki01; @Zhang05] in $\rho_{xx}$ when plotted in a density–$B$-field diagram $n_{\text{2D}}-B$, Fig. \[fig1\](b). Recently Zhang *et al.* [@Zhang06_PRB] have shown that near opposite-spin LL crossings the rings “break” at low enough temperatures ($70 {\,\mbox{mK}}$). NMR measurements [@Zhang06_PRL] near the broken edge C show a high degree of spin polarization, which points to a ferromagnetic instability of the 2DEG. For tilted $B$ fields some of rings “shrink”, fully collapsing for angles above a critical value [@ZhangSU4-07].
![(a) Two-subband GaAs well and schematic fan diagram with LL crossings from distinct subbands. The ABCD loop gives rise to ringlike structures in the calculated $n_{\text{2D}}$–$B$ diagram of $\rho_{xx}$ (b) for $\nu=4$. This ring “breaks” at lower temperatures (c) due to quantum Hall ferromagnetic transitions, thus displaying a high spin polarization within (d) [@Zhang06_PRL]. For increasing $B$ field tilt angles $\theta$, the ring shrinks (i.e., A and C move closer) and fully collapses (A=C) at $\theta_c$ (e), in agreement with the data [@ZhangSU4-07] (cf. empty and solid circles).[]{data-label="fig1"}](figura1){width="7.8cm"}
Here we use spin density functional theory (SDFT) [@dft] together with a linear response model [@ando-uemu-gerh] to investigate the shape and spin polarization of the ringlike structures in realistic quantum wells with two subbands at various filling factors $\nu$, as a function of the temperature and tilt angle $\theta$ of the $B$ field. We find that exchange-correlation (XC) effects are crucial to *quantitatively* describe the experiments: in particular (i) the $\nu=4$ ring breakup at low temperatures \[Fig. \[fig1\](c)\] follows from quantum Hall ferromagnetic phase transitions [@Zhang05; @Zhang06_PRB], Fig. 2. The calculated spin polarization \[Fig. \[fig1\](d)\] within the broken ring reaches $50\%$, consistent with NMR data [@Zhang06_PRL]. (ii) The shrinkage of the $\nu=4$ ring for increasing $\theta$ and its full collapse at $\theta = \theta_c$ \[Fig. \[fig1\](e)\] [@ZhangSU4-07] arise from the interplay between the anticrossings due to the inter-LL couplings and the exchange field, Fig. 3. We note that only rings formed from consecutive LLs, for which inter-LL coupling is operative, collapse for increasing $\theta$.
The quantum phase transitions we find here are not specific to the $\nu=4$ ring. They are general and should also occur for $\nu=6$ and others, but for distinct ranges of parameters. Other 2DEG systems, e.g., formed in Mn-based wells [@mn-wells], can also show peculiar ring structures.
*System.* We consider the structures of Zhang *et al.* [@Zhang06_PRL; @Zhang05; @Zhang06_PRB; @ZhangSU4-07]: a wide $240{\,\mbox{\AA}}$ GaAs square quantum well with Al$_{0.3}$Ga$_{0.7}$As barriers and symmetric $\delta$-doping (Si) with $240{\,\mbox{\AA}}$ spacers \[Fig. \[fig1\](a)\]. The electron density in the well is controlled by a gate voltage, as in an ideal capacitor model [@Yamaguchi06]. At zero bias $n_{\text{2D}} = 8.1 \times 10^{11} {\,\mbox{cm}}^{-2}$. The electron mobility $\mu_e = 4.1 \times 10^5
{\,\mbox{cm}}^2/{\,\mbox{Vs}}$ is assumed constant for the entire $B$ field and gate voltage ranges.
*Kohn-Sham problem.* The Kohn-Sham implementation of density-functional theory maps the problem of fully interacting electrons onto a non-interacting Schroedinger equation – the KS equation – with electrons in an effective single-particle potential [@dft]. For magnetic fields $B$ tilted $\theta$ with the 2DEG normal, this reads $$(H_{\parallel} + H_{z}^{\sigma_z} + \delta H_{\theta})\psi = \varepsilon\psi,
\label{eq1}$$ with $$\begin{aligned}
H_{\parallel} &=& \dfrac{P_x^2}{2m}+\dfrac{1}{2}m\omega_c^2(x-x_0)^2, \label{eq2} \\
H_{z}^{\sigma_z} &=& \dfrac{P_z^2}{2m}+\dfrac{1}{2}m\omega_p^2 z^2+ \dfrac{1}{2}g_e\mu_B \sigma_z B+v_{eff}^{\sigma_z}(z), \label{eq3} \\
\delta H_{\theta} &=& \omega_p z P_x, \label{eq4}\end{aligned}$$ where $m$ ($0.067m_0$) is the effective mass, $g_e$ ($-0.44$) the bulk g-factor, $P_{x,y,z}$ the $x,y,z$ components of the electron momentum operator, $\omega_c = eB\cos\theta/m$ the cyclotron frequency, $\sigma_z=\pm$ (or $\uparrow,\downarrow$), $\omega_p = eB\sin\theta/m$, $x_0 = - \ell_0^2 P_y/\hbar$, $\ell_0^2 = \hbar/eB\cos\theta$ the magnetic length and $$v_{eff}^{\sigma_z}(z) = v_c(z) + v_H(z; [n]) +
v_{xc}^{\sigma_z}(z; [n_{\uparrow},n_{\downarrow}]), \label{eq5}$$ $v_c(z)$ is the structural well potential. The Hartree potential $v_H(z; [n])$ is obtained self-consistently from Poisson’s equation. For the XC potential $v_{xc}^{\sigma_z}(z; [n_{\uparrow},n_{\downarrow}])$, we use the PW92 parametrization [@PW92] of the local-spin-density approximation (LSDA) [@AXC]. Here we have approximated the electron density $n(x,y,z)$ by its average over the $xy$ plane $n(z)=n_\uparrow(z)+n_\downarrow(z)$ [@Freire07PRL]. This renders both the Hartree and the XC potentials dependent upon only $z$.
*Perpendicular $B$ field*. For $\theta=0^\circ$, $\omega_p = 0 \Rightarrow\delta H_{\theta}=0$, the KS equation is separable in the $xy$ and $z$ variables and has eigenfunctions $\psi_{i,n,k_{y}}^{\sigma_{z}}(x,y,z) = \frac{1}{\sqrt{L_y}} \exp(\mathrm{i}k_{y}y) \varphi_{n}(x) \chi_{i}^{\sigma_{z}}(z)$ (Landau gauge), with $\varphi_{n}(x)$ being the *n*-th harmonic-oscillator eigenfunction centered at $x_{0}=-\hbar k_{y}/m\omega_{c}$ and $k_{y}$ the electron wave number along the $y$ axis; $L_{y}$ is a normalizing length. The KS eigenenergies are $\varepsilon_{i,n}^{\sigma_z}=\varepsilon_{n} +\varepsilon_i^{\sigma_z}+g_e\mu_B \sigma_z B/2$ (“Landau fan diagram”), with $\varepsilon_{n} = \left(n+1/2\right)\hbar\omega_c$, $n = 0, 1,...$ the LL energies (degeneracy $n_B=eB/h$) and $\varepsilon_i^{\sigma_z}$ the quantized levels obeying $H_z^{\sigma_z}\chi_i^{\sigma_z}=\varepsilon_i^{\sigma_z}\chi_i^{\sigma_z}$, $i=0,1,...$. with a self-consistently calculated chemical potential $\mu$.
*Tilted $B$ field.* For $\theta > 0^\circ$ the KS equation is not separable because $\delta H_{\theta} \sim \sin\theta zP_x \neq 0$. However, since $\delta H_{\theta} \ll H_0 = H_{\parallel}+H_{z}^{\sigma_z}$ and only couples consecutive LLs from distinct subbands, we can obtain the KS solutions $\tilde {\psi}(x,y,z;\theta)$ as an expansion in terms of the eigenfunctions $\phi_{i,n,k_{y}}^{\sigma_{z}}(x,y,z;\theta)$ of $H_0$. We perform this expansion *at every iteration* of our self-consistent scheme. We obtain good results by truncating the expansion for energies greater than $\mu + k_BT$. This LL coupling leads to anticrossings of the KS energies for equal-spin LLs, which ultimately make the ring shrink for tilted fields, Fig. 3.
*Linear-response $\rho_{xx}$*. By *assuming* that the KS eigenvalues $\varepsilon_{i,n}^{\sigma_z}$ represent the eigenenergies of the actual (Fermi-liquid) quasi-particles in our 2DEG, we use them in a Kubo-type formula [@ando-uemu-gerh] to calculate the conductivity tensor $\sigma$. For instance, within the self-consistent Born-approximation with short-range scatterers [@ando-uemu-gerh] $\sigma_{xx} = \frac{e^2}{\hbar \pi^2} \int\limits_{-\infty}^{\infty}
\left( -\frac{\partial f(\varepsilon)}{\partial \varepsilon} \right)
\notag \sum_{i,n,\sigma_z} \left(n+\frac{1}{2}\right)\exp \left[ -
\left( \frac{\varepsilon-\varepsilon_{i,n}^{\sigma_z}}
{\Gamma_{\mathrm{ext}}} \right)^2 \right] \mathrm{d} \varepsilon$, $\Gamma_{ext}$ denotes the width of the extended-state region within the broadened density of states and $f(\varepsilon)$ the Fermi function. We obtain the resistivity from $\rho=\sigma^{-1}$.
*Spin-polarized rings.* Figures \[fig1\](b) and \[fig1\](c) show our calculated $n_{\text{2D}}-B$ diagram of $\rho_{xx}$ for two different temperatures $T = 340{\,\mbox{mK}}$ and $T = 70{\,\mbox{mK}}$, respectively, near the $\nu=4$ ring. Similarly to the experiment of Ref. [@Zhang06_PRB], we find that the $\nu=4$ ring “breaks” at the opposite spin LL crossings (points *A* and *C*) at lower temperatures [@intensity], Fig. \[fig1\](c). Figure \[fig1\](d) shows the corresponding $n_{\text{2D}}-B$ diagram of the spin-polarization $\xi=(n_{\text{2D}}^\uparrow - n_{\text{2D}}^\downarrow)/n_{\text{2D}}$, $n_{\text{2D}}=n_{\text{2D}}^\uparrow + n_{\text{2D}}^\downarrow$. Interestingly, we find a high spin polarization ($\sim 50$ %) within the $\nu=4$ ring. This high polarization and, more importantly, its abrupt variation at the opposite-spin crossings *A* and *C* signal quantum Hall ferromagnetic phase transitions. For $T = 340{\,\mbox{mK}}$ the spin polarization of the $\nu=4$ ring (not shown), though high, varies smoothly at the opposite spin crossings. We note that the high spin polarization $\xi$ within the ring points to quantum Hall ferromagnetism, being also consistent with resistively-detected NMR data available [@Zhang06_PRL]; however, we contend that the high $\xi$ *and* the discontinuities of $\rho_{xx}$ at the crossings A and C \[Fig. 1(c)\] constitute the signature for the quantum Hall ferromagnetic instability.
![(a) Longitudinal resistivity $\rho_{xx}$ calculated for $n_{\text{2D}} = 7.3 \times 10^{-11}{\,\mbox{cm}}^{-2}$ \[see horizontal dashed line in Figs. 1(b)-1(c)\] for $T = 70 {\,\mbox{mK}}$ (dotted line) and $T = 340 {\,\mbox{mK}}$ (solid line, slightly shifted upwards for clarity). Note the distinctive spike in the higher temperature $\rho_{xx}$ at $B \approx 7.6 {\,\mbox{T}}$. (b) and (c) show the corresponding Landau fan diagrams for low and high temperatures, respectively. Note that for $T = 70 {\,\mbox{mK}}$, the LLs show discontinuities near $B = 7.6 {\,\mbox{T}}$. These discontinuities suppress the $\rho_{xx}$ spike at $B \approx 7.6{\,\mbox{T}}$ because the corresponding levels do not actually cross the chemical potential $\mu$ – they suddenly jump over it. At higher temperatures the Landau diagram is continuous and the spike appears in $\rho_{xx}$.[]{data-label="fig2"}](landaulevels2){width="8cm"}
The contrast between the low and high temperature results is more clearly seen in Fig. \[fig2\], which shows $\rho_{xx}$ for $n_{\text{2D}} = 7.3 \times 10^{11}{\,\mbox{cm}}^{-2}$ at $340{\,\mbox{mK}}$ and $70{\,\mbox{mK}}$. The spike near $B = 7.6{\,\mbox{T}}$ \[see arrow in Fig. \[fig2\](a)\] comes from the left edge of the $\nu = 4$ ring (point *A* in Fig. \[fig1\]) and is suppressed at $T = 70{\,\mbox{mK}}$. The Landau fan diagrams for both temperatures differ substantially only around this region \[see arrows in Figs. \[fig2\](b)-(c)\]. At $T = 70{\,\mbox{mK}}$ the diagram shows an abrupt transition and the chemical potential $\mu$ jumps to the spin-down state of the lower subband, thus suppressing the $\rho_{xx}$ spike. Note also the exchange enhancement of the spin splittings in Fig. 2(b)-(c) when $\mu$ lies essentially between the spin-split LLs.
A relevant parameter in our simulations is the LL broadening $\Gamma$. For short-range scatterers, the electron mobility $\mu_e$ and the LL broadening are related by $\Gamma= \Gamma_0 \sqrt{B/\mu_e}$ [@Ando83], with $\Gamma_0 = (2/\pi)^{1/2}\hbar e/m$. We use $\Gamma_{70} = 0.130\sqrt{B}{\,\mbox{meV}}$ and $\Gamma_{340} = 0.150\sqrt{B}{\,\mbox{meV}}$ to simulate the ring structures at $T = 70{\,\mbox{mK}}$ and $340 {\,\mbox{mK}}$, respectively \[see Figs. \[fig1\](b)-(c)\] [@broadening]. Note that the temperature-dependent $\Gamma_0$ differs from the one determined from the zero voltage $\mu_e$, $\Gamma = 0.210\sqrt{B} {\,\mbox{meV}}$. A strong dependence of $\mu_e$ on the gate voltage (or density) is reported in [@EllenbergerPHD] for parabolic two-subband wells, which also show ringlike structures [@Ellenberger06; @EllenbergerPHD]. Hence treating $\Gamma_0$ as a fitting parameter is somewhat justifiable here.
*Ring shrinking and collapse.* In the experiment of Ref. [@ZhangSU4-07] the authors show the projection of the $\rho_{xx}$ side crossings \[points A & C in Fig. \[fig1\]\] onto the $B_{\bot} = B\cos{\theta}$ axis as a function of $\theta$. In Fig. \[fig1\](e) we compare the experimental data (black dots) with the results of a non-interacting model (solid line) and our SDFT + Kubo approach (circles). The solid line is obtained from a non-interacting model (discussed in detail in Ref. [@non-int-model]) with effective parameters that fit the $\theta = 0^\circ$ data. This simple model illustrates qualitatively the effects of the inter-LL coupling on the ring collapse, Fig. \[fig3\](a)-(b): the anticrossings \[near D & B\] between LLs with the same spin increase with $\theta$, thus making the opposite spin LL crossings \[points A & C\] move closer in energy, effectively shrinking the ring. Note, however, that the collapsing angle for this non-interacting model ($3.46^{\circ}$) is about half of the experimental value $\theta_c^{exp} \approx 6^{\circ}$ [@ZhangSU4-07].
Our many-body calculation, on the other hand, agrees well with the data \[empty circles, Fig. \[fig1\](e)\] [@footnote-error]. Here, as the tilt angle increases, a competition sets in between the inter-LL coupling and the exchange-enhanced spin splittings of the LLs. While the inter-LL coupling tends to shrink the rings \[anticrossings, Figs. 3(a)-(b)\], the enhanced spin splittings make the rings larger \[e.g., Figs. 2(b)-(c) with no inter-LL coupling (i.e., $\theta=0^\circ$)\]. This interplay “delays” the ring collapse, Figs. 3(c)-(d); here we find $\theta_c^{theory} \approx 6^\circ$ – in agreement with [@ZhangSU4-07].
![Non-interacting (a)-(b) and interacting (c)-(d) Landau fan diagrams near the $\nu = 4$ ring for several tilt angles, showing the anti-crossings (B & D) due to the inter-LL coupling, Eq. (\[eq4\]). For increasing $\theta$ the anti-crossings near D & B become larger thus making A & C move closer and the ring shrink. The non-interacting model [@non-int-model] gives a qualitative picture of the ring collapse, while our many-body calculation (c)-(d) ($T = 70 {\,\mbox{mK}}$) provides a quantitative description and shows the relevance of competing XC effects, see Fig. 1(e).[]{data-label="fig3"}](zPx2){width="8"}
*Some remarks.* Experimentally [@Zhang06_PRB] all crossings of the $\nu = 4$ ring are broken at low temperature \[points A, B, C & D\]. In our approach the quantum Hall phase transitions occur only at opposite-spin LL crossings \[points A & C\]. Interestingly, we note that the inter-LL coupling makes the ring break near the same-spin LL crossings even for very small angles \[see D & B in Figs. 3(c)-(d)\]. In Ref. [@non-int-model] we have also obtained the $n_{\text{2D}}-B$ map of $\rho_{xx}$ for a non-interacting model and find ring breakups near D & B. However, the actual ring breakups near D & B at $\theta=0^\circ$ could also be related to the derivative discontinuity of the XC functionals [@Leeor2008], which is absent in local (LSDA) and semi-local (e.g., generalized gradient approximations) functionals [@Perdew1983; @Sham1983]. This discontinuity appears as a jump in $n_{\text{2D}}$ at the threshold for the second subband occupation at $B=0$ [@ProettoPRL07]. The results of Ref. [@ProettoPRL07] suggest that orbital functionals (e.g., exact-exchange) [@Leeor2008] may give rise to phase transitions at the crossings B & D. Clearly more work is needed here. A Hartree-Fock analysis in model bilayer systems [@Jungwirth00] suggests that quantum Hall ferromagnetic instabilities can occur at same-spin pseudospin LL crossings [@Jungwirth00].
*Summary.* We have combined SDFT and linear response to investigate magnetotransport in 2DEGs formed in two-subband wells [@Zhang06_PRB; @Zhang06_PRL; @ZhangSU4-07]. Our calculated $n_{\text{2D}}-B$ maps of $\rho_{xx}$ show ringlike structures. At low temperatures the $\nu=4$ ring breaks due to quantum Hall ferromagnetic phase transitions. The $n_{\text{2D}}-B$ diagram of the 2DEG spin polarization $\xi$ shows the ring to be $\sim 50$% spin polarized. For tilted $B$ fields, the $\nu=4$ ring shrinks and fully collapses at a critical angle $\theta_c\approx 6^\circ$, in excellent agreement with the data [@ZhangSU4-07]. The interplay between the equal-spin LL anti-crossings and the XC effects are crucial here. A direct experimental evidence of our prediction of a high $\xi$ in the ring is still lacking; the resistively-detected NMR data of Ref. [@Zhang06_PRL] only shows signals near point C. We hope our work stimulates further investigations in the literature.
*Acknowledgments.* GJF acknowledges useful conversations with X. C. Zhang and T. Ihn. This work was supported by FAPESP and CNPq.
[36]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, **, vol. of ** (, , ).
, ****, ().
, ** (, , ).
An early Hartree-Fock calculation by Jungwirth and MacDonald \[Phys. Rev. B **63**, 035305 (2000)\] has established a classification scheme for QHF in terms of subband (pseudo spin), LL, and spin quantum numbers.
Piazza *et al.* Nature **402**, 638 (1999), Eom *et al.* Science **289**, 2320 (2000), De Poortere *et al.* Science **290**,1546 (2000), Smet *et al.* Phys. Rev. Lett. **86**, 2412 (2001), Jaroszy[ń]{}ski *et al.* Phys. Rev. Lett. **89**, 266802 (2002).
See, e.g., De Portere *et al.* and Jaroszy[ń]{}ski *et al.* in [@qhf-exp].
T. Jungwirth and A. H. MacDonald, Phys. Rev. Lett. **87**, 216801 (2001).
, ****, ().
K. Muraki, T. Saku, and Y. Hirayama, Phys. Rev. Lett. **87**, 196801 (2001).
X. C. Zhang *et. al.*, Phys. Rev. Lett. **95**, 216801 (2005).
X. C. Zhang, I. Martin, and H. W. Jiang, Phys. Rev. B **74**, 073301 (2006).
X. C. Zhang, G. D. Scott, and H. W. Jiang, Phys. Rev. Lett. **98**, 246802 (2007); G. P. Guo *et al.*, Phys. Rev. B **81**, 041306R (2010).
G. P. Guo *et al.*, Phys. Rev. B **78**, 233305 (2008)
W. Kohn and P. Vashishta, in *Theory of Inhomogeneous Electron Gas*, edited by S. Lundqvist and N. H. March (Plenum, New York, 1983). The current DFT (CDFT) generalizes the DFT to finite magnetic fields, see: G. Vignale and M. Rasolt, Phys. Rev. Lett. **59**, 2360 (1987).
T. Ando and Y. Uemura, J. Phys. Soc. Jpn. **36**, 959 (1974); R. R. Gerhardts, Surf. Sci. **58**, 227 (1976).
*Single-subband* Mn-based quantum wells have a very non-linear fan diagram (due to the s-d exchange interaction), with LLs from the *same* subband crossing and forming closed loops (see Ref. [@Freire07PRL] and Jaroszynski *et al.* in [@qhf-exp]).
M. Yamaguchi *et al.,* J. App. Phys. **100**, 113523 (2006).
J. P. Perdew and Y. Wang, Phys. Rev. B **45**, 13244 (1992); other parametrizations \[e.g., S.H. Vosko *et al.*, Can. J. Phys. **58**, 1200 (1980)\] give similar results.
The XC parametrization of Ref. [@PW92] does not account for the paramagnetic current density, hence the effective vector potential is zero ($A_{xc} = 0$), see Helbig *et al.* \[Phys. Rev. B **77**, 245106 (2008)\]. Low temperature corrections to XC functionals scales with $t^2 = (T/T_F)^2$ (see S. Tanaka and S. Ichimaru, Phys. Rev. B **39**, 1036 (1989) and §2.5 in [@dft]), $T_F$: Fermi temperature. These can be safely neglected here, since our temperatures ($t \lesssim 10^{-3}$) and density parameters ($r_s \lesssim 5$) lie far below the ranges ($t \sim 0.3$ and $r_s \sim 25$) where corrections are relevant; see U. Gupta and A. K. Rajagopal, Phys. Rev. A **22**, 2792 (1980), and M. W. C. Dharma-wardana and F. Perrot, Phys. Rev. Lett. **90**, 136601 (2003).
The $(n+1/2)$ “short range” factor [@ando-uemu-gerh] in $\sigma_{xx}$ enhances $\rho_{xx}$ on the $n_{\text{2D}}-B$ plane for higher LL indices $n$. We have also simulated scatterers of slowly varying type by removing this factor. Our results are essentially unaltered.
, ****, ().
Following Ref. [@Ando83], we find that near LL crossings the density of states deviates somewhat from the gaussian of individual LLs; however, our results do not change in any significant way in the range of parameters studied.
, PhD thesis, Swiss Federal Institute of Technology Zurich, 2006. ().
C. Ellenberger *et al.*, Phys. Rev. B **74**, 195313 (2006).
G. J. Ferreira and J. C. Egues, J. Supercond. Nov. Magn., **23**, 19 (2010).
The error in the SDFT + Kubo calculation \[Fig. 1(e)\] arises not only from the approach used, but also from extracting “by inspection” the distance AC from Fig. 1(c).
, ****, ().
, ****, ().
, ****, ().
, ****, ().
|
---
abstract: 'The interaction of obliquely incident surface gravity waves with a vertical flexible permeable membrane wave barrier is investigated in the context of three-dimensional linear wave-structure interaction theory. A general formulation for wave interaction with permeable submerged vertical membrane is given. The analytic solution of the physical problem is obtained by using eigenfunction expansion method, and boundary element method has been used to get the numerical solution. In the boundary element method, since the boundary condition on the membrane is not known in advance, membrane motions and velocity potentials are solved simultaneously. From the general formulation of the submerged membrane barrier, the performance of bottom-standing, surface-piercing and fully extended membrane wave barriers are analyzed for various wave and structural parameters. It is found that the efficiency of the submerged, surface-piercing and bottom-standing membrane wave barriers can be enhanced in waves for certain design conditions. From the analysis of various membrane configurations and parameters, it can be concluded that permeable membrane wave barrier can function as a very effective breakwater if it is properly designed.'
address: |
Department of Ocean Engineering and Naval Architecture,\
Indian Institute of Technology, Kharagpur -721 302, India
author:
- 'S. Koley'
- 'T. Sahoo'
title: Scattering of oblique waves by permeable vertical flexible membrane wave barriers
---
Flexible membrane, Surface gravity wave, Porous-effect parameter, Eigenfunction expansion method, Boundary element method.
Introduction {#sec:1}
============
In recent decades, flexible porous breakwaters have been considered as better alternatives to the conventional fixed rigid breakwaters for providing protection from wave attack at locations where protection is required on temporary basis. These types of structures are more suitable where bottom soil foundation is poor as these types of structures do not required proper foundation or strong supports. Moreover, flexible permeable barriers are cost-effective, quickly deployable, lightweight, portable, reusable and environmental friendly. Due to the porosity, these structures can dissipate wave energy at a higher rate which in turn reduces wave forces on the structure. Moreover, a characteristic of flexibility is usually included in these temporary barriers in order to minimize the wave impact on them. In addition, partial flexible barriers allow the free water circulation, transportation of sediment and safe passage of ocean current. Often, permeable and flexible structures are used to reduce wave resonance inside the harbor along with both the reflected and the transmitted wave heights during wave scattering.
[@williams1991flexible] have investigated the performance of a flexible, floating beam-like structure anchored to the seabed and having a small buoyancy chamber at the top of structure. They have solved the physical problem numerically by using the boundary integral equation method and validated the results by carried out small-scale physical model tests. [@kim1996flexible] studied the interaction of water waves with a vertical flexible membrane in the context of two-dimensional linear water wave theory. [@cho1997performance] have studied the interaction of oblique incident waves with a tensioned vertical flexible membrane hinged at the sea floor and attached to a rigid cylindrical buoy at its top. By using Darcy’s fine-pore model, [@cho2000interactions] studied the interaction of monochromatic incident waves with a horizontal flexible porous membrane in the context of 2D linear hydroelastic theory. To restore the wetlands habitat, [@williams2003flexible] proposed to use flexible porous wave barrier to protect cordgrass seedlings form wave action during the initial stage of growth following plating. [@kumar2006wave] have analyzed the performance of a vertical flexible porous breakwater in two-layer fluid of finite depth. Further, [@kumar2007wave] have carried out an analysis to investigate the scattering of water waves by a vertical flexible porous membrane pinned both at the free surface and the sea bed in a two-layer fluid of finite depth. [@karmakar2013scattering] used least square approximation method to study the scattering of surface gravity waves by multiple surface-piercing flexible permeable membrane wave barriers. Using system of Fredholm integral equation approach, [@koley2015oblique] and [@kaligatla2015trapping] studied the interaction of surface gravity waves by a floating flexible porous plate in water of finite and infinite depths. Recently, using the same approach as used in [@koley2015oblique], [@koley2016membrane] analyzed the hydroelastic response of a floating horizontal flexible porous membrane.
In the present study, scattering of obliquely incident surface gravity waves with a submerged permeable vertical flexible membrane barrier in water of finite depth is investigated in the context of three-dimensional linear water wave theory. As special cases of the submerged barrier, the effectiveness of the bottom-standing as well as surface-piercing and complete membrane barrier are analyzed. The solution of the boundary value problem associated with each barrier configuration is obtained (i) analytically by using eigenfunction expansion method and (ii) numerically by using boundary element method. Finally, both the results are compared for accuracy and convergence. The boundary element method is developed based on discrete membrane dynamic model and simple-source distribution over the entire fluid boundaries. To understand the efficiency of the proposed system, the reflection and transmission coefficients, the wave forces acting on the structure and the structural displacements of the membrane have been plotted and analyzed for various values of wave and structural parameters.
Mathematical formulation {#sec:2}
========================
In the present manuscript, oblique wave scattering by a submerged flexible porous membrane is studied in water of finite depth under the assumption of small amplitude water wave theory and structural responses. The physical problems are analyzed in the three-dimensional Cartesian coordinate system with the positive $y$-axis being vertically downwards and the horizontal plane $y=0$ represents undisturbed free surface. The fluid domain is infinitely extended in the $x$-$z$ horizontal direction as $-\infty<x, z<\infty$ except the flexible porous membrane as in Fig. \[fig:1\]a. A thin vertical flexible porous membrane occupies the region $x=0, a<y<b, -\infty<z<\infty$ in the fluid domain. For notational convenience, $L_m=\left(a, b\right)$ refers to the membrane segment and $L_g=\left(0, a\right) \cup \left(b, h\right)$ refers to the gap region. The fluid is modeled using the Airy’s water wave theory, and the motion of the fluid is assumed to be of simple harmonic in time with the angular frequency $\omega$. Further, it is assumed that surface waves are incident upon the vertical membrane by making an angle $\theta$ with the $x$- axis. These assumptions ensure the existence of velocity potential $\Phi \left( x,y,z,t\right)$ and is of the form $\Phi \left( x,y,z,t\right) = \mathrm{Re}\left\{ \phi_j (x, y) e^{\mathrm{i}\left(\beta_0 z - \omega t\right)}\right\}$ with $\beta_0=k_0 \sin \theta$ being the $z$-component of the plane progressive wave incident upon the membrane and the subscripts $j=1, 2$ referring to the fluid domains 1 and 2 as shown in Fig. \[fig:1\]a. In the $j^{\mathrm{th}}$ fluid region, the spatial velocity potential $\phi_j$ satisfies the partial differential equation $$\label{eq:1}
\left(\frac{\partial^{2}}{\partial x^{2}} +
\frac{\partial^{2}}{\partial y^{2}}-\beta_0^2\right)\phi_j = 0.$$ The free surface boundary condition is given by $$\label{eq:2}
\frac{\partial\phi_j}{\partial y} + K \phi_j=0,~~\mbox{on}~~y=0,$$ where $K=\omega^2/g$ with $g$ being the acceleration due to gravity. The bottom boundary condition is given by $$\label{eq:3}
\displaystyle \frac{\partial \phi_j}{\partial y}=0,~~\mbox{on}~~y=h,$$
The membrane barrier is modeled as string of uniform mass density $m_m$ acting under uniform tension with both the ends being fixed. The motion of the membrane barrier is assumed to be uniform in the longitudinal direction and the barrier is deflected horizontally with a displacement of the form $\zeta (y,z,t) = \mbox{Re}\left\{\chi (y) e^{\mathrm{i}(\beta_0 z-\omega t)}\right\}$ with $\chi (y)$ being the complex deflection amplitude and is assumed to be small compared to the water depth. The equation of motion of the membrane deflection $\chi (y)$, acted upon by the dynamic pressure is given by $$\label{eq:4}
\overline{T} \left(\frac{d^2}{dy^2}-\beta_0^2\right) \chi + m_m \omega^2 \chi=-\mathrm{i}\rho \omega \left[\phi_1(0, y) - \phi_2(0, y)\right],~~\mbox{for}~~x=0, y\in L_{m},$$ where $\overline{T}$ is the membrane tension, $\rho$ is the density of the water and $m_m=\rho_m d_m$ is the uniform mass per unit length of the membrane with thickness $d_m$ and density $\rho_m$. Further, the boundary condition on the flexible porous membrane with porous-effect parameter $G_0$ (see [@yu1995diffraction]) is given by $$\label{eq:5}
\frac{\partial \phi_j}{\partial x}=\mathrm{i} k_0 G_0 \left[\phi_1(0, y) - \phi_2(0, y)\right] - \mathrm{i} \omega \chi,~~\mbox{for}~~x=0, y\in L_{m}.$$ It may be noted that in Eq. (\[eq:5\]), $k_0$ is the wave number associated with the plane progressive wave satisfying the dispersion relation $\omega^2=gk_0\tanh k_0h$. Assuming that the membrane is fixed at both the ends, the vanishing of the membrane deflection yields $$\label{eq:6}
\chi=0, \quad \mbox{at}\quad y=a, b$$ Further, the continuity of pressure and velocity in the gap region yield $$\label{eq:7}
\phi_1=\phi_2, \quad \frac{\partial \phi_1}{\partial x}=\frac{\partial \phi_2}{\partial x},~~\mbox{for}~~x=0, y\in L_g.$$ Finally, the far-field boundary conditions are given by $$\label{eq:8}
\left\{\begin{array}{l}
\phi_1\left(x, y\right) \rightarrow \phi_0 \left(x, y\right) + R \phi_0 \left(-x, y\right) \quad \mbox{as} x \rightarrow -\infty, \vspace{0.2cm}
\\
\phi_2\left(x, y\right) \rightarrow T \phi_0 \left(x, y\right) \quad \mbox{as} x \rightarrow \infty,
\end{array}\right.$$ where $R$ and $T$ are the complex coefficients associated with the reflected and transmitted waves. In Eq. (\[eq:8\]), $\phi_0 (x,y)$ is the incident wave potential and is given by $$\label{eq:8a}
\displaystyle \phi_0 \left(x,y\right)=-\frac{\mathrm{i}gH}{2\omega} \frac{\cosh\left(k_0(h-y)\right)}{\cosh k_0h} e^{\mathrm{i}\alpha_0 x},$$ with $\alpha_0=k_0 \cos \theta$ and $H$ is the incident wave amplitude. Next, the analytic and numerical solution technique associated with the aforementioned boundary value problems will be discussed in two different subsequent Sections.
Analytic method of solution {#sec:3}
===========================
Using the eigenfunction expansion method, the velocity potential in each region is expanded in terms of appropriate eigenfunctions. The spatial velocity potentials $\phi_j$ for $j=1, 2$ satisfying Eq. (\[eq:1\]) along with Eqs. (\[eq:2\]), (\[eq:3\]) and (\[eq:8\]) are expressed as $$\begin{aligned}
&&\phi_1(x,y)=-\frac{\mathrm{i}gH}{2\omega} \left(e^{\mathrm{i}\alpha_0 x} + A_0 e^{-\mathrm{i}\alpha_0 x}\right) I_0(y) + \sum_{n=1}^{\infty} A_n e^{\alpha_n x} I_n(y), \label{eq:9}\\
&&\phi_2(x,y)=-\frac{\mathrm{i}gH}{2\omega} B_0 e^{\mathrm{i}\alpha_0 x} I_0(y) + \sum_{n=1}^{\infty} B_n e^{-\alpha_n x} I_n(y), \label{eq:10}\end{aligned}$$ where $A_0$, $B_0$, $A_n$ and $B_n$ for $n=1, 2, 3,...$ are the arbitrary complex constants to be determined and the eigenfunctions $I_0(y)$ and $I_n(y)$ are given by $$\label{eq:11}
I_0 (y)=\frac{\cosh\left(k_0(h-y)\right)}{\cosh k_0h},\quad I_n (y)=\frac{\cos \left(k_n(h-y)\right)}{\cos k_nh}\;\;\mbox{for}\;\; n=1, 2, 3, ...$$ with $\alpha_n=\sqrt{\beta_0^2+k_n^2}$. It may be noted that the wave numbers $k_n$ satisfies $k_n \tan k_n h=-K$ for $n=1, 2, 3, ...$. Applying the continuity of normal velocity as in Eq. (\[eq:7\]) into Eqs. (\[eq:9\])-(\[eq:10\]), it is obtained that $$\label{eq:12}
1-A_0=B_0, \quad A_n=-B_n$$ Using the relations (\[eq:12\]) into Eqs. (\[eq:9\]) and (\[eq:10\]), and substituting the expressions of $\phi_1$ and $\phi_2$ into the right-hand side of the forcing term in Eq. (\[eq:4\]), the complex amplitude of the structural displacement $\chi(y)$ is expressed as $$\label{eq:13}
\chi(y)=A e^{\delta y}+B e^{-\delta y}+a_0A_0 I_0(y)+\sum_{n=1}^{\infty} a_n A_n I_n(y),$$ where $\delta=\mathrm{i}\sqrt{B^\prime}$ with $B^\prime=\left(m_m \omega^2/\overline{T}\right)-\beta_0^2$. In Eq. (\[eq:13\]), $A$ and $B$ are arbitrary constants need to be determined, and $a_0$ and $a_n$ are given by $$\label{eq:14}
a_0=-\frac{\rho g H}{\overline{T}} \frac{1}{\left(k_0^2 + B^\prime\right)}, \quad a_n=\frac{2 \mathrm{i} \rho \omega}{\overline{T}} \frac{1}{\left(k_n^2 - B^\prime\right)},$$ Now, by using continuity of pressure as in Eq. (\[eq:7\]) in Eq. (\[eq:5\]), we get $$\label{eq:15}
\mathrm{i}k_0 G_0 \left(\phi_1 - \phi_2\right) =
\left\{\begin{array}{l}
0 \qquad\qquad\quad \mbox{on}\; x=0,\; y \in L_g, \vspace{0.2cm}
\\
\displaystyle \frac{\partial \phi_j}{\partial x}+\mathrm{i}\omega\chi \quad\; \mbox{on}\; x=0,\;y \in L_m.
\end{array}\right.$$ Substituting for $\phi_1$, $\phi_2$, $\partial \phi_1/\partial x$ and $\chi$ from Eqs. (\[eq:9\]), (\[eq:10\]) and (\[eq:13\]) into Eq. (\[eq:15\]) and utilizing the relationships as in (\[eq:12\]), we obtain $$\label{eq:16}
\sum_{n=0}^{\infty} c_n A_n I_n =
\left\{\begin{array}{l}
0 \qquad\; \mbox{on}\; x=0,\; y \in L_g, \vspace{0.2cm}
\\
f(y) \quad\; \mbox{on}\; x=0,\;y \in L_m.
\end{array}\right.$$ where $$\label{eq:17}
f(y)=\mathrm{i}\omega \left(Ae^{\delta y}+Be^{-\delta y}\right) + g_0 I_0 + \sum_{n=0}^{\infty} A_n b_n I_n,$$ with $$\label{eq:18}
g_0=\frac{\alpha_0 g_0 H}{2\omega}, \;\; b_0=\mathrm{i}\omega a_0 - \frac{\alpha_0 g_0 H}{2\omega}, \;\; b_n=\mathrm{i}\omega a_n + \alpha_n,\;\; c_0=\frac{k_0 g H G_0}{\omega},\;\;\mbox{and}\;\;c_n=2\mathrm{i}k_0G_0.$$ Using the orthogonal properties of $I_n (y)$ for $n=0, 1, 2,...$ and truncating the infinite series after $N$ terms, it is derived from Eqs. (\[eq:16\]) - (\[eq:17\]) that $$\begin{aligned}
&&c_0 A_0 \langle {I_0, I_0} \rangle=\int_{L_m} I_0 f(y) dy, \label{eq:19}
\\
&&c_n A_n \langle {I_n, I_n} \rangle=\int_{L_m} I_n f(y) dy, (n=1, 2,...,N), \label{eq:20}\end{aligned}$$ where $\displaystyle \langle {I_n, I_m} \rangle=\int_0^h I_n I_m dy$ with $\langle {I_n, I_m} \rangle=0,\;\;\mbox{for}\;\; n\neq m$. Now, Eqs. (\[eq:19\]) and (\[eq:20\]) give a system of $N+1$ equations. To determine the rest of the unknowns, the conditions prescribed at the membrane edges has to be used. Substituting the expression of $\chi(y)$ as in Eq. (\[eq:13\]) into Eq. (\[eq:6\]), we get the remaining two equations $$\begin{aligned}
&&A e^{\delta a} + B e^{-\delta a} + a_0A_0 \frac{\cosh \left(k_0(h-a)\right)}{\cosh k_0h}+\sum_{n=0}^{N} a_n A_n \frac{\cos \left(k_n (h-a)\right)}{\cos k_nh}=0, \label{eq:21}
\\
&&A e^{\delta b} + B e^{-\delta b} + a_0A_0 \frac{\cosh \left(k_0(h-b)\right)}{\cosh k_0h}+\sum_{n=0}^{N} a_n A_n \frac{\cos \left(k_n (h-b)\right)}{\cos k_nh}=0. \label{eq:22}\end{aligned}$$ The system of equations (\[eq:19\]) - (\[eq:22\]) are solved to obtain the required unknowns.
Numerical method of solution {#sec:4}
============================
In the present Section, a numerical solution based on boundary element method (see [@koley2014oblique], [@behera2015wave] and [@koley2015interaction] for details) is developed for the boundary value problem as discussed in Sec. \[sec:2\]. The schematic diagram for the computational domains used in boundary element method (BEM) is given in Fig. \[fig:1\]b. Here, for the sake of clarity, various boundary conditions discussed in Sec. \[sec:2\] are rewritten in the computational boundaries for easy reference in the BEM. The free surface boundary condition as in Eq. (\[eq:2\]) is rewritten as $$\label{eq:23}
\frac{\partial\phi_j}{\partial \emph{\textbf{n}}} + K \phi_j=0,~~\mbox{on}~~\Gamma_{fj},$$ where $\partial/\partial \emph{\textbf{n}}$ denotes the normal derivative. The bottom boundary condition as in Eq. (\[eq:3\]) is rewritten as $$\label{eq:24}
\displaystyle \frac{\partial \phi_j}{\partial \emph{\textbf{n}}}=0,~~\mbox{on}~~\Gamma_{bj}.$$ Further, the far-field boundary condition as in Eq. (\[eq:8\]) is rewritten as $$\label{eq:25}
\left\{\begin{array}{l}
\displaystyle \frac{\partial \left(\phi_1 - \phi_0\right)}{\partial \emph{\textbf{n}}} + \mathrm{i} \alpha_0 \left(\phi_1 - \phi_0\right) =0,~~\mbox{on}~~\Gamma_{c1}, \vspace{0.2cm}
\\
\displaystyle \frac{\partial \phi_2} {\partial \emph{\textbf{n}}} - \mathrm{i} \alpha_0 \phi_2=0,~~\mbox{on}~~\Gamma_{c2}.
\end{array}\right.$$ It may be noted that theoretically far-field boundary boundary conditions are satisfied at $x\pm\infty$. However, for the sake of computation, the far-field boundaries $\Gamma_{cj}$ for $j=1, 2$ in Eq. (\[eq:25\]) are taken three times water depth away from the membrane so that the effect of the local wave modes vanishes on $\Gamma_{cj}$. The continuity of pressure and normal velocity at the gap regions $\Gamma_{gj}$ for $j=1, 2$ are given as $$\label{eq:26}
\left\{\begin{array}{l}
\displaystyle \phi_1=\phi_2,\qquad\mbox{on}~~\Gamma_{gj} \vspace{0.2cm}
\\
\displaystyle \frac{\partial \phi_1} {\partial \emph{\textbf{n}}}=-\frac{\partial \phi_2} {\partial \emph{\textbf{n}}},~~\mbox{on}~~\Gamma_{gj}.
\end{array}\right.$$ Moreover, the kinematic boundary condition on the membrane as in Eq. (\[eq:5\]) is rewritten as $$\label{eq:27}
\displaystyle \frac{\partial \phi_1}{\partial \emph{\textbf{n}}}=- \frac{\partial \phi_2}{\partial \emph{\textbf{n}}}
=\mathrm{i}k_0G_0 \left(\phi_1-\phi_2\right) - \mathrm{i}\omega\chi,~~\mbox{on}~~\Gamma_{m}.$$ Applying Green’s integral theorem to the velocity potential $\phi \left(x, y\right)$ and the free space Green’s function $G\left(x,y;x_0,y_0\right)$, an integral equation is obtained as $$\label{eq:28}
-\left(\begin{array}{c} \phi(x,y)
\\
\frac{1}{2}\phi(x,y)
\end{array}\right)
=\int_{\Gamma}\left(\phi\frac{\partial G} {\partial
\emph{\textbf{n}}}-G\frac{\partial\phi}{\partial \emph{\textbf{n}}}\right)\,
d\Gamma,~~\left(\begin{array}{c}
\mbox{if}\,(x,y)\,\in\Omega\,\mbox{but not on}~~ \Gamma
\\
\mbox{if}~~ (x,y)\,\mbox{on}~~ \Gamma
\end{array}\right).$$ where the free space Green’s function $G(x,y;x_0,y_0)$ is of the form (See [@koley2015oblique] for details) $$\label{eq:29}
G\left(x,\, y\,;\, x_{0},\, y_{0}\right)=-\frac{\mathrm{K_0}\left(\beta_0 r\right)} {2\pi},\qquad
r=\sqrt{\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}},$$ where $\mathrm{K_0}$ is the modified zeroth-order Bessel function of the second kind with $r$ being the distance from the field point $(x, y)$ to the source point $(x_{0}, y_{0})$. The properties and asymptotic behavior of $\mathrm{K_0}\left(\beta_0 r\right)$ can be found in [@koley2014oblique]. Using the boundary conditions (\[eq:23\]) - (\[eq:27\]) into Eq. (\[eq:28\]), the integral equations in each fluid domain are obtained as $$\begin{aligned}
\frac{1}{2}\phi_1&+&\int_{\Gamma_{f1}}\left(\frac{\partial G}{\partial \emph{\textbf{n}}}+KG \right) \phi_1 d\Gamma
+\int_{\Gamma_{g1}+\Gamma_{g2}}\left(\phi_{2}\frac{\partial G}{\partial \emph{\textbf{n}}}+ G\frac{\partial\phi_{2}}{\partial \emph{\textbf{n}}}\right) d\Gamma
+\int_{\Gamma_{m}}\left(\frac{\partial G}{\partial \emph{\textbf{n}}}- \mathrm{i}k_0 G_0 G \right) \phi_{1}d\Gamma
+\mathrm{i}k_0 G_0 \int_{\Gamma_{m}} G \phi_2 d\Gamma \nonumber\\
&+&\mathrm{i}\omega \int_{\Gamma_{m}}\chi G d\Gamma
+\int_{\Gamma_{b1}}\phi_1 \frac{\partial G}{\partial \emph{\textbf{n}}} d\Gamma
+\int_{\Gamma_{l}}\left(\frac{\partial G}{\partial \emph{\textbf{n}}}+\mathrm{i}\alpha_0 G \right) \phi_1 d\Gamma
=\int_{\Gamma_{l}}\left(\frac{\partial \phi_0}{\partial \emph{\textbf{n}}}+\mathrm{i}\alpha_0 \phi_0 \right) G d\Gamma, \label{eq:30}
\\
\frac{1}{2}\phi_2&+&\int_{\Gamma_{f2}}\left(\frac{\partial G}{\partial \emph{\textbf{n}}}+K G \right) \phi_2 d\Gamma +\int_{\Gamma_{g1}+\Gamma_{g2}}\left(\phi_{2}\frac{\partial G}{\partial \emph{\textbf{n}}}- G\frac{\partial\phi_{2}}{\partial \emph{\textbf{n}}}\right) d\Gamma
+\int_{\Gamma_{m}}\left(\frac{\partial G}{\partial \emph{\textbf{n}}}- \mathrm{i}k_0 G_0 G \right) \phi_{2}d\Gamma
+\mathrm{i}k_0 G_0 \int_{\Gamma_{m}} G \phi_1 d\Gamma \nonumber\\
&-&\mathrm{i}\omega \int_{\Gamma_{m}}\chi G d\Gamma
+\int_{\Gamma_{b2}}\phi_2 \frac{\partial G}{\partial \emph{\textbf{n}}} d\Gamma
+\int_{\Gamma_{r}}\left(\frac{\partial G}{\partial \emph{\textbf{n}}}-\mathrm{i}\alpha_0 G \right) \phi_2 d\Gamma
=0,\label{eq:31}\end{aligned}$$ In Eqs. (\[eq:30\]) and (\[eq:31\]), all the boundary conditions of $\phi_1$ and $\phi_2$ except the dynamic membrane boundary condition as in Eq. (\[eq:4\]) has been used. Since the body boundary condition is not known in advance in contrast to the rigid body hydrodynamics, Eqs. (\[eq:30\]) and (\[eq:31\]) cannot be solved without coupling with the membrane equation of motion given by Eq. (\[eq:4\]). To solve Eqs. (\[eq:30\]) and (\[eq:31\]), the entire boundary is discretized into a finite number of segments, called boundary elements. On each boundary element, the velocity potential and its normal derivative are assumed to be constants and the influence coefficients $\int G$ and $\int \partial G/\partial \emph{\textbf{n}}$ are evaluated numerically using Gauss-Legendre quadrature formula (see [@au1982numerical] and [@koleythesis] for details). Utilizing Eqs. (\[eq:4\]), the discrete form of the membrane equation of motion for the $j^{\mathrm{th}}$ element is given by (see [@kim1996flexible]) $$\label{eq:32}
\frac{\mathrm{i}\rho\omega}{\overline{T}} \left(\phi_{1j}-\phi_{2j}\right) \Delta_j - \frac{\left(\chi_j-\chi_{j-1}\right)}{\Delta_j^m} + \frac{\left(\chi_{j+1}-\chi_{j}\right)}{\Delta_{j+1}^m}
=-\left(\frac{m_m \omega^2}{\overline{T}} - \beta^2_0\right) \Delta_j \chi_j,$$ where $\Delta_j$ is the length of the $j^{\mathrm{th}}$ segment and $\Delta_j^m=\left(\Delta_j+\Delta_{j+1}\right)/2$. Using this kind of discrete model, membrane edge conditions can be easily implemented. So, the discretized form of the integral equations (\[eq:30\]) and (\[eq:31\]) together with (\[eq:32\]) give a system of $\left(N+N_m-2\right)$ number of equations consist of $\phi$ and $\partial \phi/\partial \emph{\textbf{n}}$ and $\chi$ with $N$ and $N_m$ being the total number of boundary elements on the total boundaries of regions 1 and 2 of the physical domain, and on the membrane surface respectively. It may be noted that, the common boundaries of regions $1$ and $2$ are considered only one time to count $N$ and in the two boundary elements near to the edges of the membrane, the edges condition as in Eq. (\[eq:6\]) is used. These system of $\left(N+N_m-2\right)$ linear algebraic equations are finally solved to get the required unknowns. The convergence criteria of the numerical solutions and related discussions are given in [@kim1996flexible], and the details are deferred here.
Results and Discussions {#sec:5}
=======================
In this Section, a MATLAB program is developed to analyze the effects of different wave and structural parameters on wave scattering. In all the figures, lines and symbols correspond to the analytic and numerical solutions respectively. In the current study, the values of the parameters $G_0=0.25+0.25\mathrm{i}$, $m_1=m_m/(\rho h)=0.01$, $T_1=\overline{T}/\left(\rho g h^2\right)=0.4$, $\theta=30^{\circ}$, $H=h/10$ are kept fixed unless it is mentioned. The reflection and transmission coefficients are computed using the formulae $$K_r=\left|\frac{2\omega R}{gH}\right|,\quad K_t=\left|\frac{2\omega T}{gH}\right|$$ The non-dimensional horizontal wave force acting on the membrane $K_f$ is given by the formula $$K_f=\frac{\omega}{g h^2}\left| \int_{L_m} \left[\phi_1(0, y) - \phi_2(0, y)\right]\right|.$$ The non-dimensional membrane deflection $\varsigma$ is given by the formula $$\varsigma=\left|\frac{\chi}{h}\right|.$$ Hereafter, several numerical results have been plotted and analyzed to show the effectiveness of a permeable flexible membrane as a wave barrier.
In Figs. \[fig:2\](a) and (b), the reflection and transmission coefficients are plotted as a function of non-dimensional wave number $k_0h$ and angle of incidence $\theta$ for various values of normalized membrane tension $T_1$ and porous-effect parameter $G_0$ respectively for a submerged membrane. From Fig. \[fig:2\](a), it is observed that with an increase in membrane tension, reflection coefficient decreases and reverse pattern is observed for transmission coefficient. In case of Fig. \[fig:2\](b), it is seen that with an increase in the angle of incidence $\theta$, reflection coefficient $K_r$ decreases and reverse pattern is observed for transmission coefficient $K_t$. Moreover, for a particular value of $\theta$, with an increase in porosity of the membrane, wave reflection decreases. This may be due to the reason that with an increase in porosity of the structure, more wave energy is dissipated by the structure.
In Figs. \[fig:3\](a) and (b), horizontal wave force $K_f$ acting on a submerged membrane versus (a) non-dimensional wave number $k_0h$ is plotted for different values of membrane tension $T_1$, and (b) angle of incidence $\theta$ for different values of porous-effect parameter $G_0$. The general pattern of the wave forces $K_f$ are similar to that of reflection coefficient $K_r$ as in Fig. \[fig:2\]. By comparing Fig. \[fig:3\] with that of Fig. \[fig:2\], it can be concluded that optima in wave reflection corresponds to that of wave force acting on the membrane.
In Fig. \[fig:4\](a), the reflection and transmission coefficients $K_r$ and $K_t$ versus non-dimensional wave number $k_0h$ are plotted for various values of the membrane tension $T_1$ for a bottom-standing membrane barrier. The general pattern of $K_r$ and $K_t$ are similar to that of Fig. \[fig:2\](a) in case of submerged membrane. However, a comparison between Figs. \[fig:2\](a) and \[fig:4\](a) reveals that maxima in the wave reflection and correspondingly minima in the wave transmission occurs for smaller values of $k_0h$ in case of bottom-standing membrane compared to that of submerged membrane. On the other hand, Fig. \[fig:4\](b) shows the displacement profiles for a bottom-standing membrane for various values of porous-effect parameter $G_0$. As the membrane is fixed at both ends, there are no deflection of the membrane near both the ends and maximum deflection occurs near the center of the membrane. It is also observed from Fig. \[fig:4\](b) that the deflection of the membrane decreases as the absolute value of the porous-effect parameter $G_0$ increases. This may be due to the fact that with an increase in the structural porosity, wave impact on the membrane reduces which in turn reduces the deflection of the membrane.
In Fig. \[fig:5\], the reflection and transmission coefficients $K_r$ and $K_t$ versus non-dimensional wave number $k_0h$ are plotted for various values of the porous-effect parameter $G_0$, and (b) membrane length $b/h$ for a surface-piercing impermeable membrane barrier. A comparison with Figs. \[fig:2\](a) and \[fig:4\](a) demonstrates that more wave reflection and less transmission occurs in case of surface-piercing permeable membrane barrier compared to that of the bottom-standing and submerged membrane barrier. On the other hand, Fig. \[fig:5\](b) reveals that near $k_0h \approx 0$, full wave reflection and zero transmission occurs in case of complete impermeable membrane barrier. However, reverse pattern is observed in case of partial impermeable membrane barrier irrespective of membrane length $b/h$. This phenomena occurs due to the occurrence of wave diffraction in the gap region below the surface-piercing membrane wave barrier
Conclusions {#sec:6}
===========
The interaction of obliquely incident waves with vertical flexible porous membranes of different configurations are investigated using the linear water wave theory and small amplitude structural responses. Solutions for different types of barrier configurations are obtained using eigenfunction expansion-based analytic and BEM-based numerical method. It is observed that results obtained by the two methods are in good agreement. The effectiveness of the surface-piercing membrane barrier over the bottom-standing as well as submerged membrane are analyzed from the reflection and transmission coefficients. The inclusion of structural porosity to the membrane wave barrier significantly reduces the wave forces acting on the barrier as well as the deflection of the barrier. With appropriate choice of tensile force and porous-effect parameter, a permeable flexible membrane can act as an effective wave barrier.
[9]{}
Williams, A. N., Geiger, P. T., McDougal, W. G. (1991). Flexible floating breakwater. Journal of waterway, port, coastal, and ocean engineering, 117(5), 429-450.
Kim, M. H., Kee, S. T. (1996). Flexible-membrane wave barrier. I: Analytic and numerical solutions. Journal of waterway, port, coastal, and ocean engineering, 122(1), 46-53.
Cho, I. H., Kee, S. T., Kim, M. H. (1997). The performance of flexible-membrane wave barriers in oblique incident waves. Applied ocean research, 19(3), 171-182.
Cho, I. H., Kim, M. H. (2000). Interactions of horizontal porous flexible membrane with waves. Journal of waterway, port, coastal, and ocean engineering, 126(5), 245-253.
Williams, A. N., Wang, K. H. (2003). Flexible porous wave barrier for enhanced wetlands habitat restoration. Journal of engineering mechanics, 129(1), 1-8.
Kumar, P. S., Sahoo, T. (2006). Wave interaction with a flexible porous breakwater in a two-layer fluid. Journal of engineering mechanics, 132(9), 1007-1014.
Kumar, P. S., Manam, S. R., Sahoo, T. (2007). Wave scattering by flexible porous vertical membrane barrier in a two-layer fluid. Journal of fluids and structures, 23(4), 633-647.
Karmakar, D., Bhattacharjee, J., Soares, C. G. (2013). Scattering of gravity waves by multiple surface-piercing floating membrane. Applied Ocean Research, 39, 40-52.
Koley, S., Kaligatla, R. B., Sahoo, T. (2015). Oblique wave scattering by a vertical flexible porous plate. Studies in Applied Mathematics, 135(1), 1-34.
Kaligatla, R. B., Koley, S., Sahoo, T. (2015). Trapping of surface gravity waves by a vertical flexible porous plate near a wall. Zeitschrift f[ü]{}r angewandte Mathematik und Physik, 66(5), 2677-2702.
Koley, S., Sahoo, T. (2016). Oblique wave scattering by horizontal floating flexible porous membrane. Meccanica, 1-14.
Yu, X. (1995). Diffraction of water waves by porous breakwaters. Journal of waterway, port, coastal, and ocean engineering, 121(6), 275-282.
Koley, S., Behera, H., Sahoo, T. (2014). Oblique wave trapping by porous structures near a wall. Journal of Engineering Mechanics, 141(3), 04014122.
Behera, H., Koley, S., Sahoo, T. (2015). Wave transmission by partial porous structures in two-layer fluid. Engineering Analysis with Boundary Elements, 58, 58-78.
Koley, S., Sarkar, A., Sahoo, T. (2015). Interaction of gravity waves with bottom-standing submerged structures having perforated outer-layer placed on a sloping bed. Applied Ocean Research, 52, 245-260.
Au, M. C., Brebbia, C. A. (1982). Numerical prediction of wave forces using the boundary element method. Applied Mathematical Modelling, 6(4), 218-228.
Koley, S. (2016). Integral equation and allied methods for wave interaction with porous and flexible structures. PhD thesis, Indian Institute of Technology Kharagpur, url: http://www.idr.iitkgp.ac.in/xmlui/handle/123456789/6190
|
---
abstract: 'We analyse Keck spectra of 24 candidate globular clusters (GCs) associated with the spiral galaxy NGC 2683. We identify 19 bona fide GCs based on their recession velocities, of which 15 were suitable for stellar population analysis. Age and metallicity determinations reveal old ages in 14 out of 15 GCs. These old GCs exhibit age and metallicity distributions similar to that of the Milky Way GC system. One GC in NGC 2683 was found to exhibit an age of $\sim$3 Gyr. The age, metallicity and $\alpha$-element abundance of this centrally located GC are remarkably similar to the values found for the galactic centre itself, providing further evidence for a recent star formation event in NGC 2683.'
author:
- |
Robert N. Proctor$^{1}$[^1], Duncan A. Forbes$^{1}$, Jean P. Brodie$^{2}$, Jay Strader$^{2}$\
$^1$ Centre for Astrophysics & Supercomputing, Swinburne University, Hawthorn, VIC 3122, Australia\
$^2$ Lick Observatory, University of California, Santa Cruz, CA 95064, USA\
title: Keck spectroscopy of globular clusters in the spiral galaxy NGC 2683
---
globular clusters: general – galaxies: individual: NGC 2683 – galaxies: star clusters.
Introduction {#intro}
============
Despite their historical importance in understanding the formation processes of our own Galaxy (Eggen, Lynden-Bell & Sandage 1962; Searle & Zinn 1978; Mackey & Gilmore 2004; Forbes, Strader & Brodie 2004), detailed studies of the stellar populations of globular cluster (GC) systems in spiral galaxies beyond the Local Group are somewhat limited. It is important that this be rectified, not only to inform formation models of spiral galaxies, but also to constrain formation models of other morphological types. For example, Ashman & Zepf (1992) proposed that the GC systems of elliptical galaxies represent the merged systems of spiral galaxies plus the addition of newly formed red (metal-rich) GCs. Bedregal et al. (2006) have argued that the GC systems of S0s are consistent with faded spirals. A better understanding of GC systems in spirals, with a range of types and luminosities, is needed to test these ideas.
Imaging studies of GC systems exist for about a dozen spirals (e.g. Kissler-Patig et al. 1999; Larsen, Forbes & Brodie 2001; Goudfrooij et al. 2003). When sufficient numbers of GCs are present, they reveal a bimodal colour distribution (similar to those seen in elliptical galaxies) with the red (metal-rich) subpopulation associated with the galaxy bulge component (see Forbes, Brodie & Larsen 2001). Spectroscopic studies of GCs in spirals beyond the Milky Way and M31 (Burstein et al. 1984; Beasley et al. 2004) are even more limited. Schroder et al. (2002) investigate the stellar population properties of 16 individual GCs in M81. Similar, or smaller, numbers have been investigated in M104 (Larsen et al. 2002), NGC 253 and NGC 300 (Olsen et al. 2004), and M33 (Chandar et al. 2006). These studies generally find old ages with a wide range of metallicities for the GCs. Some GC systems reveal bulk rotation, while others do not, but small numbers and the lack of edge-on systems make such analyses uncertain.
----------- ------------ ---------- ----------- ---------------------- ------------ --------------------- ---------------- -------
Galaxy Hubble Distance M$_V$ Bulge r$_{\rm{eff}}$ Disc Scale Mass HI N$_{GC}$ S$_N$
Type (Mpc) (mag) (kpc) (kpc) (10$^9$M$_{\odot}$)
Milky Way S(B)bc$^1$ – -20.9$^1$ 2.5$^3$ 5.0$^5$ 4.0$^5$ 160$\pm$20$^1$ 0.70
M31 Sb$^1$ 0.78 -21.2$^1$ 2.4$^3$ 6.4$^5$ 3.0$^5$ 400$\pm$55$^1$ 1.32
NGC 2683 Sb$^2$ 7.2 -20.3 2.5$^4$ 1.7$^2$ 0.6$^2$ 120$\pm$40$^6$ 0.90
----------- ------------ ---------- ----------- ---------------------- ------------ --------------------- ---------------- -------
Based on the HST Advanced Camera for Surveys (ACS) imaging study of Forde et al. (2007), we have obtained Keck telescope spectra of GC candidates in the nearby, edge-on Sb spiral NGC 2683. A variety of distance estimates exist in the literature for NGC 2683. Here we adopt the surface brightness fluctuation distance modulus of Tonry et al. (2001) modified by the correction found by Jensen et al. (2003). This is gives m–M = 29.28 $\pm$ 0.36 or 7.2 $\pm$ 1.3 Mpc which lies near the midpoint of the literature estimates. With a luminosity of M$_V$ = –20.31 mag it has a lower luminosity (by a factor of 2) than the Milky Way or M31. We note that, although possessing a Hubble type and bulge size similar to the Milky Way and M31, NGC 2683 exhibits a disc size and HI gas mass that are significantly smaller. Rhode et al. (2007) show that the extent of the GC system of NGC 2683 is also rather small, with the projected density of the system falling to background levels within $\sim$8 kpc. This can be compared to the Milky Way GC system in which a fraction of the GC system lies outside $\sim$30 kpc. Some properties of NGC 2683 are compared to those of the Milky Way and M31 in Table \[gal\_params\].
In Section \[obs\] we present our observations and data reduction methods. The measurement and analysis of recession velocities and Lick indices are given in Section \[spec\_anal\]. The results of our chemical and kinematic analysis of the sample is outlined in Section \[results\]. Our conclusions are presented in in Section \[disc\].
Observations and data reductions {#obs}
================================
Spectra of 24 GC candidates around NGC 2683 were obtained with the Low Resolution Imaging Spectrometer (LRIS; Oke 1995) on the Keck I telescope. Candidate selection, based on ACS imaging data, is detailed in Forde et al. (2007). Briefly, our spectroscopic sample was selected from amongst the brightest of GC candidates. The candidates were chosen to represent both red and blue subpopulations. While not a full statistical sample, the candidates are therefore representative of the GC system as a whole. It should also be noted that, with GCs only partially resolved in the HST imaging, it is to be expected that the sample will include some foreground stars.
Spectral observations were obtained in 2005 February 07–08 with an integration time of $16 \times 1800$s = 8 hours. Seeing was $\sim$ 1 arcsec on both nights. A 600 lines-per-mm grating blazed at 4000 Åwas used on the blue side, resulting in a wavelength range of 3300 – 5900 Å and a FWHM spectral resolution of $\sim3.3$ Å. The spectra were not flux calibrated.
Data reduction was carried out using standard IRAF[^2] commands. The tracing of spectra and background-subtraction was done using the command [*apall*]{}. Comparison lamp spectra were used for wavelength calibration (mostly based on 8 Hg lines). Zero–point corrections of up to 1.5 Å were performed on the science spectra using the bright \[OI\] skyline at 5577 Å. The 16 individual spectra of each GC candidate were then average-combined with 3-$\sigma$ clipping. A sample of the GC spectra are shown in Fig. \[spectra\]. The backgrounds subtracted from the spectra of the GC candidates – admixtures of background galaxy light and sky – were retained for analysis of the galaxy rotation curve. These were average combined and then sky-subtracted. The removal of the sky was achieved by identifying the GC candidate with the lowest background level; gc01 – a candidate lying at large radial and azimuthal distances from the galaxy centre (see Fig. \[position\]; right). The candidate was also found to have signal-to-noise $<$1 Å$^{-1}$, and therefore probably contained no object. The background of this candidate, which lies in the halo of the galaxy, is therefore the least contaminated by either galaxy or globular cluster light. Indeed, cross-correlation of the spectrum of the background of this candidate with the solar spectrum gave a recession velocity less than 1 km s$^{-1}$, indicating very little contamination from NGC 2683. By scaling this spectrum by the total flux in the bright OI skyline at 5577 Å, estimates of the sky levels in the backgrounds of other candidates could be made and subtracted. The residuals from this process are therefore estimates of the spectra of the background galaxy. These were used to measure a rotation curve for the galaxy, but were not subject to stellar population analysis.
We also measured the gas kinematics in NGC 2683 using the bright \[OIII\]$\lambda$5007 Å emission-lines evident in most of the galaxy spectra. We have therefore been able to measure GC, stellar *and* gaseous recession velocities from the majority of slitlets in the mask.
![The distribution of GC candidates with respect to the galaxy NGC 2683 is shown. The red diamond marks the centre of the galaxy, while the line identifies the major axis. Red stars are candidates identified as being stars. Blue stars represent candidates contaminated by OB associations. Filled symbols are GCs for which stellar population analysis was performed. Open circles are candidates whose signal-to-noise was too low for stellar population analysis. The open circle on the extreme right is the candidate used for sky estimation (Section \[obs\]).[]{data-label="position"}](comb_position.ps){width="8cm"}
During the reductions, three GC candidates (gc02, gc15 and gc24) were identified by their spectra to be stars. Four others (gc01, gc13, gc20 and gc23) were found to have signal-to-noise ratios below 10 $\AA^{-1}$. A visual inspection of the Forde et al. (2007) imaging identified two other candidates to be contaminated by stars in OB associations in NGC 2683 itself (see Fig. \[gc12\]). All nine of the candidates identified above were therefore excluded from our stellar population analysis. The sample therefore contains 19 GCs suitable for recession velocity (RV) analysis and 15 GCs suitable for stellar population analysis. Details are given as Notes in Table \[gc\_params\].
![ACS imaging showing three GC candidates (from top to bottom; gc05, gc11 and gc12). Each image (B band; left and I band; right) is 10 arcsec ($\sim$360 pc at 7.2 Mpc) on a side. Candidates gc05 and gc12 are clearly projected against OB associations in NGC 2683 itself, and contamination appears highly likely. On the other hand, there is no evidence for an OB association affecting the young GC candidate; gc11.[]{data-label="gc12"}](thumbs.ps){width="9cm"}
---- --------- --------- ------ --------------- --------------- --------------- ------------------------------------------------------------------------------------------------------------------- --- ------- ------- ----------- ------
ID RA Dec Note RV$_{GC}$ RV$_{Gal}$ RV$_{Gas}$ r z B$_0$ I$_0$ (B-I)$_0$ Size
(J2000) (J2000) (km s$^{-1}$) (km s$^{-1}$) (km s$^{-1}$) (“) & (“) & (mag) & (mag) & (mag) & (pc)\
gc01 & 8:52:27.8& 33:24:40.4&S/N$<$1 (Sky) & – & – & – & – & – & – & – & – & –\
gc02 & 8:52:27.8& 33:24:40.4&Star & – & – & – & 147.4 & 96.3 & 20.97 & 19.10 & 1.87 & 1.24\
gc03 & 8:52:37.9& 33:23:24.4& – & 368(3) & 559(7) & 525(20) & 111.7 & -46.9 & 21.12 & 19.52 & 1.60 & 1.92\
gc04 & 8:52:35.3& 33:24:31.4& – & 602(16) & 552(1) & 590(20) & 87.3 & 23.5 & 22.87 & 20.86 & 2.02 & 2.27\
gco5 & 8:52:35.5& 33:24:39.7&Contam & 283(14) & 546(2) & 593(20) & 79.7 & 27.6 & 23.28 & 21.39 & 1.89 & 2.66\
gc06 & 8:52:38.2& 33:24:07.5& – & 491(3) & 563(2) & 530(20) & 78.6 & -19.0 & 22.14 & 20.44 & 1.70 & 2.31\
gc07 & 8:52:35.3& 33:25:22.5& – & 557(14) & 532(4) & 497(20) & 59.6 & 51.2 & 22.83 & 21.33 & 1.51 & 2.86\
gc08 & 8:52:35.6& 33:25:35.5& – & 560(4) & 396(5) & 437(20) & 66.2 & 39.3 & 21.24 & 19.72 & 1.52 & 4.17\
gc09 & 8:52:41.0& 33:24:51.7& – & 388(4) & 481(2) & 471(20) & 22.5 & -12.6 & 20.95 & 18.98 & 1.97 & 2.96\
gc10 & 8:52:43.5& 33:24:16.8& – & 317(15) & 447(31)& 477(20) & 25.0 & -59.4 & 22.84 & 20.86 & 1.98 & 2.67\
gc11 & 8:52:42.0& 33:25:02.9& – & 430(3) & 461(8) & 460(20) & 5.7 & -13.5 & 21.33 & 19.82 & 1.51 & 5.55\
gc12 & 8:52:42.3& 33:25:11.3&Contam& 432(3) & 432(3) & 424(20) & -2.9 & -10.2 & 21.88 & 20.00 & 1.89 & 3.09\
gc13 & 8:52:43.6& 33:24:59.4&S/N$<$10 & – & 500(27)& – & -6.0 & -30.2 & 22.77 & 20.89 & 1.88 & 2.04\
gc14 & 8:52:44.4& 33:25:18.6& – & 304(5) & 426(2) & 424(20) & -26.6 & -23.7 & 21.27 & 19.43 & 1.84 & 2.22\
gc15 & 8:52:42.1& 33:26:30.8&Star & – & 351(5) & 386(20) & -57.3 & 47.8 & 20.86 & 18.47 & 2.39 & 1.01\
gc16 & 8:52:42.8& 33:26:45.3& – & 217(5) & 344(5) & – & -73.8 & 51.8 & 22.05 & 19.97 & 2.08 & 2.96\
gc17 & 8:52:43.1& 33:26:54.1& – & 435(5) & 385(7) & – & -82.7 & 55.4 & 21.57 & 19.84 & 1.73 & 9.66\
gc18 & 8:52:46.7& 33:25:52.9& – & 244(6) & 365(2) & – & -71.2 & -19.8 & 22.48 & 20.99 & 1.49 & 2.18\
gc19 & 8:52:47.7& 33:25:54.1& – & 352(4) & 371(4) & 304(20) & -80.9 & -27.8 & 20.85 & 19.20 & 1.65 & 1.89\
gc20 & 8:52:48.0& 33:26:26.1&S/N$<$10 & 247(21) & 295(2) & 237(20) & -106.2 & -7.8 & 23.61 & 21.62 & 1.99 & 1.77\
gc21 & 8:52:50.4& 33:25:46.9& – & 711(5) & 259(5) & 329(20) & -99.8 & -56.8 & 21.89 & 20.50 & 1.40 & 2.52\
gc22 & 8:52:49.0& 33:26:30.9& – & 307(49) & 307(2) & 329(20) & -118.5 & -13.3 & 23.01 & 21.54 & 1.47 & 2.67\
gc23 & 8:52:50.0& 33:26:45.9&S/N$<$10 & 456(43) & 285(4) & – & -137.9 & -11.5 & 23.35 & 21.38 & 1.97 & 3.07\
gc24 & 8:52:53.0& 33:27:15.3&Star & – & 166(85)& 252(20) & -185.2 & -17.3 & 22.09 & 20.58 & 1.52 & 1.31\
---- --------- --------- ------ --------------- --------------- --------------- ------------------------------------------------------------------------------------------------------------------- --- ------- ------- ----------- ------
Index Offset Error
------------- -------- -------
H$\delta_A$ 0.373 0.254
H$\delta_F$ 0.007 0.127
CN$_1$ -0.001 0.011
CN$_2$ 0.005 0.012
Ca4227 0.298 0.091
G4300 0.142 0.211
H$\gamma_A$ -0.460 0.170
H$\gamma_F$ 0.011 0.059
Fe4383 0.583 0.188
Ca4455 0.420 0.091
Fe4531 0.180 0.108
C4668 -0.846 0.073
H$\beta$ 0.032 0.123
Fe5015 0.696 0.140
Mg$_1$ 0.027 0.005
Mg$_2$ 0.052 0.003
Mgb -0.115 0.050
Fe5270 0.086 0.083
Fe5335 0.384 0.114
Fe5406 -0.025 0.068
: Index offsets required to match the Lick system (Section \[meas\]). Errors are taken as the error on the mean (i.e. rms/$\sqrt{6}$).[]{data-label="c2l"}
Spectral analysis {#spec_anal}
=================
In the following we outline the spectral analysis from which we measure recession velocities and stellar population properties. We also make use of the HST photometric measurements of Forde et al. (2007) (see Table \[gc\_params\]).
Kinematics {#kins}
----------
The recession velocities of GC candidates and their galaxy backgrounds were determined by cross-correlation against six high signal-to-noise stellar templates using the IRAF command *fxcor*. The heliocentric velocities of the templates themselves were measured by cross-correlation against a high resolution solar spectrum. The average of the values of RV derived from comparison to the six template stars was taken as the measured value, while the rms scatter was taken as the error. The RVs of the galaxy’s gas were also measured in the background galaxy spectra. This was achieved by the fitting of Gaussians to the bright \[OIII\]$\lambda$5007 emission lines evident in most galaxy spectra.
The results of this analysis are presented in Section \[results\]. We next detail the determination of the properties of the stellar populations in our sample of GCs using Lick indices.
Measurement and analysis of Lick indices {#meas}
----------------------------------------
We measured Lick indices using the definitions of Trager et al. (1998) and Worthey & Ottaviani (1997). Indices were measured after convolving the spectra with the Gaussians required to broaden to the wavelength-dependent Lick resolution (Worthey & Ottaviani 1997). Lick indices and their associated errors are shown in Table \[indices\]. Calibration to the Lick system was performed using 6 Lick standard stars. The additive corrections required to match the Lick system and their errors are given in Table \[c2l\].
The measured indices were then compared with SSP models. We elected to use the recent models of Lee & Worthey (2005) combined with Houdashelt (2002) sensitivities to abundances ratios. We detail the method by which the SSP models are combined with the Houdashelt sensitivities in Mendel et al. (2007), in which we show that this combination reproduces the ages, metallicities and ‘$\alpha$’–abundance ratios of Galactic globular clusters extremely well. This gives us confidence in making direct comparisons of our results with the Galactic globular cluster system (Section \[results\]).
The comparisons to SSP models were carried out using the $\chi^2$-fitting procedure of Proctor & Sansom (2002) (see also Proctor et al. 2004a,b and Proctor et al. 2005) to measure the derived parameters; log(age), \[Fe/H\], \[Z/H\] and \[E/Fe\] (a proxy for the ‘$\alpha$’–abundance ratio; see Thomas et al. 2003 for details). Briefly, the technique for deriving these parameters involves the simultaneous comparison of as many observed indices as possible to models of single stellar populations (SSPs). The best fit is found by minimising the deviations between observations and models in terms of the observational errors, i.e. $\chi$. We have shown this approach to be relatively robust with respect to many problems which are commonly experienced in the measurement of spectral indices and their errors. These include poor or no flux calibration, poor sky subtraction and poor calibration to the Lick system. The method is similarly robust with respect to many of the uncertainties in the SSP models used in interpretation of the measured indices; e.g. the second parameter effect in horizontal branch morphologies and the uncertainties associated with the Asymptotic-Giant Branch. It was shown in Proctor et al. (2004a) and Proctor et al. (2005) that the results derived using the $\chi^2$ technique are, indeed, significantly more reliable than those based on only a few indices.
The process by which the candidate spectra were compared to the models was iterative. First, fits were obtained for all the candidates using all the available indices. The patterns of deviations from the fits obtained was then used to identify individual indices that matched the models poorly (see Fig. \[chis\]). These included the H$\delta$, CN indices for which flux levels were generally too low for accurate determination and Mg$_1$ and Mg$_2$ indices which suffer from flux calibration sensitivity. These indices were excluded from the analysis and the fits performed again. These fits were carried out using a clipping procedure in which indices deviating from the model fit by more than 3$\sigma$ were excluded, and the fit performed again. Many of these poorly fitting indices could be associated with known problems, e.g. the contamination of the Mgb index by the 5202 Åsky-line in low signal-to-noise candidates. Indices that are excluded on this basis are in parentheses in Table \[indices\]. On average, after all exclusions, 10 indices were used in each of the final fits.
For each GC in the sample, errors in the derived parameters (log(age), \[Fe/H\], \[E/Fe\] and \[Z/H\]) were estimated using 50 Monte-Carlo realisations. Best-fit model indices were perturbed by Gaussians, the width of which were set equal to the observational errors added in quadrature to the errors in offset to the Lick system (Table \[c2l\]). Error estimates in the derived parameters are therefore highly sensitive to the estimates of index errors. The process also makes no allowance for the *correlated* components of the observational index errors, such as velocity dispersion, flux calibration and background subtraction errors. The errors are modelled instead as purely random Gaussian distributions. As a consequence, our error estimates must be considered to include both random *and* systematic errors.
Results
=======
Results of recession velocity analysis {#rv_res}
--------------------------------------
The results of our analysis of recession velocities are given in Table \[gc\_params\] and are presented in Fig. \[comb\_galrot\]. The value assumed for the galactic centre (442.8 km s$^{-1}$) was estimated such that the value of the least-squares fit to the stellar RVs with radial distance (solid line in Fig. \[comb\_galrot\]) passes through 0.0 km s$^{-1}$ at a radial distance of 0.0 arcsec. Note that candidates identified as stars and the single object with signal-to-noise $<$1 are omitted from these and all subsequent plots of GC data. We therefore present recession velocities for 19 GCs.
The stellar and gas emission-line data (Fig. \[comb\_galrot\]) clearly show an increasing rotation speed with increasing radial distance from the galactic centre, and can be seen to be essentially cylindrical (i.e. there is little scatter and no particular trend in RV with distance above or below the least-square fit). The figure also shows the rotation of gas and stars to be in very good agreement.
It is evident that we do not reach the radii at which rotation is observed to flatten. However, our results are nevertheless consistent with the rotation of Casertano & van Gorkom (1991) and Broeils & van Woerden (1994), who find similar rotation curves with a flattening/peak lieing just beyond the range probed by our data. The apparent dip in stellar and gaseous RVs at radial distance 65 arcsec in the rotation profile of Barbon & Capaccioli (1975) is also present at similar radii in our data (Fig. \[comb\_galrot\]; lower left), although we note that they find a significantly steeper rotation curve than Casertano & van Gorkom (1991), Broeils & van Woerden (1994) or ourselves. Our ‘dip’ is also significantly deeper than that observed by Casertano & van Gorkom, with both stars and gas rotating in the opposite sense to the rest of the galaxy at similar radii.
The RVs measured in the GC spectra are also shown in Fig. \[comb\_galrot\]. GC recession velocities are generally consistent with those of the stars and gas. However, we lack sufficient numbers to unambiguously identify rotation in the GC system.\
Results of stellar population analysis
--------------------------------------
The results of our age and metallicity determinations are given in Table \[tab\_agez\] and plotted in Fig. \[agez\]. Candidates with signal-to-noise $<$10 are excluded from our analysis, leaving 15 GCs suitable for stellar population analysis.
In Fig. \[agez\] our results are compared to the values for Galactic GCs from de Angeli et al. (2005) and Pritzl, Venn & Irwin (2005). It is shown in Mendel et al. (2007) that ages and metallicities derived from Lee & Worthey (2005) SSP models agree extremely well with Galactic GC measurements from colour-magnitude diagrams and high resolution spectral studies. For \[Fe/H\], Mendel et al. find only a 0.028$\pm$0.024 dex average offset between the value derived from Lee & Worthey SSPs models and the Harris (1996) values for 42 Galactic GCs. A similar offset ($\sim$–0.024$\pm$0.021 dex or –0.28$\pm$0.24 Gyr) was found in the comparison of the derived ages with the data from de Angeli et al. (2005). Finally, the average values of \[E/Fe\] derived by Mendel et al. (2007) for Galactic GCs are offset from the Pritzl et al. (2005) values by –0.024$\pm$0.02 dex (T. Mendel 2007; private communication). The Mendel et al. (2007) results are consequently fully consistent with the literature data. We therefore have reasonable confidence in our comparison of the ages and metallicities of GCs of NGC 2683 with those of the Milky Way.\
Consistency checking
--------------------
However, before interpreting the ages and metallicity estimates, we sought to gain further confidence in our results by using them to predict the B-I colours of our GC sample (using the SSP models of Bruzual & Charlot 2003) for comparison to the observed HST colours (Forde et al. 2007). The comparison is shown in Fig. \[colour\_comp\]. The predictions compare quite favourably with the observed values, particularly given the $\sim$0.15 mag overestimation of predicted (B–I) colour found by Pierce et al. (2005, 2006) in similar studies. This is believed to be primarily the effect of the poor modelling of the horizontal branch (see also Strader & Smith 2007). Scatter should also be expected to be relatively high in our study due to the highly variable internal extinction in NGC 2683.
There is, however, one clearly aberrant GC – gc11; an apparently young GC (Table \[tab\_agez\]). We note that the spectrum of this GC is clearly different from other GCs of the same colour in a sense consistent with the derived younger age and higher metallicity, i.e. similar Balmer line strengths and stronger metal lines (see Fig. \[spectra\]). It is clear from Fig. \[colour\_comp\] that this effect is not the result of extinction. However, the proximity of this GC to the galactic centre makes contamination by the background galaxy a concern. We therefore experimented with adding galaxy light back into the GC spectrum and then subjecting the resultant spectrum to our age/metallicity analysis. We found that when 50% of the galaxy light was recombined with the GC spectrum the derived age increased by 0.2 dex (i.e to 5 Gyr), while the metallicity fell by a similar amount, resulting in a similar predicted colour. This is both a relatively small change (for a relatively large amount of galaxy contamination) *and* is in the opposite sense to that required to explain the young age by galaxy contamination. We therefore conclude that background contamination in the spectroscopic analysis is unlikely to be the cause of the observed young age of this GC, or the discrepancy with its predicted colour. The cause of the discrepancy between observed and predicted colours therefore remains unknown.\
![Metallicity–age (top) and $\alpha$-element abundance–metallicity (bottom) relations for NGC 2683 GCs. Values for the Galactic GC system are shown as small solid symbols. These are from de Angeli et al. (2005) (top) and Pritzl et al. (2005) (bottom). The relation for local Galactic stars is shown as a dashed line in the bottom plot for reference only. GCs in NGC 2683 with (B-I)$_0>1.8$ are shown as red squares, those with (B-I)$_0\leq1.8$ as blue circles. Arrows indicate GCs whose derived age equals the maximum modelled by Lee & Worthey (2005). One GC (gc11; solid blue star) exhibits age, \[Fe/H\] and \[E/Fe\] similar to the those measured in the galactic centre by Proctor & Sansom (2002) (open black stars). Combined systematic and random errors from our Monte Carlo analysis (Section \[meas\]) are indicated in each plot.[]{data-label="agez"}](comb_agez1.ps){width="8cm"}
Stellar population parameters
-----------------------------
Having gained some confidence in our measured stellar parameters we now return our attention to the age and metallicity estimates.
Our stellar population analysis identifies a single young GC, with a derived age of 3.3 Gyr. This is similar to the *luminosity-weighted* age of 4.7 Gyr found for the galactic centre by Proctor & Sansom (2002). The central \[Fe/H\]=–0.03$\pm$0.09 and \[E/Fe\]=0.20$\pm$0.04 found in Proctor & Sansom (2002) are also similar to the values found for this young GC (–0.17$\pm$0.04 and 0.16$\pm$0.03 respectively; Fig. \[agez\]). This suggests the possibility that this GC formed in the same event that fuelled the central star-burst.
We find the remaining 14 of 15 GCs to possess ages older than 10 Gyr (Fig. \[agez\]). In five cases we find an age equal to the oldest age modelled by Lee & Worthey (2005). It is apparent that the scatter in GC age estimates is smaller than the error given by our Monte-Carlo analysis (Section \[meas\]). We take this to be a combination of three effects; i) the error includes both random *and* systematic errors, ii) the scatter is slightly suppressed by the GCs hitting the oldest age, iii) a slight over-estimation of observational errors is also a possibility (see Section \[meas\]).
The 14 GCs found to be old span a broad range of metallicities (Fig. \[agez\]), similar to that observed in other spiral galaxy GC systems (Burstein et al. 1984; Beasley et al. 2004; Schroder et al. 2002; Larsen et al. 2002; Olsen et al. 2004). They also span a similar range to the Milky Way GC system (de Angeli et al. 2005; Fig. \[agez\]).
------ ------- ------------ ------------- ------------- -------------
ID Age Log(age) \[Fe/H\] \[E/Fe\] \[Z/H\]
(Gyr) (Gyr)
gc03 11.9 1.08(0.15) -1.40(0.13) 0.21(0.08) -1.20(0.16)
gc04 11.2 1.05(0.13) -0.99(0.17) 0.09(0.08) -0.90(0.15)
gc06 11.9 1.08(0.20) -1.57(0.33) 0.21(0.14) -1.38(0.24)
gc07 11.9 1.08(0.18) -1.63(0.26) 0.24(0.11) -1.40(0.22)
gc08 10.0 1.00(0.08) -1.95(0.09) 0.24(0.15) -1.73(0.13)
gc09 10.0 1.00(0.18) -0.72(0.17) 0.05(0.07) -0.68(0.21)
gc10 10.0 1.00(0.18) 0.16(0.18) -0.25(0.24) -0.08(0.24)
gc11 3.3 0.53(0.02) -0.17(0.03) 0.16(0.02) -0.03(0.03)
gc14 10.0 1.00(0.07) -0.53(0.12) 0.01(0.11) -0.53(0.08)
gc16 11.2 1.05(0.07) -0.71(0.10) 0.14(0.10) -0.58(0.12)
gc17 11.2 1.05(0.14) -1.47(0.12) 0.15(0.05) -1.33(0.09)
gc18 10.0 1.00(0.10) -1.59(0.24) 0.36(0.25) -1.25(0.16)
gc19 11.9 1.08(0.15) -1.26(0.11) 0.12(0.05) -1.15(0.12)
gc21 11.2 1.05(0.04) -2.55(0.19) 0.24(0.11) -2.33(0.18)
gc22 11.9 1.08(0.26) -2.05(0.56) 0.27(0.27) -1.80(0.52)
------ ------- ------------ ------------- ------------- -------------
Fig. \[agez\] also shows a comparison of \[E/Fe\] values from our study to the \[$\alpha$/Fe\] results of Pritzl et al. (2005). We show in Mendel et al. (2007) that, for Galactic GCs, there is good agreement between \[E/Fe\] from Lick studies using Lee & Worthey (2005) models, and the \[$\alpha$/Fe\] results of Pritzl, Venn & Irwin (2005). The data suggest a slightly lower \[E/Fe\] in NGC 2683 than in the Milky Way, but a larger, high signal-to-noise sample is required before we can draw any firm conclusions.\
Radial metallicity distribution
-------------------------------
The final step in our analysis is to consider the radial distribution in GC metallicities. To this end, a plot of \[Fe/H\] with radial distance along the major axis is presented in Fig. \[raddist\].
Radial trends in GC metallicity with galactocentric radius are expected in a dissipative formation scenario. The GC systems of both M31 (Barmby et al. 2000) and Milky Way (Armandroff, Da Costa & Zinn 1992) have been found to exhibit little, or no, overall radial metallicity gradient. However, Harris (2000) shows that weak trends in metallicity with radius *are* present when red and blue GC subpopulations are considered separately. More recently, Lee et al. (2007) showed that trends between metallicity and orbital parameters are present in the sub-sample of the Milky Way population that excludes many blue GCs with extreme horizontal-branch morphologies. Citing the extreme horizontal-branch GCs as probable accreted (and stripped) dwarf galaxies, Lee et al. (2007) conclude that the ‘normal’ GCs show clear signs of dissipational collapse.
We find no evidence for trends with azimuthal distance (perpendicular to the galaxy rotation plane) in NGC 2683, although we note the extremely limited range of our data. Our data *do*, on the other hand, suggest a trend of decreasing GC \[Fe/H\] with increasing distance along the major axis of NGC 2683 with logarithmic slope of –1.7 (Fig. \[raddist\]). However, the data for both red and blue GCs of NGC 2683 are also consistent with the Harris (2000) trends for the average \[Fe/H\] with radius in Galactic GCs. In the Milky Way, there is no significant trend in the GC system as a whole, but individually both red and blue sub-populations show weak trends with radius of logarithmic slope –0.3, albeit with considerable scatter. We also note that our sample for NGC 2683 contains no blue GCs within $\sim$2 kpc, and no red GCs beyond $\sim$3 kpc, while the photometry of Forde et al. (2007) clearly shows that both red and blue GCs are present throughout the radius range covered by our data. Therefore, we suspect that the apparent trend is the result of the lack of observations of red GCs at large radii and blue GCs at small radii, and is consequently simply a sampling issue. A definitive description of this aspect of the GC system of NGC 2683 must, however, await a more extensive study.
Conclusions {#disc}
===========
We have analysed the recession velocities and stellar populations of a small sample of GCs in the spiral galaxy NGC 2683 and compared the results with the Galactic GC system.
Our stellar population analysis identified one GC, located near the centre of NGC 2683, with the relatively young age of 3.3$\pm$0.5 Gyr. The age, metallicity and \[E/Fe\] of this young GC appear remarkably similar to the values found by Proctor & Sansom (2002) for the centre of NGC 2683 itself. This result therefore suggests the possibility that this GC and the recent burst in the central regions were formed at the same time, from the same gas supply, and provide further evidence for a star-formation event in NGC 2683 about $\sim$3 Gyr ago.
The stellar population parameters of the 14 *old* globular clusters in our sample show many similarities to the old globular clusters of the Milky Way. The metallicity distribution spans a similar range to those found in studies of the Milky Way and other spiral galaxy systems, i.e from $\sim$–2.5 to 0.0 dex. The data for NGC 2683 GCs are also consistent with the trends in \[Fe/H\] with radius observed in red and blue Galactic GC subpopulations.\
Acknowledgements
================
We thank Soeren Larsen for help preparing the slit mask and Kieran Forde for providing information prior to publication. We also thank Lee Spitler for assistance with the photometric analysis. Part of this research was funded by NSF grant AST-02-06139 The data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, Caltech, under contract with the National Aeronautics and Space Administration. We also thank the Australian Research Council for funding that supported this work.\
References
==========
Armandroff T.E., Da Costa G.S., Zinn R., 1992, AJ, 104, 164\
Ashman K.M., Zepf, S.E., 1992, ApJ, 384, 50\
Barbon R., Capaccioli M., 1975, A&A, 42, 221\
Barmby P., Huchra J.P., Brodie J.P., Forbes D.A., Schroder L.L., Grillmair C.J. 2000, AJ, 119, 727\
Beasley M.A., Brodie J.P., Strader J., Forbes D.A., Proctor R.N., Barmby P., Huchra J.P., 2004, AJ, 128, 1623\
Bedregal A.G., Aragón-Salamanca A., Merrifield M.R., 2006, MNRAS, 373, 1125\
Broeils A.H., van Woerden H., 1994, A&AS, 107, 129\
Bruzual A.G., Charlot S., 2003, MNRAS, 344, 1000\
Burstein D., Faber S.M., Gaskell C.M., Krumm N., 1984, ApJ, 287, 586\
Casertano S., van Gorkom J.H., 1991, AJ, 101, 1231\
Chandar R., Puzia T.H., Sarajedini A., Goudfrooij P., 2006, ApJ, 646L, 107\
Courteau S., van den Bergh S., 1999, AJ, 118, 337\
De Angeli F., Piotto G., Cassisi S., Busso G., Recio-Blanco A., Salaris M., Aparicio A., Rosenberg A., 2005, AJ, 130, 116\
Eggen O.J., Lynden-Bell D., Sandage A.R., 1962, ApJ, 136, 748\
Forbes D.A., Beasley M.A., Bekki K., Brodie J.P., Strader J., 2003, Science, 301, 1217\
Forbes D.A., Brodie J.P., Larsen S.S., 2001, ApJ, 556 , 83\
Forbes D.A., Strader J., Brodie J.P., 2004, AJ, 127, 3394\
Forde et al., 2007, in preparation\
Gilmore G., King I., van der Kruit P., 1989, Proceedings of the 19th Advanced Course of the Swiss Society of Astronomy and Astrophysics (SSAA), Saas-Fee, Leysin, Vaud, Switzerland, 13-18 March, 1989, Geneva: Observatory, 1989, edited by Buser, Roland, p334\
Goudfrooij P., Strader J., Brenneman L., Kissler-Patig M., Minniti D., Huizinga J.E., 2003, MNRAS, 343, 665\
Harris W.E., 2000, “Star Clusters”, in “28th Saa-Fee Advanced Course for Astrophysics and Astronomy”\
Houdashelt M.L., Trager S.C., Worthey G., Bell R.A., 2002, Elemental Abundances in Old Stars and Damped Lyman-α Systems, 25th meeting of the IAU, Joint Discussion 15, 22 July 2003, Sydney, Australia\
Ibata R.A., Gilmore G., Irwin M.J., 1995, MNRAS, 277, 781\
Jensen J.B., Tonry J.L., Barris B.J., Thompson R.I., Liu M.C., Rieke M.J., Ajhar E.A., Blakeslee J.P., 2003, ApJ, 583, 712\
Kent S.M., 1985, ApJS, 59, 115\
Kissler-Patig M., Ashman K.M., Zepf S.E., Freeman K.C., 1999, AJ, 118, 197\
Lanfranchi G.A., Matteucci F., 2004, MNRAS, 351, 1338\
Larsen S.S., Forbes D.A., Brodie J.P., 2001, MNRAS, 327, 1116\
Larsen S.S., Brodie J.P., Beasley M.A., Forbes D.A., 2002, AJ, 124, 828\
Lee H., Worthey G., 2005, ApJS, 160, 176\
Lee Y-K., Gim H.B., Casetti-Dinescu D.I., ApJ, 661, L52\
Mackey A.D., Gilmore G.F., 2004, MNRAS, 355, 504\
Martin N.F., Ibata R.A., Bellazzini M., Irwin M.J., Lewis G.F., Dehnen W., 2004, MNRAS, 348, 12\
Mendel J.T., Proctor R.N., Forbes D.A., 2007, MNRAS, 379, 1618\
Merrifield M.R., Kuijken K., 1999, A&A, 345, 47\
Oke J.B., Cohen J.G., Carr M., Cromer J., Dingizian A., Harris F.H., Labrecque S., Lucinio R., Schaal W., Epps H., Miller J., 1995, PASP, 107, 375\
Olsen K.A.G., Miller B.W., Suntzeff N.B., Schommer R.A., Bright J., 2004, AJ, 127, 2674\
Pierce M., Brodie J.P., Forbes D.A., Beasley M.A., Proctor R.N., Strader J., 2005, MNRAS, 358., 419\
Pierce M., Bridges T., Forbes D.A., Proctor R.N., Beasley M.A., Gebhardt K., Faifer F.R., Forte J.C., Zepf S.E., Sharples R., Hanes D.A., 2006, MNRAS, 368. 325\
Pritzl B.J., Venn K.A., Irwin M., 2005, AJ, 130, 2140\
Proctor R.N., Sansom A.E., 2002, MNRAS, 333, 517\
Proctor R.N., Forbes D.A., Beasley M.A., 2004a, MNRAS, 355, 1327\
Proctor R.N., Forbes D.A., Hau G.K.T., Beasley M.A., De Silva G.M., Contreras R., Terlevich A.I., 2004b, MNRAS, 349, 1381\
Proctor R.N., Forbes D.A., Forestell A., Gebhardt K., 2005, MNRAS, 362, 857\
Rhode K.L., Zepf S.E., Kundu A., Larner A.N., astro-ph/0708.1166\
Schroder L.L., Brodie J.P., Kissler-Patig M., Huchra J,P., Phillips A.C., 2002, AJ, 123, 2473\
Searle L., Zinn R., 1978, ApJ, 225, 357\
Strader J., Smith G., 2007, ApJ, submitted\
Thomas D., Maraston C., Bender R., 2003, MNRAS, 339, 897\
Tonry J.L., Dressler A., Blakeslee J.P., Ajhar E. A., Fletcher A.B., Luppino G.A., Metzger M.R., Moore C.B., 2001, ApJ, 546, 681\
Trager S.C., Worthey G., Faber S.M., Burstein D., Gonzarlez J.J., 1998, ApJS, 116, 1\
van den Bergh S., 1999, A&ARv, 9, 273\
Worthey G., Ottaviani D.L., 1997, ApJS, 111, 377\
Lick indices
============
------ --------- --------- ------------- ------------- ---------- --------- ---------- ---------- ---------- ---------- --------- ---------- ---------- ----------
GC Ca4227 G4300 H$\gamma_A$ H$\gamma_F$ Fe4383 Ca4455 Fe4531 C4668 H$\beta$ Fe5015 Mgb Fe5270 Fe5335 Fe5406
$\AA$ $\AA$ $\AA$ $\AA$ $\AA$ $\AA$ $\AA$ $\AA$ $\AA$ $\AA$ $\AA$ $\AA$ $\AA$ $\AA$
gc03 0.613 2.496 (-0.773) 1.357 2.356 0.878 1.667 0.856 2.407 (3.319) (0.789) 1.459 1.354 0.473
0.158 0.308 (0.284) 0.150 0.382 0.195 0.285 0.413 0.202 (0.396) (0.195) 0.230 0.278 0.203
gc04 0.826 (2.314) -1.802 (0.650) 4.043 (2.170) 3.129 3.086 0.849 3.717 1.704 1.537 1.434 1.024
0.252 (0.497) 0.484 (0.283) 0.644 (0.335) 0.513 0.776 0.347 0.706 0.327 0.388 0.457 0.341
gc05 0.491 4.129 -1.598 1.097 (0.016) 1.410 2.178 3.120 2.710 5.260 (3.361) 2.823 2.081 (2.848)
0.297 0.515 0.543 0.316 (0.800) 0.400 0.596 0.898 0.375 0.847 (0.406) 0.481 0.558 (0.388)
gc06 0.558 1.760 1.102 1.957 (-0.892) 0.026 (4.640) -0.698 (2.973) – (2.502) (2.594) (3.365) 0.975
0.272 0.498 0.457 0.268 (0.756) 0.368 (0.505) 0.823 (0.334) – (0.349) (0.401) (0.459) 0.360
gc07 0.674 1.421 (2.184) 2.137 1.481 (1.445) 1.969 1.189 (1.944) (-0.967) 1.359 0.712 (2.443) (1.234)
0.259 0.493 (0.455) 0.273 0.706 (0.354) 0.567 0.853 (0.357) (0.811) 0.367 0.445 (0.508) (0.392)
gc08 0.270 (2.277) 1.095 (2.398) 1.525 0.543 1.005 (-0.971) 2.817 1.963 (0.889) 0.527 (-0.296) 0.478
0.139 (0.277) 0.245 (0.121) 0.335 0.170 0.249 (0.354) 0.179 0.341 (0.159) 0.196 (0.239) 0.168
gc09 0.987 4.497 (-1.567) (1.870) 4.216 1.480 2.695 (3.496) (3.054) 4.700 2.082 1.809 2.001 1.139
0.146 0.288 ( 0.274) (0.138) 0.347 0.176 0.261 (0.375) (0.192) 0.374 0.177 0.213 0.250 0.183
gc10 (2.085) (4.412) -6.486 (-0.820) 5.984 1.888 3.711 (-0.545) (3.132) 4.335 (5.628) 3.803 3.382 (-1.105)
(0.266) (0.571) 0.672 (0.397) 0.806 0.417 0.621 (1.018) (0.378) 0.908 (0.402) 0.496 0.589 (0.508)
gc11 (1.282) 4.757 -3.718 0.044 4.424 1.341 2.790 4.384 2.302 (4.093) 3.153 2.577 – 1.314
(0.101) 0.226 0.193 0.081 0.221 0.110 0.143 0.160 0.136 (0.192) 0.080 0.109 – 0.092
gc14 (2.066) (8.325) (-6.007) -0.872 3.526 1.497 2.655 2.910 (2.760) (7.069) (6.047) (3.092) 2.487 (2.594)
(0.152) (0.310) (0.349) 0.198 0.427 0.219 0.314 0.454 (0.214) (0.407) (0.181) (0.232) 0.278 (0.193)
gc16 (1.335) 4.365 -4.644 -0.772 4.343 1.316 1.977 (0.293) 1.886 (5.444) (1.715) 2.616 1.721 1.147
(0.177) 0.350 0.364 0.211 0.454 0.236 0.354 (0.530) 0.237 (0.463) (0.227) 0.255 0.311 0.226
gc17 0.812 2.337 -0.019 (1.640) 1.692 (0.951) 1.631 (0.399) (2.379) 2.857 1.007 (1.963) (1.882) 0.660
0.134 0.277 0.247 (0.124) 0.331 (0.167) 0.241 (0.340) (0.177) 0.324 0.151 (0.178) (0.220) 0.159
gc18 0.602 (4.593) -1.077 0.433 1.818 1.729 (-0.295) (-3.244) (3.752) (6.285) (6.797) 1.817 (2.002) 0.362
0.191 (0.358) 0.379 0.223 0.553 0.259 (0.452) (0.660) (0.273) (0.597) (0.260) 0.364 (0.429) 0.340
gc19 0.347 2.742 -0.704 1.413 2.508 0.893 2.543 (1.323) 1.885 3.133 1.415 (2.099) 1.694 0.657
0.168 0.319 0.299 0.160 0.404 0.208 0.307 (0.454) 0.214 0.426 0.205 (0.241) 0.292 0.217
gc21 0.414 0.368 2.799 3.054 1.560 0.614 0.488 (-3.468) 3.386 1.508 (1.695) -1.085 1.114 0.011
0.250 0.470 0.406 0.228 0.629 0.326 0.533 (0.810) 0.316 0.732 (0.347) 0.451 0.510 0.392
gc22 0.554 1.470 2.364 (3.414) 1.936 1.382 -1.338 – – (-0.760) – (-2.791) (-3.424) (3.388)
0.350 0.795 0.672 (0.393) 1.015 0.570 0.967 – – (1.420) – (0.840) (1.076) (0.642)
------ --------- --------- ------------- ------------- ---------- --------- ---------- ---------- ---------- ---------- --------- ---------- ---------- ----------
[^1]: rproctor@astro.swin.edu.au
[^2]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation
|
---
abstract: 'This paper introduces a new equation for rewriting two unit fractions to another two unit fractions. This equation is useful for optimizing the elements of an Egyptian Fraction. Parity of the elements of the Egyptian Fractions are also considered. And lastly, the statement that all rational numbers can be represented as Egyptian Fraction is re-established.'
author:
- |
Keneth Adrian P. Dagal\
`kendee2012@gmail.com`\
title: A New Operator for Egyptian Fractions
---
Introduction
============
Unit fractions are fractions of the form $\frac{1}{n}$ for all integers $n > 1$. We define the set $U_f$ to be the collection of all unit fractions. $$U_f = \left\{ \frac{1}{n} \mid n\in \mathbb{Z^+}-\{ 1 \}\right \}$$
The power set of $U_f$, denoted by $\mathcal{P}(U_f)$, is the set that contains all possible subset of $U_f$. We define the set $X$ be an arbitrary element of $\mathcal{P}(U_f)$ and we *[partition]{} the set $\mathcal{P}(U_f)$ in terms of each set’s cardinality ($\mid X\mid$) into three sets defined as:*
$$\mathcal{P}(U_f) = \mathcal{P}(U_f)_{\mid X\mid < 2} \cup \mathcal{P}(U_f)_{\mid X\mid \geq 2} \cup \mathcal{P}(U_f)_{\mid X\mid = \infty}$$
The set $\mathcal{P}(U_f)_{\mid X\mid < 2}$ contains the null set and the sets that contains each unit fraction. So, the set $\mathcal{P}(U_f)_{\mid X\mid < 2}$ is of no interest. For the set $\mathcal{P}(U_f)_{\mid X\mid = \infty}$, this is of great interest since our fundamental operation is to add all the elements of $X \in \mathcal{P}(U_f)_{\mid X\mid = \infty}$. So, we can classify all infinite series in the set $X \in \mathcal{P}(U_f)_{\mid X\mid = \infty}$ into either divergent or convergent. Before proceeding to some examples for the set $\mathcal{P}(U_f)_{\mid X\mid = \infty}$, we define some notations for simplicity.\
For convenience, we define the function $S: X \rightarrow \mathbb{R}$ be $$S(X) = \sum_{x \in X }{x}.$$ And we define the set $N$ be $N= \{n \mid n= x^{-1} \text { for all } x\in X\}.$ Equivalently, $X = \{x \mid x= n^{-1} \text { for all } n\in N\}$. With this, we can redefine the function $S$ as $S:N \rightarrow \mathbb{R}.$ For example, we have $N = \mathbb{Z^+}-\{1\}$. Therefore, $S(N) = \infty$. This is known as the *Harmonic Series* (without the term 1) which is known to be divergent.\
For simplicity, we use the Riemann zeta function ($\zeta(s)$) and limit the domain of $s$ in $\mathbb{Z^+}$ to illustrate the function $S$ for some $X \in \mathcal{P}(U_f)_{\mid X\mid = \infty}.$ It is known that $\zeta(1)=\infty$, $\zeta(2)=\frac{\pi^2}{6} \approx 1.645$. For the set $N= \{n \mid n= q^2 \text{ for all integers } q\geq 2\}$, $S(N)= \frac{\pi^2}{6} -1$. Setting that aside, the set $\mathcal{P}(U_f)_{\mid X\mid \geq 2}$ is our major concern.
The function $S$ is said to be an Egyptian fraction if $$S(X) = \sum_{x \in X }{x}.$$ for all $X \in \mathcal{P}(U_f)_{\mid X\mid \geq 2}$.
It has already been established that *Every positive rational number can be represented by an Egyptian Fraction*. We attempt to re-established it in the next section.
The Inverse of the Function S
=============================
In this section, we focus on the function $S$ where the domain is $\mathcal{P}(U_f)_{\mid X\mid \geq 2}$. In this domain, the function then becomes $S: X \rightarrow \mathbb{Q^+}$ where $\mathbb{Q^+}$ is the set of positive rational number
$$\mathbb{Q^+} = \left\{ \frac{a}{b} \mid a, b\in \mathbb{Z^+} \wedge (a,b)= 1 \right\}$$
The notation $(a,b)= 1$ means that $a$ and $b$ are relatively prime. We are interested in the inverse of the function $S$ ( which is not anymore a function) since we can have several $X$’s for a particular element in $\mathbb{Q^+}$. Our question is: *Are all elements of $\mathbb{Q^+}$ defined for $S^{-1}$ ?*\
To answer the question above, we *[partition]{} the set $\mathbb{Q^+}$ into two subsets, namely:*
$$\mathbb{Q^{\geq}} = \left\{ \frac{a}{b} \mid \frac{a}{b}\in \mathbb{Q^+} \wedge a \geq b \right\}$$
$$\mathbb{Q^{<}} = \left\{ \frac{a}{b} \mid \frac{a}{b}\in \mathbb{Q^+} \wedge a < b \right\}$$
One important known theorem is given below:
[Division Algorithm [@burton]]{}
Given $a$ and $b$, with $b \neq 0$, there exists unique integers $q$ and $r$ such that $$a= bq+r$$ and $ 0 \leq r < \mid b \mid.$
With the previous theorem, we can focus on the set $\mathbb{Q^{<}}$ instead of $\mathbb{Q^+}$ since each element in $\mathbb{Q^{\geq}}$ can be written as $$\frac{a}{b} = q +\frac{r}{b}$$ such that $q$ is an integer and $\frac{r}{b} \in \mathbb{Q^{<}} $. It is sufficient to show that each element in $\mathbb{Q^{\geq}}$ is defined for $S^{-1}$ by showing $ S^{-1}$ is defined for $\mathbb{Z^+}-\{1\}$ and $\mathbb{Q^{<}}$, and whenever $X_1 \cup X_2$ for all $X_1$ in $\mathbb{Z^+}-\{1\}$ and $X_2$ in $\mathbb{Q^{<}}$ , $X_1 \cap X_2= \emptyset$.
We start with $\frac{r}{b} \in \mathbb{Q^{<}} $.
The first splitting recursive equation is what we call the *greedy algorithm for Egyptian fractions.*
[Greedy Algorithm]{}
Let $\frac{a}{b} \in \mathbb{Q^{<}}$, $a=a_0$, $b=b_0$, $u_{i+1}=\ceil{b_i/a_i}$, $a_{i+1}= a_i\cdot u_{i+1}-b_i$, and $b_{i+1}=b_i \cdot u_{i+1}$ and the recurrence relation below: $$\frac{a_i}{b_i} =\frac{1}{u_{i+1}} + \frac{a_{i}\cdot u_{i+1}-b_i}{b_i \cdot u_{i+1}}$$.
Initialize at $i=0$. While $a_i \nmid b_i$, add 1 to $i$, and use the recurrence relation until $a_i \mid b_i$. As such, the smallest fraction in the expansion of $\frac{a}{b}$ is $\frac{1}{u_n}$ with $i=n-2$. And the resulting expansion of $\frac{a}{b}$ is $\sum_{i=1}^n\frac{1}{u_i}.$
Some proofs of the theorem above is given in [@gc].
The relation $S^{-1}: \mathbb{Q^{<}} \rightarrow X$ is well-defined.
Let $c= \frac{a}{b} \in \mathbb{Q^{<}} $ and $1/x_i \in X$ for $i=1,2,3, \cdots, t-1, t$. Clearly, $t$ is the cardinality of set $X$. By greedy algorithm, existence of the finite set $X$ is immediate wherein the $n$’s are the $u_i$’s.
[[@botts]]{}
Let $\mathbb{X}$ be the set that contains all $X$’s in the relation $S^{-1}: 1 \rightarrow X$. $\mid\mathbb{X}\mid =\infty$.
Botts(1967) had proven this theorem and explained in detail the structure of the denominators at each stage of the chain reaction.
The relation $S^{-1}: \mathbb{Q^{\geq}} \rightarrow X$ is well-defined.
By theorem 2.1, Each $\frac{a}{b} \in \mathbb{Q^{\geq}}$ can be written as $$\frac{a}{b} = q +\frac{r}{b}$$ for integers $q$ and $r$. Thus by theorem 2.3 and 2.4, we have
$$\frac{a}{b} = \sum_{}^{}\frac{1}{l} + \sum_{}^{}\frac{1}{m}$$ for some collection of $l$ and $m$ in $\mathbb{Z^+} -\{1\}$. To guarantee that the expansion have all unique $l$’s and $m$’s, we start with the $m$’s. We know that by theorem 2.3, we have all unique $m$’s. All we need to do now is to guarantee that all $l$’s are not equal to any $m$, and each $l$ is unique in the sum. To do this, we let $\frac{1}{m_k}$ be the smallest term in the expansion of $\frac{r}{b}$. By theorem 2.4, we can start to expand $1$ at any starting point, $l_i\in \mathbb{Z^+} -\{1\}$ and we let this $l_i$ be equal to $ m_k+1$. And thus generate the expansion for 1. And since $q$ can be written as sum of 1’s, we can redo the process by simply making the smallest term in the first expansion of 1, say $\frac{1}{l_k}$, and make a new expansion for the next 1 of the expansion of $q$ by starting at $\frac{1}{l_k+1}$.
In conclusion, the answer to our question is: Yes, all elements of $\mathbb{Q^+}$ are defined for the relation $S^{-1}$.
Operations for Egyptian Fractions
=================================
The function $S$ is a many-to-one function that is why $S^{-1}$ is not a function. In this section, we focus on X’s such that for $X_1$ and $X_2$ in $\mathcal{P}(U_f)_{\mid X\mid \geq 2}$, $S(X_1)= S(X_2)$. In this manner, we operate on the elements of $X$ to produce another $X'$ such that $S(X)= S(X')$ and use the notation $\mathbb{O}$ as the operator on X, $\mathbb{O}: X \rightarrow X'$.
Let $X$ be the original Egyptian Fraction and $X'$ be the new Egyptian Fraction from $X$ such that $S(X)= S(X')$.
- If $\mid X \mid < \mid X' \mid$, then the operator $\mathbb{O}$ is said to be a splitter,
- If $\mid X \mid = \mid X' \mid$, then the operator $\mathbb{O}$ is said to be a rewriter, and
- If $\mid X \mid > \mid X' \mid$, then the operator $\mathbb{O}$ is said to be a merger.
We start with *splitter* operator $\mathbb{O}$, $$\frac{1}{n}= \frac{1}{n+1} + \frac{1}{n(n+1)}$$ for $X$ such that $S(X)=1$.For simplicity of notation we can write the splitter operator ( or equation) above as $$(n) = (n+1, n(n+1)).$$
For example,
$$1= \frac{1}{2}+\frac{1}{3}+\frac{1}{6}$$
$$1= \frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{12}$$
The first $N =\{2,3,6\}$ and $N'=\{2,4,6,12\}$. It can be seen that $N \cap N' = \{2,6\}$ which are the elements that did not change when operated. Evidently, 3 becomes 4 and 12 which came about using the splitter operator above with $n=3$.
In fact, the aforementioned equation is only a special case of $$(ab) = (a(a+b) ,b(a+b))$$ when $a=1$ and $b=n$. And for odd denominators, we have $$1= \frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{9}+\frac{1}{11}+\frac{1}{15}+\frac{1}{35}+\frac{1}{45}+\frac{1}{231} \cite{intef}.$$ In Knott [@Knott], Ben Thurston (May 2017) emailed Knott two other simple formulas::
$$(ab) = (a(a+b) ,b(a+b))$$ $$(abc) = (a(ab+bc+ac) ,b(ab+bc+ac), c(ab+bc+ac))$$
Generally, it is easy to see that
$$(\prod_{i=1}^mx_i)= (x_1\cdot z,x_2 \cdot z, \cdots , x_{m-1}\cdot z, x_m \cdot z)$$ where $$z =\sum_{j=1}^m\frac{1}{x_j}(\prod_{i=1}^mx_i)$$
To illustrate, we let $x_i=i+1$ for $i=1,2,3,4,5$. Thus $$(720) = (7200,4800,3600,2880,2400)$$ Simplifying the above equation, $$\frac{1}{3}= \frac{1}{10}+\frac{1}{12}+\frac{1}{15}+\frac{1}{20}+\frac{1}{30}.$$
The methods illustrated above are splitter operators from one term to at least two terms. The latest example $(3)= (10,12,15,20,30)$ is from one term to five terms. The next section includes the parity condition on splitting the fraction into $2$ or $3$ parts (least number of parts needed for splitting).
Splitter Operator with Parity Condition
=======================================
In the work of Knott [@Knott], the case for splitting even to two even Egyptian fractions is given below which he called the Expand Even Rule,
$$(2n) = (2(n+1),2n(n+1))$$ for integers $n \geq 2$.
This came from the splitting equation $$(n) = (n+1, n(n+1))$$ multiplied both sides by $\frac{1}{2}$. The reason for such action is to expand by preserving evenness since for $$(n) = (n+1, n(n+1)).$$ Denote $e$ for even and $o$ for odd, we have all the possible cases below:
$n$ $n+1$ $n(n+1)$
----- ------- ----------
e o e
o e e
Clearly, the splitting equation $(n) = (n+1, n(n+1))$ is not a *parity-preserving splitting equation*.
An operator $$(a_1, a_2,..., a_{n-1}, a_n) = (b_1, b_2,...,b_{m-1}, b_{m})$$ is said to be a parity-preserving operator ( or equation) if all of the entries are of the same parity.
To illustrate the above definition, we take the splitter operator $(n) = (n+1, n(n+1))$ where $a_1 = n$, $b_1 = n+1$, and $b_2= n(n+1)$.
The equation $(2n) = (2(n+1),2n(n+1))$ is clearly a parity-preserving equation. In this equation, all terms are even. For odd, we have the parity-preserving splitting equations below which are given by Peter in [@SplitEq].
If $k$ is odd, $$\frac{1}{2k+1}= \frac{1}{3k+2} + \frac{1}{6k+3}+\frac{1}{18k^2+21k+6}$$
If $k$ is even, $$\frac{1}{2k+1}= \frac{1}{3k+3} + \frac{1}{6k+3}+\frac{1}{6k^2+9k+3}$$
We investigate the equations above by redefining and stating them formally:
Let $n=2k+1$, $a=3$, $b= 3k+2$, and $c=k+1$ for positive integers $k$. If $k$ is odd, then $(n)= (b,an,abn)$.Otherwise for even $k$, $(n)= (b+1,an,a\cdot(b+1)\cdot(n))$.
We split $\frac{1}{n}$ to three equal parts: $(n)= (3n,3n,3n)$, so $a=3$.The middle part is $3n$, which is odd since $n=2k+1$ for positive integers $k$.Now, let $(n)= (b,3n,b \cdot 3n)$. Finding $b$, we have $$\frac{1}{b}+\frac{1}{b\cdot 3n}= \frac{2}{3n}$$ Solving the equation, $3n+1=2b$, then $$b = \frac{3(2k+1)+1}{2}= 3k+2$$ Therefore $b$ is odd only if $k$ is odd. So, if $k$ is even, then $b+1$ must be odd.
A table below gives a summary of the previous theorem:
$n$ $k$ $b$ $b+1$ $an$ $abn$ $a\cdot(b+1)\cdot n$
----- ----- ----- ------- ------ ------- ----------------------
o o o e o o e
o e e o o e o
The equation above splits an odd $n$ to three odd terms because it is impossible to split odd $n$ into two parts. Also, the table illustrates that for odd $n$ where the $k$ in $n=2k+1$, If $k$ is odd, then all entries in $(b,an,abn)$ are odd. And if $k$ is even, then all entries in $(b+1,an,a\cdot(b+1)\cdot(n))$ are odd.
There exists no odd positive integers $a$ and $b$ greater than 1 for odd number $n$ such that $(n)= (a,b)$.
Suppose there exists odd $a$ and $b$ for odd $n$ such that $(n)= (a,b)$. Thus, $(n)= (2k_1+1,2k_2+1)$ for positive integers $k_1, k_2$. Since $n$ is an integer, then $2(k_1+k_2+1) {\mid}(2k_1+1)(2k_2+1)$. But this is false since $2{\nmid}ab$.
A Rewriter for Egyptian Fractions
==================================
So far, what we have is to split an unit fraction to at least two parts. In this section, we introduce an equation that can rewrite two unit fractions into two another unit fractions which can be seen as an operator.
Let $d$ and $q$ be positive integers where $q > 1$. If $r = q+d$, and $s = qr-d$, then $$(s)(rs)=(qr)(qs)$$
$$\begin{split}
\frac{1}{qr}+\frac{1}{qs} & = \frac{1}{r}\left(\frac{1}{q}+\frac{r}{qs}\right) \\
& = \frac{1}{r}\left(\frac{1}{q}+\frac{d}{qs}+\frac{1}{s}\right)\\
& = \frac{1}{r}\left(\frac{qr-d+d}{qs}+\frac{1}{s}\right)\\
& = \frac{1}{r}\left(\frac{qr}{qs}+\frac{1}{s}\right)\\
& = \frac{1}{r}\left(\frac{r}{s}+\frac{1}{s}\right)\\
& = \frac{1}{s}+\frac{1}{rs}\\
\end{split}$$
The terms in the equation above are $s$, $rs$ ,$qr$, $qs$. With these, we explore the related inequality for the terms.
Let $d$ and $q$ be positive integers where $q > 1$, $r = q+d$, and $s = qr-d$, then $$q < r < s < qr < qs < rs.$$
First, $q< r$ is true, then we establish $r < s$. We start with the fact that $1 < 9$, then $(d^2+6d +1) < (d^2+6d+9)$. And then, $$(d^2+6d +1) < (d+3)^2$$ $$\sqrt{(d^2+6d +1)} < d+3$$ $$1-d + \sqrt{(d^2+6d +1)} < 4$$ $$\frac{1-d + \sqrt{(d^2+6d +1)}}{2} < 2$$
Note that $q$ is at least 2. Thus, $\frac{1-d + \sqrt{(d^2+6d +1)}}{2} < 2 \leq q$ And then, $$\frac{1-d + \sqrt{(d^2+6d +1)}}{2} < q$$ $$\frac{-(d-1) + \sqrt{(d-1)^2 - 4(-2d)}}{2} < q$$ $$0 < q^2 + (d-1)q -2d$$ $$0 < (q-1)(q+d)-d$$ $$0 < qr-r+d$$ $$r < qr-d$$ $$r < s$$
With this,all other related inequalities are straightforward.
To end this section, we show the parity table of the equation above.
$d$ $q$ $r$ $s$ $qr$ $qs$ $rs$
----- ----- ----- ----- ------ ------ ------
o o e o e o e
o e o o e e o
e o o o o o o
e e e e e e e
The table generated in this paper follows the Boolean algebra such that $e=0$ and $o=1$. In addition, we have a special theorem below about odd parity.
Let $r = q+d$, and $s = qr-d$. The splitting equation $(s)(rs)=(qr)(qs)$ is an odd parity preserving equation if and only if the integer $q > 1$ is odd, and the value of $d$ is a positive even number.
We begin the proof by assuming the equation to be an odd parity preserving equation. As such, we define the function $p(t) = e$ if the expression $t$ is even, and $p(t) = o$ if $t$ is odd. Thus, $$p(s)=p(rs)=p(qr)=p(qs) = o.$$ Since $p(qr)$= o then, $p(q) = p(r)=o$. And since $p(r) = p(q+d) = o$, then $p(q)$ and $p(d)$ must have a different parity. But since, $p(q)$ is odd, then $p(d)$ must be even. As for the other direction, if $p(q)=o$ and $p(d)=e$, then $p(r) = o$, $p(s) = p(qr-d) =p(q)\cdot p(r)-p(d) =o\cdot o - e =o$. And since we know that $ o\cdot o = o$, then $p(rs)=p(qr)=p(qs) = o$.
Ultimately, we generate five examples of the previous theorems
$d$ $q$ $r$ $s$ $qr$ $qs$ $rs$
----- ----- ----- ----- ------ ------ ------
1 2 3 5 6 10 15
2 3 5 13 15 39 65
2 5 7 33 35 165 231
4 3 7 17 21 51 119
4 5 9 41 45 205 369
Rewriting some examples in the table in a conventional form, we have the following identities:
$$\frac{1}{6}+\frac{1}{10}= \frac{1}{5}+\frac{1}{15}$$ $$\frac{1}{15}+\frac{1}{39}= \frac{1}{13}+\frac{1}{65}$$ $$\frac{1}{21}+\frac{1}{51}= \frac{1}{17}+\frac{1}{119}$$
Acknowledgement
===============
The author would like to thank Peter and Ross Millikan for answering my questions and sharing useful materials at math.stackexchange.com. The author would like to thank Jose Arnaldo Dris for valuable conversation and unwavering support in preparing this manuscript.
[1]{}
Burton, David M. (2010). Elementary Number Theory. McGraw-Hill. pp. 17–19. ISBN 978-0-07-338314-9.
Botts, Truman. (1967). A Chain Reaction Process in Number Theory, Mathematics Magazine, Vol. 40, No. 2, pages 55-65
G. Carlo, “discrete mathematics - Fractions in Ancient Egypt,” Mathematics Stack Exchange, 02-Aug-2013. \[Online\]. Available: https://math.stackexchange.com/questions/458238/fractions-in-ancient-egypt. \[Accessed: 25-Mar-2020\].
K. A. Dagal, “elementary number theory - On A Splitting Equation of an Egyptian fraction to Egyptian fractions such that all produced fractions have odd denominators.,” Mathematics Stack Exchange, 18-Mar-2020. \[Online\]. Available: https://math.stackexchange.com/questions/3585135/on-a-splitting-equation-of-an-egyptian-fraction-to-egyptian-fractions-such-that. \[Accessed: 23-Mar-2020\].
K. A. Dagal, “number theory - Egyptian fraction representation of $1$ where all denominators of the fractions are odd.,” Mathematics Stack Exchange, 17-Mar-2020. \[Online\]. Available: https://math.stackexchange.com/questions/3584240/egyptian-fraction-representation-of-1-where-all-denominators-of-the-fractions. \[Accessed: 23-Mar-2020\].
R. Knott, “Egyptian Fractions,” www.maths.surrey.ac.uk, 21-Feb-2020. \[Online\]. Available: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html. \[Accessed: 23-Mar-2020\].
|
---
abstract: 'We explore two-dimensional sigma models with $(0,2)$ supersymmetry through their chiral algebras. Perturbatively, the chiral algebras of $(0,2)$ models have a rich infinite-dimensional structure described by the cohomology of a sheaf of chiral differential operators. Nonperturbatively, instantons can deform this structure drastically. We show that under some conditions they even annihilate the whole algebra, thereby triggering the spontaneous breaking of supersymmetry. For a certain class of Kähler manifolds, this suggests that there are no harmonic spinors on their loop spaces and gives a physical proof of the Höhn–Stolz conjecture.'
author:
- |
Junya Yagi\
\
Center for Frontier Science, Chiba University\
Chiba 263–8522 Japan\
`yagi@cfs.chiba-u.ac.jp`
title: 'Chiral Algebras of $(0,2)$ Models'
---
Introduction
============
Supersymmetric sigma models in two dimensions have played central roles in a number of important physical and mathematical developments during the past few decades. A key concept underlying much of the developments is that of the chiral rings of sigma models with $(2,2)$ supersymmetry [@Lerche:1989uy]. These finite-dimensional cohomology rings are basic ingredients of topological sigma models [@Witten:1988xj; @Witten:1991zz], and intimately connected to, among other things, Gromov–Witten invariants [@MR809718; @Witten:1988xj], Floer homology [@MR933228; @MR948771; @MR965228; @MR987770; @MR1001276], and mirror symmetry [@Dixon:1987bg; @Lerche:1989uy; @Hori:2000kt].
While the chiral rings of $(2,2)$ models are clearly very interesting, we also know that some of the beautiful structures of two-dimensional supersymmetric sigma models arise in essentially infinite-dimensional contexts. For example, the elliptic genera encode infinite series of topological invariants of the target space [@Witten:1986bf; @Witten:1987cg]. It is then natural to ask whether there are infinite-dimensional analogs of the chiral rings. The answer is “yes.” They are the *chiral algebras* of sigma models with $(0,2)$ supersymmetry [@Vafa:1989pa; @Witten:1993jg].
The chiral algebra of a $(0,2)$ model is the cohomology of local operators with respect to one of the supercharges, graded by the right-moving R-charge and equipped with operator product expansions (OPEs). As a consequence of $(0,2)$ supersymmetry, its elements vary holomorphically on the worldsheet and form a structure analogous to the chiral algebra of a conformal field theory (CFT), the operator product algebra of holomorphic fields. It is well known that one can twist $(2,2)$ models to obtain topological field theories characterized by their chiral rings. For $(0,2)$ models, one can perform a similar quasi-topological twisting which turns them into holomorphic field theories characterized by their chiral algebras.
Classically, the chiral algebra of a twisted $(0,2)$ model is isomorphic, as a graded vector space, to the direct sum of the Dolbeault cohomology groups of a certain infinite series of holomorphic vector bundles over the target space. Quantum mechanically, this isomorphism gets deformed by quantum corrections. Like the chiral rings, the chiral algebra is independent of the choice of the metric on the target space, hence can be computed in the large volume limit where the theory is weakly coupled and the path integral localizes to instantons. But unlike the chiral rings, it receives perturbative corrections as well as instanton corrections, because the contributions from bosonic and fermionic fluctuations do not cancel due to the lack of left-moving supersymmetry. This complication leads to the interesting subject of perturbative chiral algebras.
At the level of perturbation theory, the physics of a sigma model is governed by the local geometry of the target space. On the other hand, we can always deform the target space metric to make it locally flat without affecting the chiral algebra. Combining these observations, we reach a surprising conclusion: the perturbative chiral algebra can be described locally by a sigma model with flat target space, therefore reconstructed by gluing locally defined *free* theories globally over the target space.
This fact was exploited by Witten [@Witten:2005px] to show that in the absence of left-moving fermions, the perturbative chiral algebra of a twisted $(0,2)$ model can be formulated as the cohomology of a sheaf of chiral differential operators on the target space, a notion introduced earlier in mathematics by Malikov et al. [@Malikov:1998dw]. In this picture, the moduli of the perturbative chiral algebra are encoded in the different possible ways of gluing relevant free CFTs, whereas the anomalies of the theory manifest themselves in the obstructions to doing so consistently. When left-moving fermions are present and take values in the tangent bundle of the target space, it was shown by Kapustin [@Kapustin:2005pt] that the perturbative chiral algebra is given by the cohomology of the chiral de Rham complex [@Malikov:1998dw]. The theory of perturbative chiral algebras has been further developed along these lines by Tan [@Tan:2006qt; @Tan:2006by; @Tan:2006zg; @Tan:2007bh].
Nonperturbatively, instantons can change the picture radically [@Witten:2005px; @Tan-Yagi-1; @Tan-Yagi-2; @MR2415553]. A particularly striking example is the model with no left-moving fermions whose target space is the flag manifold $G/B$ of a complex simple Lie group $G$. The perturbative chiral algebra of this model is infinite-dimensional and has the structure of a ${\widehat{\mathfrak{g}}}$-module of critical level [@Malikov:1998dw; @MR1042449; @MR2290768]. In the presence of instantons, however, the equation $1 = 0$ holds in the cohomology and the chiral algebra *vanishes*.
One clue to the existence of such a nonperturbative phenomenon lies in a conjecture made independently by Höhn and Stolz [@MR1380455] in the mid 1990s. The Höhn–Stolz conjecture asserts that the elliptic genus of a supersymmetric sigma model with no left-moving fermions vanishes if the target space $M$ admits a Riemannian metric of positive Ricci curvature.
Stolz gave a heuristic argument for his conjecture based on the geometry of the loop space $\CL M$, the space of smooth maps from the circle $S^1$ to $M$. It goes as follows. Let us assume that the scalar curvature of $\CL M$ is given, at each loop $\gamma \in \CL M$, by the integral of the Ricci curvature of $M$ along $\gamma$. Then $\CL M$ has positive scalar curvature if $M$ has positive Ricci curvature. By analogy with the Lichnerowicz theorem, this would imply that $\CL M$ has no harmonic spinors. Meanwhile, supersymmetric states of the theory may be identified with harmonic spinors on $\CL
M$. Hence the theory would have no supersymmetric states, and since the elliptic genus counts the number of bosonic supersymmetric states minus the number of fermionic ones at each energy level, it would vanish then.
If Stolz’s reasoning is correct, the positivity of the Ricci curvature implies not only the vanishing of the elliptic genus, but also the spontaneous breaking of supersymmetry. Flag manifolds have positive Ricci curvature, so supersymmetry should be broken in the models into these spaces. Supersymmetry breaking is indeed triggered whenever the chiral algebra vanishes—instantons tunnel between infinitely many perturbative supersymmetric states and lift all of them at once.
In fact, what happens for the flag manifold model is a special case of a more general phenomenon: the chiral algebra of a $(0,2)$ model with no left-moving fermions vanishes nonperturbatively if the target space is a compact Kähler manifold with positive first Chern class and contains an embedded ${\mathbb{P}}^1$ with trivial normal bundle.
This vanishing theorem—or rather, “theorem” with quotation marks—is the main result of this paper, which we will “prove” by a physical argument. As a “corollary,” the “theorem” implies that supersymmetry is spontaneously broken. Therefore, for this particular class of Kähler manifolds, it suggests that there are no harmonic spinors on their loop spaces and gives a physical proof of the Höhn–Stolz conjecture.
The paper is organized as follows. In Section 2, we introduce the chiral algebras of $(0,2)$ models and discuss their general properties. Section 3 is devoted to the sheaf theory of perturbative chiral algebras. Finally, in Section 4, we establish the vanishing “theorem” and explain its relation to supersymmetry breaking and the geometry of loop spaces.
Chiral Algebras of $(0,2)$ Models {#CA}
=================================
We now begin our study of the chiral algebras of $(0,2)$ models. The focus of this section is on general properties of these algebras that do not depend on specific details of the target space geometry.
$(0,2)$ Models {#CA-models}
--------------
Two-dimensional sigma models have $(0,2)$ supersymmetric extensions if the target space is strong Kähler with torsion (strong KT) [@Witten:2005px]. A hermitian manifold is called strong KT if the $(1,1)$-form $\omega$ associated to the hermitian metric satisfies ${\partial}{{\bar\partial}}\omega = 0$. Kähler manifolds are strong KT since the Kähler form satisfies ${\partial}\omega = {{\bar\partial}}\omega = 0$. In this paper we will only consider $(0,2)$ models with Kähler target spaces. Let us review how these models are constructed.
Let $\Sigma$ be a Riemann surface and $X$ a Kähler manifold of complex dimension $d$. The bosonic sigma model with worldsheet $\Sigma$ and target space $X$ is a quantum field theory of maps $\phi\colon \Sigma \to X$. Imposing $(0,2)$ supersymmetry requires the introduction of two right-moving fermions, $\psi_+$ and ${\bar\psi}_+$. These are worldsheet spinors with values respectively in the holomorphic and antiholomorphic tangent bundles of $X$: $$\psi_+ \in \Gamma(\Kb_\Sigma^{1/2} \otimes \phi^*T_X) \, , \qquad
{\bar\psi}_+ \in \Gamma(\Kb_\Sigma^{1/2} \otimes \phi^*{{\wb@rl{T}}}_X) \, .$$ Here $\Kb_\Sigma^{1/2}$ is a square root of the antiholomorphic canonical bundle of $\Sigma$. The simplest $(0,2)$ model is constructed with this minimally $(0,2)$ supersymmetric field content.
In the field space, $(0,2)$ supersymmetry is realized as the transformation $$\label{susy}
\begin{alignedat}{3}
\delta\phi^i &= -\epsilon_-\psi_+^i \, , &\qquad
\delta\phi^\ib &= \epsilonb_-{\bar\psi}_+^\ib \, , \\
\delta\psi_+^i &= i\epsilonb_-{\partial}_\zb\phi^i \, , &
\delta{\bar\psi}_+^\ib &= -i\epsilon_-{\partial}_\zb\phi^\ib \, ,
\end{alignedat}$$ where $\epsilon_-$ and $\epsilonb_-$ are sections of $\Kb_\Sigma^{-1/2}$. We define the right-moving supercharges $Q_+$ and ${{\wb@rl{Q}}}_+$ so that $-i\epsilon_- Q_+ + i\epsilonb_- {{\wb@rl{Q}}}_+$ acts by this transformation. The supercharges then satisfy $$\label{screl}
\begin{gathered}
\{Q_+, Q_+\} = \{{{\wb@rl{Q}}}_+, {{\wb@rl{Q}}}_+\} = 0 \, , \\
\{Q_+,{{\wb@rl{Q}}}_+\} = -i{\partial}_\zb = \frac{1}{2} (H - P) \, ,
\end{gathered}$$ and generate the $(0,2)$ supersymmetry algebra together with the generators $H$, $P$ of translations, $M$ of rotations, and $F_R$ of the right-moving ${\mathrm{U}}(1)$ R-symmetry. Under the last symmetry, $\psi_+$ has charge $-1$ and ${\bar\psi}_+$ has charge $+1$; thus $Q_+$ has charge $-1$ and ${{\wb@rl{Q}}}_+$ has charge $+1$.
To construct a $(0,2)$ supersymmetric action, we choose a Kähler metric $g$ on $X$ and make the operator $g_{i\jb} \psi_+^i
{\partial}_z\phi^\jb$ of R-charge $-1$. Then the action $$\label{S1}
S = \int_\Sigma d^2z
\{{{\wb@rl{Q}}}_+, g_{i\jb} \psi_+^i {\partial}_z\phi^\jb\}$$ is invariant under the R-symmetry and, by virtue of the relation ${{\wb@rl{Q}}}_+^2 = 0$, under the symmetry generated by $i\epsilonb_-{{\wb@rl{Q}}}_+$ provided that $\epsilonb_-$ is antiholomorphic and so commutes with the ${\partial}_z$ inside. This action is also invariant under $-i\epsilon_-{{\wb@rl{Q}}}_+$ for antiholomorphic $\epsilon_-$, as becomes clear if we rewrite it as $$\label{S2}
S = \int_\Sigma d^2z
\{Q_+, g_{i\jb} {\partial}_z\phi^i {\bar\psi}_+^\jb\} \, .$$ Expanding the anticommutators and using the Kähler condition, one can check that the two expressions and both coincide with $$\label{S}
S = \int_\Sigma d^2z (
g_{i\jb} {\partial}_\zb\phi^i {\partial}_z\phi^\jb
+ ig_{i\jb} \psi_+^i D_z{\bar\psi}_+^\jb) \, .$$ Here the covariant derivative $D_z$ is the ${\partial}$ operator coupled to the pullback of the Levi-Civita connection $\Gamma$ on $X$. Explicitly, $D_z{\bar\psi}_+^\ib = {\partial}_z{\bar\psi}_+^\ib + {\partial}_z\phi^\jb
\Gamma_{\jb\kb}^\ib{\bar\psi}_+^\kb$.
We can add a topological invariant to the action. For a closed two-form $B$ on $X$, the functional $$\label{SB}
S_B = \int_\Sigma \phi^* B$$ depends only on the cohomology class of $B$ and the homotopy class of $\phi$. As such, it is invariant under any continuous transformations, especially the supersymmetry transformation and the R-symmetry. The topological invariant $S_B$ vanishes at the level of perturbation theory where one deals with homotopically trivial maps, but affects the dynamics nonperturbatively.
We have obtained a $(0,2)$ supersymmetric action. To complete the construction, we need to make sure that a sensible quantum theory based on this action exists. It turns out that $X$ must satisfy two topological conditions for that. First, $X$ must be spin, or equivalently, its first Chern class must be even: $$\label{spin}
c_1(X) \equiv 0 \quad (\text{mod 2}) \, .$$ As we will see, this condition ensures that the fermion parity $(-1)^{F_R}$ is well defined. Second, the second Chern character ${\mathop{\mathrm{ch}}\nolimits}_2(X) = c_1(X)^2/2 - c_2(X)$, which is also a half of the first Pontryagin class $p_1(X)$, must be zero: $$\label{p1}
\frac{1}{2} p_1(X) = 0 \, .$$ This is the condition for the absence of sigma model anomaly [@Moore:1984dc; @Moore:1984ws], the obstruction to finding a well-defined path integral measure. From the viewpoint of the loop space $\CL X $, the first condition means that $\CL X$ is orientable [@Witten:1985mj; @MR0816738], while the second condition (given the first) is interpreted as the condition for $\CL X$ to admit spinors [@Killingback:1986rd].
There is also a geometric condition. The renormalization group generates a flow of the target space metric. At the one-loop level, the metric $g(\mu)$ renormalized at an energy scale $\mu$ obeys the equation $$\label{RG}
\mu \frac{d}{d\mu} g_{i\jb}(\mu)
= \frac{1}{2\pi} R_{i\jb}\bigl(g(\mu)\bigr) \, ,$$ where $R_{i\jb}(g)$ is the Ricci curvature of $g$ [@Friedan:1980jf; @Friedan:1980jm; @AlvarezGaume:1981hn; @Callan:1985ia]. From this equation, we see that $g(\mu)$ gets larger as $\mu$ gets larger if the Ricci curvature is positive. Since the theory is weakly coupled when the target space has large volume, we have asymptotic freedom in this case. In contrast, if the Ricci curvature is negative, the theory is strongly coupled for large $\mu$ and does not have a well-defined ultraviolet limit. Hence, for the microscopic theory to exist, the Ricci curvature should be semipositive. When the above conditions are satisfied, the action plus defines the simplest version of $(0,2)$ models. The supercharges are given by $$Q_+ = \oint \! d\zb \, g_{i\jb} \psi_+^i {\partial}_\zb\phi^\jb \, , \qquad
{{\wb@rl{Q}}}_+ = \oint \! d\zb \, g_{i\jb} {\partial}_\zb\phi^i {\bar\psi}_+^\jb$$ and satisfy the reality condition $Q_+^\dagger = {{\wb@rl{Q}}}_+$.
If one has a holomorphic vector bundle $E$ over $X$, one can extend the model by adding left-moving fermions with values in $E$. This extended model is called the heterotic model. Since the left-movers contribute to sigma model anomalies in the opposite way as the right-movers do, their presence changes the anomaly cancellation condition to $$\frac{1}{2} p_1(X) = \frac{1}{2} p_1(E) \, .$$ This is trivially satisfied if $E = T_X$, in which case the model actually has $(2,2)$ supersymmetry. It is also possible to add superpotentials [@Witten:1993yc]. For brevity, we will not consider these extensions in this paper. We refer to Tan [@Tan:2006qt] for the perturbative aspects of the heterotic model.
Chiral Algebras {#CA-CA}
---------------
Having formulated $(0,2)$ models, let us introduce the notion of their chiral algebras. Among the two supercharges we will use ${{\wb@rl{Q}}}_+$ exclusively, so simply write $Q = {{\wb@rl{Q}}}_+$. Then $Q^\dagger = Q_+$.
Consider the action of $Q$ on local operators $\CO$ given by supercommutator $[Q, \CO\}$. The $Q$-action increases the R-charge of $\CO$ by one, and squares to zero. Given a $(0,2)$ model, therefore, we can define the $Q$-cohomology of local operators graded by the R-charge.
Since ${\partial}_\zb \propto H - P$ is $Q$-exact by the $(0,2)$ supersymmetry algebra, it acts trivially in the $Q$-cohomology: if $\CO$ is $Q$-closed, then ${\partial}_\zb\CO$ is $Q$-exact. Thus $Q$-cohomology classes vary holomorphically on $\Sigma$. Moreover, two classes can be multiplied by $[\CO] \cdot [\CO'] = [\CO\CO']$. From these facts, it follows that the $Q$-cohomology of local operators inherits a holomorphic OPE structure from the underlying theory: $$\label{OPE}
[\CO(z)] \cdot [\CO'(w)]
\sim \sum_k c_k(z - w) [\CO_k(w)] \, .$$ The coefficient functions $c_k(z - w)$ are holomorphic away from the diagonal $z = w$ where there can be poles.
The holomorphic $Q$-cohomology of local operators, equipped with this natural OPE structure, is the chiral algebra of the $(0,2)$ model. We will denote the chiral algebra by $\CA$, and its R-charge $q$ subspace by $\CA^q$. As is clear from the construction, the chiral algebra of a $(0,2)$ model has the structure of a chiral algebra in the sense of CFT, except that the grading by conformal weight is missing. We will see later that the chiral algebra carries a similar (but possibly reduced to ${\mathbb{Z}}_n$) grading after the theory is twisted.
The chiral algebra forms a closed sector of a $(0,2)$ model in the following sense. Consider the $n$-point function of $Q$-closed local operators: $$\label{nPF}
{\bigl\langle \CO_1(z_1,\zb_1) \dotsm \CO_n(z_n,\zb_n) \bigr\rangle} \, .$$ If one of the operators is $Q$-exact, $\CO_i = [Q, \CO_i'\}$, then the $n$-point function becomes $\pm{\langle [Q, \CO_1 \dotsm \CO_i' \dotsm
\CO_n\ \rangle}}$. Computed with a $Q$-invariant action and path integral measure, this is the integral of a “$Q$-exact form” on the field space and vanishes. The $n$-point function thus depends only on the $Q$-cohomology classes $[\CO_i]$. In particular, it is a holomorphic (more precisely, meromorphic) function of the insertion points.
So far, we have considered the chiral algebra of a fixed $(0,2)$ model, defined by a fixed $Q$-invariant action. Of course, different choices of the action lead to different chiral algebras in general. Imagine deforming the theory by perturbing the action: $$S \to S + \delta S \, .$$ For this deformation to preserve the $Q$-invariance, $\delta S$ must be $Q$-closed. If, however, $\delta S$ is $Q$-exact, the chiral algebra remains unchanged. To see this, express the matrix elements of $Q$ as path integrals on a cylinder of infinitesimal length, with a contour integral of the supercurrent sandwiched between various boundary conditions. One can show that for a $Q$-exact perturbation, the matrix elements of $Q$ in the deformed theory are equal to those of $Q + [Q, \delta A]$ in the original theory for some operator $\delta A$. But this latter operator is related to $Q$ by the conjugation $$Q \to e^{-\delta A} Q e^{\delta A} \, ,$$ hence defines an isomorphic chiral algebra. The $n$-point function is also unchanged because the perturbation just introduces $Q$-exact insertions.
Looking back at the action of our model, we see that deformations of the target space metric just give $Q$-exact perturbations. Therefore, the chiral algebra is independent of the Kähler structure of the target space. It does depend on the complex structure, however. For this enters the very definition of the supersymmetry transformation.
Instantons {#CA-instantons}
----------
An important property of the chiral algebra is that it receives contributions only from instantons and small fluctuations around them. In the case of our model, instantons obey $$\{Q, \psi_+^i\} = {\partial}_\zb\phi^i = 0 \, ,$$ so they are holomorphic maps from $\Sigma$ to $X$. It is this localization principle that makes the chiral algebra effectively computable.
To prove the localization, we rescale the target space metric so that it becomes very large. In this large volume limit, the path integral localizes to the zeros of the bosonic action $$\int_\Sigma d^2z \, g_{i\jb} {\partial}_\zb\phi^i {\partial}_z\phi^\jb \, ,$$ namely holomorphic maps, or instantons, as promised. Incidentally, there is another situation in which the same localization arises. That is when the path integral computes the correlation function of $Q$-closed operators [@Witten:1991zz]. This situation is not really relevant for us, though. In order to determine the chiral algebra, we have to ask, in the first place, whether a given local operator is $Q$-closed or not.
The space $\CM$ of holomorphic maps from $\Sigma$ to $X$ is called the instanton moduli space. It decomposes into disconnected components labeled by the homology class $\beta$ that the image of $\Sigma$ represents in $X$: $$\CM = \bigoplus_{\beta \in H_2(X,{\mathbb{Z}})} \CM_\beta \, .$$ Correspondingly, the path integral splits into distinct sectors each of which integrates over the neighborhood of a single connected component of $\CM$. In sigma model perturbation theory, one expands in the inverse volume of the target space in the zero-instanton sector. Instantons of $\beta = 0$ are constant maps, whose moduli space $\CM_0 {\cong}X$.
When $c_1(X) \neq 0$, instantons induce two important nonperturbative effects. One is the violation of the R-charge conservation, which breaks the R-symmetry down to a discrete subgroup. The other is the appearance of powers of a dynamical scale $\Lambda$ generated via dimensional transmutation, which allows objects of different scaling dimensions to show up as quantum corrections.
The anomaly in the R-symmetry is due to a nontrivial transformation of the fermionic path integral measure under this symmetry. The complex conjugate of ${\bar\psi}_+$ is a section of $K_\Sigma^{1/2} \otimes
\phi^*T_X$, while $\psi_+$ can be identified with a $(0,1)$-form with values in the same bundle. Thus, we can define $\Db = D_\zb d\zb$ and expand these fields in the eigenmodes of $\Db^*\Db$ and $\Db\Db^*$: $$\psi_+ = \sum_s b_0^s v_{0,s} + \sum_n b^n v_n \, , \qquad
{\bar\psi}_+ = \sum_r c_0^r \ub_{0,r} + \sum_n c^n \ub_n \, ,$$ Here $\ub_{0,r}$, $v_{0,s}$ are zero modes, $\ub_n$, $v_n$ nonzero modes, and $b_0^s$, $c_0^r$, $b^n$, $c^n$ anticommuting coefficients. The fermionic path integral measure is the formal product $$\label{FPM}
\prod_{r, s, n} db_0^s \, dc_0^r \, db^n dc^n \, .$$ The nonzero mode part of the measure is neutral under the R-symmetry because of the pairing of nonzero modes. The zero mode part has R-charge equal to the number of $\psi_+$ zero modes minus the number of ${\bar\psi}_+$ zero modes, that is, minus the index of the $\Db$ operator. For a compact Riemann surface $\Sigma$, the index is given by $$\label{index}
\int_\Sigma \phi^*c_1(X)$$ which (recalling $c_1(X) \equiv 0$ mod $2$) is equal to $2k$ for some integer $k$. The R-charge is violated by this amount in an instanton background. We see that the R-symmetry is broken to a ${\mathbb{Z}}_{2n}$ subgroup in the presence of instantons, where $n$ is the greatest common divisor of the integers $k$.
Instantons produce powers of $\Lambda$ when $c_1(X) \neq 0$ because the $B$ field is renormalized as $$\label{BREN}
[B(\mu)] = [B_0] + \ln\frac{\mu}{\Lambda} c_1(X) \, ,$$ with $[B_0]$ a fixed class in $H^2(X,{\mathbb{C}})$. We will derive this formula at the end of this section. Accordingly, instanton corrections violating R-charge by $2k$ units are weighted by the topological factor $$\label{e-SB}
e^{-S_B}
=\Bigl(\frac{\Lambda}{\mu}\Bigr)^{2k}
\exp\Bigl(\int_\Sigma \phi^* B_0\Bigr) \, .$$ An extra factor of $\mu^{2k}$ should come from somewhere else to cancel the dependence on $\mu$. Altogether, these corrections are proportional to $\Lambda^{2k}$ and can relate objects of scaling dimensions differing by $2k$.
For instanton corrections to be under control, we should choose $B_0$ such that the factor is exponentially suppressed for nonconstant holomorphic maps. The Kähler form is a good choice, for example.
Twisting {#CA-twisting}
--------
The action of our model is invariant under supersymmetries whose transformation parameters are antiholomorphic sections of the bundle $\Kb_\Sigma^{-1/2}$. If $\Sigma$ is topologically nontrivial, this bundle may not admit any global antiholomorphic sections. In that case the supersymmetries exist at best locally on $\Sigma$, hence so does the chiral algebra. However, what we need to define the chiral algebra is really one of the two supersymmetries, not both. So it would be nice if we can somehow globalize one in return for giving up the other. This is achieved by twisting the theory.
Let us modify the spins of the fermionic fields so that $\psi_+$ becomes a $(0,1)$-form on $\Sigma$, while ${\bar\psi}_+$ becomes a zero-form. To make this point clear, we rename them as $$-\psi_+^i \to \rho_\zb^i \, , \qquad
-i{\bar\psi}_+^\ib \to \alpha^\ib \, .$$ The action of $Q$ is then given by $$\label{Q}
\begin{alignedat}{3}
[Q,\phi^i] &= 0 \, , &\qquad
[Q,\phi^\ib] &= \alpha^\ib \, , \\
\{Q,\rho_\zb^i\} &= -{\partial}_\zb\phi^i \, , &
\{Q,\alpha^\ib\} &= 0 \, .
\end{alignedat}$$ Thus $Q$ becomes a scalar and generates a global supersymmetry. On the other hand, $Q_+$ becomes a $(0,1)$-form and in general does not exist globally. In this way, we obtain a theory with global supersymmetry on any Riemann surface $\Sigma$, described by the action $$S = \int_\Sigma d^2z
\{Q, g_{i\jb} \rho_\zb^i {\partial}_z\phi^\jb\}
+ \int_\Sigma \phi^* B \, .$$ Discarding half the original supersymmetries also allows us to consider complex target spaces which may or may not be Kähler.
Besides globalizing the supersymmetry, the twisting does one more important thing: it makes the components $T_{z\zb}$ and $T_{\zb\zb}$ of the energy-momentum tensor $Q$-exact. This is because the global supersymmetry commutes with infinitesimal diffeomorphisms $\delta z =
v^z$, $\delta \zb = v^\zb$ up to a quantity involving ${\partial}_\zb v^z$, but not ${\partial}_\zb v^\zb$, ${\partial}_z v^z$, or ${\partial}_z v^\zb$. Hence, one can take the variation of the action inside the $Q$-commutator when computing these components. Quantum mechanically the action gets renormalized, but the renormalization can be done by $Q$-exact local counterterms and the conclusion is unchanged.[^1] The generators $T_{z\zb}$, $T_{\zb\zb}$ being $Q$-exact, antiholomorphic reparametrizations act trivially in the $Q$-cohomology. Therefore, after the twisting the chiral algebra has no antiholomorphic degrees of freedom—it defines a holomorphic field theory.
A local operator $\CO$ is said to have dimension $(n,m)$ if inserted at the origin it transforms as $$\CO(0) \to \lambda^{-n} \lambdab^{-m} \CO(0)$$ under the rescaling $z \to \lambda z$, $\zb \to \lambdab\zb$. An immediate consequence of the decoupling of antiholomorphic degrees of freedom is that the chiral algebra is supported by local operators with $m = 0$, because those with $m \neq 0$ transform nontrivially under antiholomorphic rescalings. The holomorphic dimension, $n$, is then equal to the spin $n - m$, so it is an integer and protected from quantum corrections (assuming that they are small). Thus, the chiral algebra of the twisted model is graded by the dimension as well as the R-charge.
If $c_1(X) \neq 0$, the grading by the R-charge is violated by instantons as we discussed already. The same is true of the grading by dimension. Instanton corrections are accompanied with powers of $\Lambda^2$. When multiplying a local operator in the untwisted model, $\Lambda^2$ is best thought of as a section of $K_\Sigma
\otimes \Kb_\Sigma$ having dimension $(1,1)$. In the twisted model, it is more natural think of it as a section of $K_\Sigma$ with dimension $(1,0)$, absorbing the change in the dimension of the fermionic path integral measure. (This can be deduced by considering the case where $K_\Sigma^{1/2}$ is trivial and so the twisted and untwisted models are isomorphic.) Instanton corrections violating R-charge by $2k$ units are proportional to $\Lambda^{2k}$, so violate dimension by $k$. The grading by dimension is therefore reduced to ${\mathbb{Z}}_n$ nonperturbatively when that by the R-charge is reduced to ${\mathbb{Z}}_{2n}$.
With the grading by dimension at hand, we can now write the OPE of the chiral algebra in the form $$\label{OPE-dim}
[\CO_i(z)] \cdot [\CO_j(w)]
\sim \sum_{n=0}^\infty \sum_k
\frac{\Lambda^{2n} c^{(n)}_{ijk} [\CO_k(w)]}{(z - w)^{n_i + n_j - n_k - n}}
\, ,$$ where $[\CO_i]$ have dimension $n_i$ and $c^{(n)}_{ijk}$ are constants. The $Q$-cohomology group of dimension $n$ vanishes for $n
< 0$ since there are no local operators of negative dimension classically.
The $Q$-cohomology classes of dimension zero are special: they have regular OPEs, as one can see by setting $n_i = n_j = 0$ above and noting $n_k \geq 0$. Hence they form an ordinary ring, much like the chiral rings of $(2,2)$ models. This ring is called the chiral ring of the $(0,2)$ model [@Witten:2005px; @MR2281544].
We have seen advantages of the twisting. It has one drawback, however. Twisting the fermions amounts to tensoring with $\Kb_\Sigma^{-1/2}$ to which the ${\partial}$ operator couples. This changes the anomaly cancellation condition: $\Sigma$ and $X$ must now satisfy $$\frac{1}{2} p_1(X) = \frac{1}{2} c_1(\Sigma) c_1(X) = 0 \, .$$ Here $c_1(\Sigma)$ and $c_1(X)$ are pulled back to $\Sigma \times X$. Thus, the twisting introduces an additional anomaly which affects the choice of $\Sigma$. If $c_1(X) \neq 0$, we must choose $\Sigma$ with $c_1(\Sigma) = 0$, that is to say, $K_\Sigma$ must be trivial. This was actually implicit in our discussion when we interpreted $\Lambda^2$ as a nowhere vanishing section of $K_\Sigma$.
Classical Chiral Algebra and Quantum Deformations {#CA-CQCA}
-------------------------------------------------
To determine the chiral algebra of a given model, what one can usually do is to first identify its elements in the classical limit, then include quantum corrections order by order. What guarantees the validity of this procedure is the following principle: small quantum corrections can only annihilate, and never create, $Q$-cohomology classes.
This statement is justified as follows. Local operators that are not $Q$-closed at a given order cannot become $Q$-closed by higher order corrections. So the kernel of $Q$ can never become larger. At the same time, the image of $Q$ can never become smaller, because adding quantum corrections order by order defines an injection from the image of $Q$ at a given order to the image of $Q$ at the next order. Then the cohomology, which is the kernel modulo the image, can only become smaller.
In this respect, the chiral algebra is similar to the space of supersymmetric ground states (of supersymmetric quantum mechanics, say). There, small quantum corrections can lift supersymmetric ground states by giving a small positive energy, but not create ones because they cannot push positive energy states down to zero energy as long as there is a gap in the beginning. Likewise, small quantum corrections can “lift” $Q$-cohomology classes by making them $Q$-exact or no longer $Q$-closed, but not create ones because of a “gap” to the $Q$-closedness or non-$Q$-exactness.
The analogy goes further. Supersymmetry yields a one-to-one correspondence between the bosonic positive energy states and the fermionic ones, so supersymmetric ground states are always lifted in boson-fermion pairs. The same principle applies here: small quantum corrections always annihilate $Q$-cohomology classes in boson-fermion pairs. More precisely, one can show that when a $Q$-cohomology class $[\CO]$ is annihilated by quantum corrections at some order, there is another class $[\CO']$ that is annihilated together at the same order either via the relation $[Q, \CO\} \propto \CO'$ or $\CO \propto [Q,
\CO'\}$.[^2] Keeping the above principles in mind, let us study the chiral algebra of the twisted model in a little more detail.
The chiral algebra is supported by local operators whose antiholomorphic dimension $m = 0$. Since $\rho$ and $\zb$-derivatives of any fields have $m > 0$, these do not enter the chiral algebra. Furthermore, we can replace ${\partial}_z\alpha$ by other fields using the equation of motion $D_z\alpha^\ib = 0$. Thus, relevant local operators are linear combinations of operators of the form $$\label{LO}
\CO(\phi,\phib)_{j\dotsm k\dotsm \lb\dotsm \mb\dotsm \ib_1 \dotsm \ib_q}
{\partial}_z\phi^j \dotsm {\partial}_z^2\phi^k \dotsm
{\partial}_z\phi^\lb \dotsm {\partial}_z^2\phi^\mb \dotsm
\alpha^{\ib_1} \dotsm \alpha^{\ib_q} \, .$$ Identifying $\alpha^\ib$ with the differential $d\phi^\ib$, we can regard such operators with R-charge $q$ and dimension $n$ as $(0,q)$-forms with values in a certain holomorphic vector bundle $V_{X,n}$ over $X$.
For example, an $n = 0$ operator $\CO_{\ib_1\dotsm\ib_q}
\alpha^{\ib_1} \dotsm \alpha^{\ib_q}$ is a $(0,q)$-form on $X$, hence $V_{X,0} = 1$, the trivial bundle of rank $1$. At $n = 1$, there are two types of operators, $\CO_{j\ib_1\dotsm\ib_q} {\partial}_z\phi^j
\alpha^{\ib_1} \dotsm \alpha^{\ib_q}$ and $\CO^j{}_{\ib_1\dotsm\ib_q}
g_{j\kb} {\partial}_z\phi^\kb \alpha^{\ib_1} \dotsm \alpha^{\ib_q}$. These are $(0,q)$-forms with values respectively in $T_X^{*}$ and $T_X$, thus $V_{X,1} = T_X \oplus T_X^{*}$. At $n = 2$, we have five types, giving $V_{X,2} = T_X \oplus T_X^{*}\oplus S^2T_X \oplus (T_X
\otimes T_X^{*}) \oplus S^2T_X^{*}$. Here $S^k$ is the symmetric $k$th power. In general, $V_{X,n}$ is given by the series $$\label{Vn}
\sum_{n = 0}^\infty q^n V_{X,n}
= \bigotimes_{k = 0}^\infty S_{q^k}(T_X)
\bigotimes_{l = 0}^\infty S_{q^l}(T_X^{*}) \, ,$$ where $S_t(V) = 1 + tS(V) + t^2 S^2(V) + \dotsb$. We will write $$V_X = \bigoplus_{n = 0}^\infty V_{X,n}.$$
At the classical level, we can easily find the action of $Q$ on these operators. Take an $n = 1$ example, $\CO_{j\ib_1\dotsm\ib_q}
{\partial}_z\phi^j \alpha^{\ib_1} \dotsm \alpha^{\ib_q}$. Acting on it with $Q$ gives $\alpha^\ib {\partial}_\ib \CO_{j\ib_1\dotsm\ib_q} {\partial}_z\phi^j
\alpha^{\ib_1} \dotsm \alpha^{\ib_q}$. On an $n = 1$ operator of the other type, $\CO^j{}_{\ib_1\dotsm\ib_q} g_{j\kb} {\partial}_z\phi^\kb
\alpha^{\ib_1} \dotsm \alpha^{\ib_q}$, acting with $Q$ gives $\alpha^\ib {\partial}_\ib\CO^j{}_{\ib_1\dotsm\ib_q} g_{j\kb} {\partial}_z\phi^\kb
\alpha^{\ib_1} \dotsm \alpha^{\ib_q}$ plus $\CO^j{}_{\ib_1\dotsm\ib_q}
g_{j\kb} D_z\alpha^\kb \alpha^{\ib_1} \dotsm \alpha^{\ib_q}$, but the latter vanishes by the equation of motion. From these examples, we see that $Q$ acts as the ${{\bar\partial}}$ operator.[^3] Classically, the $q$th $Q$-cohomology group of dimension $n$ is therefore isomorphic to the $q$th Dolbeault cohomology group $H_{{\bar\partial}}^q(X, V_{X,n})$, and $$\label{Acl}
\CA {\cong}\bigoplus_{q = 0}^d \bigoplus_{n = 0}^\infty H_{{\bar\partial}}^q(X, V_{X,n})$$ as graded vector spaces. It is clear from this formula that the chiral algebra is generally infinite-dimensional.
Quantum mechanically, the action of $Q$ is deformed by quantum corrections. Because of this deformation, some of the classical $Q$-cohomology classes can disappear.
A notable example is the disappearance of the energy-momentum tensor. Classically, our model is conformally invariant and $T_{zz}$ is $Q$-closed on-shell. We expect that it is no longer $Q$-closed perturbatively if the Ricci curvature is nonzero, since conformal invariance is broken at one loop in that case. To determine $[Q,T_{zz}]$, we note that $Q$ commutes with ${\partial}_z \propto H + P$. Thus $$\Bigl[Q, \oint \! dz \, T_{zz} - \oint \! d\zb \, T_{z\zb}\Bigr]
= \oint \! dz [Q, T_{zz}]
= 0 \, ,$$ where we have used the fact that $T_{z\zb}$ is $Q$-exact. This suggests $$[Q, T_{zz}] = {\partial}_z\theta$$ for some $\theta$. Presumably, $\theta$ is a $Q$-closed local operator of R-charge one and dimension one, constructed from the target space metric. To leading order, $Q = {{\bar\partial}}$ and for a generic choice of the metric there is only one possibility: $$\theta \propto R_{i\jb} {\partial}_z\phi^i\alpha^\jb \, .$$ So, as expected, $T_{zz}$ would cease to be $Q$-closed unless the Ricci curvature vanishes. In Section \[PCA-c1\], we will compute $[Q, T_{zz}]$ explicitly in perturbation theory and find that it is indeed proportional to ${\partial}_z(R_{i\jb} {\partial}_z\phi^i\alpha^\jb)$ up to higher order corrections.
If $c_1(X) = 0$, then $\theta$ is $Q$-exact and we can add corrections to $T_{zz}$ that make it $Q$-closed again. But if $c_1(X) \neq 0$, there is no way to do this, so $[T_{zz}]$ is annihilated together with $[{\partial}_z\theta]$ by perturbative corrections. The chiral algebra is a little exotic in this case: it is analogous to the chiral algebras of CFTs, but lacks the energy-momentum tensor and hence invariance under holomorphic reparametrizations.
Except for the possible lack of the energy-momentum tensor, perturbatively the structure of the chiral algebra is not very different from its classical description. The basic fact about perturbative corrections is that they are local on the target space, because one only considers fluctuations around constant maps in perturbation theory. Moreover, the gradings by the R-charge and dimension are not violated perturbatively. Thanks to these properties, at the perturbative level the action of $Q$ still defines differential complexes $$\dotsb {\longrightarrow}V_{X,n} \otimes \wedge^q {{\wb@rl{T}}}_X^{*}\stackrel{Q}{{\longrightarrow}} V_{X,n} \otimes \wedge^{q+1} {{\wb@rl{T}}}_X^{*}{\longrightarrow}\dotsb \, ,$$ and the chiral algebra is given by the direct sum of their cohomology groups. How to compute these cohomology groups using a sheaf of free CFTs on $X$ is the subject of the next section.
Beyond perturbation theory, the physics is no longer local on the target space. Rather, it is local on the instanton moduli space $\CM$ since quantum fluctuations localize to instantons. So presumably the chiral algebra can be formulated nonperturbatively as a cohomology theory on $\CM$, but exactly how this should be done is not clear. At any rate, in principle one can always compute the instanton corrections to the action of $Q$ by path integral and determine the exact chiral algebra. Instanton effects often lead to surprising results. Later we will see examples where the whole chiral algebra is annihilated by instanton corrections.
Let us tie up the loose ends from Section \[CA-instantons\] by explaining why the $B$ field should be renormalized as asserted there. Consider the case where the twisted and untwisted models are isomorphic. Suppose that instantons annihilate perturbative $Q$-cohomology classes $[\CO]$ and $[\CO']$ through a relation of the form $[Q,\CO\} \sim \CO'$. If this relation violates R-charge by $2k$ units, the left-hand side contains $2k$ more $\alpha$ fields than the right-hand side does, while the both sides have antiholomorphic dimension equal to zero. Viewed in the untwisted model, this means that these instantons relate operators whose antiholomorphic dimensions differ by $k$. Since the dynamical scale $\Lambda$ is the only dimensional parameter available and has antiholomorphic dimension $1/2$, to match the scaling dimensions a factor of $\Lambda^{2k}$ must appear in the right-hand side. For that, the $B$ field must obey the renormalization group equation .
Sheaf Theory of Perturbative Chiral Algebras {#PCA}
============================================
As we have explained in the last section, the chiral algebra can be understood as a quantum deformation of the Dolbeault cohomology of an infinite-dimensional holomorphic vector bundle over the target space. At the perturbative level, there is an alternative formulation of the chiral algebra which involves a sheaf of free CFTs. The goal of this section is to develop the sheaf theory of perturbative chiral algebras.
Perturbative Chiral Algebra from Free CFTs
------------------------------------------
The chiral algebra of the twisted model is classically the Dolbeault cohomology of the holomorphic vector bundle $V_X$. By the Čech–Dolbeault isomorphism, this cohomology is isomorphic to the cohomology of the sheaf of holomorphic sections of $V_X$. Perturbatively $Q$ gets corrected, but still acts as a differential operator on the target space due to the locality of sigma model perturbation theory. In such a situation, there is an analog of the Čech–Dolbeault isomorphism: the perturbative $Q$-cohomology is isomorphic to the cohomology of the sheaf of perturbatively $Q$-closed sections of $V_X$. The proof is completely parallel to the classical case.[^4]
This “Čech–$Q$ isomorphism” may seem to have little practical value. To compute the sheaf cohomology, first of all one needs to know the general form of perturbatively $Q$-closed local sections of $V_X$. But this requires understanding beforehand how perturbative corrections deform the classical expression $Q = {{\bar\partial}}$ precisely, which is generally very hard if not impossible. However, we can circumvent this difficulty if we adopt a different approach.
Let us recast the Čech–$Q$ isomorphism in a slightly more abstract form. Choose a good cover $\{U_\alpha\}$ on $X$; thus all nonempty finite intersections of $U_\alpha$ are diffeomorphic to ${\mathbb{C}}^d$. On each $U_\alpha$, the space of perturbatively $Q$-closed sections of $V_X$ is isomorphic to the chiral algebra of the twisted model into $U_\alpha$. For, the zeroth $Q$-cohomology group is isomorphic to this space, while the higher $Q$-cohomology groups vanish as $U_\alpha$ is topologically trivial. Since Čech cohomology is defined by taking the direct limit as the open cover becomes finer and finer, and every open cover has a refinement by a good cover, it follows that the perturbative chiral algebra can be computed via the cohomology of the sheaf of chiral algebras $\CAh$ on $X$: $$\CA^q {\cong}H^q(X,\CAh) \, .$$ Rewriting the Čech–$Q$ isomorphism this way makes it clear that the perturbative chiral algebra can be formulated without reference to any globally defined metric on the target space.
When one computes the cohomology of $\CAh$, say using a good cover $\{U_\alpha\}$, one can endow $U_\alpha$ with any metric. In practice, we will put $g_{i\jb} = \delta_{i\jb}$ on all the $U_\alpha$. The point is that the twisted models with the flat target spaces $U_\alpha$ are free theories—their chiral algebras receive no quantum corrections.
In the free twisted model $Q = {{\bar\partial}}$ exactly, so sections of $\CAh(U_\alpha)$ are represented by local operators of the form $\CO(\phi, {\partial}_z\phi, \dotsc, {\partial}_z\phib, {\partial}_z^2\phib, \dotsc)$. Introducing bosonic fields $\beta_i$ of dimension one and $\gamma^i$ of dimension zero by $$\beta_i = 2\pi \delta_{i\jb}{\partial}_z\phib^\jb \, , \qquad
\gamma^i = \phi^i \, ,$$ we can conveniently write them as $\CO(\gamma, {\partial}_z\gamma, \dotsc,
\beta, {\partial}_z\beta, \dotsc)$. These are observables of the free $\beta\gamma$ system, a free CFT with action $$S = \frac{1}{2\pi} \int_\Sigma d^2z \, \beta_i {\partial}_\zb\gamma^i$$ whose OPEs are $$\beta_i(z) \gamma^j(w) \sim -\frac{\delta_i^j}{z - w} \, , \qquad
\beta_i(z) \beta_j(w) \sim 0 \, , \qquad
\gamma^i(z) \gamma^j(w) \sim 0 \, .$$ Hence, $\CAh$ may be considered as a sheaf of free $\beta\gamma$ systems. We conclude that the perturbative chiral algebra can be reconstructed by gluing free $\beta\gamma$ systems over $X$ and computing the Čech cohomology.
This construction is exact to all orders in perturbation theory since the spaces $\CAh(U_\alpha)$ are free of quantum corrections. Where did the perturbative corrections go? They are now encoded in the transition functions ${\widehat{f}}_{\alpha\beta}$ for the sheaf $\CAh$. Instead of equipping flat metrics on the $U_\alpha$, one may as well start with a globally defined, curved metric on $X$. One can then flatten it over each $U_\alpha$ to obtain a free $\beta\gamma$ system. In this process the perturbative corrections disappear locally, but the transition functions change by ${\widehat{f}}_{\alpha\beta} \to
e^{-A_\alpha} {\widehat{f}}_{\alpha\beta} e^{A_\beta}$ for some operators $A_\alpha$. These operators carry the information on the perturbative corrections.
But then, how can we find the $A_\alpha$? Here we come back to the original problem: to do that, we need detailed knowledge of the perturbative corrections. Instead, what we can do is take a collection of locally defined free $\beta\gamma$ systems and glue them using various choices of the transition functions. In doing so, we are effectively parametrizing the theory by the way this gluing is done. This is the strategy we will adopt.
Gluing $\beta\gamma$ Systems
----------------------------
We are thus led to the problem of listing the possible sets of transition functions for $\CAh$. The first step is to classify the automorphisms of the free $\beta\gamma$ system. Automorphisms are generated by currents of dimension one. There are two types of such currents.
Let $V$ be a holomorphic vector field on ${\mathbb{C}}^d$. Then $J_V =
-V^i(\gamma) \beta_i$ is a good current. (Here and below, normal ordering is implicit in expressions containing both $\beta_i$ and $\gamma^i$.) From the OPEs $$J_V(z) \gamma^i(w) \sim \frac{V^i(w)}{z - w} \, , \qquad
J_V(z) \beta_i(w) \sim -\frac{{\partial}_iV^j \beta_j(w)}{z - w} \, ,$$ we see that $J_V$ generates the infinitesimal diffeomorphism $\delta\gamma = V$ and $\beta$ transforms as a $(1,0)$-form under this symmetry. We denote the corresponding conserved charge by $K_V$.
With a holomorphic one-form $B$ on ${\mathbb{C}}^d$, we can also make $J_B =
B_i(\gamma) {\partial}_z\gamma^i$. This has the OPEs $$J_B(z) \gamma^i(w) \sim 0 \, , \qquad
J_B(z) \beta_i(w)
\sim -\frac{B_i(w)}{(z - w)^2}
+\frac{C_{ij} {\partial}_z\gamma^j(w)}{z - w}$$ with $C = {\partial}B$, so generates the transformation $\delta\beta =
-i_{{\partial}_z\gamma} C$. For any closed holomorphic two-form $C$, there is a holomorphic one-form $B$ such that $C = {\partial}B$. Moreover, $\delta\beta = 0$ if and only if $C = 0$. The automorphisms of this type are therefore labeled by the closed holomorphic two-forms $C$. We denote their charges by $K_C$.
One can readily work out the commutators between the conserved charges. Computing the relevant OPEs, one finds $$\begin{split}
[K_V, K_{V'}] &= K_{[V,V']} + K_{C(V,V')} \, , \\
[K_V, K_C] &= K_{\CL_V C} \, , \\
[K_C, K_{C'}] &= 0 \, .
\end{split}$$ Here $C(V,V') = {\partial}_i{\partial}_k V^l {\partial}_j{\partial}_l V'^k d\phi^i \wedge
d\phi^j$. The last two of these relations show that the $K_C$ generate an abelian subalgebra on which holomorphic reparametrizations act naturally.
Now, suppose that the complex manifold $X$ is built up by gluing open patches $U_\alpha {\cong}{\mathbb{C}}^d$ using transition functions $f_{\alpha\beta}$. Transition functions for $\CAh$ are constructed by lifting $f_{\alpha\beta}$ to automorphisms ${\widehat{f}}_{\alpha\beta}$ of the $\beta\gamma$ system compatible with the cocycle condition. So we choose ${\widehat{f}}_{\alpha\beta}$ such that they act on $\gamma$ by $f_{\alpha\beta}$ and satisfy ${\widehat{f}}_{\alpha\beta} {\widehat{f}}_{\beta\gamma}
{\widehat{f}}_{\gamma\alpha} = 1$ on $U_\alpha \cap U_\beta \cap U_\gamma$.
Naively, one might think that the cocycle condition is satisfied if one picks a cocycle $\{C_{\alpha\beta}\}$ of closed holomorphic two-forms and let ${\widehat{f}}_{\alpha\beta}$ act on $\beta$ by the pullback $f_{\alpha\beta}^*$ followed by $\exp(K_{C_{\alpha\beta}})$. The situation is actually more complicated. The commutator between two $K_V$s differs from the expected form by a $K_C$ term, implying $${\widehat{f}}_{\alpha\beta} {\widehat{f}}_{\beta\gamma} {\widehat{f}}_{\gamma\alpha}
= \exp(K_{C_{\alpha\beta\gamma}})$$ for some $C_{\alpha\beta\gamma}$. We need to adjust ${\widehat{f}}_{\alpha\beta}$ so that $C_{\alpha\beta\gamma}$ disappear. The freedom at our disposal is to transform ${\widehat{f}}_{\alpha\beta} \to
\exp(K_{C'_{\alpha\beta}}) {\widehat{f}}_{\alpha\beta}$, which shifts $$\label{C2C+D}
C_{\alpha\beta\gamma}
\to C_{\alpha\beta\gamma} + (\delta C')_{\alpha\beta\gamma} \, .$$ Unless $C_{\alpha\beta\gamma}$ can be canceled by an appropriate choice of $C'_{\alpha\beta}$, the gluing cannot be carried out consistently. In other words, there may be an obstruction to the existence of a sheaf of $\beta\gamma$ systems.
The obstruction has a natural physical interpretation. It is not hard to show that $C_{\alpha\beta\gamma}$ are totally antisymmetric in $\alpha$, $\beta$, $\gamma$ and obey $(\delta
C)_{\alpha\beta\gamma\delta} = 0$, hence define a cocycle. The freedom then means that the obstruction is encoded in the cohomology class $$[C_{\alpha\beta\gamma}] \in H^2(X, \Omega^{2,\mathit{cl}}_X) \, ,$$ where $\Omega^{2,\mathit{cl}}_X$ is the sheaf of closed holomorphic two-forms on $X$. It can be shown [@MR1748287] that this class is mapped to $p_1(X) \in H^4(X,{\mathbb{R}})$ under the Čech–Dolbeault isomorphism. Thus, the obstruction vanishes if and only if $p_1(X) =
0$. This is the condition for the perturbative cancellation of sigma model anomaly. Since the obstruction arises in lifting the diffeomorphisms used to construct the underlying manifold $X$, in this way we see that the sigma model anomaly causes a gravitational anomaly on the target space. The cancellation of the anomaly by adjusting ${\widehat{f}}_{\alpha\beta}$ is a kind of the Green–Schwarz mechanism.
Given $f_{\alpha\beta}$, the choice of ${\widehat{f}}_{\alpha\beta}$ is not unique. Suppose that we choose different transition functions, ${\widehat{f}}'_{\alpha\beta}$. Then the two sets of transition functions are related by ${\widehat{f}}'_{\alpha\beta} = \exp(K_{C'_{\alpha\beta}})
{\widehat{f}}_{\alpha\beta}$ for some cocycle $\{C'_{\alpha\beta}\}$. If this cocycle is exact, $C'_{\alpha\beta} = (\delta C'')_{\alpha\beta}$, then we have ${\widehat{f}}'_{\alpha\beta} = \exp(-K_{C''_\alpha}) {\widehat{f}}_{\alpha\beta}
\exp(K_{C''_\beta})$ and these define the same gluing. Hence, inequivalent choices of the transition functions are parametrized by $$H^1(X, \Omega^{2,\mathit{cl}}_X) \, .$$
The origin of this moduli space can be understood as follows. For any closed form $\CH$ of type $(3,0) \oplus (2,1)$, locally we can find a $(2,0)$-form $T$ such that $\CH = dT$.[^5] Thus we have the short exact sequence $$0 {\longrightarrow}\Omega^{2,\mathit{cl}}_X
{\longrightarrow}\CA^{2,0}_X
\stackrel{d}{{\longrightarrow}} \CZ_X^{3,0} \oplus \CZ_X^{2,1}
{\longrightarrow}0 \, ,$$ where $\CA^{p,q}_X$ and $\CZ_X^{p,q}$ are respectively the sheaves of $(p,q)$-forms and closed $(p,q)$-forms on $X$. Since $H^p(X,
\CA^{2,0}_X) = 0$ for $p > 0$, the long exact sequence of cohomology implies $$H^1(X, \Omega^{2,\mathit{cl}}_X)
{\cong}H^0(X, \CZ_X^{3,0} \oplus \CZ_X^{2,1})/dH^0(X, \CA^{2,0}_X) \, .$$ This is the space of closed forms $\CH$ of type $(3,0) \oplus (2,1)$ modulo those that can be written as $\CH = dT'$ with a globally defined $(2,0)$-form $T'$. It actually parametrizes the conformally invariant $Q$-closed term $$\int_\Sigma d^2z \{Q, T_{ij} \rho_\zb^i {\partial}_z\phi^j\}
= \int_\Sigma d^2z \, \alpha^\kb \CH_{\kb ij} \rho_\zb^i {\partial}_z\phi^j
+ i \! \int_\Sigma \phi^* T \, ,$$ which depends on $T$ only through $\CH$ perturbatively. We can add this term to our action. Doing so deforms the chiral algebra unless $T$ is globally defined. Still, we can always set $T$ to zero locally by subtracting a globally defined $T'$. Combined with flattening the metric, this turns the action locally into the free theory form. Therefore, the effect of this term can be treated—and is necessarily included—in the present framework.
Up to this point, we have implicitly worked in some coordinate neighborhood of $\Sigma$. The sheaf $\CAh$ of chiral algebras in this case (when $\Sigma {\cong}{\mathbb{C}}$) is known as a sheaf of chiral differential operators [@Malikov:1998dw; @MR1748287]. In order to reconstruct the chiral algebra globally on $\Sigma$, one has to glue sheaves of chiral differential operators patch by patch over $\Sigma$. This amounts to gluing free $\beta\gamma$ systems over $\Sigma \times
X$. The obstruction thus takes values in $$H^2(\Sigma \times X, \Omega^{2,\mathit{cl}}_{\Sigma \times X}).$$ A part of it depends on both $\Sigma$ and $X$, and corresponds to the $c_1(\Sigma) c_1(X)/2$ anomaly. The moduli of the sheaf of chiral algebras are parametrized by $$H^1(\Sigma \times X, \Omega^{2,\mathit{cl}}_{\Sigma \times X}) \, .$$
Conformal Anomaly {#PCA-c1}
-----------------
Previously, we claimed that the chiral algebra lacks invariance under holomorphic reparametrizations when $c_1(X) \neq 0$, arguing that in that case perturbative corrections would annihilate the classical $Q$-cohomology class $[T_{zz}]$ together with another class $[{\partial}_z\theta]$ via the relation $[Q, T_{zz}] = {\partial}_z\theta$. However, we could not really exclude the possibility that $\theta$ turns out to be zero and $T_{zz}$ remains $Q$-closed. Let us check that this does not happen using the tool developed in this section.
Let $\{U_\alpha\}$ be a good cover of $X$. On each $U_\alpha$, we put a free $\beta\gamma$ system with energy-momentum tensor $T_\alpha =
-\beta_{\alpha i} {\partial}_z\gamma_\alpha^i$. The OPE $$J_V(z) T_\alpha(w)
\sim -\frac{{\partial}_i V^i(w)}{(z - w)^3}
- \frac{({\partial}_z{\partial}_i V^i + V^i\beta_i)(w)}{(z - w)^2}
- \frac{1}{2} \frac{{\partial}_z^2({\partial}_i V^i)(w)}{z - w}$$ shows that $T_\alpha$ transform as $\delta T_\alpha = -{\partial}_z^2{\partial}_i
V^i/2$ under infinitesimal diffeomorphisms $\delta\gamma = V$. The finite form of this transformation is [@MR1748287] $$T_\beta - T_\alpha
= -\frac{1}{2} {\partial}_z^2 \log \det
\frac{{\partial}\gamma_\beta}{{\partial}\gamma_\alpha} \, .$$ Here ${\partial}\gamma_\beta/{\partial}\gamma_\alpha$ is the Jacobian matrix. We define a cocycle $\{\theta_{\alpha\beta}\}$ by $$\theta_{\alpha\beta}
= -\frac{1}{2} {\partial}_z \log \det
\frac{{\partial}\gamma_\beta}{{\partial}\gamma_\alpha}$$ so that it satisfies $T_\beta - T_\alpha =
{\partial}_z\theta_{\alpha\beta}$. Via the Čech–$Q$ isomorphism, this equation translates to $[Q, T_{zz}] = {\partial}_z\theta$ for some local operator $\theta$.
The explicit form of $\theta$ can be found as follows. Write $\theta_{\alpha\beta} = W_\beta - W_\alpha$, where $W_\alpha$ is the quantity $W = {\partial}_z\phi^i {\partial}_i \log \det g/2$ evaluated in the coordinate patch $U_\alpha$. Since $W_\alpha$ transform by holomorphic transition functions, ${{\bar\partial}}W$ is globally defined. This gives $\theta$ as a global section of the sheaf of free $\beta\gamma$ systems, for which $Q = {{\bar\partial}}$. Noting $R_{i\jb} = -{\partial}_i{{\bar\partial}}_\jb
\log \det g$, we find $$\label{theta}
\theta = -\frac{1}{2} R_{i\jb} {\partial}_z\phi^i \alpha^\jb \, .$$ As a globally defined local operator of the original theory, this formula gets higher order corrections since $Q = {{\bar\partial}}$ only to leading order. The form of $\theta$ is in accord with the previous discussion.
We introduced the perturbative $Q$-cohomology class $[\theta]$ through the action of $Q$ on $T_{zz}$. We can also construct it as follows, mimicking the definition of the first Chern class. Let $f_{\alpha\beta}$ be transition functions of $K_X$. The cocycle condition $f_{\alpha\beta} f_{\beta\gamma} f_{\gamma\alpha} = 1$ implies that $(\delta \log f)_{\alpha\beta\gamma}$ are integer multiples of $2\pi i$. Thus, by applying $\delta\log/2\pi i$ on $f_{\alpha\beta}$, we obtain an element of $H^2(X,{\mathbb{Z}})$. It is $c_1(X)$. To obtain an element of the chiral algebra, we apply $-{\partial}_z\log/2$ instead. This gives $-{\partial}_z\log f_{\alpha\beta}/2 =
\theta_{\alpha\beta}$, which represents $[\theta] \in H^1(X,\CAh)$.
${\mathbb{P}}^1$ Model {#PCA-CP1}
----------------------
To conclude the discussion of the sheaf theory approach, let us compute the perturbative chiral algebra of the twisted model with target space $X = {\mathbb{P}}^1$ for the first few dimensions. We will work locally on $\Sigma$.
The target space ${\mathbb{P}}^1 {\cong}{\mathbb{C}}\cup \{\infty\}$ is covered by two patches, $U = {\mathbb{P}}^1 \setminus \{\infty\}$ with coordinate $\gamma$ and $U' = {\mathbb{P}}^1 \setminus \{0\}$ with coordinate $\gamma'$, related to each other by $$\label{gamma'}
\gamma' = \frac{1}{\gamma} \, .$$ Classically $\beta$ transforms as $\beta' = -\gamma^2 \beta$, but this formula gets corrected quantum mechanically. The quantum transformation law turns out to be $$\label{beta'}
\beta' = -\gamma^2 \beta + 2{\partial}_z\gamma \, .$$ The additional term is needed to keep the $\beta\beta$ OPE regular under this transformation. Since the moduli space $H^1({\mathbb{P}}^1,
\Omega^{2,\mathit{cl}}_{{\mathbb{P}}^1}) = 0$, this is essentially the only way to glue the two free $\beta\gamma$ systems.
We first look at the zeroth $Q$-cohomology group $\CA^0$. The elements of $\CA^0$ are represented by global sections of the sheaf $\CAh$ of chiral algebras on ${\mathbb{P}}^1$.
At dimension zero, relevant local operators are holomorphic functions. Since a holomorphic function on a compact complex manifold must be constant, the dimension zero subspace of $\CA^0$ is one-dimensional and generated by the cohomology class $[1]$, represented by the identity operator $1$.
At dimension one, the possible local operators are those of the form $B(\gamma){\partial}_z\gamma$ and $V(\gamma)\beta$, where $B$ is a holomorphic one-form and $V$ is a holomorphic vector field. There are no global holomorphic one-forms on ${\mathbb{P}}^1$. For holomorphic vector fields, we have three independent ones, ${\partial}$, $-\gamma{\partial}$, and $-\gamma^2{\partial}$. Hence, at the classical level we have three cohomology classes, represented by $\beta$, $-\gamma\beta$, and $-\gamma^2\beta$. These survive to the perturbative chiral algebra. Their quantum counterparts are $$\begin{split}
J_- &= \beta = -\gamma'^2\beta' + 2{\partial}_z\gamma' \, , \\
J_3 &= -\gamma\beta = \gamma'\beta' \, , \\
J_+ &= -\gamma^2\beta + 2{\partial}_z\gamma = \beta' \, ,
\end{split}$$ generating the affine Lie algebra $\widehat{\mathfrak{sl}}_2$ at the critical level $-2$: $$\begin{split}
J_3(z) J_3(w) &\sim -\frac{1}{(z - w)^2} \, , \\
J_3(z) J_\pm(w) &\sim \pm\frac{J_\pm(w)}{(z - w)^2} \, , \\
J_+(z) J_-(w) &\sim -\frac{2}{(z - w)^2} + \frac{2J_3(w)}{z - w} \, .
\end{split}$$ The existence of these currents in the perturbative chiral algebra is a reflection of the fact that ${\mathbb{P}}^1$ admits an ${\mathrm{SL}}_2$-action.
We now turn to the first $Q$-cohomology group $\CA^1$. The elements of $\CA^1$ are represented by sections of $\CAh(U \cap U')$ that cannot be written as the difference of a section of $\CAh(U)$ and a section of $\CAh(U')$. Extended over the whole target space ${\mathbb{P}}^1$, such sections necessarily have poles at both $0$ and $\infty$.
Meromorphic functions with poles at $0$ and $\infty$ can always be split into a part regular at $0$ and a part regular at $\infty$. Thus the dimension zero subspace of $\CA^1$ is zero.
At dimension one, we can try operators of the form $\beta/\gamma^n$ which have a pole at $0$. However, they are all regular at $\infty$. The other possibilities are ${\partial}_z\gamma/\gamma^n$. Requiring they have a pole at $\infty$, we find that only ${\partial}_z\gamma/\gamma$ can represent a nontrivial cohomology class. Indeed, it does. This cohomology class is $[\theta]$ since ${\partial}_z\gamma/\gamma =
-{\partial}_z\log({\partial}\gamma'/{\partial}\gamma)/2$.
At dimension two, sections with poles at both $0$ and $\infty$ are linear combinations of ${\partial}_z^2\gamma/\gamma$, ${\partial}_z^2\gamma/\gamma^2$, $({\partial}_z\gamma)^2/\gamma$, $({\partial}_z\gamma)^2/\gamma^2$, $({\partial}_z\gamma)^2/\gamma^3$, and $\beta{\partial}_z\gamma/\gamma$. Among these, the combinations ${\partial}_z^2\gamma/\gamma^2 - 2({\partial}_z\gamma)^2/\gamma^3$ and $\beta{\partial}_z\gamma/\gamma + ({\partial}_z\gamma)^2/\gamma^3$ are regular at $\infty$, so vanish in the cohomology. (In verifying this assertion, one should keep in mind that the latter operator is normal ordered.) Moreover, ${\partial}_z({\partial}_z\gamma/\gamma)$ also vanishes due to the perturbative relation $[Q, T_{zz}] = {\partial}_z\theta$. Thus the dimension two subspace of $\CA^1$ is at most three-dimensional. From the cohomology classes we already have, we can construct three: $[J_-
\theta]$, $[J_3 \theta]$, and $[J_+ \theta]$.
Let us summarize. The dimension zero subspace of $\CA^0$ is generated by $[1]$ and the dimension one subspace of $\CA^1$ is generated by $[\theta]$, whereas the dimension one subspace of $\CA^0$ is generated by $[J_-]$, $[J_3]$, $[J_+]$ and the dimension two subspace of $\CA^1$ is generated by $[J_- \theta]$, $[J_3 \theta]$, $[J_+ \theta]$. Therefore, for the first two nontrivial dimensions, we find an isomorphism $\CA^0 {\cong}\CA^1$ given by the map $$[\CO] \longmapsto [\CO\theta].$$ It has been shown [@Malikov:1998dw] that this isomorphism persists in higher dimensions. This is as though $[1]$ and $[\theta]$ are vacua of conformal field theory—both of them are annihilated by ${\partial}_z$—and the elements of $\CA^0$ are creation operators acting on these classes to generate the rest of the $Q$-cohomology classes. It is remarkable that such a structure emerges despite the lack of conformal invariance. In order to understand where this structure comes from, we must go beyond perturbation theory.
Nonperturbative Vanishing of Chiral Algebras {#vanishing}
============================================
In the previous sections we have seen that quantum corrections deform the classical chiral algebra, but perturbatively the deformation can be understood withing the framework of a cohomology theory on the target space. This is because in perturbation theory, one considers fluctuations localized around constant maps.
Instantons are not quite like constant maps, but have a finite size in the target space. Their presence may therefore lead to deformations of different kinds. In this section, we will see a particularly striking example: instantons annihilate all of the perturbative $Q$-cohomology classes, making the chiral algebra trivial nonperturbatively. The existence of such a phenomenon was first predicted by Witten [@Witten:2005px], and subsequently confirmed by Tan and the author [@Tan-Yagi-1]; see Arakawa and Malikov [@Arakawa:2009cb] for a mathematical interpretation of Witten’s prediction. Here we generalize the results of [@Tan-Yagi-1; @Tan-Yagi-2; @MR2415553] and establish a vanishing “theorem” for chiral algebras. We then explain how the vanishing of the chiral algebra implies the spontaneous breaking of supersymmetry and the absence of harmonic spinors on the loop space of the target space.
${\mathbb{P}}^1$ Model, with Instantons
---------------------------------------
The simplest example of a vanishing chiral algebra is provided by the ${\mathbb{P}}^1$ model which we studied in Section \[PCA-CP1\]. As we saw there, the perturbative chiral algebra of the ${\mathbb{P}}^1$ model has the structure of a Fock space: $[1]$ and $[\theta]$ play the role of “ground states,” on which infinite towers of bosonic and fermionic “excited states” are constructed by acting with “creation operators,” namely bosonic $Q$-cohomology classes. In view of this suggestive structure, one may expect that instantons “tunnel” between the “ground states” and “lift” them out of the chiral algebra. This is indeed the case.
We now show that the perturbative $Q$-cohomology classes $[1]$ and $[\theta]$ are annihilated together by instantons via the relation $$\label{Qt1}
\{Q, \theta\} \propto 1 \, .$$ This relation says that the equation $1 = 0$ holds in the $Q$-cohomology. Therefore, the chiral algebra of the ${\mathbb{P}}^1$ model vanishes nonperturbatively. A half of the perturbative $Q$-cohomology classes are annihilated because their representatives become $Q$-exact. For classes $[\CO]$ in this category, we have $\{Q,\CO\theta\} \propto \CO$. Thus $[\CO]$ are annihilated together with $[\CO\theta]$, and the latter should constitute the other half of the perturbative chiral algebra. This explains the observed Fock space structure.
Since $c_1({\mathbb{P}}^1) = 2$, instantons of degree $k$ (which wrap the target space $k$ times) violate R-charge by $2k$ and dimension by $k$. Then the instanton corrections to the action of $Q$ on $\theta$ take the form $$\{Q,\theta\} = \sum_{k=1}^\infty \Lambda^{2k} e^{-kt_0} \CO_k \, ,$$ with $\CO_k$ being local operators of R-charge $2-2k$ and dimension $(1-k,0)$. Here $t_0$ is the topological invariant $S_B$ evaluated for instantons of degree one at the dynamical scale $\Lambda$. There are no local operators of negative dimension, hence $\CO_k = 0$ for $k
> 1$. The remaining operator, $\CO_1$, is perturbatively $Q$-closed. From the fact that for R-charge zero and dimension zero, $[1]$ is the only perturbative $Q$-cohomology class and there are no perturbatively $Q$-exact local operators, it follows $\CO_1 \propto 1$. So if we can show $\{Q,\theta\} \neq 0$, we establish the relation $\{Q, \theta\}
\propto 1$.
To see whether $\{Q,\theta\}$ is zero or not, we put it on a disk (embedded in the worldsheet) and evaluate the path integral for suitable boundary conditions. For our purpose, we can compactify the disk to the sphere and consider those boundary conditions that can be represented by vertex operators at $\infty$. We will therefore compute the correlation function $$\label{QtO}
{\Bigl\langle \CO(\infty) \oint \! d\zb \, G(\zb) \theta(0) \Bigr\rangle}$$ on $\Sigma = {\mathbb{P}}^1$ for some local operators $\CO$. We anticipate that $\{Q,\theta\}$, expressed here as the contour integral of the supercurrent $G$ around $\theta$, will be replaced by $1$ in the final result. To obtain a nonvanishing answer, then, we should take $\CO$ to be local operators of R-charge zero and dimension zero, that is, functions on $X$.
The contributions to this correlation function should come from instantons of degree one. These are biholomorphic maps from $\Sigma =
{\mathbb{P}}^1$ to $X = {\mathbb{P}}^1$, whose moduli space $\CM_1 {\cong}{\mathrm{PGL}}_2({\mathbb{C}})$. At $\phi_0 \in \CM_1$, the number of ${\bar\psi}_+$ and $\psi_+$ zero modes are respectively given by the zeroth and first Hodge numbers of the bundle $K_{\Sigma}^{1/2} \otimes \phi_0^*T_X {\cong}\CO_{{\mathbb{P}}^1}(1)$. Notice that we have untwisted the theory before compactifying, because the twisted ${\mathbb{P}}^1$ model is anomalous on ${\mathbb{P}}^1$. Since $h^0(\CO_{{\mathbb{P}}^1}(1)) = 2$ and $h^1(\CO_{{\mathbb{P}}^1}(1)) = 0$, there are two ${\bar\psi}_+$ zero modes and no $\psi_+$ zero modes. These zero modes can be absorbed by the ${\bar\psi}_+$ fields in $G$ and $\theta$ without bringing down interaction terms. Then, up to the ratio of the bosonic and fermionic determinants, the correlation function is computed to leading order by dropping quantum fluctuations and integrating over the fermion zero modes as well as the instantons: $$\label{intGtO}
e^{-S_B(\phi_0)} \! \int \! d\CM_1 \, dc_0^1 \, dc_0^2 \,
\CO(\infty)
\oint \! d\zb \, G(\zb) \theta(0) \Bigr|_{\phi = \phi_0} \, .$$ Here $d\CM_1$ is the measure on $\CM_1$, and $c_0^1$, $c_0^2$ are the zero mode coefficients of the mode expansion of ${\bar\psi}_+$. We can ignore subleading contributions.
Of course, we must evaluate the contour integral before dropping quantum fluctuations. (Without short distance singularities this would vanish!) It turns out that this seemingly straightforward task is actually very tricky. We are looking for an antiholomorphic single pole $1/\zb$ in the OPE $$\label{Jt}
G(\zb) \theta(0)
= (g_{\phi\phib} {\partial}_{\zb}\phi {\bar\psi}_+)(\zb)
\Bigl(-\frac{1}{2} R_{\phi\phib} {\partial}_z\phi{\bar\psi}_+\Bigr)(0) \, .$$ It may appear that one can obtain such a pole by contracting ${\partial}_{\zb}\phi$ with $R_{\phi\phib}$. However, this does not work because the residue is just the classical action of $Q$ on $\theta$ and vanishes. We must find additional antiholomorphic poles that emerge nonperturbatively.
At this point, we recall that the fermionic fields take values in the pullback of the tangent bundle of $X$ by the bosonic field. Thus, the eigenmodes in which they are expanded depend on the bosonic field, which is itself subject to quantum fluctuations. As a result, the fermion modes—even the zero modes—can produce short distance singularities when the bosonic field is present at the same location.
We can try to extract this bosonic dependence of the fermionic fields as follows. Consider a tubular neighborhood $\CN_1$ of $\CM_1$, diffeomorphic to the normal bundle of $\CM_1$ in the space of maps from $\Sigma$ to $X$. Let $\{x^\alpha\}$ be local coordinates on $\CM_1$ and parametrize the normal directions by coordinates $\{y^\beta\}$ such that $y^\beta = 0$ on $\CM_1$. For $\phi(z,\zb;x,y) \in \CN_1$, we denote its projection to $\CM_1$ by $\phi_0(z;x)$. An instanton $\phi_0 \in \CM_1$ maps the points of $\Sigma = {\mathbb{P}}^1$ to the points of $X = {\mathbb{P}}^1$ in a one-to-one manner, so we can invert $\phi_0(z;x)$ to obtain $z(\phi_0;x)$ and write $\phi(z, \zb; x, y) = \phi(\phi_0(z;x), \phib_0(\zb;x); x, y)$. Computing $[iQ,\phib]$ with this last expression, we find $$\label{psib}
{\bar\psi}_+
= \frac{{\partial}\phib}{{\partial}\phib_0} [iQ,\phib_0] + \dotsb
= \frac{{\partial}_\zb\phib}{{\partial}_\zb\phib_0} {\bar\psi}_{+0}(\phi_0) + \dotsb
\, ,$$ where ${\bar\psi}_{+0}(\phi_0) = [iQ,\phib_0]$ is the zero mode part of ${\bar\psi}_+$ evaluated at $\phi_0$. So we have extracted, partially, the dependence of ${\bar\psi}_+$ on the bosonic fluctuations.
In fact, the leading term of the formula is all we need. The reason is that the fermion nonzero modes will be discarded in our computation and, up to the nonzero modes and the equation of motion for the bosonic field, the leading term represents the zero mode part of ${\bar\psi}_+$. Indeed, it correctly reduces to ${\bar\psi}_{+0}(\phi_0)$ at $\phi = \phi_0$ and the equation of motion implies $$D_z\Bigl(\frac{{\partial}_\zb\phib}{{\partial}_\zb\phib_0} {\bar\psi}_{+0}(\phi_0)\Bigr)
= R^{\phib}{}_{\phib\phi\phib} \frac{{\partial}_z\phib}{{\partial}_\zb\phib_0}
\psi_+ {\bar\psi}_+ {\bar\psi}_{+0}(\phi_0) \, ,$$ which vanishes if the fermion nonzero modes are dropped since there are no $\psi_+$ zero modes.
As desired, ${\partial}_\zb\phi$ from $G$ can now be contracted with ${\partial}_\zb\phib$ in the ${\bar\psi}_+$ field from $\theta$ to produce an antiholomorphic double pole. This gives $$\label{Gt}
\oint \! d\zb \, G(\zb) \theta(0)
=
\Bigl(\frac{i}{2} R_{\phi\phib}
\frac{{\partial}_z\phi}{{\partial}_\zb\phib_0}
{\partial}_\zb{\bar\psi}_+ {\bar\psi}_{+0}(\phi_0)\Bigr)(0) \, .$$ The result may look strange, but will become more natural after the ${\bar\psi}_+$ zero modes are integrated out.
To proceed, we need to specify the path integral measure. On instantons, the action of $Q$ is realized as the superconformal transformation $\zb \mapsto \zb + \epsilonb_-(c_0^1 + c_0^2 \zb)$. Thus the zero mode part of ${\bar\psi}_+$ can be expanded as $${\bar\psi}_{+0}(\phi_0)
= c_0^1 {\partial}_\zb\phib_0 + c_0^2 \zb{\partial}_\zb\phib_0 \, .$$ If we choose $d\CM_1$ to be conformally invariant, it is $Q$-invariant up to terms involving $dc_0^1$ or $dc_0^2$. Then the product $d\CM_1
\, dc_0^1 \, dc_0^2$ is a $Q$-invariant measure since $dc_0^1 \,
dc_0^2$ is $Q$-invariant by itself. There is a unique conformally invariant measure on $\CM_1$ up to a factor. In terms of the points $X_0$, $X_1$, $X_\infty \in X$ to which $0$, $1$, $\infty \in \Sigma$ are mapped by instantons, it is given by $$\label{dM1}
d\CM_1
= \frac{d^2\!X_0 \, d^2\!X_1 \, d^2\!X_\infty}
{|X_0 - X_1|^2 |X_1 - X_\infty|^2 |X_\infty - X_0|^2} \, .$$ The parametrization of $\CM_1$ by $X_0,$ $X_1$, $X_\infty$ provides a compactification of $\CM_1$ to $({\mathbb{P}}^1)^3$, but $d\CM_1$ is singular on the compactified moduli space.
Let us return to the integral . After the integration over the ${\bar\psi}_+$ zero modes, the contour integral $$\label{GtRicci}
\int \! dc_0^1 \, dc_0^2 \oint \! d\zb \, G(\zb)
\theta(0)\Bigr|_{\phi = \phi_0}
= \Bigl(\frac{i}{2} R_{\phi\phib} {\partial}_z\phi_0 {\partial}_\zb\phib_0\Bigr)(0) \, .$$ This is the pullback of the Ricci form by instantons. Using the formula $${\partial}_z\phi_0(0)
= \frac{(X_\infty - X_0)(X_0 - X_1)}{X_1 - X_\infty} \, ,$$ what is left can be written as $$\label{XXX}
\frac{i}{2} e^{-S_B(\phi_0)} \! \int_{{\mathbb{P}}^1} d^2X_0 \, R_{\phi\phib}(X_0, \Xb_0)
\int_{({\mathbb{P}}^1)^2} d^2X_1 \, d^2X_\infty
\frac{\CO(X_\infty, \Xb_\infty)}{|X_1 - X_\infty|^4} \, .$$ The $X_0$-integral is the evaluation of $2\pi c_1(X)$ on $[X]$, which gives $4\pi$. The $X_1$-integral diverges, reflecting the noncompactness of $\CM_1$. One way to regularize it is to impose a lower bound $l > 0$ on the distance between $X_1$ and $X_\infty$ measured with the target space metric. For $l \ll 1$, this restricts the domain of $X_1$ to the region where $g_{\phi\phib}(X_\infty,\Xb_\infty) |X_1 - X_\infty|^2 \geq l^2$. The regularized integral is $$\int_{{\mathbb{P}}^1} \frac{d^2X_1}{|X_1 - X_\infty|^4}
= \frac{\pi}{l^2} g_{\phi\phib}(X_\infty, \Xb_\infty) \, .$$ From this and the renormalization group equation for the $B$ field, we find that the integral contains a factor $(\Lambda/l\mu)^2 e^{-t_0}$. We can take a limit such that $\mu \to
\infty$ and $l \to 0$ keeping $l\mu$ finite. The end result is $$\Lambda^2 e^{-t_0}
\int_{{\mathbb{P}}^1} g_{\phi\phib} d^2X_\infty \, \CO \, ,$$ up to an overall numerical factor.
We see that the resulting $X_\infty$-integral is performed over $\CM_0
{\cong}X$ with respect to a natural volume form. Therefore, the one-instanton computation of the correlation function has reduced to the zero-instanton computation of the expected one-point function: $${\bigl\langle \CO(\infty) \{Q, \theta(0)\} \bigr\rangle}
\propto \Lambda^2 e^{-t_0} {\bigl\langle \CO(\infty) \bigr\rangle} \, .$$ This shows $\{Q, \theta\} \propto 1$.
Vanishing “Theorem” {#NV-VT}
-------------------
The vanishing of the chiral algebra is not special to the ${\mathbb{P}}^1$ model. It seems to be a feature shared by many $(0,2)$ models. In fact, we have the following vanishing “theorem.” Let $X$ be a compact spin Kähler manifold with $p_1(X)/2 = 0$ and $c_1(X) > 0$. Suppose that there is an embedding ${\mathbb{P}}^1 \subset X$ with trivial normal bundle. Then, the chiral algebra of the $(0,2)$ model with target space $X$ vanishes nonperturbatively in the absence of left-moving fermions.
As before, we will establish the vanishing by demonstrating that instantons annihilate the perturbative $Q$-cohomology classes $[1]$ and $[\theta]$ together. Since $c_1(X) > 0$, every instanton violates R-charge by $2k$ and dimension by $k$ for some integer $k > 0$. Then $[\theta]$ can only be annihilated by instantons with $k = 1$, in which case it is paired with a class of R-charge zero and dimension zero. Such a class must be proportional to $[1]$ on the compact complex manifold $X$. Thus, it suffices to show $\{Q,\theta\} \neq
0$.
Consider the correlation function again. The leading contributions to this function come from instantons that give precisely two ${\bar\psi}_+$ zero modes and no $\psi_+$ zero modes. These are embeddings ${\mathbb{P}}^1 \hookrightarrow X$ whose normal bundle is trivial. (The normal bundle $N_{C/X}$ of a curve $C \subset X$ is a holomorphic vector bundle over $C$ defined by the short exact sequence $0 \to T_C \to T_X|_C \to N_{C/X} \to 0$.) To see this, note that given instantons wrapping a ${\mathbb{P}}^1 \subset X$ once, there are already the right amount of zero modes from the tangent direction. So there should be no additional zero modes from the normal directions, which is the case if and only if the normal bundle of the ${\mathbb{P}}^1$ is trivial.
But if the normal bundle is trivial and the fermion zero modes come only from the tangent direction, the contribution from each such ${\mathbb{P}}^1$ can be computed essentially in the same way as in the ${\mathbb{P}}^1$ model; the result is a function on $X$ supported on the ${\mathbb{P}}^1$. By assumption, there is at least one ${\mathbb{P}}^1$ that makes a contribution, hence the correlation function is nonzero.[^6] It follows that $\{Q, \theta\} \neq 0$ and the chiral algebra vanishes.
One may wonder how the existence of a single ${\mathbb{P}}^1$ with trivial normal bundle, whose contribution to $\{Q,\theta\}$ is confined on the ${\mathbb{P}}^1$ itself, can possibly tell something about the behavior of $\{Q,\theta\}$ in the other region of $X$. This point can be understood if we consider deformations of the ${\mathbb{P}}^1$. Infinitesimal deformations are given by holomorphic sections of the normal bundle. Since the normal bundle is trivial, the ${\mathbb{P}}^1$ in question can be moved in every direction in the target space. Then the family of ${\mathbb{P}}^1$s generated by deformations sweep out the whole target space, and their contributions can add up to a nonzero constant.
Flag Manifold Model
-------------------
An example in which the above deformation argument is beautifully demonstrated is the flag manifold $G/B$ of a complex simple Lie group $G$ with Borel subgroup $B$ [@Tan-Yagi-2; @MR2415553]. For $X =
G/B$, we have $p_1(X)/2 = 0$ and $c_1(X) = 2(x_1 + \dotsb + x_r)$ with $r = {\mathop{\mathrm{rank}}\nolimits}G$. Thus the $G/B$ model is well defined and has the R-symmetry broken to ${\mathbb{Z}}_2$ nonperturbatively. The simplest case is when $G = {\mathrm{SL}}_2$, for which $G/B {\cong}{\mathbb{P}}^1$.
What makes the $G/B$ model interesting is that its perturbative chiral algebra contains currents generating the affine Lie algebra ${\widehat{\mathfrak{g}}}$ of critical level [@Malikov:1998dw; @MR1042449; @MR2290768]. The critical level makes an appearance here because the Sugawara construction must fail; otherwise, it would contradict the fact that the chiral algebra lacks the energy-momentum tensor when $c_1(X) \neq
0$. Nonperturbatively, these currents disappear from the chiral algebra along with everything else.
Pick a ${\mathbb{P}}^1 \subset X$ and call it $C$. Choose linearly independent normal vectors $V_1$, $\dotsc$, $V_d \in N_{C/X}|_{g_0}$ at a point $g_0 \in C$. Under $g_0 \mapsto gg_0 \in C$, these are mapped to $g_*V_1$, $\dotsc$, $g_*V_d \in N_{C/X}|_{gg_0}$ which are again linearly independent. Varying $g$, we obtain a global frame of $N_{C/X}$. Therefore $N_{C/X}$ is trivial, and the chiral algebra vanishes. In this example, the $G$-action generates a family of ${\mathbb{P}}^1$s with trivial normal bundle that covers the target space.
Supersymmetry Breaking and Geometry of Loop Spaces
--------------------------------------------------
The vanishing of the chiral algebra of a $(0,2)$ model has important implications for the dynamics of the theory and the geometry of the loop space $\CL X$ of the target space: supersymmetry is spontaneously broken and there are no harmonic spinors on $\CL X$. These conclusions are obtained by studying the $Q$-cohomology of states rather than operators.
Since $Q$ has R-charge one and satisfies $Q^2 = 0$, one can consider the $Q$-cohomology graded by the R-charge in the Hilbert space of states. This is naturally a module over the chiral algebra: on elements $[{|\Psi\rangle}]$ of the $Q$-cohomology of states, $[\CO] \in
\CA$ acts by $$[\CO] \cdot [{|\Psi\rangle}] = [\CO{|\Psi\rangle}] \, .$$ As a graded vector space, it is isomorphic to the space of supersymmetric states, the kernel of $\{Q,Q^\dagger\} \propto H - P$ which is generally infinite-dimensional since $P$ is not bounded. If $c_1(X) = 0$, it is further isomorphic to the $Q$-cohomology of local operators by the state-operator correspondence.
To unravel the geometric meaning of the $Q$-cohomology of states, take $\Sigma$ to be a cylinder $S^1 \times {\mathbb{R}}$ with coordinates $(\sigma,
\tau)$ and regard $\tau$ as time. The theory may now be viewed as supersymmetric quantum mechanics on $\CL X$, so let us canonically quantize it and see what we get. The fermionic fields are quantized (for $z = \sigma + i\tau$ and with respect to a local orthonormal frame) to obey $$\label{CAR}
\{\psi_+^a(\sigma, \tau), {\bar\psi}_+^\bb(\sigma', \tau)\}
= \delta^{ab} \delta(\sigma - \sigma') \, .$$ This is a loop space version of the Clifford algebra, with the continuous index $\sigma$ parametrizing the direction along the loop. States are thus spinors on $\CL X$. The supercharge is quantized as $$\label{Qquant}
Q = -i\int \! d\sigma \, {\bar\psi}_+^\ib
\Bigl(\frac{D}{\delta\phi^\ib}
- B_{\ib j} {\partial}_\sigma\phi^j
- B_{\ib\jb} {\partial}_\sigma\phi^\jb\Bigr) \, ,$$ where $D/\delta\phi$ is the covariant functional derivative on $\CL
X$. From this expression, we see that $Q$ is almost a half of the Dirac operator on $\CL X$. But not quite, since it has extra pieces coupled to ${\partial}_\sigma$.
We can eliminate these extra pieces without changing the $Q$-cohomology. To do this, we define a functional $\CA_B\colon \CL X
\to {\mathbb{C}}$ as follows.[^7] First, we pick a base loop in each connected component of $\CL X$. Then, given $\phi \in \CL X$, we choose a homotopy ${\widehat{\phi}}\colon [0,1]
\times S^1 \to X$ from the base loop of the relevant component to $\phi$. Finally, we set $$\label{h}
\CA_B(\phi) = \int_{[0,1] \times S^1} {\widehat{\phi}}^* B \, .$$ Under a variation of the end point $\phi \to \phi + \delta\phi$, this functional changes by $$\label{deltah}
\delta \CA_B
= \int \! d\sigma \bigl(\delta\phi^i
(B_{ij} {\partial}_\sigma\phi^j
+ B_{i\jb}{\partial}_\sigma\phi^\jb)
- \delta\phi^\ib
(B_{\ib j}{\partial}_\sigma\phi^j
+ B_{\ib\jb} {\partial}_\sigma\phi^\jb)\bigr) \, .$$ Writing $Q_0$ for a half of the Dirac operator obtained by dropping the extra pieces from $Q$, we have $$\label{Qh}
Q = e^{-\CA_B} Q_0 e^{\CA_B} \, .$$ Therefore, the $Q$-cohomology of states is isomorphic to the $Q_0$-cohomology, which is the cohomology of the spinor bundle over $\CL X$.
Now suppose that the chiral algebra is trivial: $[1] = 0$. Then $$[{|\Psi\rangle}] = [1] \cdot [{|\Psi\rangle}] = 0$$ for any $Q$-cohomology classes $[{|\Psi\rangle}]$, so the $Q$-cohomology of states is also trivial. Since supersymmetric states are in one-to-one correspondence with the $Q$-cohomology classes, there are none. In other words, supersymmetry is spontaneously broken. Furthermore, we just saw that the $Q$-cohomology is nothing but the cohomology of the spinor bundle over $\CL X$. Hence, there are no harmonic spinors on $\CL X$ either.
[10]{} W. Lerche, C. Vafa, and N. P. Warner, [*[Chiral rings in $N=2$ superconformal theories]{}*]{}, [Nucl. Phys.]{} [**B 324**]{} (1989) 427.
E. Witten, [*[Topological sigma models]{}*]{}, [Comm. Math. Phys.]{} [ **118**]{} (1988) 411.
E. Witten, [*Mirror manifolds and topological field theory*]{}, in [*Essays on mirror manifolds*]{}, pp. 120–158. Int. Press, Hong Kong, 1992. [[hep-th/9112056]{}](http://xxx.lanl.gov/abs/hep-th/9112056).
M. Gromov, [*Pseudoholomorphic curves in symplectic manifolds*]{}, [Invent. Math.]{} [**82**]{} (1985) 307.
A. Floer, [*A relative [M]{}orse index for the symplectic action*]{}, [Comm. Pure Appl. Math.]{} [**41**]{} (1988) 393.
A. Floer, [*The unregularized gradient flow of the symplectic action*]{}, [Comm. Pure Appl. Math.]{} [**41**]{} (1988) 775.
A. Floer, [*Morse theory for [L]{}agrangian intersections*]{}, [J. Diff. Geom.]{} [**28**]{} (1988) 513.
A. Floer, [*Symplectic fixed points and holomorphic spheres*]{}, [Comm. Math. Phys.]{} [**120**]{} (1989) 575.
A. Floer, [*Witten’s complex and infinite-dimensional [M]{}orse theory*]{}, [J. Diff. Geom.]{} [**30**]{} (1989) 207.
L. J. Dixon, [*Some world-sheet properties of superstring compactifications, on orbifolds and otherwise*]{}, in [*Superstrings, unified theories and cosmology*]{}, pp. 67–126. World Scientific, Singapore, 1988.
K. Hori and C. Vafa, [*[Mirror symmetry]{}*]{}, [[hep-th/0002222]{}](http://xxx.lanl.gov/abs/hep-th/0002222).
E. Witten, [*[Elliptic genera and quantum field theory]{}*]{}, [Comm. Math. Phys.]{} [**109**]{} (1987) 525.
E. Witten, [*Index of [D]{}irac operators*]{}, in [*Quantum fields and strings: a course for mathematicians, vol. 1*]{}, pp. 475–511. Amer. Math. Soc., Providence, RI, 1999.
C. Vafa, [*[Superstring vacua]{}*]{}, in [*High Energy Physics and Cosmology*]{}, pp. 145-177. World Scientific, Singapore, 1989.
E. Witten, [*[On the Landau–Ginzburg description of N=2 minimal models]{}*]{}, [Int. J. Mod. Phys.]{} (1994) 4783, \[[[ hep-th/9304026]{}](http://xxx.lanl.gov/abs/hep-th/9304026)\].
E. Witten, [*Two-dimensional models with [$(0,2)$]{} supersymmetry: perturbative aspects*]{}, [Adv. Theor. Math. Phys.]{} [**11**]{} (2007) 1.
F. Malikov, V. Schechtman, and A. Vaintrob, [*[Chiral de Rham complex]{}*]{}, [Comm. Math. Phys.]{} [**204**]{} (1999) 439, \[[[math/9803041]{}](http://xxx.lanl.gov/abs/math/9803041)\].
A. Kapustin, [*[Chiral de Rham complex and the half-twisted sigma-model]{}*]{}, [[hep-th/0504074]{}](http://xxx.lanl.gov/abs/hep-th/0504074).
M.-C. Tan, [*[Two-dimensional twisted sigma models and the theory of chiral differential operators]{}*]{}, [Adv. Theor. Math. Phys.]{} [**10**]{} (2006) 759, \[[[ hep-th/0604179]{}](http://xxx.lanl.gov/abs/hep-th/0604179)\].
M.-C. Tan, [*[The half-twisted orbifold sigma model and the chiral de Rham complex]{}*]{}, [Adv. Theor. Math. Phys.]{} [**12**]{} (2008) 547, \[[[hep-th/0607199]{}](http://xxx.lanl.gov/abs/hep-th/0607199)\].
M.-C. Tan, [*[Equivariant cohomology of the chiral de Rham complex and the half-twisted gauged sigma model]{}*]{}, [Adv. Theor. Math. Phys.]{} [**13**]{} (2009) 3, \[[[ hep-th/0612164]{}](http://xxx.lanl.gov/abs/hep-th/0612164)\].
M.-C. Tan, [*[Two-dimensional twisted sigma models, the mirror chiral de Rham complex, and twisted generalised mirror symmetry]{}*]{}, [JHEP]{} [**07**]{} (2007) 13, \[[[ arXiv:0705.0790]{}](http://xxx.lanl.gov/abs/0705.0790)\].
M.-C. Tan and J. Yagi, [*Chiral algebras of [$(0,2)$]{} sigma models: beyond perturbation theory*]{}, [[ arXiv:0801.4782]{}](http://xxx.lanl.gov/abs/0801.4782).
M.-C. Tan and J. Yagi, [*Chiral algebras of $(0,2)$ sigma models: beyond perturbation theory – II*]{}, [[ arXiv:0805.1410]{}](http://xxx.lanl.gov/abs/0805.1410).
M.-C. Tan and J. Yagi, [*Chiral algebras of [$(0,2)$]{} models: beyond perturbation theory*]{}, [Lett. Math. Phys.]{} [**84**]{} (2008) 257.
B. L. Fe[ĭ]{}gin and E. V. Frenkel, [*Affine [K]{}ac–[M]{}oody algebras and semi-infinite flag manifolds*]{}, [Comm. Math. Phys.]{} [**128**]{} (1990) 161.
E. Frenkel, [*Lectures on the [L]{}anglands program and conformal field theory*]{}, in [*Frontiers in number theory, physics, and geometry. II*]{}, pp. 387–533. Springer, Berlin, 2007. [[hep-th/0512172]{}](http://xxx.lanl.gov/abs/hep-th/0512172).
S. Stolz, [*A conjecture concerning positive [R]{}icci curvature and the [W]{}itten genus*]{}, [Math. Ann.]{} [**304**]{} (1996) 785.
G. W. Moore and P. C. Nelson, [*[Anomalies in nonlinear sigma models]{}*]{}, [Phys. Rev. Lett.]{} [**53**]{} (1984) 1519.
G. W. Moore and P. C. Nelson, [*[The Aetiology of sigma model anomalies]{}*]{}, [Comm. Math. Phys.]{} [**100**]{} (1985) 83.
E. Witten, [*Global anomalies in string theory*]{}, in [*Symposium on anomalies, geometry, topology*]{}, pp. 61–99. World Scientific, Singapore, 1985.
M. F. Atiyah, [*Circular symmetry and stationary-phase approximation*]{}, Astérisque [**131**]{} (1985) 43.
T. P. Killingback, [*[World sheet anomalies and loop geometry]{}*]{}, [Nucl. Phys.]{} [**B 288**]{} (1987) 578.
D. Friedan, [*[Nonlinear models in $2 + \varepsilon$ dimensions]{}*]{}, Phys. Rev. Lett. [**45**]{} (1980) 1057.
D. H. Friedan, [*[Nonlinear models in $2 + \varepsilon$ dimensions]{}*]{}. Ph.D. thesis, University of California at Berkeley, August 1980. Published as Lawrence Berkeley Laboratory Publication LBL-11517 and Ann. Phys. **163** (1985) 318.
L. Alvarez-Gaume, D. Z. Freedman, and S. Mukhi, [*[The background field method and the ultraviolet structure of the supersymmetric nonlinear sigma model]{}*]{}, [Ann. Phys.]{} [**134**]{} (1981) 85.
C. G. Callan, Jr., E. J. Martinec, M. J. Perry, and D. Friedan, [*[Strings in background fields]{}*]{}, [Nucl. Phys.]{} [**B 262**]{} (1985) 593.
E. Witten, [*[Phases of $N = 2$ theories in two dimensions]{}*]{}, [Nucl. Phys.]{} [**B 403**]{} (1993) 159, \[[[hep-th/9301042]{}](http://xxx.lanl.gov/abs/hep-th/9301042)\].
A. Adams, J. Distler and M. Ernebjerg, [*Topological heterotic rings*]{}, Adv. Theor. Math. Phys. [**10**]{}, (2006) 657, \[[[hep-th/0506263]{}](http://xxx.lanl.gov/abs/hep-th/0506263)\]. V. Gorbounov, F. Malikov, and V. Schechtman, [*Gerbes of chiral differential operators*]{}, Math. Res. Lett. [**7**]{} (2000) 55.
T. Arakawa, F. Malikov, [*A Vertex algebra attached to the flag manifold and Lie algebra cohomology*]{}, AIP Conf. Proc. [**1243**]{} (2010) 151. \[[[arXiv:0911.0922]{}](http://xxx.lanl.gov/abs/0911.0922)\].
K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, [*Mirror symmetry*]{}, vol. 1 of [Clay Mathematics Monographs]{}. American Mathematical Society, Providence, RI, 2003. With a preface by Vafa.
[^1]: This is not a precise statement. When $\Sigma$ is curved, one needs to introduce a worldsheet metric to impose a meaningful cutoff length. This produces an anomalous term in $T_{z\zb}$ proportional to the Ricci curvature of $\Sigma$ [@Callan:1985ia]. Such a c-number anomaly does not affect the structure of the chiral algebra.
[^2]: To prove this, let $\epsilon$ be a small parameter that controls the strength of quantum effects, and $\CO$ a local operator that is nontrivial in the $Q$-cohomology to order $\epsilon^{k-1}$. First, suppose that $\CO$ is no longer $Q$-closed at order $\epsilon^k$; thus $[Q, \CO\} = \epsilon^k \CO'$ for some $\CO'$, and no corrections to $\CO$ can make it $Q$-closed again. Then $\CO'$ is $Q$-closed, but cannot be $Q$-exact at order $\epsilon^l$ for any $l < k$. For if there exists $\CO''$ such that $\epsilon^l
\CO' = [Q, \CO''\} + O(\epsilon^{l+1})$, then $[Q, \CO -
\epsilon^{k-l} \CO''\} = O(\epsilon^{k+1})$ and $\CO -
\epsilon^{k-l} \CO''$ is $Q$-closed to order $\epsilon^k$. Next, suppose that $\CO$ becomes $Q$-exact at order $\epsilon^k$; thus $\epsilon^k \CO = [Q, \CO'\} + O(\epsilon^{k+1})$ for some $\CO'$. Then $\CO'$ is $Q$-closed to order $\epsilon^{k-1}$, but cannot be $Q$-exact to the same order. For if there exist $\CO''$ and $\CO'''$ such that $\epsilon^l \CO' = [Q, \CO''\} + \epsilon^{l+1}
\CO'''$ for some $l < k$, then $\epsilon^{k-1} \CO = [Q, \CO'''\} +
O(\epsilon^k)$ and $\CO$ is already $Q$-exact at order $\epsilon^{k-1}$.
[^3]: General local operators of dimension greater than one are constructed using covariant derivatives. On such operators, $Q$ acts by ${{\bar\partial}}$ plus terms involving the curvature of the target space. Classically, these additional terms which vanish for a flat metric do not change the $Q$-cohomology. This point will become clear when we develop a sheaf theory approach to the perturbative chiral algebra in Section \[PCA\].
[^4]: The key ingredients are the $Q$-Poincaré lemma and the existence of partitions of unity on the sheaf of $(0,q)$-forms with values in $V_X$. The former follows from the ${{\bar\partial}}$-Poincaré lemma since small quantum corrections cannot create $Q$-cohomology classes. As for the latter, “quantum” partitions of unity can always be constructed by renormalizing “classical” partitions of unity.
[^5]: By the Poincaré lemma, locally $\CH = d(U + V)$ for some $(2,0)$-form $U$ and ${{\bar\partial}}$-closed $(1,1)$-form $V$. By the ${{\bar\partial}}$-Poincaré lemma, locally $V = {{\bar\partial}}W$ for some $(1,0)$-form $W$. Then $T = U + V - dW$.
[^6]: In case the contributions from all the instantons add up to zero, we can insert delta functions supported on a particular ${\mathbb{P}}^1$ at various points on $\Sigma$ so that the target space effectively becomes that ${\mathbb{P}}^1$. The correlation function is still nonzero.
[^7]: This functional is not single valued if the cohomology class $[B]$ does not vanish on some two-cycles. However, one can always go to a covering space of $\CL X$ in which it becomes single valued, and define the theory in this space. See [@MR2003030] for more discussion.
|
---
abstract: 'Based on results of E. DiBenedetto and D. Hoff we propose an explicit finite difference scheme for the one dimensional Generalized Porous Medium Equation $\partial_t u=\partial_{xx}^2 \Phi(u)$. The scheme allows to track the moving free boundaries and captures the hole filling phenomenon when two free boundaries collide. We give an abstract convergence result when the mesh parameter $\Delta x\to 0 $ without any error estimates, and invesigate numerically the convergence rates.'
author:
- 'Léonard Monsaingeon [^1]'
bibliography:
- './biblio.bib'
title: 'An algorithm for one-dimensional Generalized Porous Medium Equations: interface tracking and the hole filling problem'
---
Introduction
============
We consider the numerical approximation of nonnegative solutions $u(x,t)\geq 0$ to one-dimensional degenerate diffusion equations of the Generalized Porous Medium Equation type $$\partial_t u=\partial^2_{xx} \Phi(u),\qquad t\geq 0,x\in {\mathbb{R}}.
\tag{GPME}
\label{eq:GPME_u}$$ The nonlinearity $\Phi(s)$ is normalized as $\Phi(0)=0$, is monotone increasing for $s>0$, and satisfies the structural condition $$1<a\leq \frac{s\Phi'(s)}{\Phi(s)} \leq b
\tag{$\Gamma_{a,b}$}
\label{eq:strucural_ab}$$ for some constants $a,b$. This roughly means that nonlinearities in the class $\Gamma_{a,b}$ behave in between two pure powers $s^a,s^b$ for $1<a\leq b$, which is a generalization of the celebrated Porous Medium Equation (PME) $\partial_tu=\Delta u^m$ for $m>1$. Moreover, $a>1$ implies that $\Phi(s)/s$ is monotone increasing and $\lim\limits_{s\to 0^+}\frac{\Phi(s)}{s}=\Phi'(0)=0$. Writing $\partial_{xx}^2\Phi(u)=\partial_x(\Phi'(u)\partial_x u)$ the equation clearly degenerates at the levelset $\{u=0\}$, which results in the so-called *finite speed of propagation*: if the initial data $u^0(x)$ is compactly supported then $u(\,.\,,t)$ remains compactly supported for all $t>0$, see [@dPV91]. Thus free-boundaries $\Gamma(t)=\partial \operatorname{supp} u(\,.\,,t)$ separate $\{u =0\}$ from $\{u>0\}$. In order to understand their propagation it is more convenient to use the *pressure* variable, defined as $$v:=\Psi(u),\qquad \Psi(s):=\int_0^s\frac{\Phi'(z)}{z}dz.$$ The pressure formally solves $$v_t=\sigma(v)\partial^2_{xx}v+|\partial_xv|^2 ,
\label{eq:GPME_v}$$ where $$\sigma(v)=\Phi'(u)=\Phi'\circ \Psi^{-1}(v).$$ The structural assumption implies that $(a-1)v\leq \sigma(v)\leq (b-1)v$, and $v,\sigma(v),\Phi'(u),\Phi(u)/u$ are comparable in the sense that the ratio of any two of them is bounded away from zero and from above. As a consequence $u=v=\sigma(v)=0$ at the free-boundaries, and formally discarding the $\sigma(v)\partial_{xx}^2v$ term we see that $\partial_t v=|\partial_x v|^2$ at any free-boundary point. This suggests that the free-boundary curves $\zeta(t)=\partial \operatorname{supp}v(\,.\,,t)$ should propagate with local speed $d\zeta/dt=-\partial_{x}v(\zeta(t),t)$, provided that these quantities make sense. As a consequence the speed of propagation should be bounded as soon as the pressure is Lipschitz in the space variable. Degenerate diffusion equations such as have attracted considerable attention in the last decades. We refer the reader to [@Va07; @DK86; @DK07; @S83; @DB93] and references therein for the Cauchy problem and regularity theory, and to [@A70; @CVW87; @CW90; @DR03; @dPV91] for the theory of free-boundaries. In order to track the free-boundaries we shall work exclusively in the pressure framework rather than with , and we restrict in the whole paper to Lipschitz-continuous and compactly supported initial pressure $$0\leq v^0(x)\leq M,\qquad \operatorname{Lip}(v^0)\leq \gamma_0.$$ Because and satisfy a comparison principle [@Va07] we expect that $0\leq v(x,t)\leq M$ for all times and the behaviour of $\Phi(s)$ should therefore be irrelevant for large $r=\Psi(s)\geq M$. As a consequence we relax and only assume throughout the whole paper $$\sigma\in \mathcal{C}^1([0,\infty),{\mathbb{R}}^+)\cap\mathcal{C}^2({\mathbb{R}}^+,{\mathbb{R}}^+),\qquad \sigma(0)=0,\qquad \sigma'>0,$$ and $$\forall \,r\in[0,M]:\qquad
0<s_1(M)\leq \sigma'(r)\leq S_1(M)
\quad\mbox{and}\quad
|\sigma''(r)|\leq S_2(M)
\label{eq:structural_condition_sigma}$$ for structural $s_1,S_1,S_2$.
This condition on $\sigma(r)$ can be translated into conditions on the original $\Phi(s)$ nonlinearity through $r=\Psi(s)$, for example $\sigma'(r)=s\Phi''(s)/\Phi'(s)$. In the case of the pure PME nonlinearity $\Phi(s)=s^m$ one can compute explicitly $v=\Psi(u)=mu^{m-1}/(m-1)$ and $\sigma(v)=(m-1)v$, thus $s_1=S_1=(m-1)$ and $S_2=0$ in . As a consequence the above structural assumptions for $\sigma$ can be viewed as some PME-like behaviour condition in bounded intervals.
Because of gradient jumps at the free-boundaries no classical solutions can exist if $v^0$ has compact support, and we shall use the following weak formulation:
A function $0\leq v\in\mathcal{C}({\mathbb{R}}\times [0,T])$ is a weak solution of with initial datum $v^0(x)$ if $\partial_x v\in L^2({\mathbb{R}}\times(0,T))$ and $$\begin{aligned}
& \int\limits_{{\mathbb{R}}}v(x,\tau)\varphi(x,\tau)\mathrm{d}x-\int\limits_{{\mathbb{R}}}v^0(x)\varphi(x,0)\mathrm{d}x\nonumber\\
&
\qquad+ \int\limits_{0}^{\tau}\int\limits_{{\mathbb{R}}}\left\{-v \partial_t\varphi + \sigma(v)\partial_x v\partial_x\varphi+\Big(1-\sigma'(v)\Big)|\partial_x v|^2\varphi\right\}\mathrm{d}x\,\mathrm{d}t=0
\label{eq:weak_formulation_v_IBP0}
\end{aligned}$$ for all $0\leq \tau\leq T$ and test functions $\varphi\in \mathcal{C}^{\infty}_c({\mathbb{R}}\times[0,T])$. \[defi:weak\_sols\_v\]
The equivalence between the density $u$ and pressure $v$ formulations with $v=\Psi(u)$ is well known [@A69], and any weak solution $v$ in the sense of Definition \[defi:weak\_sols\_v\] automatically gives a weak solution $u=\Psi^{-1}(v)$ to in some sense. As already mentioned we only work in the pressure variable, hence we refrain from giving a precise definition of weak solutions for and refer the reader e.g. to [@Va07; @dPV91]. Note that we impose here continuity at $t=0^+$, so that the initial data are taken in a strong sense.\
The problem of numerical approximation to in dimension one goes back to [@GJ71], where a finite difference approach was first proposed to compute numerical solutions of $\partial_t v=f(x,t,v)\partial_{xx}^2v+|\partial_x v|^2$ but free-boundaries were not accurately tracked. Later in [@TM83] a scheme allowing to track the interfaces was implemented for the pure PME nonlinearity $\Phi(s)=s^m$, but the authors were not able to prove convergence of the interface curves. Almost simultaneously, DiBenedetto and Hoff proposed in [@DBH84] an explicit finite-difference interface-tracking algorithm for the pure PME nonlinearity, and established rigorous error estimates for the solution and interfaces. In [@DBH84; @GJ71; @TM83] only the case of initial data $v^0$ consisting in a single patch is considered, i-e with when the initial support only has one connected component $\operatorname{supp}v^0=[\zeta_l(0),\zeta_r(0)]$. In this case the free-boundaries can be represented by two continuous left/right curves $\zeta_{lr}(t)$ with $\operatorname{supp}v(\,.\,,t)=[\zeta_l(t),\zeta_r(t)]$ for all $t\geq 0$. It is well known [@dPV91 Corollary 1.5] that due to the diffusive nature of the problem $\operatorname{supp}v(\,.\,,t)$ is noncontracting in time, and as a consequence $\zeta_l$ and $\zeta_r$ are monotone nonincreasing and nondecreasing respectively. In addition to this simple setting we will also consider the so-called *hole-filling problem* when the initial support has two connected components at positive distance from each other, in which case the internal hole eventually fills and the internal interfaces disappear in finite time (see section \[section:two\_patches\] for a detailed description of the problem). A finite elements method was recently employed in [@QZ09] to investigate the hole-filling and related problems, with satisfactory qualitative results but no rigorous convergence result.
Closely following [@DBH84], we propose in this paper an extension of DiBenedetto and Hoff’s algorithm to general nonlinearities, allowing to track the interfaces and solve past the hole-filling time. As in [@DBH84] the algorithm reproduces at the discrete level all the properties satisfied by the solutions of at the continuous level. More precisely: initial $\gamma_0$ Lipschitz regularity, nonnegativity, and $L^{\infty}$ bounds are preserved along the time evolution, solutions are $1/2$ Hölder continuous in time, and satisfy a generalized Aronson-Bénilan estimate $\partial_{xx} v(\,.\,,t)\geq\underline{z}(t)\approx -C(1+1/t)$ in the sense of distributions $\mathcal{D}'({\mathbb{R}})$ for all fixed $t>0$. For the pure PME nonlinearity $\Phi(s)=s^m$ the latter semi-convexity property was first proved in [@AB79] in the optimal form $\partial_{xx} v(x,t)\geq-1/(m+1)t$, and is fundamental for the regularity and propagation theories. The scheme relies on the following splitting method: inside the support $\{v>0\}=\{\sigma(v)>0\}$ is formally parabolic, hence a classical finite difference scheme can be used with an extra ${\varepsilon}$-viscosity stabilizing term. As already discussed one formally expects the hyperbolic propagation law $d\zeta/dt=-\partial_x v$ at the free-boundaries $x=\zeta(t)$, and thus enforcing the discrete equivalent allows to track the interfaces. Technically speaking this interface condition is in fact applied at the discrete level in some neighborhood of the interface curves. The neighborhood has thickness of the same order $\mathcal{O}(\Delta x)$ as the space mesh $\Delta x$, and can therefore be viewed as a numerical boundary layer.\
The paper is organized as follows: in Section \[section:one\_patch\] we describe the scheme for general nonlinearities when the initial data consists in a single patch (i-e has connected initial support). Imposing a suitable stability condition $\Delta t=\mathcal{O}(\Delta x^2)$ on the mesh parameters we establish discrete a priori bounds, including a generalized Aronson-Bénilan estimate (Lemma \[lem:Aronson\_Benilan\_estimate\]). These a priori estimates then allow us to prove convergence of the approximate solutions and interface curves when $h=(\Delta x,\Delta t)\to 0$. In Section \[section:two\_patches\] we show that the scheme can be extended to study the hole-filling problem. We construct a numerical approximation to the filling time and show that our scheme really captures the hole-filling phenomenon, in the sense that it allows to keep computing a consistent approximation to the solution past the filling time. In Section \[section:num\_exp\_comments\] we present a numerical experiments and investigate the order of convergence.
As already mentioned Section \[section:one\_patch\] is an adaptation of [@DBH84] to general nonlinearities but requires significant technical modifications, in particular for the generalized Aronson-Bénilan estimate (Lemma \[lem:Aronson\_Benilan\_estimate\]). To the best of our knowledge all the results in Section \[section:two\_patches\] are new, even for the pure PME nonlinearity.
The scheme for one patch only {#section:one_patch}
=============================
Throughout the whole paper we fix mesh parameters $\Delta x,\Delta t$ and write $\{x_k\}_{k\in{\mathbb{Z}}}=\{k\,\Delta x\}$, $\{t^n\}_{n\geq 0}=\{n\,\Delta t\}$, as well as $v^n_k\approx v(x_k,t^n)$ and $\zeta_{lr}^n\approx\zeta_{lr}(t^n)$. Given a “single patch” compactly supported initial datum $v^0$ $$0\leq v^0(x)\leq M,\qquad
\operatorname{Lip}(v^0)\leq \gamma_0,
\qquad
\operatorname{supp}v^0=[\zeta_l(0),\zeta_r(0)],$$ we first initialize $$v^0_k:=v^0(x_k)
\qquad \mbox{and}\qquad
\zeta_{l,r}^0:=\zeta_{l,r}(0).$$ Given an approximate solution $v_k^n$ and interfaces $\zeta_{l,r}^n$ at time $t^n$, we define $$K_l(n):=\min\{k\in {\mathbb{Z}}:\,x_{k-1}\geq \zeta_l^n\},
\qquad
K_r(n):=\max\{k\in {\mathbb{Z}}:\,x_{k+1}\leq \zeta_r^n\}$$ and $$0\leq s^n_l:=x_{K_l(n)}-\zeta_l^n,\qquad 0\leq s^n_r:=\zeta_r^n-x_{K_r(n)}.$$ We shall often speak of $x_k\in [x_{K_l(n)},x_{K_r(n)}]$ as the (numerical) support at time $t^n$, while $x_k\in[\zeta_l^n,x_{K_l(n)}]$ and $x_k\in[x_{K_r(n)},\zeta_r^n,]$ will be referred to as the (numerical) left and right boundary layers. Observe that by construction these boundary layers have thickness $\Delta x\leq s_l^n,s_r^n\leq 2\Delta x$, see Figure \[fig:FIG1\]. The interfaces at time $t^{n+1}$ are next computed as $$\frac{\zeta_l^{n+1}-\zeta_l^n}{\Delta t}=-\frac{v_{K_l(n)}^n}{s_l^n},
\qquad\qquad
\frac{\zeta_r^{n+1}-\zeta_r^n}{\Delta t}=-\frac{v_{K_r(n)}^n}{s_r^n},
\label{eq:propagation_interfaces}$$ thus reproducing the propagation law $d\zeta/dt=-\partial_x v$ at the free-boundaries. We will prove in Lemma \[lem:Linfty\_Lipschitz\_estimate\] that $v_k^n\geq 0$, and therefore $\zeta_{l}^{n+1}\leq \zeta_l^n$ and $\zeta_{r}^{n+1}\geq \zeta_r^n$. This monotonicity translates the noncontractivity of the support at the discrete level. We also define for later use $$\left(s^{n}_l\right)':=x_{K_l(n)}-\zeta_l^{n+1}\geq s^n_l,\qquad \left(s^{n}_r\right)':=\zeta_r^{n+1}-x_{K_r(n)}\geq s_r^n.$$ Carefully note that $\left(s^{n}_{lr}\right)'\neq s^{n+1}_{lr}$ and that $\zeta^{n}_{lr}$ needs not be integer meshpoints, see Figure \[fig:FIG1\].
(1,0.70707072) (0,0)[![right numerical boundary layer[]{data-label="fig:FIG1"}](FIG1.pdf "fig:"){width="\unitlength"}]{} (0.92520172,0.04161024)[(0,0)\[lb\]]{} (0.20724699,0.04058544)[(0,0)\[lb\]]{} (0.56787001,0.04191117)[(0,0)\[lb\]]{} (0.70581748,0.04160674)[(0,0)\[lb\]]{} (0.01925342,0.32353758)[(0,0)\[lb\]]{} (0.0059475,0.45515401)[(0,0)\[lb\]]{} (0.01888279,0.66697707)[(0,0)\[lb\]]{} (0.39631151,0.2460684)[(0,0)\[lb\]]{} (0.45548712,0.14654446)[(0,0)\[lb\]]{} (0.54257394,0.53529407)[(0,0)\[lb\]]{} (0.38510369,0.04028106)[(0,0)\[lb\]]{}
The solution $v^{n+1}_k$ is then updated inside the support by enforcing $$k\in[K_l(n),K_r(n)]:\qquad
\frac{v^{n+1}_k-v^n_k}{\Delta t}=(\sigma(v^n_k)+{\varepsilon})\frac{v_{k-1}^{n}-2v_{k}^{n}+v_{k+1}^{n}}{\Delta x^2}+\left|\frac{v_{k+1}^{n}-v_{k-1}^{n}}{2\,\Delta x}\right|^2,
\label{eq:scheme}$$ where ${\varepsilon}>0$ is a fixed artificial viscosity parameter to be chosen later. Observe that is not applied across the interfaces but only in the numerical support, where is formally in the parabolic regime since $\{v>0\}=\{\sigma(v)>0\}$.
Inside the boundary layers of thickness $\left(s^n\right)'$ the solution is interpolated as $$\qquad v_{k}^{n+1}:=\left\{
\begin{array}{cl}
v_{K_l(n)}^{n+1}\frac{x_k-\zeta_l^{n+1}}{x_{K_l(n)}-\zeta_l^{n+1}} \qquad & x_k\in[\zeta^{n+1}_l,x_{K_l(n)-1}]\\
v_{K_r(n)}^{n+1}\frac{\zeta_r^{n+1}-x_k}{\zeta_r^{n+1}-x_{K_r(n)}} & x_k\in[x_{K_r(n)+1},\zeta^{n+1}_r]
\end{array}
\right.,
\label{eq:def_linear_interpolation}$$ and finally we set $$v_k^{n+1}:=0\qquad \mbox{for }x_k\notin[\zeta_l^{n+1},\zeta_r^{n+1}].$$ The interpolation is consistent with the well known linear behaviour of the pressure variable across the moving free boundaries [@Va07 Theorem 15.24], see Lemma \[lem:linear\_growth\_interface\] later on.
According to $v_k^n$ is exactly linear in the boundary layers. As a consequence also reads $\frac{\zeta_l^{n+1}-\zeta_l^n}{\Delta t}=-\frac{v_{K_l(n)}-v_{K_l(n)-1}}{\Delta x}$ and $\frac{\zeta_r^{n+1}-\zeta_r^n}{\Delta t}=-\frac{v_{K_r(n)+1}-v_{K_r(n)}}{\Delta x}$, again reproducing the propagation law $d\zeta/dt=-\partial_x v$. \[rmk:propagation\_law\_interpolation\]
Throughout the whole paper and without further mention we impose the following Courant-Fredriech-Lewis stability condition
$$ \begin{array}{c}
\frac{\Delta t}{\Delta x^2}:=\beta\leq \frac{1}{2\Big( \sigma(M)+{\varepsilon}\Big)
+\gamma_0\Delta x\Big(4+3S_1(M)\Big)
+ \gamma_0^2\Delta x^2 S_2(M)/2}\\
\gamma_0\Delta x\Big(27+9s_1(M)+3S_1(M)+\Delta xS_2(M)/4\Big)\leq {\varepsilon}\leq\mathcal{O}(\Delta x)
\end{array}
\label{eq:CFL}
\tag{CFL}$$
with $\|v^0\|_{L^{\infty}({\mathbb{R}})}\leq M$, $\operatorname{Lip}(v^0)\leq \gamma_0$, and $s_1(M),S_1(M),S_2(M)\geq 0$ as in .
A priori discrete estimates {#subsection:estimates_one_patch}
---------------------------
Defining the discrete downwind and centered spatial derivatives $$w^n_k:=\frac{v^n_k-v^n_{k-1}}{\Delta x},\qquad \overline{w}^n_k:=\frac{v^n_{k+1}-v^n_{k-1}}{2\Delta x},$$ the first discrete estimate reads
Assume that $0\leq v^0_k\leq M$ with $|w^0_k|\leq \gamma_0$. Then for all $k,n$ there holds $$0\leq v^n_k\leq M\quad \mbox{and}\quad |w^n_k|\leq \gamma_0.$$ \[lem:Linfty\_Lipschitz\_estimate\]
We write $\beta=\Delta t/\Delta x^2$ and abbreviate $\sigma_k^n:=\sigma(v_k^n)$. Arguing by induction on $n$ our statement holds for $n=0$ by assumption on the initial datum.\
[**Step 1: positivity and $l^{\infty}$ stability.**]{} Noting that $\frac{v_{k+1}^n-v_{k-1}^n}{2\Delta x}=\frac{w_{k+1}^n+w_{k}^n}{2}$ it is easy to rewrite inside the support $x_k\in[x_{K_l(n)},x_{K_r(N)}]$ as $$v_k^{n+1} = (1-2a)v_k^n + (a-b)v_{k-1}^n +(a+b)v_{k-1}^n$$ with $$a:=\beta(\sigma_k^n+{\varepsilon})
\quad\mbox{and}\quad
b:=\beta \Delta x(w_{k+1}^n+w_{k}^n)/4 .
\label{eq:def_ab_Linfty_Lipschitz}$$ By the induction hypothesis and monotonicity of $\sigma$ the condition implies $$0\leq \beta {\varepsilon}\leq a\leq \beta(\sigma(M)+{\varepsilon})\leq \frac{1}{2},\qquad |b|\leq \Delta x\beta\gamma_0/2\leq \beta{\varepsilon}\leq a,$$ thus $v_k^{n+1}$ is a convex combination of $v_{k-1}^n,v_{k}^n,v_{k+1}^n\in [0,M]$. In particular $0\leq v_{k}^{n+1}\leq M$ for $k\in[K_l(n),K_r(n)]$, and by clearly $0\leq v^{n+1}_k\leq M$ everywhere. [**Step 2: Lischitz bounds in the support.**]{} Consider any $k\in[K_l(n)+1, K_r(n)]$, so that $v^{n+1}_k,v^{n+1}_{k-1}$ are both computed using , which we recast in the form $$v_{k}^{n+1}=v_k^n+\beta \Delta x\left(\sigma_k^n+{\varepsilon}\right)(w^n_{k+1}-w^n_{k})+\beta \Delta x^2/4\left(w^n_{k+1}+w_{k-1}^n\right)^2.
\label{eq:finite_diff_v_1}$$ Subtracting the corresponding equation for $v_{k-1}^{n+1}$ and dividing by $\Delta x$,straightforward manipulations lead to $$\begin{aligned}
w_k^{n+1}&= w_k^n +\Delta t \Bigg[\left(\frac{\sigma_k^n+\sigma_{k-1}^n}{2}+{\varepsilon}\right) \frac{ w_{k+1}^n-2w_{k}^n+w_{k-1}^n}{\Delta x^2}\nonumber\\
& \hspace{2cm}+ \left(\mathfrak{S}_{v} w_k^n+ 2 \frac{w_{k+1}^n+2w_{k}^n+w_{k-1}^n}{4}\right)\frac{w_{k+1}^n-w_{k-1}^n}{2 \Delta x}\Bigg]
\label{eq:diff_wk}\end{aligned}$$ with $$\mathfrak{S}_v:=\frac{\sigma_k^n-\sigma_{k-1}^n}{v_k^n-v_{k-1}^n} = \frac{\sigma(v_k^n)-\sigma(v_{k-1}^n)}{v_k^n-v_{k-1}^n}\approx \sigma'(v(x_k,t^n)).$$ Formula is the discrete equivalent of $$w=\partial_x v:\qquad \partial_t w=\sigma(v)\partial_{xx}^2w+\big[\sigma'(v)w+2w\big]\partial_x w,
\label{eq:PDE_w}$$ which is formally obtained differentiating w.r.t. $x$. Considering as a linear parabolic equation $\partial_t w=a\partial_{xx}^2v+b\partial_xw$ with no zero-th order coefficient, we see that $w=\partial_x v$ formally satisfies the maximum principle. Thus the initial $\gamma_0$-Lipschitz bounds for $v^0$ should be preserved for $t\geq 0$ as in our statement.
In order to make this maximum principle rigorous at the discrete level we rewrite as $$w_k^{n+1}=(1-2a)w_k^n+(a-b)w_{k-1}^n+(a+b)w_{k+1}^n,
\label{eq:w_k^n+1}$$ with now $$a=\beta\left(\frac{\sigma_k^n+\sigma_{k-1}^n}{2}+{\varepsilon}\right)
\quad \mbox{and} \quad
b=\beta \Delta x\left(\frac{\mathfrak{S}_v w_k^n}{2}+\frac{w_{k+1}^n+2w_{k}^n+w_{k-1}^n}{4}\right).
\label{eq:def_ab}$$ By the induction hypothesis $0\leq v_k^n\leq M$ and the structural assumption we get $0\leq \mathfrak{S}_v\approx \sigma'(v_k^n)\leq S_1(M)$, and the condition implies $$0\leq\beta {\varepsilon}\leq a \leq \beta\Big(\sigma(M)+{\varepsilon}\Big)\leq 1/2\quad \mbox{and} \quad |b|\leq \beta \Delta x \gamma_0\left(\frac{S_1(M)}{2}+1\right) \leq \beta {\varepsilon}\leq a.$$ From we see that $w_{k}^{n+1}$ is a convex combination of $w_{k-1}^{n},w_{k-1}^{n},w_{k+1}^{n}$ and we conclude that $|w_{k}^{n+1}|\leq \gamma_0$ as claimed.\
[**Step 3: Lischitz bounds close to the interfaces.**]{} The computations at the left and right interfaces are identical, so we only deal with the right one and write $K=K_r(n)$, $s^n=\zeta_r^n-x_{K}$ and $(s^n)'=\zeta_r^{n+1}-x_{K}$ for simplicity. By construction of the scheme $v_k^{n+1}$ is linear for $x_k\in[x_{K},\zeta^{n+1}]$ and zero for $x_k\geq \zeta^{n+1}$. In particular $w_{K+1}^{n+1}\leq w_k^{n+1}\leq 0$ for all $x_k\geq x_{K+1}$ and it is clearly enough to estimate $|w_{K+1}^{n+1}|$. From we see that $w^{n+1}_{K+1}=-\frac{v_K^{n+1}}{(s^n)'}$, and exploiting we get $$\begin{aligned}
w_{K+1}^{n+1} & =-\frac{1}{(s^n)'}\Bigg{[}v_K^n + \Delta t\left(\sigma_K^n+{\varepsilon}\right)\frac{v_{K+1}^n-2v_K^n+v_{K_1}^n}{\Delta x^2}+\Delta t\left(\frac{v_{K+1}^n-v_{K-1}^n}{2\Delta x}\right)^2 \Bigg{]}\nonumber\\
& =-\frac{1}{(s^n)'}\Bigg{[}v_K^n + \beta\Delta x \left(\sigma_K^n+{\varepsilon}\right)(w_{K+1}^n-w_K^n)\nonumber\\
& \hspace{2cm} +\Delta t \left\{(w_{K+1}^n)^2 -\frac{w_K^n+3w_{K+1}^n}{4}(w_{K+1}^n-w_K^n)\right\} \Bigg{]}.
$$ According to Remark \[rmk:propagation\_law\_interpolation\] we have $(s^n)'-s^n=\zeta^{n+1}-\zeta^n=-w_{K+1}^n\Delta t$, and since $v_K^{n}=-w_{K+1}^ns^n$ we get $v_K^n+\Delta t (w_{K+1}^n)^2=-w_{K+1}^n (s^n)'$. Substituting in the previous expression gives $$w_{K+1}^{n+1}=w_{K+1}^n+c(w_{K}^n-w_{K+1}^n)
\label{eq:w_k^n+1_interface}$$ with $$c=\frac{\beta \Delta x}{(s^n)'}\left((\sigma_K^n+{\varepsilon})-\Delta x\frac{3w_{K+1}^n+w_{K}^n }{4}\right).
\label{eq:def_c}$$ Using the induction hypothesis, the condition, and $(s^n)'\geq s^n \geq \Delta x$ yields $$0\leq \beta\frac{\Delta x}{(s^n)'}\left({\varepsilon}-\gamma_0 \Delta x\right)\leq c\leq \beta\left(\sigma(M)+{\varepsilon}+\gamma_0\Delta x\right)\leq 1,$$ thus by $|w_{K+1}^{n+1}|\leq \gamma_0$ as the convex combination of $w_{K}^{n},w_{K+1}^{n}$ and the proof is complete.
As a consequence the interfaces propagate with finite speed:
For all $t^n\geq 0$ there holds $\left|\frac{\zeta^{n+1}_{lr}-\zeta^n_{lr}}{\Delta t}\right|\leq \gamma_0$ and $$\zeta_l(0)-\gamma_0t^n\leq \zeta^n_l\leq \zeta_l(0)\leq \zeta_r(0)\leq \zeta_r^n\leq \zeta_r(0)+\gamma_0t^n.$$ \[lem:discrete\_speed\_interfaces\]
By Remark \[rmk:propagation\_law\_interpolation\] $|(\zeta^{n+1}-\zeta^n)/\Delta t|=|-w_{K(n)\pm1}^n|$ so our statement immediately follows by Lemma \[lem:Linfty\_Lipschitz\_estimate\] and the pinning $\zeta_{ln}^0=\zeta_{lr}(0)$. The monotonicity is a consequence of with $v_k^n\geq 0$.
In the next auxiliary lemma we construct the lower bound to be used in the generalized Aronson-Bénilan estimate $\partial^2_{xx} v\geq \underline{z}(t)$ by means of a certain ODE:
Let $\Lambda:=\gamma_0^2S_2(M)$ and $F(z):=\Lambda z +(2+s_1(M))z^2$ with $s_1,S_2$ as in . There is a function $\underline{z}(t):{\mathbb{R}}^+\to {\mathbb{R}}$ such that $\frac{d\underline{z}}{dt}=F(\underline{z})$ with $\lim\limits_{t\searrow 0}\underline{z}(t)=-\infty$. Moreover $\underline{z}$ is monotone increasing and concave, $\underline{z}(t)\leq \underline{z}(\infty)=-\Lambda/(2+s_1(M))$, and $\underline{z}(t)\sim -\frac{1}{(2+s_1(M))t}$ when $t\searrow 0$. \[lem:def\_ODE\_z\]
Observe that $F(z)$ is a quadratic polynomial with $F(-\Lambda/(2+s_1(M)))=0$. Picking any $t_0>0,z_0<-\Lambda/(2+s_1(M))$ and solving $dz/dt=F(z)$ with $z(t_0)=z_0$ it is easy to see that $z$ is monotone increasing in $(\underline{T},\infty)$ with blow-up in finite time $z(\underline{T})=-\infty$ and $z(\infty)=-\Lambda/(2+s_1(M))$. Shifting $\underline{z}(t):=z(t+\underline{T})$ gives the sought solution, and all the qualitative properties follow from a straightforward phase portrait analysis.
The generalized Aronson-Bénilan estimate then takes the form
Let $\underline{z}(t)$ as in Lemma \[lem:def\_ODE\_z\]. Then for all $k,n$ there holds $$Z_k^n:=\frac{Av_k^n}{\Delta x^2}=\frac{v_{k-1}^n-2v_{k}^n+v_{k+1}^n}{\Delta x^2}\geq \underline{z}(t^n).
\label{eq:Aronson_Benilan}$$ \[lem:Aronson\_Benilan\_estimate\]
Since $\underline{z}$ is monotone increasing with $\underline{z}(0)=-\infty$ the time $t^N=\max\{t^n:\,\underline{z}(t^n)\leq -2\gamma_0/\Delta x\}$ is well defined and positive, provided that $\Delta x,\Delta t$ are small enough. By Lemma \[lem:Linfty\_Lipschitz\_estimate\] we have $Z_{k}^n=\frac{w_{k+1}^n-w_{k}^n}{\Delta x}\geq -2\gamma_0/\Delta x$ and our estimate automatically holds if $t_n\leq t^N$. We argue now by induction on $n\geq N$.\
[**Step 1: estimate in the support.**]{} Consider first any $k\in[K_l(n)+1,K_r(n)-1]$, so that $v_{k-1}^{n+1},v_k^{n+1},v_{k+1}^{n+1}$ are all computed from the finite difference equation . Applying the second order difference operator $A$ to and dividing by $\Delta x^2$, straightforward algebra leads to $$\begin{aligned}
Z_{k}^{n+1} =& Z_k^n + \Delta t \Bigg{[}(\mathfrak{S}+{\varepsilon})\frac{AZ_{k}^n}{\Delta x^2}
+2 \left(\mathfrak{S}_x+W_{1}\right)\left(\frac{Z_{k+1}^n-Z_{k-1}^n}{2\Delta x}\right)
\nonumber\\
& \hspace{3cm}
+ \mathfrak{S}_{vv}(W_{2})^2\frac{Z_{k-1}^n+2Z_k^n+Z_{k+1}^n}{4}
\nonumber\\
&\hspace{2cm}
+\Bigg{\{}\mathfrak{S}_v Z_k^n\frac{Z_{k-1}^n+2Z_k^n+Z_{k+1}^n}{4}
+ 2 \left(\frac{Z_{k-1}^n+2Z_k^n+Z_{k+1}^n}{4}\right)^2 \Bigg{\}}\Bigg{]}
\label{eq:discrete_PDE_Z}\end{aligned}$$ with $$\begin{aligned}
& \mathfrak{S} := \frac{\sigma_{k-1}^n+2\sigma_{k}^n+\sigma_{k+1}^n}{4}\approx \sigma(v(x_k,t^n)),
\qquad
\mathfrak{S}_x := \frac{\sigma_{k+1}^n-\sigma_{k-1}^n}{2\Delta x}\approx \partial_x \sigma(v(x_k,t^n)),\\
& \mathfrak{S}_v := \frac{1}{2}\left(\frac{\sigma_{k+1}^{n}-\sigma_{k}^{n}}{v_{k+1}^{n}-v_{k}^{n}} + \frac{\sigma_{k}^{n}-\sigma_{k-1}^{n}}{v_{k}^{n}-v_{k-1}^{n}}\right)\approx \sigma'(v(x_k,t^n)),\\
&\mathfrak{S}_{vv} :=2\frac{(v_k^n-v_{k-1}^n)\sigma_{k+1}^n-(v_{k+1}^n-v_{k-1}^n)\sigma_{k}^n + (v_{k+1}^n-v_{k}^n)\sigma_{k-1}^n}{(v_{k+1}^n-v_{k}^n)(v_{k}^n-v_{k-1}^n)(v_{k+1}^n-v_{k-1}^n)}\approx \sigma''(v(x_k,t^n)),\end{aligned}$$ and $$W_{1} :=\frac{\overline{w}_{k-1}^n+2\overline{w}_{k}^n+\overline{w}_{k+1}^n}{4}\approx\partial_xv(x_k,t^n),
\qquad
W_{2} :=\overline{w}_{k}^n\approx\partial_xv(x_k,t^n)$$ (recall that we write $\sigma_k^n=\sigma(v_k^n)$ and $\overline{w}_{k}^n=(v_{k+1}^n-v_{k-1}^n)/2\Delta x$). Note that is nothing but the discrete equivalent of $$\partial_t z=\sigma(v)\partial_{xx}^2 z
+2\Big{[}\partial_x \sigma(v)+\partial_x v\Big{]}\partial_x z
+\Big{[} \sigma''(v)|\partial_x v|^2\Big{]}z
+\Big{[}\sigma'(v)+2\Big{]}z^2
\label{eq:PDE_z=vxx}$$ for $z=\partial_{xx}^2 v$, which is obtained differentiating twice $\partial_t v=\sigma(v)\partial_{xx}^2v+|\partial_x v|^2$ w.r.t. $x$. Let us give a formal proof that $z=\partial^2_{xx}v\geq \underline{z}(t)$ at the continuous level: since $0\leq v(x,t)\leq M$ we have $0<s_1(M)\leq \sigma'(v)$ and $|\sigma''(v)|\leq S_2(M)$, and recall that $|\partial_x v|\leq \gamma_0$. Using the definition of $\underline{z}(t)$ in Lemma \[lem:def\_ODE\_z\] it is easy to check that $\underline{z}(t)$ is a subsolution of . Since $\underline{z}(0)=-\infty\leq z(x,0)$ the comparison principle should give $z(x,t)\geq\underline{z}(t)$. In order to reproduce this formal computation at the discrete level let us first rewrite as $$\begin{aligned}
Z_{k}^{n+1}= & \left[1-2\beta(\mathfrak{S}+{\varepsilon}) + \frac{\beta \Delta x^2(W_{2})^2\mathfrak{S}_{vv}}{2}\right]Z_k^{n}
\nonumber\\
& +\beta\left[(\mathfrak{S}+{\varepsilon}) + \frac{\Delta x^2(W_{2})^2\mathfrak{S}_{vv}}{4} - \Delta x(\mathfrak{S}_x+W_{1})\right]Z_{k-1}^{n}
\nonumber\\
& +\beta\left[(\mathfrak{S}+{\varepsilon}) + \frac{\Delta x^2(W_{2})^2\mathfrak{S}_{vv}}{4} + \Delta x(\mathfrak{S}_x+W_{1})\right]Z_{k-1}^{n}
\nonumber\\
& +\frac{\beta \Delta x^2}{8}\left(Z_{k-1}^n+2Z_{k}^n+Z_{k+1}^n\right)\left[Z_{k-1}^n+2(1+\mathfrak{S}_v)Z_{k}^n+Z_{k+1}^n\right].
\label{eq:discrete_Zk^n+1}\end{aligned}$$ We show now satisfies the discrete comparison principle, in the sense that $Z_{k}^{n+1}$ is non-decreasing in the three arguments $Z_{k-1}^n,Z_{k}^n,Z_{k+1}^n$. To this end we first note that $$|Z_{k}^n|=|(w_{k+1}^n-w_{k}^n)/\Delta x|\leq 2\gamma_0/\Delta x,$$ and by our structural hypotheses and Lemma \[lem:Linfty\_Lipschitz\_estimate\] it is easy to check that $$\begin{array}{ccc}
0\leq \mathfrak{S}\leq \sigma(M)
&
|\mathfrak{S}_x|\leq S_1(M)\gamma_0,
&
0\leq \mathfrak{S}_v\leq S_1(M),\\
|\mathfrak{S}_{vv}|\leq S_2(M),
&
|W_{1}|\leq \gamma_0,
&
|W_{2}|\leq \gamma_0.
\end{array}$$ Thus by the condition $$\begin{aligned}
\frac{\partial Z_{k}^{n+1}}{\partial Z_k^n} & = 1 -2\beta (\mathfrak{S}+{\varepsilon})+\frac{\beta \Delta x^2(W_{2})^2\mathfrak{S}_{vv}}{2}\\
& +\frac{\beta \Delta x^2}{8}\Big{[}2\Big{(}Z_{k-1}^n+2(1+\mathfrak{S}_v)Z_{k}^n+Z_{k+1}^n\Big{)}
+ 2(1+\mathfrak{S}_v)\Big{(}Z_{k-1}^n+2Z_{k}^n+Z_{k+1}^n\Big{)}
\Big{]}\\
& \geq 1-\beta\left[2\Big(\sigma(M)+{\varepsilon}\Big)+\frac{\Delta x^2 \gamma_0^2 S_2(M)}{2}+\gamma_0\Delta x\Big(4+3S_1(M)\Big)\right]\geq 0,
$$ $$\begin{aligned}
\frac{\partial Z_{k}^{n+1}}{\partial Z_{k-1}^n} & = \beta\Bigg{[}(\mathfrak{S}+{\varepsilon}) + \frac{\Delta x^2(W_{2})^2\mathfrak{S}_{vv}}{4} - \Delta x(\mathfrak{S}_x+W_{1})\\
& \hspace{1cm}+\frac{\Delta x^2}{8}\Big\{(Z_{k-1}^n+2Z_{k}^n+Z_{k+1}^n)+(Z_{k-1}^n+2(1+\mathfrak{S}_v)Z_{k}^n+Z_{k+1}^n))\Big\}\Bigg{]}\\
& \geq \beta\left[{\varepsilon}-\gamma_0\Delta x\left\{\frac{\Delta x\gamma_0S_2(M)}{4}+\Big(S_1(M)+1\Big)+\frac{1}{2}\Big(4+S_1(M)\Big)\right\}\right]\geq 0,
$$ and similarly $\frac{\partial Z_{k}^{n+1}}{\partial Z_{k+1}^n}\geq 0$. By the induction hypothesis we see that $Z_{k}^{n+1}$ is greater or equal to the right-hand side of evaluated with $Z_{k-1}^n,Z_{k}^n,Z_{k+1}^n\geq \underline{z}(t^n)$, and using the structural assumptions $\mathfrak{S}_v\geq s_1(M)$ and $|\mathfrak{S}_{vv}|\leq S_2(M)$ we get $$\begin{aligned}
Z_{k}^{n+1}& \geq \underline{z}(t^n)+ \beta \Delta x^2(W_{2})^2\mathfrak{S}_{vv}\,\underline{z}(t^n)+\beta \Delta x^2(2+\mathfrak{S}_v)\underline{z}^2(t^n)
\\
& \geq \underline{z}(t^n)+ \Delta t\left[\gamma_0^2S_2(M)\underline{z}(t^n)+(2+s_1(M))\underline{z}^2(t^n)\right].\end{aligned}$$ In the righ-hand side we recognize $\underline{z}(t^n)+\Delta t F(\underline{z}(t^n))$ with $F$ as in Lemma \[lem:def\_ODE\_z\]. Since by construction $\dot{\underline{z}}=F(\underline{z})$ and $\underline{z}$ is concave we finally get $$Z_{k}^{n+1}\geq \underline{z}(t^n)+\dot{\underline{z}}(t^n)[t^{n+1}-t^n]\geq \underline{z}(t^{n+1})$$ as required.\
[**Step 2: estimate close to the interfaces.**]{} We only establish the AB estimate across the right interface and boundary layer, and write again $\zeta=\zeta_r$ and $K=K_r(n)$ to keep the notations light (the argument is identical to the left). Recall that for $x_k\in[x_{K},\zeta^{n+1}]$ the next step $v_{k}^{n+1}\geq 0$ is linearly interpolated by , and $v_{k}^{n+1}=0$ for $x_k\geq \zeta^{n+1}$. As a consequence $Av_k^{n+1}\geq 0$ for $k>K$ and is trivially satisfied there as $Z_k^{n+1}\geq 0>\underline{z}(t^{n+1})$. Hence we only need to look at $k=K$. By definition of $K=K_r(n)$ we see that $w_{K}^{n+1}$, $w_{K+1}^{n+1}$ satisfy and , namely $$w_K^{n+1}=w_K^{n}+(a+b)\Delta x Z_K^{n}-(a-b)\Delta x Z_{K+1}^{n}
\quad\mbox{and}\quad
w_{K+1}^{n+1}=w_{K+1}^n-c\Delta xZ_K^n$$ with $a,b$ as in with $k=K$ and $c$ as in . Subtracting and dividing by $\Delta x$ we get that $Z_{K}^{n+1}=(w_{K+1}^{n+1}-w_{K}^{n+1})/\Delta x$ can be expressed as $$Z_{K}^{n+1}=(1-a-b-c)Z_{K}^n + (a-b)Z_{K-1}^n.
\label{eq:Z_K^n+1_interface}$$ We claim as in step 1 that the right-hand side is nondecreasing in $Z_{K}^n,Z_{K-1}^n$. Indeed we already showed in the proof of Lemma \[lem:Linfty\_Lipschitz\_estimate\] that $a-|b|\geq 0$, and recalling that $(s^n)'=\zeta^{n+1}-x_{K(n)}\geq\zeta^{n}-x_{K(n)} \geq \Delta x$ we compute $$\begin{aligned}
1-a-b-c &= 1 - \beta\left(\frac{\sigma_K^n+\sigma_{K-1}^n}{2}+{\varepsilon}\right)\\
& \hspace{1cm}-\beta \Delta x\left(\frac{1}{2}\frac{\sigma_K^n-\sigma_{K-1}^n}{v_K^n-v_{K-1}^n} w_K^n+\frac{w_{K-1}^n+2w_{K}^n+w_{K+1}^n}{4}\right)\\
&\hspace{2cm}-\frac{\beta \Delta x}{s_n'}\left((\sigma_K^n+{\varepsilon})-\Delta x\frac{3w_{K+1}^n+w_{K}^n}{4}\right)\\
& \geq 1-\beta\left[
\Big(\sigma(M)+{\varepsilon}\Big) + \Delta x\left(\frac{S_1(M)\gamma_0}{2}+\gamma_0\right)
+ \Big(\sigma(M)+{\varepsilon}+\gamma_0\Delta x\Big)
\right]\geq 0,
$$ where the last inequality follows by the condition. Before evaluating with $Z_{K}^n,Z_{K-1}^n\geq \underline{z}(t^n)$ we first recall from Lemma \[lem:discrete\_speed\_interfaces\] that the interfaces propagate with discrete speed at most $\gamma_0$, and that by the CFL condition $\Delta t=\mathcal{O}(\Delta x^2)$. In particular $\Delta x \leq s_n\leq (s^n)'=s_n+(\zeta^{n+1}-\zeta^n)\leq 2\Delta x +\mathcal{O}( \Delta x^2)$, thus $\Delta x\leq (s^n)'\leq 3\Delta x$ for small $\Delta x$ and $$\begin{aligned}
1-2b-c & = 1-\beta \Delta x\left(\frac{\sigma_K^n-\sigma_{K-1}^n}{v_K^n-v_{K-1}^n}w_K + \frac{w_{K-1}^n+2w_{K}^n+w_{K+1}^n}{2}\right)\\
& \hspace{3cm}-\frac{\beta \Delta x}{(s^n)'}\left((\sigma_K^n+{\varepsilon})-\Delta x\frac{3w_{K+1}^n+w_{K}^n}{4}\right)\\
& \leq 1 + \beta\gamma_0\Delta x\big( S_1(M)+2 \big)-\frac{\beta {\varepsilon}}{3} +\beta \gamma_0 \Delta x\\
& \leq 1-\Delta t\big(2+s_1(M)\big)\frac{3\gamma_0}{\Delta x}$$ by the condition (${\varepsilon}\geq c\gamma_0\Delta x$). For small $\Delta x,\Delta t$ and by definition of $t^N=\max\{t^{n'}:\,F(t^{n'})\leq -2\gamma_0/\Delta x\}$ it is easy to check that $F(t^N)\sim -2\gamma_0/\Delta x$, and because $\underline{z}$ is increasing and our induction is on $n\geq N$ we can assume that $-3\gamma_0/\Delta x<-2\gamma_0/\Delta x\approx \underline{z}(t^N)\leq \underline{z}(t^n)$ hence $$1-2b-c \leq 1+\Delta t\big(2+s_1(M)\big)\underline{z}(t^*)\leq 1+\Delta t\big(2+s_1(M)\big)\underline{z}(t^n).$$ Evaluating with $Z_{k-1}^n,Z_{k}^n,Z_{k+1}^n\geq \underline{z}(t^n)$ thus gives $$\begin{aligned}
Z_{K}^{n+1}\geq (1-2b-c)\underline{z}(t^n)
& \geq \underline{z}(t^{n})+\Delta t\big(2+s_1(M)\big)\underline{z}^2(t^n)\\
& \geq \underline{z}(t^{n})+\Delta t\Big[\Lambda \underline{z}(t^n) + \big(2+s_1(M)\big)\underline{z}^2(t^n)\Big]\\
& =\underline{z}(t^{n})+\Delta t F(\underline{z}(t^n)),\end{aligned}$$ and we conclude by concavity of $\underline{z}$ as in step 1.
For the pure PME nonlinearity $\Phi(s)=s^m$ one has $\sigma(r)=(m-1)r$ and therefore $s_1(M)=s_1=(m-1)$ and $S_2(M)=0$ in . Tthe ODE for $\underline{z}$ then becomes $\dot{z}=(m+1)z^2$, thus $\underline{z}(t)=-1/(m+1)t$ in Lemma \[lem:def\_ODE\_z\] and we recover the optimal Aronson-Bénilan estimate $\partial^2_{xx} v\geq -1/(m+1)t$. For general nonlinearities the optimal estimate [@CP82] takes the form $\partial_{xx}^2 v\geq -h(v)/t$ for some structural function $h$ related to $\Phi$. Unfortunately we were not able to reproduce the optimal computations at the discrete level, and we shall be content here with our lower bound $\partial_{xx}^2 v\geq \underline{z}(t)\sim-C(1+1/t)$.
There is $C=C(v^0)>0$ only such that $$\forall n\geq 0,\,k\notin [K_l(n),K_r(n)]:\qquad \left|\frac{v_k^{n+1}-v_k^n}{\Delta t}\right|\leq C.$$ \[lem:dv/dt\_interface\]
The argument is identical to [@DBH84 Lemma 2.4].
Combining Lemma \[lem:Aronson\_Benilan\_estimate\] and Lemma \[lem:dv/dt\_interface\] we get
There is $C=C(v^0)>0$ such that $$\sum\limits_k\left|\frac{Av_k^n}{\Delta x^2}\right|\Delta x +\sum\limits_k\left|\frac{v_{k}^{n+1}-v^n_k}{\Delta t}\right|\Delta x\leq C\left(1+\frac{1}{t^n}+T\right)
\label{eq:L1-bounds_vxx_vt}$$ for all $t^n\leq T$. \[cor:L1-bounds\_vxx\_vt\]
From Lemma \[lem:Aronson\_Benilan\_estimate\] and $-C(1+1/t)\leq \underline{z}(t)\leq 0$ we see that $$\left|\frac{Av_k^n}{\Delta x^2}\right| \leq \frac{Av_k^n}{\Delta x^2}+2|\underline{z}(t^n)|\leq \frac{Av_k^n}{\Delta x^2} +C\left(1+\frac{1}{t^n}\right).$$ Multiplying by $\Delta x$ and summing over $k$’s with $v^n_k=0$ outside an interval of length $C(1+t^n)$ (Lemma \[lem:discrete\_speed\_interfaces\]) we get the first part of the estimate $$\sum\limits_k\left|\frac{Av_k^n}{\Delta x^2}\right|\Delta x\leq C\left(1+\frac{1}{t^n}\right)(1+t^n)\leq C(1+1/t^n+T).$$ Inside the support $k\in[K_l(n),K_r(n)]$ the time derivative can be estimated from as $$\begin{aligned}
\left|\frac{v_k^{n+1}-v_k^n}{\Delta t}\right| & =\left|\Big(\sigma(v_k^n)+{\varepsilon}\Big)\frac{Av_k^n}{\Delta x^2}+\left(\frac{v_{k+1}^n-v_{k-1}^n}{2\Delta x}\right)^2\right|\\
& \leq \Big(\sigma(M)+{\varepsilon}\Big)\left|\frac{Av_k^n}{\Delta x^2}\right|+\gamma_0^2\leq C\left(\left|\frac{Av_k^n}{\Delta x^2}\right|+1\right),\end{aligned}$$ and inside the boundary layers of thickness $\Delta x\leq s^n\leq 2\Delta x$ the time derivative is estimated by Lemma \[lem:dv/dt\_interface\]. Multiplying by $\Delta x$ and summing over $k$’s as before gives the second part of the estimate and ends the proof.
We end this section with uniform Höder estimates in time up to $t=0^+$, which are inherited from the initial Lipschitz regularity for $v^0(x)$.
For any $T>0$ there is $C=C(T,v^0)>0$ such that $$|v_{k}^n-v_k^m|\leq C|t^n-t^m|^{1/2}$$ for all $t^n,t^m\in[0,T]$. \[prop:Holder\]
The proof is almost identical to [@DBH84 Lemma 2.7], and the argument is a discrete version of that in [@G76]. However we will need to make sure in Section \[section:two\_patches\] that the proof carries out for the hole-filling problem so we give nonetheless the full details for the sake of completeness.
We argue locally in cylinders $$Q=[x_{k_0}-r,x_{k_0}+r]\times[t^{n_0},t^{n_1}],$$ where $x_{k_0}$ and $0\leq t^{n_0}\leq t^{n_1}\leq T$ are fixed and $r$ is a multiple of $\Delta x$ to be adjusted.\
[**Step 1:**]{} letting $$\begin{array}{c}
H:=\max\limits_{n_0\leq n\leq n_1}\,|v_{k_0}^n-v_{k_0}^{n_0}|,\qquad c:= 2\Big(\sigma(M)+{\varepsilon}\Big)+\gamma_0 r\\
V_k^n:= v_k^n-v_{k_0}^{n_0}-\gamma_0 r -\frac{H}{r^2}\left[(x_k-x_{k_0})^2+c(t_n-t_{n_0})\right],
\end{array}$$ we claim that $$V_k^n\leq 0\qquad \mbox{for all }(x_k,t^n)\in Q.
\label{eq:V_k^n_leq_0}$$ Arguing by induction on $n$, holds for $n=n_0$ as $V_{k}^{n_0}\leq \left|v_{k}^{n_0}-v_{k_0}^{n_0}\right|-\gamma_0r\leq 0$ since $|x_k-x_{k_0}|\leq r$ and $|w^n_k|\leq \gamma_0$. For the induction step we consider three cases: (i) $x_k\notin[\zeta_l^{n+1},\zeta_r^{n+1}]$, (ii) $x_k$ is inside the boundary layer, and (iii) $x_k$ is inside the numerical support where holds.
In the first case we have $v_k^{n+1}=0$ and our claim immediately holds by definition of $V_{k}^{n+1}$ with $v_{k_0}^{n_0}\geq 0$. For (ii) we have $x_k\in[x_{K(n)},\zeta^{n+1}]$, and we have already shown that $(s^n)'=|\zeta^{n+1}-x_{K(n)}|\leq 3\Delta x$ for small $\Delta x$. In the boundary layer $v_{k}^{n+1}$ is computed by linear interpolation with slope $|w_k^{n+1}|\leq \gamma_0$ and therefore $$V_{k}^{n+1}\leq v_{k}^{n+1}-\gamma_0\leq \gamma_0 .3\Delta x-\gamma_0 r\leq 0$$ provided that $r\geq 3\Delta x$, which will be ensured in step 2. In the last case (iii) we consider the linearized operator $L$ of , whose action on any sequence $a_k^n$ is defined as $$La_k^{n+1}:=\frac{a_k^{n+1}-a_k^n}{\Delta t}-\Big(\sigma(v_k^n)+{\varepsilon}\Big)\frac{A a_k^n}{\Delta x^2}-\left(\frac{v_{k+1}^n-v_{k-1}^n}{2\Delta x}\right)\left(\frac{a_{k+1}^n-a_{k-1}^n}{2\Delta x}\right).$$ Applying $L$ to $V_k^{n+1}$ with $Lv_k^{n+1}=0$ as in , it is easy to compute $$\begin{aligned}
LV_{k}^{n+1} & =\frac{H}{r^2}\left[-c + 2\Big(\sigma(v_k^n)+{\varepsilon}\Big)+\left(\frac{v_{k+1}^n-v_{k-1}^n}{2\Delta x}\right)\left(x_k-x_{k_0}\right)\right] \\
& \leq \frac{H}{r^2}\left[-c+2\Big(\sigma(M)+{\varepsilon}\Big)+\gamma_0r\right]\leq 0\end{aligned}$$ by definition of $c$. The inequality $LV_k^{n+1}\leq 0$ can then be rewritten as $$V_{k}^{n+1}\leq (1-2a)V_{k}^n+(a-b)V_{k-1}^n+(a+b)V_{k+1}^{n}$$ with coefficients $a,b$ exactly as in . We already showed in the proof of Lemma \[lem:Linfty\_Lipschitz\_estimate\] that $0\leq a \leq 1/2$ and $|b|\leq a$. In particular the above right-hand side is a convex combination of $V_{k-1}^n,V_{k}^n,V_{k+1}^n$, thus $V_{k}^{n+1}\leq 0$ as desired.\
[**Step 2.**]{} Choosing $k=k_0$ in $V_k^n\leq 0$ we see that $v_{k_0}^n-v_{k_0}^{n_0}\leq \gamma_0r+\frac{cH}{r^2}|t^{n_1}-t^{n_0}|$, and in a similar way we get the same upper bound for $v_{k_0}^{n_0}-v_{k_0}^{n}$. Taking the maximum over $n\in [n_0,n_1]$ and writing $s=|t^{n_1}-t^{n_0}|$ we see by definition of $H$ that $$H\leq \gamma_0 r+\frac{cH}{r^2}s.
\label{eq:H_leq}$$ Choose now $r$ to be a multiple of $\Delta x$ such that $$r_1+3\Delta x\leq r\leq r_1+4\Delta x,$$ where $r_1>0$ is the largest root of $$\rho^2-2cs=\rho^2-2\gamma_0s\rho+4\Big(\sigma(M)+{\varepsilon}\Big)s=0.$$ In particular $3\Delta x \leq r$ as required in step 1, and it is easy to check that $r_1\lesssim Cs^{1/2}$ when $s\leq T$. Moreover $cs/r^2\leq 1/2$ and give $$H/2\leq \gamma_0r \leq \gamma_0(r_1+4\Delta x)\leq C(s^{1/2}+\Delta x).$$ Now $s=|t^{n_1}-t^{n_0}|$ and $\Delta x =\sqrt{\Delta t/\beta}\leq \beta^{-1/2}|t^{n_1}-t^{n_0}|^{1/2}$, so finally $$|v_{k_0}^{n_1}-v_{k_0}^{n_0}|\leq H\leq C|t^{n_1}-t^{n_0}|^{1/2}$$ and the proof is complete.
Convergence of the approximate solution and interfaces {#subsection:CV_one_patch}
------------------------------------------------------
Denoting the mesh parameters $h=(\Delta t,\Delta x)$ and $L_k^n,U_k^n$ the lower and upper triangles in Figure \[fig:FIG2\],
(1,0.70707072) (0,0)[![linear interpolation domains[]{data-label="fig:FIG2"}](FIG2.pdf "fig:"){width="\unitlength"}]{} (0.90972295,0.04297841)[(0,0)\[lb\]]{} (0.02188702,0.24294591)[(0,0)\[lb\]]{} (0.00984916,0.46509914)[(0,0)\[lb\]]{} (0.02517007,0.62049697)[(0,0)\[lb\]]{} (0.22134711,0.03939664)[(0,0)\[lb\]]{} (0.57453424,0.04158532)[(0,0)\[lb\]]{} (0.46291044,0.3097013)[(0,0)\[lb\]]{} (0.29766345,0.39396634)[(0,0)\[lb\]]{}
we first define the continuous and piecewise linear interpolation $$v_h(x,t):=
\left\{
\begin{array}{ll}
v_k^n+(x-x_k)\frac{v_{k+1}^n-v_k^n}{\Delta x}+(t-t^n) \frac{v_{k+1}^{n+1}-v_{k+1}^n}{\Delta t}, &(x,t)\in L_k^n\\
v_{k+1}^{n+1}+(x-x_{k+1})\frac{v_{k+1}^{n+1}-v_k^{n+1}}{\Delta x}+(t-t^{n+1}) \frac{v_{k}^{n+1}-v_{k}^n}{\Delta t}, &(x,t)\in U_k^n
\end{array}
\right..
\label{eq:def_interpolation_vh_meshpoints}$$
We also interpolate the interfaces by the piecewise linear curves $$\zeta_{h,lr}(t):= \zeta^n_{lr}+(t-t^n)\frac{\zeta_{lr}^{n+1}-\zeta_{lr}^{n}}{\Delta t}, \qquad t\in [t^n,t^{n+1}].
\label{eq:def_interpolation_interface_meshpoints}$$ If $Q_T={\mathbb{R}}\times (0,T)$ the estimates from Section \[subsection:estimates\_one\_patch\] can be summarized as $$0 \leq v_h(x,t)\leq M
\quad\mbox{and}\quad
|\partial_x v_h(x,t)|\leq \gamma_0
\qquad\mbox{a.e. in }Q_T,
\label{eq:estimate_Linfty_Lipschitz_vh}$$ $$\forall t_1,t_2\in[0,T]:\qquad |v_h(x,t_1)-v_h(x,t_2)| \leq C(T,v^0)|t_1-t_2|^{1/2},
\label{eq:estimate_Holder_vh}$$ $$\forall\ 0<t\leq T:\qquad
\int\limits_{{\mathbb{R}}}\left|\partial_{xx}^2 v_h(\,.\,,t)\right| +\int\limits_{{\mathbb{R}}}\left|\partial_tv_h(\,.\,,t)\right|\leq C\left(1+\frac{1}{t}+T\right)
\label{eq:estimate_vxx_vt_measures}$$ as measures in ${\mathbb{R}}$, and $$\left|\frac{d\zeta_{h,lr}}{dt}\right|\leq \gamma_0
\mbox{ and }
\operatorname{supp}v_h(\,.\,,t)\subseteq [\zeta_l(0)-\Delta x-\gamma_0t,\zeta_r(0)+\gamma_0t+\Delta x]
\mbox{ for a.e. }t\in [0,T]
\label{eq:estimate_interface_vh}$$ (Lemma \[lem:Linfty\_Lipschitz\_estimate\], Proposition \[prop:Holder\], Lemma \[lem:Aronson\_Benilan\_estimate\], and Lemma \[lem:discrete\_speed\_interfaces\]). The extra $\Delta x$ is needed in because $\zeta^n$ needs not be an integer meshpoint, while $v_h$ is only interpolated from the $(x_k,t^n)$ nodes. It is well known [@DK86] that the Cauchy problem has a unique solution. As in [@DBH84 Theorem 3.3] the main convergence result then reads:
Let $v$ be the unique solution to with initial datum $v^0$, and $\zeta_{l,r}$ the corresponding interfaces with $\operatorname{supp}v(\,.\,,t)=[\zeta_l(t),\zeta_r(t)]$. Then $$\begin{aligned}
v_h\to v \qquad & \mbox{uniformly in }\overline{Q_T},\label{eq:CV_vh_v_one_patch}\\
\partial_x v_h \to \partial_x v \qquad & \mbox{in }L^p(Q_T)\mbox{ for all }p\in[1,\infty),\label{eq:CV_vhx_vx_Lp_one_patch}\\
\zeta_{h,lr}\to \zeta_{lr}\qquad &\mbox{uniformly in }[0,T]\label{eq:CV_zetah_zeta_one_patch}\end{aligned}$$ when $h=(\Delta x,\Delta t)\to 0$. \[theo:CV\_solution\_interfaces\_single\_patch\]
The rest of this section is devoted to the proof of Theorem \[theo:CV\_solution\_interfaces\_single\_patch\], which closely follows [@DBH84].\
For we show that there is at least one subsequence $v_{h'}$ converging to some limit $v^*$, and that for any such converging subsequence the limit $v^*$ is a solution to the Cauchy problem. By uniqueness $v^*=v$ and standard separation arguments this implies that the whole sequence $v_h\to v$ as in our statement.
By - with the upper bound for the supports, we can extract a subsequence $\{h'\}\subseteq\{h\}$ such that $v_{h'}\to v^*$ uniformly in $\overline{Q}_T$ for some limit $v^*\in\mathcal{C}(\overline{Q}_T)$. For any fixed $t>0$ we see by that $\partial_{x}v_h(\,.\,,t)$ is bounded in $BV({\mathbb{R}})$ (bounded variation) uniformly in $h'$. By standard compactness [@AFP00] in BV spaces there is a further subsequence $\partial_x v_{h''}(\,.\,,t)\to w^*$ in $L^1({\mathbb{R}})$. By continuity we get $w^*(\,.\,)=\partial_x v^*(\,.\,,t)$, so by uniqueness and separation we conclude that $\partial_xv_{h'}(\,.\,,t)\to \partial_xv^*(\,.\,,t)$ for all $t>0$. An easy application of Lebesgue’s dominated convergence theorem with uniform bounds $|\partial_x v_h|\leq \gamma_0$ gives strong $L^{p}(Q_T)$ convergence for all $p\in[1,\infty)$ as in our statement.
We check now that the limit $v^*$ is indeed a solution to the Cauchy problem in the sense of Definition \[defi:weak\_sols\_v\]. Since $v_h^0(x)\to v^0(x)$ uniformly in ${\mathbb{R}}$ and $v^*$ is continuous up to $t=0$ the initial trace will be taken in the strong sense, and it is enough to check that $$\int\limits_{{\mathbb{R}}}v^*(x,\,.\,)\varphi(x,\,.\,)\Big|_{t_0}^{t_1}\mathrm{d}x
+ \int\limits_{t_0}^{t_1}\int\limits_{{\mathbb{R}}}\left\{-v^* \partial_t\varphi + \sigma(v^*)\partial_x v^*\partial_x\varphi+\Big(1-\sigma'(v^*)\Big)|\partial_x v^*|^2\varphi\right\}\mathrm{d}x\,\mathrm{d}t=0.
\label{eq:weak_formulation_v_IBP}$$ for all $0<t_0\leq t_1\leq T$ and test functions $\varphi\in\mathcal{C}^{\infty}_c(\overline{Q}_T)$. This weak formulation formally follows from $\partial_tv=\sigma(v)\partial_{xx}^2 v+|\partial_x v|^2$ after multiplying by $\varphi$ and integration by parts. Let now $\varphi_k^n:=\varphi(x_k,t^n)$, set $N_0:=\lfloor t_0/\Delta t\rfloor$ and $N_1:=\lfloor t_1/\Delta t\rfloor$, and consider the approximate Riemann sum$$S:=\sum\limits_{n=N_0}^{N_1-1}\left\{
\sum\limits_k\left[
\frac{v_k^{n+1}-v_k^n}{\Delta t}-\big(\sigma(v_k^n)+{\varepsilon}\big)\frac{Av_k^n}{\Delta x^2}-\left|\frac{v_{k+1}^n-v_{k-1}^n}{2\Delta x}\right|^2
\right]\varphi_k^n\Delta x
\right\}\Delta t.$$ By construction of our scheme the summand in $S$ is identically zero for $x_k\notin[\zeta_l^n,\zeta_r^n]$ and $x_k\in [x_{K_l(n)},x_{K_r(n)}]$. In the remaining boundary layers, which have thickness at most $s^n=|\zeta^n-x_{K(n)}|\leq 2\Delta x$ and where $v_{k}^n$ is linear, we have $|(v_k^{n+1}-v_k^n)/\Delta t|=\mathcal{O}(1)$ by Lemma \[lem:dv/dt\_interface\] and $(\sigma(v_k^n)+{\varepsilon})\frac{Av_k^n}{\Delta x^2}=\mathcal{O}(\Delta x)\frac{w_{k+1}^n-w_k^n}{\Delta x}=\mathcal{O}(1)$. Here we used in particular that the artificial viscosity ${\varepsilon}=\mathcal{O}(\Delta x)$. Thus we see that $S\to 0$ when $h'\to 0$. Summing by parts in $S$ one can get $S=S'\to 0$, where $S'$ is the discrete $\Delta x\Delta t$ Riemann sum corresponding to . Using then the definition of the interpolation $v_{h'}$ in terms of $v_k^n$, the strong convergence $v_{h'}\to v^*$, the Lipschitz and Hölder regularity of $v_{h'}$ and the test function $\varphi$, it is easy to express $S'$ as the sum of $\mathrm{dx}\mathrm{dt}$ integrals over all triangles $L_k^n,U_k^n$, plus a remainder $o(1)$, and then send $h'\to 0$ in order to retrieve the weak formulation for $v^*$ (note that $\sigma\in\mathcal{C}^1([0,\infty))$ and therefore $\sigma'(v_h)\to \sigma'(v)$ uniformly). We refer to [@DBH84 pp. 480] for the details.\
Turning now to the uniform convergence of the interfaces, we only argue for the right one and write $\zeta^n=\zeta^n_r,\zeta_h=\zeta_{h,r}$ and $K(n)=K_r(n)$ for simplicity (the proof for the left interface is exactly similar). From we see that $\zeta_{h'}$ is bounded in $W^{1,\infty}(0,T)$, so up to extraction of a further sequence if needed we may assume that $\zeta_{h'}\to\zeta^*$ uniformly in $[0,T]$ for some $\zeta^*$. This limit $\zeta^*$ is moreover monotone nondecreasing in $t$ with $\zeta^*(0)=\zeta(0)$, as the uniform limit of the nondecreasing functions $\zeta_h$ with $\zeta_h(0)=\zeta(0)$. We shall prove that the limit agrees with the true interface $\zeta^*=\zeta$, and the same separation argument as before will then show that the whole sequence actually converges.
Following again [@DBH84] we first need a technical result ensuring that, at a point $(\zeta^*(t_0),t_0)$ where the limit $\zeta^*$ is moving with positive speed, then $v^*(\,.\,,t_0)$ grows at least linearly in an interior neighborhood $[\zeta^*(t_0)-\delta,\zeta^*(t_0)]$:
Let $v^*,\zeta^*=\lim v_{h'},\zeta_{h'}$ as above and $\underline{z}(t)$ as in Lemma \[lem:def\_ODE\_z\]. Then
1. For any $0<t_0<t_0+\eta\leq T$ and $\delta>0$ there holds $$\int_{t_0}^{t_0+\eta}v^*(\zeta^*(s)-\delta,s)\,\mathrm{d}s
\geq
\delta\big(\zeta^*(t_0+\eta)-\zeta^*(t_0)\big)-\delta^2\eta\underline{z}(t_0)
\label{eq:integrated_linear_growth_interface}$$
2. If $0<t_0<T$ is such that $d\zeta^*/dt(t_0)$ exists and is positive, then there is $\delta_0>0$ and $c>0$ such that $$v^*(\zeta^*(t_0)-\delta,t_0)\geq c\delta
\label{eq:linear_growth_interface}$$ for all $\delta\in[0,\delta_0] $.
\[lem:linear\_growth\_interface\]
This is somehow the converse statement of a well known fact for the so-called *waiting-time phenomenon*: if $(\zeta(t_0),t_0)$ is a free-boundary point and the pressure grows at least linearly in $x$ in an interior neighborhood$\{v>0\}\cap B_r(\zeta(t_0))\times \{t_0\}$ then the free-boundary starts to move immediately (see e.g. [@Va07 Theorem 15.19] for a stronger statement and simple proof in dimension $d=1$ for the pure PME nonlinearity). This explanation is of course an educated guess, as we do not know at this stage that $\zeta^*=\lim\zeta_{h'}$ is really the interface. Also note in (ii) that $\zeta^*\in W^{1,\infty}(0,T)$ is differentiable a.e., and that the statement fails if $d\zeta^*/dt(t_0)=0$.
We first give a formal proof, keeping in mind that at the discrete level we enforced $d\zeta/dt=-\partial_xv$ at the interface and that the AB estimate $\partial_{xx}v(x,t)\geq \underline{z}(t)$ holds. Taking $h'\to 0$ we thus expect $d\zeta^*/dt(t_0)=-\partial_xv^*(\zeta^*(t_0),t_0)$, so that $v^*$ should indeed grow at least linearly $\partial_xv(\zeta^*(t_0),t_0)<0$ whenever the interface is moving $d\zeta^*/dt(t_0)>0$. In fact (ii) rigorously follows from (i): for whenever $\zeta^*$ is differentiable at $t_0$ with $d\zeta^*/dt(t_0)>0$ then dividing by $\eta\to 0$ and discarding the $\delta^2=o(\delta)$ term for small $\delta>0$ yields with $c\approx d\zeta^*/dt(t_0)>0$. Let us therefore also give a formal proof of (i): all regularity issues left aside and assuming that $v^*(\zeta^*(t),t)=0$, $d\zeta^*/dt=-\partial_xv^*(\zeta^*(t),t)$ and $\partial_{xx}
^2v(x,t)\geq \underline{z}(t)$ as expected, we first integrate by parts and use the generalized Aronson-Bénilan estimate to estimate $$\begin{aligned}
v^*(\zeta^*(s)-\delta,s) & =
\underbrace{v^*(\zeta^*(s),s)}_{=0}-\int_{\zeta^*(s)-\delta}^{\zeta^*(s)}\partial_xv^*(x,s)\mathrm{d}x\\
& =-\int_{\zeta^*(s)-\delta}^{\zeta^*(s)}\left(\partial_xv^*(\zeta^*(s),s)-\int_x^{\zeta^*(s)}\partial_{xx}^2v^*(y,s)\mathrm{d}y\right)\mathrm{d}x\\
& \geq\int_{\zeta^*(s)-\delta}^{\zeta^*(s)}\underbrace{-\partial_xv^*(\zeta^*(s),s)}_{=+d\zeta^*/dt(s)}\mathrm{d}x
+ \int_{\zeta^*(s)-\delta}^{\zeta^*(s)}\left(\int_x^{\zeta^*(s)}\underline{z}(s)\mathrm{d}y\right)\mathrm{d}x\\
& \geq \delta\frac{d\zeta^*}{dt}(s)+\int_{\zeta^*(s)-\delta}^{\zeta^*(s)}\delta\underline{z}(s)\,\mathrm{d}x=\delta\frac{d\zeta^*}{dt}(s)+\delta^2\underline{z}(s).\end{aligned}$$ Recalling also that $\underline{z}(t)$ is monotone increasing and integrating from $t_0$ to $t_0+\eta$ we conclude that $$\begin{aligned}
\int_{t_0}^{t_0+\eta}v^*(\zeta^*(s)-\delta,s)\,\mathrm{d}s & \geq
\int_{t_0}^{t_0+\eta}\left(\delta\frac{d\zeta^*}{dt}(s)+\delta^2\underline{z}(s)\right)\mathrm{ds}\\
& \geq \int_{t_0}^{t_0+\eta}\left(\delta\frac{d\zeta^*}{dt}(s)+\delta^2\underline{z}(t_0)\right)\mathrm{ds}\\
&= \delta\big(\zeta^*(t_0+\eta)-\zeta^*(t_0)\big)-\delta^2\eta\underline{z}(t_0)\end{aligned}$$ as desired.\
Following [@DBH84 Lemma 3.4] we now briefly sketch how to get (i) rigorously, from which (ii) will follow as already explained. For fixed $\delta,\eta,t_0>0$ let $p=\lfloor \delta/\Delta x\rfloor$, $q=\lfloor \eta/\Delta x\rfloor$, and $N=\lfloor t_0/\Delta t\rfloor$. Recalling that $\frac{\zeta^{n+1-\zeta^n}}{\Delta t}=-\frac{v_{K(n)+1}^n-v_{K(n)}^n}{\Delta x}$ and summing by parts instead of integrating by parts as above, an explicit computation gives the discrete equivalent of $$\sum\limits_{n=N}^{N+q-1}v_{K(n)-p}\Delta t\geq p\Delta x\big(\zeta^{N+q}-\zeta^N)\big)-(p\Delta x)^2(q\Delta t)\underline{z}(t^N).$$ Sending $h'\to 0$ with uniform convergence $v_{h'}\to v^*$, $\zeta_{h'}\to \zeta^*$ and $x_{K(n)}\to \zeta^*(t)$ for $n=\lfloor t/\Delta t\rfloor$ finally allows to retrieve and the proof is complete.
Back to the proof of , we recall that we only need to establish $\lim \zeta_{h'}=\zeta^*=\zeta$. From we have $v_{h'}(x,t)=0$ for all $x\geq \zeta_{h'}(t)+\Delta x$. As a consequence $v^*(x,t)=\lim v_{h'}(x,t) =0$ for all $x\geq \zeta^*(t)$, which shows by definition of $\zeta(t)=\zeta_r(t)=\sup\{x:\,v(x,t)>0\}$ that $\zeta^*(t)\geq \zeta(t)$. Assuming by contradiction that there is $t_1>0$ for which $\zeta^*(t_1)>\zeta(t_1)$, we claim that there is $t_0\in (0,t_1)$ such that $\zeta^*(t_0)>\zeta(t_0)$ and $d\zeta/dt(t_0)>0$.
For if not, then arguing backwards in time starting from $t_1$ it is easy to see that either $\zeta^*(t)=cst=\zeta^*(t_1)$ for all $t\in[0,t_1]$, or there is $t_2\in(0,t_1)$ such that $\zeta^*(t)=cst=\zeta^*(t_1)$ for all $t\in[t_2,t_1]$ with $\zeta^*(t_2)=\zeta(t_2)$. The first case would contradict $\zeta^*(0)=\zeta(0)$ since $\zeta^*(t_1)>\zeta(t_1)\geq \zeta(0)$. In the second case, $\zeta\leq \zeta^*$ and the monotonicity of $\zeta$ show that $\zeta^*(t)=\zeta(t)=cst=\zeta^*(t_1)$ for all $t\in[t_2,t_1]$, thus contradicting $\zeta^*(t_1)>\zeta(t_1)$.
For any such $t_0$ Lemma \[lem:linear\_growth\_interface\] gives then $v(\zeta^*(t_0)-\delta)\geq c\delta >0$ for small $\delta$’s, and in particular choosing $0<\delta< \zeta^*(t_0)-\zeta(t_0)$ small enough there is a point $x_0=\zeta^*(t_0)-\delta>\zeta(t_0)$ such that $v(x_0,t_0)\geq c\delta>0$. This finally contradicts $\zeta(t_0)=\sup\{x:\,v(x,t_0)>0\}$ and ends the proof of Theorem \[theo:CV\_solution\_interfaces\_single\_patch\].
The hole-filling problem {#section:two_patches}
========================
In this section we consider the so-called hole-filling problem. We choose two compactly supported “patches” $\hat{v}^0(x),\check{v}^0(x)$ such that: (i) both $\hat{v}^0,\check{v}^0$ are $\gamma_0$-Lipschitz, (ii) $0\leq \hat{v}^0(x),\check{v}^0(x)\leq M$, and (iii) $\operatorname{supp}\hat{v}^0$ is at positive distance from $\operatorname{supp}\check{v}^0$ with $$\hat{\zeta}_l(0)<\hat{\zeta}_r(0)<\check{\zeta}_l(0)<\check{\zeta}_r(0).$$ Defining $$v^0:=\max\{\hat{v}^0,\check v^0\}$$ this means that $\operatorname{supp}v^0=\operatorname{supp}\hat{v}^0\cup \operatorname{supp}\check{v}^0$ has an internal hole of width $d_0=\check{\zeta}_l(0)-\hat{\zeta}_r(0)>0$ between $\operatorname{supp}\hat{v}^0$ and $\operatorname{supp}\check{v}^0$. Let $v(x,t),\hat{v}(x,t),\check{v}(x,t)$ be the solution of the Cauchy problem with initial data respectively $v^0(x),\hat{v}^0(x),\check{v}^0(x)$. We are interested here in computing a numerical approximation to $v(x,t)$. By noncontraction of the supports we know that $\hat{\zeta}_r(t)$ is nondecreasing, $\check{\zeta}_l(t)$ is nonincreasing, and because the interfaces propagate with finite speed at most $\gamma_0$ (which also follows from Section \[section:one\_patch\]) the first time when the supports touch $$T^*=\sup\Big\{t\geq 0:\quad \hat{\zeta}_r(t)<\check{\zeta}_l(t)\Big\}\leq \infty$$ is positive (possibly infinite). By uniqueness this implies that $$v=\max\{\hat{v},\check{v}\} \qquad\mbox{in }[0,T^*),$$ so for $t\in[0,T^*)$ the support of $v$ still has an internal hole of width $d(t)=\check{\zeta}_l(t)-\hat{\zeta}_r(t)>0$. A well-known property of GPME is that “once an interface starts moving it never stops”, see e.g. [@Va07 Lemma 14.20] in any dimension for the pure PME nonlinearity and [@Va07 Corollary 15.23] for a simple proof in dimension one. Since the internal interfaces were at positive distance at time $0$ this implies that, if and when they meet in finite time $\hat{\zeta}_r(T^*)=x^*=\check{\zeta}_l(T^*)$, at least one of the internal interfaces has started moving (otherwise the two would not meet) and is therefore still moving with positive speed. As a consequence at least one of the patches $\hat{v},\check{v}$ becomes instantaneously positive at $x=x^*$ for $t>T^*$, the comparison principle then implies $v(x^*,t)\geq\max\{\hat{v}(x^*,t),\check{v}(x^*,t)\}>0$, and the hole eventually disappears at $t=T^*$. Once the hole has filled the internal interfaces disappear, $\operatorname{supp}
v(\,.\,,
t)$ becomes a connected interval $[\zeta_l(t),\zeta_r(t)]$ containing the whole $[\hat{\zeta}_l(T^*),\check{\zeta}_r(T^*)]$, and $v$ does not equal $\max\{\hat{v},\check{v}\}$ anymore. In section \[section:one\_patch\] we described how to compute the approximate solution and interfaces when the initial datum consists in a single patch, which is exactly our assumption for each of $\hat{v}^0,\hat{v}^0$ separately. Using the results in the previous section we can therefore construct an approximation to each of the corresponding solutions $\hat{v},\check{v}$ and track all the resulting interfaces. We explain below how this previous one-patch algorithm can be naturally extended to the above case of two initial patches, while tracking all the interfaces (internal and external), detecting the hole-filling with accuracy, and solving past this time.
We discuss here the case of two patches only for the ease of exposition, but the argument is easily adapted to any arbitrary number of initial patches at positive distance one from each other.
Roughly speaking, the algorithm goes as follows: starting from $\hat{v}^0_k,\check{v}^0_k$, construct two independent sets of approximate solutions and interfaces $(\hat{v}_k^n,\hat{\zeta}^n_{l,r})$ and $(\hat{v}_k^n,\hat{\zeta}^n_{l,r})$ applying the one-patch scheme from Section \[section:one\_patch\] separately to each patch. As long as the internal interfaces do not meet keep solving, and define $v_k^n=\max\{\hat{v}_k^n,\check{v}_k^n\}$. If the interfaces meet at $t=t^N$ then stop tracking the internal interfaces $\hat{\zeta}_r, \check{\zeta}_l$, define the external interfaces $\zeta^N_l:=\hat{\zeta}_l^N,\zeta^N_r:=\check{\zeta}_r^N$, and resume the computation applying the one-patch scheme to $v^{n}_k$ starting from $v^N_k$ at time $t^N$. More precisely,
Initialize $\hat{v}^0_k:=\hat{v}^0(x_k)$, $\check{v}^0_k:=\check{v}^0(x_k)$, $v^0_k:=\max\{\hat{v}^0_k,\check{v}^0_k\}$, as well as $\hat{\zeta}_{l,r}^0:=\hat{\zeta}_{l,r}(0)$, $\check{\zeta}_{l,r}^0:=\check{\zeta}_{l,r}(0)$, and $\zeta^0_l:=\hat{\zeta}_l^0$, $\zeta^0_r:=\check{\zeta}^0_r$. For fixed $T>0$ and while $t^n\leq T$, do:
1. \[item:algo\_no\_hitting\] Apply the one-patch algorithm from section \[section:one\_patch\] separately to $\hat{v}^{n},\hat{\zeta}_{l,r}^{n}$ and $\check{v}^{n},\check{\zeta}_{l,r}^{n}$ in order to predict $\hat{v}^{(n+1)'},\hat{\zeta}_{l,r}^{(n+1)'}$ and $\check{v}^{(n+1)'},\check{\zeta}_{l,r}^{(n+1)'}$. If the predicted internal interfaces are at least $\Delta x$ away $\check{\zeta}_l^{(n+1)'}-\hat{\zeta}_r^{(n+1)'}>\Delta x$, update $(n+1)'\to(n+1)$, set $v^{n+1}_k:=\max\{\hat{v}^{n+1}_k,\check{v}^{n+1}_k\}$, and repeat step 1. Otherwise define the numerical filling time $\tilde{T}^*:=t^n$, the external interfaces $\zeta_l^n:=\hat{\zeta}_l^n$ and $\zeta^n_r:=\check{\zeta}_r^n$, and go to step 2.
2. \[item:algo\_after\_hitting\] Apply the one-patch algorithm from section \[section:one\_patch\] to $v^{n},\zeta^{n}_{l,r}$ in order to construct $v^{n+1},\zeta^{n+1}_{l,r}$, and repeat Step 2.
\[algo:two\_patches\]
Note that because all the interfaces propagate with numerical speed at most $\gamma_0$ and the internal ones are at initial distance $d(0)>0$, Step \[item:algo\_no\_hitting\] will be applied at least for $t^n\leq d(0)/2\gamma_0$ hence $\tilde{T}^*\geq d(0)/2\gamma_0$ uniformly in the mesh parameters. In case the hitting does occur for some $\tilde{T}^*\leq T$ then the numerical internal interfaces are not defined for later times.
A priori estimates
------------------
We show here that all the previous estimates discrete are preserved across and after the filling time, including the $L^{\infty}$, Lipschitz, and Hölder bounds as well as the generalized Aronson-Bénilan estimate. In particular we will obtain that the pressure $v$ stays $\gamma_0$-Lipshitz across the filling time, which is well known to hold in dimension one only (formally because $w=\partial_x v$ satisfies a maximum principle as in the proof of Lemma \[lem:Linfty\_Lipschitz\_estimate\]). As in the previous section we impose the condition on the mesh parameters $\Delta x,\Delta t,{\varepsilon}$.
Let $v_k^n$ be the (two-patches) discrete solution constructed with Algorithm \[algo:two\_patches\], and $\underline{z}(t)<0$ as in Lemma \[lem:def\_ODE\_z\]. Then $$0\leq v_k^n\leq M,
\quad
\left|\frac{v_{k}^n-v_{k-1}^n}{\Delta x}\right|\leq \gamma_0,
\quad
\left|v_{k}^n-v_k^m\right|\leq C|t^n-t^m|^{1/2},
\quad
\frac{Av_k^n}{\Delta x^2}\geq \underline{z}(t^n)$$ hold for all $k$ and $t^n,t^m\in[0,T]$.\[prop:a\_priori\_estimates\_two\_patches\]
If no hole filling occurs our statement immediately follows from the results in Section \[section:one\_patch\], as $v_k^n$ coincides with either $\hat{v}_k^n$ or $\check{v}_k^n$, depending on which side of the internal hole one is looking at. Thus we may assume that internal interfaces meet at $t=t^N$.
For times $t^n\leq t^N$ the patches $0\leq \hat{v}_k^n,\check{v}_k^n\leq M$ are $\gamma_0$-Lipschitz (Lemma \[lem:Linfty\_Lipschitz\_estimate\]) so clearly $v^n_k=\max\{\hat{v}_k^n,\check{v}_k^n\}$ satisfies the same bounds for all $t^n\leq t^N$, and in particular at $t=t^N$. By definition $v^n_k$ is then constructed for $t^n\geq t^N$ by applying the one-patch scheme to solve the discrete Cauchy problem starting from the initial data $v^N_k$ at time $t^N$. Since $v_k^{N}$ satisfies the desired bounds we conclude by Lemma \[lem:Linfty\_Lipschitz\_estimate\] that $v_k^n$ satisfies the same $L^{\infty}$ and $\gamma_0$-Lipschitz estimates for all $t^n\geq t^N$.
Regarding now the Hölder continuity in time, we check that the proof of Proposition \[prop:Holder\] still applies. In Step 1 ($V_k^n\leq 0$ in $Q$ by induction on $n\in [n_0,n_1]$) the initialization $n=n_0$ only requires $\gamma_0$-Lipschitz bounds, which is true here. For the induction step we distinguished three cases: (i) $x_k$ is outside of the support with $v_k^n=0$ , (ii) $x_k$ is within one of the boundary layers, and (iii) when $v^{n+1}_k$ is constructed applying the finite difference scheme . All three cases are easily checked here with two patches: (i) and (ii) are identical, and (iii) also works here since $v^{n+1}_k$ is in fact constructed applying the finite difference equation to either one of the two patches before the filling time and to the unique patch afterward. Step 2 is identical, since it relies only on structural considerations and the previous $L^{\infty}$ and Lipschitz bounds.
We finally turn to the AB estimate. By definition of the hitting time $t^N$ we have that $\check{\zeta}_l^n-\hat{\zeta}_r^n>\Delta x$ stay strictly $\Delta x$ away from each other for $t^n\leq t^N$, so that there is always at least one integer mesh point in the hole. Since $v_k^n\geq 0$ everywhere and $v_k^n=0$ in the hole it is easy to check that $Av_k^n\geq 0$ for all $x_k$ such that $\hat{\zeta}^n_r-\Delta x\leq x_k\leq \check{\zeta}_l^n+\Delta x$, hence the AB estimate is trivially satisfied there (recall that $\underline{z}(t)<0$). Now outside the hole $Av_k^n$ equals either $A\hat{v}_k^n$ or $A\check{v}_k^n$, hence the AB estimate holds for all $t^n\leq t^N$ and including at $t=t^N$. Now for $t^n\geq t^N$ the solution $v^{n}_k$ is constructed applying the one-patch algorithm with initial datum $v^N_k$ at time $t^N$, which satisfies the AB estimate. By Lemma \[lem:Aronson\_Benilan\_estimate\] we conclude that the estimate also holds for all $t^n\geq t^N$ and the proof is complete.
Convergence of the approximate solutions and interfaces
-------------------------------------------------------
For fixed $T>0$ we denote $Q_T={\mathbb{R}}\times(0,T)$ and $h=(\Delta x,\Delta t)$ as before. As in section \[subsection:CV\_one\_patch\] we define $v_h$ to be continuous and piecewise linear in all triangle $L_k^n,U_k^n$ according to . The external $\zeta_{h,lr}$ and internal $\hat{\zeta}_{h,r},\check{\zeta}_{h,l}$ interfaces are defined to be piecewise linear as in . Note that $\zeta_{h,lr}$ are defined up to $t=T$, while $\hat{\zeta}_{h,r}\leq \check{\zeta}_{h,l}$ are only defined up to the (numerical) filling time $$T^*_h:=\max\{t^n:\quad \check{\zeta}^n_l-\hat{\zeta}_r^n>\Delta x\}
\label{eq:def_Th}$$ (see Algorithm \[algo:two\_patches\]). If no filling is numerically detected before the end of the computation we simply do not define $T^*_h$. In any case $\hat{\zeta}_{h,r},\check{\zeta}_{h,l}$ are respectively monotone nondecreasing and nonincreasing as long as they exist.
For fixed $T>0$ the numerical solution $v_h$ converges uniformly in $\overline{Q}_T$ to the unique solution $v$ when $h\to 0$. \[theo:CV\_solution\_double\_patch\]
Note that the proof of for the case of one patch only in Theorem \[theo:CV\_solution\_interfaces\_single\_patch\] only relies on: (i) the discrete estimates on $v_h$ uniformly in $h$ allowing to get compactness both for $v_h$ and $\partial_x v_h$, (ii) uniqueness for the Cauchy problem, (iii) the consistence of the finite difference equation inside the support, and (iv) the fact that all quantities involved in are of order $\mathcal{O}(1)$ inside the numerical boundary layers, see section \[subsection:CV\_one\_patch\] for the details. By Proposition \[prop:a\_priori\_estimates\_two\_patches\] this remains true in the case of two patches, thus allowing to conclude as in the proof of Theorem \[theo:CV\_solution\_interfaces\_single\_patch\].
The uniform convergence of the interfaces is now more delicate, as we need to distinguish between cases depending on whether the hole fills or not before the computation time $T$ . Roughly speaking, as long as the interfaces make sense the convergence follows as in the case of one patch only. We prove in particular that, if and when the *numerical* filling occurs at time $t=T^*_h$, then $T^*_h$ is indeed a good approximation to the theoretical filling time $T^*$:
Fix $T>0$ and let $T^*$ be the theoretical hole-filling time. Then
1. If $T^*<T$ then there is a small $\delta_0>0$ such that the numerical hitting occurs at times $T^*_h\leq T-\delta_0$ for all $h\leq h_0$, and there exists $\lim\limits_{h\to 0}T^*_h=T^*$. Moreover $$\|\zeta_{h,l}-\zeta_l\|_{L^{\infty}(0,T)}+\|\zeta_{h,r}-\zeta_r\|_{L^{\infty}(0,T)}\to 0$$ and $$\|\hat\zeta_{h,r}-\hat{\zeta}_r\|_{L^{\infty}(0,T^*-\eta)}+\|\check{\zeta}_{h,l}-\check{\zeta}_l\|_{L^{\infty}(0,T^*-\eta)}\to 0$$ for any small $\eta>0$ fixed.
2. If $T^*\geq T$ then for all $\eta>0$ there exists $h_0(\eta)$ such that for all $h\leq h_0$ either no numerical hitting occurs before $t=T$, or does so at $T^*_h\geq T-\eta$. In particular for small $\eta$ the internal interfaces $\hat{\zeta}_{h,r},\check{\zeta}_{h,l}$ are defined at least for $t\leq T-\eta$. Moreover $$\|\zeta_{h,l}-\zeta_l\|_{L^{\infty}(0,T)}+\|\zeta_{h,r}-\zeta_r\|_{L^{\infty}(0,T)}\to 0$$ and $$\|\hat\zeta_{h,r}-\hat{\zeta}_r\|_{L^{\infty}(0,T-\eta)}+\|\check{\zeta}_{h,l}-\check{\zeta}_l\|_{L^{\infty}(0,T-\eta)}\to 0$$ for any small $\eta>0$ fixed.
\[theo:CV\_hitting\_time\]
Practically speaking this means that if a hole-filling is detected numerically at $t=T^*_h$ then indeed $T^*_h$ is a good approximation to the theoretical filling time $T^*$, while if no hole-filling is detected before the end of the computation then one has simply not waited long enough to see the hole-filling, i-e $T^*\geq T$. In any case the numerical interfaces converge to the theoretical ones, both internal (as long as they exist) and external (up to $t=T$).
Before going into the details, it is worth pointing out that at the filling time there holds $$0\leq \check{\zeta}_{h,l}(T^*_h)-\hat{\zeta}_{h,r}(T^*_h)\leq \mathcal{O}(\Delta x).
\label{eq:zetal=zetar_hole_filling_time}$$ Indeed by we have $T^*_h=t^N$ for some $N$, which according to Algorithm \[algo:two\_patches\] is characterized by the fact that virtually computing one more step separately for each patch would result in $\check{\zeta}_l^{N+1}-\hat{\zeta}_r^{N+1}\leq \Delta x$. Recalling that any interface propagates with discrete speed at most $\gamma_0$ (Lemma \[lem:discrete\_speed\_interfaces\]) we see that indeed $0\leq \check{\zeta}_l^{N}-\hat{\zeta}_r^{N} \leq (\check{\zeta}_l^{N+1}-\hat{\zeta}_r^{N+1})+2\gamma_0\Delta t\leq \Delta x+2\gamma_0\Delta t\leq \mathcal{O}(\Delta x)$ since $\Delta t=\mathcal{O}(\Delta x^2)$.
We first show that the hole-filling always eventually occurs before the end of the computation if $h$ is small enough, i-e $T_{h}^*\leq T-\delta_0$ for some small $\delta_0>0$ as in our statement. Assuming by contradiction that this does no hold, then by definition of $T^*_h$ there is a discrete subsequence (not relabeled) such that either no numerical hitting occurs before $t=T$, or occurs for times $T^*_h\nearrow T$. In any case and by definition of the internal interfaces we can find a sequence of points $(x_h,t_h)$ such that $t_h\nearrow T$ and $x_h\in [\hat{\zeta}_{h,r}(t_h),\check{\zeta}_{h,l}(t_h)]$ with $v_h(x_h,t_h)$=0. By monotonicity of the interfaces we see that $x_h$ stays in the fixed compact set $[\hat{\zeta}_{r}(0),\check{\zeta}_{l}(0)]$, so up to extracting a further subsequence we can assume that $x_h\to x_0\in[\hat{\zeta}_{r}(0),\check{\zeta}_{l}(0)]$. By Theorem \[theo:CV\_solution\_double\_patch\] we get $$v(x_0,T)=\lim\limits_{h\searrow 0}v_h(x_h,T^*_h)=0 \quad\mbox{for some } x_0\in[\hat{\zeta}_{r}(0),\check{\zeta}_{l}(0)].$$ We argue now for the theoretical solution and interfaces in order to get a contradiction. Because $T^*<T$ and the internal interfaces start at positive distance from each other they must meet for some $x^*=\hat{\zeta}_{r}(T^*)=\check{\zeta}_{l}(T^*)\in[\hat{\zeta}_{r}(0),\check{\zeta}_{l}(0)]$. Then necessarily one of them has started moving before $t=T^*$ (otherwise they would not meet). Once an interface starts moving it never stops, so at least one of the interfaces is really moving at $t=T^*$ and thus $\hat{v}(x^*,t)>0$ or $\check{v}(x^*,t)>0$ for all $t>T^*$. By the comparison principle $v\geq \max\{\hat{v},\check{v}\}$ is positive everywhere in $[\hat{\zeta}_{r}(0),\check{\zeta}_{l}(0)]$ for all $t>T^*$, in particular for $t=T>T^*$. This finally contradicts $v(x_0,T)=0$.
We claim now that $\lim\limits_{h\searrow 0}T^*_h=T^*$. Since $0\leq T^*_h\leq T-\delta_0$ for small $h$, we can extract a subsequence such that $T^*_{h'}\to\tilde{T}^*$ for some $\tilde{T}^*<T$. We prove that necessarily $\tilde{T}^*=T^*$, which will show that the whole sequence converges. Virtually keeping applying the one-patch algorithm separately to each of the patches $\hat{v}_{h'},\check{v}_{h'}$ after $t=T^*_{h'}$, we can naturally extend $\hat{\zeta}_{h',r},\check{\zeta}_{h',l}$ to all $t\in[0,T]$. By construction of our scheme these extended interfaces, still denoted $\hat{\zeta}_{h',r},\check{\zeta}_{h',l}$ with a slight abuse of notations, coincide with the internal interfaces for $v_{h'}$ up to the numerical filling time $T^*_{h}$, after which we stop tracking the true internal interfaces but the extended ones virtually still exist up to $t=T$. Applying Theorem \[theo:CV\_solution\_interfaces\_single\_patch\] we see that the extended interfaces $\hat{\zeta}_{h',r},\check{\zeta}_{h',l} \to \hat{
\zeta}_{r},\check{\zeta}_{l}$ uniformly in $[0,T]$, where $\hat{\zeta},\check{\zeta}$ are the interfaces of each patch $\hat{v},\check{v}$ considered as two independent solutions. Since $T^*_{h'}\to \tilde{T}^*$ we get by and uniform convergence that $$\hat{\zeta}_r(\tilde{T}^*)-\check{\zeta}_l(\tilde{T}^*)=\lim\limits_{h'\to 0}\left(\hat{\zeta}_{h',r}(T^*_{h'})-\check{\zeta}_{h',l}(T^*_{h'})\right)=0.$$ Since $\hat{\zeta}_{r},\check{\zeta}_{l}$ are monotone and start at positive distance, and because once an interface starts moving it never stops, they can only meet at a unique time. By definition this time is $t=T^*$, thus $\tilde{T}^*=T^*$ and $T^*_{h}\to T^*$ as desired. Uniform convergence of the interfaces can be obtained as in the proof of Theorem \[theo:CV\_solution\_interfaces\_single\_patch\] as long as the internal interfaces exist and are tracked numerically (this is why we need to step $\eta$ away from $T^*$ as in our statement, thus ensuring that the internal interfaces are numerically defined at least for fixed time intervals $[0,T^*-\eta]$), and the proof is achieved.
We claim that a hole-filling can only be detected numerically for times $T_h^*\geq T-\eta$ close to the total computation time $T$ if $h$ is small enough (and may actually not be detected). For if not, then $T^*_{h'}\leq T-\delta_0$ for some subsequence and fixed $\delta_0>0$. Arguing exactly as in (a) we conclude that $T^*_{h'}\to T^*$, which shows in particular that $T^*\leq T-\delta_0$ and contradicts $T^*\geq T$. The convergence of the interfaces is also exactly similar to the proof of Theorem \[theo:CV\_solution\_interfaces\_single\_patch\], stepping again $\eta>0$ away from $t=T$ for the internal interfaces as in our statement.
Numerical experiments {#section:num_exp_comments}
=====================
The stability condition was imposed in order to ensure Lipschitz bounds and $L^\infty$ stability of the scheme (Lemma \[lem:Linfty\_Lipschitz\_estimate\]), but also the generalized Aronson-Bénilan estimate (Lemma \[lem:Aronson\_Benilan\_estimate\]). For numerical purposes the less stringent condition $$\beta\leq \frac{1}{2\Big(\sigma(M)+{\varepsilon}\Big)}
\quad\mbox{and}\quad
\gamma_0\Delta x\Big(1+S_1(M)/2\Big)\leq {\varepsilon}\leq \mathcal{O}(\Delta x)
\label{eq:CFL'}
\tag{CFL'}$$ suffices to guarantee the stability Lemma \[lem:Linfty\_Lipschitz\_estimate\] and seems to give satisfactory convergence (see below). Note that in contrast with this relaxed condition does not depend on $s_1(M),S_2(M)$ anymore. In any case the computationally expensive $\beta=\Delta t/\Delta x^2=\mathcal{O}(1)$ condition is necessary due to the explicit nature of the scheme. In [@H85] Hoff considered a linearly implicit version of [@DBH84] for the pure PME nonlinearity. We presented here the explicit scheme for the ease of exposition, but all the theoretical results in Sections \[section:one\_patch\] and \[section:two\_patches\] extend to general nonlinearities by considering the implicit scheme $$\frac{v_k^{n+1}-v_k^n}{\Delta t}=\sigma(v_k^n)\frac{A v_k^{n+1}}{\Delta x^2}+{\varepsilon}\frac{Av_k^n}{\Delta x^2}+\left|\frac{v_{k+1}^n-v_{k-1}^n}{2\Delta x}\right|^2.$$ In this case the stability condition becomes $\Delta t=\mathcal{O}(\Delta x)$, which is clearly the best one can hope since the propagation law $d\zeta/dt=-\partial_x v$ is intrinsically hyperbolic.\
In order to test our scheme and because no explicit solutions are known for general nonlinearities we restrict to the pure PME $\partial_tv =(m-1)v\partial_{xx}^2v+|\partial_x v|^2$, to which the Barenblatt profiles $$t\geq -t_0:\qquad V_m(x,t;C,x_0,t_0)=\frac{1}{t_0+t}\left(C(t_0+t)^{2/(m+1)}-\frac{1}{2(m+1)}|x-x_0|^2\right)_+$$ are explicit solutions for any $m>1$. Here $C>0$ is a free parameter, while $x_0,t_0$ reflect the invariance under shifts. The interfaces are then explicitly given by $$\zeta_{lr}(t)=x_0\pm \sqrt{2(m+1)C}\,(t_0+t)^{1/(m+1)}.$$ For our numerical experiment we fix $m=2$ and choose arbitrary parameters $$\hat{v}(x,t):=V_2(x,t;4/6,0,1),\qquad \check{v}(x,t):=V_2(x,t;1/6,3\sqrt[3]{2},1)$$ such that the initial supports of $\hat{v}^0(x):=\hat{v}(x,0),\check{v}^0(x):=\check{v}(x,0)$ are at positive distance from each other as in Section \[section:two\_patches\]. The exact interfaces are $$\hat{\zeta}_{lr}(r)=0\pm 2(1+t)^{1/3},\qquad \check{\zeta}_{lr}(t)=3\sqrt[3]{2}\pm (1+t)^{1/3}.$$ Starting with initial datum $v_0=\max\{\hat{v}^0,\check{v}^0\}$ the theoretical hole-filling time $T^*$ can be computed according to Section \[section:two\_patches\] by solving $\hat{\zeta}_r(t)=\check{\zeta}_l(t)
\Leftrightarrow t=T^*$, which gives explicitly $$T^*=1,\qquad x^*=\hat{\zeta}_r(T^*)=\check{\zeta}_l(T^*)=2\sqrt[3]{2}\approx 2.5198.$$
All the computations were performed on a personal computer with [Linux/Octave]{}. We only specify the value of $\Delta x$, the parameters $\Delta t,{\varepsilon}$ being then chosen respectively with the largest and smallest value allowed by . Figure \[fig:FIG\_cauchy\] shows a typical result with $\Delta x=0.01$ plotted for several values of $t$, and Figure \[fig:FIG\_interface\] illustrates the corresponding numerical interfaces. The hole filling was numerically detected for $T^*_h=1.0205$ and $x^*_h=2.5236$ (compare with $T^*=1$ and $x^*=2.5198$).
(1,0.74902471) (0,0)[![numerical solution $v_h(\,.\,,t)$ plotted for several times ($\Delta x=0.01$)[]{data-label="fig:FIG_cauchy"}](FIG_cauchy.pdf "fig:"){width="\unitlength"}]{} (0.55942783,0.07646294)[(0,0)\[rb\]]{} (0.55942783,0.1130039)[(0,0)\[rb\]]{} (0.55942783,0.14941482)[(0,0)\[rb\]]{} (0.55942783,0.18595579)[(0,0)\[rb\]]{} (0.55942783,0.22249675)[(0,0)\[rb\]]{} (0.55942783,0.25903771)[(0,0)\[rb\]]{} (0.55942783,0.29544863)[(0,0)\[rb\]]{} (0.55942783,0.3319896)[(0,0)\[rb\]]{} (0.57022107,0.05305592)[(0,0)\[b\]]{} (0.63719116,0.05305592)[(0,0)\[b\]]{} (0.70403121,0.05305592)[(0,0)\[b\]]{} (0.7710013,0.05305592)[(0,0)\[b\]]{} (0.83784135,0.05305592)[(0,0)\[b\]]{} (0.90481144,0.05305592)[(0,0)\[b\]]{} (0.73745124,0.01794538)[(0,0)\[b\]]{} (0.76065866,0.29208228)[(0,0)\[lb\]]{} (0.11911573,0.07646294)[(0,0)\[rb\]]{} (0.11911573,0.1130039)[(0,0)\[rb\]]{} (0.11911573,0.14941482)[(0,0)\[rb\]]{} (0.11911573,0.18595579)[(0,0)\[rb\]]{} (0.11911573,0.22249675)[(0,0)\[rb\]]{} (0.11911573,0.25903771)[(0,0)\[rb\]]{} (0.11911573,0.29544863)[(0,0)\[rb\]]{} (0.11911573,0.3319896)[(0,0)\[rb\]]{} (0.12990897,0.05305592)[(0,0)\[b\]]{} (0.19687906,0.05305592)[(0,0)\[b\]]{} (0.26371912,0.05305592)[(0,0)\[b\]]{} (0.33068921,0.05305592)[(0,0)\[b\]]{} (0.39752926,0.05305592)[(0,0)\[b\]]{} (0.46449935,0.05305592)[(0,0)\[b\]]{} (0.29713914,0.01794538)[(0,0)\[b\]]{} (0.32259081,0.29544863)[(0,0)\[lb\]]{} (0.55942783,0.4313394)[(0,0)\[rb\]]{} (0.55942783,0.46788036)[(0,0)\[rb\]]{} (0.55942783,0.50429129)[(0,0)\[rb\]]{} (0.55942783,0.54083225)[(0,0)\[rb\]]{} (0.55942783,0.57737321)[(0,0)\[rb\]]{} (0.55942783,0.61391417)[(0,0)\[rb\]]{} (0.55942783,0.6503251)[(0,0)\[rb\]]{} (0.55942783,0.68686606)[(0,0)\[rb\]]{} (0.57022107,0.40793238)[(0,0)\[b\]]{} (0.63719116,0.40793238)[(0,0)\[b\]]{} (0.70403121,0.40793238)[(0,0)\[b\]]{} (0.7710013,0.40793238)[(0,0)\[b\]]{} (0.83784135,0.40793238)[(0,0)\[b\]]{} (0.90481144,0.40793238)[(0,0)\[b\]]{} (0.75953654,0.6503251)[(0,0)\[lb\]]{} (0.11911573,0.4313394)[(0,0)\[rb\]]{} (0.11911573,0.46788036)[(0,0)\[rb\]]{} (0.11911573,0.50429129)[(0,0)\[rb\]]{} (0.11911573,0.54083225)[(0,0)\[rb\]]{} (0.11911573,0.57737321)[(0,0)\[rb\]]{} (0.11911573,0.61391417)[(0,0)\[rb\]]{} (0.11911573,0.6503251)[(0,0)\[rb\]]{} (0.11911573,0.68686606)[(0,0)\[rb\]]{} (0.12990897,0.40793238)[(0,0)\[b\]]{} (0.19687906,0.40793238)[(0,0)\[b\]]{} (0.26371912,0.40793238)[(0,0)\[b\]]{} (0.33068921,0.40793238)[(0,0)\[b\]]{} (0.39752926,0.40793238)[(0,0)\[b\]]{} (0.46449935,0.40793238)[(0,0)\[b\]]{} (0.29713914,0.37282185)[(0,0)\[b\]]{} (0.32259081,0.6503251)[(0,0)\[lb\]]{}
(1,0.74902471) (0,0)[![interface curves ($\Delta x=0.01$)[]{data-label="fig:FIG_interface"}](FIG_interface.pdf "fig:"){width="\unitlength"}]{} (0.11911573,0.07646294)[(0,0)\[rb\]]{} (0.11911573,0.22184655)[(0,0)\[rb\]]{} (0.11911573,0.36710013)[(0,0)\[rb\]]{} (0.11911573,0.51248375)[(0,0)\[rb\]]{} (0.11911573,0.65773732)[(0,0)\[rb\]]{} (0.12990897,0.05305592)[(0,0)\[b\]]{} (0.28491547,0.05305592)[(0,0)\[b\]]{} (0.43992198,0.05305592)[(0,0)\[b\]]{} (0.59479844,0.05305592)[(0,0)\[b\]]{} (0.74980494,0.05305592)[(0,0)\[b\]]{} (0.90481144,0.05305592)[(0,0)\[b\]]{} (0.07009103,0.38751625) (0.51729519,0.01794538)[(0,0)\[b\]]{}
In addition to an abstract convergence result as in Theorem \[theo:CV\_solution\_interfaces\_single\_patch\], DiBenedetto and Hoff also derived explicit error estimates for the pure PME nonlinearity in the form $\|\zeta_h-\zeta\|_{L^\infty(0,T)}+\|v_h-v\|_{L^{\infty}(Q_T)}\leq \mathcal{O}\left( \Delta x^{\alpha}|\log \Delta x|^{\beta}\right)$ for some structural $\alpha,\beta$ related to $m>1$, see [@DBH84 Theorem 4.1]. However their proof heavily relies on the explicit power structure $\Phi(s)=s^m$, and obtaining error estimates for general nonlinearities is a hard task that we did not carry out here due to the technical difficulties and lack of space. Figure \[fig:FIG\_error\] shows the numerical errors $E_{x}:=\left|x^*_h-x^*\right|,E_t=|T^*_h-T^*|$ and $E_{\zeta}=\|\zeta_h-\zeta\|_{L^\infty(0,T^*_h))},E_v=\|v_h-v\|_{L^{\infty}(Q_{T^*_h})}$ as a function of $\Delta x$, and quite clearly exhibits $\mathcal{O}(\Delta x^{\alpha})$ convergence rates. Thus our scheme gives a good approximation of the solution, interfaces, and coordinates of the hole-filling as predicted from Theorem \[theo:CV\_solution\_double\_patch\] and Theorem \[theo:CV\_hitting\_time\].\
(1,0.74902471) (0,0)[![errors as a function of $\Delta x$[]{data-label="fig:FIG_error"}](FIG_error.pdf "fig:"){width="\unitlength"}]{} (0.55942783,0.07646294)[(0,0)\[rb\]]{} (0.55942783,0.2289987)[(0,0)\[rb\]]{} (0.55942783,0.3816645)[(0,0)\[rb\]]{} (0.55942783,0.5343303)[(0,0)\[rb\]]{} (0.55942783,0.68686606)[(0,0)\[rb\]]{} (0.59375813,0.05305592)[(0,0)\[b\]]{} (0.74616385,0.05305592)[(0,0)\[b\]]{} (0.89856957,0.05305592)[(0,0)\[b\]]{} (0.73745124,0.01794538)[(0,0)\[b\]]{} (0.81755527,0.66345904)[(0,0)\[rb\]]{} (0.81755527,0.64005202)[(0,0)\[rb\]]{} (0.11911573,0.07646294)[(0,0)\[rb\]]{} (0.11911573,0.2289987)[(0,0)\[rb\]]{} (0.11911573,0.3816645)[(0,0)\[rb\]]{} (0.11911573,0.5343303)[(0,0)\[rb\]]{} (0.11911573,0.68686606)[(0,0)\[rb\]]{} (0.15344603,0.05305592)[(0,0)\[b\]]{} (0.30585176,0.05305592)[(0,0)\[b\]]{} (0.45825748,0.05305592)[(0,0)\[b\]]{} (0.29713914,0.01794538)[(0,0)\[b\]]{} (0.37724317,0.66345904)[(0,0)\[rb\]]{} (0.37168896,0.64005202)[(0,0)\[rb\]]{}
Acknowledgements {#acknowledgements .unnumbered}
----------------
The author was supported by the Portuguese FCT fellowship SFRH/BPD/88207/2012.
[^1]: CAMGSD, Instituto Superior Técnico, Av. Rovisco Pais 1049-001 Lisboa, Portugal `leonard.monsaingeon@ist.utl.pt`
|
[Radiative corrections to the lightest KK states in the $T^2/(Z_2\times Z_2')$ orbifold]{}[ \
]{} **A.T. Azatov[^1]**\
*Department of Physics, University of Maryland, College Park, MD 20742, USA*\
`Abstract`
We study radiative corrections localized at the fixed points of the orbifold for the field theory in six dimensions with two dimensions compactified on the $T_2/(Z_2\times Z_2')$ orbifold in a specific realistic model for low energy physics that solves the proton decay and neutrino mass problem. We calculate corrections to the masses of the lightest stable KK modes, which could be the candidates for dark matter.
Introduction
============
One of the important questions in particle physics today is the nature of physics beyond the standard model (SM). The new Large Hadron Collider (LHC) machine starting soon, experiments searching for dark matter of the universe as well as many neutrino experiments planned or under way, have raised the level of excitement in the field since they are poised to provide a unique experimental window into this new physics. The theoretical ideas they are likely to test are supersymmetry, left-right symmetry as well as possible hidden extra dimensions [@antoniadis][@dvali][@kokorelis] in nature, which all have separate motivations and address different puzzles of the SM. In this paper, I will focus on an aspect of one interesting class of models known as universal extra dimension models(UED) [@acd](see for review [@review]). These models provide a very different class of new physics at TeV (see [@higgsboson] for the constraints on size of compactification $R $ ) scale than supersymmetry. But in general UED models based on the standard model gauge group, there is no simple explanation for the suppressed proton decay and the small neutrino mass. One way to solve the proton decay problem in the context of total six space-time dimensions, was proposed in [@protonstability]. In this case, the additional dimensions lead to the new U(1) symmetry, that suppresses all baryon-number violating operators. The small neutrino masses can be explained by the propagation of the neutrino in the seventh warp extra dimension [@neutrinomass]. On the other hand we can solve both these problems by extending gauge group to the $SU(2)_L\times
SU(2)_R\times U(1)_{B-L}$[@lrs]. Such class of UED models were proposed in [@mohapat]. In this case, the neutrino mass is suppressed due to the $B-L$ gauge symmetry and specific orbifolding conditions that keep left-handed neutrinos at zero mode and forbid lower dimensional operators that can lead to the unsuppressed neutrino mass.
An important consequence of UED models is the existence of a new class of dark matter particle, i.e. the lightest KK (Kaluza-Klein) mode[@tait]. The detailed nature of the dark matter and its consequences for new physics is quite model-dependent. It was shown recently [@ken] that the lightest stable KK modes in the model [@mohapat] with universal extra dimensions could provide the required amount of cold dark matter [@feng]. Dark matter in this particular class of UED models is an admixture of the KK photon and right-handed neutrinos. In the case of the two extra dimensions, KK mode of the every gauge boson is accompanied by the additional adjoint scalar which has the same quantum numbers as a gauge boson. In the tree level approximation KK masses of this adjoint scalar and gauge boson are the same, so they both can be dark matter candidates. The paper [@ken] presented relic density analysis assuming either the adjoint scalar or the gauge boson is the lightest stable KK particle. These assumptions lead to the different restrictions on the parameter space. My goal in this work was to find out whether radiative corrections could produce mass splitting of these modes, and if they do, determine the lightest stable one. In this calculations I will follow the works [@Georgi],[@matchev]. Similar calculations for different types of orbifolding were considered in [@ponton] ($T^2/Z_4$ orbifold),and [@quiros]($M_4\times S^1/Z_2$ and $T^2/Z_2$ orbifolds).
Model
=====
In this sections we will review the basic features of the model [@mohapat]. The gauge group of the model is $SU(3)_c\times
SU(2)_L\times SU(2)_R\times U(1)_{B-L}$ with the following matter content for generation : $$\begin{aligned}
{\cal Q}_{1,-}, {\cal Q}'_{1,-}= (3,2,1,\tfrac{1}{3});&
{\cal Q}_{2,+}, {\cal Q}'_{2,+}= (3,1,2,\tfrac{1}{3});\nonumber\\
{\cal \psi}_{1,-}, {\cal \psi}'_{1,-}= (1,2,1,-1);& {\cal
\psi}_{2,+}, {\cal \psi}'_{2,+}= (1,1,2,-1); \label{matter}\end{aligned}$$
We denote the gauge bosons as $G_A$, $W^{\pm}_{L,A}$, $W^{\pm}_{R,A}$, and $B_A$, for $SU(3)_c$, $SU(2)_L$, $SU(2)_R$ and $U(1)_{B-L}$ respectively, where $A=0,1,2,3,4,5$ denotes the six space-time indices. We will also use the following short hand notations: Greek letters $\mu,\nu,\dots=0,1,2,3$ for usual four dimensions indices and lower case Latin letters $a,b,\dots=4,5$ for the extra space dimensions.
We compactify the extra $x_4$, $x_5$ dimensions into a torus, $T^2$, with equal radii, $R$, by imposing periodicity conditions, $\varphi(x_4,x_5) = \varphi(x_4+ 2\pi R,x_5) = \varphi(x_4,x_5+
2\pi R)$ for any field $\varphi$. We impose the further orbifolding conditions i.e. $ Z_2:\vec{y} \rightarrow -\vec {y}
~\mbox{and}~
Z'_2: \vec{y}~'\rightarrow -\vec{y}~'$ where $\vec y = (x_4,x_5)$, $\vec{y}~' = \vec{y} - (\pi R /2, \pi
R/2)$. The $Z_2$ fixed points will be located at the coordinates $(0,0)$ and $(\pi R,\pi R )$, whereas those of $Z_2'$ will be in $(\pi R/2,\pm \pi R/2)$. The generic field $\phi(x_\mu, x_a)$ with fixed $Z_2\times Z_2' $ parities can be expanded as: $$\begin{aligned}
\phi(+,+)=\frac{1}{2\pi R}\varphi^{(0,0)}+\frac{1}{\sqrt{2}\pi
R}\sum_{n_4+n_5~ \text{-even}}\varphi^{(n_4,n_5)}(x_\mu)
cos\left(\frac{n_4x_4 +n_5x_5
}{R}\right)\nonumber\\
\phi(+,-)=\frac{1}{\sqrt{2}\pi R}\sum_{n_4+n_5~\text{
-odd}}\varphi^{(n_4,n_5)}(x_\mu) cos\left(\frac{n_4x_4 +n_5x_5
}{R}\right)\nonumber\\
\phi(-,+)=\frac{1}{\sqrt{2}\pi R}\sum_{n_4+n_5~\text{
-odd}}\varphi^{(n_4,n_5)}(x_\mu) sin\left(\frac{n_4x_4 +n_5x_5
}{R}\right)\nonumber\\
\phi(-,-)=\frac{1}{\sqrt{2}\pi R}\sum_{n_4+n_5~\text{
-even}}\varphi^{(n_4,n_5)}(x_\mu) sin\left(\frac{n_4x_4 +n_5x_5
}{R}\right) \label{decompos}\end{aligned}$$ One can see that only the $(+,+)$ fields will have zero modes. In the effective 4D theory the mass of each mode has the form: $m_{N}^2 = m_0^2 + \frac{N}{R^2}$; with $N=\vec{n}^2=n_4^2 +
n_5^2$ and $m_0$ is the physical mass of the zero mode. We assign the following $Z_2\times Z'_2$ charges to the various fields: $$\begin{aligned}
G_\mu(+,+);\quad B_\mu(+,+); \quad
W_{L,\mu}^{3,\pm}(+,+);W^3_{R,\mu}(+,+);
W^\pm_{R,\mu}(+,-); \nonumber\\
G_{a}(-,-);\quad B_a(-,-);\quad W_{L,a}^{3,\pm}(-,-);
W^3_{R,a}(-,-); W^\pm_{R,a}(-,+). \label{gparity}\end{aligned}$$ As a result, the gauge symmetry $SU(3)_c\times SU(2)_L\times
SU(2)_R\times U(1)_{B-L}$ breaks down to $SU(3)_c\times
SU(2)_L\times U(1)_{I_{3R}}\times U(1)_{B-L}$ on the 3+1 dimensional brane. The $W^{\pm}_R$ picks up a mass $R^{-1}$, whereas prior to symmetry breaking the rest of the gauge bosons remain massless.
For quarks we choose, $$\begin{aligned}
Q_{1,L}\equiv
\left(\begin{array}{c} u_{1L}(+,+)\\ d_{1L}(+,+)\end{array}\right);
&\quad&
Q'_{1,L}\equiv
\left(\begin{array}{c} u'_{1L}(+,-)\\ d'_{1L}(+,-)\end{array}\right);
\nonumber \\
Q_{1,R}\equiv
\left(\begin{array}{c} u_{1R}(-,-)\\ d_{1R}(-,-)\end{array}\right);
&\quad&
Q'_{1,R}\equiv
\left(\begin{array}{c} u'_{1R}(-,+)\\ d'_{1R}(-,+)\end{array}\right);
\nonumber\end{aligned}$$ $$\begin{aligned}
Q_{2,L}\equiv
\left(\begin{array}{c} u_{2L}(-,-)\\ d_{2L}(-,+)\end{array}\right);
&\quad&
Q'_{2,L}\equiv
\left(\begin{array}{c} u'_{2L}(-,+)\\ d'_{2L}(-,-)\end{array}\right);
\nonumber \\
Q_{2,R}\equiv
\left(\begin{array}{c} u_{2R}(+,+)\\ d_{2R}(+,-)\end{array}\right);
& \quad& Q'_{2,R}\equiv
\left(\begin{array}{c} u'_{2R}(+,-)\\ d'_{2R}(+,+)\end{array}\right);
\label{quarks} \end{aligned}$$ and for leptons: $$\begin{aligned}
\psi_{1,L}\equiv
\left(\begin{array}{c} \nu_{1L}(+,+) \\ e_{1L}(+,+)\end{array}\right);
&\qquad&
\psi'_{1,L}\equiv
\left(\begin{array}{c} \nu'_{1L}(-,+) \\ e'_{1L}(-,+)\end{array}\right);
\nonumber \\
\psi_{1,R}\equiv
\left(\begin{array}{c} \nu_{1R}(-,-) \\ e_{1R}(-,-)\end{array}\right);
& \qquad &
\psi'_{1,R}\equiv
\left(\begin{array}{c} \nu'_{1R}(+,-) \\ e'_{1R}(+,-)\end{array}\right);
\qquad
\nonumber \\ [1ex]
\psi_{2,L}\equiv
\left(\begin{array}{c} \nu_{2L}(-,+) \\ e_{2L}(-,-)\end{array}\right);
&\qquad&
\psi'_{2,L}\equiv
\left(\begin{array}{c} \nu'_{2L}(+,+)\\ e'_{2L}(+,-)\end{array}\right);
\nonumber \\
\psi_{2,R}\equiv
\left(\begin{array}{c} \nu_{2R}(+,-)\\ e_{2R}(+,+)\end{array}\right);
& \qquad &
\psi'_{2,R}\equiv
\left(\begin{array}{c} \nu'_{2R}(-,-)\\ e'_{2R}(-,+)\end{array}\right).
\label{leptons}
\end{aligned}$$ The zero modes i.e. (+,+) fields correspond to the standard model fields along with an extra singlet neutrino which is left-handed. They will have zero mass prior to gauge symmetry breaking.
The Higgs sector of the model consists of $$\begin{aligned}
\phi &\equiv
\left(\begin{array}{cc} \phi^0_u(+,+) & \phi^+_d(+,-)\\
\phi^-_u(+,+) & \phi^0_d(+,-)\end{array}\right);\nonumber\\
\chi_L&\equiv \left(\begin{array}{c} \chi^0_L(-,+) \\
\chi^-_L(-,+)\end{array}\right); \quad
\chi_R\equiv \left(\begin{array}{c} \chi^0_R(+,+) \\
\chi^-_R(+,-)\end{array}\right),
\end{aligned}$$ with the charge assignment under the gauge group, $$\begin{aligned}
\phi &=& (1,2,2,0),\nonumber\\
\chi_L&=&(1,2,1,-1),\quad \chi_R=(1,1,2,-1).\end{aligned}$$ In the limit when the scale of $SU(2)_L$ is much smaller than the scale of $SU(2)_R$ (that is, $v_w\ll v_R$) the symmetry breaking occurs in two stages. First $SU(2)_L\times SU(2)_R\times U(1)\rightarrow
SU(2)_L\times U(1)_Y$, where a linear combination of $B_{B-L}$ and $W_{R}^3$, acquire a mass to become $Z'$, while orthogonal combination of $B_{B-L}$ remains massless and serves as a gauge boson for residual group $U(1)_Y$. In terms of the gauge bosons of $SU(2)_R$ and $U(1)_{B-L}$, we have $$\begin{aligned}
Z'_A=\frac{g_RW_{R,A}^3-g_{B-L}B_{B-L,A}}{\sqrt{g_R^2+g_{B-L}^2}},\nonumber\\
B_{Y,A}=\frac{g_RB_{B-L,A}+g_{B-L}W_{R,A}^3}{\sqrt{g_R^2+g_{B-L}^2}}.\end{aligned}$$ Then we have standard breaking of the electroweak symmetry. A detailed discussion of the spectrum of the zeroth and first KK modes was presented in [@ken]. The main result of the discussion is that in the tree level approximation only the KK modes $B_{Y,\mu}$ $B_{Y,a}$ and $\nu_2$ will be stable and can be considered as candidates for dark matter, and the relic CDM density value leads to the upper limits on $R^{-1}$ of about 400-650 Gev, and the mass of the $M_{Z'}\leq 1.5$ Tev. However, radiative corrections can split the KK masses of the $B_{Y,\mu}$ and $B_{Y,a}$, and only the lightest of them will be stable. The goal of this work is to find out which of the two modes is lighter and serves as dark matter.
Propagators
===========
To calculate the radiative corrections, we follow the methods presented in Refs. [@Georgi] and [@matchev]. We derive the propagators for the scalar, fermion, and vector fields in the $T^2/(Z_2\times Z_2')$ orbifold, $\varepsilon$ and $\varepsilon'$ are the $Z_2$ and $Z_2'$ parities respectively, so arbitrary field satisfying the boundary conditions $$\begin{aligned}
\phi(x_4,x_5)=\varepsilon\phi(-x_4,-x_5)\nonumber\\
\phi(x_4,x_5)=\varepsilon'\phi(\pi R-x_4,\pi R-x_5) \label{orb}\end{aligned}$$ can be decomposed as (we always omit the dependence on 4D coordinates) $$\begin{aligned}
\phi(x_4,x_5)&=\Phi(x_4,x_5)+\varepsilon\Phi(-x_4,-x_5)\nonumber\\
&\quad+\varepsilon'\Phi(\pi
R-x_4,\pi R-x_5 )+\varepsilon\varepsilon '\Phi(x_4-\pi R,x_5-\pi
R).\end{aligned}$$ The field $\phi$ will automatically satisfy the orbifolding conditions of Eq. (\[orb\]), and one can easily calculate $\langle 0|\phi(x_4,x_5)\phi(x'_4,x'_5)|0\rangle$ in the momentum space. This leads to the following expressions for the propagators of the scalar, gauge and fermion fields. Propagator of the scalar field is given by $$\begin{aligned}
iD=\frac{i}{4(p^2-p_a^2)}\left(1+\varepsilon_\phi\varepsilon^{\prime}_\phi
e^{i p_a(\pi R)_a}\right) \left(
\delta_{p_ap^{\prime}_a}+\varepsilon_\phi\delta_{p_a
-p^{\prime}_a}\right),\end{aligned}$$ where $ p_a(\pi R)_a\equiv\pi R( p_4+p_5)$. Propagator of the gauge boson in($\xi=1$) gauge is $$\begin{aligned}
iD_{AB}=\frac{-ig_{AB}}{4(p^2-p_a^2)}\left(1+\varepsilon_A\varepsilon_{A}
^{\prime} e^{i p_a(\pi R)_a}\right) \left(
\delta_{p_ap^{\prime}_a}+\varepsilon_A\delta_{p_a
-p^{\prime}_a}\right),\end{aligned}$$ where fields $A_a$ and $A_\mu$ will have opposite $Z_2\times
Z_2'$ parities: ($\varepsilon_\mu
=-\varepsilon_a,~\varepsilon'_\mu =-\varepsilon'_a$). The fermion propagator is given by $$\begin{aligned}
iS_{F}=\frac{i}{4(\mathord{\not\mathrel{{\mathrel{p}}}}-\mathord{\not%
\mathrel{{\mathrel{p_a'}}}})} \left( 1+\varepsilon _{\psi
}\varepsilon^{\prime}_{\psi }e^{ip_{a}(\pi R)_{a}}\right) \left(
\delta _{p_{a}p_{a}^{\prime }}+\Sigma_{45}\varepsilon _{\psi
}\delta _{p_{a}-p_{a}^{\prime }}\right),\end{aligned}$$ where we have defined $$\begin{aligned}
\spur{p_a}\equiv p_4\Gamma_4+p_5\Gamma_5.\end{aligned}$$
Radiative corrections to the fermion mass
=========================================
Now we want to find corrections for the mass of the $\nu_2$ field. First let us consider general interaction between a fermion and a vector boson, $$\begin{aligned}
\mathcal{L}_{int}=g_{6D}\overline{\psi}\Gamma_A\psi A^A,\end{aligned}$$ where $g_{6D}$ is 6 dimensional coupling constant that is related to the 4 dimensional coupling $g$ by $$\begin{aligned}
g=\frac{g_{6D}}{(2\pi R)}.\end{aligned}$$ The gauge interaction will give mass corrections due to the diagram Fig.1 (a).
 \[figferm1\]
The matrix element will be proportional to the $$\begin{aligned}
i\Sigma &=-\sum_{k_{a}}g^{2}\int \frac{d^{4}k}{(2\pi )^{4}}\frac{1}{4}\frac{1%
}{(p-k)^{2}-(p_{a}-k_{a})^{2}}
\nonumber\\
&\quad\quad\times\Gamma_A\frac{\mathord{\not\mathrel{{%
\mathrel{k}}}}- \mathord{\not\mathrel{{\mathrel{k_a'}}}}}{k^{2}-k_{a}^{2}}[%
\delta_{k^{\prime}_a
k_a}+\varepsilon_\psi\Sigma_{45}\delta_{-k^{\prime}_a,k_a}]\Gamma^A[\delta_{(p-k)_a,(p^{\prime}-k^{\prime})_a}+\epsilon^A%
\delta_{(p-k)_a,-(p^{\prime}-k^{\prime})_a}]
\label{sigma}\end{aligned}$$ The $\varepsilon_\psi$ and $\varepsilon'_\psi$, are the $Z_2\times
Z_2' $ parities of the fermion and $\varepsilon^A~,
{\varepsilon^A}'$ are the parities of the gauge boson. The sum is only over the $k_{a}$ which are allowed by the $Z_{2} \times
Z_{2}'$ parities i.e. for the ones where $1+\varepsilon _{\psi
}\varepsilon _{\psi }^{\prime }e^{ik_{a}(\pi R)_{a}}\neq 0$ There are two types of terms that can lead to the corrections of the fermion self energy, the bulk terms appearing due to the nonlocal Lorentz breaking effects and brane like terms which appear because of the specific orbifold conditions, but the bulk terms for fermion self energy graph appear to vanish (see [@matchev]), so we will concentrate our attention only on the brane like terms. In the case of our $T^2/(Z_2\times Z_2')$ orbifold they will be localized at the points $(0,0),~(\pi R,\pi R
),~(\pi R/2,\pm\pi R/2)$ (see Appendix). The numerator of the integrand simplifies to $$\begin{aligned}
\Gamma_A(\mathord{\not\mathrel{{\mathrel{k}}}}-\mathord{\not\mathrel{{%
\mathrel{k_a'}}}})\Gamma^A\varepsilon^A\delta_{(p+p^{\prime})_a,2k_a}+
\Gamma_A(\mathord{\not\mathrel{{\mathrel{k}}}}-\mathord{\not\mathrel{{%
\mathrel{k_a'}}}})\Sigma_{45}\Gamma^A\varepsilon_\psi\delta_{(p-p^{%
\prime})_a,2k_a}\nonumber\\
=4\mathord{\not\mathrel{{\mathrel{k_a'}}}}\varepsilon^\mu\delta_{p_a+p^{%
\prime}_a,2k_a}+4\mathord{\not\mathrel{{\mathrel{k_a'}}}}\varepsilon_\psi%
\Sigma_{45}\delta_{2k_a,p_a-p^{\prime}_a}\end{aligned}$$ We can then write Eq. (\[sigma\]) as $$\begin{aligned}
i\Sigma&=-\sum_{k_a'}\frac{g^2}{4}\int\frac{d^4k}{(2\pi)^4}\frac{4%
\mathord{\not\mathrel{{\mathrel{k_a'}}}} (\varepsilon^\mu\delta_{p_a+p^{%
\prime}_a,2k_a}+\varepsilon_\psi\Sigma_{45}\delta_{2k_a,p_a-p^{\prime}_a})} {%
((p-k)^{2}-(p_{a}-k_{a})^{2})(k^2-k_a^2)}\nonumber\\
&=\frac{-ig^2}{2(4\pi)^2}\ln\left(\frac{\Lambda^2}{\mu^2}\right) \left[\frac{%
\mathord{\not\mathrel{{\mathrel{p_a}}}}+\mathord{\not\mathrel{{%
\mathrel{p'_a}}}}}{2}(1+\varepsilon_\psi\varepsilon_\psi'e^{(p_a+p'_a)(\pi R)_a/2})\varepsilon^\mu+%
\varepsilon_\psi\frac{\mathord{\not\mathrel{{\mathrel{p'_a}}}}-\mathord{\not%
\mathrel{{\mathrel{p_a}}}}}{2}(1+\varepsilon_\psi\varepsilon_\psi'e^{(p_a-p'_a)(\pi
R)_a/2})\Sigma_{45}\right], \label{ptilde}\end{aligned}$$ where $\Lambda$ is a cut-off and $\mu$ is renormalization scale. After transforming to the position space we get $$\begin{aligned}
\delta\mathcal{L}&=\frac{g^2}{8(4\pi)^2}\ln\left(\frac{\Lambda^2}{\mu^2}%
\right)\left[\delta(I)\{\overline{\psi}({i\mathord{\not\mathrel{{%
\mathrel{\partial_a}}}}})(-\varepsilon_\psi \Sigma_{45}
+\varepsilon^\mu)\psi+\overline{\psi}{(i{%
\overleftarrow{\mathord{\not\mathrel{{\mathrel{\partial_a}}}}}})}(-\varepsilon_\psi
\Sigma_{45} -\varepsilon^\mu)\psi\}\right.\nonumber\\
&\quad\quad\quad+\left.\delta(II)\{\overline{\psi}({i\mathord{\not\mathrel{{%
\mathrel{\partial_a}}}}})(-\varepsilon'_\psi \Sigma_{45}
+\varepsilon^\mu)\psi+\overline{\psi}{(i{%
\overleftarrow{\mathord{\not\mathrel{{\mathrel{\partial_a}}}}}})}(-\varepsilon'_\psi
\Sigma_{45} -\varepsilon^\mu)\psi\}\right], \label{fmv}\end{aligned}$$ where $$\begin{aligned}
\delta(I)\equiv \delta(x_a)+ \delta(x_a-\pi R),~~\delta(II)\equiv
\delta(x_a-\pi R/2) +\delta(x_a+\pi R/2),\end{aligned}$$ and $\psi$ is normalized as four dimensional fermion field related to the six dimensional field by $\psi=\psi^{6D}(2\pi R)$. In our case the corrections to the self energy of the neutrino will arise from the diagrams with $W^+_R$, $Z'$, but one can see that these fields have nonzero mass coming from the breaking of $SU(2)_R$, thus in Eq. (\[ptilde\]) $\tilde p^2 \rightarrow \tilde p^2
+(1-\alpha)M^2_{W^\pm_R,Z'}$. The contribution of the diagram with $W^\pm_R$ will be $$\begin{aligned}
\delta
\mathcal{L}=\frac{g_R^2}{8(4\pi)^2}\ln\left(\frac{\Lambda^2}{\mu_{W_R}^2}\right)
\left[\delta(I)\{\overline{\nu_2^R}(-\partial_4-i\partial_5)\nu_2^L+(-\partial_4+i\partial_5)\overline{\nu_2^L}\nu_2^R\}\right.\nonumber\\
+\left.\delta(II)\{\overline{\nu_2^L}(-\partial_4+i\partial_5)\nu_2^R+(-\partial_4-i\partial_5)\overline{\nu_2^R}\nu_2^L\}\right],\end{aligned}$$ where $\mu_{W_R}^2\sim\mu^2+M^2_{W_R}$. The terms proportional to the $\delta(I)$ and $\delta(II)$ will lead to the corrections to the four dimensional action that will have equal magnitude and opposite sign, so the total correction to the fermion mass will vanish. The contribution of the diagram with $Z'$ will lead to the $$\begin{aligned}
\label{deltal} \delta
\mathcal{L}=\frac{g_R^2+g_{B-L}^2}{16(4\pi)^2}\ln\left(\frac{\Lambda^2}{\mu_{Z'}^2}\right)
\left[\delta(I)\{\overline{\nu_2^R}(-\partial_4-i\partial_5)\nu_2^L+(-\partial_4+i\partial_5)\overline{\nu_2^L}\nu_2^R\}\right.\nonumber\\
+\left.\delta(II)\{\overline{\nu_2^L}(\partial_4-i\partial_5)\nu_2^R+(\partial_4+i\partial_5)\overline{\nu_2^R}\nu_2^L\}\right],\end{aligned}$$ where $\mu_{Z'}^2\sim\mu^2+M^2_{Z'}$. Let us look on the first term of the formula (\[deltal\]), it is proportional to the $\delta(I)\overline{\nu_2^R}\partial_a\nu_2^L$, but one can see from the KK decomposition (\[decompos\]), that profiles of the $\nu_2^R(+,-)$ and $\partial_a\nu_2^L(-,+)$ are both equal to the $\text{cos}(\frac{n_4x_4+n_5x_5}{R})$ i.e. are maximal at the $\delta(I)$. The same is true for the others terms of the (\[deltal\]), thus the correction to the effective 4D lagrangian, and KK masses will be [^2] $$\begin{aligned}
\mathcal{L}_{4D}=\frac{g_R^2+g_{B-L}^2}{4(4\pi)^2}\ln\left(\frac{\Lambda^2}{\mu_{Z'}^2}\right)
\left[\left(\frac{-n_4-in_5}{R}\right)\overline{\nu_2^R}\nu_2^L+\left(\frac{-n_4+in_5}{R}\right)\overline{\nu_2^L}\nu_2^R\right]\nonumber\\
\delta m_{\nu(n_4,n_5)}
=\frac{(g_R^2+g_{B-L}^2)\sqrt{n_4^2+n_5^2}}{4R(4\pi)^2}\ln\left(\frac{\Lambda^2}{\mu_{Z'}^2}\right).
\label{mnu}\end{aligned}$$ So the correction to the mass of the first KK mode for $\nu_2$ will be: $$\begin{aligned}
\delta m_{\nu}
=\frac{(g_R^2+g_{B-L}^2)}{4R(4\pi)^2}\ln\left(\frac{\Lambda^2}{\mu_{Z'}^2}\right).
\label{mnu}\end{aligned}$$ Now we have to evaluate contribution of the diagram Fig.1 (b) $$\begin{aligned}
i\Sigma =\sum_{k_{a}}f^{2}\int \frac{d^{4}k}{(2\pi )^{4}}\frac{1}{4}\frac{1}{%
(p-k)^{2}-(p_{a}-k_{a})^{2}}\frac{\mathord{\not\mathrel{k}}-%
\mathord{\not\mathrel{k_a'}}}{k^{2}-k_{a}^{2}}[\varepsilon _{\psi
}\Sigma _{45}\delta _{p_{a}-p_{a}^{\prime },2k_{a}}+\varepsilon
_{\phi }\delta _{p_{a}+p_{a}^{\prime },2k_{a}}],\end{aligned}$$ where $f$ is the 4D Yukawa coupling , and again we will consider only the terms that are localized at the fixed points of the orbifold. $$\begin{aligned}
i\Sigma=\sum_{k_a}f^2\int \frac{d^{4}k}{(2\pi )^{4}}\frac{1}{4}%
\int_0^1d\alpha \frac{(\mathord{\not\mathrel{k}}-\mathord{\not\mathrel{k_a'}}%
)[\varepsilon _{\psi }\Sigma _{45}\delta _{p_{a}-p_{a}^{\prime
},2k_{a}}+\varepsilon _{\phi }\delta _{p_{a}+p_{a}^{\prime },2k_{a}}]}{%
[k^2-k_a^2(1-\alpha)-2(kp)\alpha +p^2\alpha-(p_a-k_a)^2\alpha]^2}\end{aligned}$$ Proceeding in the same wave as we have done for the diagram with the vector field we find $$\begin{aligned}
i\Sigma =\frac{if^{2}}{16(4\pi
)^{2}}\ln\left(\frac{\Lambda^2}{\mu^2 }\right)
\left[( \spur{p}-\spur{p_a'}+\spur{p_a})\varepsilon_\psi\Sigma_{45}(1+\varepsilon_\psi\varepsilon'_\psi
e^{(p_a-p'_a)(\pi R)_a/2})\right.+\nonumber\\
+\left.(\spur{p}-\spur{p_a'}-\spur{p_a})\varepsilon_\phi(1+\varepsilon_\psi\varepsilon'_\psi
e^{(p_a+p'_a)(\pi R)_a/2})
\right] .\end{aligned}$$ In our model we have the following Yukawa couplings $$\begin{aligned}
\overline{\psi}_1^-\Phi\psi_2^+=\overline{\nu}_1\Phi_d^0\nu_2+\overline{e}_1\Phi_d^-\nu_2+\overline{\nu}%
_1\Phi_u^+e_2+\overline{e}_1\Phi_u^0e_2\end{aligned}$$ So the corrections to the self energy of neutrino will arise from the diagrams with $\Phi_d^0$ and $\Phi_d^-$. This leads to the following corrections in the lagrangian $$\begin{aligned}
\delta {\cal L} =\frac{f^{2}}{8(4\pi
)^{2}}\ln\left(\frac{\Lambda^2
}{\mu^2}\right)[\delta(I)\{\overline{\nu_{2}^R}i
\spur{\partial}\nu_{2}^R +\overline{\nu_{2}^R}
(\partial_4+i\partial_5)\nu_{2}^L+(\partial_4-i\partial_5)\overline{\nu_{2}^L}
\nu_{2}^R\}+\nonumber\\
+\delta(II)\{-\overline{\nu_{2}^L} i\spur{\partial}\nu_{2}^L
+\overline{\nu_{2}^L}
(\partial_4-i\partial_5)\nu_{2}^R+(\partial_4+i\partial_5)\overline{\nu_{2}^R}
\nu_{2}^L\}]\end{aligned}$$
The terms proportional to $\delta(I)$ and $\delta(II)$ lead to the corrections to the four dimensional action that will cancel each other, so the total mass shift due to the diagrams with $\Phi_d^{0,-}$ will be equal to zero, thus the mass of the neutrino will be corrected only due to the diagram with the $Z'$ boson (\[mnu\]).
Corrections to the mass of the gauge boson
==========================================
As we have mentioned above the dark matter in the model [@mohapat] is believed to consist from mixture of the KK photon and right handed neutrinos, so we are interested in the corrections to the masses of the $B_{Y,a}$ and $B_{Y,\mu}$ bosons. The lowest KK excitations of the $B_{Y,a}$ and $B_{Y,\mu}$ fields correspond to $|p_4|=|p_5|=\frac{1}{R}~$, so everywhere in the calculations we set ($p_4=p_5\equiv\frac{1}{R})$. At the tree level both $B_{Y,a}$ and $B_{Y,\mu}$ fields have the same mass $\frac{\sqrt{2}}{R}$, but radiative corrections can split their mass levels, and only the lightest one of these two will be stable and could be the candidate for the dark matter. In this case the bulk corrections do not vanish by themselves but as was shown in the [@matchev] lead to the same mass corrections for the $B_{Y,\mu}$ and $B_{Y,a}$ fields.
 \[figgf\]
First we will calculate radiative corrections for the $B_{Y,\mu}$ field (calculations are carried out in the Feynman gauge $\xi=1$ ), see Fig.2 for the list of the relevant diagrams. The contribution of every diagram can be presented in the form:
Diagram A B C D
------------------ -------------- --------------- ---- -----
$(a)$ [19]{}/[3]{} -[22]{}/[3]{} 9 18
$(b)$ 0 0 -6 -12
$(c)$ [1]{}/[3]{} [2]{}/[3]{} -1 -2
$(d)$ [4]{}/[3]{} -[4]{}/[3]{} -4 -8
$(e)$ 0 0 4 8
$(f)$ 0 0 12 0
$(g)~\chi_R^\pm$ 0 0 -2 -4
$(h)~\chi_R^\pm$ -2/3 2/3 2 4
$(i)$ 0 0 0 -4
$(j)$ 0 0 0 0
: Coefficients A,B,C,D for the self energy diagrams for $B_{Y,\mu}$ from the gauge sector and $\chi_R^\pm$
$$\begin{aligned}
i\Pi_{\mu\nu}=\frac{i}{4(4\pi)^2}\frac{g^2_{B-L}g^2_R}{{g_R^2+g_{B-L}^2}}\ln
\left(\frac{\Lambda^2}{\mu^2_{W_R}}\right)\nonumber\\
\times\left[ A p^2g_{\mu\nu}+Bp_\mu p_\nu +C
\frac{p_a^2+p_a'^2}{2}g_{\mu\nu}+DM_{W_R}^2g_{\mu\nu}\right]\end{aligned}$$
The coefficients $A,B,C,D$ are listed in the Table 1. The sum of all diagrams is equal to $$\begin{aligned}
i\Pi_{\mu\nu}=\frac{i}{4(4\pi)^2}\frac{g^2_{B-L}g^2_R}{{g_R^2+g_{B-L}^2}}\ln
\left(\frac{\Lambda^2}{\mu^2_{W_R}}\right) \left[14g_{\mu\nu}
\frac{p_a^2+p_a'^2}{2}+\frac{22}{3}(p^2g_{\mu\nu}-p_\mu p_\nu)
\right].\end{aligned}$$ this leads to the following corrections to the lagrangian $$\begin{aligned}
\delta \mathcal{L}=
\left(-\frac{22}{3}(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu})-7
B_{Y\mu}(\partial^2_a B_{Y}^\mu)
\right)\left[\frac{1}{4(4\pi)^2}\frac{g^2_{B-L}g^2_R}{{g_R^2+g_{B-L}^2}}\ln
\left(\frac{\Lambda^2}{\mu^2_{W_R}}\right)\right]\left[\frac{\delta(I)-\delta(II)}{4}\right]\end{aligned}$$
These diagrams will not lead to the mass corrections due to the factor $[\delta(I)-\delta(II)]$. The field $B_Y$ also interacts with $\chi_L$ and $\phi$, because the $U(1)_Y$ charge is equal to $Q_{Y}= T^3_R+\frac{Y_{B-L}}{2}$, where $Y_{B-L}$ is $U(1)_{B-L}$ hypercharge. These diagrams will have the same structure as diagrams (g) and (h), the only difference will be that $\chi_L$ and $\phi$ will have no mass from the breaking of $SU(2)_R$. The contribution from the fields $\chi_L$ and $\phi_d^{0,+}$ will have the factor $[\delta(I)-\delta(II)]$, so only the loops with $\phi_u^{0,-}$ lead to the nonvanishing result. $$\begin{aligned}
\label{Bmu} \delta
\mathcal{L}=\frac{1}{12(4\pi)^2}\frac{g^2_{B-L}g^2_R}{{g_R^2+g_{B-L}^2}}
\ln
\left(\frac{\Lambda^2}{\mu^2}\right)\left[\frac{\delta(I)+\delta(II)}{4}\right]\left(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right)\nonumber\\
\delta
m^2_{B_{Y,\mu}}=-\frac{1}{12(4\pi)^2}\frac{g^2_{B-L}g^2_R}{{g_R^2+g_{B-L}^2}}\frac{2}{R^2}\ln
\left(\frac{\Lambda^2}{\mu^2}\right).\end{aligned}$$ Now we will calculate the mass corrections for the $B_{Y,a}$ field. At the tree level the mass matrix of the $B_{Y,a}$ arises from $(-\frac{1}{2}(F_{45})^2)$, and it has two eigenstates: massless and massive. The massless one is eaten to become the longitudinal component of the KK excitations of the $B_{Y\mu}$ field, and the massive state behaves like 4D scalar, and is our candidate for dark matter. In our case ($p_4=p_5=\frac{1}{R}$), the $(B_+\equiv\frac{B_4+B_5}{\sqrt{2}})$ is the longitudinal component of the $B_{Y\mu}$, and $B_-\equiv\frac{B_4-B_5}{\sqrt{2}}$ is the massive scalar. Nonvanishing mass corrections will arise only from the loops containing $\phi_u^{0,-}$ fields, the other terms will cancel out exactly in the same way as for the $B_{Y\mu}$ field. $$\begin{aligned}
i\Pi_{ab}=\frac{-i}{4(4\pi)^2}\frac{g^2_{B-L}g^2_R}{{g_R^2+g_{B-L}^2}}[p_c p'_c\delta_{ab}
-p_ap'_b]\ln\left(\frac{\Lambda^2}{\mu^2}\right)\nonumber\\
\label{B_a} \delta
\mathcal{L}=\frac{1}{2}(F_{45})^2\left[\frac{1}{4(4\pi)^2}\frac{g^2_{B-L}g^2_R}{{g_R^2+g_{B-L}^2}}\ln\left(\frac{\Lambda^2}{\mu^2}\right)\right]
\left[\frac{\delta(I)+\delta(II)}{4}\right]\nonumber\\
\delta
m^2_{B_{Y-}}=-\frac{1}{4(4\pi)^2}\frac{g^2_{B-L}g^2_R}{{g_R^2+g_{B-L}^2}}\frac{2}{R^2}\ln\left(\frac{\Lambda^2}{\mu^2}\right)\end{aligned}$$ Comparing equations (\[B\_a\]) and (\[Bmu\]), we see that in the one loop approximation $B_{Y-}$ will be the lighter than $B_{Y\mu}$, so our calculations predict that within the model [@mohapat], dark matter is admixture of the $B_{Y-}$ and $\nu_2$ fields. It is interesting to point out that the same inequality for the radiative corrections to the masses of the gauge bosons was found in the context of model [@ponton].
Conclusion
==========
We studied the one-loop structure in the field theory in six dimensions compactified on the $T_2/(Z_2\times Z_2')$ orbifold. We showed how to take into account boundary conditions on the $T_2/(Z_2\times Z_2')$ orbifold and derived propagators for the fermion, scalar and vector fields. We calculated mass corrections for the fermion and vector fields, and then we applied our results to the lightest stable KK particles in the model [@mohapat]. We showed that the lightest stable modes would be, $B_{Y-}$ and $\nu_2$ fields. These results are important for the phenomelogical predictions of the model.
Acknowledgments {#acknowledgments .unnumbered}
===============
I want to thank R.N.Mohapatra for suggesting the problem and useful discussion, K.Hsieh for comments. This work was supported by the NSF grant PHY-0354401 and University of Maryland Center for Particle and String Theory.
Appendix {#appendix .unnumbered}
=========
In the appendix we will show that contribution of the terms, which do not conserve magnitude of the $|p_a|$, will lead to the operators localized at the fixed points of the orbifold. We will follow the discussion presented in the work of H.Georgi,A.Grant and G.Hailu [@Georgi] and apply it to our case of $T^2/(Z_2\times Z_2')$ orbifold. So let us consider general expression. $$\begin{aligned}
\sum_{p^{\prime}_a=p_a+\frac{m_a}{R}} \frac{1}{2}
\left(1+e^{i\pi(m_4+m_5)}\right)\overline{\psi}(p^{\prime})\Gamma\psi(P)\end{aligned}$$ where $\Gamma$ is some generic operator, $\psi$ is six-dimensional fermion field, and factor $1+e^{i\pi(m_4+m_5)}$ appears because initial and final fields have the same ($Z_2\times Z_2'$) parities. The action in the momentum space will be given by $$\begin{aligned}
S=\sum_{p_a}\frac{1}{(2\pi
R)^2}\sum_{p^{\prime}_a=p_a+\frac{m_a}{R}}
\frac{1}{8}\left({1+\varepsilon_e\varepsilon'_e e^{i(\pi
R)_ap_a}}\right)\left(
{1+e^{i\pi(m_4+m_5)}}\right)\left(1+\varepsilon_i\varepsilon'_ie^{i(\pi
R)_a k_a}\right)\overline{\psi} (p^{\prime})\Gamma\psi(p),\end{aligned}$$ where $\varepsilon_{i,e},~\varepsilon'_{i,e}$ are the $Z_2$ and $Z_2'$ parities for the particles in the internal and external lines of the diagram respectively, and $k_a$ is the momentum of the internal line (we omit integration over the 4D momentum in the expression). Transforming fields $\psi$ to position space we get $$\begin{aligned}
S=\frac{1}{8(2\pi R)^2}\sum_{p_a}\sum_{
p^{\prime}_a=p_a+\frac{m_a}{R}}\int dx_a dx^{\prime}_a
e^{-ip^{\prime}_ax^{\prime}_a+ip_ax_a} \cdot
\nonumber\\
\cdot\left[(1+\varepsilon_e\varepsilon_e' e^{i(\pi
R)_ap_a})(1+\varepsilon_i\varepsilon'_i e^{i(\pi R)_a(p_a\pm
p_a\pm m_a/R )/2})
(1+e^{i\pi(m_4+m_5)})\right]\overline{\psi}(x^{\prime})\Gamma\psi(x),\end{aligned}$$ the upper and lower signs in the expression $(1+\varepsilon_i\varepsilon_i' e^{i(\pi R)_a(p_a\pm p_a\pm m_a
/R)/2})$ correspond to the $k_a=\frac{p_a\pm p'_a}{2}$ in the propagator. Now we can use identities: $$\begin{aligned}
\sum_{p_a} \frac{e^{ip_a(x_a-x^{\prime}_a)}}{(2\pi R)^2}=\delta(x_a-x^{%
\prime}_a) ,\nonumber \\
\sum_{m=-\infty}^\infty e^{\frac{imx}{R}}=\sum_{m=-\infty}^{\infty}\delta(m-%
\frac{x}{2\pi R}).\end{aligned}$$ so $$\begin{aligned}
S=\frac{1}{4}\int d^{(6)}x~ \overline{\psi}(x)\Gamma\psi(x)
\sum_{m_a} \left[\delta(m_a-\frac{x_a}{2\pi R})+
\delta(m_a-\frac{1}{2}-\frac{x_a}{2\pi R })\right. +\nonumber\\
+\left.\left
(\delta(m_a-\frac{1}{4}-\frac{x_a}{2\pi R})
+\delta(m_a+\frac{1}{4}-\frac{x_a}{2\pi R }) \right) \cdot
\begin{cases}
(\varepsilon_i\varepsilon_i')(\varepsilon_e\varepsilon_e')~
\text{for } k_a=\frac{p_a+ p'_a}{2} \\
\varepsilon_i\varepsilon_i'~ \text{for } k_a=\frac{p_a-
p'_a}{2}\end{cases} \right] \label{local}\end{aligned}$$ So the brane terms will be localized at the points $(0,0),(\pi
R,\pi R), (\pm\pi R/2,\pm\pi R/2)$.
[99]{} I. Antoniadis, Phys.Lett. [**B 246**]{}:377-384 (1990). I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. [**B 436**]{}: 257 (1998); N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. [**B 429**]{}: 263 (1998)
D. Cremades, L.E.Ibanez and F.Marchesano, Nucl. Phys. B 643 (2002) 93, C. Kokorelis, Nucl. Phys. B 677 (2004) 115,
T. Appelquist, H. C. Cheng and B. A. Dobrescu, “Bounds on universal extra dimensions,” Phys. Rev. [**D 64**]{}, 035002 (2001).
D.Hooper, S.Profumo, \[hep-ph/0701197\].
T. Appelquist, H-U. Yee, Phys.Rev. [**D 67**]{}:055002 (2003)
T. Appelquist, B.A. Dobrescu, E. Ponton, H-U. Yee, Phys.Rev.Lett. 87:181802 (2001) T. Appelquist, B.A. Dobrescu, E. Ponton, H-U. Yee, Phys.Rev. [**D 65**]{}, 105019 (2002).
J. C. Pati and A. Salam, Phys. Rev. [**D 10**]{}, 275 (1974); R. N. Mohapatra and J. C. Pati, Phys. Rev. [**D 11**]{}, 566, 2558 (1975); G. Senjanović and R. N. Mohapatra, Phys. Rev. [**D 12**]{}, 1502 (1975).
R.N.Mohapatra and A.Perez-Lorenzana, Phys.Rev. [**D 67**]{}:075015 (2003)
G. Servant and T. M. P. Tait, Nucl. Phys. [**B 650**]{}, 391 (2003).
Ken Hsieh, R.N. Mohapatra, Salah Nasri, Phys. Rev. [**D 74**]{}:066004 (2006); Ken Hsieh, R.N. Mohapatra, Salah Nasri, JHEP 0612:067 (2006)
H. C. Cheng, J. L. Feng and K. T. Matchev, Phys. Rev. Lett. [**89**]{}, 211301 (2002); K. Kong and K. T. Matchev, JHEP [**0601**]{}, 038 (2006); F. Burnell and G. D. Kribs, Phys. Rev. [**D 73**]{}, 015001 (2006); M. Kakizaki, S. Matsumoto and M. Senami, Phys. Rev.[**D 74**]{}: 023504 (2006) ; S.Matsumoto, M. Senami, Phys.Lett.B633:671-674 (2006) ; M.Kakizaki, S. Matsumoto, Y. Sato, M. Senami, Phys.Rev.[**D 71**]{}:123522 (2005) ; S. Matsumoto, J. Sato, M. Senami, M. Yamanaka, Phys.Lett.B647:466-471 (2007) ; S. Matsumoto, J. Sato, M. Senami, M. Yamanaka, arXiv:0705.0934\[hep-ph\].
Howard Georgi, Aaron K. Grant, Girma Hailu, Phys.Lett.B506:207-214,(2001) Hsin-Chia Cheng, Konstantin T. Matchev, Martin Schmaltz, Phys.Rev. [**D 66**]{}, 036005, (2002). Eduardo Ponton, Lin Wang, JHEP 0611:018(2006);B. A. Dobrescu and E. Pont´on, JHEP 0403, 071 (2004); G. Burdman, B. A. Dobrescu and E. Ponton\[hep-ph/0506334\].
G. von Gersdorff, N. Irges, M. Quiros Nucl.Phys. B 635:127-157,(2002); G. von Gersdorff, N. Irges, M. Quiros Phys.Lett. B 551:351-359,(2003) F. del Aguila, M.Perez-Victoria, J.Santiago, JHEP 0302(2003) 051; F. del Aguila, M.Perez-Victoria, J.Santiago, JHEP 0610(2006 )056;
[^1]: Email: aazatov@umd.edu
[^2]: We are assuming that at the cut off scale brane like terms are small, and that one loop brane terms are small compared to the tree level bulk lagrangian, so to find mass corrections we can use unperturbed KK decomposition and ignore KK mixing terms , in this approximation our results coincide with the results presented in [@aguilla].
|
---
abstract: 'We study the entanglement of multipartite quantum states. Some lower bounds of the multipartite concurrence are reviewed. We further present more effective lower bounds for detecting and qualifying entanglement, by establishing functional relations between the concurrence and the generalized partial transpositions of the multipartite systems.'
author:
- 'Xue-Na Zhu$^{1}$'
- 'Ming Li$^{2}$'
- 'Shao-Ming Fei$^{3}$'
title: Lower Bounds of Concurrence for Multipartite States
---
1. Introduction
================
Entanglement is a distinctive feature of quantum mechanics, and an indispensable ingredient in various kinds of quantum information processing applications such as quantum computation [@di], quantum teleportation [@teleportation], dense coding [@dense], quantum cryptographic schemes [@schemes], entanglement swapping [@swapping] and remote states preparation (RSP) [@RSP1]. These effects based on quantum entanglement have been demonstrated in many pioneering experiments.
An important theoretical challenge in the theory of quantum entanglement is to give a proper description and quantification of quantum entanglement for given quantum states. For bipartite quantum systems, entanglement of formation (EOF) [@eof] and concurrence [@concurrence; @anote] are two well defined quantitative measures of quantum entanglement. For two-qubit systems it has been proved that EOF is a monotonically increasing function of the concurrence and an elegant formula for the concurrence was derived analytically by Wootters [@wotters]. However with the increasing dimensions of the subsystems the computation of EOF and concurrence become formidably difficult. A few explicit analytic formulae for EOF and concurrence have been found only for some special symmetric states [@Terhal-Voll2000; @fjlw; @fl; @fwz; @Rungta03].
The first analytic lower bound of concurrence that can be tightened by numerical optimization over some parameters was derived in [@167902]. In [@Chen-Albeverio-Fei1; @chen] analytic lower bounds on EOF and concurrence for any dimensional mixed bipartite quantum states have been presented by using the positive partial transposition (PPT) and realignment separability criteria. These bounds are exact for some special classes of states and can be used to detect many bound entangled states. In [@breuer] another lower bound on EOF for bipartite states has been presented from a new separability criterion [@breuerprl]. A lower bound of concurrence based on local uncertainty relations (LURs) criterion is derived in [@vicente]. This bound is further optimized in [@zhang]. In [@edward; @ou] the authors presented lower bounds of concurrence for bipartite systems in terms of a different approach. It has been shown that this lower bound has a close relationship with the distillability of bipartite quantum states. In Ref. [@X.; @S.; @Li] an explicit analytical lower bound of concurrence is obtained by using positive maps, which is better than the ones in Refs. [@chen; @breuer] in detecting some quantum entanglement. These bounds give rise to a good quantitative estimation of concurrence. They are supplementary in detecting quantum entanglement for bipartite systems.
When referring to multipartite systems, we focus on multipartite concurrence, since the EOF is only defined for bipartite systems. With the increasing of the number of quantum systems, quantifying multipartite entanglement has become a much difficult task and only few results are obtained. In this paper, we first give a brief review of the lower bounds for multipartite concurrence in section 2. We present some new lower bounds of multipartite concurrence in sections 3-5. These new bounds give rise to better estimations of multipartite concurrence and are more effective in detecting multipartite entanglement. Conclusions and remarks are given in section 6.
2. Lower bounds of multipartite concurrence
============================================
We first recall the definition and some lower bounds of the multipartite concurrence. Let ${\cal {H}}_{i}$, $i=1,...,N$, be Hilbert spaces with $d_i$ dimensions. The concurrence of an $N$-partite state $|\psi\ra\in {\cal {H}}_{1}\otimes{\cal
{H}}_{2}\otimes\cdots\otimes{\cal {H}}_{N}$ is defined by [@multicon] $$\begin{aligned}
\label{xxxx}
C_{N}(|\psi\ra\la\psi|)=2^{1-\frac{N}{2}}\sqrt{(2^{N}-2)-\sum_{\alpha}{\rm
Tr}[\rho_{\alpha}^{2}]},\end{aligned}$$ where $\alpha$ labels all different reduced density matrices.
Up to constant factor (\[xxxx\]) can be also expressed in another way. Set $d_i=d, i=1,2,...,N$. The $N$-partite pure state $|\psi\ra$ is generally of the form, $$\begin{aligned}
\label{purestate}
|\psi\ra=\sum\limits_{i_{1},i_{2},\cdots,
i_{N}=1}^{d}a_{i_{1},i_{2},\cdots, i_{N}}|i_{1},i_{2},\cdots,
i_{N}\ra,\quad a_{i_{1},i_{2},\cdots, i_{N}}\in \Cb,\end{aligned}$$ with $\sum\limits_{i_{1},i_{2},\cdots,i_{N}=1}^{d}
a_{i_{1},i_{2},\cdots, i_{N}}a_{i_{1},i_{2},\cdots, i_{N}}^\ast=1$.
Let $\alpha$ and $\alpha^{'}$ (resp.$\beta$ and $\beta^{'}$) be subsets of the subindices of $a$, associated to the same sub Hilbert spaces but with different summing indices. $\alpha$ (or $\alpha^{'}$) and $\beta$ (or $\beta^{'}$) span the whole space of the given sub-indix of $a$. The generalized concurrence of $|\psi\ra$ is then given by [@anote], $$\begin{aligned}
\label{defmulticon}
C_{d}^{N}(|\psi\ra)=\sqrt{\frac{d}{2m(d-1)}\sum\limits_{p}
\sum\limits_{\{\alpha,\alpha^{'},\beta,\beta^{'}\}}^{d}
|a_{\alpha\beta}a_{\alpha^{'}\beta^{'}}-a_{\alpha\beta^{'}}a_{\alpha^{'}\beta}|^{2}},\end{aligned}$$ where $m=2^{N-1}-1$, $\sum\limits_{p}$ stands for the summation over all possible combinations of the indices of $\alpha$ and $\beta$.
For a mixed multipartite quantum state, $\rho=\sum_{i}p_{i}|\psi_{i}\ra\la\psi_{i}|\in{\cal
{H}}_{1}\otimes{\cal {H}}_{2}\otimes\cdots\otimes{\cal {H}}_{N}$, the corresponding concurrence is given by the convex roof: $$\begin{aligned}
\label{defe}
C_{N}(\rho)=\min_{\{p_{i},|\psi_{i}\}\ra}\sum_{i}p_{i}C_{N}(|\psi_{i}\ra).\end{aligned}$$
In [@gao] the lower bound of concurrence for tripartite systems has been studied by exploring the connection between the generalized partial transposition criterion and concurrence. Let ${\cal{H}}_A$, ${\cal{H}}_B$ and ${\cal{H}}_C$ be three finite dimensional Hilbert spaces associated with the subsystems $A$, $B$ and $C$, with dimensions $\dim A=m$, $\dim B=n$ and $\dim C=p$. Define that $T_{r_k}$ (resp. $T_{c_k}$), $k=A,B,C,AB,BC,AC$ to be the row (resp. column) transpositions with respect to the subsystems $k$. Consider three classes: 1) $y_i= \{c_k, r_k\}$, where $i=1,2,3$ for $k=A,B,C$ respectively; 2) $y_4= \{c_A, r_{BC}\}$, $y_5= \{c_{AB}, r_{C}\}$, $y_6=
\{c_{AC}, r_{B}\}$; 3) $y_7=\{c_A, r_B\}$, $y_8=\{c_A, r_C\}$, $y_9=\{c_B, r_C\}$.
For any $m\otimes n\otimes p$ $(m\leq n, p)$ tripartite mixed quantum state $\rho$, the concurrence $C(\rho)$ defined in (\[xxxx\]) satisfies $$\begin{aligned}
\label{thgao}
&&C_{N}(\rho)\\
&&\geq\max\{\sqrt{\frac{1}{m(m-1)}}(||\rho^{T_{y_a}}||-1),
\sqrt{\frac{1}{n(n-1)}}(||\rho^{T_{y_b}}||-1),\sqrt{\frac{1}{r(r-1)}}(||\rho^{T_{y_c}}||-1)\}.\end{aligned}$$ where $q=\min(n,mp)$ and $r=\min(p,mn), y_a=y_1$ or $y_4, y_b=y_2$ or $y_6, y_c=y_3$ or $y_5$.
In [@mintert; @aolita] the definition of multipartite concurrence defined in (\[xxxx\]) is re-expressed as $C(|\psi\ra)=\sqrt{\la\psi|\otimes\la\psi|
A|\psi\ra\otimes|\psi\ra})$, with $A=4(P_+-P_+^{(1)}\otimes\cdots\otimes P_+^{(N)})$. $P_+$ (resp. $P_-$) is the projector ont o the globally symmetric (reps. antisymmetric) space. The authors have obtained that the multipartite concurrence satisfies $$\begin{aligned}
\label{thmintert}
[C_N(\rho)]^2\geq Tr(\rho\otimes\rho V),\end{aligned}$$ with $V=4(P_+-P_+^{(1)}\otimes\cdots\otimes P_+^{(N)}-(1-2^{1-N})P_-)$.
In [@zhang; @mintert; @aolita], it is shown that the multipartite concurrence defined in (\[xxxx\]) satisfies $$\begin{aligned}
\label{upperlowerboundo}
C_{N}(\rho)\geq \sqrt{(4-2^{3-N}){\rm
Tr}\{\rho^{2}\}-2^{2-N}\sum_{\alpha}{\rm Tr}\{\rho_{\alpha}^{2}\}}.\end{aligned}$$
We derived an effective lower bound for multipartite quantum systems in [@jpa]. First for tripartite case,
\[th1\] For an arbitrary $d\times d\times d$ mixed state $\rho$ in ${\cal{H}}\otimes {\cal{H}}\otimes {\cal{H}}$, the concurrence $C(\rho)$ defined in (\[defmulticon\]) satisfies $$\begin{aligned}
\label{11o}
\tau_{3}(\rho)\equiv\frac{d}{6(d-1)}\sum_{\alpha}^{\frac{d^{2}(d^{2}-1)}{2}}\sum_{\beta}^{\frac{d(d-1)}{2}}
[({C_{\alpha\beta}^{12|3}(\rho)})^{2}
+({C_{\alpha\beta}^{13|2}(\rho)})^{2}+({C_{\alpha\beta}^{23|1}(\rho)})^{2}]\leq C^{2}(\rho),\end{aligned}$$ where $\tau_{3}(\rho)$ is a lower bound of $C(\rho)$, $$\begin{aligned}
C_{\alpha\beta}^{12|3}(\rho)=\max\{0,\lambda(1)_{\alpha\beta}^{12|3}-\lambda(2)_{\alpha\beta}^{12|3}
-\lambda(3)_{\alpha\beta}^{12|3}-\lambda(4)_{\alpha\beta}^{12|3}\},\end{aligned}$$ $\lambda(1)_{\alpha\beta}^{12|3}, \lambda(2)_{\alpha\beta}^{12|3},
\lambda(3)_{\alpha\beta}^{12|3}, \lambda(4)_{\alpha\beta}^{12|3}$ are the square roots of the four nonzero eigenvalues, in decreasing order, of the non-Hermitian matrix $\rho\widetilde{\rho}_{\alpha\beta}^{12|3}$ with $\widetilde{\rho}_{\alpha\beta}^{12|3}=S_{\alpha\beta}^{12|3}\rho^{*}S_{\alpha\beta}^{12|3}$. $C_{\alpha\beta}^{13|2}(\rho)$ and $C_{\alpha\beta}^{23|1}(\rho)$ are defined in a similar way to $C_{\alpha\beta}^{12|3}(\rho)$.
Theorem $\ref{th1}$ can be directly generalized to arbitrary multipartite case.
\[th2\] For an arbitrary $N$-partite state $\rho\in
{\cal{H}}\otimes {\cal{H}}\otimes...\otimes{\cal{H}}$, the concurrence defined in ($\ref{defmulticon}$) satisfies: $$\begin{aligned}
\label{tau}
\tau_{N}(\rho)\equiv\frac{d}{2m(d-1)}\sum_{p}\sum_{\alpha\beta}(C_{\alpha\beta}^{p}(\rho))^{2}\leq
C^{2}(\rho),\end{aligned}$$ where $\tau_{N}(\rho)$ is the lower bound of $C(\rho)$, $\sum\limits_{p}$ stands for the summation over all possible combinations of the indices of $\alpha,\beta$, $C_{\alpha\beta}^{p}(\rho)=\max\{0,
\lambda(1)_{\alpha\beta}^{p}-\lambda(2)_{\alpha\beta}^{p}
-\lambda(3)_{\alpha\beta}^{p}-\lambda(4)_{\alpha\beta}^{p}\}$, $\lambda(i)_{\alpha\beta}^{p}$, $i=1, 2, 3, 4$, are the square roots of the four nonzero eigenvalues, in decreasing order, of the non-Hermitian matrix $\rho\widetilde{\rho}_{\alpha\beta}^{p}$ where $\widetilde{\rho}_{\alpha\beta}^{p}=S_{\alpha\beta}^{p}\rho^{*}S_{\alpha\beta}^{p}$.
In [@rmp] we further obtained lower bound of multipartite concurrence by bipartite partitions of the whole quantum systems. For a pure N-partite quantum state $|\psi\ra\in {\mathcal
{H}}_{1}\otimes{\mathcal {H}}_{2}\otimes\cdots\otimes{\mathcal
{H}}_{N}$, $dim {\mathcal {H}}_{i}=d_i$, $i=1,...,N$, the concurrence of bipartite decomposition between subsystems $12\cdots M$ and $M+1\cdots N$ is defined by $$\begin{aligned}
\label{xx}
C_{2}(|\psi\ra\la\psi|)=\sqrt{2(1-{\rm Tr}\{\rho_{{1}{2}\cdots
{M}}^{2}\})},\end{aligned}$$ where $\rho_{{1}{2}\cdots {M}}^{2}={\rm Tr}_{{M+1}\cdots
{N}}\{|\psi\ra\la\psi|\}$ is the reduced density matrix of $\rho=|\psi\ra\la\psi|$ by tracing over the subsystems $M+1\cdots{N}$.
For a mixed multipartite quantum state, $\rho=\sum_{i}p_{i}|\psi_{i}\ra\la\psi_{i}| \in {\mathcal
{H}}_{1}\otimes{\mathcal {H}}_{2}\otimes\cdots\otimes{\mathcal
{H}}_{N}$, the corresponding concurrence of (\[xx\]) is then given by the convex roof: $$\begin{aligned}
\label{def1}
C_{2}(\rho)=\min_{\{p_{i},|\psi_{i}\}\ra}\sum_{i}p_{i}C_{2}(|\psi_{i}\ra\la\psi_{i}|),\end{aligned}$$ which will be called the bipartite concurrence.
The relation between the concurrences in (\[defe\]) and the bipartite concurrence in (\[def1\]) can be directly given by the following theorem.
For a multipartite quantum state $\rho\in
{\mathcal {H}}_{1}\otimes{\mathcal
{H}}_{2}\otimes\cdots\otimes{\mathcal {H}}_{N}$ with $N\geq 3$, the following inequality holds, $$\begin{aligned}
\label{obound}
C_{N}(\rho)\geq\max 2^{\frac{3-N}{2}}C_{2}(\rho),\end{aligned}$$ where the maximum is taken over all kinds of bipartite concurrence.
In terms of the lower bounds of bipartite concurrence derived from PPT, realignment of the density matrix, local uncertainty relation and the covariance matrix separability criterion in [@chen; @vicente; @zhang], and (\[obound\]), we get the following theorem.
For any N-partite quantum state $\rho$, we have: $$\begin{aligned}
\label{newlowerbound}
C_{N}(\rho)\geq2^{\frac{3-N}{2}}\max\{B1,B2,B3\},\end{aligned}$$ where $$\begin{aligned}
B1&=&\max_{\{i\}}\sqrt{\frac{2}{M_{i}(M_{i}-1)}}\left[\max(||{\mathcal
{T}}_{A}(\rho^{i})||,||R(\rho^{i})||)-1\right],\\
B2&=&\max_{\{i\}}\frac{2||C(\rho^{i})||- (1-{\rm
Tr}\{(\rho^{i}_{A})^{2}\})-(1-{\rm Tr}\{(\rho^{i}_{B})^{2}\})}
{\sqrt{2M_{i}(M_{i}-1)}},\\
B3&=&\max_{\{i\}}\sqrt{\frac{8}{M_{i}^{3}N_{i}^{2}(M_{i}-1)}}
(||T(\rho^{i})||-\frac{\sqrt{M_{i}N_{i}(M_{i}-1)(N_{i}-1)}}{2}),\end{aligned}$$ $\rho^i$ are all possible bipartite decompositions of $\rho$, $M_{i}=\min{\{d_{s_{1}}d_{s_{2}}\cdots d_{s_{m}},
d_{s_{m+1}}d_{s_{m+2}}\cdots d_{s_{N}}\}}$, $N_{i}=\max{\{d_{s_{1}}d_{s_{2}}\cdots d_{s_{m}},
d_{s_{m+1}}d_{s_{m+2}}\cdots d_{s_{N}}\}}$.
3. Improved lower bounds of the multipartite concurrence
=========================================================
In this section, we will derive a new bound for multipartite quantum systems by using the following lemma.
For a bipartite density matrix $\rho\in H_A\otimes
H_B$. one has [@zhang] $$\label{5}
1-Tr\{\rho_{AB}^2\}\geq(1-Tr\{\rho_{A}^2\})-(1-Tr\{\rho_{B}^2\}),$$ $$\label{6}
1-Tr\{\rho_{AB}^2\}\geq(1-Tr\{\rho_{B}^2\})-(1-Tr\{\rho_{A}^2\}),$$ where $\rho_{A|B}=Tr_{A}\{\rho_{B}\}$, $\rho_{B}=Tr_{B|A}\{\rho_{A}\}$.
For a multipartite quantum state $\rho\in
H_1\otimes H_2\otimes ... \otimes H_N$ with $N\geq3$, the following inequality holds: $$\label{7}
C_N(\rho)\geq\max_{\{M=1,2,...,N-1\}}\left\{\left(2^{\frac{1-N}{2}}\sqrt{2^{N-M}+2^{M}-2}\right)C_2(\rho_M)\right\},$$ where the maximum takes over all kinds of bipartite concurrences.
[**Proof.**]{} For a pure multipartite state $ \vert\varphi\rangle\in
{\cal {H}}_{1}\otimes{\cal {H}}_{2}\otimes\cdots\otimes{\cal {H}}_{N}$, one has $Tr\{\rho_{12...M}^2\}=Tr\{\rho_{M+1...N}^2\}$ for all $M={1,2,...,N-1}$.
From (\[5\]) and (\[6\]), we obtain $$1-Tr{\rho_{12...Mi_1...i_p}^2}\geq(1-Tr{\rho_{12...M}^2})-(1-Tr{\rho_{i_1...i_p}^2}),$$ and $$1-Tr{\rho_{j_1...j_qM+1...N}^2}\geq(1-Tr{\rho_{M+1...N}^2})-(1-Tr{\rho_{j_1...j_q}^2}),$$ where $M+1\leq i_1<...<i_p\leq N$, $p\leq N-M-1$ and $1\leq
j_1<...<j_q\leq M$, $q\leq M-1$.
From the above inequalities, we have $$\begin{aligned}
C_N^2(\vert\varphi\rangle
\langle\varphi\vert)&&=2^{2-N}\left[(2^N-2)-\sum_\alpha
Tr{\rho_\alpha^2}\right]
=2^{2-N}\left(\sum_{k=1}^{2^N-2}(1-Tr{\rho_k^2)}\right)\\[1mm]
&&\geq2^{2-N}\left\{(2^{N-M}-1)(1-Tr{\rho_{12...M}^2})+(2^M-1)(1-Tr{\rho_{M+1...N}^2})\right\}\\[2mm]
&&=2^{2-N}\left\{(2^{N-M}+2^M-2)(1-Tr{\rho_{12...M}^2})\right\}\\[2mm]
&&=2^{2-N}\left\{(2^{N-M}+2^M-2)\frac{C_2(|\varphi\rangle_M\langle\varphi|)}{2}\right\},\end{aligned}$$ i.e. $C_N(\vert\varphi\rangle \langle\varphi\vert)\geq\max_{\{M=1,
2,..., N-1\}} \left(2^{\frac{1-N}{2}}\sqrt{2^{N-M}+2^{M}-2}\right)
C_2(\vert\varphi\rangle_M \langle\varphi\vert).$
Assuming that $\rho=\sum_{i} p_i\vert\varphi_i\rangle \langle\varphi_i\vert$ attains the minimal decomposition of the multipartite concurrence, one has $$\begin{aligned}
&&C_N(\rho)=\sum_{i}p_iC_N(\vert\varphi_i\rangle \langle\varphi_i\vert)\\
&&\qquad\geq2^{\frac{1-N}{2}}\sqrt{2^{N-M}+2^{M}-2}\sum_ip_iC_2(\vert\varphi_i\rangle_M\langle\varphi_i\vert)\\
&&\qquad\geq2^{\frac{1-N}{2}}\sqrt{2^{N-M}+2^{M}-2}\min_{\{p_i,\vert\varphi_i\rangle\}}\sum_ip_iC_2(\vert\varphi_i\rangle_M\langle\varphi_i\vert)\\
&&\qquad=\left(2^{\frac{1-N}{2}}\sqrt{2^{N-M}+2^{M}-2}\right)C_2(\rho_M).\end{aligned}$$ Therefore we have $$C_N(\rho)\geq\max_{\{M=1,2,...,N-1\}}\left\{\left(2^{\frac{1-N}{2}}\sqrt{2^{N-M}+2^{M}-2}\right)C_2(\rho_M)\right\}.$$
4. Functional relations between concurrence and the generalized partial transpositions
=======================================================================================
Let us consider an $N$-qubit state, the generalized $W$ state, $$\label{11}
\vert\varphi\rangle=a_1\vert10...0\rangle+a_2\vert01...0\rangle+...+a_N\vert00...1\rangle.$$
For any $N$-qubit mixed state with decomposition on the generalized $W$ states, $\rho=\sum_{i}
p_i\vert\varphi_i\rangle \langle\varphi_i\vert$, such that $\vert\varphi_i\rangle$ can be written in the form (\[11\]) for all $i$, the concurrence $C(\rho)$ satisfies
$$\label{12}
C(\rho)\geq2^{1-\frac{N}{2}}\max
\left\{|\rho^{T_{\Gamma^1_\alpha}}|\
-1,\max_{M}\left\{\sqrt{\frac{2^{N-M}+2^{M}-2}{4}}(|
\mathcal{R}_{\Gamma^1_\alpha\vert\Gamma^2_\alpha}(\rho)|\
-1)\right\}\right\},$$
where $\Gamma^1_\alpha$, $\Gamma^2_\alpha$ denote two subsets of the indices $\{1, 2, ..., N\}$, $\Gamma^1_\alpha\cap\Gamma^2_\alpha=\emptyset$, $\Gamma^1_\alpha\cup\Gamma^2_\alpha=\{1, 2, ..., N\}, \alpha=1, ...,
d$, $M=(1,2,...,N-1)$ is the number of elements of $\Gamma^1_\alpha$.
[**Proof.**]{} An $N$-qubit $W$ state can be viewed as $d$ different bipartite systems. From the results for bipartite systems [@chen], these $d$ bipartite separations give rise to, respectively $$1-Tr\{\rho^2_{\Gamma^1_\alpha}\}\geq\frac{1}{2}(|
\mathcal{R}_{\Gamma^1_\alpha\vert\Gamma^2_\alpha}(\rho)|\ -1)^2,
\alpha=1, ..., d.$$
Hence $$\begin{aligned}
C(\vert\varphi\rangle \langle\varphi\vert)&&=2^{1-\frac{N}{2}}
\sqrt{d-\sum_{\alpha=1}^{d} Tr\{\rho^2_{\Gamma^1_{\alpha}}\}}\\
&&=2^{1-\frac{N}{2}}\sqrt{\frac{2d-\sum_{\alpha=1}^d Tr\{\rho^2_{\Gamma^1_\alpha}\}-\sum_{\alpha=1}^d Tr\{\rho^2_{\Gamma^2_\alpha}\}}{2}}\\
&&\geq2^{1-\frac{N}{2}}\max_{M}\sqrt{\frac{2^{N-M}+2^{M}-2}{2}\left(1-Tr\{\rho^2_{\Gamma^1_\alpha}\}\right)}\\
&&\geq2^{1-\frac{N}{2}}\max_{M}\sqrt{\frac{2^{N-M}+2^{M}-2}{4}}(|
\mathcal{R}_{\Gamma^1_\alpha\vert\Gamma^2_\alpha}(\rho)|\ -1).\end{aligned}$$
Let $\rho=\sum_{i} p_i\vert\varphi_i\rangle \langle\varphi_i\vert$ attain the minimal decomposition of the multipartite concurrence. Note that $| \mathcal{R}(\rho)|\leq\sum_{i} p_i|
\mathcal{R}(\vert\varphi_i\rangle
\langle\varphi_i\vert)|$ [@chen]. One has $$\begin{aligned}
&&C(\rho)=\sum_{i}p_iC(\vert\varphi_i\rangle \langle\varphi_i\vert)\\
&&\geq 2^{1-\frac{N}{2}}\max_{M}\sqrt{\frac{2^{N-M}+2^{M}-2}{4}}
\sum_{i}p_i(| \mathcal{R}_{\Gamma^1_\alpha\vert\Gamma^2_\alpha}(\vert\varphi_i\rangle \langle\varphi_i\vert)|\ -1)\\
&&\geq 2^{1-\frac{N}{2}}\max_{M}\sqrt{\frac{2^{N-M}+2^{M}-2}{4}}(|
\mathcal{R}_{\Gamma^1_\alpha\vert\Gamma^2_\alpha}(\rho)|\ -1),\end{aligned}$$ From which one gets (\[12\]).
5. Entanglement detecting and estimation of concurrence
========================================================
In this section, we use the above several lower bounds of multipartite concurrence to detect quantum entanglement. We will show by examples that these bounds provide a better estimation of the multipartite concurrence.
An N-partite quantum state $\rho$ is fully separable if and only if there exist $p_{i}$ with $p_{i}\geq0, \sum\limits_{i}p_{i}=1$ and pure states $\rho_{i}^{j}=|\psi_{i}^{j}\ra\la\psi_{i}^{j}|$ such that $\rho=\sum_{i}p_{i}\rho_{i}^{1}\otimes\rho_{i}^{2}\otimes\cdots\otimes\rho_{i}^{N}.$ It is easily verified that for a fully separable multipartite state $\rho$, $\tau_{N}(\rho)$ defined in (\[tau\]) is zero. Thus $\tau_{N}(\rho)>0$ indicates that there must be some kinds of entanglement inside the quantum state, which shows that the lower bound $\tau_{N}(\rho)$ can be used to recognize entanglement.
As an example we consider a tripartite quantum state [@acin], $\rho=\frac{1-p}{8}I_{8}+p|W\ra\la W|$, where $I_{8}$ is the $8\times8$ identity matrix, and $|W\ra=\frac{1}{\sqrt{3}}(|100\ra+|010\ra+|001\ra)$ is the tripartite W-state. By using the generalized correlation matrix criterion presented in [@hassan] the entanglement of $\rho$ is detected for $0.3068 < p \leq 1$. From our theorem, we have that the lower bound $\tau_{3}(\rho)>0$ for $0.2727 < p \leq 1$. Therefore our bound detects entanglement better in this case. If we replace W with GHZ state in $\rho$, the criterion in [@hassan] detects the entanglement of $\rho$ for $0.35355 < p
\leq 1$, while $\tau_{3}(\rho)$ detects, again better, the entanglement for $0.2 < p \leq 1$.
The lower bounds together with some upper bounds can be used to estimate the value of the concurrence. In [@zhang; @mintert; @aolita], it is shown that the upper and lower bound of multipartite concurrence satisfy $$\begin{aligned}
\label{upperlowerbound}
\sqrt{(4-2^{3-N}){\rm Tr}\{\rho^{2}\}-2^{2-N}\sum_{\alpha}{\rm
Tr}\{\rho_{\alpha}^{2}\}}\leq
C_{N}(\rho)\leq\sqrt{2^{2-N}[(2^{N}-2)-\sum_{\alpha}{\rm
Tr}\{\rho_{\alpha}^{2}\}]}.\end{aligned}$$
In fact we can obtain a more effective upper bound for multi-partite concurrence. Let $\rho=\sum\limits_{i}\lambda_{i}|\psi_{i}\ra\la
\psi_{i}|\in {\mathcal
{H}}_{1}\otimes{\mathcal{H}}_{2}\otimes\cdots\otimes{\mathcal
{H}}_{N}$, where $|\psi_{i}\ra$s are the orthogonal pure states and $\sum\limits_{i}\lambda_{i}=1$. We have $$\begin{aligned}
\label{newupperbound}
C_{N}(\rho)=\min_{\{p_{i},|\varphi_{i}\}\ra}\sum_{i}p_{i}C_{N}(|\varphi_{i}\ra\la\varphi_{i}|)
\leq\sum_{i}\lambda_{i}C_{N}(|\psi_{i}\ra\la\psi_{i}|).\end{aligned}$$
We now show that our upper and lower bounds can be better than that in $(\ref{upperlowerboundo})$ by detailed examples.
[*[Example 1:]{}*]{} Consider the $2\times 2\times 2$ Dür-Cirac-Tarrach states defined by [@dur]: $$\rho=\sum_{\sigma=\pm}\lambda_{0}^{\sigma}|\psi_{0}^{\sigma}\ra\la
\psi_{0}^{\sigma}|+\sum_{j=1}^{3}\lambda_{j}(|\psi_{j}^{+}\ra\la\psi_{j}^{+}|+|\psi_{j}^{-}\ra\la\psi_{j}^{-}|),$$ where the orthonormal Greenberger-Horne-Zeilinger (GHZ)-basis $|\psi_{j}^{\pm}\ra\equiv\frac{1}{\sqrt{2}}(|j\ra_{12}|0\ra_{3}\pm|(3-j)\ra_{12}|1\ra_{3})$, $|j\ra_{12}\equiv|j_{1}\ra_{1}|j_{2}\ra_{2}$ with $j=j_{1}j_{2}$ in binary notation. From theorem 2 we have that the lower bound of $\rho$ is $\frac{1}{3}$. If we mix the state with white noise, $\rho(x)=\frac{(1-x)}{8}I_{8}+x\rho,$ by direct computation we have, as shown in FIG. $\ref{fig1}$, the lower bound obtained in $(\ref{upperlowerboundo})$ is always zero, while the lower bound in $(\ref{newlowerbound})$ is larger than zero for $0.425\leq x\leq 1$, which shows that $\rho(x)$ is detected to be entangled at this situation. And the upper bound (dot line) in $(\ref{upperlowerboundo})$ is much larger than the upper bound we have obtained in $(\ref{newupperbound})$ (solid line).
Actually, our new lower bound in (\[7\]) is different from the lower bound in (\[obound\]), which can be seen from the following example.
[*[Example 2:]{}*]{} Consider the generalized GHZ state: $\vert\varphi\rangle=\cos\theta\vert00...0\rangle+\sin\theta\vert11...1\rangle.$ It is easy to obtain that $Tr\rho^2_{i_1,i_2,...,i_m}=1-2\sin^2{\theta}\cos^2{\theta}$ for all $i_1\not=i_2\not=...\not=i_m\in{1,2,...,N}$. Hence we have by definition $C(\vert\varphi\rangle)=2^{1-\frac{N}{2}}\sqrt{(2^N-2)(2\sin^2{\theta}\cos^2{\theta})}.$ By our new lower bound in (\[7\]), we get $$\begin{aligned}
C_N(\rho)&&\geq\max_{\{M=1,2,...,N-1\}}\left\{\left(2^{\frac{1-N}{2}}\sqrt{2^{N-M}+2^{M}-2}\right)C_2(\rho_M)\right\}\\
&&=\max_{\{M=1,2,...,N-1\}}\left\{\left(2^{\frac{1-N}{2}}\sqrt{2^{N-M}+2^{M}-2}\right)\sqrt{4\sin^2{\theta}\cos^2{\theta}}\right\}\\\end{aligned}$$
For example, $N=4$, we get $C_N(\vert\varphi\rangle
\langle\varphi\vert)=\sqrt{7 \sin^2{\theta}\cos^2{\theta}}$. From our bound we have $C_N(\vert\varphi\rangle
\langle\varphi\vert)\geq\sqrt{4\sin^2{\theta}\cos^2{\theta}}>\sqrt{2\sin^2{\theta}\cos^2{\theta}}$, where $\sqrt{2\sin^2{\theta}\cos^2{\theta}}$ is the bound from [@rmp].
6. Remarks and conclusions
===========================
By establishing functional relations between the concurrence and the generalized partial transpositions of the multipartite systems, we have presented some effective lower bounds for detecting and qualifying entanglement for multipartite systems. These bounds can be also served as separability criteria. They detect entanglement of some states better than some separability criteria.
Generally, to derive a lower bound of multipartite concurrence, we calculate the multipartite concurrence for pure states first. Then by using the convex property of the quantities in the calculation one can directly find a tight lower bound. Mintert et al. in [@mintert7] have derived a precise lower bound for bipartite concurrence, which detects mixed entangled states with a positive partial transpose. It would be interesting and challenging to use this approach to derive a lower bound for multipartite concurrence.
[**Acknowledgments**]{} This work is supported by the NSFC 10875081 and PHR201007107.
[99]{} See, for example, D.P. Di Vincenzo, Science 270, 255(1995).
C.H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres and W.K. Wootters, Phys. Rev. Lett. 70, 1895 (1993);\
S.Albeverio, S.M.Fei and Y.L.Yang, Phys. Rev. A 66, 012301(2002).
C.H.Bennett and S.J.Wiesner, Phys. Rev. Lett. 69, 2881(1992).
A. Ekert, Phys. Rev. Lett. 67, 661(1991);\
D. Deutsch et al, Phys. Rev. A 56, 1163(1997).
M. Żukowski, A. Zeilinger, M.A. Horne and A.K. Ekert, Phys. Rev. Lett. 71, 4287(1993);\
S. Bose, V. Vedral and P.L. Knight, Phys. Rev. A 57, 822(1998); 60, 194(1999);\
B.S. Shi, Y.K. Jiang, G.C. Guo, Phys. Rev. A 62, 054301(2000).
C.H. Bennett, D.P. DiVincenzo, P.W. Shor, J.A. Smolin, B.M. Terhal and W.K. Wootter, Phys. Rev. Lett. 87, 077902(2001);\
D. W. Leung and P. W. Shor, Phys. Rev. Lett. 90, 127905(2003).
C. H. Bennett, D. P. DiVincenzo and J. A. Smolin, et al. Phys. Rev. A 54, 3824(1996).\
M. B. Plenio and S. Virmani, Quant. Inf. Comp. 7, 1(2007).
A. Uhlmann Phys. Rev. A 62 032307(2000);\
P. Rungta, V. Bu$\breve{z}$ek, and C. M. Caves, et al. Phys. Rev. A 64, 042315(2001).
S. Albeverio and S. M. Fei, J. Opt. B: Quantum Semiclass Opt, 3, 223-227(2001).
W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
Terhal B M, Vollbrecht K G H, Phys Rev Lett, 85, 2625-2628(2000).
S.M. Fei, J. Jost, X.Q. Li-Jost and G.F. Wang, Phys. Lett. A 310, 333-338(2003).
S.M. Fei and X.Q. Li-Jost, Rep. Math. Phys. 53, 195-210(2004).
S.M. Fei, Z.X. Wang and H. Zhao, Phys. Lett. A 329, 414-419(2004).
P. Rungta and C.M. Caves, Phys Rev A 67, 012307(2003).
F. Mintert, M. Kus, A. Buchleitner, Phys. Rev. Lett. 92, 167902(2004).
K. Chen, S. Albeverio and S.M. Fei, Phys. Rev. Lett. 95, 210501(2005).
K. Chen, S. Albeverio, and S. M. Fei, Phys. Rev. Lett. 95, 040504(2005).
H.P. Breuer, J. Phys. A 39, 11847(2006).
H.P. Breuer, Phys. Rev. Lett. 97, 080501(2006).
J. I. de Vicente, Phys. Rev. A 75, 052320(2007).
C. J. Zhang, Y. S. Zhang, and S. Zhang, et al. Phys. Rev. A 76, 012334(2007).
E. Gerjuoy, Phys. Rev. A 67, 052308(2003).
Y. C. Ou, H. Fan and S. M. Fei, Phys. Rev. A 78, 012311(2008).
X.S. Li, X.H. Gao, and S.M. Fei, Phys. Rev. A **83**, 034303 (2011).
L. Aolita and F. Mintert, Phys. Rev. Lett. 97, 050501(2006);\
A. R. R. Carvalho, F. Mintert, and A. Buchleitner, Phys. Rev. Lett. 93, 230501(2004).
X. H. Gao, S. M. Fei and K. Wu, Phys. Rev. A 74, 050303(R) (2006).
F. Mintert and A. Buchleitner, Phys. Rev. Lett. 98, 140505 (2007).
L. Aolita, A. Buchleitner, and F. Mintert, Phys. Rev. A 78, 022308(2008).
M. Li, S.M. Fei and Z.X. Wang, J. Phys. A, Math. Theor.£¬42, 145303£¨2009£©.
M. Li, S.M. Fei and Z.X. Wang, Rep. Math. Phys.£¬65, 2£¨2010£©.
A. Ac$\acute{i}$n, D. Bruss, M. Lewenstein, and A. Sanpera, Phys. Rev. Lett. 87, 040401(2001).
A. S. M. Hassan and P. S. Joag, Quant. Inf. Comp. 8, 0773-0790(2008).
W. Dü r, J. I. Cirac, and R. Tarrach, Phys. Rev. Lett. 83, 3562 (1999).
F. Mintert, M. Kús, and A. Buchleitner, Phys. Rev. Lett. 92, 167902 (2004).
|
---
address: |
Theoretisch-Physikalisches Institut\
Universität Jena, Fröbelsteig 1\
D-07743 Jena, Germany
author:
- 'Christopher Ford [^1]'
title: 'MULTISCALE RENORMALIZATION AND DECOUPLING [^2]'
---
Ł ł
Introduction
============
Let us consider a very simple problem in perturbative quantum field theory, the computation of the effective potential in the four-dimensional Higgs-Yukawa model defined by the Lagrangian =(\_)\^2-m\^2\^2 -\^4+|i/+g|+Ł. Here $\L$ is a “cosmological constant” term which enters non-trivially into the renormalization group equation for the effective potential [@kast] (see also Christian Wiesendanger’s talk). It is well known how to perform a loopwise perturbative expansion of the effective potential [@jac], $V(\p)=V^{(0)}(\p)+\hbar V^{(1)}(\p)+\hbar^2V^{(2)}(\p)+...\,
$. Using dimensional regularization together with (modified) minimal subtraction gives V\^[(0)]{}()&=&\^4+m\^2\^2+Ł,\
V\^[(1)]{}()&=&-. Notice the logarithmic terms in the one-loop potential. We have a $\log \frac{m^2+\h\l\p^2}{\m^2}$ due to the “Higgs” loop and $\log \frac{g^2\p^2}{\m^2}$ associated with the fermionic contribution to the one-loop potential. The two-loop potential, $V^{(2)}$, is quadratic in these logarithms, and in general the $n$-loop potential is a $n$th order polynomial in the two logarithms. Thus, for believable perturbation theory one must not only have “small” couplings $\hbar \lambda$, $\hbar g^2$, but the two logarithms must also be small. As was explained a long time ago by Coleman and Weinberg (CW) [@cw] one must make a ($\p$-dependent) choice of $\m$ such that the logarithms are not too large. To relate the renormalized parameters at different scales one uses the renormalization group (RG). The CW procedure of RG “improving” the potential is equivalent to a resummation of the large logarithms in the perturbation series.
However, it is not too difficult to see that if $m^2+\h\l\p^2>>g^2\p^2$ (the heavy Higgs case) or $g^2\p^2>>m^2+\h\l\p^2$ (a heavy fermion) *there is no choice of $\m$ that will simultaneously render both logarithms small. Thus, we are only able to implement the CW method when $m^2+\l\p^2\sim g^2\p^2$, ie. when we have essentially a one-scale problem.*
Multiscale renormalization
===========================
We have seen that in the $\MS$ scheme the RG equation is not powerful enough to deal with multiscale problems. In a single scale problem one is able to track the relevant scale with the $\MS$ RG scale $\mu$, whereas in a 2-scale model it is not possible to track two scales with a *single RG scale. In order to deal with the 2-scale case it seems natural to seek a *2-scale version of $\MS$. This 2-scale scheme, should be as similar to $\MS$ as possible, but with *two RG scales $\ka_1$ and $\ka_2$ instead of one $\MS$ scale $\m$. Here $\ka_1$ and $\ka_2$ should “track” the Higgs and fermionic scale, respectively. Using such a 2-scale scheme one should be able to sum up the two logarithms in the perturbation series. For attempts to deal with this problem while retaining a single scale RG see refs. [@onescale].***
How do we define a 2-scale subtraction scheme? In fact, a multiscale renormalization scheme has already been proposed by Einhorn and Jones (EJ) [@ej] which has some of the properties we seek. To motivate their idea, let us look at the bare Lagrangian for our Higgs-Yukawa problem written in terms of the usual $\MS$ renormalized parameters. \_&=&Z\_(\_)\^2- Z\_Z\_[m\^2]{} m\^2 \^2- Z\_\^2 Z\_ł\^ ł\^4\
&& + Z\_|i/+ Z\_Z\_g [Z\_]{}\^ \^ g|+ Ł+(Z\_Ł-1)\^[-]{}m\^4ł\^[-1]{}, \[3\] where $\e=4-d$ is the dimensional continuation parameter, and all the $Z_.$ factors have the form $Z_.=1+\hbox{pole terms only}$. Notice that the $\MS$ RG scale $\mu$ enters eqn. (\[3\]) in three places. The EJ idea was simply to replace the three occurrences of $\m$ in (\[3\]) with three *independent RG scales $\k_1$, $\ka_2$, $\ka_3$, so that ł\_=\_1\^Z\_łł, g\_=\_2\^ Z\_g g,Ł\_=+\_3\^[-]{}(Z\_Ł-1) m\^4ł\^[-1]{}. As in standard $\MS$, the $Z_.$ factors are defined by the requirement that the effective action is finite when written in terms of the renormalized parameters and the restriction that the $Z_.$ factors have the form $Z_.=1+\hbox{pole terms only}$. Note that the $Z_.$ factors will *not be the same as the $\MS$ $Z_.$ factors (except where $\k_1=\k_2=\k_3$). In the EJ scheme, the $Z_.$’s will contain logarithms of the RG scale ratios.**
We now have three separate RG equations associated with the independent variations of the three RG scales. We also have three sets of beta functions \[5\] \_i\_=\_i i=1,2,3.and similarly for the other parameters. It is straightforward to compute the one-loop beta functions in the EJ scheme: $${}_2\be_g=\frac{5\hbar g^3}{3(4\pi)^2},\quad {}_1\be_g={}_3\beta_g=0,$$ \[6\] \_1\_ł=(3ł\^2+48g\^4),\_2\_ł=(8łg\^2-96 g\^4),\_3\_ł=0. One may be tempted now to turn these one-loop RG functions into running couplings via eqs. (\[5\]). However, if one were to compute the two-loop beta functions, one would find terms proportional to $\log \frac{\k_1}{\k_2}$, and in general the $n$-loop RG functions contain $\log^{n-1}\frac{\k_1}{\k_2}$ terms(as well as lower powers of the logarithm). Therefore, unlike in standard $\MS$ we *cannot trust the perturbative RG functions. So if we still wish to use the EJ scheme we must *somehow perform a large logarithms expansion on the *beta functions themselves.***
Another problem with the EJ prescription is that although it has two RG scales (three if you include $\k_3$ which is only relevant to the running cosmological constant) $\k_1$ and $\k_2$, they do not seem to “track” the Higgs and fermionic scales, respectively. If such a tracking were present we would expect the one-loop beta function for $\l$ to have the form \[7\] \_1\_ł=,\_2\_ł=(8łg\^2 -48g\^4). That is the contributions to ${}_1\beta_\l$ and ${}_2\beta_\l$ can be identified with contributions from the Higgs and fermion loop, respectively. So although the EJ proposal is very interesting, it is not quite what we were looking for. However, it may still be that some (possibly quite simple) modification of the EJ scheme does the job. Another possibility would be to construct a multiscale version of the Callan-Symanzik equation [@ni].
Although we are unable to define such a modified EJ scheme we can exploit the fact that *any multiscale scheme must be related to the standard $\MS$ prescription by a *finite renormalization. That is if we have a scheme with two RG scales $\k_1$ and $\k_2$ then we must [^3] have g\_&=&F\_g(g,ł;\_1,\_2,)\
ł\_&=&F\_ł(g,ł;\_1,\_2,)\
m\^2\_&=&m\^2F\_[m\^2]{}(g,ł; \_1,\_2, ), with similar relations for $\L$, $\p$ and $\psi$. Here, the $\MS$ parameters $g_\MSf$, $\l_\MSf$, etc. at scale $\m$ may be regarded as “bare” ones as opposed to the new “renormalized” 2-scale parameters $g$, $\l$, etc. The (finite) $F_.$ functions are chosen so that:**
i\) The effective action $\Gamma$, when expressed in terms of the new parameters should be independent of the $\MS$ scale $\m$.
ii\) When $\k_1=\k_2$ the 2-scale scheme should coincide with $\MS$ at that scale.
There are an infinite number of 2- scale schemes (ie. $F_.$ functions) satisfying conditions i) and ii). Each of these schemes will have different beta functions. Let us now restrict ourselves to schemes with the correct one-loop tracking behaviour. That is we assume that the one-loop beta functions for $\l$ are as in eqn. (\[7\]). This tracking assumption also fixes the one-loop RG functions for $m^2$, $\L$, $\p$ and $\psi$. The tracking assumption does *not fix the one-loop beta functions for $g$; all we can say is that \_1\_g=,\_2\_g=, \_1+\_2=. A problem with the EJ scheme was the occurrence of logarithmic terms in the higher loop RG functions. Is it possible to devise a 2-scale scheme where the beta functions have no such logarithms? The answer to this question is no, and so anyone wishing to generalize the EJ scheme must face the problem of resumming logs in the beta functions themselves. To see this consider the two RG equation for the effective potential \_iV=0,\_i=\_i+\_i\_g +\_i\_ł+ \_i\_[m\^2]{}+\_i\_Ł-\_i\_, where $i=1,2$ and the summation convention was not used in the last equation. We have the integrability condition \[11\] \[[D]{}\_1,[D]{}\_2\]=0. The point is that the absence of logs in the RG functions, ie. $[\k_i\d/\d\k_i,{\cal D}_j]=0$ is *incompatible with the integrability condition eqn. (\[11\]).**
However, it is still possible to arrange for *one of the two sets of beta functions to be independent of $\k_1/\k_2$, eg. we can take the first set of beta functions to be independent of $\k_1/\k_2$. Alternatively, we can take the second set of RG functions (tracking the fermionic scale) to be independent of $\k_1/\k_2$. Whichever of these prescriptions we adopt, we can then use the integrability condition to resum the logarithms in the other set of beta functions. Some detailed calculations of this type (though in a different model) have been given in ref. [@fowi].*
Decoupling and Conclusions
==========================
We have argued that it is possible to construct a 2-scale scheme with appropriate tracking at one-loop where one of the two sets of RG functions is independent of the RG scales. Let us consider the case where we require that the first set of beta functions (tracking the Higgs scale) is independent of $\k_1/\k_2$. Then at leading order the first set of beta functions are \[dec\] \_1\_ł=,\_1\_[m\^2]{}=,\_1\_Ł=,\_1\_=0,\_1\_g=. Clearly, these beta functions are just the usual one-loop beta functions for pure $\p^4$ theory (provided we make the choice $\e_1=0$). The second set of beta functions depend on $\k_1/\k_2$; this dependence can be computed via the integrability condition (\[11\]). Now if the fermion is much heavier than the Higgs scale we would *expect that the beta functions for the low energy theory would be exactly those given by eqn. (\[dec\]), since in this case we expect to observe a decoupling [@decouple] of the heavy fermion. Thus, it seems that the condition that the first set of beta functions is independent of the RG scales is appropriate for the heavy fermion case ($g^2\p^2>>m^2+\h\l\p^2$). Similarly, one can argue that the alternative possibility of requiring that the second set of beta functions is independent of $\k_1/\k_2$ is suited to the heavy Higgs case ($m^2+\h\l\p^2>>g^2\p^2$). Note that although in the previous section we were unable to fix the values of $\e_1$ and $\e_2$ via a one-loop tracking condition, an explicit calculation shows that the final improved potential only depends on $\e_1$ and $\e_2$ through the combination $\e_1+\e_2=
\frac{5}{3}$.*
From decoupling arguments we expect the case where the first set of beta functions has no logarithms is suited to the heavy fermion case. We believe that this approach would correctly interpolate between the heavy fermion case and the single scale regime ($m^2+\h\l\p^2\sim g^2\p^2)$, since by construction our two scale scheme collapses to $\MS$ for $\k_1=\k_2$. There is no reason to expect that this prescription would be valid in the heavy Higgs case (for this we would use the alternative possibility where the second set of beta functions is independent of the RG scales). Thus, it seems that we can deal with both the heavy fermion and heavy Higgs problem, but these require a separate treatment. It remains an open question whether it is possible to devise a simple EJ type scheme which can interpolate all the way from the heavy fermion to the heavy Higgs sector.
Acknowledgments {#acknowledgments .unnumbered}
===============
I am grateful to the organizers of RG96 for giving me the opportunity to speak at this conference. Thanks also to Denjoe O’Connor and Chris Stephens for explaining some of their ideas on multiscale problems.
References {#references .unnumbered}
==========
[99]{}
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
B. Kastening, ;\
M. Bando, T. Kugo, N. Maekaka and H. Nakano, ;\
I.L. Buchbinder, S.D. Odintsov and I.L. Shapiro, Effective Action in Quantum Gravity (Institute of Physics Publishing, Bristol, 1992). R. Jackiw . S. Coleman and E. Weinberg, . M. Bando, T. Kugo, N. Maewaka and H. Nakano, . M.B. Einhorn and D.R.T. Jones, . K. Nishijima, ; . C. Ford and C. Wiesendanger, ‘A Multi-scale Subtraction Scheme and Partial Renormalization Group Equations in the $O(N)$-symmetric $\p^4$-theory’, DIAS-STP 96-10, hep-ph/9604392, 1996. T. Appelquist and J. Carazzone, ;\
K. Symanzik, .
[^1]: email: Ford@hpfs1.physik.uni-jena.de
[^2]: Work done in collaboration with C. Wiesendanger, email: wie@stp.dias.ie
[^3]: We assume that the transformation has a trivial dependence on $m^2$.
|
---
abstract: 'The power mechanism and accretion geometry for low-power FR1 radio galaxies is poorly understood in comparison to Seyfert galaxies and QSOs. In this paper we use the diagnostic power of the [Ly$\alpha$]{}recombination line observed using the Cosmic Origins Spectrograph (COS) aboard the Hubble Space Telescope (HST) to investigate the accretion flows in three well-known, nearby FR1s: M87, NGC4696, and HydraA. The [Ly$\alpha$]{} emission line’s luminosity, velocity structure and the limited knowledge of its spatial extent provided by COS are used to assess conditions within a few parsecs of the super-massive black hole (SMBH) in these radio-mode AGN. We observe strong [Ly$\alpha$]{} emission in all three objects with similar total luminosity to that seen in BLLacertae objects. M87 shows a complicated emission line profile in [Ly$\alpha$]{} which varies spatially across the COS aperture and possibly temporally over several epochs of observation. In both NGC4696 and M87, the [Ly$\alpha$]{} luminosities $\sim10^{40}$ ergs s$^{-1}$ are closely consistent with the observed strength of the ionizing continuum in Case B recombination theory and with the assumption of near unity covering factor. It is possible that the [Ly$\alpha$]{} emitting clouds are ionized largely by beamed radiation associated with the jets. Long-slit UV spectroscopy can be used to test this hypothesis. HydraA and the several BLLac objects studied in this and previous papers have [Ly$\alpha$]{} luminosities larger than M87 but their extrapolated, non-thermal continua are so luminous that they over-predict the observed strength of [Ly$\alpha$]{}, a clear indicator of relativistic beaming in our direction. Given their substantial space density ($\sim4\times10^{-3}$ Mpc$^{-3}$) the unbeamed Lyman continuum radiation of FR1s may make a substantial minority contribution ($\sim10$%) to the local UV background if all FR1s are similar to M87 in ionizing flux level.'
author:
- 'Charles W. Danforth, John T. Stocke, Kevin France, Mitchell C. Begelman'
- Eric Perlman
title: 'Far-UV Emission Properties of FR1 Radio Galaxies [^1]'
---
Introduction
============
Low power Fanaroff-Riley (1974) class 1 (FR1) radio galaxies [$P_{20cm}\leq 10^{24-25}~\rm W\,Hz^{-1}$; @LedlowOwen96] are among the most numerous AGN, comparable in space density to the better-known (and far better-understood) Seyfert galaxies [@MauchSadler07; @Owen96; @HoUlvestad01]. M87 with its famous synchrotron jet is the FR1 class prototype with other local examples including CentaurusA, NGC4696 (the brightest galaxy in the Centaurus Cluster), and HydraA. But where Seyfert galaxies exhibit luminous accretion disk emission and broad-line-region (BLR) emission lines, these two emission mechanisms are mostly or entirely absent in FR1s, making their power source more mysterious. Adding to this mystery is the presence of FR1 low-power sources in most brightest cluster galaxies (BCGs), including the morphologically-distinct “cD” galaxies [@Lauer14]. For example, 70% of BCGs in clusters with a “cool X-ray core” have FR1 radio sources with $P_{20cm}\geq
10^{20}$ W Hz$^{-1}$ [@Burns90]; [see also @LedlowOwen96; @Stocke99; @Perlman04; @Branchesi06; @Blanton11]; i.e., FR1s are commonly found in clusters and rich groups of galaxies and often, but not exclusively in the BCG [@LedlowOwen95; @LedlowOwen96].
These BCGs are the most luminous, most massive galaxies in the Universe. Given the observed correlation between the supermassive black hole (SMBH) mass and the bulge mass or bulge velocity dispersion [@KormendyHo13], these galaxies should also possess the most massive SMBHs. Indeed, local FR1 galaxies host some of the most massive SMBHs known: e.g., M87 [$3.5^{+0.9}_{-0.7}\times10^9
~\rm M_\sun$; @Walsh13] and two rich cluster BCGs [$1.3$ and $3.7\times10^9\rm M_\sun$; listed in @KormendyHo13]. However these most massive SMBHs power very low-luminosity AGN in comparison to many found in lower-mass, lower-luminosity ellipticals and early-type spirals. Apparently these sources either have little fuel available or do not reprocess gravitational energy as effectively into X-rays, relativistic particles, and radio emission as many AGN in less-luminous galaxies. It is possible that these sources do extract enormous amounts of energy from these most-massive black holes but that most of this energy is converted into the kinetic energy of a bulk outflow whose effect is visible as giant cavities in the surrounding X-ray emitting gas [e.g., @McNamaraNulsen06; @Fabian11]. Alternatively, it is possible that for some reason the sub-parsec structures near the SMBH prevent these sources from being as luminous as AGN in spirals (i.e., Seyferts and QSOs) and even those in lower luminosity ellipticals (i.e., FR2 radio galaxies and quasars).
Theoretical modeling of FR1s [sometimes called the AGN “radio mode”; @Croton06] rely on a “radiatively inefficient accretion flow” [RIAF, or its several variants: @NarayanYi94; @NarayanYi95; @BlandfordBegelman99; @QuataertGruzinov00; @HawleyBalbus02], but its parameters are much more poorly quantified than for luminous accretion disk AGN. For example, @Allen06 [A06 hereafter] found a strong correlation between the Bondi accretion rate (the hot gas mass available at or near the Bondi radius) and the observed jet power in FR1s. While A06 use this result to suggest that Bondi accretion is a good model for an RIAF, they also derive a very high efficiency (2%) for converting the energy of the gas accretion into relativistic jet power, requiring a very efficient accretion flow from the Bondi radius ($\sim100-200$ parsecs) all the way into the SMBH. Detailed models of this process [e.g., @NarayanYi95; @BlandfordBegelman99] infer that such efficient accretion is quite unlikely and that in many circumstances a thin accretion disk and an outflowing wind can be formed as in the “ADIOS” (advection-dominated inflow-outflow solution) model of @BlandfordBegelman99. For example, @BegelmanCelotti04 argue for the “ADIOS” scenario in which the presence of an “outer accretion disk” and an outflowing disk wind will reduce the accretion flow by two orders of magnitude.
Observationally, deep [*Chandra*]{} imaging of the FR1 prototype M87 derived an accretion rate comparable to the expected Bondi rate of 0.1 $M_{\odot}$ yr$^{-1}$ [@DiMatteo03] outside the Bondi radius ($\sim100-200$ pc). However, more recent X-ray imaging spectroscopy work by @Russell15 finds an accretion flow greatly reduced below the Bondi rate inside the Bondi radius of M87.
Detailed fitting of the spectral energy distribution (SED) within $\sim$ 30 pc of the nucleus of M87 by @Prieto16 finds an SED dominated by jet emission throughout the electromagnetic spectrum excepting a small contribution from a cool accretion disk longward of $1\mu$m. This corroborates earlier work by @Perlman01 [@Perlman07]. Based on the weak, cool disk detected in the IR, @Prieto16 derive a very low accretion rate of $<6\times
10^{-5}~M_{\odot}$ yr$^{-1}$ from the small, cool accretion disk they derive to be present from the SED. This same broadband SED was also used by @Broderick15 to place similar constraints on the accretion rate and on the presence of an event horizon. @Prieto16 also derive a similarly low accretion rate from the previous HST imaging data on the [H$\alpha$]{} emission line in the ionized gas disk discovered by @Ford94. The absence of significant Faraday rotation for the M87 nuclear source at sub-mm wavelengths is also consistent with a very low accretion rate [@Kuo14]. Clearly these observational and theoretical results cause problems for the simple Bondi rate model of A06, suggesting that a more complex nuclear geometry than a spherically symmetric, hot gaseous atmosphere is necessary in M87 and other similar sources.
[lllcccl]{} M87 & 12 30 49.4 $+$12 23 27 & 0.0044 & 0.020 & 13489 & 4.4 & 2014, Mar 3\
NGC4696 & 12 48 49.2 $-$41 18 39 & 0.00987& 0.098 & 13489 & 5.2 & 2013, Dec 20\
HydraA & 09 18 05.7 $-$12 05 44 & 0.05488& 0.036 & 13489 & 4.7 & 2013, Nov 6\
1ES1028$+$511 & 10 31 18.5 $+$50 53 36 & 0.3604 & 0.012 & 12025 & 22,33& 2011, May 1\
PMNJ1103$-$2329 & 11 03 37.6 $-$23 29 30 & 0.186 & 0.078 & 12025 & 18,13& 2011, Jul 5\
Because an accretion disk and wind can create the physical conditions necessary for a broad-line-region (BLR) at sub-parsec scales, [Ly$\alpha$]{}and other UV emission lines can provide a method for probing the accretion region in FR1s much closer to the black hole than can X-ray continuum observations (i.e., 2–30 pc in nearby FR1s). Observations using the full spatial resolution of HST (0025 in the UV) are capable of resolving line and continuum emission at the level of 2 pc in M87, 4.5 pc in NGC4696 and 30 pc in HydraA. Well-sampled optical imaging of M87 in [H$\alpha$]{} [@Ford94] and imaging spectroscopy in \[\] [@Macchetto97] and [H$\alpha$]{}[@Walsh13] with the [*Space Telescope Imaging Spectrograph*]{} (STIS) yield detections of this disk to within $\sim$ 6 pc of the SMBH. UV spectroscopy can do a bit better than this so the detection of [Ly$\alpha$]{} through the 0."26 FOS aperture by @Sankrit99 provides a probe of ionized gas close to the nucleus as well ($\sim20$ pc). The detection of [Ly$\alpha$]{} emission in M87 and other FR1s verifies the presence of cooler gas in the RIAF and its observed line-width and shape can determine some specifics about the accretion; e.g., the minimum mass that has “dropped out” of a hot accretion flow. Also, the mere presence of small amounts of cool gas inside the Bondi radius supports models of a much more inefficient RIAF than originally suggested [@BlandfordBegelman99; @BegelmanCelotti04; @SikoraBegelman13], consistent with the new X-ray analysis of @Russell15 and the SED analysis of @Prieto16. Weak emission line wings also are expected in some models [@SikoraBegelman13] as well as weak, highly-ionized absorption suggestive of an outflowing wind (an “ADIOS”).
Weak optical emission lines have been observed in some FR1s [e.g., @LedlowOwen95; @Ho97; @Ho01], but observed optical line-widths are usually rather narrow ($\leq300$ [km s$^{-1}$]{}) and the $\rm
[N\,II]/H\alpha$ ratios are indicative of LINER emission [@Ho01]. There is no indication in most cases that much of this emission line gas is participating in the accretion process. Given the dramatic spatial extent of [H$\alpha$]{} emission in some of the nearest FR1s, including M87 [@Ford94; @Ford97] and NGC4696 [@Canning11], most of the observed narrow emission line gas is not nuclear at all. However, some few BLLac objects have been observed to possess broad [H$\alpha$]{} lines, including OJ287 [@SitkoJunkkarinen85], PKS0521$-$365 [@Ulrich81] and even BLLac itself [@Vermeulen95], which is undoubtedly nuclear. Recently, using the [*Cosmic Origins Spectrograph*]{} (COS) on the Hubble Space Telescope (HST), we discovered weak, and relatively narrow [Ly$\alpha$]{} emission lines in three low-$z$ BLLac objects (Stocke, Danforth & Perlman 2011; Paper 1 hereafter). The COS spectra provide [Ly$\alpha$]{} emission line luminosities ($10^{40-41}~{\rm erg ~s^{-1}}$), line widths ($FWHM=200-1000$ [km s$^{-1}$]{}) and simultaneous beamed continuum luminosities. These observed [Ly$\alpha$]{} line luminosities are orders of magnitude less than those seen in Seyfert Galaxies and are well below the line luminosities required in an ionization-bounded BLR based on the observed non-thermal continuum luminosities in these sources. But we know that the continuum is beamed in BLLacs and an uncertain beaming factor needs to be applied to estimate the luminosity of the unbeamed continuum to determine whether the cool [Ly$\alpha$]{}-emitting gas is ionization-bounded or density-bounded. If the gas surrounding the SMBH in FR1 AGN is density-bounded (the prediction of RIAF models), the [Ly$\alpha$]{} line luminosities provide an order-of-magnitude estimate of the amount of cool gas in the BLR.
In this paper we use far-UV (FUV) spectroscopic observations from HST/COS of three nearby FR1s: M87, NGC4696 and HydraA and two unpublished BLLac objects to investigate the relationship between FR1s and BLLac objects and to investigate the [Ly$\alpha$]{} and continuum luminosities in unbeamed radio galaxies. [Ly$\alpha$]{} emission lines consistent with being circum-nuclear in origin are seen in all three FR1s. Therefore, these emission lines and the unbeamed continua underlying them can be used to probe accretion physics. In Section 2 we present and describe the COS spectroscopy and data modeling techniques for these three FR1s as well as for two as-yet-unpublished BLLac objects from observing time provided to the COS Science Team. In Section 3 we describe our findings and derive basic physical parameters from emission models. In Section 4 we discuss the implications of these observations for physical models of the SMBH and circum-nuclear regions of these objects. Finally, in Section 5 we summarize our main results and discuss their importance for AGN physics and cosmology.
Observations and Data Analysis
==============================
Three FR1 galaxies (Table 1) were observed as part of HST Guest Observer (GO) program 13489 (PI: Stocke). Each was observed for a nominal two-orbit visit using the COS/G130M grating. Four exposures at different central wavelengths give continuous spectral coverage over the entire G130M range ($1140\la\lambda\la1465$) at a resolution of $R\approx17,000$ ($\Delta v\sim15$ [km s$^{-1}$]{}).
The two BLLac objects in Table 1 were observed as part of the COS Guaranteed Time Observations (GTO) programs (PI: Green). Like the previous three BLLacs for which [Ly$\alpha$]{} emission was detected by Paper 1, these were observed with both the COS/G130M and G160M gratings giving continuous spectral coverage over $1140<\lambda<1795$ Å at a resolution of $R\approx17,000$. Intervening absorption line measurements for these two BLLacs were published as part of the @Danforth16 compilation, but AGN continuum and emission-line properties are new to this work.
The data were retrieved from the Mikulski Archive for Space Telescopes (MAST) and reduced via the methods described in @Danforth10 [@Danforth16]. Briefly, each exposure was binned by three native pixels or $\sim40\%$ of a point-source resolution element to increase the signal-to-noise before coaddition. The exposures were coaligned in wavelength by cross-correlating strong Galactic absorption features in each exposure and interpolated onto a common wavelength vector and then combined using an exposure-time-weighted algorithm. More details of the COS spectral reduction process can be found in @Danforth16 and @Keeney12.
Next, the Galactic [Ly$\alpha$]{} profile was fitted to determine the Milky Way [$N_{\rm HI}$]{} which was in turn used to remove foreground reddening (calculated $E(B-V)$ values are given in Table 1). In all three FR1 cases, the foreground reddening is small. No internal source reddening was assumed for either the FR1s or the BLLacs.
Results
=======
All three FR1 galaxies show prominent [Ly$\alpha$]{} emission at very close to the systemic redshifts. Both M87 and HydraA show weak FUV continuum emission while the continuum level in NGC4696 is barely detectable in these data. In M87 the [Ly$\alpha$]{} emission line has extensive line wings to both blue- and red-shifted relative velocities. In addition, M87 shows weak emission at $\sim1240$ Å, $\sim 1340$ Å, and $\sim1410$ Å which we identify with , , and $+$\], respectively. No metal ion emission was seen in either of the other two FR1s. Both new BLLac objects (1ES1028$+$511 and PMNJ1103$-$2329) show smooth power-law continua with weak [Ly$\alpha$]{} emission at approximately the systemic redshifts. We discuss the detailed measurements of each source below.
[lcccl]{} centroid &$ 1211.93\pm0.47$ &$1220.79\pm0.05 $&$ 1228.00\pm0.07$& Å\
&$-923\pm116$ &$ 1263\pm12 $&$ 3040\pm17 $& [km s$^{-1}$]{}\
FWHM &$ 6.43\pm0.68$ &$ 5.04\pm0.08 $&$ 3.30\pm0.15 $& Å\
&$ 1600\pm170 $ &$ 1241\pm18 $&$ 813\pm37 $& [km s$^{-1}$]{}\
$I_{\rm Ly\alpha}$&$ 22\pm5 $ &$ 597\pm16$&$ 14.0\pm0.7 $& $10^{-15}\rm~erg~cm^{-2}~s^{-1}$\
$L_{\rm Ly\alpha}$&$ 0.18\pm0.15$ &$ 5.06\pm0.35$&$0.12\pm0.02$& $10^{39}\rm~erg~s^{-1}$\
continuum & &$ 1.20\pm0.03 $& & $10^{-15}\rm~erg~cm^{-2}~s^{-1}~\AA^{-1}$\
centroid &&$ 1226\pm30 $&& [km s$^{-1}$]{}\
FWHM &&$ 5.5\pm0.8 $&& Å\
&&$ 1443\pm100 $&& [km s$^{-1}$]{}\
$I_{\rm NV 1238}$&&$ 4.9\pm0.9 $&& $10^{-15}\rm~erg~cm^{-2}~s^{-1}$\
$I_{\rm NV 1242}$&&$ 3.7\pm0.7 $&& $10^{-15}\rm~erg~cm^{-2}~s^{-1}$\
doublet ratio &&$ 1.3\pm0.3 $&&\
continuum &&$ 1.27\pm0.02$&& $10^{-15}\rm~erg~cm^{-2}~s^{-1}~\AA^{-1}$\
centroid &&$ 1500\pm40 $ && [km s$^{-1}$]{}\
FWHM &&$ 6.76\pm0.88$&& Å\
&&$ 1500\pm100$ && [km s$^{-1}$]{}\
$I_{\rm CII}$ &&$ 4.59\pm0.59$&& $10^{-15}\rm~erg~cm^{-2}~s^{-1}$\
continuum &&$ 1.20\pm0.02$&& $10^{-15}\rm~erg~cm^{-2}~s^{-1}~\AA^{-1}$
M87
---
The nucleus of the FR1 prototype M87 with its famous optical/radio/X-ray jet has been observed intensely over the last two decades with HST including previous UV and optical spectra obtained with the Faint Object Spectrograph (FOS) by @Tsvetanov98 [@Tsvetanov99] and @Sankrit99. These FOS spectra of the nucleus cover approximately the same wavelength range as the COS/G130M data including the [Ly$\alpha$]{} emission in M87 at 1220.8 Å. In both the two @Sankrit99 observations and our own, the intrinsic flux and line shape of [Ly$\alpha$]{} are difficult to determine due to the presence of the Galactic damped [Ly$\alpha$]{} absorption (DLA) as well as probable absorption in M87. Regardless of the corrections which must be made due to the Galactic DLA and other absorbers in M87, @Sankrit99 find that the [Ly$\alpha$]{} emission line possesses a $\sim2000$ [km s$^{-1}$]{}-wide core and a redshifted tail that extends at least another 800 [km s$^{-1}$]{} to the red. However, the M87 absorption has unknown [$N_{\rm HI}$]{}, which makes it difficult to remove in order to determine the intrinsic [Ly$\alpha$]{} emission line properties. Absorption has been detected in two narrow components ($cz\approx980$ and 1330 [km s$^{-1}$]{}) in H&K and D by @Carter92 and @Carter97 and in several other ions ( D, , , H&K, and ) in one broad component ($cz=1134\pm22$ [km s$^{-1}$]{}) by @Tsvetanov99 with FOS. The variable absorption velocities reported as well as the partial covering of the source suggested by the and absorption, which have 1:1 doublet ratios but low optical depths [@Carter97], are strong indicators that variable absorption exists in the nucleus of M87, perhaps due to cloud proper motions.
On what is likely a slightly larger scale, HST/FOS off-nuclear optical spectra and HST/Faint Object Camera (FOC) optical imaging discovered an ionized gas disk with Keplerian motion [@Ford94; @Ford99] that connects to non-Keplerian blueshifted emission-line filaments primarily to the NW of the nucleus. @Ford99 speculate that the observed nuclear absorption lines found by @Tsvetanov98 are kinematically related to this filamentary gas since these are blueshifted with respect to the nucleus as well. @Macchetto97 used the spectroscopic capabilities of the Faint Object Camera (FOC) on HST to determine the rotation curve of the \[\] emitting gas in the nuclear disk which better determined the mass of the SMBH. Relevant to this study, no \[\] forbidden-line emission was found within 007 (5 pc) of the nucleus. Similar long-slit, optical spectra obtained with the [*Space Telescope Imaging Spectrograph*]{} (STIS) were obtained by @Walsh13 to refine the SMBH mass estimate.
Due to all these potential complications and uncertainties, we elected to deconvolve the emission and absorption components in [Ly$\alpha$]{} using an [*a priori*]{} approach. The [Ly$\alpha$]{} profile is characterized by a strong emission peak at $v_{\rm LSR}=1260$ [km s$^{-1}$]{} with a dereddened peak amplitude of $8\times10^{-14}$ [$\rm erg~cm^{-2}~s^{-1}~\AA^{-1}$]{}. This is $\sim6$ times brighter than observed with FOS by @Sankrit99 on 1997, 23 January, and nearly $10\times$ brighter than on 1997, 18 January (although these new observations are through a larger aperture; 25 with COS compared to 026 with FOS; see Section 3.1.2). Weaker emission can be seen extending over the range $1207<\lambda<1232$ Å and thus blended with the Milky Way DLA. The emission line profiles in the three COS spectra shown here, especially the [Ly$\alpha$]{} line in M87, are far too broad to be affected by any spectral smearing caused by the marginal spatial resolution in the COS aperture (see Section 3.4).
We model the [Ly$\alpha$]{} emission profile as three emission components (“blue”, “main” and “red”), a damped Galactic [Ly$\alpha$]{} absorber at $v_{LSR}\sim0$, a weak, broad absorption component to accomodate possible absorption in M87, and a linear far-UV continuum flux. The detailed flux model includes fifteen fitting parameters (three for each Gaussian emission component, three for the M87 absorber, two each for the Galactic DLA and the non-thermal continuum of M87), each allowed to vary around a reasonable range. Despite this complexity, a remarkably robust fit is determined ($\bar\chi^2=0.92$) even when fit parameters are allowed to vary over a broad range of plausible values. Best-fit quantities and $1\sigma$ fitting uncertainties are given in Table 2 and shown overplotted on the data in the left-hand panels of Figure \[fig:m87lines\].
Blue and red wings are apparent in the emission profile and these are best fit with two additional Gaussian components with centroids at $1211.93\pm0.47$ Å and $1228.00\pm0.07$ Å, respectively, modified by the Galactic DLA. If [Ly$\alpha$]{} emission, these correspond to $v_{\rm
LSR}=-922$ [km s$^{-1}$]{} and $v_{\rm LSR}=+3040$ [km s$^{-1}$]{}. We note that the blue- and red-wings of the profile [**cannot**]{} be fitted with a single very broad emission component at $v_{\rm LSR}\sim1200$ [km s$^{-1}$]{}; two Gaussian profiles offset from the main, central component by $\sim1000-2000$ [km s$^{-1}$]{} are required. Thus, if the blue and red emission are both [Ly$\alpha$]{} they are approximately symmetrically-placed around the main [Ly$\alpha$]{} line. We interpret the red component as [Ly$\alpha$]{} emission redshifted by $\sim1800$ [km s$^{-1}$]{} with respect to the main [Ly$\alpha$]{} emission in agreement with @Sankrit99. The maximum redshifted velocity observed in this broad component is $\sim3000$ [km s$^{-1}$]{} relative to the main [Ly$\alpha$]{} line.
The blue side emission is possibly either 1206 Å emission at $v_{\rm LSR}=+1540\pm140$ [km s$^{-1}$]{} or [Ly$\alpha$]{} emission blueshifted at $-922$ [km s$^{-1}$]{} LSR or $-2185$ [km s$^{-1}$]{} with respect to the main [Ly$\alpha$]{}emission peak. The larger uncertainty in the velocity centroid of this line is due to the presence of the Galactic DLA, making this component’s position and profile very dependent upon the exact assumed DLA column density (see below).
The interpretation is consistent with the velocity and line width of emission seen in (see below), albeit with significant uncertainty. However, the velocity centroid of the proposed emission is almost 300 [km s$^{-1}$]{} [*greater*]{} than the strong [Ly$\alpha$]{}emission centroid, casting doubt on this identification. Further, if this emission is , it would be much stronger than what is seen in any other AGN. In a typical Seyfert like Mrk817 [@Winter11], a line as luminous as the blue component in M87 would be visible as an asymmetry in the line profile of [Ly$\alpha$]{}. This is not observed. Further, the FUV spectum of the prototypical narrow-line Seyfert1 galaxy, IZw1 has only a reported marginal detection of emission at a level ten times weaker than the feature we observe in M87. We conclude that this feature is unlikely to be .
If this emission is interpreted as [Ly$\alpha$]{}, it has a velocity of $-922$ [km s$^{-1}$]{} LSR and a blueshifted velocity of $-2185$ [km s$^{-1}$]{} relative to the [Ly$\alpha$]{} emission peak, making it reasonably symmetrical in both velocity offset and luminosity to the redshifted [Ly$\alpha$]{} emission line which is not affected by the Galactic DLA and has been reported previously.
A dip near the peak of the main component of the [Ly$\alpha$]{} emission line is well-fit as a very broad [Ly$\alpha$]{} absorption line ($b\approx250$ [km s$^{-1}$]{}) at $v_{\rm LSR}=1490$ [km s$^{-1}$]{} ($v=+230$ [km s$^{-1}$]{} relative to the main line centroid). While similar to the broad, low-ion detections seen by @Tsvetanov99, this absorption component is at a very different heliocentric velocity ($\approx+350$ [km s$^{-1}$]{} relative to the @Tsvetanov99 detections). This absorption lies on the red side of the [Ly$\alpha$]{} peak compared to the blue side absorption seen in the epoch 1997 FOS spectrum.
The Galactic DLA absorption we fit has a derived $\log N\rm_{HI}
(cm^{-2})=20.14$, quite close to but slightly less than the Galactic $\log N\rm_{HI} (cm^{-2})=20.34-20.53$ value inferred by @Sankrit99. This value is also close to but slightly larger than the total $N_H$ found by @DiMatteo03 from a continuum fit to the X-ray emission at the M87 nucleus. If we constrain the Galactic DLA [$N_{\rm HI}$]{} to be between the two Sankrit values, our best-fit solution is driven to the minimum value (i.e., $\log
N\rm_{HI}(cm^{-2}) =20.34$) and the intrinsic main [Ly$\alpha$]{} emission component would increase by a factor of 8% in flux. However, the blue emission is much more affected by the Galactic DLA increasing; its line-flux increases by nearly a factor of two and its emission centroid shifts to $1214.04\pm0.16$ Å. At nearly 1.5 Å redward of its location in the unconstrained model (Table 2, Figure \[fig:m87lines\]), it is even more unlikely to be 1206Å and much more likely to be a blue-shifted [Ly$\alpha$]{}velocity component at $v_{lsr}=-400$ [km s$^{-1}$]{} or $v=-1670$ [km s$^{-1}$]{} with respect to the main [Ly$\alpha$]{} emission.
### Metal ion emission lines
In addition to the strong [Ly$\alpha$]{} emission, there are a number of weak emission features in the M87 spectrum. As discussed above, the emission feature near to a potential in M87 was fitted concurrently with [Ly$\alpha$]{} but is interpreted as blushifted [Ly$\alpha$]{}, not . Peaks at $\sim1244$ Å and $\sim1340$ Å are consistent with doublet and emission respectively. Best fit parameters for these lines are given in Table 2.
The 1334.5 Å emission line is modeled as a single Gaussian along with a $v_{\rm LSR}\approx0$ Galactic absorption line. The Galactic line is fitted with the parameters $v_{\rm LSR}=
+26\pm11$ [km s$^{-1}$]{}, $b=80\pm16$ [km s$^{-1}$]{}, $\log\,N (\rm cm^{-2})=15.0\pm0.1$.
The doublet ($\lambda=1238.8$, $1242.8$) was fitted assuming an identical velocity centroid and FWHM parameters for the two lines, but letting the relative line strength vary from the nominal 2:1, optically thin doublet ratio to an optically thick 1:1 ratio.
A low-contrast peak near 1400 Å is consistent with a blended complex of the $\lambda=1393.8$, $1402.7$ doublet and \] $\lambda\approx1403$ Å emission. However, given the low data quality near the edge of the COS/G130M detector and the complicated nature of the required model, no fit was attempted.
@AndersonSunyaev16 recently reported the tentative detection of [\[\]]{} $\lambda=1354.08$ emission from M87. A careful examination of the data around $\lambda=1360$ Å reanalyzed using the techniques of @Danforth16 shows a weak ($\sim5\sigma$) feature consistent with \[\] emission (Figure \[fig:m87fe21\]). We measure $F_{\rm [FeXXI]}=(1.8\pm0.8)\times10^{-16}$ erg cm$^{-2}$ s$^{-1}$, $v_{lsr}=1316\pm29$ [km s$^{-1}$]{}, and $FWHM=130\pm70$ [km s$^{-1}$]{}. The line flux value quoted here is consistent with the upper limit of ($4\times10^{-16}$ erg cm$^{-2}$ s$^{-1}$) quoted by @AndersonSunyaev16. While their estimate of line-width $\approx290$ [km s$^{-1}$]{} is much higher than we obtain, it is based on a marginal detection. A value of $FWHM=130\pm70$ [km s$^{-1}$]{} is both larger than a thermal width of $\sim50$ [km s$^{-1}$]{} for this species and provides a first estimate of the turbulence of the hot gas close to the SMBH of $\approx100$ [km s$^{-1}$]{}[@AndersonSunyaev16]. New exposures of comparable integration time to the values in Table 1 will make a definitive measurement of turbulence in the accretion flow. This will be reported at a later time.
### Line and continuum variability
@Sankrit99 reported on two epochs of nuclear FOS observations in January 1997 separated by five days in which the UV continuum remained approximately constant while the [Ly$\alpha$]{} emission line flux increased by a factor of $\approx1.6$. However, these authors caution that this apparent variability may be due to a slight displacement of the FOS aperture. Earlier FOS spectra from late 1993 into 1995 reported on by @Tsvetanov98 [@Tsvetanov99] found that the NUV $+$ optical continuum (2000-4000Å) was variable by a factor of $\sim2$ in 2.5 months, showed smaller variability on a timescale of weeks but no change over a one day period. Multi-epoch NUV imaging compiled by Madrid (2009; 30 epochs) and Perlman [et al.]{} (2011; 20 epochs) in more recent years finds $\sim20$% continuum variability over timescales of months to years. Variability at the few percent levels are possible on shorter time periods of 2-4 weeks.
While we can not be definitive about the [Ly$\alpha$]{} emission line variability in our observations relative to the FOS observations due to the much larger COS aperture, it is likely that at least some of the factor of 6–10 flux difference is due to intrinsic variability. In subsection 3.4 below we use the limited spatial resolution of the COS spectroscopy to determine that both the nuclear [Ly$\alpha$]{} line and the UV continuum are spatially extended by $\sim1$. In the STIS spatially-resolved spectroscopy of @Macchetto97 and @Walsh13 the observed flux of the optical emission lines is confined primarily to the inner 05. If the spatial extent of [Ly$\alpha$]{} is similar, then most of the luminosity of [Ly$\alpha$]{} is coming from a region not too much larger than the FOS aperture. While proof of [Ly$\alpha$]{} variability awaits future COS spectroscopy for accurate flux comparisons through the same aperture, an order of magnitude increase seems quite extreme, suggesting that at least some of the FOS-COS flux difference is intrinsic variability. If the line variability over long time intervals (months to years) is modest ($\sim20$%), similar to the observed continuum variability, the [Ly$\alpha$]{} line flux could be responding to earlier continuum variations (a.k.a. “reverberation”).
However, the red wing seen in the @Sankrit99 FOS spectrum and in Figure 1 is the most variable portion of the emission profile and is unaffected both by the Galactic DLA and the reported locations of the absorption in M87. Therefore, this emission component must possess significant intrinsic variability. The potential variability of the blue wing emission is not yet known until we obtain new epochs of COS FUV spectroscopy. The only constant emission or absorption component in the vicinity of the M87 [Ly$\alpha$]{} emission line is the foreground Galactic absorption.
If the [Ly$\alpha$]{} line variability observed in two FOS epochs is intrinsic, a very small upper limit on the size of the [Ly$\alpha$]{} emitting region can be set by $c\tau\sim10^{16}$ cm [$\sim10$ Schwarzschild radii for a $4\times10^9~M_\sun$ SMBH; @Sankrit99]. But this interpretation is inconsistent both with the rather narrow line width of [Ly$\alpha$]{} and with the observed spatial distribution of the emission (see Section 3.4). This is simply closer to the SMBH than can be accommodated by the emission line widths. A model in which the [Ly$\alpha$]{}line flux changes due to a change in the amount of cloud mass in the [Ly$\alpha$]{} emitting region (i.e., variable accretion in a density-bounded region, as was suggested in Paper 1) is also ruled out since any variability would occur on a timescale over which clouds move in and out of the emitting region at $v\leq1000$ [km s$^{-1}$]{}. This model would suggest variations only on a several year or longer timeframe, not on the short timescales observed. Apparent emission line variability due to variable absorption is a possibility as long as the line variability is modest (few percent level). Again, future FUV spectroscopy with COS can settle the important question of [Ly$\alpha$]{}variability through comparison to the spectrum presented here.
[lccl]{} Centroid &$ 1227.59\pm0.30 $&$ 1230\pm7$ & Å\
&$ 2940\pm70$ &$\sim3500$ & [km s$^{-1}$]{}\
FWHM &$3.2\pm0.8$ &$\sim5$ & Å\
&$790\pm200$ &$\sim1200$ & [km s$^{-1}$]{}\
$I_{\rm Ly\alpha}$&$14\pm9$ &$\sim2$ & $10^{-15}\rm~erg~cm^{-2}~s^{-1}$\
$L_{\rm Ly\alpha}$&$2.3\pm1.6$ &$\sim0.3$ & $10^{39}\rm~erg~s^{-1}$\
continuum &$3\pm1$ & & $10^{-17}\rm~erg~cm^{-2}~s^{-1}~\AA^{-1}$\
Centroid &$2893\pm10$ &$\sim2280 $ & [km s$^{-1}$]{}\
$\log\,N_{\rm HI}$& $14.10\pm0.08$ &$14.7\pm0.6$&\
$b$ &$87\pm11 $ &$270\pm140$ & [km s$^{-1}$]{} \[tab:ngc4696\]
[lccl]{} centroid &$ 1281.85\pm0.02$&$1281.80\pm0.02$& Å\
&$-130\pm10 $&$ -140\pm10 $& [km s$^{-1}$]{} \
FWHM &$1.44\pm0.09 $&$ 2.61\pm0.08 $& Å\
&$ 336\pm21 $&$ 610\pm19 $& [km s$^{-1}$]{}\
$I_{\rm Ly\alpha}$&$16.3\pm2.9 $&$ 29.5\pm2.9 $& $10^{-15}\rm~erg~cm^{-2}~s^{-1}$\
$L_{\rm Ly\alpha}$&$ 112\pm20 $&$ 203\pm20 $& $10^{39}\rm~erg~s^{-1}$ \
continuum &$0.58\pm0.02 $& & $10^{-15}\rm~erg~cm^{-2}~s^{-1}~\AA^{-1}$\
slope &$ 11\pm2 $& & $10^{-18}\rm~erg~cm^{-2}~s^{-1}~\AA^{-2}$ \[tab:hydraa\]
NGC4696
-------
NGC4696 shows an asymmetric [Ly$\alpha$]{} emission profile at 1227.6 Å($z=0.0090$, $cz=2710$ [km s$^{-1}$]{}) with what appears to be a narrow absorption line superimposed at 1227.3 Å. There is a very low level of continuum flux ($F<10^{-16}$ [$\rm erg~cm^{-2}~s^{-1}~\AA^{-1}$]{}); the continuum level listed in Table 3 was obtained by averaging the flux in continuum pixels between $1218<\lambda<1240$ Å. The component structure of this emission line is unclear. We interpret the narrow absorption as [Ly$\alpha$]{} at the same redshift as NGC4696. There are no Galactic, $z=0$ absorption lines expected at this wavelength and the pathlength to NGC4696 is short enough that no intervening, intergalactic absorption lines are expected. The simplest model–a single Gaussian emission component and a single Voigt absorption profile–does not give satisfactory results. Adding a second, weaker emission component helps account for the red wing of the emission, but the peak is still poorly fit. The steep blue edge of the emission suggests the presence of additional absorbing gas slightly blueshifted from the systemic [Ly$\alpha$]{} of NGC4696, possibly outflowing from the nucleus or associated with the host galaxy itself. If a single, broad absorption line is included in the fit, the profile model fits the data well. However, we caution against overinterpretation of this fit, especially the inferred blue-side absorber. The emission may be intrinsically non-Gaussian or the absorption profile may be more complicated than a single, simple Gaussian component. Fit parameters are given in Table \[tab:ngc4696\].
@AndersonSunyaev16 found a very low-significance emission feature they interpreted as [\[\]]{} associated with NGC4696. We confirm this possible detection in our more sophisticated reduction of the data (Figure \[fig:ngc4696\_fe21\]) at a significance level of $\sim2\sigma$. The possible emission line is fitted by a Gaussian with $FWHM=135\pm83$ [km s$^{-1}$]{} and $v_{lsr}=2931\pm38$ [km s$^{-1}$]{}. Our fitted flux of $F_{\rm FeXXI}=(8\pm5)\times10^{-17}$ erg cm$^{-2}$ s$^{-1}$ is consistent with the Anderson & Sunyaev 90% upper limit of $F_{\rm
FeXXI}\le2.2\times10^{-16}$ erg cm$^{-2}$ s$^{-1}$.
HydraA
------
The far-UV spectrum of HydraA shows a relatively narrow [Ly$\alpha$]{}emission peak at close to the systemic velocity. It is well-fit with a broad-plus-narrow pair of Gaussian emission components and a first-order linear continuum (Figure \[fig:hydraa\]). Some structure in the residual suggests that there may be additional components, but nothing obvious is seen.
Recent HST images of HydraA in the ultraviolet by @Tremblay15 show that the nucleus is obscured by a dust disk viewed edge-on. Thus one would expect the nuclear FUV continuum and line emission to be suppressed or absent entirely. It is therefore surprising that a FUV continuum is present, the [Ly$\alpha$]{} line emission is many times more luminous than either M87 or NGC4696 (Tables 2, 3), and that no narrow absorption line is seen superimposed on the [Ly$\alpha$]{} emission as is seen in the other two cases (Figures 1, 3). We conclude that the nuclear region must therefore not be obscured by the dust disk in HydraA.
In addition to the usual Galactic, $z=0$ absorption lines, the HydraA spectrum shows hints of 1334.5, $+$ 1304, and 1206.5 absorption at the redshift of HydraA. It is possible that this absorption arises in the dust disk bisecting the galaxy [@Tremblay15]. However, no absorption is seen at the systemic redshift, so modeling of these absorption lines would require data of much higher quality.
No metal-ion emission is seen including [\[\]]{}.
Limited angular resolution information from HST/COS
---------------------------------------------------
While $HST$/COS was not designed to provide high angular resolution imaging spectroscopy, it has been demonstrated that limited spatial information can be obtained from an analysis of a target’s cross-dispersion profile [@France11]. The angular resolution of COS in the G130M mode is $0.8-1.1$. In order to evaluate the spatial extent of the [Ly$\alpha$]{} emission line region at $\approx1\arcsec$ resolution, we compared the angular profiles of the three FR1 galaxies with 1) a known extended emission source and 2) a known point source. HST orbits in a diffuse cloud of neutral and ionized atoms; the “air glow” from the atomic recombination and resonant scattering of solar photons in this cloud produces a uniform background of hydrogen and oxygen emission that fills the COS aperture and approximates a filled-slit observation of an astronomical target. For point source comparison, we downloaded spectra of a well-studied COS calibration target, WD0308$-$565. We analyzed observations from the longest central wavelength settings available from each target ([cenwave]{}$=1318$ and $1327$) as these grating settings have the smallest intrinsic cross-dispersion heights [@Roman_duval13_COS_ISR].
Figure \[fig:xdisp\] shows the cross-dispersion profiles of the [Ly$\alpha$]{} emission lines of the three FR1 galaxies (top three panels), the FUV continuum of WD0308 (bottom panel), and the [Ly$\alpha$]{} airglow spectra measured simultaneously. The galactic/stellar profiles were extracted over a 400 dispersion-direction pixel region ($\approx4$ Å) centered on the source [Ly$\alpha$]{} emission line (offset into the continuum at 1247.8 Å for WD0308). The geocoronal [Ly$\alpha$]{} airglow profile was fitted with a Gaussian (red) and shown offset from the central cross-dispersion profile of the targets by $-50$ pixels. Gaussian fits to the galaxy/star cross-dispersion profiles are shown in green. The Gaussian FWHMs (in pixels, recall 1 pixel $\approx0.11$ for COS/G130M) are shown in the legends. The y-axis label shows the total counts per cross-dispersion extraction region in each of the targets, with the geocoronal profile scaled to the peak of the target flux. For the FR1 galaxies, the geocoronal [Ly$\alpha$]{} line-height is always between $21-22$ pixels FWHM ($2.3-2.4$), consistent with the expected instrumental profile for a filled aperture and the non-uniform primary science aperture response function of COS. The WD0308 continuum is bright enough that it has comparable flux to the airglow line, thus its geocoronal line width must be fitted with a pair of Gaussians. The FWHM of the white dwarf cross-dispersion profile is $8.4\pm0.2$ pixels (0.92), consistent with the expected cross-dispersion profile for a point source. The broader component has FWHM$=21\pm1$ pixels (2.3), consistent with a filled-aperture (bottom panel, Figure \[fig:xdisp\]).
Contrasting the filled-aperture geocoronal observations and the point source white dwarf observations with the cross-dispersion heights of the FR1 galaxies, we see that all three galaxies are in an intermediate category: they are clearly extended emission sources, but do not fully fill the COS aperture. For M87, we find a cross-dispersion line height of $11.5\pm0.1$ pixels (1.27), with $18.7\pm0.3$ pixels (2.06) and $15.7\pm0.5$ pixels (1.73) for HydraA and NGC4696, respectively. We conclude that the [Ly$\alpha$]{}-emitting regions for these FR1 galaxies are indeed extended, though do not fill the aperture.
A similar analysis of a region of the FUV continuum away from the galactic [Ly$\alpha$]{} emission signal shows an interesting result: the FUV continua in M87 and HydraA are also extended (that is, not consistent with a point-like AGN alone) and the FUV continuum in M87 shows an angular extent $20-40$% larger than the [Ly$\alpha$]{} emission region (FWHM$=14-16$ pixels over a range of continuum wavelengths and extraction region sizes). NGC4696 does not have sufficient FUV continuum flux for a cross-dispersion profile to be reliably measured.
Furthermore, in M87 the [Ly$\alpha$]{} core emission, blue wing emission, and red wing emission are centered at slightly different locations in the aperture (see Figure \[fig:tilt\]). While slight, these offsets are consistent with [Ly$\alpha$]{}-emitting material outflowing along the direction of the jet (blue wing is in the direction of the jet; red wing in the direction of the counterjet).
The inferences on the spatial extent and location of the line and continuum emission in M87 can be tested by obtaining long-slit FUV spectra of the nucleus at position angles along the jet and perpendicular to the jet using HST/STIS. The results of these observations (approved for HST cycle 24) will be reported in a future publication.
BLLac Objects
-------------
Several BLLac objects in the literature show weak [Ly$\alpha$]{} emission: Mrk421, Mrk501, and PKS2005$-$489 (Paper 1) and more recently H2356$-$309 [@Fang14]. To these we add new detections of weak [Ly$\alpha$]{} emission in PMNJ1103$-$2329 and 1ES1028$+$511. As with Paper 1, we fit power-law continua to the spectra and model the [Ly$\alpha$]{} emission profiles as single, Gaussian components.
Neither PMNJ1103$-$2329 nor 1ES1028$+$511 show high-significance [Ly$\alpha$]{} emission features like the FR1s, nor even as large an equivalent width as the previously-published BLLac objects with [Ly$\alpha$]{} emission (Paper 1). Nevertheless, a Gaussian fit in the expected location of the systemic [Ly$\alpha$]{} shows weak, broad, emission with significance levels of $\sim6\sigma$ and $\sim3\sigma$ for PMNJ1103$-$2329 and 1ES1028$+$511, respectively (Figure \[fig:new\_bllacs\]).
Of the seven BLLac objects with well-known redshifts and high S/N FUV spectra, six show intrinsic [Ly$\alpha$]{} emission. Weak, relatively narrow [Ly$\alpha$]{} emission appears to be a generic feature of both BLLac objects and their parent population FR1 radio galaxies.
Discussion
==========
The similar [Ly$\alpha$]{} luminosities of FR1s and BLLac objects is additional evidence that these two classes are the same type of object seen from different viewing angles. Specifically, the BLLacs (Mrk421, Mrk501 and PKS2005$-$489) and the FR1 (HydraA) are at comparable redshifts and have comparable [Ly$\alpha$]{} luminosities (see Table \[tab:lyaproperties\]). But the line luminosities for the observed sources vary by two orders of magnitude, an unexpected result if the ionizing continuum and distribution of [Ly$\alpha$]{} emitting clouds in these objects are both isotropic. The pure FR1s in Table \[tab:lyaproperties\], M87 and NGC4696, have at least an order of magnitude less [Ly$\alpha$]{} luminosity than any of the other AGN which have at least some beamed radiation coming in our direction. Therefore, the beamed radiation must be considered in modeling the [Ly$\alpha$]{} emission, which makes the emission mechanism and its location(s) a great deal harder to model in these objects.
[llccccc]{} M87 & 0.0022 & &$1240\pm20$ &$0.52\pm0.04 $&HST/COS & this work\
NGC4696 & 0.00987& &$790\pm200$ &$0.26\pm0.16 $&HST/COS & this work\
HydraA & 0.05488& &$610\pm20 $ &$32\pm3 $ &HST/COS & this work\
PMNJ1103$-$2329 & 0.1847 & 6.2&$ 580:$ &$4.9\pm0.4 $ &HST/COS & this work\
1ES1028$+$511 & 0.3607 & 3.3&$ 220:$ &$5.8\pm1.7 $ &HST/COS & this work\
H2356$-$309 & 0.165 & &$1340\pm320$ &$9.53\pm2.02$&HST/COS & @Fang14\
Mrk421 & 0.0300 & 9 &$300\pm30$ &$2.37\pm0.22$ &HST/COS & Paper 1\
PKS2005$-$489 & 0.0710 & 15 &$1050\pm60$&$24.9\pm1.1$ &HST/COS & Paper 1\
Mrk501 & 0.0337 & 23 &$820\pm80$ &$5.2\pm0.3 $ &HST/FOS & Paper 1\
PKS2155$-$304 & 0.116 &$<4$&&$ <11$ &HST/STIS& Paper 1 \[tab:lyaproperties\]
In a Seyfert galaxy, the line emission comes from the integrated emission of the broad and narrow emission line clouds while the continuum arises in an accretion disk. The low-level, non-thermal continuum emission in FR1s suggests that an accretion disk is not present or at least does not possess anywhere near the accretion rate seen in Seyferts [@Prieto16]. What, then, is the ionizing radiation source we detect in recombining ? And what is the distribution of the emission line clouds? Our COS spectra are suggestive of answers to these questions but are not conclusive.
[lcccl]{} $I_{1215}$&$1.37\times10^{-15}$&$5.9\times10^{-16}$&$ 4\times10^{-17}$&$\rm erg~cm^{-2} s^{-1} \AA^{-1}$\
$\alpha_\lambda $&$ 0.92 $&$ 0.85 $&$ 0.97 $&\
$I_{912} $&$1.79\times10^{-15}$&$7.6\times10^{-16}$&$ 5\times10^{-17}$&$\rm erg~cm^{-2} s^{-1} \AA^{-1}$\
ionizing flux &$ 0.0687 $&$ 0.0273 $&$ 0.0021 $&photons $\rm cm^{-2} s^{-1}$\
predicted $I_{Ly\alpha}$&$120 $ &$45$ &$ 3.0$ &$\rm 10^{-14}~erg~cm^{-2} s^{-1}$\
observed $I_{Ly\alpha} $&$ 60 $ &$4.5$ &$ 1.4$ &$\rm 10^{-14}~erg~cm^{-2} s^{-1}$\
OPF &$ 2.0 $&$ 10 $&$ 2.1 $& predicted/observed \[tab:opf\]
In Table \[tab:opf\] we use COS observations of the three FR1s to determine the factor by which the ionizing continuum extrapolated from our COS FUV spectra overpredicts the strength of [Ly$\alpha$]{} emission: the so-called over-prediction factor (OPF) of Paper 1. The OPF is the ratio between the Ly $\alpha$ luminosity predicted from the power-law fit to the ionizing continuum, to that observed in COS spectra, and it should be related to the Doppler factor $\delta = [\Gamma(1 - \beta
\cos \theta)]^{-1}$. Whereas the BLLac objects exhibit OPFs of hundreds to tens of thousands, two of the three FR1s have OPFs only slightly greater than unity. In previous FOS observations of M87 [@Sankrit99] the [Ly$\alpha$]{} luminosity was lower but the continuum flux was at the same level as shown here, leading to OPF values slightly less than unity inside the FOS aperture. The OPF of HydraA is intermediate between the very nearby FR1s and the BLLac objects as might be expected if we are seeing a portion of the the beamed continuum in that AGN. A small OPF ($\sim3$) is also seen in the prototypical Seyfert 1 galaxy Mrk817 so that the OPF values slightly greater than one in Table \[tab:opf\] likely are consistent with photo-ionization with modest covering factor for the emission-line clouds (see Paper 1).
If the ionizing continuum producing the [Ly$\alpha$]{} in these FR1s is the result of beaming, then the ionizing continuum seen by potential [Ly$\alpha$]{}emitting clouds should vary smoothly with angle away from the beaming axis. If [Ly$\alpha$]{} emitting clouds exist isotropically around the continuum source, then some clouds are illuminated by a much larger ionizing continuum than we observe. For example, it has been estimated from population statistics [@Perlman93; @UrryPadovani95] that the half power-angle of the X-ray emission in BLLacs is $\sim25$, which produces a significant fraction of beamed continuum at all angles relative to the beaming axis [see plots in the Appendix of @UrryPadovani95]. In this case the OPFs in Table \[tab:opf\] could be significant underestimates; i.e., there is much more ionizing continuum illuminating the near nuclear region than the amount of that can absorb it. Again, anisotropic distributions of ionizing flux and [Ly$\alpha$]{} emitting clouds are suggested.
M87
---
While all three FR1 objects studied here show intriguing complexity, more constraints on source size and geometry are available for M87 due to its proximity to us and its previous observations in the FUV by HST. In particular, M87 is the only one of these objects where the jet viewing angle ($\approx 15^\circ$) is well constrained [@Perlman11; @Meyer13]. Variability information is particularly important and puzzling. The [Ly$\alpha$]{}-emitting gas cannot possibly be as close in as $10^{16}$ cm (0.003 pc) to the SMBH as claimed by @Sankrit99 from their variability constraints. The Schwarzschild radius for the SMBH in M87 is $\sim10^{15}$ cm. Even at $10^{17}$ cm ($100~R_S$) the Keplerian speed is $>10^4$ [km s$^{-1}$]{}, so the lines would be an order of magnitude broader than we observe if the [Ly$\alpha$]{}-emitting gas were even this close to the SMBH. A more likely location for the [Ly$\alpha$]{} clouds is $\sim10^4~R_S$ ($\sim10^{19}$ cm; $\sim3$ pc), the inner reaches of the narrow-line region and of the observed gaseous disk. So, there is little chance that the FOS-observed variability could be due to a very small source size. The larger distance of $\sim3$ pc from the SMBH is consistent with the observed angular extent of the [Ly$\alpha$]{} components in the COS aperture as well as the observed line-width, but leaves the large variability amplitude over short time periods as a puzzle. It seems most likely that the reported 5 day [Ly$\alpha$]{} variability is spurious, due to a mis-placement of the FOS aperture as suggested by @Sankrit99. Currently there is no unambiguous indication for emission-line gas closer than 3 pc, at much larger radii than where the BLR of Seyferts exists.
The suppression of BLR emission in BLLacs was first explored by @Guilbert83 who pointed out that the steep spectrum ionizing continua seen in BLLacs will inhibit the creation of cool, $10^4$ K clouds that are typical of Seyferts and QSOs. For this mechanism to be operable in FR1s, this steep ionizing continuum should be illuminating much of the potential BLR of these AGN. Indeed, the steep X-ray spectra of BLLacs must be much less beamed (half illumination angle $\sim1$ radian) than the radio and optical continua based on their observed properties and source counts [@UrryPadovani95] when selection is made in the X-ray versus radio or optical bands. Therefore, the presence of broad-line emitting gas is [**not**]{} expected in this class of AGN, as we observe.
Unlike the narrow core emission in M87, the red and blue wings to the [Ly$\alpha$]{} emission are much broader and appear to be offset spatially from the main [Ly$\alpha$]{} emission along the jet axis (see Sec 3.4). The red wing emission also varies over the same observing epochs and is very likely intrinsic variability since there is no observed absorption in M87 at these wavelengths. Given their oppositely-trending spatial extents, the red and blue emission wings are likely to arise in gas illuminated or shocked by the jet (blue wing) and counter-jet (red wing). Spatially-resolved STIS spectra can confirm this suggestion.
FR1 and BLLac Accretion Powers
------------------------------
In the three FR1s observed by HST/COS, the [Ly$\alpha$]{} luminosities are comparable to or somewhat less than what is expected in “Case B” recombination theory assuming unity covering factor. In M87 previous epochs saw lower [Ly$\alpha$]{} luminosities while the continuum luminosities remained the same. Therefore, at the highest [Ly$\alpha$]{}luminosities observed, an ionization-bounded scenario is possible in these AGN but, even then, the amount of gas must be close to the amount needed to absorb all the available ionizing photons given the OPF $\approx1$ values found here. If the line-emitting clouds are in a density-bounded regime then, as illustrated in Paper 1 using the [Ly$\alpha$]{}line luminosities seen in BLLac objects, there is very little warm gas in this region, $10^{-4}-10^{-5}~M_\sun$. This amount is inferred by assuming that there is one hydrogen atom for every [Ly$\alpha$]{} photon emitted. A reasonable energy conversion rate of this mass (e.g., $\sim$1%) finds a very low estimated accretion rate, which is consistent with earlier estimates using a variety of methods [@Kuo14; @Russell15; @DiMatteo16].
It is also possible that the observed [Ly$\alpha$]{} gas is [*outflowing*]{} rather than infalling, which decreases the estimated accretion rate even further. This possibility is consistent with the limited red/blue wing spatial information from the COS aperture (Section 3.4). It is important to verify this assertion using the full HST spatial resolution with STIS.
There could be a significant amount of much hotter, “coronal” gas in the line-emitting region surrounding the cooler, [Ly$\alpha$]{}-emitting clouds. Using the results from @AndersonSunyaev16 on the detection of [\[\]]{}, which we confirm here, an estimate of particle density of $0.2\rm~cm^{-3}$ can be derived for a temperature of $T\approx10^7$ K based on the presence of the [\[\]]{} feature. Assuming this hot coronal gas fills the region around the SMBH out to $\sim3$ pc, this phase contains $\sim0.3~M_\sun$ of gas. These values are consistent with the recent re-analysis of [*Chandra*]{} X-ray images of M87 conducted by @Russell15. Although we have no good, direct measurement of the density of the [Ly$\alpha$]{}-emitting clouds, if the coronal gas is in pressure equilibrium with these clouds, their densities are a few $\times10^{-4}\rm~cm^{-3}$. It is reasonable to suspect that the [Ly$\alpha$]{}-emitting cloud ensemble is gas which has condensed out of this hotter phase in either an infalling or outflowing wind.
Extragalactic Ionizing Continuum Supplied by FR1s
-------------------------------------------------
If the ionizing continuum radiation extrapolated from the HST/COS spectra of M87 and NGC4696 is typical of FR1 radio galaxies, then the contribution of this class of AGN to the extragalactic UV background (UVB) in the local universe may be substantial since this AGN class is so numerous [$\sim$3% of all bright cluster ellipticals have $\log P(\rm W\,Hz^{-1})\geq22.0$; @LedlowOwen96].
The beaming geometry in FR1s also must be taken into account in estimating the amount of UVB contributed by these objects. For example, there is some evidence from BLLac population studies [e.g., @Perlman93; @UrryPadovani95; @Nieppola06; @Padovani07; @Meyer11; @Giommi13] that the amount of Doppler boosting (i.e, the relativistic $\Gamma$) varies with frequency of emitted radiation. This is also known as the re-collimating or accelerating jet model [@Ghisellini89; @UrryPadovani95; @BoettcherDermer01] for blazars. X-ray-selected BLLacs (most of which are now termed High-energy peaked BLLacs or HBLs) often lack the optical non-thermal BLLac spectrum, instead showing either a pure stellar spectrum or a mixture of an old stellar population and a non-thermal continuum. This property and their large space density means that the X-ray beam must be much broader than the optical and radio emitting beams, presumably coming from particles lower in $\Gamma$. This hypothesis is supported by the optical polarization studies conducted by @Jannuzi94 which showed that HBLs possess only modestly variable polarization amounts and relatively constant position angles of polarization as if we are viewing HBL optical beams from well off their axes. A gradient of broader to narrower emission beams (lower to higher $\Gamma$s) with increasing wavelength predicts that the UV continuum beam is somewhat narrower than the $\sim25-30$ for the X-ray emission [e.g., @Perlman93]. Due to this somewhat rudimentary understanding of the beaming geometry, here we adopt a simplistic model with a beamed ionizing flux emanating from two cones with half-angles of 20 and an unbeamed flux given by our FR1 observations emitted over the rest of the sphere.
Since the extrapolated Lyman continuum flux of HydraA it is almost two orders of magnitude more luminous than that of M87, we assume that some significant beaming is present in our direction towards this source and we do not use its ionizing flux as typical of unbeamed FR1s. Using just M87 and NGC4696 as typical for the unbeamed FR1 population, the ionizing continuum luminosity in these objects is $\sim10^{40}$ ergs s$^{-1}$. If only a small fraction of luminous early-type galaxies are FR1s, then the ionizing radiation contributed by this class is also quite small. But the evidence is otherwise; e.g., @LedlowOwen96 find that 3% of all bright cluster ellipticals have radio luminosities with $\log P_{1.4~GHz}
(\rm W\,Hz^{-1})\geq22$ while @LinMohr07 find a larger fraction of 5% with $\log P_{1.4~GHz} (\rm W\,Hz^{-1}) \geq 23$ in X-ray emitting clusters. At X-ray wavelengths @Martini06 found that $\sim2$% of all bright cluster ellipticals emit X-rays at $P_x\geq10^{42}$ ergs s$^{-1}$. In a combined X-ray/radio study of cluster AGN, @Hart09 found 6% of bright cluster ellipticals are radio sources at $\log P_{1.4~GHz} (\rm W\,Hz^{-1})\geq 23.5$ and 1% are X-ray sources at $P_x \geq 10^{42}$ ergs s$^{-1}$. Extrapolation of these results to lower radio and X-ray power levels is consistent with all bright cluster ellipticals being radio sources at $\log P_{1.4~GHz} (\rm W\,Hz^{-1}) \geq 21.4$ and X-ray sources at $P_x \geq 10^{40}$ erg s$^{-1}$. Indeed, a deep [*Chandra*]{} observation of the central region of the Perseus Cluster [@Santra07] has detected weak ($10^{40-41}$ ergs s$^{-1}$) non-thermal X-ray point sources from [*all thirteen*]{} early-type galaxies brighter than $0.2\,L^*$ in this region. In the @Santra07 X-ray imaging, there is no obvious trend in X-ray luminosity with optical luminosity, suggesting that the AGN emission in these X-ray point sources is similar in luminosity for all bright cluster ellipticals. Together, all these studies suggest that virtually all bright early-type cluster galaxies are weak FR1s and could be emitting Lyman continuum radiation at levels similar to M87. Despite consistent results from the several studies cited above, this important assumption must be confirmed in order to validate our conclusions concerning the FR1 contribution to the UVB.
The CfA galaxy luminosity function of elliptical galaxies from @Marzke94 finds that the number density of $L>0.2L*$ ellipticals is $4\times10^{-3}$ Mpc$^{-3}$ (we ignore S0 galaxies in this estimate since they appear not to harbor FR1s in general). Following the @Santra07 result as well as the extrapolations from higher power levels made in the other studies quoted above, we assume that all bright ellipticals emit Lyman continuum radiation at a level similar to M87 and NGC4696. From this assumption we derive an estimate of the unbeamed UV emission from FR1s of $\sim4\times10^{37}$ ergs s$^{-1}$ Mpc$^{-3}$. This amount is $\sim7$% of the total ionizing background at $z\approx0$ [assuming the total UV background at $z\sim0$ from @Haardt12].
If the UV luminosity of $10^{40}$ ergs s$^{-1}$ is the unbeamed continuum then beaming of these sources can add significantly to this total, perhaps as much as an additional 7% at $z\approx0$ if the standard beaming model of @UrryPadovani95 is assumed (half opening angle of $\approx20$; see above). This standard model assumption is equivalent to assuming that 1% of all FR1s are seen as beamed sources at any one location and that their mean observed Doppler boosting factor is 20 in the UV. There are a very few (roughly one in $10^6$) FR1s whose beamed non-thermal luminosities are at the luminosities of $L_x>10^{44}$ ergs s$^{-1}$ [@Morris91] typical of HBL type BLLac objects, but these contribute negligibly to the UVB total.
Overall the FR1 AGN class could contribute an amount to the ionizing background of up to 1/7th (10-15% of the total UVB) of the amount contributed by QSOs and Seyferts locally. If the above assumptions about beaming and the FR1 source population are correct, luminous ellipticals contribute non-negligibly to the amount of ionizing background inferred from source population studies at low-$z$ [@Haardt01; @Haardt12]. A sizeable contribution from FR1s could account for the discrepancy between the most recent UV background model of @Haardt12 (which has minimal contribution from normal, star forming galaxies) and the observed absorber statistics in the [Ly$\alpha$]{} forest [@Kollmeier14; @Shull15]. However, since most bright ellipticals are found in rich clusters this putative additional UV radiation could be very non-uniform, clumped around regions rich in early-type galaxies. A more detailed look at FR1 radio galaxy beaming geometries and number density statistics is required at assess these possibilities in more detail.
Support for the idea that a large contribution to the UVB could be made by FR1s/BLLacs comes from the all-sky survey of far-UV sources made by the [*Extreme Ultraviolet Explorer*]{} (EUVE). This continuum survey conducted at a wavelength closer to the Lyman limit than any other, detected more BLLacs and narrow-line Seyferts than any other AGN class [@Marshall95]. These two AGN classes share an unusual SED with a very steep soft X-ray spectrum as well as the absence of broad permitted lines in their optical and UV spectra. The three FR1s studied here share the absence of broad emission lines with these more luminous, EUV-bright AGN classes.
Conclusions & Summary
=====================
We present far-UV spectroscopic observations of three FR1 radio galaxies (M87, NGC4696, & HydraA) and two previously-unpublished BLLac objects (1ES1028$+$511 and PMNJ1103$-$2329). All three FR1 galaxies show prominent [Ly$\alpha$]{}emission at very close to the systemic redshifts and a weak FUV continuum.
[**M87**]{} is the brightest and most interesting of the objects observed. Though the COS/G130M spectral range is relatively short, we measure a power law index of $\alpha_\lambda=0.87\pm0.06$ and extrapolate the flux at the Lyman limit of $F_{912}=(1.78\pm0.04)
\times10^{-15}$ [$\rm erg~cm^{-2}~s^{-1}~\AA^{-1}$]{} at epoch 2014.
The strong [Ly$\alpha$]{} emission of M87 is flanked by red and blue wings separated from the main peak by $\sim1000-2000$ [km s$^{-1}$]{}. Weak , , and emission lines are also seen at the redshift of M87 and we confirm the very weak [\[\]]{} feature published by @AndersonSunyaev16 at a $5\sigma$ level.
With the limited spatial information available with HST/COS, we find that the emitting regions of both line and continuum are intermediate between that of a point-source and a filled COS aperture implying that both extend over regions tens of parsecs wide. Furthermore, the continuum may be slightly more extended than the line emission and the red and blue wings of the [Ly$\alpha$]{} emission are extended along the jet axis consistent with being due to outflowing material from the nucleus. The most puzzling feature of the M 87 FUV data is the possible very short ($\sim5$ day) [Ly$\alpha$]{} line variability seen in the FOS data which is inconsistent both with the spatial extent of the [Ly$\alpha$]{} emission seen in the COS aperture and the observed, rather narrow line widths of [Ly$\alpha$]{}. @Sankrit99 raise a concern that a slight displacement of the FOS aperture could have falsely created the apparent [Ly$\alpha$]{} variability. In light of the other evidence, the short-term [Ly$\alpha$]{} variability appears to be spurious although this can be confirmed or refuted by future COS and STIS FUV spectroscopy (approved HST Cycle 24 program 14277). It is expected that future long-slit STIS spectra also can trace the extents of the three components of [Ly$\alpha$]{} and the FUV continuum to provide a viable physical picture of the M87 nucleus.
[**NGC4696**]{} shows an asymmetric [Ly$\alpha$]{} profile and a very weak UV continuum. We model the emission as a strong central component and weak red wing overlaid on at least one strong absorption system at the systemic redshift. We confirm the presence of weak [\[\]]{}emission in this object as well.
[**HydraA**]{} shows a simple broad-plus-narrow [Ly$\alpha$]{} emission profile and a moderate FUV continuum. Metal-ion absorption lines in $+$ and show the presence of neutral, metal-enriched gas associated with HydraA, but it is not known if this gas is circumnuclear or associated with the host galaxy on a broader scale. The [Ly$\alpha$]{} emission profile is well-fit without intrinsic absorption. Despite the appearance of an edge-on dust disk in optical and FUV images of HydraA [@Tremblay15], the observations of strong continuum and [Ly$\alpha$]{} line emission–as well as the lack of narrow absorption seen in the other two FR1s–leads us to conclude that the dust disk misses the nuclear region itself.
The BLLac objects [**1ES1028$+$511**]{} and [**PMNJ1103$-$2329**]{} show low-significance ($3-6\sigma$) [Ly$\alpha$]{} emission features consistent with other BLLac objects from the literature (e.g., Paper 1; Fang [et al.]{} 2014). The [Ly$\alpha$]{} luminosities of these objects are comparable to that of the FR1s discussed here and in Paper 1. While it is generally accepted that FR1s are the parent population for BLLac objects (and in fact optical and X-ray imaging of BLLacs reveal that they reside in giant elliptical galaxies in clusters, as do FR 1s [see e.g., @Wurtz96; @Donato04; @Sambruna07], the emission mechanisms, source locations and geometries for the beamed and unbeamed ionizing continua and [Ly$\alpha$]{} emission lines are still unclear. As the nearest unobscured FR1 (Centaurus A is a bit closer but has a nucleus obscured by dust), M 87 provides the best opportunity to observe the nuclear structures in “radio-mode” AGN and solve the puzzle of their apparent very low accretion rates. New high spatial and spectral resolution observations in the FUV with COS and STIS will complement upcoming efforts at X-ray wavelengths and at mm wavelengths with VLBI to determine the size and geometry of the line and continuum-emitting regions in FR1s.
The observed [Ly$\alpha$]{} luminosity in M87 and NGC4696 is within a factor of a few to what is predicted by “Case B” recombination illuminated by the non-thermal continuum in these sources extrapolated to the Lyman edge. Thus the circumnuclear gas could be either in an ionization- or density-bounded regime. The ionizing continuum of HydraA overpredicts the observed [Ly$\alpha$]{} luminosity in this FR1 by a factor of ten suggesting that the observed continuum has some significant contribution from beamed radiation as well as the more isotropic radiation seen in the two other FR1s. The extrapolated FUV continuum flux of BLLac objects overpredicts the observed [Ly$\alpha$]{} line flux by factors of $10^3-10^5$. These results are consistent with all of these sources possessing an unbeamed non-thermal continuum that is the primary ionizing source for the gas that produces the observed [Ly$\alpha$]{} emission line. While both the continuum and [Ly$\alpha$]{} line are seen to vary, the substantial distance (1-10 pc) we have inferred for the [Ly$\alpha$]{} emitting region does not allow the tight connection between continuum and line variability that is seen for Seyfert galaxies [“reverberation mapping”; @Peterson14]. However, one apparently successful narrow-line-region distance determination has been obtained for NGC5548 using the \[\] 5007Å forbidden line [@Peterson13]. A similar study may be possible for M 87.
FR1 radio galaxies are very common in the modern era, comparable in space density to Seyfert galaxies. Recent work at radio and X-ray wavelengths [e.g., @Martini06; @Santra07; @LinMohr07; @Hart09] make a plausible case that virtually all bright ellipticals are AGN with ionizing luminosities comparable to or greater than what we find in M87 ($P_{UV}=10^{40}$ ergs s$^{-1}$). If it is assumed that all $L\geq0.2L^*$ ellipticals possess unbeamed non-thermal continua at this level, the contribution to the low-$z$ UVB could be substantial, perhaps as much as $\sim10$% of the extragalactic ionizing background.
We acknowledge valuable input on this project from Mike Shull and Mike Anderson. This work was made possible by HST guest observing grant GO-13489 (J. Stocke, PI).
[*Facility: HST (COS)*]{}
Allen, S. W., Dunn, R. J. H., Fabian, A. C., Taylor, G. B., & Reynolds, C. S. 2006, , 372, 21
Anderson, M. E. & Sunyaev, R., 2016, , 459, 2806
Begelman, M. C., & Celotti, A. 2004, , 352, L45
Blandford, R. D., & Begelman, M. C. 1999, , 303, L1
Böttcher, M., Dermer, C. D., 2002, , 564, L86
Branchesi, M., Gioia, I. M., Fanti, C., Fanti, R., Perley, R., 2006, å, 446, 97
Broderick, A. E., Narayan, R., Kormendy, J., Perlman, E. S., Rieke, M. J., Doeleman, S. S., 2015, , 805, 179
Burns, J. O. 1990, Astronomy, 18, 29
Canning, R. E. A., Fabian, A. C., Johnstone, R. M., Sanders, J. S., Crawford, C. S., Ferland, G. J., Hatch, N. A., 2011, , 417, 3080
Carter, D., & Jenkins, C. R. 1992, , 257, 7P
Carter, D., Johnstone, R. M., & Fabian, A. C. 1997, , 285, L20
Croton, D. J., [et al.]{} 2006, , 365, 11
Danforth, C. W., Keeney, B. A., Stocke, J. T., Shull, J. M., & Yao, Y. 2010, , 720, 976
Danforth, C. W., [et al.]{}, 2016, , 817:111
Di Matteo, T., Allen, S. W., Fabian, A. C., Wilson, A. S., & Young, A. J., 2003, , 582, 133
Di Matteo, T., Croft, R. A. C., Feng, Y., Waters, D., & Wilkins, S. 2016, (ArXiv:1606.08871)
Donato, D., Gliozzi, M., Sambruna, R. M., Pesce, J. E., 2003, å, 407, 503
Fabian, A. C., Sanders, J. S., Allen, S. W., et al. 2011, , 418, 2154
Fanaroff, B. L., & Riley, J. M. 1974, , 167, 31P
Fang, T, Danforth, C. W., Buote, D. A., Stocke, J. T., Shull, J. M., Canizares, C. R. & Gastaldello, F., 2014, , 795:57
Ford, H. C., [et al.]{} 1994, ., 435, L27
Ford, H., Tsvetanov, Z., Ferrarese, L., Kriss, G., Jaffe, W., Harms, R., & Dressel, L., 1997, IAU Colloq. 163: Accretion Phenomena and Related Outflows, ed. Wickramasinghe, Bicknell & Ferrario, 121, 620.
Ford, H., & Tsvetanov, Z. 1999, The Radio Galaxy Messier 87, 530, 278
France, K. [et al.]{} 2011, , 743, 186
Ghisellini, G. & Maraschi, L., 1989, , 340, 181
Giommi, P., Padovani, P., Polenta, G., 2013, , 431, 1914
Guilbert, P. W., McCray, R., & Fabian, A. C., 1983, , 266, 466
Haardt, F., & Madau, P. 2001, in “Clusters of Galaxies and the High Redshift Universe Observed in X-rays”, Neumann & Tran, (ArXiv:0106018)
Haardt, F., & Madau, P. 2012, , 746, 125
Harms, R. J. [et al.]{} 1994, , 435, L35
Hart, Q. N., Stocke, J. T ., & Hallman, E. J., 2009, , 705, 854
Hawley, J. F., & Balbus, S. A. 2002, , 573, 738
Ho, L. C., Filippenko, A. V., Sargent, W. L. W., & Peng, C. Y. 1997, , 112, 391
Ho, L. C., & Peng, C. Y. 2001, , 555, 650
Ho, L. C., & Ulvestad, J. S. 2001, , 133, 77
Jannuzi, B. T., Smith, P. S., & Elston, R., 1994, , 480, 130
Kaspi, S. & Behar, E., 2006, , 636, 674
Keeney, B. A., Danforth, C. W., Stocke, J. T., France, K., & Green, J. C. 2012, , 124, 830
Kollmeier, J. A., Weinberg, D. H., Oppenheimer, B. D., [et al.]{} 2014, , 789, L32
Kormendy, J., & Ho, L. C. 2013, , 51, 511
Kuo, C.Y., Asada, K., Rao, R., Nakamura, N. et al. 2014, , 783, L33
Lauer, T. R., Postman, M., Strauss, M. A., Graves, G. J., & Chisari, N. E. 2014, , 797, 82
Ledlow, M. J., & Owen, F. N. 1995, , 110, 1959
Ledlow, M. J., & Owen, F. N. 1996, , 112, 9
Lin, Y.-T. & Mohr, J. J. 2007, , 170, 71
Macchetto, F., Marconi, A., Axon, D. J., Capetti, A., Sparks, W., & Crane, P. 1997, , 489, 579
Madrid, J. P., 2009, , 137, 3864
Marshall, H. L., Fruscione, A., & Carone, T. E. 1995, , 439, 90
Martini, P., Kelson, D. D., Kim, E., Mulchaey, J. S., & Athey, A. A. 2006, , 644, 116
Marzke, R. O., Geller, M. J., Huchra, J. P. & Corwin, Jr., H. G. 1994, , 108, 437
Mauch, T., & Sadler, E. M. 2007, , 375, 931
McNamara, B. R., & Nulsen, P. E. J. 2007, , 45, 117
Meyer, E. T., Fossati, G., Georganopoulos, M., Lister, M. L., 2011, 740, 98
Meyer, E. T., Sparks, W. B., Biretta, J. A., Anderson, J., Sohn, S. T., van der Marel, R., Norman, C., Nakamura, M., 2013, , 774, 21
Morris, S. L., Stocke, J. T., Gioia, I. M., Schild, R. E., Wolter, A., Maccacaro, T., & della Ceca, R. 1991, , 380, 49
Narayan, R., & Yi, I. 1994, , 428, L13
Narayan, R., & Yi, I. 1995, , 444, 231
Nieppola, E., Tornikoski, M., Valtaoja, E., 2006, å, 445, 441
Owen, F. N., Ledlow, M. J., & Keel, W. C. 1996, , 111, 53
Padovani, P, Giommi, P., Landt, H., Perlman, E. S., 2007, , 662, 182
Perlman, E. S., Frye, C., Ebeling, H., Jones, L. R., Scharf, C. A., Horner, D., 2003, arXiv: astro-ph/0303533.
Perlman, E. S., Mason, R. E., Packham, C., Levenson, N. A., Elitzur, M., Schaefer, J. J., Imanishi, M., Sparks, W. B., Radomski, J., 2007, , 663, 808
Perlman, E. S. & Stocke, J. T., 1993, , 406, 430
Perlman, E. S., [et al.]{} 2011, , 743, 119
Perlman, E. S., Sparks, W. B., Radomski, J., Packham, C., Fisher, R. S., Pina, R., Biretta, J. A., 2001, , 561, L51
Peterson, B.M., Denney, K.D., DeRosa, G., Grier, C.J., et al. 2013, , 779, 109
Peterson, B.M., Grier, C.J., Horne, K., Pogge, R.W., et al., 2014, , 795, 149
Prieto, M.A., Fernandez-Ontiveros, J.A., Markoff, S., Espada, D., & Gonzalez-Martin, O. 2016, . 457, 3801
Quataert, E., & Gruzinov, A. 2000, , 539, 809
Roman-Duval, J. [et al.]{} 2013, “[COS/FUV Spatial and Spectral Resolution at the new Lifetime Position]{}”, Instrument Science Report COS 2013-07
Russell, H. R., Fabian, A. C., McNamara, B. R., & Broderick, A. E. 2015, , 451, 588
Sambruna, R. M., Donator, D., Tavecchio, F., Maraschi, L., Cheung, C. C., Urry, C. M., 2007, , 670, 74
Sankrit, R., Sembach, K.R. & Canizares, C.R. 1999, , 527, 733
Santra, S., Sanders, J.S., & Fabian, A.C., 2007, , 382, 895
Shull, J. M., Stevans, M., & Danforth, C. W., 2012, , 752, 162
Shull, J. M., Moloney, J., Danforth, C. W., & Tilton, E. M. 2015, , 811, 3
Sikora, M., & Begelman, M. C. 2013, , 764, L24
Sitko, M. L., & Junkkarinen, V. T. 1985, , 97, 1158
Stocke, J. T., Danforth, C. W., and Perlman, E. S., 2011, , 732, 113 (Paper 1)
Stocke, J. T., Perlman, E. S., Gioia, I. M., Harvanek, M., 1999, , 117, 1957
Tremblay, G. R., [et al.]{} 2015, , 451, 3768
Tsvetanov, Z. I., Hartig, G. F., Ford, H. C., [et al.]{} 1998, , 493, L83
Tsvetanov, Z. I., Hartig, G. F., Ford, H. C., [et al.]{} 1999, The Radio Galaxy Messier 87, 530, 307 (The Nuclear Spectrum of M87)
Ulrich, M. H. 1981, , 103, L1
Urry, C. M., & Padovani, P., 1995, PASP, 107, 803
Vermeulen, R. C., Ogle, P. M., Tran, H. D., Browne, I. W. A., Cohen, M. H., Readhead, A. C. S., Taylor, G. B., & Goodrich, R. W., 1995, , 452, L5
Walsh, J. L., Barth, A. J., Ho, L. C., & Sarzi, M., 2013, , 770, 86
Wing, J., Blanton, E., 2011, , 141, 88
Winter, L. M. [et al.]{} 2011, , 728, 28
Wurtz, R., Stocke, J. T., Ellingson, E., Yee, H. K. C., 1997, , 480, 547
[^1]: Based on observations made with the NASA/ESA [*Hubble Space Telescope*]{}, obtained from the data archive at the Space Telescope Science Institute. STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS5-26555.
|
---
abstract: 'We study the stability of two coexisting languages (Catalan and Spanish) in Catalonia (North-Eastern Spain), a key European region in political and economic terms. Our analysis relies on recent, abundant empirical data that is studied within an analytic model of population dynamics. This model contemplates the possibilities of long-term language coexistence or extinction. We establish that the most likely scenario is a sustained coexistence. The data needs to be interpreted under different circumstances, some of them leading to the asymptotic extinction of one of the languages involved. We delimit the cases in which this can happen. Asymptotic behavior is often unrealistic as a predictor for complex social systems, hence we make an attempt at forecasting trends of speakers towards $2030$. These also suggest sustained coexistence between both tongues, but some counterintuitive dynamics are unveiled for extreme cases in which Catalan would be likely to lose an important fraction of speakers. As an intermediate step, model parameters are obtained that convey relevant information about the prestige and interlinguistic similarity of the tongues as perceived by the population. This is the first time that these parameters are quantified rigorously for this couple of languages. Remarkably, Spanish is found to have a larger prestige specially in areas which historically had larger communities of Catalan monolingual speakers. Limited, spatially-segregated data allows us to examine more fine grained dynamics, thus better addressing the likely coexistence or extinction. Variation of the model parameters across regions are informative about how the two languages are perceived in more urban or rural environments.'
author:
- 'Luís F. Seoane$^{1,2,3,@,*}$, Xaquín Loredo$^{4}$, Henrique Monteagudo$^{5,6}$, and Jorge Mira$^{7,@,*}$'
title: 'Is the coexistence of Catalan and Spanish possible in Catalonia?'
---
Introduction {#sec:1}
============
Catalan is a Romance language evolved in the north-east of the Iberian peninsula after the fall of the Roman Empire [@Lleal1990; @Vila2008]. This language was linked to the political institutions emerging in that region within the Crown of Aragón. This polity included some of the south departments of modern France and the prominent county of Barcelona, often center of the government of all these territories. The Crown of Aragón expanded southwards along the Mediterranean coastline (nowadays Valencia) and beyond the Iberian peninsula bringing the Catalan language to the Balearic and other Mediterranean islands. Variants of this tongue are still spoken in Sardinia and southern France.
Castillian (nowadays Spanish) is a Romance language evolved in the north central part of the Iberian peninsula at the same time as Catalan [@Lleal1990]. Castillian was linked to the political institutions of the region, notably the Crown of Castille, which extended southwards, like Aragón, as the struggle between the Christian and Islamic forces unfolded in the Iberian peninsula during the Middle Ages. Similar historic stages can be seen in the spread of Portuguese along the Atlantic coast, while Castillian advanced across the inland territories. When the colonial era began, both Castillian and Portuguese diffused prominently through the American continent [@Penny2002; @Pharies2008].
Back in the Iberian Peninsula, the Portuguese and Castillian kingdoms remained different political entities in the long term. The Crowns of Aragón and Castille became unified forming the embryo of modern Spain (which involved also other regional kingdoms). Explicit political measurements were tried to mitigate the social split between the populations originating from the old, separate kingdoms. But, despite these efforts, the Catalan-Castillian linguistic division lasted through the creation of the Spanish nation-state. The existence of a vernacular language (Catalan) different from the official language of the Spanish kingdom (Castillian or Spanish), together with other cultural singularities, provided support for the emergence of regionalist and nationalist movements during the XIX and XX centuries in Catalonia and other Catalan-speaking regions [@Llobera2003]. After the Spanish Civil war (1936-1939), Catalan was repressed and banned from official communications [@Guibernau2004]. The restoration of democracy in Spain (symbolized by its modern constitution from 1978) awarded Catalan a co-official status in Catalonia. Different policies have ever since been articulated to protect Catalan from decline – these range from subsidized presence in the mass media to different strategies within the educative systems [@Pradilla2001].
The past and future evolution of the Spanish and Catalan languages is tightly linked to the complex political scenario of modern Spain. Secessionist Catalan movements have grown steadily during the past decades [@growingIndep], currently reaching a climax with an explicit agenda towards full independence formulated by the Catalan government. While language use is not a definitive factor, it certainly correlates with the different political actors [@Clua2014; @Woolard2008]. Understanding and forecasting the evolution of the Catalan-Spanish tongues in Catalonia from a population dynamics perspective is, hence, a relevant political goal.
These dynamics become even more important from a sociolinguistic point of view. We worry about the survival and peaceful coexistence of linguistic communities. This is the mindset that we wish to adopt in this paper. Our approach is of relevance beyond the specific example that we deal with here. The total number of languages across the globe is reported to decline and the number of endangered tongues augments [@Sutherland2003]. While this is the overall context, here we focus on the particular Catalan-Spanish coexisting dynamics. Abundant, curated data is available for this system. Also, as argued above, the ongoing political scenario makes it a very appealing study case.\
{width="\linewidth"}
The mathematical characterization of population dynamics is well rooted within the field of ecology [@Kot2001; @Turching2003]. Recent works have extended this kind of analysis to different social aspects, including the evolution of speakers of different coexisting languages [@BaggsFreedman1990; @BaggsFreedman1993; @AbramsStrogatz2003; @MiraParedes2005; @Kandler2008; @Castellano2009; @PatriarcaHeinsalu2009; @CastelloSan2013; @ZhangGong2013]. The approach is based on sets of differential equations able to reconstruct historical series of data and, hopefully, make informative predictions.
The reconstruction part of this problem has been successful in various scenarios. Relevant examples are the modeling of up to $42$ cases of language coexistence by Abrams and Strogatz [@AbramsStrogatz2003]. One of the studied cases was that of Sottish Gaelic. This was further investigated in its full complexity (which implies language dynamics across different territories) by Kandler et al. [@KandlerSteele2010]. Besides accounting for historical data, these authors make a first effort in prediction in an uncertain political environment. This is a scenario similar to ours. Forecasting in the face of unsettled (political) struggles is an ambitious goal that also calls for a warning: there is a limit to the contingencies that our mathematical models can account for, and hence no prediction can be taken as definitive. This is also an opportunity for the valuable interplay between theory, its predictions, and its prospective failures; to improve our models.
We based our work on a vast amount of official data collected by the Institut d’Estadística de Catalunya (Catalan Institute of Statistics) during the last decades [@EULP2003a; @EULP2003b; @EULP2008a; @EULP2008b; @EULPSummary; @EULP2013]. These are very rich data sets with more details available than our models can account for. Hence, important simplifications and preprocessing of the data (described below) were necessary. Ultimately, we modeled the data according to the proposal by Mira et al. [@MiraParedes2005]. This relies on a system of differential equations that track over time two monolingual populations along a bilingual one. The stability of this model has been characterized during the last years [@MiraNieto2011; @OteroMira2013; @ColucciOtero2014; @SeoaneMira2017], which makes it a powerful tool for our analysis. However, we wish to invite future contributions to examine the same or similar data using alternative equations as well.
Methods {#sec:2}
=======
Data sources and pre-processing
-------------------------------
The original data were gathered by the Institut d’Estadística de Catalunya (Idescat, [*Catalan Institute of Statistics*]{}) in the EULP (Enquesta d’Usos Lingüístics de la Població, [*Survey of Language Use by the Population*]{}) surveys [@EULP2003a; @EULP2003b; @EULP2008a; @EULP2008b; @EULPSummary; @EULP2013], which have been conducted every five years since $2003$. The relevant item for us is the self-assessment of language use, in which the respondents reported as a percentage their daily use of each language as detailed below. This item is absent in the $2003$ studies [@EULP2003a; @EULP2003b], so we only used the $2008$ and $2013$ editions [@EULP2008a; @EULP2008b; @EULP2013].
Each speaker self-assessed, as a percentage, her daily use of each of the tongues that she speaks. These included Catalan and Spanish together with Galician, Arab, Urdu, and many others arising from different migratory currents. We want to focus on the Catalan-Spanish dynamics alone. We removed all speakers that used any other language more than $30\%$ of the time. For the speakers retained, we discarded the other languages reported and normalized the data such that Catalan plus Spanish add up to $100\%$. The respondents were stratified in $5$-year intervals to generate a table (Fig. \[fig:01\][**a**]{}) in which each square is associated to the average date in which respondents were born (spanning from $1910$ to $1990$ in EULP-2008 and from $1915$ to $1995$ in EULP-2013) and the percentage of Catalan use that they reported. Hence, each entry of the table contains the estimated number of people in Catalonia of the same age that would report a same percentage of Catalan use. We treat this age-stratified data as a proxy about the proportion of Catalan speakers at the time that each group was born. This is inspired by other [*apparent-time*]{} studies [@Labov1963; @Eckert1997; @Mague2006; @Chambers2013]. This approximation is not free of criticism and has been more often used to study variants of a same language, but it is a suitable way to generate a data series from the available data.
In Catalonia more than half of the population concentrates in Barcelona and its metropolitan area, which may arguably have different dynamics from the rest of the region. Accordingly, after studying the population dynamics inferred from global time series, we repeated the analysis for these two segregated regions (Barcelona plus metropolitan area vs rest of the territory).
In every case, a table similar to that in Fig. \[fig:01\][**a**]{} constitutes our raw data. Several models discussed in the literature [@BaggsFreedman1990; @BaggsFreedman1993; @MiraParedes2005; @ZhangGong2013; @MinettWang2008; @HeinsaluLeonard2014] coarse grain language use into three broad categories: two monolinguals and a bilingual one. The model chosen for our analysis [@MiraParedes2005] (see below) does so. We imposed these divisions on our raw data by defining a [*bilingualism threshold*]{} $r$ beyond which a speaker would be considered bilingual. For example, setting $r=20$ every speaker who uses Catalan more than $80\%$ of time during a day is considered a Catalan monolingual, every speaker who uses Catalan less than $20\%$ is considered a Spanish monolingual, and every speaker employing both Catalan and Spanish within $20-80\%$ is considered bilingual. There is not a clear criterion about what threshold to use, so we conducted our analysis for all possible integer values of $r \in [1, 50]$. This covers all cases from the extreme in which anyone employing both languages is considered bilingual, to the stringent situation in which only those using both tongues half of the time score as non-monolinguals.
For each value of the bilingualism threshold we derive a full data series with a proportion $x$ of Spanish monolinguals, a proportion $b$ of bilinguals, and a proportion $y$ of Catalan monolinguals over time (Fig. \[fig:01\][**b**]{}). We built these data series for each EULP survey and combinations of them. Most analysis were conducted on all available data; here we present results for the more robust (less noisy) time series. (See Supplementary Information for details and to explore all existing results, which are consistent throughout.)
Data Availability Statement
---------------------------
The data used in this study has been keenly provided by the Institut d’Estadística de Catalunya (IDESCAT) and is available in their website (http://www.idescat.cat/en/). It can also be found in the different works referenced in this paper. Notwithstanding, the data is only available at those original sources. We do not own the original data and we have not been given permission to make public a centralized summary of it.
Model
-----
To characterize our data we use the model by Mira et al. [@MiraParedes2005; @MiraNieto2011; @OteroMira2013; @SeoaneMira2017] which considers the existence of $X$ (in this case Spanish) and $Y$ (Catalan) monolingual groups and a bilingual group $B$. These groups present fractions $x$, $y$, and $b$ of speakers respectively within a normalized population ($x + b + y = 1$). The model assumes that the probability that monolingual speakers acquire the opposite language is proportional to the prestige ($s_X$ or $s_Y$) of the other language and to the population speaking that other tongue. It is taken $s_X, s_Y \in [0, 1]$ and $s_X + s_Y = 1$ so we can focus on $s \equiv s_X$. Of all speakers acquiring a new tongue, a fraction $k$ of them retains the old one (hence becoming bilinguals) while $1-k$ of them switch and forget. The parameter $k$ is termed [*interlinguistic similarity*]{} [@MiraParedes2005; @MiraNieto2011] and measures how close is the couple of languages as perceived by the population. The probabilities of leaving or entering each group ($X$, $Y$, or $B$) result in a set of differential equations that tell us the time evolution of the linguistic population: $$\begin{aligned}
{dx \over dt} & = & c\Big[ (b + y)(1-k)s(1-y)^a \nonumber \\
&& - x\left( (1-k)(1-s)(1-x)^a + k(1-s)(1-x)^a \right) \Big], \nonumber \\
{dy \over dt} & = & c\Big[ (b + x)(1-k)(1-s)(1-x)^a \nonumber \\
&& - y\left( (1-k)s(1-y)^a + ks(1-y)^a \right) \Big].
\label{eq:01}
\end{aligned}$$ (Only two equations are needed thanks to the normalized population. Also, this is a compact version of the equations, for detailed discussion, e.g. explicit flow between groups, see [@MiraParedes2005; @MiraNieto2011; @OteroMira2013; @SeoaneMira2017].) The parameter $a$ (which has been referred to as [*volatility*]{} [@CastelloSan2013]) affects those speakers that promote language shift (termed [*attracting population*]{} [@SeoaneMira2017; @HeinsaluLeonard2014]). It confers an idea of how persistent the linguistic groups are: the lower $a$ the easier it is for all groups to lose speakers, thus rendering the system more [*volatile*]{} [@ColucciOtero2014].
The stability of this model has been thoroughly characterized as a function of its parameters [@MiraNieto2011; @OteroMira2013; @ColucciOtero2014; @SeoaneMira2017]. If $a>1$, stable solutions include scenarios in which i) the bilinguals and either one of the monolingual groups get extinct and ii) both monolingual groups survive along a bilingual group. Coexistence is usually reached for larger $k$ and relatively balanced prestiges $s_X \sim s_Y$.
Equations are a generalization of the seminal Abrams-Strogatz model [@AbramsStrogatz2003] that promoted non-linear differential equations for the study of language population dynamics (even if earlier, similar approaches existed [@BaggsFreedman1990; @BaggsFreedman1993]). The original equations did not include bilingualism on the grounds that it played a minor role for the languages under research. This is not the case in the Catalan-Spanish coexistence scenario.
Other valuable models consider bilingual situations [@BaggsFreedman1990; @BaggsFreedman1993; @ZhangGong2013; @MinettWang2008; @HeinsaluLeonard2014]. Besides our familiarity with the chosen system of equations, the stability of the alternatives has not always been studied. Some of these models do not contemplate stable, coexist languages [@MinettWang2008] or do so only after alternative parameterizations are included [@BaggsFreedman1993]. It is intensely debated whether languages can coexist steadily in an asymptotic time, but it does not seem appropriate to barren that possibility beforehand. Hence, we decided to conduct our analysis with equations that allow this scenario explicitly.
This model consists of two coupled, non-lineal differential equations with $4$ parameters $\{a, c, k, s\}$ and two initial conditions ($x(t=t^0)$, $y(t=t^0)$). To extract these parameters from the data we followed the fitting procedure described in the Supplementary Information, which basically makes a fast, heuristic least square minimization. Also in the Supplementary Information we compare the best and worst fits and provide plots of the fits from all data series and for all bilingualism thresholds. One example of a good fit is that obtained for $r=10$, shown in Fig. \[fig:01\][**c**]{}, along with an extrapolation towards the end of the XXI century.
{width="\linewidth"}
Results {#sec:3}
=======
The most important result that we extract is that the Catalan-Spanish system of coexisting languages tends, under most circumstances analyzed, to a stable state in which both languages coexist. The data also reveals a few counterintuitive insights that we examine in the next subsections. The discussion concerns mainly the parameters $k$ and $s$ extracted from adjusting the model equations to the different datasets. We can always track down the stability of the system to these two parameters and the initial conditions. The other parameters in equations ($a$ and $c$) are not so determinant regarding the stability. Their trends as a function of the bilingualism threshold are discussed in the Supporting Material.
Stability of the Catalan-Spanish system
---------------------------------------
The most relevant parameters of the model are the interlinguistic similarity ($k$) and prestige ($s$) which have intuitive interpretations owing to their roles in equations . Thanks to previous studies of the model [@MiraNieto2011; @OteroMira2013; @ColucciOtero2014] we know how to link these parameters to the stability of the system. Figs. \[fig:02\][**a-d**]{} show how the $k-s$ plane is divided into two regions: a gray area where coexistence is possible (depending on the intial conditions) and a white area where coexistence is never possible. For these plots, $a = 1.31$ (a value inherited from the original Abrams-Strogatz studies [@AbramsStrogatz2003]). For other values of $a \le 1$ a similar division of the plane happens[@OteroMira2013], and values of $(k, s)$ exist for which the dynamics are equally well explained. We could have chosen any arbitrary value $a > 1$ without losing explanatory power. Comparisons between $(k, s)$ values only make sense if $a$ is fixed. Hence, to better illustrate the results, we performed our analyses both allowing $a$ to vary and keeping it fixed at $a = 1.31$. Similar conclusions are reached in both cases (see Supplementary Information), but we focus on the fixed case now.
{width="\linewidth"}
For each data set and each bilingualism threshold we derived several collections of parameters compatible with the corresponding time series. The existence of several good fits in each case allows us to perform a statistic analysis that bootstraps the variability of the parameters. Fig. \[fig:02\][**a**]{} shows average and standard deviations for $k$ and $s$ for the different integer values of $r \in [1, 50]$. More generous definitions of bilingualism are explained by models with larger interlinguistic similarity – i.e. smaller values of $r$ lay at the right side of the $k-s$ plane and larger values of $r$ lay towards the middle-left. This range of $k$ values is explained by the differences in the bilingualism threshold $r$. Unluckily, the model and data available cannot offer a rigid constrain on the interlinguistic similarity. We note, though, that in average it does not become arbitrary low – not even for the most restrictive definitions of bilingualism in which only speakers using $50\%$ of the time each language are considered bilinguals.
Our results also indicate that the prestige of Spanish ($s \sim 0.57$) is consistently slightly larger than that of Catalan ($1-s \sim 0.43$) for any value of $r$. These numbers are sustained throughout the whole range $r\in[1, 50]$, strongly suggesting that this is a good descriptor of the Catalan-Spanish system given the data and the model.
Roughly half of the $r$ values allow for asymptotic stability (up to $r \le 22$) and another half ($r \ge 23$) strictly ban it. However, for larger values of $r$ the prestige of both languages becomes more leveled ($s$ is only slightly above $0.5$ for larger $r$) and the fits become less consistent: some of them predict the extinction of one tongue and some others predict the extinction of the other one – both usually in an asymptotic time much larger than $100$ years. This indeterminacy comes about because the system sits near a bifurcation point and the data is not enough to clarify the outcome of the competition dynamics.
Our stability analysis is complemented with an attempt at prediction towards $2030$. Such predictions must be taken with all the prudence possible: they are the results obtained with this model for the data available, and the seemingly open-ended nature of human dynamics does not allow us to have an all comprehensive understanding of the situation. However, some qualitative results outlined below are fairly consistent across datasets, fitting setups, and definitions of bilingualism (through the parameter $r$). This invites us to be confident about the general conclusions.
We registered the percentages of Spanish (Fig. \[fig:03\][**a**]{}), Bilingual (Fig. \[fig:03\][**b**]{}), and Catalan (Fig. \[fig:03\][**c**]{}) speakers predicted by the model for the year $2030$ for each of the combinations of parameters derived for each time series. These year-$2030$ predictions were binned in intervals comprising $0.05$ increments in the fraction of speakers. Fig. \[fig:03\][a-c]{} shows how, consistently, our results indicate a middle-term coexistence between Spanish and Catalan, even for those configurations of parameters that imply the eventual extinction of one of the tongues. These extinction scenarios happen with stringent definitions of bilingualism (large $r$) and present relatively balanced prestiges ($s_S \sim 0.5 \sim s_C$) indicating that even if one language must die eventually, that result will only happen asymptotically and coexistence could perhaps be granted for several generations.
Notwithstanding the bilingualism threshold, the model always predicts a less important role for Catalan language in a middle-term future. Catalan speakers towards $2030$ would always amount to less than Spanish and, more often than not, bilingual speakers. Meanwhile, Spanish stands as the dominating language for some definitions of bilingualism, and it is consistently the group projected to grow the most until 2030. Figure \[fig:03\][**d**]{} shows the expected gain of speakers for each group in $2030$, with Spanish standing out. For $r<30$, Spanish is expected to win most of its new speakers from bilinguals. For $r>30$ both bilinguals and Catalan monolinguals would lose a substantial amount of speakers to Spanish.
Analysis across different geographical areas
--------------------------------------------
The linguistic map of Catalonia contains two very distinguishable regions: On the one hand, Barcelona and its metropolitan area constitute the second largest urban hub in Spain and agglutinates more than $70\%$ of the Catalan population. This is home to large migrant groups from the rest of Spain and elsewhere (notably Pakistan and China), while Barcelona itself is a very cosmopolitan city attracting large masses of tourists. On the other hand, the rest of Catalonia (while still containing notable urban areas and some regions with large migrant populations) has a more rural character and is spread across larger territories.
To further understand the linguistic reality of the system we segregated the data in those two broad geographical regions and repeated our analyses. We found similar tendencies in the parameters $a$ and $c$ (see Supplementary Information). The interlinguistic similarity again drops as the definition of bilingualism becomes more stringent (Fig. \[fig:02\][**b**]{} and [**c**]{}), again showing the limitations of the model and existing data to constrain the value of $k$. However, note once again that the interlinguistic similarity never drops to zero (not even for the most stringent definitions of bilingualism); and note also how it peaks at roughly $k=0.9$ in non-urban areas (Fig. \[fig:02\][**c**]{}), while it comes much closer to $k=1$ in the Barcelona area, thus stressing the difference that population outside Barcelona always perceives between both languages. On the other hand, average values of prestige ($s$) are broadly consistent throughout $r\in[1, 50]$, suggesting that the model and data together are able to constrain this characteristic of the Spanish-Catalan dynamics.
The analysis of the parameter $s$ offers a counterintuitive result: In general, Spanish presents a lower prestige in Barcelona and its metropolitan area than in the rest of Catalonia. Note that Catalan is less spoken in Barcelona: Over the last $100$ years it never had more speakers than Spanish (Supplementary Fig. 1), while the rest of Catalonia does present a larger body of monolingual Catalan speakers (Supplementary Fig. 2). This would naively suggest that the perceived prestige of Catalan is lower in the urban metropolis (hence Spanish prestige would be higher), but our analysis indicates exactly the opposite: rural areas (where Spanish is less spoken) perceive Spanish as a more prestigious tongue. The catch is that while the decay of Spanish and Catalan in the Barcelona area has been roughly symmetric over the last century and favors a strong bilingual group (Supplementary Fig. 1[**a**]{}); in regions outside Barcelona, Spanish speakers have remained relatively constant across time while the larger Catalan group has been decaying in favor of bilinguals (Supplementary Fig. 2[**a**]{}). In equations , a large prestige captures precisely the ability of a smaller group to make a larger (and originally stronger) one decline.
Figs. \[fig:02\][**d-e**]{} summarize the differences between the perception that the two geographical areas have about Catalan and Spanish. The arrows in Fig. \[fig:02\][**d**]{} connect the averages $k$ and $s$ in Barcelona with the averages outside Barcelona. These arrows are replotted with their origin in $(0, 0)$ (Fig. \[fig:02\][**e**]{}) to indicate how speakers outside Barcelona not only assign a slightly lower status to Catalan, but also they perceive both languages as more different – as indicated by the $\sim 0.1$ drop in interlinguistic similarity consistent across similar definitions of bilingualism.
As we did for the aggregated data, we complemented the stability analysis by projecting the evolution of the model into the future until $2030$. Again, both for the metropolitan and non-metropolitan areas, most configurations of the model are compatible with middle-term coexistence of the tongues. When the definition of bilingualism is more rigorous (larger $r$) the asymmetries (either in prestige or initial conditions) become relevant and, in some cases, are capable of substantial gains and losses in number of speakers within the projected time. Within Barcelona, Catalan would be the endangered language (Supplementary Fig. 3). For almost every definition of bilingualism, Spanish would draw most of its new speakers right away from Catalan monolinguals (Supplementary Fig. 3[**d**]{}).
Counter intuitively again, outside the metropolis, for large $r$ Catalan would be able to gain a large number of speakers from Spanish (Supplementary Fig. 4). Despite the larger prestige of Spanish in those regions, this would be possible due to the still large Catalan monolingual population outside the Barcelona area. For very low $r$ in regions outside Barcelona, the large monolingual support of Catalan would fade as the projections predict a larger presence of Spanish monolingual speakers. For intermediate $r$, Catalan would grow notably, but extracting speakers from the bilingual group rather than from the Spanish monolinguals (Supplementary Fig. 4[**d**]{}).
Discussion {#sec:4}
==========
In this paper we analyzed the system of Spanish and Catalan coexistence using recent and thorough data surveys and up-to-date models based on non-linear equations. There is a gap between the theoretical developments and the empirical data available [@SeoaneMira2017]. The former often rely on concepts (e.g. [*bilingual speakers*]{}) that have a clear definition within the model but that are difficult to pin down empirically. Given the complex and subjective nature of the problem under research it is necessary to rely on the self-assessment of linguistic qualities – in this case, percentage of language use.
We wanted to conduct our analysis in the most general way possible given the data. We considered a series of [*bilingualism thresholds*]{} (encoded by $r\in[1, 50]$) and did not assume that any of these thresholds constitutes [*the right definition of bilingualism*]{}. Instead, we performed our analysis for all possible scenarios. This could potentially produce a wealth of models with antagonistic predictions, hence frustrating any robust conclusion. This happens often in complex systems that sit midway between competing forces – which is the case here. Instead, our analysis renders a consistent picture across different data sources and for most different definitions of the bilingual group. This picture is that of a long term coexistence between Spanish and Catalan in Catalonia, always along a bilingual group, and with a dominating role for Spanish while the group of monolingual Catalan speakers declines lightly.
Further details hinge on the bilingualism threshold employed. For roughly half of the definitions of bilingualism ($r \le 22$) it is predicted that both languages will survive. The monolingual Catalan group is expected to be smaller than the Spanish one towards 2030 (Fig. \[fig:03\]), disregarding of $r$. Most of the population would be bilingual for $r \le 22$. For very stringent definitions of bilingualism ($r>40$) some of the models predict the a huge loss of Catalan speakers before 2030, but coexistence between both languages is still the most common outcome. Similar results are obtained when segregating the data between Barcelona plus metropolitan area versus rest of Catalonia. Coexistence of both tongues remains the most persistent outcome, but for strict definitions of bilingualism ($r>40$) large losses of Catalan speakers in Barcelona and of Spanish speakers outside Barcelona become are likely within a few decades.
These mid-term predictions leave considerable room for action. Consequently, they are also daring and should be subjected to continuous revision as new data becomes available. The correctness of our analyses relies on some assumptions: i) The data collected so far is reliable and significant about how the situation might evolve. Consequently, ii) social and political circumstances shall not vary considerably in the future. Unluckily there are no studies about how notable political events affect the smooth dynamics of the system. (This is also true for all other models of language shift [@BaggsFreedman1990; @BaggsFreedman1993; @AbramsStrogatz2003; @MiraParedes2005; @Kandler2008; @Castellano2009; @PatriarcaHeinsalu2009; @CastelloSan2013; @ZhangGong2013].) Should the socio-political stage change drastically (e.g. if Catalonia would become an independent state, a possibility debated nowadays), our analysis might become outdated. iii) We never assume that we are using [*the right theory*]{}. Our equations might not be correct, so indeed this exercise should help us validate the model – even if some predictions lay far in the future.
In this paper we also quantified the perceived prestige of both languages and their interlinguistic similarity. The former cannot be hugely constrained by the model since we lack a definitive measure of bilingualism and, for this dataset, $k$ changes widely with $r$. Results for the prestige parameter were relatively consistent across $r\in[1, 50]$ suggesting that we have successfully captured $s$ for the languages involved. Also, spatially segregated data reveals interesting differences across regions – notably the higher prestige of Spanish in the areas that, historically, had more Catalan speakers. In those regions also the perceived difference across languages is larger. Both these observations hold despite the variation of $r$, strongly suggesting that they are real features of the system.
It could be thought that an objection to our model and others similar is that the parameters are abstract and difficult to relate to more concrete features. Notwithstanding our abstractions, the parameters $s$ and $k$ have definite causal consequences in terms of population flows in our equations. Both steady states and the dynamical unfolding of the equations are intimately linked to their numerical values. Measures under different circumstances (e.g. values of $r$, or geographically stratified data) can be compared to each other and sound conclusions can be extracted. In this sense, both $k$, $s$, and other parameters carry meaningful information about the Catalan-Spanish system. We assess these quantities indirectly (by fitting the data to our model), but perceived prestige or similarity between tongues and other sociolinguistic characteristics can be directly reported in future surveys. These more concrete quantities can then be correlated to our parameters, thus helping us bridge the gap between theory and empirical data in the social sciences.
Mathematical models of language contact situations give us hints about the important factors that could reverse the current predictions – notably, the perception of bilingualism and the geographic distribution of the population [@SeoaneMira2017]. The former is clear from our analysis and has been discussed theoretically by Heinsalu et al. [@HeinsaluLeonard2014]. A key for stability is thus bolstering a strong bilingual group capable of capturing speakers faster than any monolingual group. To achieve this, it is relevant that bilingual individuals reach a preponderant role within their society. Failing to establish a lasting bilingual group guarantees that the competition will result in an extinct language; and the most likely scenario, given the data, would be the decline of Catalan.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are very thankful to the Institut d’Estadística de Catalunya (IDESCAT), for its collaboration supplying the data that made possible this research. We also acknowledge the attention of the Xarxa Cruscat, that responded effectively to our demands of information concerning sociolinguistic aspects of Catalonia. This work was partially funded by the Galician Royal Academy ([*Real Academia Galega*]{}), to which we also acknowledge a lasting insfrastructural support. Seoane wishes to acknowledge the members of the Complex Systems Lab at the Pompeu Fabra University (especially Prof. Ricard Solé and Dr. Salvador Durán) for useful comments and discussion.
Author contributions {#author-contributions .unnumbered}
====================
All authors participated in the writing of this document. L.F.S. and J.M. coordinated the research and decided what mathematical analyses to conduct. L.F.S. implemented the mathematical and computational aspects of these analyses, elaborated the figures, and designed and implemented the fitting technique. X.L. curated the data and performed preliminary statistical analysis on it. H.M. provided qualitative knowledge about the linguistic reality of the system under research.
[10]{}
Lleal, C. [*La formación de las lenguas romances peninsulares*]{}. (Barcanova, SA., 1990)
Vila i Moreno, F. X. Catalan in Spain in [*Multilingual Europe: Factus and Policies*]{} (eds. Extra, G. & Gorter, D.) 157-183 (Walter de Gruyter, 2008).
Penny, R. J. [*A history of Spanish Language*]{} (Cambridge University Press, 2002).
Pharies, D.A. [*A brief history of the Spanish language*]{} (University of Chicago Press, 2008).
Llobera, J.R. [*De Catalunya a Europa: fonaments de la identitat nacional*]{} (Anagrama-Empúries, 2003).
Guibernau i Berdún, M. [*Catalan nationalism: Francoism, transition and democracy*]{} (Routledge, 2004).
Pradilla, M.A. The Catalan-speaking Communities in [*Multilingualism in Spain. Sociolinguistic and Psycholinguistic Aspects of Linguistic Minority Gropus*]{} (ed. Turell, T.) 303-323 (Iberoamericana Editorial, 2001).
Centro d’Estudis d’Opinió. Baròmetre d’Opinió Política. 2a onada 2016. REO 826. (Generalitat de Catalunya, 2016).
Clua i Fainé, M. Identidad y política en Cataluña: el auge del independentismo en el nacionalismo catalán actual. [*Quaderns-e de l’Institut Català d’Antropologia*]{} [**19**]{}(2), 79-99 (2014).
Woolard, K. Language and identity choice in Catalonia: The interplay of contrasting ideologies of linguistic authority in [*Lengua, nación e identidad: la regulación del plurilingüísmo en España y América Latina*]{} (eds. Süselbeck, K., Mühlschlegel, U., & Masson, P.) 303-323 (Iberoamericana Editorial, 2008).
Sutherland, W.J. Parallel extinction risk and global distribution of languages and species. [*Nature*]{} [**423**]{}(6937), 276-279 (2003).
Kot, M. [*Elements of mathematical ecology.*]{} (Cambridge University Press, 2001).
Turchin, P. [*Complex population dynamics: a theoretical/empirical synthesis (Vol. 35).*]{} (Princeton University Press, 2003).
Baggs, I. & Freedman, H. I. A mathematical model for the dynamical interactions between a unilingual and bilingual populations: Persistence versus extinction. [*J. Math. Sociology*]{} [**16**]{}, 51-75 (1990).
Baggs, I. & Freedman, H. I. Can the Speakers of a Dominated Language Survive as Unilinguals?: A Mathematical Model of Bilingualism. [*Mathl. Comput. Modelling*]{} [**18**]{}(6), 9-18 (1993).
Abrams, D. M. & Strogatz, S. H. Modelling the dynamics of language death. [*Nature*]{} [**424**]{}, 900 (2003).
Mira, J. & Paredes, Á. Interlinguistic similarity and language death dynamics. [*Europhys. Lett.*]{} [**69**]{}, 1031-1034 (2005).
Kandler, A. & Steele, J. Ecological models of language competition. [*Biol. Theor.*]{} [**3**]{}(2), 164-173 (2008).
Castellano, C. Fortunato, S., & Loreto, V. Statistical physics of social dynamics. [*Rev. Mod. Phys.*]{} [**81**]{}, 591-646 (2009).
Patriarca, M. & Heinsalu, H. Influence of geography on language competition. [*Physica A*]{} [**388**]{}(2-3), 174-186 (2009).
Castelló, X., Loureiro-Porto, L., & San Miguel, M. Agent-based models of language competition. [*International Journal of the Sociology of Language*]{} [**2013**]{}(221), 21-51 (2013).
Zhang, M. & Gong, T. Principles of parametric estimation in modeling language competition. [*Proc. Nat. Acad. Sci.*]{} [**110**]{}(24), 9698-9703 (2013).
Kandler, A., Unger, R., & Steele, J. Language shift, bilingualism and the future of Britain’s Celtic languages. [*Philos. T. Roy. Soc. B*]{} [**365**]{}(1559), 3855-3864 (2010).
Generalitat de Catalunya, Direcció General de Política Lingüística. [*Estadística d’usos lingüístics a Catalunya 2003. (Survey of language use in Catalonia 2003.)*]{} (Generalit de Catalunya and Institut d’Estadística de Catalunya, 2004).
Torres, J., Vila, F. X., Fabà, A., Bretxa, V., & Pradilla, M. À. [*Estadística sobre els usos lingüístics a Catalunya 2003. Llengua i societat a Catalunya en els inicis del segle XXI. (Statistics on language use in Catalonia 2003. Language and society in Catalonia at the beginning of the XXI century.)*]{} (Generalit de Catalunya. Secretary for linguistic policy, 2005).
Generalitat de Catalunya. [*Enquesta d’usos lingüístics de la població 2008: anàlisi. Volum I. Les llengües a Catalunya: coneixeiments, usos, transmissió i accituds lingüístics. (Survey of language use of the population 2008: analysis. Volume I. Languages in Catalonia: knowledge, usage, transmission, and attitudes towards language.)*]{} (Generalit de Catalunya, 2011).
Generalitat de Catalunya. [*Enquesta d’usos lingüístics dela població 2008: anàlisi. Volum II. Els factors clau de les llengües a Catalunya: edadt, ocupació, classe social, lloc de naixement, territori i aprenentatge de català. (Survey of language use of the population 2008: analysis. Volume II. Key aspects of the languages in Catalonia: age, occupation, social class, place of birth, territory, and learning of Catalan.)*]{} (Generalit de Catalunya, 2013).
Generalitat de Catalunya. [*Estadística d’usos lingüístics de la població a Catalunya (2003, 2008, 2013). (Statistics of linguistics uses of the population in Catalonia (2003, 2008, 2013).*]{} (Generalitat de Catalunya, 2014).
Generalitat de Catalunya, Direcció General de Política Lingüística. [*Anàlisi de l’Enquesta d’usos lingüístics de la població 2013. Resum dels factors clau. (Analysis of the survey of language use of the population 2013. Summary of key aspects.)*]{} (Generalit de Catalunya, 2015).
Mira, J., Seoane, L. F., & Nieto, J. J. The importance of interlinguistic similarity and stable bilingualism when two languages compete, [New J. of Phys.]{} [**13**]{}, 033007 (2011).
Otero-Espinar, M. V., Seoane, L. F., Nieto, J. J., & Mira, J. An analytic solution of a model of language competition with bilingualism and interlinguistic similarity, [*Phys. D*]{} [**264**]{}, 17-26 (2013).
Colucci, R., Mira, J., Nieto, J. J., & Otero-Espinar, M. Coexistence in exotic scenarios of a modified Abrams–Strogatz model, [*Complexity*]{} [**21**]{}(4), 86-93 (2016).
Seoane, L. F. & Mira, J. Modeling the life and death of competing languages from a physical and mathematical perspective. In [*Bilingualism and Minority Languages in Europe: Current trends and development*]{}. (eds Lauchlan, F. & Parafita Couto, M.C.) 70-93 (Cambridge Scholars Press, 2017).
Labov, W. The social motivation of a sound change. [*Word*]{} [**19**]{}(3), 273-309 (1963).
Eckert, P. Age as a sociolinguistic variable. In [*The handbook of sociolinguistics*]{} (ed. Coulmas, F.) 151-167 (Blackwell, 1997).
Magué, J. P. Semantic Changes in Apparent Time. In [*Annual Meeting of the Berkeley Linguistics Society*]{} [**32**]{}(1), 227-235 (2006).
Chambers, J. K. Patterns of variation including change. In [*The Handbook of Language Variation and Change*]{} 297-324 (2013).
Minett, J. W. & Wang, W. S.-Y. Modeling endangered languages: The effects of bilingualism and social structure. [*Lingua*]{} [**118**]{}, 19-45 (2008).
Heinsalu, E., Patriarca, M., & Léonard, J. The role of bilinguals in language competition,. [*Advances in Complex Systems*]{} [**17**]{}(1), 1450003 (2014).
Shalizi, C. [*Advanced data analysis from an elementary point of view*]{} (Cambridge University Press, 2013).
|
---
abstract: 'We show that Bertini theorems hold for $F$-signature. In particular, if $X \subseteq \bP^n$ is quasi-projective with $F$-signature greater than $\lambda$ at all points $x \in X$, then for a general hyperplane $H \subseteq \bP^n$ the $F$-signature of $X \cap H$ is greater than $\lambda$ at all points $x \in X \cap H$.'
address:
- |
Department of Mathematics\
University of Utah\
Salt Lake City\
UT 84112 Escuela de Matemática\
Universidad de Costa Rica\
San José 11501\
Costa Rica
- |
Department of Mathematics\
University of Utah\
Salt Lake City\
UT 84112
- |
Department of Mathematics\
University of Illinois at Chicago\
Chicago\
IL 60607
author:
- 'Javier Carvajal-Rojas'
- Karl Schwede
- Kevin Tucker
bibliography:
- 'MainBib.bib'
title: 'Bertini theorems for $F$-signature'
---
[^1] [^2] [^3]
Introduction
============
A common tool for studying a quasi-projective algebraic variety $X \subseteq \bP^n_k$, $k = \overline{k}$, is to perform induction on dimension by intersecting with a general hyperplane $H$. When doing this, you want the resulting intersection $X \cap H$ to have similar properties to the original variety $X$. Bertini’s theorem accomplishes exactly this: the classical result asserts that if $X$ is smooth then so is $X \cap H$ for a general choice of $H$ [@Hartshorne II, Theorem 8.18], [@KleimanBertiniAndHisTheorems]. Many classes of singularities also satisfy this property. For example, in characteristic zero if $X$ is log terminal (respectively log canonical), then so is $X \cap H$ [@KollarMori Lemma 5.17]. Even more generally the multiplier ideal of a divisor pair restricts to the multiplier ideal of the intersection $$\mJ(X, \Delta) |_{X \cap H} = \mJ(X \cap H, \Delta_{X \cap H}),$$ see [@LazarsfeldPositivity2 Example 9.5.9]. In characteristic zero, Bertini theorems can be generalized to the case where $H$ is a general member of a base point free linear system.
In characteristic $p > 0$, the situation is more complicated. It is essential that $H$ is a general member of a very ample linear system (or something close to that) if you expect Bertini-type results to hold. Since strongly $F$-regular and $F$-pure singularities are analogous to log terminal and log canonical singularities respectively [@HaraWatanabeFRegFPure], it is natural to expect that the corresponding Bertini-results hold. In [@SchwedeZhangBertiniTheoremsForFSings], this is exactly what was shown.
If $(X, \Delta)$ is a strongly $F$-regular (resp. sharply $F$-pure) pair such that $X \subseteq \bP^n_k$ is quasi-projective and $k = \overline{k}$ is of characteristic $p > 0$, then $(X \cap H, \Delta|_{X \cap H})$ is also strongly $F$-regular (resp. sharply $F$-pure) for a general choice of hyperplane $H \subseteq \bP^n_k$.
However, the corresponding result for test ideals is false:
For any $p > 0$ and $n \geq 3$, there exists a $\bQ$-divisor $\Delta$ on $X = \bA^3_k$, where $k = \overline{k}$ is of characteristic $p > 0$, such that $$\tau(X, \Delta)|_{H} \neq \tau(X \cap H, \Delta|_{X \cap H})$$ for a general hyperplane $H \subseteq \bA^3$.
It is then natural to ask about other types of $F$-singularities in characteristic $p > 0$. For example the behavior of $F$-rational singularities under restriction to general hyperplanes is still unknown. In this paper show that the above sort of Bertini-theorem holds for $F$-signature $s({\mathscr O}_{X,x})$ in the following sense.
Suppose that $X \subseteq \bP^n_k$ is a normal quasi-projective variety, $k = \overline{k}$ is of characteristic $p > 0$, and $\Delta \geq 0$ is a $\bQ$-divisor. Suppose that $\lambda \geq 0$ is a number such that the $F$-signature is bigger than $\lambda$, $$s({\mathscr O}_{X,x}, \Delta) > \lambda$$ for all $x \in X$. Then for a general hyperplane $H \subseteq \bP^n_k$, $$s({\mathscr O}_{X \cap H, x}, \Delta|_{X \cap H}) > \lambda$$ for all $x \in X \cap H$.
We actually prove a slightly stronger result by weakening the hypothesis that $X \subseteq \bP^n_k$ and we also make statements about the locus $U$ where $s({\mathscr O}_{X,x}, \Delta) > \lambda$ for all $x \in U$.
Recall that $F$-signature measures how strongly $F$-regular a variety or pair is. Explicitly, if $R$ is finite type over $k = \overline{k}$, then $R$ is regular if and only if $R^{1/p^e}$ is a locally free $R$-module by [@KunzCharacterizationsOfRegularLocalRings]. The $F$-signature refines this, by definition $s(R)$ is a number that indicates what percentage of $R^{1/p^e}$ is locally free asymptotically as $e$ goes to $\infty$. Thus $1 \geq s(R) \geq 0$ and
- $s(R) = 1$ if and only if $R$ is regular [@HunekeLeuschkeTwoTheoremsAboutMaximal] ([@YaoModulesWithFFRT]) and
- $s(R) > 0$ if and only if $R$ is strongly $F$-regular [@AberbachLeuschke].
The $F$-signature should be thought of some sort of local volume of the singularity.
We prove our main result by relying on the axiomatic Bertini framework as introduced in [@CuminoGrecoManaresiAxiomatic]. In particular, to show the type of result in our Main Theorem, it suffices to show the following two properties for a property of singularities $\sP$ (such as $s({\mathscr O}_{X,x}) > \lambda$):
- If $\phi : Y \to Z$ is a flat morphism with regular fibers and $Z$ is $\sP$, then $Y$ is $\sP$ too.
- Let $\phi : Y \to S$ be a morphism of finite type where $Y$ is excellent and $S$ is integral with generic point $\eta$. If $Y_{\eta}$ is geometrically $\sP$, then there exists an open neighborhood $U$ of $\eta$ in $S$ such that the fibers $Y_{s}$ are geometrically $\sP$ for each $s \in U$. (In fact, it suffices to check this for $S = (\bP^n_k)^*$, the space of hyperplanes).
Property (A1) was already proven for $F$-signature in [@YaoObservationsAboutTheFSignature]; in , we generalize this result to the context of pairs and give a new proof in the classical non-pair setting. In we show that property (A2) holds for $F$-signature. [*Acknowledgements:*]{} The authors thank Patrick Graf and Yongwei Yao for stimulating discussions. Work on this project was conducted in CIRM (Luminy) and Oberwolfach.
Preliminaries
=============
$F$-signature
-------------
Throughout this article, we shall assume all schemes $X$ are Noetherian, separated, and have prime characteristic $p>0$. If $x \in X$, we let $k(x)$ denote the residue field of the local ring ${\mathscr O}_{X,x}$. We let $F^e \colon X \to X$ denote the $e$-iterated Frobenius endomorphism or $p^e$-th power map. We say $X$ is $F$-finite if $F^e$ is a finite morphism, in which case $X$ is automatically excellent and has a dualizing complex [@KunzOnNoetherianRingsOfCharP; @Gabber.tStruc].
When $X = \Spec(A)$ is affine, we often conflate scheme-theoretic and ring-theoretic notation. In particular, $F^e \colon A \to A$ denotes the $e$-iterated Frobenius, and for an $A$-module $M$ we write $F^e_*M$ for $\Gamma\bigl(\Spec(A), F^e_*\widetilde{M}\bigr)$ where $\widetilde{M}$ is the associated quasi-coherent sheaf on $\Spec(A)$. In other words, $F^e_*M$ is the $A$-module arising from $M$ via restriction of scalars for $F^e$. In case $A$ is reduced, we also identify $F^e$ with the inclusion $A \subseteq A^{1/p^e}$, and shall at times use $M^{1/p^e}$ to denote $F^e_*M$ accordingly.
Recall that an $A$-module inclusion $M_1 \to M_2$ is said to be pure if $M_1 \otimes_A N \to M_2 \otimes_A N$ remains injective for any $A$-module $N$. An inclusion $A \to M$ where $M$ is a finitely generated $A$-module is pure if and only if it is split, *i.e.* admits an $A$-module section. If $(A,{\mathfrak{m}})$ is local, $A \to M$ is pure if and only if $E_A(k) \to M \otimes_A E_A(k)$ is injective, where $E_A(k)$ is an injective hull of the residue field $k = A/{\mathfrak{m}}$. We write $\ell_A( \blank)$ for the length of an $A$-module, omitting the subscript at times to simplify notation.
If $(A,{\mathfrak{m}})$ is an excellent local ring of dimension $d$, the $e$-th Frobenius degeneracy ideal $$I_e(A) = \bigl\langle a \in A \mid A \xrightarrow{1 \mapsto F^e_* a} F^e_*A \mbox{ is not a pure $A$-module inclusion} \bigr\rangle$$ is an ideal of $A$, and the $F$-signature is $$s(A) = \lim_{e \rightarrow \infty} \frac{1}{p^{ed}} \ell_A\bigl( A / I_e(A) \bigr).$$
Recall the following results on $F$-signature, see [@HunekeLeuschkeTwoTheoremsAboutMaximal; @YaoModulesWithFFRT; @AberbachLeuschke].
Suppose $(A, {\mathfrak{m}})$ is an excellent local ring of dimension $d$.
1. The limit defining the $F$-signature $s(A)$ exists, and moreover $$\ell_A\bigl( A / I_e(A) \bigr) = s(A) \cdot p^{ed} + O\bigl(p^{e(d - 1)}\bigr).$$
2. The $F$-signature $s(A) \leq 1$, and $s(A) = 1$ if and only if $A$ is regular.
3. The $F$-signature $s(A) \geq 0$, and $s(A) > 0$ if and only if $A$ is strongly $F$-regular. In this case, $A$ is necessarily a Cohen-Macaulay normal domain.
The $F$-signature is also known to satisfy additional properties in the $F$-finite setting, such as semi-continuity.
[@PolstraFsigSemiCont; @PolstraTuckerCombined] For an $F$-finite domain $A$, the $F$-signature determines a lower semi-continuous function $$Q \in \Spec(A) \mapsto s(A_Q)$$ on $\Spec(A)$.
Moreover, if $(A,{\mathfrak{m}})$ is an $F$-finite local ring of dimension $d$, note that one can alternately describe the degeneracy ideals as $$\begin{aligned}
I_e(A) &= \bigl\langle a \in A \mid A \xrightarrow{1 \mapsto F^e_* a} F^e_*A \mbox{ is not a split $A$-module inclusion} \bigr\rangle \\
&= \bigl\langle a \in A \mid \phi(F^e_*a) \in {\mathfrak{m}}\mbox{ for all } \phi \in \Hom_A(F^e_*A,A) \bigr\rangle,\end{aligned}$$ and the $F$-signature can be viewed as giving an asymptotic measure of the number of splittings of the $e$-iterated Frobenius. In particular, if $(A, {\mathfrak{m}})$ is an $F$-finite local domain, we have $$s(A) = \lim_{e \rightarrow \infty} \frac{\operatorname{frk}_A (A^{1/p^e})}{\rank_A (A^{1/p^e})}.$$ where $\operatorname{frk}_A(\blank)$ denotes free rank. Recall that, for arbitrary (and not necessarily local) $A$, the free rank of an $A$-module $M$ is the maximal rank $\operatorname{frk}_A(M)$ of a free $A$-module quotient of $M$.
One can generalize the interpretation of $F$-signature for $F$-finite rings beyond the local setting as well. To make this more precise, recall first the following result of Kunz.
[@KunzOnNoetherianRingsOfCharP] If $A$ is a reduced equidimensional $F$-finite ring, the function $$Q \in \Spec(A) \mapsto \bigl[k(Q)^{1/p}:k(Q)\bigr] \cdot p^{\height Q}$$ is constant on $\Spec(A)$. In particular, if $A$ is a domain, $$\rank_A(A^{1/p^e}) = \bigl[k(Q)^{1/p^e}:k(Q)\bigr] \cdot p^{e \height Q}$$ for any $e \geq 0$ and $Q \in \Spec(A)$.
We recall a recent result globalizing $F$-signature.
[@DeStefaniPolstraYaoGlobalizingFinvariants] If $A$ is a reduced equidimensional $F$-finite ring, and $\gamma \in \ZZ_{\geq 0}$ with $p^{\gamma} = \bigl[k(Q)^{1/p}:k(Q)\bigr] \cdot p^{\height Q} $ for all $Q \in \Spec(A)$, then the limit $$s(A) = \lim_{e \rightarrow \infty} \frac{\operatorname{frk}_A\bigl(A^{1/p^e}\bigr)}{p^{e \gamma}}$$ exists and equals $\min \{ s(A_Q) \mid Q \in \Spec(A) \} = \min \{ s(A_{\mathfrak{m}}) \mid {\mathfrak{m}}\in \max \Spec(A) \}$.
Divisors
--------
In this subsection we review the definitions and properties of the $F$-signature of divisor pairs.
If $(A,{\mathfrak{m}})$ is a normal excellent local domain of dimension $d$ and $D$ is an effective Weil divisor on $\Spec(A)$, the $e$-th Frobenius degeneracy ideal along $D$ is $$I_e(A,D) = \bigl\langle a \in A \mid A \xrightarrow{1 \mapsto F^e_* a} F^e_*\bigl(A (D)\bigr)\mbox{ is not a pure $A$-module inclusion} \bigr\rangle.$$ If $\Delta$ is an effective $\Q$-divisor on $\Spec(A)$, the $F$-signature of $(A, \Delta)$ is $$s(A, \Delta) = \lim_{e \rightarrow \infty} \frac{1}{p^{ed}} \ell_A\Bigl( A \big/ I_e\bigl(A, \lceil (p^e - 1) \Delta \rceil \bigr) \Bigr).$$
\[lem:perturb\] Suppose $(A,{\mathfrak{m}})$ is a normal excellent local domain of dimension $d$ and $\Delta$ is an effective $\Q$-divisor on $\Spec(A)$. Let $\{ D_e \}_{e > 0}$ be a sequence of Weil divisors on $\Spec(A)$ with bounded difference from $\bigl\{ \lceil (p^e -1) \Delta \rceil \bigr\}_{e > 0}$ independent of $e > 0$. In other words, there exists an effective Cartier divisor $C$ such that $$- C \leq D_e - \lceil (p^e -1) \Delta \rceil \leq C$$ for all $e > 0$. Then $$s(A, \Delta) = \lim_{e \rightarrow \infty} \frac{1}{p^{ed}} \ell_A\bigl( A / I_e(A, D_e) \bigr).$$
This is essentially the same argument as [@BlickleSchwedeTuckerFSigPairs1 Lemma 4.17] and [@PolstraTuckerCombined Theorem 4.13], and so we omit it.
[@BlickleSchwedeTuckerFSigPairs1; @PolstraTuckerCombined] Suppose $(A,{\mathfrak{m}})$ is a normal excellent local domain of dimension $d$ and $\Delta$ is an effective $\Q$-divisor on $\Spec(A)$.
1. The limit defining the $F$-signature $s(A, \Delta)$ exists, and moreover $$\ell_A\Bigl( A \big/ I_e\bigl(A, \lceil (p^e - 1) \Delta \rceil \bigr) \Bigl) = s(A, \Delta) \cdot p^{ed} + O\bigl(p^{e(d - 1)}\bigr).$$
2. The $F$-signature $s(A, \Delta) \geq 0$, and $s(A, \Delta) > 0$ if and only if $(A, \Delta)$ is strongly $F$-regular.
If $A$ is an $F$-finite normal excellent domain of dimension $d$ and $D$ is an effective Weil divisor on $\Spec(A)$, one can define the *free rank of $A^{1/p^e}$ along $D$* $$\operatorname{frk}_{A}^D\bigl(A^{1/p^e}\bigr)$$ to be the maximal rank $a_e(D)$ of a simultaneous free $A$-module quotient of $A^{1/p^e}$ and $\bigl(A(D)\bigr)^{1/p^e}$. In other words, $a_e(D)$ is the largest non-negative integer such that there is a commuting diagram $$\xymatrix{
& \bigl(A(D)\bigr)^{1/p^e} \ar@{->>}[dr] & \\
A^{1/p^e} \ar@{->>}[rr] \ar[ru]^{\subseteq} & & A^{\oplus a_e(D)}.
}$$ In case $(A,{\mathfrak{m}})$ is local, we have that $\operatorname{frk}_{A}^D\bigl(A^{1/p^e}\bigr) = \ell_A\bigl( A / I_e(A, D) \bigr)$, and once more this leads to a recent global interpretation of the $F$-signature along a divisor.
[@DeStefaniPolstraYaoGlobalizingFinvariants] \[thm.globaldivisorfsig\] Let $A$ be an $F$-finite normal excellent domain of dimension $d$, and $\gamma \in \ZZ_{\geq 0}$ with $p^{\gamma} = \bigl[k(Q)^{1/p}:k(Q)\bigr] \cdot p^{\height Q} $ for all $Q \in \Spec(A)$. Suppose $\Delta$ is an effective $\Q$-divisor on $\Spec(A)$. The $F$-signature along $\Delta$ determines a lower semi-continuous function $$Q \in \Spec(A) \mapsto s(A_Q, \Delta)$$ on $\Spec(A)$. Moreover, the limit $$s(A, \Delta) = \lim_{e \rightarrow \infty} \frac{\operatorname{frk}_A^{\lceil p^e \Delta \rceil}\bigl(A^{1/p^e}\bigr)}{p^{e \gamma}}$$ exists and equals $\min \{ s(A_Q, \Delta) \mid Q \in \Spec(A) \} = \min \{ s(A_{\mathfrak{m}}, \Delta) \mid {\mathfrak{m}}\in \max \Spec(A) \}$.
In light of , and following [@DeStefaniPolstraYaoGlobalizingFinvariants], we also make the following global definition.
For a normal $F$-finite scheme $X$ and effective $\bQ$-divisor $\Delta$ we set $$s(X, \Delta) = \min \{ s({\mathscr O}_{X,x}, \Delta) \mid x \in X \} = \min \{ s({\mathscr O}_{X,x}, \Delta) \mid x \in X \mbox{ a closed point} \}.$$ When $X = \Spec A$ is affine, we write $s(A, \Delta)$ for $s(X, \Delta)$.
Divisors and families {#subsec.DivisorsAndFamilies}
---------------------
Finally we discuss the correspondence between $\bQ$-divisors and $p^{-e}$-linear maps in the relative setting of $A \subseteq R$ (or in other words, for families). What follows is contained in [@PatakfalviSchwedeZhangFFamilies] although we work in a less general setting.
\[set.RelativeSetting\] Suppose that $A$ is an $F$-finite regular domain and suppose we have $A \subseteq R$ a flat finite type extension of rings with geometrically[^4] normal fibers. Additionally assume that for some choice of $\omega_A$, $$\label{eq.FShriekCompatible}
\tag{$\dagger$}
F^! \omega_A \cong \omega_A.$$ This always holds for rings essentially of finite type over a Gorenstein semi-local ring.
For any $A$-algebra $B$, we write $R_{B} = R \otimes_A B$. Frequent values of $B$ include $A^{1/p^e}$, the fraction field $K := K(A)$ and $k(Q)$, the residue field of a point $Q \in \Spec A$.
We make some quick observations.
\[lem.BaseChangeIsNormalInSetting\] In the setting of , each $R_{A^{1/p^e}}$ is a normal integral domain, as are $R_{K^{1/p^e}}$ and $R_{K^{\infty}}$ as well.
$A^{1/p^e} \to R_{A^{1/p^e}}$ is flat with normal fibers over a regular base, and hence $R_{A^{1/p^e}}$ is normal by [@MatsumuraCommutativeRingTheory Theorem 23.9]. Since $R \to R_{A^{1/p^e}}$ is purely inseparable and $R_{A^{1/p^e}}$ is reduced, it follows that $R_{A^{1/p^e}}$ is a domain. Localizing, we have that $K \to R_{K}$ also has geometrically normal fibers, and the same argument gives that $R_{K^{1/p^e}}$ and $R_{K^\infty}$ are normal domains as well.
In the setting of , for each $Q \in \Spec A$ and $x \in \Spec R_{K(Q)} \subseteq \Spec R$ a point of codimension $1$ on the fiber, we have that $R_x$ is regular and thus $\Delta$ is $\bQ$-Cartier at $x$. In particular, we can restrict $\Delta|_{\Spec R_{k(Q)}}$ to any fiber.
Choose a codimension 1 point $x \in \Spec R_{K(Q)}$, in other words a codimension one point of a fiber over $Q \in \Spec(A)$. In particular, $(R_{K(Q)})_x $ is normal and hence regular. It follows that $R_x$ is also regular since $R_{K(Q)}$ is obtained from $R$ by killing a regular sequence and localizing.
We now discuss the correspondence between divisors and maps in .
[[@PatakfalviSchwedeZhangFFamilies 2.8–2.11]]{.nodecor} \[lem.RelativeDivisorMapCorresponds\] Suppose that $A$ is an $F$-finite regular domain and suppose we have $A \subseteq R$ a flat finite type extension of rings with geometrically[^5] normal fibers. Then for every $R_{A^{1/p^e}}$-linear map $$\phi : R^{1/p^e} \to R_{A^{1/p^e}}$$ which generates $\Hom_{R_{A^{1/p^e}}}(R^{1/p^e}, R_{A^{1/p^e}})$ at the generic point of every fiber, there exists a corresponding $\bZ_{(p)}$-divisor[^6] on $\Spec R$ $$\Delta_{\phi} \sim_{\bQ} -K_{R/A}$$ which does not contain any fiber in its support.
Conversely, given an effective $\bZ_{(p)}$-divisor $\Delta \sim_{\bQ} -K_{R/A}$ on $\Spec R$ whose support does not contain any fiber, we can construct a map $\phi : R^{1/p^e} \to R_{A^{1/p^e}}$ such that $\Delta_{\phi} = \Delta$.
Finally, we recall the interaction between divisors and maps behaves under base change. While not crucial for the following statement, in this paper we restrict ourselves to base changes which are either flat or restriction to a fiber followed by a flat base change, which is easier to work with than the generality of [@PatakfalviSchwedeZhangFFamilies].
[[@PatakfalviSchwedeZhangFFamilies Lemma 2.21]]{.nodecor} \[lem.RelativeDivisorsAndBaseChange\] In the setting of assume that $\Delta = \Delta_{\phi}$ is constructed as in . For any regular $A$-algebra $B$ satisfying , let $\pi : \Spec R_B \to \Spec R$ denote the canonical map. Set $\phi_B := \phi \otimes_{A^{1/p^e}} B^{1/p^e}$ to be the base changed map $$\phi_B : (R_B)^{1/p^e} = R^{1/p^e} \otimes_{A^{1/p^e}} B^{1/p^e} \to R_{A^{1/p^e}}\otimes_{A^{1/p^e}} B^{1/p^e} = R_{B^{1/p^e}}.$$ In this case, $$\Delta_{\phi_B} = \pi^* \Delta = \pi^* \Delta_{\phi}.$$
Frequently $B = A^{1/p^d}$ in which case the based changed map $\phi_B$ in is simply $$\phi_{A^{1/p^d}} : \bigl(R_{A^{1/p^d}}\bigr)^{1/p^e} \to R_{A^{1/p^{e+d}}}.$$
$F$-signature transformation for regular fibers {#sec.YaoProofForPairs}
===============================================
In this section, we will be concerned with the behavior of the $F$-signature under flat local extensions, building on the following result of Y. Yao.
Suppose that $(A, \fram) \subseteq (R, \frn)$ is a flat local extension of excellent local rings of characteristic $p > 0$. Then if $R/\fram R$ is regular, we have $$s(A) = s(R).$$
Our goal is to generalize the above result to the context of divisor pairs $(R, \Delta)$, for which we will first need to give a variation on the proof of the original statement. We begin with some preliminary lemmas.
Suppose that $(A,\fram) \subseteq (R, \frn)$ is a flat local extension of local rings. If $x_1, \ldots, x_\delta \in R$ are a regular sequence on $R / \fram R$, then $ R / \langle x_1, \ldots, x_\delta \rangle$ is a flat $A$-algebra. Moreover, $x_1, \ldots, x_\delta \in R$ are a regular sequence on $M \otimes_A R$ for any finitely generated $A$-module $M$, and lastly for any $t \geq 0$ the $R$-module inclusion $$R / \langle x_1^t, \ldots, x_\delta^t \rangle \xrightarrow{1 \mapsto \left[\frac{1}{x_1^t\cdots x_\delta^t} \right]} H^\delta_{\langle x_1, \ldots, x_\delta \rangle}(R)$$ is pure as an inclusion of $A$-modules.
See [@MatsumuraCommutativeAlgebra Corollary 20.F, page 151] or [@HochsterHunekeFRegularityTestElementsBaseChange Lemma 7.10]. For the final statement, note that it suffices to check purity after tensoring with finitely generated $A$-modules, where injectivity follows from the previous regular sequence assertion.
The following was used in Hochster and Huneke’s original study of $F$-regularity and base change.
[@HochsterHunekeFRegularityTestElementsBaseChange Lemma 7.10] \[lem.HHTensorLocalCMBaseChange\] Let $(A,\fram) \subseteq (R, \frn)$ be a flat local extension of local rings. Suppose $R / \fram R$ is regular and $x_1, \ldots, x_\delta \in R$ give a regular system of parameters of $R / \fram R$. If $E_A$ is an injective hull of $A / \fram$ over $A$ with socle generated by $u$, then $E_R = H^\delta_{\langle x_1, \ldots, x_\delta \rangle} (R) \otimes_A E_A$ is an injective hull of $R / \frn$ over $R$ with socle generated by $\left[\frac{1}{x_1\cdots x_\delta}\right] \otimes u$.
Now we give a new proof of Yao’s result.
If $x_1, \ldots, x_\delta \in R$ give a regular system of parameters of $R / \fram R$, then by we have $E_R = H^\delta_{\langle x_1, \ldots, x_\delta \rangle} (R) \otimes_A E_A$ with socle generated by $v = \left[\frac{1}{x_1\cdots x_\delta}\right] \otimes u$. Consider now $R^{1/p^e}\otimes_R E_R$, so that $I_e(R)^{1/p^e} = \Ann_{R^{1/p^e}}(1 \otimes v)$. We may identify $R^{1/p^e} \otimes_R H^\delta_{\langle x_1, \ldots, x_\delta \rangle} (R) = \bigl(H^\delta_{\langle x_1, \ldots, x_\delta \rangle} (R)\bigr)^{1/p^e}$ and with $1 \otimes \left[\frac{1}{x_1\cdots x_\delta}\right] \leftrightarrow \left[\frac{1}{x_1^{p^e}\cdots x_\delta^{p^e}}\right]^{1/p^e} $. Using that $$\left( R \bigl/ \bigl\langle x_1^{p^e}, \ldots, x_\delta^{p^e} \bigr\rangle \right)^{1/p^e} \xrightarrow{1 \mapsto \left[\frac{1}{x_1^{p^e}\cdots x_\delta^{p^e}} \right]^{1/p^e}} \left( H^\delta_{\langle x_1, \ldots, x_\delta \rangle}(R) \right)^{1/p^e}$$ is pure as an inclusion of $A^{1/p^e}$-modules, this gives further identifications $$\begin{array}{ccc}
R^{1/p^e}\otimes_R E_R & = & \left( H^\delta_{\langle x_1, \ldots, x_\delta \rangle}(R) \right)^{1/p^e} \otimes_{A^{1/p^e}} \left( A^{1/p^e} \otimes_A E_A\right)
\\
1 \otimes v & \leftrightarrow & \left[\frac{1}{x_1^{p^e}\cdots x_\delta^{p^e}} \right]^{1/p^e} \otimes (1 \otimes u)
\\ \mbox{} &&
\\
& \supseteq & \left( R \bigl/ \bigl\langle x_1^{p^e}, \ldots, x_\delta^{p^e} \bigr\rangle \right)^{1/p^e} \otimes_{A^{1/p^e}} \left( A^{1/p^e} \otimes_A E_A\right)
\\
& \leftrightarrow & 1 \otimes (1 \otimes u).
\end{array}$$ But since $A^{1/p^e} \to \Bigl( R \bigl/ \bigl\langle x_1^{p^e}, \ldots, x_\delta^{p^e} \bigr\rangle \Bigr)^{1/p^e} $ is flat, it follows that the annihilator of $1 \otimes (1 \otimes u)$ over $\left( R \bigl/ \bigl\langle x_1^{p^e}, \ldots, x_\delta^{p^e} \bigr\rangle \right)^{1/p^e} $ is the expansion of $\Ann_{A^{1/p^e}}\left(1 \otimes u \in A^{1/p^e}\otimes_A E_A\right) = I_e(A)^{1/p^e}$. In other words, we have shown $I_e(R)^{1/p^e} = \bigl( I_e(A)R + \bigl\langle x_1^{p^e}, \ldots, x_\delta^{p^e} \bigr\rangle \bigr)^{1/p^e}$. Thus, using the flatness of $A \to R \bigl/ \bigl\langle x_1^{p^e}, \ldots, x_\delta^{p^e} \bigr\rangle $ once again, it follows $$\begin{aligned}
\ell_R\left(\frac{R}{I_e(R)}\right) &= \ell_R\left(\frac{R}{ I_e(A)R + \bigl\langle x_1^{p^e}, \ldots, x_\delta^{p^e} \bigr\rangle}\right) = \ell_A \left( \frac{A}{I_e(A)}\right) \ell_R \left( \frac{R}{\fram R + \bigl\langle x_1^{p^e}, \ldots, x_\delta^{p^e} \bigr\rangle}\right) \\
&= p^{e \delta} \ell_A \left( \frac{A}{I_e(A)}\right)\ell_R \left( \frac{R}{\fram R + \langle x_1, \ldots, x_\delta \rangle}\right) = p^{e \delta} \ell_A \left( \frac{A}{I_e(A)}\right).\end{aligned}$$ Since $\dim R = \dim A + \delta$, the desired equality now follows after dividing by $p^{e \dim R}$ and taking limits.
We now generalize the above proof to the context of pairs. We break off the main technical step into a lemma.
\[lem.IeTransformADtoRD\] Suppose that $(A, \fram) \subseteq (R, \frn)$ is a flat local extension of normal local rings of characteristic $p > 0$ and write $f : \Spec R \to \Spec A$ for the induced map. For any effective Weil $D$ on $\Spec A$ and $e > 0$, define $$\begin{aligned}
{c}
I_e(A, D) & = \langle a \in A \mid A \to A(D)^{1/p^e} \mbox{ with } 1 \mapsto a^{1/p^e} \mbox{ is not $A$-pure} \rangle \\
I_e(R, f^* D) & = \langle r \in R \mid R \to R(f^* D)^{1/p^e} \mbox{ with } 1 \mapsto r^{1/p^e} \mbox{ is not $R$-pure} \rangle.\end{aligned}$$
Then if $R/\fram R$ is regular, $$\ell_A\left( \frac{A}{I_e(A, D)} \right) = p^{e(\dim R - \dim A)} \ell_R \left( \frac{R}{I_e(R, f^* D)} \right).$$
If $x_1, \ldots, x_\delta \in R$ give a regular system of parameters of $R / \fram R$, we have that $E_R = H^\delta_{\langle x_1, \ldots, x_\delta \rangle} (R) \otimes_A E_A$ with socle generated by $v = \left[\frac{1}{x_1\cdots x_\delta}\right] \otimes u$. Consider now $R(f^*D)^{1/p^e}\otimes_R E_R$, so that $I_e(R, f^*D)^{1/p^e} = \Ann_{R^{1/p^e}}(1 \otimes v)$. Using that $R(f^*D) = R\otimes_A A(D)$ and the same identifications made in the proof above, we see that $$\begin{array}{ccc}
R(f^*D)^{1/p^e}\otimes_R E_R & = & \left( H^\delta_{\langle x_1, \ldots, x_\delta \rangle}(R) \right)^{1/p^e} \otimes_{A^{1/p^e}} \left( A(D)^{1/p^e} \otimes_A E_A\right)
\\
1 \otimes v & \leftrightarrow & \left[\frac{1}{x_1^{p^e}\cdots x_\delta^{p^e}} \right]^{1/p^e} \otimes (1 \otimes u)
\\ \mbox{} &&
\\
& \supseteq &\Bigl( R \bigl/ \bigl\langle x_1^{p^e}, \ldots, x_\delta^{p^e} \bigr\rangle \Bigr)^{1/p^e} \otimes_{A^{1/p^e}} \left( A(D)^{1/p^e} \otimes_A E_A\right)
\\
& \leftrightarrow & 1 \otimes (1 \otimes u).
\end{array}$$ But since $A^{1/p^e} \to \Bigl( R \bigl/ \bigl\langle x_1^{p^e}, \ldots, x_\delta^{p^e} \bigr\rangle \Bigr)^{1/p^e} $ is flat, it follows that the annihilator of $1 \otimes (1 \otimes u)$ over $\Bigl( R \bigl/ \bigl\langle x_1^{p^e}, \ldots, x_\delta^{p^e} \bigr\rangle \Bigr)^{1/p^e} $ is the expansion of $\Ann_{A^{1/p^e}}\left(1 \otimes u \in A(D)^{1/p^e}\otimes_A E_A\right) = I_e(A,D)^{1/p^e}$. In other words, we have shown $I_e(R, f^*D)^{1/p^e} = \bigl( I_e(A,D)R + \bigl\langle x_1^{p^e}, \ldots, x_\delta^{p^e} \bigr\rangle \bigr)^{1/p^e}$. Thus, using the flatness of $A \to R\bigl/\bigl\langle x_1^{p^e}, \ldots, x_\delta^{p^e} \bigr\rangle$ once again, it follows $$\begin{aligned}
\ell_R\left(\frac{R}{I_e(R,f^*D)}\right) &= \ell_R\left(\frac{R}{ I_e(A,D)R + \bigl\langle x_1^{p^e}, \ldots, x_\delta^{p^e} \bigr\rangle}\right) \\
&= \ell_A \left( \frac{A}{I_e(A,D)}\right) \ell_R \left( \frac{R}{\fram R + \bigl\langle x_1^{p^e}, \ldots, x_\delta^{p^e} \bigr\rangle}\right) \\
& = p^{e \delta} \ell_A \left( \frac{A}{I_e(A,D)}\right)\ell_R \left( \frac{R}{\fram R + \langle x_1, \ldots, x_\delta \rangle}\right) = p^{e \delta} \ell_A \left( \frac{A}{I_e(A,D)}\right)\end{aligned}$$ as desired.
We now can prove the main result of the section.
\[thm.FsignatureStableFlatMapRegularFiber\] Suppose that $(A, \fram) \subseteq (R, \frn)$ is a flat local extension of normal local rings of characteristic $p > 0$ and write $f : \Spec R \to \Spec A$ the induced map. Suppose further that $\Delta \geq 0$ is a $\bQ$-divisor on $\Spec A$. Then if $R/\fram R$ is regular, we have $$s(A, \Delta, \fram) = s(R, f^* \Delta, \frn).$$
We will first apply to $D = \lfloor p^e \Delta \rfloor$. We see that $f^* D = f^* \lfloor p^e \Delta \rfloor \leq \lfloor p^e f^* \Delta \rfloor$. Hence, recalling that $d = \dim R$ and applying both and , $$\begin{aligned}
s(R, f^* \Delta) & = {\lim_{e \rightarrow \infty} {\frac{1}{p^{de}} \ell_R\Big(R\big/I_e\big(R, \lceil (p^e-1) f^* \Delta\rceil \big)\Big)}} = {\lim_{e \rightarrow \infty} {\frac{1}{p^{de}} \ell_R\Big(R\big/I_e\big(R, \lfloor p^e f^* \Delta\rfloor \big)\Big)}} \\
& \leq {\lim_{e \rightarrow \infty} {\frac{1}{p^{de}} \cdot \ell_R\Big(R\big/I_e\big(R, f^* \lfloor p^e \Delta\rfloor \big)\Big) }} = { \lim_{e \rightarrow \infty} {\frac{1}{p^{(d - \delta)e}} \ell_A\Big(A\big/I_e\big(A, \lfloor p^e \Delta \rfloor\big)\Big) }}\\
& = s(A, \Delta).\end{aligned}$$
On the other hand, if we choose $D = \lceil p^e \Delta \rceil$, then $f^* D = f^* \lceil p^e \Delta \rceil \geq \lceil p^e f^* \Delta \rceil$ and arguing as above gives $s(R, f^* \Delta) \geq s(A, \Delta)$. This completes the proof.
$F$-signature of general fibers {#sec.FsignatureOfGeneralFibers}
===============================
Before proving Bertini-type theorems, we need one more result. We need to show that if $A \subseteq R$ is a finite type extension of rings such that the perfectified generic fiber has $F$-signature greater than $\lambda$, then so do *most* of the closed fibers.
\[set.A2Setting\] We assume that $A \subseteq R$ is a flat finite type morphism of Noetherian $F$-finite integral domains with fraction fields $K = \Frac(A) \subseteq L = \Frac(R)$. Suppose further that $A$ is regular and that $A \subseteq R$ has geometrically normal fibers. Further assume that $\Delta \geq 0$ on $\Spec R$ is a $\bQ$-divisor whose support does not contain any fiber.
We will not universally assume this setting in this section, but we will always be able to reduce to it. In order to motivate the main result of this section, we first give an easy proof of a weaker statement.
\[prop.A2ForVeryGeneral\] In the setting of , further suppose that $A$ is finite type over an uncountable algebraically closed field of characteristic $p > 0$. If $$s(R_{K^{\infty}, x}) \geq \lambda$$ for all $x \in \Spec R_{K^{\infty}}$, then for a very general[^7] closed point $Q \in \Spec A$ with residue field $k(Q)$, $$s(R_{k(Q),x}) \geq \lambda$$ for all $x \in \Spec R_{k(Q)}$.
By [@DeStefaniPolstraYaoGlobalizingFinvariants Theorem 4.13], for each $e > 0$, and by below, we can spread out our splitting and obtain some $a_e, d_e$ and $0 \neq g_e \in A$ so that there is a surjection $$\label{eq.SurjectivityAfterSpreadingOut}
R^{1/p^{e}}_{A[1/g_e]^{1/p^{e+d_e}}} \to R_{A[1/g_e]^{1/p^{e+d_e}}}^{\oplus a_e}$$ and so that $$\lambda \leq \min_{x \in \Spec R_{K^{\infty}}} \{s(R_{K^{\infty},x})\} = \lim_{e \rightarrow \infty} {\frac{a_e}{p^{e \dim R}}}.$$
Since our $Q$ is very general, $Q \notin V(g_e)$ for any $e$. Hence we have surjections $A[1/g_e] \to k(Q)$ for all $e$. We now apply $$\blank \otimes_{{A[1/g_e]}^{1/p^{e+d_e}}} k(Q)^{1/p^{e+d_e}}$$ to which yields a surjective map $$R^{1/p^{e}}_{k(Q)^{1/p^{e+d_e}}} \to R_{k(Q)^{1/p^{e+d_e}}}^{\oplus a_e}.$$ But $k(Q)$ is perfect and so this can be identified with a surjective map $$(R_{k(Q)})^{1/p^{e}} \to R_{k(Q)}^{\oplus a_e}.$$ The result follows.
We do not expect this result to hold for simply general fibers; see [@Monskypointquartics1998] for an example where the analagous Hilbert-Kunz statement for general fibers does not hold.
We now need the following result of Pérez, the third author, and Yao.
[@PerezYaoTucker] \[thm.uniformbound\] For every Noetherian ring $A$ of characteristic $p > 0$, and every finitely generated $A$-algebra $R$, and every finitely generated $R$-module $M$, there exists a positive constant $C$ with the following property: for all primes $Q \in \Spec(A)$, all regular $k(Q)$-algebras $\Gamma$, and all $P \in \Spec(R_\Gamma := R\otimes_A \Gamma)$, and all $e \geq 1$ , we have that $\ell_{R_{\Gamma}}\bigl(M_\Gamma / P^{[p^e]}(M_\Gamma)\bigr) \leq C p^{e \dim(M_\Gamma)}$ where $M_\Gamma := M \otimes_A \Gamma$.
The next result is the technical heart of the section. We state and prove it first in the non-pairs setting and then explain how to generalize it to pairs in a proposition which follows it.
\[prop.uniformconverginfamily\] Suppose we are in the setting of . There exists a positive constant $C$ and $0 \neq g \in A$ with the following property: for all $Q \in \Spec\bigl(B := A[g^{-1}]\bigr)$, all $d > 0$, all $x \in \Spec \big( R_{k(Q)^{1/p^d}} \big)$, and all $e > 0$, we have $$\left| s\bigl(R_{k(Q)^{1/p^d},x}\bigr) - \frac{a_e\bigl(R_{k(Q)^{1/p^d},x}\bigr)}{\rank_{R_{k(Q)^{1/p^d},x}}\bigl(R_{k(Q)^{1/p^d},x}\bigr)^{1/p^e}} \right| \leq \frac{C}{p^e}.$$
Let $\delta = \dim R_K = \dim R_{K^{\infty}} $, so that $\rank_{R_{K^{\infty}}} (R_{K^{\infty}})^{1/p^e} = p^{e \delta}$. Since $R_{A^{1/p^{e+d}}} \to R_{A^{1/p^{e+d}}}^{1/p^e}$ base changes to $R_{K^{\infty}} \to R_{K^{\infty}}^{1/p^e}$ for any $e, d > 0$, we see that $$\rank_{R_{A^{1/p^{e+d}}}} R_{A^{1/p^{e+d}}}^{1/p^e} = p^{e \delta}$$ as well. Note that $A^{1/p^d} \subseteq R_{A^{1/p^d}}$ is also flat, and for any $Q \in \Spec(A)$ and $x \in \Spec\big(R_{k(Q)^{1/p^d}}\big)$, we have that $\height x - \height Q = \dim\big(R_{k(Q)^{1/p^d},x}\big)$. Using that $\Frac\big(R_{A^{1/p^{e+d}}}\big) = L_{K^{1/p^{e+d}}}$ as $R_{A^{1/p^{e + d}}}$ is a domain, we also compute $$\begin{aligned}
p^{e\delta} = \rank_{R_{A^{1/p^{e + d}}}}R^{1/p^{e}}_{A^{1/p^{e + d}}} = \Big[L^{1/p^e}_{K^{1/p^{e+d}}}: L_{K^{1/p^{e+d}}}\Big] &= \frac{\Big[L^{1/p^e}_{K^{1/p^{e + d}}}: L_{K^{1/p^{d}}}\Big]}{\Big[L_{K^{1/p^{e+d}}}:L_{K^{1/p^d}}\Big]} \\[0.5em]
& = \frac{\Big[L^{1/p^e}_{K^{1/p^{e + d}}}: L_{K^{1/p^{d}}}\Big]}{\big[K^{1/p^{e}}:K\big]}\\[0.5em]
& = \frac{\big[k(x)^{1/p^e}:k(x)\big] }{\big[k(Q)^{1/p^e}:k(Q)\big]} \cdot p^{e(\height x - \height Q)}\end{aligned}$$
whence $$\big[k(Q)^{1/p^e}:k(Q)\big] \cdot p^{e\delta} = \rank_{R_{k(Q)^{1/p^d},x}}\big(R_{k(Q)^{1/p^d},x}\big)^{1/p^e}.$$
Form right exact sequences $$\label{eq:sescee}
(R_{A^{1/p}})^{\oplus p^{\delta}} \xrightarrow{\alpha_1} R^{1/p} \to M_1 \to 0$$ $$\label{eq:sesmap}
R^{1/p} \xrightarrow{\alpha_2} (R_{A^{1/p}})^{\oplus p^{\delta}} \to M_2 \to 0$$ of $R_{A^{1/p}}$-modules so that both $M_1, M_2$ are torsion. Take $0 \neq c \in R_{A^{1/p}}$ that kills both; replacing $c$ with $c^p$ if necessary, we may further assume $0 \neq c \in R$. The image of $U = \Spec R[1/c] \subseteq \Spec R$ in $\Spec A$ is open [@stacks-project Tag 01UA] and contains the image of the generic point. Thus, after inverting an element of $A$, we may assume $c$ does not vanish along any fiber. In other words, for any $Q \in \Spec A$ and $x \in \Spec R_{k(Q)^{1/p^d}}$, the image of $c$ in $R_{k(Q)}$ is non-zero, and hence also in $R_{k(Q)^{1/p^d},x}$.
Applying $\blank \otimes_{A^{1/p}} k(Q)^{1/p^{d+1}}$ to the sequences above gives that $$\big(R_{k(Q)^{1/p^{d+1}}}\big)^{\oplus p^{\delta}} \to R^{1/p}_{k(Q)^{1/p^{d+1}}} \to M_1 \otimes_{A^{1/p}} k(Q)^{1/p^{d+1}} \to 0$$ $$R^{1/p}_{k(Q)^{1/p^{d+1}}} \to \big(R_{k(Q)^{1/p^{d+1}}}\big)^{\oplus p^{\delta}} \to M_2 \otimes_{A^{1/p}} k(Q)^{1/p^{d+1}} \to 0$$ are right exact sequences of $R_{k(Q)^{1/p^{d+1}}}$-modules. But we have that $R_{k(Q)^{1/p^{d+1}}}$ is a free $R_{k(Q)^{1/p^d}}$-module of rank $\big[k(Q)^{1/p}:k(Q)\big]$, so we may view these as sequences of $R_{k(Q)^{1/p^d}}$-modules and localize at $x \in \Spec R_{k(Q)^{1/p^d}}$ to give the right exact sequences of $R_{k(Q)^{1/p^d},x}$-modules $$\big(R_{k(Q)^{1/p^d},x}\big)^{\oplus p^{\delta}[k(Q)^{1/p}:k(Q)]} \xrightarrow{\psi_1} \big(R_{k(Q)^{1/p^d},x}\big)^{1/p} \to \Big(M_1 \otimes_{A^{1/p}} k(Q)^{1/p^{d}}\Big)_x^{\oplus [k(Q)^{1/p}:k(Q)]} \to 0$$ $$\big(R_{k(Q)^{1/p^d},x}\big)^{1/p} \xrightarrow{\psi_2} \big(R_{k(Q)^{1/p^d},x}\big)^{\oplus p^{\delta}[k(Q)^{1/p}:k(Q)]} \to \Big(M_2 \otimes_{A^{1/p}} k(Q)^{1/p^{d}}\Big)_x^{\oplus [k(Q)^{1/p}:k(Q)]} \to 0$$ so that the summands of the quotients $\big(M_i \otimes_{A^{1/p}} k(Q)^{1/p^{d}}\big)_x$ for $i=1,2$ are killed by the image of $c$ in $R_{k(Q)^{1/p^d},x}$. If $P$ is the maximal ideal of $R_{k(Q)^{1/p^d},x}$ and $\ell(\blank)$ denotes length over $R_{k(Q)^{1/p^d},x}$, applying (for $A^{1/p} \to R_{A^{1/p}}$ with the $R_{A^{1/p}}$-modules $M_1, M_2$), we have that there is a positive constant $C'$ so that $$\begin{aligned}
\ell\left( \frac{\big(M_i \otimes_{A^{1/p}} k(Q)^{1/p^{d}}\big)_x}{P^{[p^e]}\big(M_i \otimes_{A^{1/p}} k(Q)^{1/p^{d}}\big)_x} \right) \leq \frac{C'}{p^e} p^{e \cdot \dim R_{k(Q)^{1/p^d},x}} = \frac{C'}{p^e} \frac{ [k(x)^{1/p}:k(x)] p^{(e+1) \dim R_{k(Q)^{1/p^d},x}}}{[k(Q)^{1/p}:k(Q)] p^\delta}\end{aligned}$$ for $i = 1,2$ and all $e > 0$. Using the well-known properties $$\begin{array}{c} \displaystyle
I_e\big(R_{k(Q)^{1/p^d},x}\big)^{[p]} \subseteq I_{e+1}\big(R_{k(Q)^{1/p^d},x}\big) \\
\displaystyle \phi\Big(I_{e+1}\big(R_{k(Q)^{1/p^d},x}\big)^{1/p}\Big) \subseteq I_e\big(R_{k(Q)^{1/p^d},x}\big)
\end{array}$$ for all $\phi \in \Hom_{R_{k(Q)^{1/p^d},x}}\Big(\big(R_{k(Q)^{1/p^d},x}\big)^{1/p},R_{k(Q)^{1/p^d},x}\Big)$, the maps $\psi_1, \psi_2$ induce $$\begin{aligned}
&\Big(R_{k(Q)^{1/p^d},x} \big/ I_e\big(R_{k(Q)^{1/p^d},x}\big)\Big)^{\oplus p^{\delta}[k(Q)^{1/p}:k(Q)]} \xrightarrow{\psi_{1,e}} \Big(R_{k(Q)^{1/p^d},x} \big/ I_{e+1}\big(R_{k(Q)^{1/p^d},x}\big)\Big)^{1/p}\\
&\Big(R_{k(Q)^{1/p^d},x} \big/ I_{e+1}\big(R_{k(Q)^{1/p^d},x}\big)\Big)^{1/p} \xrightarrow{\psi_{2,e}} \Big(R_{k(Q)^{1/p^d},x} \big/ I_e\big(R_{k(Q)^{1/p^d},x}\big)\big)^{\oplus p^{\delta}[k(Q)^{1/p}:k(Q)]}
\end{aligned}$$ with $\coker \psi_{i,e}$ a quotient of $\coker \psi_i$ killed by $P^{[p^e]}$ for $i = 1,2$. Taking lengths and dividing by $$[k(x)^{1/p}:k(x)] p^{(e+1) \dim R_{k(Q)^{1/p^d},x}}$$ gives $$\left| \frac{\ell \left( \displaystyle \frac{R_{k(Q)^{1/p^d},x}}{I_e\big(R_{k(Q)^{1/p^d},x}\big)} \right)}{p^{e \cdot\dim R_{k(Q)^{1/p^d},x}}} - \frac{\ell \left( \displaystyle \frac{R_{k(Q)^{1/p^d},x}}{I_{e+1}\big(R_{k(Q)^{1/p^d},x}\big)} \right)}{p^{(e+1)\dim R_{k(Q)^{1/p^d},x}}} \right| \leq \frac{C'}{p^{e + \delta}}$$ so that the proposition follows from [@PolstraTuckerCombined Lemma 3.5] with $C = 2C'/p^\delta$.
As mentioned above, we need to generalize the above to the context of pairs.
\[prop.uniformconverginfamilydeltas\] Suppose we are in the setting of . There exists a positive constant $C$ and $0 \neq g \in A$ with the following property: for all $Q \in \Spec(B := A[g^{-1}])$, all $d > 0$, all $x \in \Spec(R_{k(Q)^{1/p^d}})$, and all $e > 0$, we have $$\left| s(R_{k(Q)^{1/p^d},x}, \Delta_{Q,d}) - \frac{a_e^{\Delta_{Q,d}}(R_{k(Q)^{1/p^d},x})}{\rank_{R_{k(Q)^{1/p^d},x}}(R_{k(Q)^{1/p^d},x})^{1/p^e}} \right| \leq \frac{C}{p^e}$$ where $\Delta_{Q,d} = \Delta|_{\Spec(R_{k(Q)^{1/p^d}})}$.
The desired result follows the argument in , with modifications we now describe to account for the addition of $\Delta$. Choose $0 \neq c' \in R$ so that $\Div_R(c') \geq p \Delta$. After inverting an element of $A$, we may assume $c'$ does not vanish along any fiber and thus $\Div_R(c')|_{\Spec R_{k(Q)}} \geq p \Delta|_{\Spec R_{k(Q)}}$ on fibers as well. In particular, for any $\phi \in \Hom_{R_{k(Q)^{1/p^d},x}}((R_{k(Q)^{1/p^d},x})^{1/p},R_{k(Q)^{1/p^d},x})$ and $\psi(\blank) = \phi( (c')^{1/p} \cdot \blank)$, we have that $\Delta_\psi \geq \Delta_{Q,d}$ and $\Div_R(c')|_{\Spec R_{k(Q)^{1/p^d}}} \geq p \Delta_{Q,d}$ where $\Delta_{Q,d} = \Delta|_{\Spec R_{k(Q)^{1/p^d}}}$. Replace $\alpha_1, \alpha_2$ in the right exact sequences \[eq:sescee\] and \[eq:sesmap\] with their premultiples $$(R_{A^{1/p}})^{\oplus p^\delta} \xrightarrow{\cdot c'} (R_{A^{1/p}})^{\oplus p^\delta} \xrightarrow{\alpha_1} R^{1/p}$$ $$R^{1/p} \xrightarrow{\cdot (c')^{1/p}} R^{1/p} \xrightarrow{\alpha_2} (R_{A^{1/p}})^{\oplus p^\delta},$$ respectively. In [@PolstraTuckerCombined proof of Theorem 4.12], the properties $$\begin{array}{c} \displaystyle
c'(I_e(R_{k(Q)^{1/p^d},x}, \lceil (p^e-1)\Delta_{Q,d}\rceil)^{[p]}) \subseteq I_{e+1}(R_{k(Q)^{1/p^d},x}, \lceil (p^{e+1}-1)\Delta_{Q,d}\rceil) \\
\displaystyle \phi((c'I_{e+1}(R_{k(Q)^{1/p^d},x},\lceil (p^{e+1}-1)\Delta_{Q,d}\rceil))^{1/p}) \subseteq I_e(R_{k(Q)^{1/p^d},x},\lceil (p^e-1)\Delta_{Q,d}\rceil)
\end{array}$$ are shown to hold. The proof of can now be traced through without further modification. The corresponding maps $\psi_i$ satisfy the analogs of the above properties with respect to the ideals $I_e(R_{k(Q)^{1/p^d},x}, \lceil (p^e-1)\Delta_{Q,d}\rceil)$ and pass to maps $\psi_{i,e}$ on the quotients, with $\coker \psi_{i,e}$ a quotient of $\coker \psi_i$ killed by $P^{[p^e]}$ for $i = 1,2$. In particular, the constant $C'$ derived in the proof of from once more gives $$\left| \frac{\ell \left( \displaystyle \frac{R_{k(Q)^{1/p^d},x}}{I_e(R_{k(Q)^{1/p^d},x}, \lceil (p^e-1)\Delta_{Q,d}\rceil)} \right)}{p^{e(\dim R_{k(Q)^{1/p^d},x})}} - \frac{\ell \left( \displaystyle \frac{R_{k(Q)^{1/p^d},x}}{I_{e+1}(R_{k(Q)^{1/p^d},x}, \lceil (p^{e+1}-1)\Delta_{Q,d}\rceil)} \right)}{p^{(e+1)(\dim R_{k(Q)^{1/p^d},x},)}} \right| \leq \frac{C'}{p^{e + \delta}}$$ so that once more the proposition follows from [@PolstraTuckerCombined Lemma 3.5] with $C = \frac{2C'}{p^\delta}$.
\[lem.SplitAtPerfectionImpliesSplit\] In the setting of , suppose that there is a surjective $R_{K^{\infty}}$-linear map $$\label{eq.BaseChangeSurjectionImplies}
(R_{K^{\infty}})^{1/p^e} \to R_{K^{\infty}}^{\oplus a_e}$$ for some $a_e > 0$. Then for some $d_e > 0$ and $0 \neq g \in A$, setting $B = A[1/g]$, there is a surjective $R_{B^{1/p^{e+d_e}}}$-linear map $$R^{1/p^e} \otimes_B B^{1/p^{e+d_e}} = R^{1/p^{e}}_{B^{1/p^{e+d_e}}} \to R_{B^{1/p^{e+d_e}}}^{\oplus a_e}$$ which tensors with $\otimes_{B^{1/p^{e+d_e}}} K^{\infty}$ to recover .
Furthermore, suppose there exists a Weil divisor $\Delta$ on $\Spec R$ (still in ) such that each component projection $\rho : R_{K^{\infty}}^{1/p^e} \to R_{K^{\infty}}$ corresponds to a $\bQ$-divisor $\Delta_{\rho} \geq \xi^* \Delta$ (where $\xi : \Spec R_{K^{\infty}} \to \Spec R$ is the canonical map). In this case we can choose our $g$ such that the map $$R^{1/p^{e}}_{B^{1/p^{e+d_e}}} \to R_{B^{1/p^{e+d_e}}}^{\oplus a_e}$$ also has the property that each component projection $\gamma : R^{1/p^{e}}_{B^{1/p^{e+d_e}}} \to R_{B^{1/p^{e+d_e}}}$ corresponds to a $\bQ$-divisor $\Delta_{\gamma}$ on $\Spec R_{B^{1/p^{d_e}}}$ such that $\Delta_{\gamma} \geq \eta^* \Delta$ (where $\eta : \Spec R_{B^{1/p^{d_e}}} \to \Spec R$ is the canonical map).
First notice since we are planning to invert an element of $A$, we may assume that $\omega_A \cong A$. Furthermore, any future $B$ satisfies the same property. Note also that $R_{K^{\infty}}$ is a normal domain by . We have $(R_{K^{\infty}})^{1/p^e} \cong R^{1/p^e} \otimes_{K^{1/p^e}} K^{\infty}$ and so we can view our initial map as an $R_{K^{\infty}}$-linear map, and in particular a $K^{\infty}$-linear map $$\phi : (R^{1/p^e})_{K^{\infty}} \to R_{K^{\infty}}^{\oplus a_e}.$$ In other words, we are simply identifying relative and absolute Frobenius over a perfect field. Fix $x_1, \dots, x_t$ a generating set for $R^{1/p^e}$ over $R_{A^{1/p^e}}$. By base change, the images of those elements are also a generating set for $R^{1/p^e}_{K^{\infty}}$ over $R_{K^{\infty}}$ or for any intermediate base change. We may assume that all of the $\phi(x_i)$ land inside $R_{K^{1/p^{e+d_e}}}^{\oplus a_e} \hookrightarrow R_{K^{\infty}}^{\oplus a_e}$ for some $d_e > 0$. Note the $\phi(x_i)$ generate $\phi\left(R^{1/p^e}_{K^{1/p^{e+d_e}}}\right)$ as a $R_{K^{1/p^{e+d_e}}}$-module. This implies that $$\phi\left(R^{1/p^e}_{K^{1/p^{e+d_e}}}\right) \subseteq R_{K^{1/p^{e+d_e}}}^{\oplus a_e}$$ and hence we have a map (which we also call $\phi$) $$\phi : R^{1/p^e}_{K^{1/p^{e+d_e}}} \to R_{K^{1/p^{e+d_e}}}^{\oplus a_e}.$$ Since this map becomes surjective after the faithfully flat base change to $K^{\infty}$, it is surjective.
By the same argument as above, we may find a denominator $g'$ so that $$\phi\left(R^{1/p^e}_{A^{1/p^{e+d_e}}[1/g']}\right) \subseteq R_{A^{1/p^{e+d_e}}[1/g']}^{\oplus a_e}$$ which produces a map $$\phi : R^{1/p^e}_{A^{1/p^{e+d_e}}[1/g']} \to R_{A^{1/p^{e+d_e}}[1/g']}^{\oplus a_e}.$$ We do not know that this map is surjective but the cokernel is zero if we tensor with $R_{K^{1/p^{e+d_e}}}$. Inverting another element $g''$, setting $g = g' g''$ and $B = A[g^{-1}]$ we may assume that $$\phi : R^{1/p^e}_{B^{1/p^{e+d_e}}} \to R_{B^{1/p^{e+d_e}}}^{\oplus a_e}$$ is surjective as desired.
Now we move on to the statement involving $\Delta$. We begin in exactly the same way and produce a surjective map $$\phi : R^{1/p^e}_{B^{1/p^{e+d_e}}} \to R_{B^{1/p^{e+d_e}}}^{\oplus a_e}$$ for some $d_e > 0$ where $B=A[1/g]$. We need to show that the component projection maps $\gamma$ coming from $\phi$ produce divisors $\Delta_{\gamma}$ on $\Spec R_{B^{1/p^{d_e}}}$ via such that $\Delta_{\gamma} \geq \eta^* \Delta$ where $\eta : \Spec R_{B^{1/p^{d_e}}} \to \Spec R$ is the canonical map. Consider the following diagram where all of these maps are labeled. $$\xymatrix{
\Spec R_{K^{\infty}} \ar@/_2pc/[rr]_-{\xi} \ar[r]^-{\zeta} & \Spec R_{B^{1/p^{d_e}}} \ar[r]^-{\eta} & \Spec R
}$$ We also know that $\zeta^* \Delta_{\gamma} = \Delta_{\rho}$ by since $\gamma$ base changes to a projection $\rho$. Since $\Delta_{\rho} \geq \xi^* \Delta = \zeta^* \eta^* \Delta$, we see that $\zeta^* \Delta_{\gamma} \geq \zeta^* \eta^* \Delta$. Since $\Delta$ has no vertical components neither does $\eta^* \Delta$. Therefore because $\Delta_{\gamma} \geq 0$, we conclude that $\Delta_{\gamma} \geq \eta^* \Delta$ as desired.
\[thm.GeneralFiberImpliesMostSpecialFibers\] In the setting of , further suppose that $A$ is finite type over a perfect field of characteristic $p > 0$. If $$s(R_{K^{\infty}}, \Delta_{K^{\infty}}) > \lambda$$ then there exists an open dense $U \subseteq \Spec A$ such that for any closed point $Q \in U$, $$s(R_{k(Q)}, \Delta_{k(Q)}) > \lambda.$$
Inverting an element of $A$ if necessary, we may choose a positive constant $C$ as in . By [@DeStefaniPolstraYaoGlobalizingFinvariants], fix $0 < \epsilon \ll 1$ such that $s(R_{K^{\infty}, x}, \Delta_{K^{\infty}}) > \lambda + 2\epsilon$ for all $x \in \Spec R_{K^\infty}$. Pick $e \gg 0$ so that $C/p^e < \epsilon$, so that we have $$a_e^{\Delta_{K^{\infty}}}\big(R_{K^\infty,x}\big) \Big/ \rank_{R_{K^\infty,x}}\big(R^{1/p^e}_{K^\infty,x}\big) > \lambda + \epsilon.$$ By [@DeStefaniPolstraYaoGlobalizingFinvariants Theorem 4.22] and by , after inverting an element of $A$ we may assume there is a $d \geq 0$ and a surjective $R_{A^{1/p^{e+d}}}$-linear map $$R_{A^{1/p^{e+d}}}^{1/p^e} \to R_{A^{1/p^{e+d}}}^{\oplus a_e} \text{, where $a_e := a_e^{\Delta_{K_{\infty}}}(R_{K^\infty})$}$$ satisfying the divisorial condition on projections from . Applying $\blank \otimes_{A^{1/p^{e+d}}} k(Q)^{1/p^{e+d}}$ for maximal $Q \in \Spec(A)$ gives a surjection $$R_{k(Q)^{1/p^{e+d}}}^{1/p^e} \to R_{k(Q)^{1/p^{e+d}}}^{\oplus a_e}.$$ of $R_{k(Q)^{1/p^{e+d}}}$-modules where still $a_e = a_e^{\Delta_{K_{\infty}}}(R_{K^\infty})$. Note the projections corresponding to this map also have the property that their corresponding divisors are $\geq \Delta_Q := \Delta|_{R_{k(Q)^{1/p^d}}}$ by ..
Since $A$ is finite type over a perfect field and $Q$ is maximal, $k(Q)$ is also perfect and so $k(Q)^{1/p^{e+d}} = k(Q)^{1/p^e} = k(Q)$. It also follows that $$\rank_{R_{K^{\infty}}}(R_{K^{\infty}}^{1/p^e}) = \rank_{R_{k(Q)}}(R_{k(Q)}^{1/p^e})$$ since $A \subseteq R$ is flat and of finite type and $A$ is $F$-finite.
Therefore we have a surjection $$(R_{{k(Q)},x})^{1/p^e} \to R_{k(Q),x}^{\oplus a_e}$$ showing that $$\frac{a_e^{\Delta_{Q}}(R_{k(Q),x})}{\rank_{R_{k(Q),x}}(R_{k(Q),x})^{1/p^e}} > \lambda + \epsilon.$$ Thus, it follows once again from that $$s(R_{k(Q),x}, \Delta_Q) > \lambda$$ for all $x \in \Spec R_{k(Q)}$ as desired.
Bertini theorems for $F$-signature
==================================
In this section we conclude by proving our Bertini theorems for $F$-signature. We first recall the main result of [@CuminoGrecoManaresiAxiomatic] and the very slight generalization to the context of pairs of [@SchwedeZhangBertiniTheoremsForFSings].
Suppose $\sP$ is a local property for locally Noetherian schemes (respectively pairs $(X, \Delta \geq 0)$).
- \[prop.A1\] Whenever $\phi : Y \to Z$ is a flat morphism with regular fibers and $Z$ (resp. $(Z, \Delta)$) is $\sP$, then $Y$ (resp. $(Y, \phi^* \Delta)$) is $\sP$ too.
- \[prop.A2\] Let $\phi : Y \to S$ be a morphism of finite type where $Y$ is excellent and $S$ is integral with generic point $\eta$. If $Y_{\eta}$ (resp. $(Y_{\eta}, \Delta|_{Y_{\eta}}$) is geometrically $\sP$, then there exists an open neighborhood $U$ of $\eta$ in $S$ such that $Y_{s}$ (resp. $(Y_s, \Delta|_{Y_s})$) is geometrically $\sP$ for each $s \in U$.
- $\sP$ is open on schemes $X$ (resp. pairs $(X, \Delta)$) of finite type over a field. \[prop.A3\]
[[@CuminoGrecoManaresiAxiomatic Theorem 1]]{.nodecor} \[thm.CuminoGrecoManaresi\] Let $X$ be a scheme of finite type over an algebraically closed field $k$, let $\phi : X \to \bP^n_k$ be a morphism with separable generated residue field extensions. Suppose $X$ (resp. $(X, \Delta)$) has a property $\sP$ satisfying conditions (A1) and (A2). Then there exists a nonempty open subscheme $U$ of $(\bP_k^n)^*$ such that $\phi^{-1}(H)$ has property $\sP$ for each hyperplane $H \in U$.
\[rem.A2forClosedPointsAndVeryGeneralEnough\] In the proof of , when using (A2), $S$ is (an open subset) of $(\bP_k^n)^*$ and $\phi^{-1}(s) = Y_s$ are fibers that are exactly equal to the hyperplane sections. In particular, one may additionally assume that $S$ is of finite type over an algebraically closed field and we only need to verify (A2) for the closed fibers.
Suppose that $k = \overline{k}$ is uncountable and consider the following weakening of (A2):
- Let $\phi : Y \to S$ be a morphism of finite type where $S$ is integral of finite type over $k$, with generic point $\eta$. If $Y_{\eta}$ (resp. $(Y_{\eta}, \Delta|_{Y_{\eta}}$) is geometrically $\sP$, then for a very general closed point $s \in S$ we have that $Y_{s}$ (resp. $(Y_s, \Delta|_{Y_s})$) is geometrically $\sP$ for each $s \in U$. \[prop.B2\]
If (A1) and (B2) hold for $\sP$, then it immediately follows that the weakening of holds for *very general* hyperplane sections.
[[@CuminoGrecoManaresiAxiomatic Corollary 2]]{.nodecor} \[cor.OpenSubsetRestriction\] Let $k = \overline{k}$, $V \subseteq \bP^n_k$ be a closed subscheme (resp. and let $\Delta$ be a $\bQ$-divisor on $V$) and let $\sP$ be a local property satisfying (A1).
1. If $\sP$ satisfies (A2), and $V$ (resp. $(V, \Delta)$) is $\sP$, then the general hyperplane section of $V$ (resp. $(V, \Delta)$) satisfies $\sP$.
2. If $k$ is uncountable, $\sP$ satisfies (B2), and $V$ (resp. $(V, \Delta)$) is $\sP$, then the very general hyperplane section of $V$ (resp. $(V, \Delta)$) satisfies $\sP$.
3. Suppose $\sP$ satisfies (A1), (A2) and (A3), and set $\sP(V)$ to be the $\sP$ locus of $V$ then $$\sP(V \cap H) \supseteq \sP(V) \cap H$$ for a general hyperplane $H$.
Combining this machinery with our work of the previous sections, we immediately obtain the main result of the paper.
\[thm.Main\] Suppose that $\psi : X \to \bP^n_k$ is a map of varieties over $k = \overline{k}$ with separably generated residue field extensions (for example, a closed embedding) and that $\Delta \geq 0$ is a $\bQ$-divisor on $X$. Suppose that $\lambda \geq 0$.
1. \[thm.Main.1\] Suppose $s({\mathscr O}_{X,x}, \Delta_x) > \lambda$ for all closed points $x \in X$. Choose a general hyperplane $H \subseteq \bP^n_k$, and set $Y = \psi^{-1}(H)$. Then $$s({\mathscr O}_{Y,y}, \Delta_y|_Y) > \lambda$$ for all closed points $y \in Y$.
2. \[thm.Main.2\] Suppose $\psi : X \subseteq \bP^n_k$ is a closed embedding. Let $U_X \subseteq X$ be the subset of points $x \in X$ such that $s({\mathscr O}_{X,x}, \Delta_x) > \lambda$. For any hyperplane $H \subseteq \bP^n_k$ let $U_{H \cap X}$ denote the set of points $x \in X \cap H$ such that $s({\mathscr O}_{H,x}, \Delta_x|_H) > \lambda$. Then for $H$ a general hyperplane in $\bP^n_k$ $$U_{H \cap X} \supseteq U_X \cap H.$$
3. \[thm.Main.3\] Suppose additionally that $k$ is uncoutnable, and that $s({\mathscr O}_{X,x}, \Delta_x) \geq \lambda$ for all closed points $x \in X$. Choose a very general hyperplane $H \subseteq \bP^n_k$, and set $Y = \psi^{-1}(H)$. Then $$s({\mathscr O}_{Y,y}, \Delta_y|_Y) \geq \lambda$$ for all closed points $y \in Y$.
For part we consider the condition $\sP$ that $s({\mathscr O}_{X,x}, \Delta) > \lambda$. We apply using the fact that properties (A1) and (A2) are satisfied by and respectively (see also ).
For part , we simply use and use the fact that $s({\mathscr O}_{X,x}, \Delta) > \lambda$ is an open condition by semicontinuity [@PolstraFsigSemiCont; @PolstraTuckerCombined] so that (A3) is satisfied.
Part either follows from by considering a sequence of $\lambda_i = \lambda - 1/i$ or alternately can be directly proven via by replacing property (A2) with (B2), which was verified in .
[^1]: The first named author was supported in part by the NSF FRG Grant DMS \#1265261/1501115 and NSF CAREER Grant DMS \#1252860/1501102
[^2]: The second named author was supported in part by the NSF FRG Grant DMS \#1265261/1501115 and NSF CAREER Grant DMS \#1252860/1501102
[^3]: The third named author was supported in part by NSF Grant DMS \#1602070 and a fellowship from the Sloan foundation
[^4]: Here we mean that the fibers are normal after any base change, including inseparable ones.
[^5]: Here we mean that the fibers are normal after any base change, including inseparable ones.
[^6]: A $\bQ$-divisor in which no denominators to contain $p$.
[^7]: Meaning outside a countable union of proper closed subsets of $\Spec A$
|
---
bibliography:
- 'IEEEabrv.bib'
- 'includes/ref.bib'
---
|
---
bibliography:
- 'PLoS.bib'
---
[**Can Google searches help nowcast and forecast unemployment rates in the Visegrad Group countries?** ]{}\
Jaroslav Pavlicek $^{1}$ and Ladislav Kristoufek$^{1,2,3,\ast}$\
**[1]{} Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod Vodarenskou vezi 4, Prague 8, 182 08, Czech Republic, EU\
**[2]{} Institute of Economic Studies, Charles University, Opletalova 26, 110 00, Prague, Czech Republic, EU\
**[3]{} Warwick Business School, University of Warwick, Coventry, West Midlands, CV4 7AL, United Kingdom, EU\
$\ast$ E-mail: kristouf@utia.cas.cz******
Abstract {#abstract .unnumbered}
========
Online activity of the Internet users has been repeatedly shown to provide a rich information set for various research fields. We focus on the job-related searches on Google and their possible usefulness in the region of the Visegrad Group – the Czech Republic, Hungary, Poland and Slovakia. Even for rather small economies, the online searches of their inhabitants can be successfully utilized for macroeconomic predictions. Specifically, we study the unemployment rates and their interconnection to the job-related searches. We show that the Google searches strongly enhance both nowcasting and forecasting models of the unemployment rates.
Introduction {#introduction .unnumbered}
============
Online activity has become an inherent part of the modern society and the way of living among its members. The Internet provides a vast amount of information to its users as well as an aid and assistance in times of need. During the current financial and following economic and production crises, most of the developed as well as developing economies have been hit by an economic downturn which is usually tightly connected with a growing unemployment. Job loss can be a very traumatizing experience with long lasting impact on one’s life. Seeking a new job then becomes an integral part of an everyday life. In the current digitalized era, the job seeking does not restrict itself to job offices but the seekers (as well as potential employers) more frequently turn to the Internet as a source of information and new possibilities. As such, the job seekers leave a digital track of their activity.
Analysis and examination of various patterns of the online activity have become a fruitful branch of research in the last years with some exciting applications such as elections [@Metaxas2012], investment allocation [@mondria2010; @kristoufek2013a], private consumption [@Vosen2011] and consumers’ behavior [@Goel2010], future orientation [@Preis2012a], earnings announcements [@drake2012], diseases spreading [@polgreen2008; @ginsberg2009; @Carneiro2009; @Seifter2010; @Dugas2012], and economics and finance [@Preis2010; @varian2012; @Bordino2012; @kristoufek2013b; @preis2013; @Moat2013; @Curme2014]. Turning back to the unemployment and its possible examination utilizing the online activity of the Internet users, there has been some research done in the area as well focusing primarily on the Google engine search queries. The first study focusing on the possible connection between Google searching activity and unemployment rates examining the series in Germany shows usefulness of adding search queries data into the models [@askitas2009]. Following research [@varian2009b; @bughin2011; @varian2013] analyzes connection between the queries and claims for unemployment benefits in the USA and the unemployment rate itself has been studies as well [@damuri2010; @damuri2012]. Even job search activity index based on the Google search data has been developed [@baker2011]. Most of these studies focus on the US economy and its modeling while the other economies are studied rather marginally [@chadwick2012; @fondeur2012].
Here, we focus on possible connection between job-related search queries on the Google search engine and the unemployment rate in countries of the so-called Visegrad Group (the Czech Republic, Hungary, Poland and Slovakia). Our contributions lay in the following. First, we focus on a set of countries which would be normally treated as a marginal one and thus not much studied. However, if the utility of the online search activity (and specifically the Google searching) is to be claimed, its efficiency should be shown not only on developed and well covered countries but also on the smaller ones and the results might prove useful to all policy makers even in such regions. Second, we provide a careful and step-by-step procedure to the unemployment modeling focusing not only on simple correlations but also nowcasting, forecasting and causality. And third, a cross-countries comparison is delivered which is rather unique in comparable studies focusing primarily on one specific country.
Results {#results .unnumbered}
=======
The unemployment rates have undergone quite heterogenous evolution in the analyzed countries (Fig. \[fig\_U\]). In the Czech Republic, the rate ranged between 4% and 9% between years 2004 and 2013. Initially, there was a significant downward trend from year 2004 to 2008 when the rate dropped from 9% to 4%. As the recession hit the Czech Republic in 2008, the rate started to rise to reach its new maximum of 8.5% in 2010. Since then, the unemployment rate fluctuated between 7% and 8.5%. The Hungarian unemployment rate was steadily rising from the year 2004 to 2010 where it reached its new maximum of nearly 12%. After that the rate fluctuated for almost 3 years between 10.5% and 12% to start declining in the year 2013. The unemployment in Poland experienced a steady decline from the astronomical rate of nearly 22% in the year 2004 to 6% in 2009. However, as the recession hit Poland, the unemployment rate began rising again. With some minor fluctuations, it smoothly increased to the current level of approximately 10%. And in Slovakia, the unemployment rate seems to have a similar pattern as the one of the Czech Republic, although on a different scale. In 2004, Slovakia had an unemployment rate of almost 20%. This rate linearly decreased to 8% in 2009. With the hit of recession, the unemployment rate quickly escalated to 16% around which it has been fluctuating until today.
The evolution of the Google searches is illustrated in Fig. \[fig\_Google\]. There are evident seasonal patterns in all four series. Hungary is characterized by quite regularly increasing trend in the Google searches whereas Slovakia shows the opposite and the remaining two analyzed series remain quite stable in time. Even though there seems to be some connection between the Google searches and the unemployment rates for the Czech Republic and Hungary visible by the naked eye, we can hardly claim any relationship without a proper analysis.
Basic relationship {#basic-relationship .unnumbered}
------------------
As the initial step, we present the results of the stationarity tests which tell us whether we should analyze the original series or some of their transformations. In Tab. \[table:stat\], we show the results of the ADF and KPSS tests (see the Methods section for more details) for the original as well as the logarithmic series and their first differences. The outcome is quite straightforward as we do not reject unit roots for either of the original series (or their logarithmic transformation for the Google searches, we do not examine the logarithmic transformation for the unemployment time series as these are already in the percentage representation). Further testing, which is not reported here, shows no cointegration relationship between the unemployment and the search queries series so that we need to proceed with the first differences of the series. For most of the cases, we support stationarity of the first differences. In the analysis, we further proceed with the first differences of the unemployment rate and the first logarithmic differences of the Google searches. We opt for this combination as the pair of percentage representation and logarithmic transformation allows for a straightforward interpretation as an elasticity, i.e. as a proportional relationship.
For the very basic relationship between the unemployment rate and the intensity of the job-related searches on Google, we study the following equation $$\label{eq:basic}
\begin{aligned}
\Delta \text{UR}_t &= \alpha_0 + \alpha_1 \Delta \log(\text{GI})_t + \varepsilon_{t}
\end{aligned}$$ where $\Delta \text{UR}_t$ and $\Delta \log(\text{GI})_t$ stand for the first difference of an unemployment rate at time $t$ and the first logarithmic difference of the Google searches at time $t$, respectively, for a given country, and $\varepsilon_t$ is an error term.
The elasticity between the Google searches and unemployment rate from Eq. \[eq:basic\] is estimated at 0.5538 (with the $p$=value of 0.0533), 0.2056 (0.0726), 0.3317 (0.2163) and 0.4630 (0.0062) for the Czech Republic, Hungary, Poland and Slovakia, respectively, with the heteroskedasticity and autocorrelation consistent (HAC) standard errors. The proportional relationship thus varies across the analyzed countries but it remains positive for all four and statistically significant for three out of four (at least at the 10% significance level). Specifically, the relationship is very strong for the Czech Republic and Slovakia with the value around 0.5. This shows that the changes in the unemployment rate are well projected into the online search queries for the vacancies and job-related terms. Studying the connection between these two variables thus seems promising and worth further utilization and investigation.
Nowcasting {#nowcasting .unnumbered}
----------
Macroeconomic time series, such as the unemployment rates, have a special property which is not present for financial series or other series in natural sciences – they are available with a pronounced lag. This is due to the data processing and collection which usually take several months and even after such period, there are sometimes corrections to the reported values. Such characteristic makes a series, which is available immediately without any lag and which is strongly correlated with the variable of interest, very useful for forecasting the present value of the variable without waiting for several months. Such forecasting the present is usually referred to as “nowcasting”.
In the previous section, we have shown that the Google searches for job-related terms are significantly correlated with the unemployment rate which makes the search queries potentially useful for nowcasting of the unemployment. As a nowcasting model, we consider the following one
$$\label{eq:ARnow}
\begin{aligned}
\Delta \text{UR}_t &= \beta_0 + \sum_{i=3}^{12}{\beta_{i} \Delta \text{UR}_{t-i}} + \sum_{j=0}^{12}{\gamma_{j} \Delta \log(\text{GI})_{t-j}} + \varepsilon_{t}
\end{aligned}$$
where the unemployment rate is assumed to be available with a three months lag. We again consider the differenced series due to stationarity issues discussed above. Both series are kept to the lag of 12 months which controls for the seasonal pattern in both the series.
The results of the nowcasting models are summarized in Tab. \[table:now\]. There we show the adjusted $R^2$ ($\bar{R}^2$) as a measure of the models’ quality controlling for the number of explanatory variables. We observe that for all countries, the inclusion of the Google series enhances the model strongly. The $\bar{R}^2$ increases by approximately a third for all countries but Poland for which it increases slightly less. Nonetheless, inclusion of the search queries improves the model for all countries significantly as is reported by the $F$-statistics for the insignificance of the searches. All series are jointly significant even at the 1% level.
Forecasting & Causality {#forecasting-causality .unnumbered}
-----------------------
The nowcasting results are very promising and they illustrate usefulness of the Google searches series. However, we are also interested whether such usefulness is mainly due to the unavailability of the unemployment data or whether the search queries data provide additional informative value as well. To do so, we also undergo a standard forecasting exercise where we practically hypothesize what would happen were the unemployment data available straightaway. If the Google series improve even such hypothetical model, we conclude that the search queries data bring additional information to the model in addition to being strongly correlated with the changes in the unemployment rate by itself.
For the forecasting exercise, we utilize the standard vector autoregressive model (VAR, see the Methods section for more details). The specific model takes the following form $$\label{eq:VAR}
\begin{aligned}
\Delta \text{UR}_t &= \beta_{01} + \sum_{i=1}^{12}{\beta_{1i} \Delta \text{UR}_{t-i}} + \sum_{i=1}^{12}{\gamma_{1i} \Delta \log(\text{GI})_{t-i}} + \varepsilon_{1t} \\
\Delta \log(\text{GI})_t &= \beta_{02} + \sum_{i=1}^{12}{\beta_{2i} \Delta \text{UR}_{t-i}} + \sum_{i=1}^{12}{\gamma_{2i} \Delta \log(\text{GI})_{t-i}} + \varepsilon_{2t}
\end{aligned}$$ and it is compared to a simple autoregressive model of unemployment $$\label{eq:AR}
\begin{aligned}
\Delta \text{UR}_t &= \delta_{0} + \sum_{i=1}^{12}{\delta_{1i} \Delta \text{UR}_{t-i}} + \nu_{t}.
\end{aligned}$$
For the comparison purposes, we use two measures of the forecasting quality – root mean squared error and mean absolute error (RMSE and MAE, respectively, see the Methods section for more details). These measures are very straightforward – the lower they are the better performing the model is. In addition, we utilize the Diebold-Mariano test [@diebold1995] which compares the forecasting performance of two models with the null hypothesis of the models performing the same (see the Methods section for more details). The model is estimated on the series between January 2004 and December 2012 and the forecasting period is set between January and December 2013.
The summary of the forecasting performances is given in Tab. \[table:fore\]. There we can see that for all countries, the forecasting performance of the models increases strongly with the addition of the Google searches. This is further supported by the results of the Diebold-Mariano test which gives significant results, i.e. the model using the Google data outperforms the ones without them, for all countries at at least the 5% significance level. The online search data thus evidently provide an additional informative value to the unemployment modeling.
As the last step of the analysis, we provide a causality examination. We are thus interested in the specific relationship between the two analyzed series. Concretely, we examine whether the increasing unemployment causes people to look up the job-related terms more, or the increased online activity signalizes potential tensions on the job market, or both ways, or none. To do so, we utilize the Granger causality framework (see the Methods section for more details) which is built on the VAR analysis. The results are summarized in Tab. \[table:fore\]. Note that the null hypothesis of the Granger causality is “no Granger causality”. Therefore, if the null hypothesis is rejected, the causality is claimed to be found. The findings are quite homogenous. For three out of four countries (Hungary being the exception), we report causality in both directions. The influence thus goes from both directions and the series strongly influence each other.
Discussion {#discussion .unnumbered}
==========
Online activity of the Internet users has been proven useful in various fields. Nowcasting the unemployment rate is one of these fields. Contrary to the prevailing trend in the literature focusing on the well-developed (Western) countries, we have focused on utilizing the job-related Google searches in the Visegrad Group countries, i.e. the Czech Republic, Hungary, Poland and Slovakia. Even though the data availability and utilization of the Internet might not be as widespread in the region as one would expect for the developed countries, we have shown that in fact the online searches provide a very strong basis for the unemployment modeling.
In summary, we have shown that the basic dynamics of the Google searches for the job-related terms closely follows the one of the unemployment rates. Further, we have utilized this idea to successfully nowcast the unemployment rates using the current and lagged values of the Google searches. Such results have been shown to be caused not simply by the fact that the unemployment rates are not immediately available but also by the additional informative value of the online searches. Our findings indicate that the information left online by the Internet users can be easily utilized even for small or medium countries such as the ones of the Visegrad Group.
Methods {#methods .unnumbered}
=======
Data {#data .unnumbered}
----
The monthly unemployment data for the Czech Republic, Hungary, Poland and Slovakia have been obtained from the Eurostat database. The basis of unemployment measurement among the EU countries lies in the EU Labour Force Survey (EU LFS) – a continuous and harmonized household survey, which is in accordance with the EU legislation carried out in each member state. The monthly data from Eurostat are estimates based on the results of EU LFS. Since there are no legal obligations for the EU countries to deliver monthly data, these data are often interpolated/extrapolated using national survey or registered unemployment data.
According to Eurostat, an unemployed person is defined as someone aged between 15 and 74 without work during the reference week who is available to start working within two weeks and who has actively sought employment at some time during the last four weeks. In our analysis, we use the general (both sex, 15-74 years old) raw (not seasonally adjusted) unemployment rate. We do this since we do not know the method used for the seasonal adjustment and the Google data are not seasonally adjusted either.
The Google search queries data have been downloaded from the Google Trends webpage. As languages of the studied countries differ, we have looked for various terms. As the Czech, Polish and Slovakian are all Slavonic languages, the searched words are very similar or even the same. For Czech, we searched for “práce”, for Polish “praca” and for Slovakian “práce” as well. For Hungarian, we used term “állás”. For the Slavonic languages, the terms are equivalent to “job” or “work”, and for Hungarian, it is close to “job” or “work” but rather in a sense of looking for it. The term “állás” provides better results than a more straightforward “munka” which would be closer to a more standard meanings of terms “job” or “work’.
The weekly series obtained from the Google Trends site have been transformed to the monthly series on a basis of the number of days in the month basis. All series, both of the unemployment rate and the Google searches, are studied between January 2004 and December 2013.
Stationarity {#stationarity .unnumbered}
------------
Stochastic process $\{z_t\}$ is stationary if for every collection of time indices $1 \leq t_1 < t_2 < t_m$, the joint probability distribution of ($x_{t_1}$, $x_{t_2}$, ..., $x_{t_m}$) is the same as the joint probability distribution of ($x_{t_{1+h}}$, $x_{t_{2+h}}$, ..., $x_{t_{m+h}}$) for all integers $h\geq 1$ [@wooldridge2008]. To test for stationarity, we utilize the Augmented Dickey-Fuller (ADF) test [@dickey1979] and the KPSS test [@kwiatkowski1992]. The tests have opposite null hypotheses so that they provide a complementary pair which is commonly used for stationarity testing.
In the ADF procedure [@dickey1979], the OLS regression is run on $$\label{eq:adf}
\Delta z_t = \alpha_0 + \theta z_{t-1} + \gamma t + \Delta z_{t-1} + \Delta z_{t-2} + \cdots + \Delta z_{t-p} + \varepsilon_t$$ in order to perform the test, where $\alpha_0$ and $\gamma t$ are an intercept and a time trend, respectively, and $p$ represents the lag order. The null hypothesis under which the series contains a unit root is found for $$H_0: \theta = 0$$ against the alternative $$H_A: \theta < 0.$$ The ADF test statistics is then computed as usual $t$-statistics, which, however, follows a more complicated distribution under the null hypothesis. Due to the relative short time series, we set the number of lags arbitrarily to three.
The null hypothesis of the KPSS test [@kwiatkowski1992] is opposite to the one of the [ADF]{} test, i.e. the KPSS test has the null hypothesis of stationarity. The test is based on the [OLS]{} regression of the series $\{z_t\}$ $$\label{eq:kpss}
z_t = \alpha_0 + \gamma t + k\sum_{i=0}^{t}{\xi_i} + \varepsilon_t$$\
where $\alpha_0$ and $\gamma t$ again represent an intercept and a time trend, respectively, and $\xi_i$ are independent and identically distributed random variables with a zero mean and a unit variance. The null hypothesis of stationarity is found for $$H_0: k = 0$$ against the alternative $$H_A: k \neq 0.$$ The KPSS test statistic is defined as $$KPSS = \frac{\sum_{t=1}^{n}{S_t^2}}{n^2\hat{\omega}_T^2}$$ where $S_t$ is partial sum of residuals $$S_t = \sum_{i=1}^{t}{\hat{\varepsilon}_i}$$ and $\hat{\omega}_T^2$ is an estimator of the spectral density at a frequency zero.
Vector autoregression {#vector-autoregression .unnumbered}
---------------------
Vector autoregression (VAR) is simply a system of temporally dependent series. More precisely, denote the number of variables $k$ and the length of the series $T$, then VAR of order $p$ is generally represented by equation $$\label{eq:var}
y_t = \alpha + A_1 y_{t-1} + A_2 y_{t-2} + \cdots + A_p y_{t-p} + \varepsilon_t$$ where $y_t$ and $\varepsilon_t$ are $k \times T$ matrices representing the studied series and residuals, respectively, $\alpha$ represents a vector of constants and $A_i$ are time invariant matrices replacing the traditional $\beta_i$ coefficients. The selection of appropriate lag order $p$ is usually based on a specific information criterion.
In the VAR framework, the Granger causality concept is usually used as well. The causality testing simply stems in testing the joint significance of one of the variables in the equation for some other variable. The testing procedure is thus an $F$-test for joint significance of a specific variable. In needs to be noted that such causality is strictly statistical and it should be always treated with caution.
Forecasting {#forecasting .unnumbered}
-----------
To compare forecasting accuracy of the proposed models, we utilize three measures – mean absolute error (MAE), root mean squared error (RMSE) and the Diebold-Mariano test [@diebold1995].
MAE measures the average value of absolute losses. In other words, it gives an average deviation of forecast from realized value in absolute terms. It is given by the equation $$\label{eq:mae}
MAE = \frac{1}{T} \sum_{i=1}^{T} |f_i - y_i| = \frac{1}{T} \sum_{i=1}^{T} a_i$$ where $f_i$ stands for the predicted value, $y_i$ is the actual value and $a_i = |f_i - y_i|$.
RMSE is quite similar to the mean absolute error as it is simply a square root of the mean squared error, and it is defined as $$\label{eq:rmse}
RMSE = \sqrt{\frac{1}{T} \sum_{i=1}^{T} (f_i - y_i)^2} = \sqrt{\frac{1}{T} \sum_{i=1}^{T} s_i}$$ where $f_i$ stands for the predicted value $y_i$ is the actual value and $s_i = (f_i - y_i)^2$.
Diebold & Mariano [@diebold1995] propose a test to compare the predictive accuracy of two competing forecasts. Let $\{\varepsilon^1_t\}_{t_0}^T$ and $\{\varepsilon^2_t\}_{t_0}^T$ be the sequences of forecast errors losses from two competing forecasting measures by particular loss function (e.g. absolute error loss as $a_i$ in Eq. \[eq:mae\] or squared error loss as $s_i$ in Eq. \[eq:rmse\]). The null and alternative hypotheses are then stated as $$H_0: \mathbb{E}\{\varepsilon^1_t\}_{t_0}^T = \mathbb{E}\{\varepsilon^2_t\}_{t_0}^T$$ $$H_A: \mathbb{E}\{\varepsilon^1_t\}_{t_0}^T \neq \mathbb{E}\{\varepsilon^2_t\}_{t_0}^T.$$ The Diebold-Mariano test assesses the accuracy based on the loss differential $$d_t = \{\varepsilon^1_t\}_{t_0}^T - \{\varepsilon^2_t\}_{t_0}^T$$ and the underlining null $$H_0: \mathbb{E}\{d_t\} = 0.$$ The Diebold-Mariano statistics is then $$S = \frac{\bar{d}}{\sqrt{\widehat{LRV}_{\bar{d}}/T}}$$ where $\bar{d}$ is the mean loss differential $$LRV_{\bar{d}} = \gamma_0 + 2 \sum_{j=1}^{\infty}{\gamma_j}, \;\; \gamma_j = \text{cov}(d_t, d_{t-j})$$ and $\widehat{LRV}_{\bar{d}}$ is a consistent estimate of the asymptotic (long-run) variance of $\sqrt{T}\bar{d}$. Under the null hypothesis, the testing statistic goes to a standard normal distribution so that $S \stackrel{A}{\sim} N(0,1)$ [@diebold1995].
Acknowledgments {#acknowledgments .unnumbered}
===============
The research leading to these results has received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement No. FP7-SSH-612955 (FinMaP) and the Czech Science Foundation project No. P402/12/G097 “DYME – Dynamic Models in Economics”.
Figures {#figures .unnumbered}
=======
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![**Unemployment rate in the Visegrad countries.** The group of countries is evidently quite heterogenous in the unemployment rates. The Hungarian rate starts at the lowest level but increases stably during the whole period. The Czech rate begins at quite low levels and decreases up to the outbreak of the financial crisis when the rate surges up until 2010 after which it remains quite stable. The Polish and Slovakian rates commence at very high levels of unemployment which go down again up until the outbreak of the crisis after which they change the trends similarly to the Czech rate.\[fig\_U\]](Fig_U_all "fig:"){width="4.5in"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![**Google search queries for the job-related terms in the Visegrad countries.** The patterns are again quite heterogenous and the connection between the Google searches and the unemployment rates can be observed for the Czech and Hungarian rates. For the other two, the connection is not visible by the naked eye. Detailed treatment of the interconnections is given in the Results section of the text.\[fig\_Google\]](Fig_Google_all "fig:"){width="4.5in"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Tables {#tables .unnumbered}
======
[l |cccc]{} &Czech Rep. & Hungary & Poland & Slovakia\
\
Unemployment&-1.6066&-1.78611&-2.6739$^{\ast}$&-2.6438$^{\ast}$\
- first difference&-5.3860$^{\ast\ast\ast}$&-4.5134$^{\ast\ast\ast}$&-3.5267$^{\ast\ast\ast}$&-4.2349$^{\ast\ast\ast}$\
Google&-2.6399$^{\ast}$&-1.76434&-2.1745&-1.4327\
- logarithm&-2.6504$^{\ast}$&-2.0239&-2.3280&-0.9082\
- difference&-11.2221$^{\ast\ast\ast}$&-10.2869$^{\ast\ast\ast}$&-11.1560$^{\ast\ast\ast}$&-10.5487$^{\ast\ast\ast}$\
- logarithmic difference&-11.3131$^{\ast\ast\ast}$&-10.7993$^{\ast\ast\ast}$&-11.0750$^{\ast\ast\ast}$&-10.5391$^{\ast\ast\ast}$\
\
Unemployment&0.5399$^{\ast\ast}$&2.5995$^{\ast\ast\ast}$&1.7946$^{\ast\ast\ast}$&0.7507$^{\ast\ast\ast}$\
- first difference&0.1932&0.2848&0.6708$^{\ast\ast}$&0.5673$^{\ast\ast}$\
Google&0.4208$^{\ast\ast}$&2.5994$^{\ast\ast\ast}$&1.3281$^{\ast\ast\ast}$&2.2879$^{\ast\ast\ast}$\
- logarithm&0.4048$^{\ast}$&2.6349$^{\ast\ast\ast}$&1.3596$^{\ast\ast\ast}$&2.2123$^{\ast\ast\ast}$\
- difference&0.0730&0.1406&0.1358&0.0436\
- logarithmic difference&0.0818&0.1201&0.1362&0.0821\
Czech Rep. Hungary Poland Slovakia
-- ---------------- ------------ --------- -------- ----------
without Google 0.2796 0.3590 0.4673 0.1605
with Google 0.3763 0.4469 0.5521 0.2193
$F$-stat 5.8507 2.5828 2.5251 2.7135
$p$-value 0.0000 0.0089 0.0057 0.0031
: Nowcasting summary[]{data-label="table:now"}
Czech Rep. Hungary Poland Slovakia
-- ---------------- ------------ --------- --------- ----------
no Google 0.3686 0.3156 0.1990 0.1907
Google 0.3216 0.2742 0.1467 0.1359
change -12.75% -13.11% -26.28% -28.73%
no Google 0.3100 0.2183 0.1437 0.1638
Google 0.2744 0.1889 0.1164 0.1009
change -11.50% -13.49% -19.01% -38.41%
test statistic 2.4160 2.8870 2.0220 1.7750
$p$-value 0.0078 0.0019 0.0216 0.0379
test statistic 6.0311 1.1163 3.8285 3.1555
$p$-value 0.0000 0.3614 0.0002 0.0012
test statistic 2.5545 1.4951 2.5635 2.5641
$p$-value 0.0073 0.1469 0.0071 0.0071
: Forecasting and causality summary[]{data-label="table:fore"}
|
---
abstract: 'Galaxy observations and N-body cosmological simulations produce conflicting dark matter halo density profiles for galaxy central regions. While simulations suggest a cuspy and universal density profile (UDP) of this region, the majority of observations favor variable profiles with a core in the center. In this paper, we investigate the convergency of standard N-body simulations, especially in the cusp region, following the approach proposed by . We simulate the well known Hernquist model using the SPH code Gadget-3 and consider the full array of dynamical parameters of the particles. We find that, although the cuspy profile is stable, all integrals of motion characterizing individual particles suffer strong unphysical variations along the whole halo, revealing an effective interaction between the test bodies. This result casts doubts on the reliability of the velocity distribution function obtained in the simulations. Moreover, we find unphysical Fokker-Planck streams of particles in the cusp region. The same streams should appear in cosmological N-body simulations, being strong enough to change the shape of the cusp or even to create it. Our analysis, based on the Hernquist model and the standard SPH code, strongly suggests that the UDPs generally found by the cosmological N-body simulations may be a consequence of numerical effects. A much better understanding of the N-body simulation convergency is necessary before a ’core-cusp problem’ can properly be used to question the validity of the CDM model.'
author:
- 'A. N. Baushev'
- 'L. del Valle'
- 'L.E. Campusano'
- 'A. Escala'
- 'R.R. Muñoz'
- 'G.A. Palma'
title: 'Cusps in the center of galaxies: a real conflict with observations or a numerical artefact of cosmological simulations?'
---
Introduction
============
Results of N-body simulations come into increasing conflict with observations of the dark matter (DM) distribution in the central regions of dwarf galaxies. Astronomical observations favor relatively soft cored density profiles [@mamon2011; @deblok2001; @bosma2002; @oh2011; @governato2012; @tollerud2012; @delpopolo2016]. On the contrary, N-body simulations of cold dark matter tell us that dark matter halos have a universal shape, independent of the halo mass and initial density fluctuation spectrum, and that the central universal density profile (hereafter UDP) is cuspy. The first works on the subject proposed the Navarro-Frenk-White profile (hereafter NFW) that behaves as $\rho\propto r^{-1}$ at the center. Later simulations [@mo09; @navarro2010] favor the Einasto profile with a finite central density. However, the obtained Einasto index is so high (typically $n\sim 5-6$) that the profile is still cuspy and close to the NFW one.
For a time, there was a hope that the ’core-cusp problem’ would disappear once the baryon contribution is taken into account. However, recent simulations including baryon matter have rather amplified the problem [@diversity]: apart from the profile disagreement, a more fundamental difficulty was found. Of course, the presence of baryons in simulations changes the density profile, but it remains almost universal for all the halos, while the profiles of real galaxies are extremely varied. The conflict between simulations and observations might suggest that the cold DM paradigm is wrong. However, before reaching this conclusion, the accuracy and convergence of the simulations should be scrutinized. For instance, the overestimation of the energy exchange between the test bodies that may occur in the N-body simulations leads to the cusp formation [@15]. If the energy evolution during the halo formation is limited, then the density profile of the formed halo resembles more closely the observed one [@16].
As an example, the overestimation of the particle energy exchange may be due to the unphysical pair collisions of the test bodies. Its importance may be characterized by the relaxation time [@bt eqn. 1.32] $$\tau_r =\dfrac{N(r)}{8\ln\Lambda}\cdot\tau_d \label{relaxation_time}$$ where $N(r)$ is the number of test bodies inside a sphere of radius $r$, $\ln\Lambda$ is the Coulomb logarithm, $\tau_d=(6\pi G\bar\rho(r))^{-1/2}$ is the characteristic dynamical time of the system at radius $r$, $\bar\rho(r)$ is the average density inside $r$. Equation \[relaxation\_time\] has two important consequences. First, $\tau_r$ depends on the smoothing radius of the N-body simulations only through $\Lambda$, i.e., only logarithmically [@13]. Therefore, the influence of the unphysical collisional relaxation cannot be decreased much by the smoothing of the test body potentials. Second, since the number of dark matter particles is huge ($\sim 10^{60}$, if dark matter consists of elementary particles), the collisional relaxation plays no role in nature, being a purely numerical effect.
The algorithm stability is the critical point of N-body simulations: the Miller’s instability makes the Liapunov time comparable with the dynamical time of the system [@miller1964]. Even if we take into account the specificity of N-body algorithms, like the potential smoothing, the instability development time is much shorter than $\tau_r$ and remains comparable with the dynamical time at the given radius $\tau_d(r)$ [@valluri2000; @hut2002]. However, different N-body codes, with various versions of the Poisson solvers, integration algorithms etc., lead to final halos with the above-mentioned UDP, which is almost the same and close to NFW. Therefore, it is widely believed that the universal profile is physically meaningful and that it describes real halos, even though the orbits of individual test bodies have no physical significance [@bt section 4.7.1(b)]. The aim of this paper is to question this opinion.
Indeed, the convergency criteria of N-body simulations used at present are exclusively based on the density profile stability. [@power2003] found that the cusp of the UDP remains stable at least until $t=1.7\tau_r$ and then a core forms. On this basis [@power2003] supposed that the core formation is the first sign of the collision influence and offered the most extensively used criterion for simulation convergency $t<1.7\tau_r$. The acceptance that the collisions have no effect even if the simulation time exceeds $\tau_r$ seems surprising. However, later convergency tests (also based only on the stability of the density profile) suggested even softer criteria [@hayashi2003; @klypin2013]. In this paper we perform a more sophisticated convergency test, going beyond the density profile analysis and considering the full array of the dynamical parameters of the particles.
Calculations
============
The main idea
-------------
In order to test the N-body convergency, we follow the method offered in [@13]. We simulate the well-known Hernquist model with the density profile $\rho(r)=Ma/[2\pi r(r+a)^3]$ (where $a$ is the scale radius and $M$ is the total halo mass), and with the isotropic velocity distribution at each point [@hernquist1990]. The model is spherically symmetric and fully stable, i.e., the density and velocity profiles should not change with time. We chose the Hernquist model because it is close to the NFW and behaves exactly as the NFW ($\rho\propto r^{-1}$) in the central region, but it has a known analytical solution for the stationary velocity distribution, contrary to the NFW one. The region of the cusp ($r<a$) is of main concern to us.
Since the gravitational potential $\phi(r)$ is constant, the specific energy ${\epsilon}=\phi(r)+v^2/2$ and the specific angular momentum $\vec K$ of each particle should be conserved. Instead of ${\epsilon}$, it will be more convenient to use the apocenter distance of the particle $r_0$ (i.e. the maximum distance on which the particle can move off the center, which can be found from the implicit equation ${\epsilon}=\phi(r_0)+K^2/2r_0$). Being an implicit function of the integrals of motion ${\epsilon}$ and $K$, $r_0$ is an integral of motion as well. Thus, any time variation of ${\epsilon}$, $\vec K$, or $r_0$ is necessarily a numerical effect, and we may judge the simulation convergency following the behavior of these quantities.
We need to clarify two important points of our work. Some properties of the perfectly symmetrical model we consider (like the exact conservation of the angular momentum for every particle) are unstable and not realistic for real astrophysical DM halos that are always triaxial as a result of tidal perturbations etc. The application of perfectly spherical models to real systems may give rise to false conclusions [@pontzen2015]. However, the use of the spherical model for our purposes is well founded. We are not considering the task of comparison of simulation results with observations. Our aim is just to check if the ’N-body matter’ behaves as a collisionless matter, which is the principle question of the dark matter modelling.
Second, there is a frequent belief that it is much easier to converge on the spherically averaged density distribution than on the full properties of the phase space distribution function. Indeed, we need not correctly reproduce each individual particle trajectory. Moreover, it is not even necessarily desirable since real dark matter halos are not spherically symmetric and therefore host chaotic orbits. However, it would be completely wrong to disregard the phase evolution of the system or consider its evolution as a ’second order effect’ with respect to the density profile shape. As we will show in the *Results: the simulation convergency* section, correct simulations of the energy and angular momentum of each particle (contrary to individual particle trajectories) are of critical importance for correct simulations of the density profile.
The simulations
---------------
We simulate a single separate Hernquist halo. The aim of this work is to perform a sophisticated test of the standard convergency criteria, therefore we do not try to model any real astronomical object. Since the standard N-body units [@nbodyunits] are used, the results are independent on the choice of $a$ and halo mass. However, we choose some values of the parameters, for illustrative purposes. Let us set $a=100$ [pc]{}, which roughly corresponds to the well-known dwarf spheroidal satellite of the Milky Way, Segue 1. This is one of the most popular objects for the indirect dark matter search, since it is close to the Solar System; its present-day mass can be estimated as $3\cdot 10^7 M_\odot$ [@12]. Segue 1 experienced strong tidal disruption, and we do not know its initial mass. We consider two limiting cases. In the body of the paper we accept the halo mass $M=10^9 M_\odot$, which is comparable with the present-day mass of a larger dwarf satellite, Fornax [@walker2011] and almost certainly exceeds the initial mass of Segue 1. Thus we consider the case of a compact and very dense dwarf spheroidal galaxy. However, since all the simulations are performed in the dimensionless N-body units, a reader may easily extend the results for any value of $M$. If $a$ is fixed, the only value that is sensitive to the choice of $M$ is time: all the time intervals scale as $\Delta t\propto M^{-0.5}$ (while the ratios of time intervals remain the same). As an illustration, we also considered the case of $M=10^7 M_\odot$, which is certainly lower, then the present-day mass of Segue 1. The only difference is that all the time intervals get ten times larger, and we everywhere specify the values corresponding to $M=10^7 M_\odot$ in the footnotes. Anticipating events, we say that the results shown in all the figures in this paper are not sensitive at all to the choice of $M$.
We use $N=10^6$ test bodies [^1]. They are placed randomly, in accordance with the analytically obtained space and velocity distributions [@hernquist1990]. The relaxation time at $r=a$ is $\tau_r(a)\simeq 8.8\cdot 10^{16}\text{s}\simeq 2.8\cdot 10^9$ [years]{}. Therefore, we make $200$ snapshots with the time interval $\Delta t=10^{15}\text{s}\simeq 30$ [mln. years]{}, covering the time from $0$ to $t_{max}=2\cdot 10^{17}\text{s}\simeq 6.5\cdot 10^9$ [years]{} (for the case of the halo mass $M=10^7 M_\odot$, $\tau_r(a)\simeq 2.8\cdot 10^{10}$ [years]{}, $\Delta
t=10^{16}\text{s}\simeq 300$ [mln. years]{}, $t_{max}=2\cdot 10^{17}\text{s}\simeq 6.5\cdot
10^{10}$ [years]{}). We record the positions and the velocities of each particle on each snapshot.
We evolve the system using one of the most extensively employed in cosmological simulations SPH codes, Gadget-3, an update version of Gadget-2 [@springel2001; @springel2005]. The gravitational interactions in Gadget-3 are computed using a hierarchical tree [@tree1; @tree2]. In this algorithm the space is divided in different cells and the gravitational force acting on a particle is computed using a direct summation for particles that are in the same cell and by means of multipole (up to the quadrupole) expansion for the particles that are in a different cell. The minimum distance between particles to be part of a different cell is controlled by a tree opening criterion. Gadget-3 uses the Barnes-Hut tree opening criterion for the first force computation. This criterion is controlled by an opening angle $\mu$, which determines the maximum ratio between the distance to the center of mass of the cell ($d$) and the size of the cell ($l$). If the cell is too close to the particle, $d/l$ will be greater than $\mu$, and new cells have to be opened to maintain the accuracy on the force computation. In the further evolution of the system a dynamical updating criterion (controlled by the fractional error $f_{acc}$) is used. We use the standard set of parameters $\mu=0.7$ and $f_{acc}=0.005$, as suggested by [@springel2005; @springel2008; @boylan-kolchin2009]. These values lead to a relative force error that is roughly constant in the simulation $\sim 0.5$%. We chose the softening radius $0.02a=2$ [pc]{}, in accordance with [@kampen2000; @hayashi2003].
Results: the integrals of motion
================================
Initially we convert each of the $201$ snapshots into the center-of-mass frame of references at the moment when a snapshot is made.
First of all, we try to reproduce the results of [@power2003]. The density profile indeed remains quite stable, and then a core in the center appears. Exactly following [@power2003], we consider the moment $t_{20\%}$ when the mass inside some radius drops on $20\%$ comparing to the initial value as the moment of the core formation. The ratio of $t_{20\%}$ to the relaxation time $\tau_r$ at the same radius $r$ is represented in Fig. \[17fig1\]. We see that our data by and large confirm the results of [@power2003], the core really appears at $t\simeq 2\tau_r$.
Before proceeding any further, two important comments relating to all the subsequent text should be made. First, our convergency tests are mainly oriented on the radius interval $[0.25a;1.5a]$ where they are the most precise. This choice of the working interval might appear strange at first sight: typically the convergency problems occur much closer to the halo center. However, if we had chosen a realistic area $(r\le 0.01a)$, then it would have contained only $\sim 100$ test particles, and the statistic would have been poor. On the other hand, the density profile between $0.25a$ and $0.75a$ remains much the same as in the center, since a power-law profile $\rho\propto r^{-1}$ is self-similar. The lower border of the region under consideration $r=0.25a$ is defined by our choice of the timestep $\Delta t=10^{15}$ [s]{} (for the case of the halo mass $M=10^7 M_\odot$, $\Delta
t=10^{16}\text{s}\simeq 300$ [mln. years]{}). At $r=0.1a$, $\tau_r\simeq\Delta t$, and the Hernquist profile is certainly corrupted by the collisions even on the first timestep. However, as we will see from the discussion of fig. \[17fig4\], the core formation becomes visible in phase portrait at much larger distances than in the density profile itself. Therefore, only the results related to $r\ge 0.25a$ can be totally trusted.
Second, we want to consider variations of the integrals of motion as a function of radius. However, each particle contributes to the density profile on an interval between its pericenter radius $r_{min}$ and apocenter radius $r_0$. Hereafter we will consider $r_0$ as the characteristic radius corresponding to the particle. Indeed, if the particle orbit is elongated, the particle spends almost all the time near the apocenter, in accordance with the Kepler’s second law. On the contrary, if the orbit is circular, the particle moves along almost uniformly, but its radius always remains close to $r_0$.
In order to study the behavior of the integrals of motion (theoretically they should conserve), we order all $10^6$ particles according to their $r_0$ in the initial snapshot, and then divide the particles into 200 groups of 5000 particles each. All the particles in the same group have similar $r_0$, and the group may be characterized by the average initial $\overline{r_0}$ of its members. We calculate $\Delta r_0/r_0=(r_0(i+1)-r_0(i))/r_0$ and $\Delta K/K_{circ}=(K(i+1)-K(i))/K_{circ}$ for each particle on each timestep. Here $i$ is the number of the snapshot, $K_{circ}$ is the angular momentum corresponding to the circular orbit at $r_0$; apparently, this is the maximum value of $K$ any particle with the apocenter distance $r_0$ may possess. Then we find the root-mean-squares of $\Delta r_0/r_0$ and $\Delta K/K_{circ}$ averaged over each group and for each snapshot. Our analysis shows that the root-mean-squares do not significantly depend on time until the moment when the core forms at the radius corresponding to $\overline{r_0}$ of the group. Therefore, we then average the root-mean-squares of $\Delta r_0/r_0$ and $\Delta K/K_{circ}$ over all the timesteps where the core had not formed yet. We denote the values averaged in such a complex manner by $\langle\widehat{\Delta} r_0/r_0\rangle$ and $\langle\widehat{\Delta}
K/K_{circ}\rangle$.
The dependance of $\langle\widehat{\Delta} K/K_{circ}\rangle$ (squares) and $\langle\widehat{\Delta} r_0/r_0\rangle$ (crosses) from the dimensionless radius $\overline{r_0}/a$ is represented in Fig. \[17fig2\]. We see that even in a single time step $\Delta
t=10^{15}\text{s}\simeq 30$ [mln. years]{} (for the case of the halo mass $M=10^7 M_\odot$, $\Delta
t=10^{16}\text{s}\simeq 300$ [mln. years]{}) the integrals (that should be constant) vary significantly. Fig. \[17fig3\] represents the values $\frac{K_{circ}}{\tau_r}\left\langle\frac{\widehat{\Delta} K}{\Delta t}\right\rangle^{-1}$ (squares) and $\frac{1}{\tau_r}\left\langle\frac{\widehat{\Delta} r_0}{r_0\Delta
t}\right\rangle^{-1}$ (crosses) that are the ratios of the time intervals in which an average particle totally ’forgets’ its initial values of $K$ and $r_0$ to the relaxation time $\tau_r(r)$. Everywhere in the region of reliability ($r\ge 0.25a$) the ratios are much less than $1$.=20000
It means that the particles totally ’forget’ their integrals of motion in a time much shorter than $\tau(r)$. In general one could not expect a reliable simulation of the velocity distribution at $t\sim\tau(r)$ under such conditions.
Results: the simulation convergency
===================================
Thus, the integrals of motion of the particles are not conserved at all, while the density profile remains stationary in quite good agreement with the theory in our simulations (we should mention, however, that the absence of the collision influence up to almost $2$ relaxation times looks surprising [@13]. The same stability and reproducibility of the cusps in cosmological modelling leads to the wide acceptance of the idea that, though no significance can be attached to the trajectories of individual particles in the N-body simulations, the cuspy density profile is meaningful and should correctly describe the profiles of real halos. Let us use our results to illustrate the vulnerability of the profile stability as the only convergency criterion of the N-body simulations.
Indeed, if a Hernquist halo consists of real DM (we suppose that it is cold and noninteracting), the values of ${\epsilon}$, $\vec K$, and $r_0$ of each particle must conserve, the particle distribution function $f$ should depend only on ${\epsilon}$ and $K$ [@bt] and obey the collisionless kinetic equation $df/dt=0$. It means that there are no particle fluxes in the phase space $({\epsilon},K)$.
However, figures \[17fig2\] and \[17fig3\] doubtlessly reveal an intensive energy and angular momentum exchange between the particles, i.e., the test bodies interact. Then the system may be described by the Fokker-Planck (hereafter FP) equation [@ll10] $$\dfrac{df}{dt}=\frac{\partial}{\partial q_\alpha}\left\{{\tilde A_\alpha}
f+\frac{\partial}{\partial q_\beta}[B_{\alpha\beta}f]\right\}
\label{fokker_planck}$$ where $q_\alpha$ is an arbitrary set of generalized coordinates, $${\tilde A_\alpha}=\dfrac{\overline{\delta q_\alpha}}{\delta t}\qquad
B_{\alpha\beta}=\dfrac{\overline{\delta q_\alpha\delta q_\beta}}{2\delta
t}
\label{fokker_planck_coefficients}$$ We may choose $q_1={\epsilon}$, $q_2=K$, and figure \[17fig2\] shows that at least coefficients $B_{11}$ and $B_{22}$ in the equation (\[fokker\_planck\]) differ essentially from zero. Thus we model real DM halos that are believed to be collisionless, by a system of test bodies governed by the kinetic equation with a significant collisional term, i.e., by an [*essentially collisional*]{} equation.
An important point must be underscored: the density profile in our simulation indeed holds its shape (close to the NFW one in the center), in gratifying agreement with the theoretical predictions. The variations of the integrals of motion, that we found, mainly touches on the velocity distribution of the particles. Together with the UDP stability in cosmological simulations, it can produce a dangerous illusion that N-body simulations might adequately model the density profiles of dark matter structures, despite the fact that the velocity distribution was distorted. We should emphasize that it cannot be true. Indeed, let us consider a stationary spherically symmetric halo for the sake of simplicity. The particle distribution in the phase space $f d^3x d^3v$ is a function of only the particle energy ${\epsilon}$ and three components if its angular momentum $\vec K$ [@bt]. If the velocity distribution of the particles is anisotropic in each point (which is the case under consideration in this paper), $f$ depends only on the particle energy $f(ep)=f(\phi(r)+v^2/2)=f(\phi(r_0)+K^2/(2r_0^2))$. The particle speed distribution at some radius $r$ is therewith equal to $4\pi v^2 f(\phi(r)+v^2/2) dv$, and the density is $\int 4\pi v^2
f(\phi(r)+v^2/2) dv$. These relationships clearly demonstrate the impossibility of a reliable determination of the density profile without a reliable determination of the velocity distribution. The distributions over $\vec v$ and $r$ are not just bound, there are actually a sort of projections of the same distribution $f$ on the velocity or space coordinates. Apparently, this conclusion is very general and does not depend on the assumption about the spherical symmetry that we made.
We are able to compare the results with the theoretical prediction and check their agreement in the model case that we consider. However, it is impossible in the case of real cosmological simulations. Therefore, any numerical effect influencing on the velocity distribution $f(\vec v)$ or on the integrals of motion of the test particles puts the density profiles obtained in the simulations in doubt. Moreover, the fact that the cuspy profile $\rho\propto r^{-1}$ turns out to be very stable in our simulation, despite of $r_0$ and $K$ variations, is probably not a coincidence, but a direct result of the numerical effects we discuss.
Indeed, the fact that we model collisionless systems with the test bodies governed by an essentially collisional equation is surprising [*per se*]{}, but the main consequence is that the profile stability does not guarantee the simulation correctness. The FP streams in the phase space created by the particle interaction may form stable density profiles (corresponding to the stationary solutions of the Fokker-Planck equation), but these profiles and their persistence are at odds with the behavior of real collisionless systems. As the first and crude illustration, the collisions lead to the contraction of the central region of any realistic profile and finally to the core collapse. The density profile outside the core approaches a power law $\rho\propto
r^{-2.23}$ and then remains quite stable for a long time [@bt]. Of course, this distribution is already formed by the unphysical test body collisions, and the immutability of the $\rho\propto
r^{-2.23}$ profile says nothing either about the simulation convergency or about the behavior of real collisionless systems.
The core collapse appears at $t\gg\tau_r$ and has nothing to do with the Hernquist or the UDP profiles. However, the Fokker-Planck equation has an another stationary solution close to the NFW one [@evans1997; @13].
A question appears: if we obtain a stable cuspy density profile, how can we differentiate cusps correlating with the properties of real collisionless systems from the solutions created by the numerical effects? In a collisionless system, the values of $r_0$ of the particles in the cusp should remain constant. If collisions are significant, the values of $r_0$ should experience a random walking, and the particles move up and down in the cusp forming a downward stream (of the particles with decreasing $r_0$) and an upward stream (of the particles with increasing $r_0$). For the cusp to be stable, the streams should compensate each other, which corresponds to a stationary solution of the Fokker-Planck equation. Thus, if the cusp is created by the FP diffusion, we should see two significant streams of particles with decreasing and increasing $r_0$, and the streams should compensate each other in order to provide the cusp stability.
We chose two adjacent (i.e., divided by a single $\Delta t$) snapshots at the beginning of the simulations, in order to minimize the core formation effects. For an array of radii $r$, we calculated the number $\Delta N_{+}(r)$ of particles that had $r_0<r$ at the first snapshot and $r_0>r$ at the second one, and the number $\Delta N_{-}(r)$ of particles that had $r_0>r$ at the first snapshot and $r_0<r$ at the second one. Of course, $\Delta N_{+}(r)=\Delta N_{-}(r)=0$ in the collisionless case, since $r_0$ is an integral of motion.
Fig. \[17fig4\] represents $\Delta N_{+}(r)$ (squares) and $\Delta N_{-}(r)$ (crosses) divided by the total number of the particles $N(r)$ inside $r$. As we can see, the FP streams exist, though they compensate well each other outside of $r=0.4a$. As we approach the center, the upward stream becomes increasingly stronger than the downward one. This is the first sign of the core formation, that is still invisible in the density profile at this radius, being already quite clear at the phase picture of the system.
A question appears: are the discovered fluxes $\Delta N_{+}(r)$ and $\Delta N_{-}(r)$ real and important? May they be just a small noise, produced by particles near the boundary, crossing and recrossing it and thus giving the impression of flows that do not exist? One can readily see that this is not the case. First of all, $\Delta N_{+}(r)$ and $\Delta N_{-}(r)$ apparently give only the lower bounds on the upward and downward FP streams: the value of $r_0$ of a particle could have crossed $r$ an odd number of times (and then it is counted only once) or an even number of times (and then it is not counted at all). Since we count each particle no more than once on a timestep, we totaly avoid the recrossing effect.
Second, the flows are just too strong to be just a noise. For instance, Fig. \[17fig4\] shows that, though $\Delta N_{+}(r)$ and $\Delta N_{-}(r)$ are just the lower bounds on the streams, $\Delta N_{+}(r)\simeq\Delta N_{-}(r)\simeq 2\cdot 10^4$ at $r=a$, i.e. $\sim 2 \%$ of the total halo mass crosses this radius because of this unphysical effect on each timestep. This is approximately the total number of particles in the layer of thickness $\sim a/12$ around the radius $r=a$. The value $a/12$ by far exceeds the smoothing radius or any reasonable numerical noise that may occur in the computing scheme.
The surprisingly high intensity of the Fokker-Planck diffusion is the main result of this work. Approximately $8\%$ of particles are renewed even inside $r=a$. It means that in only $10\Delta
t\simeq 300$ [mln. years]{} (for the case of the halo mass $M=10^7 M_\odot$, $10\Delta t\simeq 3\cdot
10^{9}$ [years]{}), i.e., in $~5\%$ of the simulation time, all the particles inside the sphere $r=a$ (which contains a quarter of the total mass of the system) can be substituted by a purely numerical effect. The fractions of particles inside radius $r$ that can be carried away or in by the upward and downward Fokker-Planck streams in the Power’s time $1.7\tau_r$ are $1.7\tau_r\frac{\Delta
N_{+}(r)}{N(r)\Delta t}$ and $1.7\tau_r\frac{\Delta N_{-}(r)}{N(r)\Delta t}$. Fig. \[17fig5\] shows that they always significantly exceed $1$. The cusp in our simulations was created in the initial conditions, but its shape is similar to the UDP. We can see that the unphysical FP streams are strong enough to arbitrarily change the shape of the cusp (and therefore the shape is defined by the FP diffusion rather than by the properties of the collisionless system) and even to create it.
Another argument in support of the numerical nature of the cusps in cosmological simulations is the profile universality. The similarity contradicts the observational results [@diversity], but is quite natural if the cusps are formed by the FP diffusion. An UDP-like stationary solution is innate for the FP equation: the suppositions in [@evans1997] and [@13] are quite different, but the results are similar. The properties of the solution of the FP equation are totally defined by only a few coefficients $\tilde A_\alpha$ and $B_{\alpha\beta}$ that can be similar for different N-body codes using similar algorithms, and are almost certainly the same within the same simulation. As a result, the resulting halos are also self-similar, while nature is much more variable.
The second important conclusion of this section is that the profile stability cannot be used as the simulation convergency criterion: the first unquestionable signs of the influence of the test particle interaction appear in the phase portrait much earlier than the density profile evolution and the beginning of the core formation.
The third conclusion is that, since the variations of $K$ and $r_0$ are very significant in Fig. \[17fig2\] even at $r>4a$, where the role of collisions or potential softening is minor, the integral variations there are most likely due to the potential calculating algorithm. But whatever the reason of the variations may be, the ill effect on the simulations is the same from the point of view of the kinetic equation: variations if the integrals of motion reveal the collisional influence and suggest that the system behavior is no longer described by the correct collisionless equation. A convergence study with varying the opening angle $\mu$ of the Barnes-Hut tree opening criterion, softening scale, as well as other parameters of the gravitational force computation, is essential to understanding the origin of the non-conservation of integrals of motion and find the optimal parameter set to decrease these undesirable numerical effects.
A much better understanding of the N-body simulation convergency is necessary to cast doubts on the CDM model on the basis of ’cusp vs. core’ contradiction.
Acknowledgements
================
The work is supported by the CONICYT Anillo project ACT-1122 and the Center of Excellence in Astrophysics and Associated Technologies CATA (PFB06). We used the HPC clusters Docorozco (FONDECYT1130458) and Geryon(2) (PFB06, QUIMAL130008 and Fondequip AIC-57).
[34]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ****, (), .
, , , ****, (), .
, , , ****, ().
, ****, (), .
, , , , , ****, (), .
, , , , , , , , , ****, (), .
, , , , , , , , , , , ****, (), .
, ****, (), .
, , , , , , , , ****, (), .
, , , , , , , , , ****, (), .
, , , , , , , , , , , ****, (), .
, ****, (), .
, ****, (), .
, ** (, ).
, ****, ().
, ****, (), .
, ****, (), .
, , , , , , , , ****, (), .
, , , , , ****, (), .
, , , , , (), .
, ****, ().
, , , , , , , ****, (), .
, ****, ().
, , , ****, (), .
, ****, (), .
, , , ****, (), .
, ****, (), .
, ****, ().
, ****, ().
, , , , , , , , , ****, (), .
, , , , , ****, (), .
, (), .
, ** ().
, ****, (), .
[^1]: All the data, as well as results of simulations of a Plummer sphere of mass $10^{12}M_\odot$ we used as an auxiliary test model, are publicly available at [*http://www.das.uchile.cl/anton*]{}
|
---
abstract: 'Autonomous vehicles are heavily reliant upon their sensors to perfect the perception of surrounding environments, however, with the current state of technology, the data which a vehicle uses is confined to that from its own sensors. Data sharing between vehicles and/or edge servers is limited by the available network bandwidth and the stringent real-time constraints of autonomous driving applications. To address these issues, we propose a point cloud feature based cooperative perception framework (F-Cooper) for connected autonomous vehicles to achieve a better object detection precision. Not only will feature based data be sufficient for the training process, we also use the features’ intrinsically small size to achieve real-time edge computing, without running the risk of congesting the network. Our experiment results show that by fusing features, we are able to achieve a better object detection result, around 10% improvement for detection within 20 meters and 30% for further distances, as well as achieve faster edge computing with a low communication delay, requiring 71 milliseconds in certain feature selections. To the best of our knowledge, we are the first to introduce feature-level data fusion to connected autonomous vehicles for the purpose of enhancing object detection and making real-time edge computing on inter-vehicle data feasible for autonomous vehicles.'
author:
- 'Qi Chen, Xu Ma, Sihai Tang, Jingda Guo, Qing Yang, Song Fu'
bibliography:
- 'sample-base.bib'
title: 'F-Cooper: Feature based Cooperative Perception for Autonomous Vehicle Edge Computing System Using 3D Point Clouds'
---
Introduction
============
Connected autonomous vehicles (CAV) provide a promising solution to improving road safety. This relies on vehicles being able to perceive road conditions and detect objects precisely in real-time. However, accurate and real-time perception is challenging in the field. It involves processing high-volume and continuous data streams from various sensors with strict timing requirements. Moreover, the perception accuracy of a vehicle is often affected by the limited view and scope of the sensors. Edge computing can help CAVs achieve better situational awareness via combining and processing information collected from multiple CAVs with more powerful machine learning technologies [@wang2018cavbench; @shi2016edge]. The ultimate goal of integrating edge computing and CAVs is to efficiently analyze massive amount of data in real time under limited network bandwidth.
An autonomous vehicle edge computing system consists of three layers: vehicle, edge, and cloud [@vedge]. Each autonomous vehicle is equipped with onboard edge device(s) that integrates the needed functional modules for autonomous driving, including localization, perception, path planning, and vehicle control. Autonomous vehicles can communicate with roadside edge servers, and eventually reach the cloud through wireless networks, e.g., the dedicated short range communication (DSRC) [@dsrc], 5G or millimeter-wave communication technologies [@va2016millimeter]. This provides a perfect opportunity to develop a cooperative perception system in which vehicles exchange their data with nearby edge servers. It is here that data are fused and processed to further extend the individual vehicle’s perception range; beyond line-of-sight and beyond field-of-view.
Motivation
----------
Having a single source of data input for autonomous vehicles is risky in real-world environments, as sensors are just another component of the vehicle that is susceptible to failure. In addition, sensors are also limited by their physical capabilities such as scan frequency, range, and resolution. Perception gets even worse when sensors are occluded, as shown in Fig. \[fig:acclusion\].
{width="0.95\linewidth"}
\[fig:acclusion\]
Among related works on cooperative perception for autonomous vehicles [@rauch2012car2x; @cho2014multi], we find that their main focus is on improving the individual vehicle’s precision, overlooking benefits from cooperative perception. Potential issues involved in cooperative perception, such as accuracy of local perception results, impact on networks, format of data to be exchanged, and data fusion on edge servers, are not addressed. When it comes to 3D object detection, Lidar is one of the most important components of autonomous driving vehicles as it generates 3D point clouds to capture the 3D structures of scenes. This data gives precise location in 3D space with respect to the LiDAR, and by extension, the car.
Based on our best acknowledge, the state of the art 3D object detection precision based on monocular LiDAR (Light Detection and Ranging) data comes from VoxelNet [@zhou2018voxelnet], SECOND [@yan2018second] and PointRCNN [@shi2018pointrcnn], etc. For example, PointRCNN achieves 75.76% mAP (mean average precision) on the KITTI moderate benchmark [@geiger2012we], and 85.94%, 68.32% on easy and hard benchmarks, respectively. That implies the simple fusion of object detection results from different cars would yield errors. Although fusing raw LiDAR data from two vehicles can improve the car detection precision [@qi2019cooper], it is challenging to send the huge amount of LiDAR data generated by autonomous vehicles in real time. Solutions that increase vehicle’s perception precision as well as maintaining or reducing computational complexity and turnaround time are rare in the literature.
Proposed Solution
-----------------
We propose a method that improves the autonomous vehicle’s detection precision without introducing much computational overhead. An useful insight is that modern object detection techniques for autonomous vehicles, both image based [@ren2015faster; @liu2016ssd] and 3D LiDAR data based [@yan2018second; @qi2018frustum], commonly adopt a convolutional neural network (CNN) [@simonyan2014very; @he2016deep] to process raw data, and leverage a region proposal network (RPN) [@ren2015faster] to detect objects. We argue that the capacity of feature maps is not fully explored, especially for 3D LiDAR data generated on autonomous vehicles, as the feature maps are used for object detection only by single vehicles.
To this end, we introduce a [*feature based cooperative perception (F-Cooper) framework*]{} that realizes an end-to-end 3D object detection leveraging feature-level fusion to improve detection precision. Our F-Cooper framework supports two different fusion schemes: voxel feature fusion and spatial feature fusion. While the former achieves almost the same detection precision improvement when compared to the raw-data level fusion solution [@qi2019cooper], the latter offers the ability to dynamically adjust the size of feature maps to be transmitted. A unique characteristic of F-Cooper is that it can be deployed and executed on in-vehicle and roadside edge systems.
Aside from being able to improve detection precision, data needed for feature fusion is only one hundredth of the size of the original data. For a typical LiDAR sensor, each LiDAR frame contains about 100,000 points, which is about 4 MB. Such huge amount data would become a severe burden for any existing wireless network infrastructure. In stark contrast to the large volume of raw LiDAR data, the size of features generated by a CNN can be as low as 200 Kb after compression techniques is applied. Empirical evidences from our experiments demonstrate that transmitting these features only takes dozens of milliseconds, which makes real-time edge computing feasible. Such a negligible cost also enables feature-level fusion to become an ideal choice for connected autonomous vehicles to improve detection precision while keeping a reasonable communication time.
Main Contributions
------------------
To the best of our knowledge, we are the first to propose feature map fusion based 3D object detection for connected autonomous vehicles on the edge. Through our experimentation and analysis, we have proved that not only does feature fusion provide an enhanced perception, it also allows for data to be compressed without losing detection value.
With this data compression factor, we are able to state with confidence that our feature fusion strategies are suited for autonomous vehicles On-Edge. Due to the fact that vehicles have a limited amount of computational resources on-board, we look towards the edge for more powerful and reliable computational power. Should an autonomous vehicle require extra perception, it only needs to send its compressed feature data and receive either a detection result or a compressed, fused feature map, or even both. By cutting out the computational step, we are effectively pushing the heavy workload onto the edge and mitigating any downsides to data sharing.
As proven in our experiments, both the data size and network transmission time are small enough that even in the most congested areas, vehicle data transmission will still be smooth. Both voxel feature fusion and spatial feature fusion perform better than the baseline for single vehicles without fusion, both the fusion and non-fusion baseline are derived from the same detection model. While spatial feature fusion data can be dynamically adjusted for a smaller compression size than voxel feature fusion data, the latter is capable of detection improvement on par with raw-data level fusion [@qi2019cooper]. With each strategy holding its own special advantages, we believe that our F-Cooper framework makes a substantial contribution that allows improvement no matter whether deployed in-vehicle or on roadside edge systems.
The remainder of this paper is organized as follows. Section 2 analyzes the properties of features to see if features are fit for fusion. Section 3 explains how our feature based methods work and outlines their place in F-Cooper. Section 4 tests our methods in fusion scenarios and evaluates suitability for on-edge deployment. Section 5 and 6 discuss previous works and related studies. Finally, Section 7 concludes this paper.
Towards Feature based Fusion of Vehicle Data
============================================
Convolutional Feature Maps
--------------------------
{width="0.9\linewidth"}
\[fig:cnn\]
With 3D points cloud data, the details for the location of each point are used to calculate the relationship between a car and its surrounding environment. Each frame in 3D points cloud data is processed in the same way, and one common key step in the process is to generate feature maps from points cloud data. Due to the popularity of CNN based solutions to object detection for autonomous driving vehicles [@rajaram2016refinenet; @li2019stereo; @chen2016monocular], in this paper, we focus on the feature maps generated by the convolutional layers in CNN networks.
{width="0.9\linewidth"}
As a CNN network processes raw 3D points cloud data [@ren2017object], we are able to extract the processed feature maps from the CNN. These feature maps provide the essential information for object detection. Fig. \[fig:cnn\] depicts the convolutional layers in a classical CNN. First, we send the original data as input to a convolutional layer which is composed of several filters with each filter generating a feature map. All these generated feature maps are considered as the output of the first layer and will be sent to the second convolutional layer as input data. Recursively, previous layer’s outputs are fed as input into the next layer.
**Features for Fusion**
-----------------------
Features are a well established and integrated part of any CNN, and due to the nature of CNN, it is opaque by nature. When working with feature maps, we need to ensure that coincident issues are dealt with and explored. For example, depending on the specifications of the convolutional layers in a CNN network, the resulting voxel features may be located equal-distant from other voxels, making lossless fusion impossible to achieve without additional run-time cost. To confirm the usefulness of features for fusion, we must answer the following three essential questions. (1) Do features possess the necessary means for fusion? (2) Are we able to communicate the data between autonomous vehicles effectively through features? (3) If features satisfy both of the two prior requirements, then how hard is it for us to obtain feature maps from autonomous vehicles? To analyse these questions and their implications, we provide an in-depth analysis in the sections below.
### **Fusion Characteristics**
Inspired by the works that have been dedicated to fusing feature maps generated by different layers, such as Feature Pyramid Network (FPN) [@lin2017feature] and Cascade R-CNN [@cai2018cascade], we find that it is possible to detect objects in different feature maps. For example, FPN adopts a top-down pyramid structure feature maps for detection. These networks are very adept in compounding the efficiency of feature fusion.
Taking the inspiration from these works, we make the assumption that cars compatible for fusion will use the same detection model. This is important as we see only the most reliable detection model being used for self driving. With this assumption in place, we now look at the fusion characteristics.
As feature maps are available from the CNN, we are able to ensure that all extracted feature maps are obtained with the same format and data type. Next, as feature maps extracted from 3D points cloud also contain location data, we are able to fuse the feature maps from different autonomous vehicles as long as there exists a single point of overlap in between the two vehicles. However, when we faced the issue of equal-distant location alignment, we needed to adjust our fusion algorithm to accommodate such situations. To achieve this goal, we let each car send its GPS and IMU data to allow for the transformation calculations towards point clouds fusion, i.e., transforming the view seen by a sender to the view seen by a receiver. We are clear that GPS and IMU cannot provide enough accurate details to perspective transformation. However, there are existing methods that allow for accurate alignment of two vehicles into the same 3D space. We will discuss this further in the discussion section.
### **Compression and Transmission**
Another advantage of feature maps over raw data is the transmission process between vehicles. Raw data might come in many different formats, they all achieve a single purpose, and that is to preserve the original state of the data captured. For example, LiDAR data taken from a driving session would store all the points cloud along the path of the driving session. However, this storage format records unnecessary data along with the essential data; **feature maps avoid this issue**. As the raw data is being processed by the CNN network, all the extraneous data is being filtered out by the network, leaving behind only information that is potentially capable of being used for object detection by the network. These feature maps are stored in sparse matrices, which only store the data deemed useful, with a 0 stored in the matrix for any data filtered out.
The data size advantage can be further compounded through lossless compression such as the gzip compression method as seen in [@gzip]. Adding in the nature of sparse matrix, we are able to combine the two to achieve compressed feature data that is no bigger than 1 MB, making features a great option for deploying On-Edge fusion.
### **Generic and Inherent Properties**
All autonomous driving vehicles must base their decisions on the data that their sensors generate. The raw data is generated from the physical sensors on the vehicle before getting forwarded to the onboard computing device. From there, the raw data is fed through a CNN based deep learning network to process the raw data and ultimately make the driving decisions.
During this process, we are able to pull the extracted features for sharing. By doing so, we are effectively able to obtain the feature maps of the raw data without needing extra computation time or power from the onboard computing device. With the CNN based network being used by almost all known autonomous driving vehicles to date, the feature extraction is generic and does not require further processing before fusion.
Thanks to the means by which autonomous vehicles process data, we are able to directly take the processed feature maps from the raw LiDAR points cloud data for the purpose of fusion, as this inherently provides location data. As long as the LiDAR sensor has been calibrated to the standards needed for autonomous driving, then we should have a feature map that is capable of retaining the relative locations of all things in relation to the vehicle.
F-Cooper: Feature based Cooperative Perception
==============================================
Inspired by the advantages of feature map fusion in 2D object detection and the feature maps generated by 3D object detection based on LiDAR data, we propose the Feature based Cooperative Perception (F-Cooper) framework for 3D object detection. Our F-Cooper fuses feature maps generated from two LiDAR data sources oriented in two different aspects. Fusing feature maps (rather than raw data) will not only address privacy issues, but also greatly reduce the network bandwidth requirement. In F-Cooper, we present two schemes for feature fusion: Voxel Feature Fusion (VFF) and Spatial Feature Fusion (SFF). As shown in Fig. \[fig:architecture\], the first scheme directly fuses the feature maps generated by the Voxel Feature Encoding (VFE) layer, while the second scheme fuses the output spatial feature maps generated by the Feature Learning Network (FLN) [@zhou2018voxelnet]. SFF can be viewed as an enhanced version of VFE, i.e., SFF extracts spatial features locally from voxel features available on individual vehicles before they are transmitted into the network.
**Voxel Features Fusion**
-------------------------
As with pixels in a bitmap, a voxel represents a value on a regular cube in three-dimensional space. Within a voxel, there may be zero or several points cloud generated by a LiDAR sensor. For any voxel containing at least one point, a voxel feature can be generated by the VFE layer of VoxelNet [@zhou2018voxelnet].
\[fig:voxel\_feature\_maps\]
Suppose the original LiDAR detection area is divided into a voxels grid. Of these voxels, we will obtain a vast majority of empty voxels with the remaining ones containing critical information. All non-empty voxels are transformed by a series of full connection layers and converted into a fixed-size vector, with a length of 128. The fixed-sized vector is often referred to as feature map. An example feature map derived from 3D point cloud data is shown on the right part of Fig. \[fig:3d\_feature\_maps\]. For example, as shown in Fig. \[fig:voxel\_feature\_maps\], only four voxels are non-empty amongst the twelve voxels present, and each of the four selected voxels becomes a 128-dimensional vector.
{width="0.9\linewidth"}
\[fig:voxel\_fusion\]
For memory/compute efficiency, we save the features of non-empty voxels into a hash table where the voxel coordinates are used as the hash keys. As our focus is primarily on autonomous driving, we only store non-empty voxels into our hash table. Combining the fact that our 3D point clouds are of outside driving scenarios, which yields around a few thousand voxels, searching the hash table for voxel fusion becomes an non-factor in the overall speed of our framework. In VFF, we explicitly combine the features of all voxels from two inputs, as depicted in Fig. \[fig:voxel\_fusion\]. Specifically, the Voxel $3$ from Car 1 and Voxel $5$ from Car 2 share the same calibrated location. While the two cars are located in different locations physically, they share the same calibrated 3D space, with different offsets indicating the relative physical location of each car in said 3D calibrated space. To this end, we employ the element-wise $maxout$ scheme to fuse Voxel 3 and Voxel 5.
Taking inspiration from convolutional neural networks, using maxout [@goodfellow2013maxout] for latent scale selection, we extract the obvious features while suppressing the features that does not contribute to detection in 3D space, thus achieving lower data size. In our experiments, we use the $maxout$ to decide which feature is most prominent when comparing the data in between vehicles. We denote these two voxel features as $V_3$ and $V_5$, respectively, and $V^i$ as the $i$-th element in the voxel. The fused features $V$ can be presented as follows. $$V^i=max\left ( V_{3}^{i},V_{5}^{i} \right ), \forall i=1,\cdots ,128$$
The key insight behind our $maxout$ strategy is that it emphasizes important features and removes trivial ones. Also, as $maxout$ is a simple floating-point operation, it introduces no extra parameters. Such a negligible additional computational overhead can be ignored when compared to the overall improvement on object detection precision.
Naturally, we expect voxels from two cars are able to be perfectly matched. However, this is impractical for real-world applications, even slight bias between voxels would explicitly lead to mismatches. Here, we showcase four different mismatched situations in Fig. \[fig:voxel\_fusion\]. The green dot $C_3$ indicates the center of the voxel 3 from Car 1 and the diamonds $C_{5a}, C_{5b}, C_{5c}, C_{5d}$ denote the possible centers of the voxel 5 from Car 2. In case (a), the center of Voxel 5, denoted as $C_{5a}$, falls within Voxel 3. In case (b), the center $C_{5b}$ falls on one side of the voxel 3, meaning Voxel 5 connects with two voxels from Car 1. In case (c), $C_{5c}$ falls along an edge of Voxel 3, which implies Voxel 5 intersects with four voxels from Car 1. In case (d), $C_{5d}$ falls on a corner point of Voxel 3 and connects with eight voxels. For case (a), we fuse the voxel 3 and voxel 5 directly using maxout. For cases (b,c,d), we fuse Voxel 5 with all the connected voxels from Car 1, and give the results to the connected voxels, respectively.
**Spatial Feature Fusion**
--------------------------
VFF needs to consider the features of all voxels from two cars, which involves a large amount of data exchanged between vehicles. To further reduce the network traffic, as well as keeping the benefits of feature based fusion, we design a spatial feature fusion (SFF) scheme. Compared to VFF, SFF fuses spatial feature maps, which are sparser when compared to voxel features and thus more easily compressed for communication. Fig. \[fig:architecture\] intuitively showcases the relationship between VFF and SFF. Different from VFF, we pre-process the voxel features on each vehicle to get the spatial features. Next we fuse the two source spatial features together and forward the fused spatial features to a RPN for region proposal and object detection.
{width="0.9\linewidth"}
\[fig:3d\_feature\_maps\]
As shown in Fig. \[fig:3d\_feature\_maps\], the spatial feature maps of a LiDAR frame is generated by the Feature Learning Network [@zhou2018voxelnet]. The output of the feature learning network is a sparse tensor, which has a shape of $128 \times 10 \times 400 \times 352$. In order to integrate all the voxel features, we adopt three 3D convolutional layers, and sequentially obtain smaller feature maps with more semantic information and a size of $64 \times 2 \times 400 \times 352$. However, the generated features cannot fit into the required shape of the conventional region proposal network. To this end, we must reshape the outputs to the 3D feature maps of size $128\times400\times352$ before we can forward them to RPN. For SFF, we generate a bigger detection range with size $W\times H$, where $W>W_1, H>H_1$. Next we fuse the overlapped regions while retaining the original features in the non-overlapped regions. Suppose a GPS records the real-world location of Car 1 as ($x_1,y_1$) and Car 2 as ($x_2,y_2$), then we can get the position of the left-top corner. And if $\left(x_2+H_1,y_2-\frac{W_1}{2}\right)$ belongs to Car 2’s feature maps with the left-top corner being representative of the feature maps of Car 1, then it is easy for us to determine the overlapped areas. Similar to VFF adopting the $maxout$ strategy, we also employ $maxout$ for SFF to fuse the overlapped spatial features.
{width="0.9\linewidth"}
\[fig:spatial\_fusion\]
As indicated in Fig. \[fig:spatial\_fusion\], the top left corner of Car 2’s feature maps can be presented as $\left(x_2+H_1,,y_2-\frac{W_1}{2}\right)$ if the car moves towards left. Suppose the corner point falls in the region of Car 1’s feature map, then we can fuse the overlapped features in the same manner as the voxel fusion strategy.
Finally, we adopt region proposal network to propose potential regions on the fused feature maps. Paradigm II in Fig. \[fig:architecture\] holistically showcases the pipeline of our SFF.
Recent work like SENet [@hu2018squeeze] indicates that different channels share different weights. That is to say some channels in feature maps contribute more toward classification/detection while other channels being redundant or unneeded. Inspired by this, we opt to select partial channels, out of all 128 channels, to transport. We assume that autonomous vehicles are assembled with the same well-trained detection model as in real-world applications. After extensive experimentation, we demonstrate that transporting part of channels can further reduce the time consumption of transmission while keeping the comparable detection precision in our experimental analysis in Section 4.
**Object Detection Using Fused Features**
-----------------------------------------
For detecting vehicles, we feed the synthetic feature maps to a Region Propose Network (RPN) for object proposal. Next a loss function is applied for network training.
### Region Proposed Network
As indicated in Fig. \[fig:architecture\], once we get the spatial feature maps, regardless of whether we adopt voxel fusion paradigm or spatial fusion paradigm, we send it to the region proposal networks (RPN) [@zhou2018voxelnet]. After passing through the RPN network, we will obtain two generated outputs for a loss function (Section 3.3.2): (1) a probability score $p\in\left [ 0,1 \right ]$ of the proposed region of interests, and (2) the locations of proposed regions $P=\left ( P_x,P_w,P_z,P_l,P_w,P_h,P_\theta \right )$, where $\left ( P_x,P_y,P_z\right )$ indicates the center of the proposed region and $\left ( P_l,P_w,P_h,P_\theta\right )$ means the length, width, height and rotation angle, respectively.
### Loss Function
The loss function is comprised of two parts: classification loss $L_{cls}$ and regression loss $L_{reg}$.
Suppose a 3D ground-truth bounding box can be presented as $G=\left ( G_x,G_y,G_z,G_l,G_w,G_h,G_\theta \right )$, where $\left ( G_x, G_y, G_z\right )$ represents the central point of the box, and $\left (G_l,G_w,G_h,G_\theta\right)$ denotes the length, width, height and yaw rotation angle, respectively. Our proposed method will generate a vector $P$ to represent the predicted 3D box. In order to minimize the loss between our prediction and the ground truth, we regress our predicted boxes by minimizing the differences $\left ( \Delta x,\Delta y,\Delta z,\Delta l,\Delta w,\Delta \theta \right )$ [@girshick2014rich] as: $$\begin{split}
\Delta x=\frac{G_x-P_x}{P_d},
\Delta y=\frac{G_y-P_y}{P_d},
\Delta z=\frac{G_z-P_z}{P_h}\\
\Delta l=\log \left ( \frac{G_l}{P_l} \right ),
\Delta w=\log \left ( \frac{G_w}{P_w} \right ),
\Delta h=\log \left ( \frac{G_h}{P_h} \right )\\
\Delta \theta=G_\theta-P_\theta\\
\end{split}$$ where $P_d=\left (\left ( P_l \right ) ^2+\left ( P_w \right ) ^2 \right )^{\frac{1}{2}}$ is the dialog of length and width .
Suppose our model proposes $N_{pos}$ positive anchors and $N_{neg}$ negative anchors, we define the loss function as follows: $$\begin{split}
L &= \alpha\frac{1}{N_{neg}}\sum_{i=1}^{N_{neg}}L_{cls}\left ( p_{neg}^i ,0\right )\\
&+ \beta \frac{1}{N_{pos}}\sum_{i=1}^{N_{pos}}L_{cls}\left ( p_{pos}^i ,1\right )\\
& +\frac{1}{N_{pos}}\sum_{i=1}^{N_{neg}}L_{reg}\left ( P^i ,G^i\right )
\end{split}$$ where $p_{neg}^i$ and $p_{pos}^i$ are the probability of positive anchors and negative anchors respectively, and $N_{neg}$ and $N_{pos}$ denote the number of proposed negative and positive anchors respectively. In regression loss, $G^i$ indicates the $i$th ground truth while $P^i$ means the corresponding predicted anchor. We use $\alpha$ and $\beta$ to balance these three losses. We employ a binary cross entropy loss for classification Loss and Smooth-L1 loss function [@girshick2015fast; @ren2015faster].
Performance Evaluation
======================
\[fig:lidar\_detect1\]
Datasets
--------
KITTI [@geiger2012we] is a well-known vision benchmark suite project which contains labeled data that allows for autonomous vehicles to train detection models and evaluate detection precision .
As we focus on 3D object detection, we use the 3D Velodyne point cloud data provided by the KITTI dataset. The cloud point data provides 100K points per frame and is stored in a binary float matrix. The data includes 3D location of each point and associated reflectance information. However, as KITTI data is recorded from single vehicles, we must utilize different time segments from the same recording to emulate data generated from two vehicles. As a result, KITTI data is only suitable for certain test scenarios.
To address this issue, we equip two vehicles, named Tom & Jerry (T&J), with necessary sensors, such as LiDARs (Velodyne VLP-16), cameras (Allied Vision Mako G-319C), radars (Delphi ESR 2.5), IMU&GPSes (Xsens MTi-G-710 kit), and edge computing devices (Nvidia Drive PX2) to gather desired data on the campus of our institution. Our vehicles have 16-beam Velodyn LiDAR sensors that store data in binary raw Ethernet packets. As our vehicles can move independently of each other, we are able to test the entire gamut of scenarios in a real-world environment with our two vehicles.
Both datasets provide data that allows 3D object detection. Moreover, the LiDAR data provided contains ample data for us to extract feature maps from the CNN network.
Test Scenarios
--------------
From these two datasets, we are able to fully test or simulate an array of different common scenarios such as those detailed below.
*Road intersections.* One of the most common places for cars to congregate and thus cause occlusion is a busy road intersection. As the optical based LiDAR and camera sensors are blocked by vehicles in front of them, the information becomes severely limited. Due to this fact, we included this scenario as one of our test cases.
*Multi-lane roads.* Another common place is a multi-lane road. Such roads feature the combination of high speed driving and T-junctions, both of which are prone to accidents. To ensure our F-Cooper framework is capable in such extreme situations, we also included this scenario in our experiments.
*Campus parking lots.* Last but not least, as our main objective is to enhance perception through fusion, we need to test our framework in a crowded situation with many obstacles. As congested zones are best represented by a crowded parking lot, we choose busy campus parking lots as our main test scenario to evaluate the accuracy of F-Cooper in a real-life environment.
Experiment Setup
----------------
To evaluate F-Cooper, over 200 sets of data were collected and tested in our experiments. We separate our tests into four categories, based on the methods used to process the LiDAR data, methods (1) through (3) are derived from the same detection model: (1) Non-fusion as baseline, (2) F-Cooper with VFF, (3) F-Cooper with SFF, and (4) raw point clouds fusion method - Cooper [@qi2019cooper]. Feature fusion takes place in random cases of the above four categories with a heavier focus on busy campus parking lots as it is the most difficult scenario due to significant occlusions. Within each category, we further divide our experiments by considering the distances between objects and the sensing vehicle. We treat objects that are within 20 meters away from a vehicle as high-priority objects and those beyond 20 meters as low-priority objects in the parking lot environment.
As our LiDAR sensor has only 16 beams, the resulting point cloud data is relatively sparse, compared to higher-end LiDAR sensors. To mitigate the negative impacts of sparse data, we limit the detection range to \[0,70.4\] by \[-40,40\] by \[-3,1\] meters along the X, Y, and Z axles. We do not use data beyond the detection ranges. In addition to the vehicle’s detection range, we also set the voxel size as ${v_{D}} = 0.4$ meter, ${v_{H}} = 0.2$ meter, ${ v_{W}} = 0.2$ meter, and thus $D_1 = 10$, $H_1 = 400$, and $W_1 = 352$. In our experiments, the F-Cooper framework runs on a computer with a GeForce GTX 1080 Ti GPU.
\[fig:lidar\_detect2\]
**Top-Level Evaluation of F-Cooper**
------------------------------------
To evaluate F-Cooper, we analyze each component individually as well as against other frameworks. Starting with VFF, we can see the results of fusion from two cars in Fig. \[fig:lidar\_detect1\] and Fig. \[fig:lidar\_detect2\], with data receiving vehicle (Car 1) and data transmitting vehicle (Car 2). In the figure, we have the LiDAR representation with the detection results on the top and the right-camera in the middle and the left-camera at the bottom. Both the baseline detection and the fusion detection use 0.5 as a confidence threshold, meaning if the confidence level is above this score, we mark the boundary box for that object. As we can see in column (c) of Fig. \[fig:lidar\_detect1\] and Fig. \[fig:lidar\_detect2\], we have the voxel fusion result on the top and the raw data fusion result on the bottom. Through all of our marked bounding boxes, we have distinguished them in three levels of importance to the receiving car: yellow, green and red. The cars marked in yellow represent those that have already been detected by the receiving car originally. The cars marked in green represent those detected by only the sender and not the receiver. The cars marked in red represent those undetected by neither the sender nor the receiver but detected after feature fusion.
Taking a closer look at Fig. \[fig:lidar\_detect1\], which details two cars driving forward in parallel, we can see that Car 1 was able to detect four vehicles while Car 2 was able to detect three vehicles. However, in both cases, neither Car 1 nor Car 2 was able to detect cars further away. This was caused by a combination of factors such as occlusion and distance. Through VFF, we are able to detect cars previously occluded to Car 1 or was completely undetected by either cars. Similarly, Fig. \[fig:lidar\_detect2\] depicts two cars approaching each other from opposing directions. In this figure, we can see that Car 1 detects three vehicles while Car 2 detects four. However, when SFF was conducted, we can see that spatial fusion only enhanced perception by two detections for Car 1 where as raw data fusion enhanced detection by three. A closer inspection reveals that the right most new detection from SFF was not detected in the raw data fusion. From this comparison, we can see that while VFF is similar in precision to raw data fusion, SFF is able to perform better for near cars when compared to VFF.
**Detection Precision Analysis**
--------------------------------
Having taken an overview of the precision of our two feature fusion methods, we dive into the details of how each method performs against each other as well as against the baseline, and the Cooper approaches [@qi2019cooper].
The data that we use for this analysis comes from both datasets to test all of our listed scenarios. In all our experiments, we report our results using Intersection over Union (IoU) threshold at 0.7 for vehicles. Then, we calculate the precision by comparing the detected vehicles with the ground truth.
-------------------- --------- ------- ------- ------- ------- ------- ------- ------- ------- -- -- -- -- -- -- --
Scenario Dataset
Near Far Near Far Near Far Near Far
Multi-lane roads KITTI 63.22 22.37 77.46 58.27 50.00 57.14 77.46 71.42
Road Intersections T&J 78.37 19.60 80.21 72.37 73.68 53.33 80.21 72.37
Parking Lot1 T&J 58.33 33.33 66.67 62.54 66.67 33.33 66.67 70.58
Parking Lot2 T&J 66.67 18.85 72.25 46.42 72.25 25.00 75.50 50.00
Parking Lot3 T&J N/A 45.81 N/A 66.41 N/A 66.41 N/A 66.41
Parking Lot4 T&J 100 N/A 100 48.83 100 33.33 100 48.83
-------------------- --------- ------- ------- ------- ------- ------- ------- ------- ------- -- -- -- -- -- -- --
\[table:compare\]
In Table \[table:compare\], we observe that in our baseline test, baseline without fusion on Car 1 achieve a good “Near” detection precision for the road scenarios but fall off sharply in precision in their “Far” detection. As mentioned before, the “Near” and “Far” cut off is $20$ meters from the car as the center. Next, looking at how the baseline performs in parking lot scenarios where the most occlusions take place, we can see that again, the “Near” precision is much better than the “Far”. This is understandable as the lasers reach out further, it returns a much sparser point cloud.
Moving on to our method testing, we compare the precision against both the baseline and raw fusion [@qi2019cooper]. It should be noted that we only compared against fusion methods instead of non-fusion detection methods as the former yields a meaningful comparison whereas the latter is not relatable in the same context. For our road scenarios, we see that VFF achieves a similar precision to Cooper (a raw data fusion solution). **This signifies that VFF is as capable as raw data fusion method for near object detection, but without collecting others point clouds.**
Interestingly, as we look at the SFF precision, we can see a drastic difference in between the “Near” and “Far” precision. While SFF does not outperform VFF, it was still able to perform better than the baselines in most scenarios. However, it must be noted that SFF is more sparse than both voxel features and raw data by a considerable margin. When we factor in the fact that spatial features are derived from the voxel features, it is normal to have the better precision in the regions where the data is naturally denser, i.e., “Near” the vehicle where the LiDAR point cloud data is the densest.
To distinguish the differences in how well VFF and SFFs perform in the “Near” and “Far” categories respectively, Fig. \[figure:cdf\] shows the cumulative distribution functions of increase in detection precision. Additionally, in the “Far” category, VFF was able to achieve a 40% detection precision increase for almost 85% of the time; it is also able to increase detection precision by 60% for 30% of the time. Looking at SFF, we do see an increase in detection precision, however, it is not as great of an increase as VFF shows. When it comes to the “Near” category, however, SFF was able to perform as well as VFF if not better in some cases. In Fig. \[figure:cdf\], SFF and VFF are both at 50% chance to increase detection precision by 20%. But, as we look deeper, we find that SFF outperforms VFF slightly at 30% chance to increase precision by 30%.
{width="0.33\textheight"}
\[figure:cdf\]
We conclude from this test that our detection range is able to be extended with an overall average increase in detection precision due to the extra features being harvested. As our features may target the same object multiple times, the detection confidence scores also see a notable increase. The reason why detection results become more precise after fusion is due to the points from different cars becoming fused, thus making the originally sparse data representation of a 3D object less sparse and more “outlined”. This allows for the detection model to have a higher precision. Moreover, as single cars have a limited range on their LiDAR beams, multi-car fusion allows for points in the distance to be registered by the receiving car. Through fusion, the missing points in the distance are provided by the other cars, and thus allowing for the recipient car to enhance its detection results. Our detection precision may increase even further with more vehicles sharing data, solving the issue of missing detection on some of our target cars.
\[figure:GPS\_det\]
**Sensitivity and Resilience**
------------------------------
As feature fusion relies heavily on location information for fusion, alignment has a big impact on the final detection precision of the fusion. To understand the sensitivity and resilience of F-Cooper, we will not only study missed detections, but also compare the changes of confidence level of each detected target.
In real world situations, all sensors are built within a specific acceptable error tolerance. However, these small discrepancies in between different sensors may cause the same object in 3D space to be labeled at slightly different locations by different cars. As SFF is by nature sensitive to the position of the features, we need to deal with this phenomenon in our fusion. When we integrate our GPS and IMU data, we observe yields of less than 10 cm of positional error [@imugps]. Additionally, when we explained the nature of voxel and spatial fusion in Sections 3.1 and 3.2, we noticed the discrepancies in location based data fusion. To test the resilience of our fusion methods against sensor drift, we conducted procedural artificial skewing of our GPS readings as seen in Fig \[figure:GPS\_det\].
In Fig. \[figure:GPS\_det\], we have part (a) showing the scenarios and part (b) and (c) displaying the effects of GPS drifting on VFF and SFF. First, in both VFF and SFF, we can see that there are two tables, one with a drift of 0 meters and the other with a drift of 0.1 meters to simulate drift. The target cars are then separated into “Far” and “Near” groups with respect to the location of each vehicle, “Far” is shaded dark while “Near” is not shaded.
When we focus on the missed detections, the experiment results indicate location drifting does not significantly affect the detection accuracy of SFF. On the other hand, if we look at the confidence score of each detected target, we find that VFF strategy is not too sensitive to GPS drifting. Taking all of the changes from all of our target scores of VFF, the average of increase and decrease in our confidence scores balance out, indicating that GPS drifting slightly affects VFF. Considering the same scores of SFF, we see that the average of change becomes worse, when compared to VFF, indicating that SFF is more sensitive to GPS drifting than VFF.
During our experimentation, SFF performed worse than VFF in the “Far” category. After careful investigation on our experimental setup and methodology, we concluded the following: Compared to raw point cloud fusion and voxel feature fusion, spatial feature fusion is relatively low in feature map resolution. This factor is exponentially amplified during detection for objects in “Far” category as well as for detection of small objects. In retrospect, we realize that for feature extraction on small object, we are even more susceptible to location distance. Furthermore, smaller objects may suffer from missing features after extraction. In point cloud data, when fusing from different angles and perspectives, we are at a higher risk of merging features from different aspects, therein causing a detection conflict. We believe that to overcome this issue, we need to propose a voxel feature extraction method that allows for surgical extraction of features from point clouds.
**F-Cooper On-Edge**
--------------------
Even though point clouds can be simplified to coordinate values, we still need to consider the gap between data generated by autonomous vehicles and the limited wireless networking resources, such as the limited bandwidth provided by DSRC.
Due to this limitation, we cannot simply transmit raw data for the purpose of fusion, as that would congest the network as well as consume valuable on-board computing resources. With F-Cooper, we are able to eliminate this limitation.
### Transmission
First, both VFF and SFF are fusion methods that allow for enhanced perception, with VFF achieving close to raw data fusion and SFF achieving better “Near” detection results than our baseline. Second, both of our feature fusion methods allow for a final compression size of less than 1 MB, which is well within DSRC limits.
{width="0.33\textheight"}
\[figure:channel\]
Due to the limitations of DSRC, F-Cooper restricts the frequency of data exchange between vehicles to 1$Hz$ (1 fusion per second). Given the nature of 3D detection and the situations that we envision, it is not necessary to have a continuous stream of data of more than 1$Hz$ to achieve enhanced precision. For the majority of cases, only one frame of data is needed to provide crucial supplement to the recipient vehicle. In the case of obstructed views, the feature fusion on a single frame, from different perspectives, will be enough to provide ample warning.
With VFF achieving better results, why do we still need SFF? To answer this question, we analyze the impact of different spatial feature maps on the detection results. As shown in Fig. \[figure:channel\], derived from Fig. \[fig:lidar\_detect2\], we have the indexes of channels used in SFF as well as their respective detection precision for each of the 5 vehicles in the scene. We have 0-127 channels representing full feature maps usage, while 55-99 channels representing the range of key channels contributing the most to SFF results, 95-99 channels represent a minimal set of required channels to obtain a reasonable detection result. **This finding is crucial for deploying fusion strategies on the edge.**
### Computation
Due to the small number of channels being used to detect vehicles, we are able to reduce the amount of data that needs to be encoded for compression and transmission. Looking at Fig. \[figure:size1\] and Fig. \[figure:size2\], we have the graphs depicting both the data volume and processing time for each of our fusion strategies.
{width="0.33\textheight"}
\[figure:size1\]
From Fig. \[figure:size1\] we see that the raw point cloud data is about 2 MB when taken directly from our defined LiDAR range as mentioned in the experiment setup section. Similarly, the original data volume for spatial feature is 72.1 MB and 1 to 1.3 MB for voxel feature. However, both voxel and spatial data is capable of being compressed to less than 1 MB as shown in the figure. When combining the data from Fig. \[figure:channel\] and Fig. \[figure:size1\], we can see that with a 55-99 channel SFF compressed, we achieve the highest compression results for all five cases, the average of which is 250 KB. Additionally, if we are to use 95-99 channel SFF, then the end result will achieve an even higher compression. At the same time, SFF is capable of achieving a similar precision while being capable of a far better compression. With this, we can now analyze in Fig. \[figure:size2\] for how this strategy fares in time consumption.
Firstly, it should be noted that as vehicles are communicating with each other for data transmission and computation, they are eating up valuable computational resources, so to achieve the best result when it comes to augmenting their perception based on the data from nearby vehicles, edge computing becomes the most important factor.
As shown in Fig. \[figure:size2\], the total time used for both the raw fusion and SFF strategies are both close to the 1 second mark. Here, the total time we state includes the time for both data processing/transmission and object detection. This can become quite the issues when compounding this factor with the fact that a single vehicle may need to process the same request from other vehicles at the same time, causing a waste of computational resources. However, when we cut down the total time to just the transmission time needed for the vehicle to transmit and receive the result to and from an edge node, then we have a very feasible method of reliably enhancing perception with no downsides, especially since transmitting features to an edge computing node will not compromise any privacy.
{width="0.33\textheight"}
\[figure:size2\]
Hence, our fastest strategies only requiring less than a tenth of a second to send and receive results from an edge computing device; the vehicle will only be responsible for sending the data needed for feature fusion without needing to consume computation resources on decoding, fusing and computing the results from other vehicles.
Summary of Experimental Analysis
--------------------------------
We adopted an On-Edge end to end framework, F-Cooper, and achieved a satisfactory collaborative perception towards enhancing detection. Both of our strategies, VFF and SFF, performed better than our single car detection results in almost all of our tests. In addition to better precision, our methods were also lightweight and versatile enough to be deployed in On-Edge systems without adjustments to the current infrastructure of autonomous vehicles.
We also discover that F-Cooper can be leveraged to achieve a reasonable tradeoff in a vehicular edge computing system, considering not only latency and prediction compensation but also data size and network bandwidth. In our experiments, F-Cooper helps detect more objects that are unclear in the distance. This allows for a less constrained latency range as the fusion allows for distant objects to be detected before the car in question reaches that point in space. In addition, with regards to CNN channel selection and compression, our resulting data sizes make low latency transfers a possibility. We endeavor to continue researching even more powerful methods in future works. Lastly, we are only simulating the latency on DSRC channels as that is the most immediate networking medium. However, there are also 5G and millimeter-wave vehicular communications techniques [@va2016millimeter] coming into play, allowing for much smaller latency. Latency is a massive issue, and we are not able to solve the real time challenges fully with our current methodology, but we will continue to strive in our future works.
Related Work
============
The exploration of object data fusion has prevailed for years. Usually, data fusion methods can be grouped into 3 categories: low level, feature level and high level data fusion [@shi2018leveraging]. In the era of high level fusion, several works are conducted to fuse the detection results in pursuit of improving detection precision. The work by [@rauch2012car2x] exploits a high level sensor data fusion architecture named Car2X-based perception. Their pioneering work delivers one vehicle’s consistent results for fusion with the results generated by the host vehicle. High level fusion on multi-sensors has been well investigated to facilitate the development of 3D object detection. [@cho2014multi] proposed to detect and track moving objects using fused results from multiple sensors. Recently, Crowd sourcing, which has been learned in an automated manner [@qiu2018towards], has shown competitive perception precision. Sensors from various vehicles are typically crowd-sourced, as cooperators, to provide wider spatial coverage as well as disambiguation. However, their inability to explore undetected objects and the lack of semantic information communication caused the limited success of cooperative perception system. To this end, Qi *et al.* presented Cooper [@qi2019cooper], which fuses original calibrated raw LiDAR data from multiple vehicles to improve 3D detection precision in a low-level data fusion method.
Though Data fusion has been adopted in many areas, such as object detection and object tracking [@luo2018fast], the idea of fusing data from multiple sources data On-Edge has been explored by only a few authors. An inspiring work is [@satyanarayanan2017edge], where the authors developed a shared real-time situational awareness system by aggregating crowd sourcing and edge computing together. Another related work is [@collaborative], which employ collaborative learning On-Edge computing. However, the challenges that edge computing needs to face in the specific application of object detection are not mentioned in this paper.
Our fusion strategy is different from previous feature-level data fusion methods. For example, [@lin2017feature] fuses features from different convolutional layers in one detection model, [@chen2017multi; @liang2018deep] fuse features from different sensors within one veihcle. In pursuit of better representation ability, we fuse processed LiDAR features from multiple vehicles. We argue that fusing features from different perspectives is a better solution to improve detection precision. Similar to our work, AVR [@qiu2018avr] extends vision of multiple vehicles by communicating short range stereo camera data. The method uses metadata for localization in 3D map, allowing for a much more precise calibration. Unlike aforementioned works, we present a feature level data fusion method in pursuit of lightweight On-Edge deployment for connected autonomous vehicles. Our methods are fully suited for On-Edge deployment since the amount of transmitted data is significantly reduced and it effectively takes the advantage of On-Edge computing capacity. Finally, our method is based on intermediate features, which can detect more possible objects than high-level data fusion.
Discussions
===========
While it is faster to implement high-level data fusion, there is a fundamental flaw associated with this action. As high-level fusion is fusing object detection results from individual cars, we cannot avoid the issue of what if no car senses enough information to detect a critical object. An example would be if car A and car B both detect half of an object, but neither can detect the whole object due to missing half of the point cloud data. Because neither detected the object, the high-level fusion result will exclude the object. Another issue involved in data fusion is perspective transformation, in which a receiver needs to estimate its position relative to a sender, so the sender’s data can be mapped into the receiver’s local coordinate system. Existing solutions, e.g., AVR (augmented vehicular reality) [@qiu2018avr], have been proposed for precise fusion. AVR, with an offline sparse 3D map as the benchmark, can provide an accurate relative localization among vehicles, and thus increase data fusion precision.
Although it is outside the scope of the paper, the fusion of information from different vehicles at the edge opens the door to security vulnerabilities. A prime example can be a malicious vehicle sending phantom vehicle information. This might benefit the malicious vehicle by making space for itself through sending fake information. However, to the general public, this poses a serious driving hazard as they could potentially incur an accident from trying to avoid the phantom vehicle. In addition, we must acknowledge that a vehicle might be unintentionally malicious due to the potential of faulty sensors. This poses the question of how does a vehicle trust the information provided by another vehicle.
Towards these two issues, we assume that all sources are valid and trustworthy for experimentation purposes; however, these issues must be addressed. One possible approach is to have the edge perform the fusion and check the past history of how trustworthy of individual vehicles, and to have the edge perform authentication of newly registered vehicles.
Conclusions
===========
In this paper, we proposed F-Cooper, which provides both a new framework for applications On-Edge servicing autonomous vehicles as well as new strategies for 3D fusion detection. Through experiment testing and analysis, we conclude that not only does F-Cooper perform at the same level as Cooper, but it also has the added benefits of being more lightweight and computationally inexpensive. Both voxel features and spatial features have their separate advantages and special uses. Compounded with their great fusion detection enhancing capabilities, both strategies are well suited for autonomous vehicles On-Edge.
Voxel feature fusion out performs spatial feature fusion, but likewise, spatial feature fusion can be adjusted to be more suited for compression and data transfer. As both methods achieve a high detection perception enhancement over the baseline, both are viable for fusion. When we consider the size difference between raw data generated by each autonomous vehicle and only features from the 3D LiDAR data, it becomes clear that the latter is much more suited towards networks with a limited bandwidth.
When we apply F-Cooper to real-world scenarios, our experimental results on both the data volume and transmission time fall well within acceptable range for On-Edge computation and communication. Thus, from our evaluation, we believe that our proposed F-Cooper framework will add improvement to connected autonomous vehicle system, no matter where or how it is deployed for either in-vehicle or on roadside edge computing.
Acknowledgment {#acknowledgment .unnumbered}
==============
The work is supported by National Science Foundation (NSF) grants NSF CNS-1761641 and CNS-1852134. We thank the anonymous reviewers for their many suggestions for improving this paper. In particular we thank our shepherd Ramesh Govindan at the University of Southern California, who have read the previous versions of this paper and provided valuable feedback on our work.
|
---
abstract: |
We show that the continua $\II_u$ and ${\HH^*}$ are non-chainable and have span nonzero. Under ${\mathsf{CH}}$ this can be strengthened to surjective symmetric span nonzero.
We discuss the logical consequences of this.
address: |
Faculty EEMCS\
TU Delft\
Postbus 5031\
2600 GA Delft\
the Netherlands
author:
- 'K. P. Hart'
- 'B. J. van der Steeg'
title: 'Span, chainability and the continua ${\HH^*}$ and $\II_u$'
---
Time-stamp: <\#1 \#2>[\#2]{}
Introduction {#sect-intro}
============
Chainable (or arc-like) continua are ‘long and thin’; in an attempt to capture this idea in metric terms Lelek introduced, in [@L1], the notion of span. Chainable continua have span zero, which is useful in proving that certain continua are not chainable. The converse, a conjecture by Lelek in [@L2], is one of the main open problems in continuum theory today. While the particular value of the span of a continuum depends on the metric chosen, the distinction between span zero and span nonzero is a topological one. As chainability is a topological notion as well, Lelek’s theorem and conjecture are meaningful in the class of all Hausdorff continua. We investigate the chainability and span of several continua that are closely connected to the Čech-Stone compactification of the real line.
Preliminaries {#sect-prelim}
=============
Various kinds of span
---------------------
The kinds of span that we consider in this paper are, in the metric case, defined as suprema of distances between the diagonal of the continuum and certain subcontinua of the square. The following families of subcontinua feature in these definitions:
$S(X)$
: the symmetric subcontinua of $X^2$, i.e., those that satisfy $Z=Z^{-1}$;
$\Sigma(X)$
: the subcontinua of $X^2$ that satisfy $\pi_1[Z]=\pi_2[Z]$; and
$\Sigma_0(X)$
: the subcontinua of $X^2$ that satisfy $\pi_2[Z]\subseteq\pi_1[Z]$.
Here, $\pi_1$ and $\pi_2$ are the projections onto the first and second coordinates respectively. It is clear that $S(X)\subseteq\Sigma(X)\subseteq\Sigma_0(X)$ and hence that $s(X)\le\sigma(X)\le\sigma_0(X)$, where
1. $s(X)=\sup\bigl\{d(\Delta(X),Z):Z\in S(X)\bigr\}$;
2. $\sigma(X)=\sup\bigl\{d(\Delta(X),Z):Z\in \Sigma(X)\bigr\}$; and
3. $\sigma_0(X)=\sup\bigl\{d(\Delta(X),Z):Z\in \Sigma_0(X)\bigr\}$.
These numbers are, respectively, the *symmetric span*, the *span* and the *semi-span* of $X$.
If one uses, in each definition, only the continua $Z$ with $\pi_1[Z]=X$ then one gets the *surjective symmetric span*, $s^*(X)$, the *surjective span*, $\sigma^*(X)$, and the *surjective semi-span*, $\sigma_0^*(X)$, of $X$ respectively. The following diagram shows the obvious relationships between the six kinds of span. $$\label{eq:1}
\begin{CD}
s(X) @>>> \sigma(X) @>>> \sigma_0(X) \\
@AAA @AAA @AAA \\
s^*(X) @>>> \sigma^*(X) @>>> \sigma_0^*(X) \\
\end{CD}$$ Topologically we can only distinguish between a span being zero or nonzero. A span is zero if and only if every continuum from its defining family intersects the diagonal. This defines span zero (or span nonzero) for the six possible types of span in general continua.
Below we will show that for the continua ${\HH^*}$ and $\II_u$ all six kinds of span are nonzero. Diagram (\[eq:1\]) shows that it will be most difficult to show that $s^*$ is nonzero (or dually that it would be hardest to show that $\sigma_0$ is zero). Indeed, we will give successively more difficult proofs that the various spans are nonzero, where we traverse the diagram from top right to bottom left.
The need for these different proofs lies in their set-theoretic assumptions. We need nothing beyond ${\mathsf{ZFC}}$ to show that $\sigma^*({\HH^*})$ and $\sigma(\II_u)$ are nonzero; to show that the other spans (in particular $s^*$) are nonzero we shall need the Continuum Hypothesis (${\mathsf{CH}}$).
Chainability
------------
A continuum is *chainable* if every open cover of it has an open refinement that is a chain cover, where ${\mathcal{C}}=\{C_1,\ldots,C_m\}$ *chain cover* if $C_i\cap C_j$ is nonempty if and only if $|i-j|\le 1$.
One readily shows that every chainable continuum has span zero, whatever kind of span one uses. This follows from the fact that chainability is a hereditary property of continua and from the following theorem whose proof we give for completeness sake.
\[thm:chainable.spanzero\] Every chainable continuum has surjective semi-span zero.
Let $X$ be a chainable continuum and let $Z$ be a subcontinuum of $X^2$ that is disjoint from $\Delta(X)$. Let $\mathcal{U}$ be a finite open cover of $X$ such that $U^2\cap Z=\emptyset$ for all $U\in\mathcal{U}$. Next let $\{V_1,V_2,\ldots,V_n\}$ be an open chain cover that refines $\mathcal{U}$. Define open sets $O_1$ and $O_2$ in $X^2$ by $$O_1=\bigcup\{V_i\times V_j:i<j\},\quad O_2=\bigcup\{V_i\times
V_j:i>j\}.$$ Then $Z\subset O_1\cup O_2$ and $Z\cap O_1\cap O_2=\emptyset$. As $Z$ is connected, it is contained in one of $O_1$ or $O_2$, say $Z\subseteq O_2$. Then $\pi_1[Z]\subseteq \bigcup_{i<n}V_i$ and $\pi_2[Z]\subseteq \bigcup_{i>1}V_i$. This means that neither $\pi_1[Z]$ nor $\pi_2[Z]$ is equal to $X$.
The continua $\II_u$ and ${\HH^*}$
----------------------------------
In this paper we will be investigating the different kinds of span and the chainability of the continua $\II_u$ and ${\HH^*}$. These two spaces are related to one another. Following [@Mio] and [@Ha], we will use the space $\MM=\omega\times\II$ in our investigation of the spaces $\II_u$ and ${\HH^*}$, where $\II$ denotes the unit interval $[0,1]$.
The map $\pi:\MM\rightarrow\omega$ given by $\pi(n,x)=n$ is perfect and monotone, as is its Čech-Stone extension $\beta\pi$. The preimage of an ultrafilter $u\in\beta\omega^*$ is a continuum and denoted by $\II_u$.
Given any sequence $\langle x_n\rangle_{n\in \omega}$ in $\II$ and any $u\in\omega^*$ there is a unique point, denoted $x_u$, in $\II_u$ such that for every $\beta\MM$-neighborhood $O$ of $x_u$, the set $\{n\in\omega:(n,x_n)\in O\}$ is an element of $u$, i.e., $x_u$ is the $u$-limit of the sequence $\langle (n,x_n)\rangle_{n\in \omega}$. These points form a dense set $\CC_u$ of cut points of $\II_u$, for details see [@Ha]. The set $\CC_u$ is in fact the ultrapower of $\II$ by the ultrafilter $u$, i.e., the set ${{}^\omega \II}$ modulo the equivalence relation $x\sim_uy$ defined by $\{n:x_n=y_n\}\in u$.
The continuum $\II_u$ is irreducible between the points $0_u$ and $1_u$ (defined in the obvious way) and as it has a natural pre-order $\le_u$ defined by $x\le_u y$ iff every subcontinuum of $\II_u$ that contains $0_u$ and $y$ also contains $x$. The equivalence classes under the equivalence relation “$x\le_uy$ and $y\le_ux$” are called layers and the set of layers is linearly ordered by $\le_u$. The points of $\CC_u$ provide one-point layers, the restriction of $\le_u$ to this set coincides with the ultrapower order defined by $\{n:x_n\le y_n\}\in u$. We shall freely use interval notation, allowing non-trivial layers as end points.
If $\langle x_n\rangle_{n\in\omega}$ is a strictly increasing sequence in $\II_u$ then its supremum $L$ is a non-trivial layer. Because $\beta\MM\setminus\MM$ is an $F$-space the closure of $\{x_n:n\in\omega\}$ is homeomorphic to $\beta\omega$; by upper semicontinuity the remainder (which is a copy of $\omega^*$) must be contained in $L$. We call such a layer a countable-cofinality layer.
The continuum ${\HH^*}$ is the remainder of the Čech-Stone compactification $\beta\HH$, where $\HH$ is the half line $[0,\infty)$. Let $q:\MM\rightarrow\HH$ be given by $q(n,x)=n+x$, then $q$ is a perfect map and its Čech-Stone extension $\beta
q:\beta\MM\rightarrow\beta\HH$ maps $\MM^*$ onto ${\HH^*}$. Again for properties of ${\HH^*}$ and its relation to $\II_u$ see [@Ha].
The span of ${\HH^*}$
=====================
In this section we show that the surjective (semi-)span of ${\HH^*}$ is nonzero. The following theorem more than establishes this.
\[thm:hstar-not-fpp\] There exists a fixed-point free autohomeomorphism of ${\HH^*}$.
Let $f:\HH\to\HH$ be the map defined by $f:x\mapsto x+1$. It is clear that $\beta f$ maps ${\HH^*}$ onto ${\HH^*}$. The restriction $f^*=\beta f\restr{\HH^*}$ is a fixed-point free autohomeomorphism of ${\HH^*}$.
To see that $f^*$ is an autohomeomorphism consider $g:\HH\to\HH$ defined by $g(x)=\max\{0,x-1\}$. From the fact that $f\bigl(g(x)\bigr)=x$ and $g\bigl(f(x)\bigr)=x$ for $x\ge1$ it follows that $f^*\circ g^*$ and $g^*\circ f^*$ are the identity on ${\HH^*}$.
That $f$ is fixed-point free on ${\HH^*}$ follows by considering the following closed cover $\{F_0,F_1,F_2,F_3\}$ of $\HH$, defined by $F_i=\bigcup_n[2n+\frac i2,2n+\frac{i+1}2]$. Observe that $f^*[F_i^*]=F_{i+2\bmod4}^*$ and that $F_i^*\cap F_{i+2\bmod4}^*$ is always empty, so that $f^*(x)\neq x$ for $x\in{\HH^*}$.
\[cor:sigmastar.Hstar.nonzero\] $\sigma^*({\HH^*})$ is nonzero.
The graph of $f^*$ is a continuum in ${\HH^*}\times{\HH^*}$ that is disjoint from the diagonal and whose projection on each of the axes is ${\HH^*}$.
Later we shall see that under ${\mathsf{CH}}$ even $s^*({\HH^*})$ is nonzero.
By Theorem \[thm:chainable.spanzero\] we also know that ${\HH^*}$ is not chainable. The reader may enjoy showing that the four open sets $U_0$, $U_1$, $U_2$ and $U_3$ defined by $$U_i=\bigcup_{n<\omega}(8n+2i, 8n+2i+3)$$ induce an open cover of ${\HH^*}$ without a chain refinement.
More fixed-point free homeomorphisms
------------------------------------
We use the description of indecomposable subcontinua from [@DH1] to show that many subcontinua of ${\HH^*}$ have fixed-point free autohomeomorphisms.
We use the shift-map $\sigma:\omega\to\omega$, defined by $\sigma(n)=n+1$, and its extension to $\beta\omega$. We note that $\sigma$ is an autohomeomorphism of $\omega^*$. We also write $u+1$ for $\sigma(u)$ and $u-1$ for $\sigma^{-1}(u)$.
For $F\subseteq\omega^*$ we put $\MM_F=\bigcup_{u\in F}\II_u$ and $C_F=\beta q[\MM_F]$. We say that $F$ is $\sigma$-invariant if $u+1,u-1\in F$ whenever $u\in F$. Clearly then, if $F$ is $\sigma$-invariant then $f^*\restr C_F$ is an autohomeomorphism of $C_F$, where $f^*$ is the autohomeomorphism of ${\HH^*}$ defined in the proof of Theorem \[thm:hstar-not-fpp\].
From [@DH1] we quote the following: $C_F$ is a subcontinuum whenever $F$ is closed, $\sigma$-invariant and not the union of two disjoint proper closed $\sigma$-invariant subsets. In that case $C_F$ is indecomposable if and only if $F$ is dense-in-itself.
From [@DH1] we also quote: if $K$ is an indecomposable subcontinuum of ${\HH^*}$ then there is a strictly increasing sequence $\langle a_n\rangle_n$ in $\HH$ that diverges to $\infty$ and such that $K=q_a[C_F]$ for some closed dense-it-itself $\sigma$-invariant subset $F$ of $\omega^*$ that is not the union of two disjoint proper closed $\sigma$-invariant subsets and where $q_a:{\HH^*}\to{\HH^*}$ is induced by the piecewise linear self-map of $\HH$ that sends $n$ to $a_n$.
We can combine all this into the following theorem.
\[thm:all.indec.not.fpp\] Every indecomposable subcontinuum of ${\HH^*}$ has a fixed-point free autohomeomorphism (and hence surjective span nonzero).
The span of $\II_u$
===================
In this section we show that $\II_u$ has span nonzero for any ultrafilter $u$; the next section will be devoted to the surjective versions of span.
The following theorem, akin to Theorem \[thm:hstar-not-fpp\] and with a similar proof, provides a continuum witnessing that $\II_u$ has nonzero span.
\[thm:layer-not-fpp\] Every countable-cofinality layer has a fixed-point free autohomeomorphism.
This follows from Theorem \[thm:all.indec.not.fpp\] but for later use we give a direct construction, which establishes a bit more, namely that the interval $[0_u,L]$ has a fixed-point free continuous self-map.
We prove the theorem for one particular layer but the argument is easily adapted to the general case.
For $m\in\omega$ put $x_m=1-2^{-m}$; then $\{x_m\}_{m<\omega}$ is a strictly increasing sequence in $\II$ that converges to $1$ and with $x_0=0$. Let $x_{m,u}$ denote the point of $\II_u$ that corresponds to the constant sequence $\{x_m\}_{n\in\omega}$ in $\II$. Then $\{x_{m,u}\}_{m\in\omega}$ is a strictly increasing sequence in $\II_u$; let $L$ denote the limit of this sequence, a non-trivial layer of $\II_u$.
We define a map $f:\II_u\rightarrow\II_u$ by defining it on $\MM$, taking its Čech-Stone extension and restricting that to $\II_u$.
1. Let $f\restr\II_0$ be equal to the identity.
2. For all $n\ge 1$ let $f\restr\II_n$ be the piecewise linear map that maps $(n,x_m)$ to $(n,x_{m+1})$ for all $m<n$ and the point $(n,1)$ to itself.
The Čech-Stone extension of the map $f$ maps $[0_u,L]$ homeomorphically onto $[x_{1,u},L]$. It is not hard to see that $\beta f$ maps the interval $[x_{m,u},x_{m+1,u}]$ of $\II_u$ homeomorphically onto $[x_{m+1,u},x_{m+2,u}]$ for all $m\in\omega$. This implies that $\beta f$ maps $[0_u,L)$ homeomorphically onto $[x_{1,u},L)$. The fact that $[0_u,L]=\beta[0_u,L)$ now establishes the claim.
We let $h$ denote the restriction of $\beta f$ to $[0_u,L]$. The fact that $[0_u,L]=\beta[0_u,L)$ also establishes the following claim.
The restriction $h\restr L$ maps $L$ homeomorphically onto $L$.
To see that $h$ has no fixed points we argue as in the proof of Theorem \[thm:hstar-not-fpp\].
For every $m$ let $a_m$ be the mid point of the interval $(x_m,x_{m+1})$. Note that the map $f$ maps $(n,a_m)$ onto the point $(n,a_{m+1})$ whenever $m<n$. Define the following closed subsets $F_i$ for $i=0$, $1$, $2$ and $3$: $$\tabskip0pt plus\displaywidth
\halign to\displaywidth{$#$\tabskip0pt&${}#{}$&$#$\hfil&#&
\hfil$#$&${}#{}$&$#$\tabskip0pt plus\displaywidth\cr
F_0&=&\bigcup_n\bigl(\{n\}\times\bigcup_{m<n}[x_{2m},a_{2m}]\bigr),&\qquad &
F_2&=&\bigcup_n\bigl(\{n\}\times\bigcup_{m<n}[x_{2m+1},a_{2m+1}]\bigr),\cr
\noalign{\vskip 2pt}
F_1&=&\bigcup_n\bigl(\{n\}\times\bigcup_{m<n}[a_{2m},x_{2m+1}]\bigr),&\qquad &
F_3&=&\bigcup_n\bigl(\{n\}\times\bigcup_{m<n}[a_{2m+1},x_{2m+2}]\bigr).\cr}$$ Note that the closure in $\beta\MM$ of the union of the $F_i$’s contains the interval $[0_u,L]$ of $\II_u$. Also note that the closed set $F_i$ is mapped onto the closed set $F_{i+2\bmod4}$, so $f[F_i]\cap F_i=\emptyset$. As in the proof of Theorem \[thm:hstar-not-fpp\] this implies that $h$ has no fixed points.
As before we get the following corollaries.
The surjective span of $L$ is nonzero, hence $\sigma(\II_u)$ is nonzero.
The surjective semi-span of $[0_u,L]$ is nonzero.
It will be more difficult to prove the same for $\II_u$.
The surjective spans of $\II_u$ and ${\HH^*}$ {#sec:surj-spans}
=============================================
Using the map from the previous section and the retraction we get from the next theorem we will show that under ${\mathsf{CH}}$ there exists a fixed-point free continuous self map of $\II_u$; as the map is not onto this only implies that the surjective semi-span of $\II_u$ is nonzero. However, the special structure of $\II_u$ will allow us to build, using the graph of this map, a symmetric subcontinuum of $\II_u^2$ that will witness $s^*(\II_u)\neq0$; it will then also be possible to show that $s^*({\HH^*})$ is nonzero.
We retain the notation from the previous section but we write $a_m=x_{m,u}$ for ease of notation and we recall that layer $L$ is the supremum, in $\II_u$, of the set $\{a_m:m\in\omega\}$. The following theorem is what makes the rest of this section work.
\[retraction\] $L$ is a retract of $[L,1_u]$.
Before we prove the theorem we give the promised consequences.
The continuum $\II_u$ does not have the fixed-point property.
Let $h:[0_u,L]\to[0_u,L]$ be the map constructed in the proof of Theorem \[thm:layer-not-fpp\] and let $r:[L,1_u]\to L$ be the retraction from Theorem \[retraction\]. Extending $r$ by the identity on $[0_u,L]$ yields a retraction $r^*$ from $\II_u$ onto $[0_u,L]$. The composition $h\circ r^*$ is then a fixed-point free continuous self-map of $\II_u$.
The surjective semi-span of $\II_u$ is nonzero.
The graph of $h\circ r^*$ is a witness.
We now show how to make $s^*(\II_u)$ nonzero.
\[thm:sssI-u.neq.0\] The surjective symmetric span of $\II_u$ is nonzero.
Let $G$ be the graph of $h\circ r^*$. We complete $G$ to symmetric continuum by adding the following continua: $\{1_u\}\times[0_u,L]$, $[h(0_u),1_u]\times\{0_u\}$, $G^{-1}$, $[0_u,L]\times\{1_u\}$, and $\{0_u\}\times[h(0_u),1_u]$. It is straightforward to check that the union $Z$ is a continuum (each continuum meets its successor) that is symmetric and projects onto each axis. As none of the pieces intersects the diagonal we get a witness to $s^*(\II_u)$ being nonzero.
The surjective symmetric span of ${\HH^*}$ is nonzero.
We begin by taking the graph $F$ of the map $f$ from Theorem \[thm:hstar-not-fpp\] and its inverse $F^{-1}$; unfortunately the union $F\cup F^{-1}$ is not connected, as $F$ and $F^{-1}$ are disjoint. To connect them we take one ultrafilter $u$ on $\omega$ and observe that the image $q[\II_u]$ connects the ultrafilters $u$ and $u+1$. The image $K=(q\times q)[Z]$, where $Z$ is from the proof of Corollary \[thm:sssI-u.neq.0\] meets both $F$ (in $(u,u+1)$) and $F^{-1}$ (in $(u+1,u)$). The union $F\cup K\cup F^{-1}$ is a witness to $s^*({\HH^*})\neq 0$.
Proof of Theorem \[retraction\]
-------------------------------
We will construct the retraction by algebraic, rather than topological, means. Let ${\mathcal{R}}$ be the family of finite unions of closed intervals of $\II$ with rational endpoints. For every $f\in{{}^\omega {\mathcal{R}}}$ we define the closed subset $A_f$ of $\MM$ by $$A_f=\bigcup_{n<\omega}\{n\}\times f(n).$$ These sets form a lattice base for the closed sets of $\MM$, i.e., it is a base for the closed sets and closed under finite unions and intersections. It is an elementary exercise to show that disjoint closed sets in $\MM$ can be separated by disjoint closed sets of the form $A_f$. This implies that the closures ${\operatorname{cl}}{A_f}$ form a lattice base for the closed sets of $\beta\MM$. It follows that ${\mathcal{B}}=\{{\operatorname{cl}}{A_f}\cap L:f\in{{}^\omega {\mathcal{R}}}\}$ is a base for the closed sets of $L$ and similarly that ${\mathcal{C}}=\{{\operatorname{cl}}{A_f}\cap [L,1_u]:f\in{{}^\omega {\mathcal{R}}}\}$ is a base for $[L,1_u]$.
Theorem 1.2 from [@DH] tells us that in order to construct a retraction from $[L,1_u]$ onto $L$ it suffices to construct a map $\varphi:{\mathcal{B}}\to{\mathcal{C}}$ that satisfies
1. $\varphi(\emptyset)=\emptyset$, and if $F\neq\emptyset$ then $\varphi(F)\neq\emptyset$;
2. if $F\cup G=L$ then $\varphi(F)\cup\varphi(G)=[L,1_u]$;
3. if $F_1\cap\cdots\cap F_n=\emptyset$ then $\varphi(F_1)\cap\cdots\cap\varphi(F_n)=\emptyset$; and
4. $\varphi(F)\cap L= F$.
The retraction $r:[L,1_u]\to L$ is then defined by $r(x)={}$‘the unique point in $\bigcap\{F:x\in\varphi(F)\}$’. The first three conditions ensure that $r$ is well-defined, continuous and onto; the last condition ensures that $r\restr L$ is the identity.
There is a decreasing $\omega_1$-sequence $\langle b_\alpha\rangle_{\alpha<\omega_1}$ of cut points in $\II_u$ such that $L=\bigcap_{m,\alpha}[a_m,b_\alpha]$: by [@Ha]\*[Lemma 10.1]{} such a sequence must have uncountable cofinality and by ${\mathsf{CH}}$ the only possible (minimal) length then is $\omega_1$. For each $\alpha$ choose a sequence $\langle b_{\alpha,n}\rangle_{n\in\omega}$ in $\II$ such that $b_\alpha=b_{\alpha,u}$.
Again by ${\mathsf{CH}}$ we list ${{}^\omega {\mathcal{R}}}$ in an $\omega_1$-sequence $\langle f_\alpha\rangle_{\alpha<\omega_1}$. We will assign to each $f_\alpha$ a $g_\alpha\in{{}^\omega {\mathcal{R}}}$ in such a way that ${\operatorname{cl}}{A_{f_\alpha}}\cap L \mapsto
{\operatorname{cl}}{A_{g_\alpha}}\cap[L,1_u]$ defines the desired map $\varphi$.
The assignment will be constructed in a recursion of length $\omega_1$, where at stage $\alpha$ we assume the conditions (1)–(4) are satisfied for the $A_{f_\beta}$ and $A_{g_\beta}$ with $\beta<\alpha$ and choose $g_\alpha$ in such a way that they remain satisfied for $\beta\le\alpha$. At every stage we will list $\alpha$ in an $\omega$-sequence; this means that it suffices to consider the case $\alpha=\omega$ only.
We need a few lemmas that translate intersection properties in ${\mathcal{B}}$ and ${\mathcal{C}}$ to ${\mathcal{R}}$.
\[lemma:AfcapL=0\] ${\operatorname{cl}}{A_f}\cap L=\emptyset$ if and only if there are $m$ and $\alpha$ such that the set $\bigl\{n:f(n)\cap[a_{m,n},b_{\alpha,n}]=\emptyset\bigr\}$ belongs to $u$.
By compactness ${\operatorname{cl}}{A_f}\cap L=\emptyset$ if and only if there are $m$ and $\alpha$ such that ${\operatorname{cl}}{A_f}\cap [a_m,b_\alpha]=\emptyset$ and the latter is equivalent to $\{n:f(n)\cap[a_{m,n},b_{\alpha,n}]=\emptyset\}\in u$ again by compactness and the formula $${\operatorname{cl}}{A_f}\cap [a_m,b_\alpha]=
\bigcap_{U\in u}
{\operatorname{cl}}\Bigl(
\bigcup_{n\in U}\{n\}\times\bigl(f(n)\cap[a_{m,n},b_{\alpha,n}]\bigl)
\Bigr).
\qedhere$$
\[lemma:Af=Ag\] ${\operatorname{cl}}A_f\cap L={\operatorname{cl}}A_g\cap L$ if and only if there are $m$ and $\alpha$ such that the set $\bigl\{n:f(n)\cap[a_{m,n},b_{\alpha,n}]=g(n)\cap[a_{m,n},b_{\alpha,n}]\bigr\}$ belongs to $u$.
The ‘if’ part is clear. For the ‘only if’ part let $D$ be the set of all mid points of all maximal intervals in $A_f\setminus A_g$; then ${\operatorname{cl}}D\subseteq{\operatorname{cl}}A_f\setminus{\operatorname{cl}}A_g$ and so ${\operatorname{cl}}D\cap L=\emptyset$. Observe that $D=A_h$ for some $h$, so there are $m$ and $\alpha$ as in Lemma \[lemma:AfcapL=0\] for $D$. By convexity, for each $n$ the interval $[a_{m,n},b_{\alpha,n}]$ meets at most two of the maximal intervals in $f(n)\setminus g(n)$ — one, $I_n$, at the top and and one, $J_n$, at the bottom. The two sequences $\langle i_n\rangle_{n\in\omega}$ (bottom points of the $I_n$) and $\langle j_n\rangle_{n\in\omega}$ (top points of the $J_n$) determine cut points $i_u$ and $j_u$ of $\II_u$, which cannot belong to $L$. Therefore we can enlarge $m$ and $\alpha$ such that $\bigl\{n:i_n,j_n\notin[a_{m,n},b_{\alpha,n}]\bigr\}$ is in $u$. A convexity argument will now establish that $\bigl\{n:\bigl(f(n)\setminus g(n)\bigr)\cap[a_{m,n},b_{\alpha,n}]
=\emptyset\bigr\}$ belongs to $u$. The same argument, interchanging $f$ and $g$ will yield our final $m$ and $\alpha$.
\[A-h-cover-L\] $L\subset{\operatorname{cl}}{A_f}$ if and only if there are $m$ and $\alpha<\omega_1$ such that the set $\bigl\{n:[a_{m,n},b_{\alpha,n}]\subseteq f(n)\bigr\}$ belongs to $u$.
Apply Lemma \[lemma:Af=Ag\] to $f$ and the constant function $n\mapsto\II$.
Now we are ready to perform the construction of $g_\omega$, given subsets $\{f_k\}_{k\le\omega}$ and $\{g_k\}_{k<\omega}$ of ${{}^\omega {\mathcal{R}}}$ such that the map ${\operatorname{cl}}{A_{f_k}}\cap L \mapsto {\operatorname{cl}}{A_{g_k}}\cap[L,1_u]$ ($k<\omega$) satisfies the conditions (1)–(4) from our list.
The conditions that need to be met are
- $L\cap{\operatorname{cl}}A_{f_\omega}=L\cap{\operatorname{cl}}A_{g_\omega}$;
- if $L\subseteq {\operatorname{cl}}A_{f_k}\cup {\operatorname{cl}}A_{f_\omega}$ then $[L,1_u]\subseteq {\operatorname{cl}}A_{g_k}\cup {\operatorname{cl}}A_{g_\omega}$; and
- if $F\subseteq\omega$ is finite and $L\cap{\operatorname{cl}}A_{f_\omega}\cap\bigcap_{l\in F}{\operatorname{cl}}A_{f_l}=\emptyset$ then $[L,1_u]\cap{\operatorname{cl}}A_{g_\omega}\cap\bigcap_{l\in F}{\operatorname{cl}}A_{g_l}=\emptyset$.
The first condition takes care of (1) and (4) in our list, except possibly when ${\operatorname{cl}}A_{f_\omega}\cap L=\emptyset$ but in that case it suffices to let $g_\omega$ be the constant function $n\mapsto\emptyset$. The second and third condition ensure (2) and (3) respectively. There is one more condition that we need to keep the recursion alive; it is needed to take care of combinations of (b) and (c): if $L\subseteq {\operatorname{cl}}A_{f_k}\cup {\operatorname{cl}}A_{f_\omega}$ and $L\cap{\operatorname{cl}}A_{f_\omega}\cap\bigcap_{l\in F}{\operatorname{cl}}A_{f_l}=\emptyset$ then we must have room to be able to ensure that both $[L,1_u]\subseteq {\operatorname{cl}}A_{g_k}\cup {\operatorname{cl}}A_{g_\omega}$ and $[L,1_u]\cap{\operatorname{cl}}A_{g_\omega}\cap\bigcap_{l\in F}{\operatorname{cl}}A_{g_l}=\emptyset$. Note that the antecedent implies that, in the subspace $L$, the intersection $L\cap\bigcap_{l\in F}{\operatorname{cl}}A_{f_l}$ is contained in the interior of $L\cap{\operatorname{cl}}A_{f_k}$. A moment’s reflection shows that we need
- if $L\cap\bigcap_{l\in F}{\operatorname{cl}}A_{f_l}$ is contained in ${\operatorname{int}}_L L\cap{\operatorname{cl}}A_{f_k}$ then $[L,1_u]\cap\bigcap_{l\in F}{\operatorname{cl}}A_{g_l}$ is contained in ${\operatorname{int}}_{[L,1_u]} [L,1_u]\cap{\operatorname{cl}}A_{f_k}$.
For every $k$ as in (b) choose $m_k$ and $\alpha_k$ as per Lemma \[A-h-cover-L\] such that $U_k=
\bigl\{n:[a_{m_k,n},b_{\alpha_k,n}]\subseteq f_k(n)\cup f_\omega(n)\bigr\}$ belongs to $u$. Likewise, for every $F$ as in (c) choose $m_F$ and $\alpha_F$ as per Lemma \[lemma:AfcapL=0\] such that $U_F=\bigl\{n:[a_{m_F,n},b_{\alpha_F,n}]\cap f_\omega(n)\cap
\bigcap_{l\in F} f_l(n)=\emptyset\bigr\}$ belongs to $u$. And, finally, for every pair $(F,k)$ as in (d) (with $F$ finite but with $k\le\omega$ in this case) choose $m_{F,k}$ and $\alpha_{F,k}$, and $U_{F,k}\in u$ such that for every $n\in U_{F,k}$ we have $[a_{m_{F,k},n},b_{\alpha_{F,k},n}]\cap\bigcap_{l\in F}f_l(n)
\subseteq{\operatorname{int}}f_k(n)$ and $[a_{m_{F,k},n},1]\cap\bigcap_{l\in F}g_l(n)\subseteq{\operatorname{int}}g_k(n)$ (the latter only if $k<\omega$ of course).
We fix an ordinal $\alpha$ larger than the $\alpha_k$, $\alpha_F$ and $\alpha_{F,k}$ by $\alpha$ and use it instead in the definitions of the sets $U_k$, $U_F$ and $U_{F,k}$ — they will still belong to $u$. Next take a decreasing sequence $\langle V_p\rangle_{p\in\omega}$ of elements of $u$ such that $V_p$ is a subset of
- $U_k$ whenever $k<p$;
- $U_F$ whenever $F\subseteq p$; and
- $U_{F,k}$ whenever $F\subseteq p$ and $k<p$ or $k=\omega$.
In addition we can, and will, assume that whenever $F\subseteq p$ and $L\cap\bigcap_{l\in F}{\operatorname{cl}}A_f=\emptyset$ then $[b_\alpha,1]\cap\bigcap_{l\in F}g_l(n)=\emptyset$ — that this is possible follows from the assumption that (c) holds for $\max F$.
Now were are truly ready to define $g_\omega$. If $n\notin V_0$ define $g_\omega(n)=\II$. In case $n\in V_p\setminus V_{p+1}$ observe first that if $k<p$ is as in (b) and $F\subseteq p$ is as in (c) then $(F,k)$ is as in (d) so that certainly $$[a_{m_{F,k}},1]\cap\bigcap_{l\in F}g_l(n)\subseteq{\operatorname{int}}g_k(n).\eqno(*)$$ Define $g_\omega(n)$ as the union of $f_\omega(n)\cap[0,b_\alpha(n)]$ and an element $h(n)$ of ${\mathcal{R}}$ that is a subset of $[b_\alpha(n),1]$ and satisfies
- $h(n)\cup g_k(n)\supseteq[b_\alpha(n),1]$ whenever $k<p$ is as in (b);
- $h(n)\cap\bigcap_{l\in F}g_l(n)=\emptyset$ whenever $F\subseteq p$ is as in (c); and
- $h(n)\supseteq [b_{\alpha,n},1]\cap\bigcap_{l\in F}g_l(n)$ whenever $(F,\omega)$ is as in (d).
This is possible because of $(*)$ and because $\bigcap_{l\in F}g_l(n) \cap\bigcap_{l\in G}g_l(n)=\emptyset$ whenever $F$ is as in (c) and $(G,\omega)$ is as in (d). This gives us just enough room to choose $h(n)$.
It is now routine to verify that all conditions on $g_\omega$ are met $u$-often: e.g., if $F\subseteq\omega$ is finite and $L\cap{\operatorname{cl}}A_{f_\omega}\cap\bigcap_{l\in F}{\operatorname{cl}}A_{f_l}=\emptyset$ then $[a_{m_F,n},1]\cap g_\omega(n)\cap\bigcap_{l\in F}g_l(n)=\emptyset$ for all $n\in V_p$, where $p=1+\max F$.
Further considerations
----------------------
The proof in the previous section can be used to show that, under ${\mathsf{CH}}$, all other layers of the continuum $\II_u$ are retracts of $\II_u$. If the layer is a point then this is clear. If the layer $L$ is non-trivial then the cofinality of $[0_u,L)$ and the coinitiality of $(L,1_u]$ are $\omega_1$. It is then a matter of making the proof of Theorem \[retraction\] symmetric to get our retraction $r:\II_u\to L$. The details can be found in [@vanderSteeg2003].
The fixed-point free homeomorphism $h:L\to L$ from Theorem \[thm:all.indec.not.fpp\] can then be used to construct another witness to $s^*(\II_u)\neq0$, almost exactly as in the proof of Theorem \[thm:sssI-u.neq.0\].
Remarks
=======
The results of this paper grew out of an attempt to find non-metric counterexamples to Lelek’s conjecture. The fairly easy proof, indicated after Corollary \[cor:sigmastar.Hstar.nonzero\], that ${\HH^*}$ is not chainable, which also works for layers of countable cofinality lead us to consider $\II_u$ as a possible candidate.
A secondary goal was to convert any non-metric counterexample into a metric one by an application of the Löwenheim-Skolem theorem ([@Hodges1997]\*[Section 3.1]{}) to its lattice of closed sets. This produces a countable sublattice with exactly the same (first-order) lattice-theoretic properties; its Wallman representation space, see [@W], is a metrizable continuum with many properties in common with the starting space, e.g., covering dimension unicoherence, (hereditary) indecomposability, …, see [@vanderSteeg2003]\*[Chapter 2]{} for a comprehensive list.
The results of this paper cast doubt of the possibility of adding (non-)chainability and span (non)zero (of any kind) to this list. The reason for this is that the family ${\mathcal{R}}_u=\{{\operatorname{cl}}A_f\cap\II_u:f\in{{}^\omega {\mathcal{R}}}\}$ is isomorphic to the ultrapower of ${\mathcal{R}}$ (from the proof Theorem \[retraction\]) by the ultrafilter $u$; this follows in essence from the equivalence of ${\operatorname{cl}}A_f\cap\II_u={\operatorname{cl}}A_g\cap\II_u$ and $\{n:f(n)=g(n)\}\in u$. By the Łos Ultraproduct Theorem ([@Hodges1997]\*[Theorem 8.5.3]{}) we see that ${\mathcal{R}}$ and ${\mathcal{R}}_u$ have the same first-order lattice theoretic properties yet their Wallman representations, $\II$ and $\II_u$ respectively, differ in chainability and in various kinds of span (all kinds if ${\mathsf{CH}}$ is assumed).
Chainability is a property that can be read off from a lattice base for the closed sets (or dually for the open sets): using compactness one readily shows that a continuum is chainable iff every basic open cover has a chain refinement from the base. Thus we deduce that chainability is not a first-order property of the lattice base.
For span (non)zero there are two possibilities: it cannot be read off from a base or, if it can be, it is not a first-order property of the lattice base.
Questions
=========
The remarks in the previous section suggest lots of questions. We mention the more important ones.
Is there a non-metric counterexample to any one version of Lelek’s conjecture?
It should be noted that, as mentioned in [@Da], H. Cook has shown that the dyadic solenoid has symmetric span zero.
In spite of the results on $\II$ and $\II_u$ it is still possible that the Löwenheim-Skolem method may convert a non-metric counterexample into a metric one. The reason for this is that ${\mathcal{R}}_u$ is special base for the closed sets of $\II_u$ and not an elementary sublattice of its lattice of closed sets.
If $L$ is an elementary sublattice of the full lattice of closed sets of the continuum $X$, does its Wallman representation inherit (non-)chainability and or span (non)zero from $X$?
Section 3.7 of [@vanderSteeg2003] gives a positive answer for very special sublattices, but unfortunately except for span zero. Further, more specialized, questions can be found in that reference.
The corollaries in Section \[sec:surj-spans\] were derived from Theorem \[retraction\], which needed ${\mathsf{CH}}$ in its proof. This clearly suggests the question whether a more insightful analysis of the structure of the $\II_u$ and the use of more intricate combinatorics will make the use of ${\mathsf{CH}}$ unnecessary.
Can one show in ${\mathsf{ZFC}}$ only that all spans of ${\HH^*}$ and $\II_u$ are nonzero?
It would already be of interest if one could find at least one $u$ such that all spans of $\II_u$ are nonzero.
We have shown implicitly that the fixed-point property like chainability and span zero in that $\II$ has it but $\II_u$ does not, at least under ${\mathsf{CH}}$.
Is there in ${\mathsf{ZFC}}$ at least one $u$ such that $\II_u$ does not have the fixed-point property?
|
---
abstract: 'Sylvester doubles sums, introduced first by Sylvester (see [@S1; @S2]), are symmetric expressions of the roots of two polynomials $P$ and $Q$. Sylvester’s definition of double sums makes no sense if $P$ an $Q$ have multiple roots, since the definition involves denominators that vanish when there are multiple roots. The aims of this paper are to give a new definition for Sylvester double sums making sense if $P$ and $ Q$ have multiple roots, which coincides with the definition by Sylvester in the case of simple roots, to prove the fundamental property of Sylvester double sums, i.e. that Sylvester double sums indexed by $(k,\ell)$ are equal up to a constant if they share the same value for $k+\ell$, and to prove the relationship between double sums and subresultants, i.e. that they are equal up to a constant. In the simple root case, proofs of these properties are already known (see [@ALPP; @AHKS; @RS]). The more general proofs given here are using generalized Vandermonde determinants and a new symmetric multivariate Hermite interpolation as well as an induction on the length of the remainder sequence of $P$ and $Q$.'
author:
- 'Marie-Françoise Roy'
- Aviva Szpirglas
bibliography:
- 'references.bib'
title: 'Sylvester double sums, subresultants and symmetric multivariate Hermite interpolation'
---
[**Keywords**]{}: subresultants, Sylvester double sums, multivariate Hermite interpolation, generalized Vandermonde determinants
Introduction {#introduction .unnumbered}
============
The first aim of this paper is to provide a definition for Sylvester double sums making sense if $P$ and $ Q$ have multiple roots, which is done using quotients of Vandermonde determinants involving variables, and substitutions. When the structure of the multiplicities of the roots of $P$ and $Q$ is known, we obtain a direct expresion of the Sylvester double sums in terms of generalized Vandermonde determinants.
The second aim of the paper is to prove, in the general case, the fundamental property for Sylvester double sums, i.e. that Sylvester double sums indexed by $(k,\ell)$ are equal up to a constant if they share the same value for $k+\ell$. In order to prove this fundamental property, it is convenient to define more general objects, the [*multi Sylvester double sums*]{}. We introduce a new multivariate symmetric Hermite interpolation and use it to study the properties of multi Sylvester double sums. The strategy then consists in proving the fundamental property for multi Sylvester double sums and obtaining the result for Sylvester double sums as a corollary by identifying coefficients.
The third aim of the paper is to prove the relationship between double sums and subresultants, i.e. that they are equal up to a constant. Our strategy is based on an induction on the length of the remainder sequence of $P$ and $Q$.
Our more general proofs are new even in the special case when the roots of the polynomials are simple.
The idea of introducing a multivariate symmetric Hermite interpolation and using multi Sylvester double sums was inspired by [@KSV]’s use of multivariate symmetric Lagrange interpolation and introduction of multi Sysvtester double sums in the context of simple roots.
The content of the paper is the following.
In Section \[sec:def\] we give a general definition for Sylvester double sums, valid also when there are multiple roots, and prove that it coincides with Sylvester’s definition in the special case where all roots are simple (Proposition \[theoreme0\]).
In Section \[sec:DSandVandermonde\] we consider generalized Vandermonde determinants and use them to give a new formula for Sylvester double sums when the structure of multiplicities is known (Proposition \[lemme2bis\]).
In Section \[sec:DSandHermite\], we introduce an Hermite interpolation for multivariate symmetric polynomials (Proposition \[interpolationdata\]).
In Section \[sec:fonda\] we study multi Sylvester double sums. We give their definition in subsection \[subsec:defmulti\] In subsection \[subsec:mult\] we compute the multi Sylvester double sums and Sylvester double sums for indices $(k,\ell)$ with $k+\ell\ge \deg(Q)$. In subsection \[subsec:fonda\] we prove that multi Sylvester double sums and Sylvester double sums indexed by $k,\ell$, depend only (up to a constant) on $j=k+\ell$ (Theorem \[theo4mult\] and Theorem \[theo4\]).
In Section \[sec:remainders\] we give a relationship between Sylvester double sums of $(P,Q)$ and Sylvester double sums of $(Q,R)$ where $R$ is the opposite of the remainder of $P$ by $Q$ in the Euclidean division (Proposition \[prorecurrence\]).
Finally we prove in Section \[sec:subresdblsum\] that Sylvester double sums coincide (up to a constant) with subresultants, by an induction on the length of the remainder sequence of $P$ and $Q$ (Theorem \[theoreme2\]).
Sylvester double sums {#sec:def}
=====================
We give a general definition for Sylvester double sums, valid also when the polynomials have multiple roots, and prove that it coincides with Sylvester’s definition in the special case where all roots are simple (Proposition \[theoreme0\]).
Basic notations and definitions
-------------------------------
Let $\mathbb{K}$ be a field of characteristic $0$.
Let $\BA$ be a finite list of elements of $\mathbb{K}$.
We denote $\BA' \subset_a \BA$ when $\BA'$ is a sublist of $\BA$ with $a$ elements (i.e. the list $\BA'$ is ordered by the restriction of the order on the list $\BA$).
Let $\BB$ be another finite list of elements of $\mathbb{K}$.
We denote $$\Pi(\BA,\BB)=\prod_{\genfrac{}{}{0pt}{}{x\in \BA}{y\in \BB}}(x-y).$$ Note that $\Pi(\BA, \BB)$ is independant on the order of $\BA$ and $\BB$.
We abbreviate $\Pi(\{x\},\BB)$ and $\Pi(\BA,\{y\})$ to $\Pi(x,\BB)$ and $\Pi(\BA,y)$ respectively.
Note that $\Pi(\BA,\BB)$ is the classical resultant of the monic polynomials $\Pi(X,\BA)$ and $\Pi(X,\BB)$.
\[vderm\] The [Vandermonde vector ]{}of length $i$ of $x\in \mathbb{K}$, denoted by $v_i(x)$, is $$v_i(x)=\left[
\begin{array}{c}
1\\
x\\
\vdots\\
\vdots\\
x^{i-1}\\
\end{array}
\right].$$
Let $\BA=(x_1,\ldots,x_i)$ be a finite ordered list of elements of $\mathbb{K}$. The Vandermonde matrix $\mathcal{V}(\BA)$ is the $i \times i$ matrix having as column vectors $v_i(x_1),\ldots, v_i(x_i)$. The Vandermonde determinant $ V(\BA)$ is the determinant of the Vandermonde matrix $\mathcal{V}(\BA)$. It is well known that $$V(\BA)= \prod_{i \ge k > j \ge 1} (x_k - x_j) .$$
By ${\bf B} \|{\bf A}$ the we denote the list obtained by concatening ${\bf B}$ and ${\bf A}$.
The following result is obvious.
\[tresutile\] $$V({\bf B} \|{\bf A})=V({\bf A}) \Pi ({\bf A},{\bf B}) V({\bf B}).$$ and, as a special case, given a variable $U$, $$V({\bf B} \|U)=\Pi (U,{\bf B})V ({\bf B}).$$
Definition of Sylvester double sums
-----------------------------------
Let $\BP=(x_1,\ldots,x_p)$ and $\BQ=(y_1,\ldots,y_q)$ be two finite ordered sets of element of $\mathbb{K}$ and $P=\Pi(X,\BP)$, $Q=\Pi(X,\BQ)$
The Sylvester double sum of $(P,Q)$ of index $k \in\mathbb{N}, \ell \in \mathbb{N} $ is usually defined as the following polynomial in $\mathbb{K}[U]$:
$$\label{simple}
\sum_{\genfrac{}{}{0pt}{}{\BK \subset_k \BP}
{\BL \subset_\ell \BQ}} \Pi (U,\BK) \Pi (U,\BL)
\frac{\Pi (\BK, \BL) \Pi (\BP \setminus \BK,
\BQ \setminus \BL)}{\Pi (\BK, \BP \setminus \BK)
\Pi (\BL, \BQ \setminus \BL)}$$
(see [@S1; @S2]).
This definition of Sylvester double sums makes no sense if $P$ and $ Q$ have multiple roots, since some of the quantities $\Pi (\BK, \BP \setminus \BK)$ (resp. $ \Pi (\BL, \BQ \setminus \BL)$) at the denominator are equal to $0$.
In this section, we give a general definition of Sylvester double sums, valid even if $P$ and $ Q$ have multiple roots and prove that it coincides with the classical one when all these roots are simple.
Let $\BX=(X_1,\ldots,X_p)$ and $\BY=(Y_1,\ldots,Y_q)$ be two ordered sets of indeterminates.
Given $\BX'\subset_k\BX$ (resp. $\BY'\subset_\ell\BY$ ), we denote $s_{\BX'}$ (resp. $s_{\BY'}$ ) the signature of the permutation $\sigma_{\BX'}$ (resp. $\sigma_{\BY'}$) putting the elements of $\BX$ (resp. $\BY$) in the order $(\BX\setminus\BX')\|\BX'$ (resp.$(\BY\setminus\BY')\|\BY'$).
For any $k \in\mathbb{N}, \ell \in\mathbb{N}$, we define the polynomial $F^{k,\ell}(\BX,\BY)(U)$ in $K[\BX,\BY,U]$ $$\label{defA} F^{k,\ell}(\BX,\BY)(U) =
\sum_{\genfrac{}{}{0pt}{}{\BX' \subset_k \BX}
{\BY' \subset_\ell \BY}} s_{\BX'} s_{\BY'} V ((\BY \setminus \BY') \|
(\BX\setminus \BX'))V (\BY' \| \BX' \| U)$$
Note that if $k>p$ or $\ell>q$ then $F^{k,\ell}(\BX,\BY)(U)=0$.
\[lemme1\] The polynomial $F^{k,\ell}(\BX,\BY)(U)$ is antisymmetric in the variables $\BX$ and in the variables $\BY$.
For any permutation $\sigma$ of the ordered set $\BX$, we call also $\sigma$ the action of $\sigma$ on a polynomial $F$ in $K[\BX,\BY,U]$, i.e $\sigma(F)(\BX,\BY)=F(\sigma(\BX),\BY)$. Denoting $s$ the signature of $\sigma$ we want to prove $$\label{anti}
\sigma(F^{k,\ell})(\BX,\BY)(U)=s F^{k,\ell}(\BX,\BY)(U).$$ It is enough to prove (\[anti\]) for a transposition exchanging two sucessive elements, of signature $-1$.
So, let $\tau$ be the transposition exchanging $X_i$ and $X_{i+1}$. We want to prove $$\label{anti1}
\tau(F^{k,\ell})(\BX,\BY)(U)=-F^{k,\ell}(\BX,\BY)(U).$$
We denote by $\tau(\BX)$ the ordered set obtained from $\BX$ by exchanging $X_i$ and $X_{i+1}$. Given $\BX'\subset_k\BX$, we denote by $\tau(\BX')$ the ordered set $\tau(\BX')\subset_k \tau(\BX)$ (i.e. $\tau(\BX')$ is ordered by the restriction of the order on $\tau(\BX)$) and by $\bar \BX'$ the ordered set $\tau
(\BX')\subset_k \BX$ (i.e. $\bar \BX'$ and $\tau
(\BX')$ have the same elements but $\bar \BX'$ is ordered by the restriction of the order on $\BX$).
Denote $$F^{\BX',\BY'}=s_{\BX'} s_{\BY'}V ((\BY \setminus \BY') \|
(\BX\setminus \BX'))V (\BY' \| \BX' \|U) .$$
We have 3 cases to consider.
- If $X_i\in \BX'$ and $X_{i+1}\in \BX'$ then $\tau(\BX \setminus \BX')= \BX \setminus \BX'$ and $$\begin{array}{rcl}
\tau(F^{\BX',\BY'})&=&s_{\BX'} s_{\BY'}
V ((\BY \setminus \BY') \|
\tau(\BX \setminus \BX'))V ( \BY'\| \tau(\BX') \|U) \\
&=&s_{\BX'} s_{\BY'}V ((\BY \setminus \BY') \|
( \BX \setminus \BX'))V ( \BY' \| \tau(\BX') \|U) \\
&=&-F^{\BX',\BY'}.
\end{array}$$
- If $X_i\notin \BX'$ and $X_{i+1}\notin \BX'$ then $\tau(\BX')=\BX'$ and $$\begin{array}{rcl}
\tau(F^{\BX',\BY'})&=&s_{\BX'} s_{\BY'}
V ((\BY \setminus \BY') \|
\tau(\BX \setminus \BX'))V ( \BY'\| \tau(\BX') \|U) \\
&=&s_{\BX'} s_{\BY'}V ((\BY \setminus \BY') \|
\tau( \BX \setminus \BX'))V ( \BY' \| \BX' \|U) \\
&=&-F^{\BX',\BY'}.
\end{array}$$
- If $X_i \in \BX'$ and $X_{i+1} \notin \BX'$ , or $X_i \notin \BX'$ and $X_{i+1} \in \BX'$, then $\sigma_{\bar \BX'}=\tau \circ \sigma_{\BX'}$, $\tau(\BX')=\bar \BX'$ and $\tau(\BX \setminus \BX')=\BX \setminus \bar \BX'$ so that $$\begin{array}{rcl}
\tau(F^{\BX',\BY'})&=&s_{\BX'} s_{\BY'} V ( (\BY \setminus \BY') \|
\tau(\BX\setminus \BX'))V (\BY' \| \tau(\BX') \|U)\\
&=&- s_{\bar \BX'} s_{\BY'} V ((\BY\setminus \BY') \|
(\BX \setminus \bar \BX'))V (\BY' \| \bar \BX' \|U)\\
&=&-F^{\bar \BX',\BY'}
\end{array}$$ and $$\begin{array}{rcl}
\tau(F^{\bar \BX',\BY'})&=&s_{\bar \BX} s_{\BY'}V (( \BY \setminus \BY') \|
\tau ( \BX \setminus \bar\BX'))V ( \BY'\|\tau(\bar\BX') \|U) \\
&=&-s_{\BX} s_{\BY'}V ((\BY \setminus \BY') \|
(\BX \setminus \BX'))V (\BY' \| \BX' \|U) \\
&=&-F^{\BX',\BY'},
\end{array}$$ From which we deduce $$\tau
\left(F^{\BX',\BY'}+F^{\bar \BX',\BY'}\right)=-\left(F^{\bar \BX',\BY'}+F^{\BX',\BY'}\right).$$
So, we get (\[anti1\]).
The exchange between two elements of $\BY$ can be treated similarly.
\[div\] If $A(\BX,\BY)$ in $K[\BX,\BY]$ is antisymmetric with respect to the variables $\BX$, then $A(\BX,\BY)=S(\BX,\BY) V(\BX)$ where $S\in K[\BX,\BY]$ is symmetric with respect to the variables $\BX$.
If $A(\BX,\BY)$ is antisymmetric with respect to $\BX$ then, for any $j<k$, denote $\tau_{j,k}(\BX)$ the ordered set of variables obtained by transposing $X_j$ and $X_k$.
$$\frac{A(\BX,\BY)-A(\tau_{j,k}(\BX,\BY))}{X_j-X_k}=2\frac{A(\BX,\BY)}{X_j-X_k}$$ is a polynomial. So $A(\BX,\BY)=S(\BX,\BY) V(\BX) $ and $S(\BX,\BY)$ is a symmetric polynomial with respect to $\BX$.
Applying Lemma \[div\] and Proposition \[lemme1\] we denote $S^{k,\ell}(\BX,\BY)(U)$ the symmetric polynomial with respect to the indeterminates $\BX$ and with respect to the indeterminates $\BY$ satisfying $$\label{defS}
S^{k,\ell}(\BX,\BY)(U)=\frac{F^{k,\ell}(\BX,\BY)(U)}{V(\BX)V(\BY)}.$$
Given two monic univariate polynomials $P$ and $Q$ of degree $p$ and $q$ we denote $\BP=(x_1,\ldots,x_p)$ and $\BQ=(y_1,\ldots,y_q)$ ordered lists of the roots of $P$ and $Q$ in an algebraic closure $\C$ of $\mathbb{K}$, counted with multiplicities.
The [generalized Sylvester double sum of $(P,Q)$]{} for the exponents $k,\ell\in\mathbb{N}\times\mathbb{N}$ is defined by $$\Sylv^{k,\ell}(P,Q)(U)=S^{k,\ell}(\BP,\BQ)(U).$$
Note that this definition does not depend on the order given for the roots of $P$ and $Q$.
This definition of generalized Sylvester double sums for monic polynomials coincides with the usual definition of Sylvester double sums when the polynomials $P$ and $Q$ have no multiple roots, as we see now in Proposition \[theoreme0\].
\[theoreme0\] If $P,Q$ have only simple roots, $$\Sylv^{k,\ell}(P,Q )(U)=\sum_{\genfrac{}{}{0pt}{}{\BK \subset_k \BP}
{\BL \subset_\ell \BQ}} \Pi (U,\BK) \Pi (U,\BL)
\frac{\Pi (\BK, \BL) \Pi (\BP \setminus \BK,
\BQ \setminus \BL)}{\Pi (\BK, \BP \setminus \BK)
\Pi (\BL, \BQ \setminus \BL)}$$
$\displaystyle{ \sum_{\genfrac{}{}{0pt}{}{\BK \subset_k \BP}
{\BL \subset_\ell \BQ}} \Pi (U,\BK) \Pi (U,\BL)
\frac{\Pi (\BK, \BL) \Pi (\BP \setminus \BK,
\BQ \setminus \BL)}{\Pi (\BK,\BP \setminus \BK)
\Pi (\BL, \BQ \setminus \BL)}}=$ $$\begin{array}{cl}
=& \displaystyle{\sum_{\genfrac{}{}{0pt}{}{\BK \subset_k \BP}
{\BL \subset_\ell \BQ}} \Pi (U,\BK) \Pi (U,\BL)
\frac{V(\BL) \Pi (\BK, \BL) V(\BK) V(\BQ \setminus \BL) \Pi (\BP \setminus \BK,\BQ \setminus \BL)V(\BP \setminus \BK)}{V(\BK)\Pi (\BK,\BP \setminus \BK) V(\BP \setminus \BK)
V(\BL) \Pi (\BL,\BQ \setminus \BL) V(\BQ \setminus \BL) }}\\
=& \displaystyle{ \sum_{\genfrac{}{}{0pt}{}{\BK \subset_k \BP}
{\BL \subset_\ell \BQ}}s_{\BK} s_{\BL} \frac{V (\BL \| \BK \|
U) V ((\BQ \setminus \BL)\| (\BP \setminus \BK))}{V (\BP)
V (\BQ)}}\\
=& \displaystyle{\frac{F^{k,\ell}(\BP,\BQ)(U)}{V(\BP)V(\BQ)}}=S^{k,\ell}(\BP,\BQ)(U)\\\\
=&\Sylv^{k,\ell}(P,Q)(U)
\end{array}$$ applying Lemma \[tresutile\].
Generalized Vandermonde determinants and Sylvester double sums {#sec:DSandVandermonde}
==============================================================
We consider generalized Vandermonde determinants (also called sometimes confluent Vandermonde determinants, see [@LT; @HJ]) and connect them with the Sylvester double sums (Proposition \[lemme2bis\]).
\[notmultiset\] Let $P$ be a polynomial of degree $p$ with coefficients in a field $\mathbb{K}$. Let $(x_1,\ldots,x_m)$ be an ordered set of the distinct roots of $P$ in an algebraic closure $\C$ of $\mathbb{K}$, with $x_i$ of multiplicity $\mu_i$, and let $\BP$ be the multiset of roots of $P$, represented by the ordered set $$\BP=(x_{1,0},\ldots,x_{1,\mu_1-1},\ldots,x_{m,0},\ldots,x_{m,\mu_m-1}),$$ with $x_{i,j}=(x_i,j)$ for $0\leq j\leq \mu_i-1$, $\sum_{i=1}^m \mu_i=p.$
Let $Q$ be a polynomial of degree $q$ with coefficients in $\mathbb{K}$. Let $(y_1,\ldots,y_n)$ be an ordered set of the distinct roots of $Q$ in $\C$ with $y_i$ of multiplicity $\nu_i$, , for $i=1,\ldots,n$. Let $\BQ$ be the ordered multiset of its root, represented by the ordered set $$\BQ=(y_{1,0},\ldots,y_{1,\nu_1-1},\ldots,y_{n,0},\ldots,y_{n,\nu_n-1}),$$ with $y_{i,j}=(y_i,j)$ for $0\leq j\leq \nu_i-1$, $\sum_{i=1}^n \nu_i=q.$
We introduce an ordered set of variables $X_\BP=(X_{1,0}, \ldots,X_{1,\mu_1-1},\ldots,X_{m,0},\ldots,X_{m,\mu_m-1})$ and an ordereed set of variables $Y_\BQ=(Y_{1,0}, \ldots,Y_{1,\nu_1-1},\ldots,Y_{n,0},\ldots,Y_{n,\nu_n-1})$.
For a polynomial $f(X_\BP,X_\BQ)$ we denote $f(\BP,\BQ)$ the result of the substitution of $X_{i,j}$ by $x_i$ and $Y_{i,j}$ by $y_i$.
Given $f$ a polynomial depending on the variable $U$, we denote $$f^{[i]}=\frac1{i!}\frac{\partial^{i} f}{\partial U^i }.\label{derdermult}$$
Let $\BK\subset_k \BP$, $\BL \subset_\ell \BQ$ and $\BU=(U_1,\ldots,U_u)$ an ordered set of $u$ indeterminates.
The [generalized Vandermonde matrix]{} ${\cal V}[\BL\|\BK\|\BU)$ is the $(\ell+k+u) \times (\ell+k+u)$ matrix having as column vectors the $\ell$ columns $v^{[j]}_{k+\ell+u}(y_i)$ for $y_{i,j}\in \BL$ followed by the $k$ columns $v^{[j]}_{k+\ell+u}(x_i)$ for $x_{i,j}\in \BK$ followed by the $u$ columns $v_{k+\ell+u}(U_i)$ (using notation (\[vderm\]) and notation (\[derdermult\])).
The [generalized Vandermonde determinant]{} $V[\BL\|\BK\|\BU)$ is the determinant of ${\cal V}[\BL\|\BK|\BU)$.
- In the particular case $u=0$ we denote $V[\BL\|\BK]$ the corresponding determinant.
- In the particular case $k=p,\ell=u=0$ we denote $V[\BP]$ the corresponding determinant .
- Similarly, in the particular case $k=0,\ell=q,u=0$ we denote $V[\BQ]$ the corresponding determinant.
[The peculiar notation $V[\BL\|\BK\|\BU)$ with one square bracket to the left and one parenthesis to the right is here to indicate that the column $v_{k+\ell+u}^{[j]}(x_i)$ indexed by $x_{i,j}\in \BK$ and $v_{k+\ell+u}^{[j]}(y_i)$ indexed by $y_{i,j}\in\BL$ have been derivated, while there is no derivation with respect to the columns indexed by the variables in $\BU$.]{}
While the classical Vandermonde determinant $V(\BP)$ is null when $P$ has multiple roots, we have the following result for the generalized Vandermonde determinant.
The generalized Vandermonde determinant $ V[\BP]$ is equal to $$V[\BP]
=\prod_{1\leq i<j\leq m }(x_j-x_i)^{\mu_i\mu_j}.$$
The proof is done by induction on $p$.
If $p=1$, $V[\BP]=1$.
Suppose that $$\displaystyle{ V[\BP]
=\prod_{1\leq i<j\leq m }(x_j-x_i)^{\mu_i\mu_j}}.$$
The polynomial $F(U)=V[\BP\| U)$ is of degree $p$, with leading coefficient $V[\BP]$ and satisfies the property $$\mbox{ for all }1\leq i\leq m, \mbox{ for all }0\leq j < \mu_i,
\begin{array}{lcl}
F^{[j]}(x_i)&=&0,
\end{array}$$ So, $$F(U)=V[\BP]\prod_{i=1}^{m}(U-x_i)^{\mu_i}=\prod_{1\leq i<j\leq m }(x_j-x_i)^{\mu_i\mu_j}\prod_{i=1}^{m}(U-x_i)^{\mu_i}.$$
Consider $T(U)=(U-x)P(U)$.
$-$ First case: $x$ is not a root of $P$. Let $\BT$ the ordered set (obtained by adding $x$ at the end of $\BP$) of roots of the polynomial $T$, so $x=x_{m+1}$ is a root of $T$ with multiplicity 1. Then $$V[\BT]=F(x)=\prod_{1\leq i<j\leq m }(x_j-x_i)^{\mu_i\mu_j}\prod_{i=1}^{m}(x-x_i)^{\mu_i}=\prod_{1\leq i<j\leq m+1 }(x_j-x_i)^{\mu_i\mu_j}.$$
$-$ Second case: $x$ is a root of $P$. So there exists $1\leq j\leq m$ such that $x=x_j$ , and $x_j$ is a root of multiplicity $\mu_j+1$ of $T$.
Let $\BT$ the ordered set of roots of the polynomial $T$ obtained by inserting $x_{j,\mu_j}=(x_j,\mu_j)$ after $x_{j,\mu_j-1}$ in $\BP$. Then $$\begin{aligned}
V[\BT]&=(-1)^{\mu_{j+1}+\cdots+\mu_m}F^{[\mu_j]}(x_j)\\
&=(-1)^{\mu_{j+1}+\cdots+\mu_m}V[\BP]\displaystyle{\prod_{\genfrac{}{}{0pt}{}{i=1}{i\not= j}}^{m}(x_j-x_i)^{\mu_i}}\\
&=\displaystyle{\prod_{1\leq i<j\leq m }(x_j-x_i)^{\mu_i(\mu_j+1)}}\qedhere\end{aligned}$$
[If $\BK\subset_k \BP$, it can happen that $V[\BK]=0$. Taking for example $\BP=(x_{1,0},x_{1,1},x_{2,0},x_{2,1})$ and $\BK=(x_{1,1},x_{2,1})$, it is easy to check that $ V[\BK]=0$. ]{}
From now on, and till Section \[sec:remainders\], $P$ and $Q$ are monic polynomials,
The following proposition makes the link between generalized Vandermonde determinants and Sylvester double sums.
We denote $s_{\BK}$ (resp. $s_{\BL}$) the signature of the permutation $\sigma_{\BK}$ (resp. $\sigma_{\BL}$) obtained by putting the elements of $\mathbf{P}$ (resp. $\BQ$) in the order $(\BP\setminus\BK)\|\BK$ (resp. $(\BQ\setminus\BL)\|\BL$).
\[lemme2bis\] $$\Sylv^{k,\ell}(P,Q)(U)= \sum_{\genfrac{}{}{0pt}{}{\BK \subset_k \BP}
{\BL \subset_\ell \BQ}} s_{\BK} s_{\BL}\frac{
V[(\BQ \setminus \BL) \|
(\BP \setminus \BK)] V[\BL \| \BK \| U) } {V[\BP] V[\BQ]}$$
In the proof of Proposition \[lemme2bis\], we use the following notation \[derivations\] and Lemma \[exemple2\].
\[derivations\] For any polynomial $f$ depending on the variables $X_\BP$, and $\BK \subset_k \BP$, denote $\partial^{[\BK]}f$ the polynomial defined by induction on $r$ as follows. $$\partial^{[\emptyset]}f=f$$ If $\BK=\BK'\|(x_{i,j})$, $$\partial^{[\BK]}f=\frac1{j!}\frac{\partial^{j}\partial^{[\BK']}f}{\partial X_{i,j}^j }.$$
Similarly, for any polynomial $f$ depending on the variables $Y_\BQ$, and $\BL \subset_\ell \BQ$, denote $\partial^{[\BL]}f$ the polynomial defined by induction on $s$ as follows. $$\partial^{[\emptyset]}f=f$$ If $\BL=\BL'\|(y_{i,j})$, $$\partial^{[\BL]}f=\frac1{j!}\frac{\partial^{j}\partial^{[\BL']}f}{\partial X_{i,j}^j }.$$
Note that $$V[\BL\|\BK\|\BU)=f(\BK,\BL,\BU)$$ with $f(X_\BK,Y_\BL,\BU)=\partial^{[\BL]}\partial^{[\BK]}V(Y_\BL\|X_\BK\|\BU)$.
\[exemple2\] $$\partial^{[\BP]}\left(V(X_\BP)f(X_\BP)\right)(\BP)= V[\BP] f(\BP)$$
We first note that $$\partial^{[\BP]}\left(V(X_\BP)f(X_\BP)\right)=\partial^{[\BP]}(V(X_\BP))f(X_\BP)+\sum_r V_r(X_\BP) f_r(X_\BP)$$ where $V_r(X_\BP)$ (resp. $f_r(X_{\BP}^{})$) is obtained from $V(X_\BP)$ (resp. from $f(X_\BP)$) by partial derivations, one variable $X_{i,j}$ at least being derived less than $j$ times (resp. at least one time). Denoting $X_{i,j}$ the first variable which is being derived less than $j$ times in $V_r(X_\BP)$, we define $j'$ as the order of derivation of $X_{i,j}$ in $V_r(X_\BP)$. We notice that $V_r(X_\BP)$ is the determinant of a matrix with two equal columns, the one indexed by ${i,j'}$ and the one indexed by ${i,j}$. Hence $V_r(\BP)=0$. This proves the claim.
\[Proof of Proposition \[lemme2bis\]\]
Since $$F^{k,\ell}(X_\BP,Y_\BQ)(U)=V(X_\BP)V(Y_\BQ)S^{k,\ell}(X_\BP,Y_\BQ)(U),$$ using Lemma \[exemple2\] we obtain $$\partial^{[\BQ]}\partial^{[\BP]}F^{k,\ell}(\BP,\BQ)(U)= V[\BP] V[\BQ]S^{k,\ell}(\BP,\BQ)(U)= V[\BP] V[\BQ]\Sylv^{k,\ell}(P,Q)(U).$$ On the other hand, denoting $$h_{\BK,\BL}(X_\BP,Y_\BQ)(U)=
V (Y_{\BQ \setminus \BL} \|
X_{\BP \setminus\BK})V (Y_\BL\| X_\BK \| U),$$ we have $$\partial^{[\BQ]}\partial^{[\BP]} h_{\BK,\BL}(\BP,\BQ)(U)= V[(\BQ \setminus \BL) \| (\BP \setminus \BK)]V[\BL \| \BK \| U).$$
Since $$F^{k,\ell}(X_\BP,Y_\BQ)(U)=\sum_{\genfrac{}{}{0pt}{}{\BK \subset_k \BP}{\BL \subset_\ell \BQ}}s_\BK s_\BL h_{\BK,\BL}(X_\BP,Y_\BQ)(U)$$ we get $$\begin{aligned}
\partial^{[\BQ]}\partial^{[\BP]}F^{k,\ell}(\BP,\BQ)(U)&=\sum_{\genfrac{}{}{0pt}{}{\BK \subset_k \BP}
{\BL \subset_\ell \BQ}} s_{\BK} s_{\BL} V[(\BQ \setminus \BL )\| (\BP \setminus \BK)]V[\BL \| \BK \| U)\qedhere\end{aligned}$$
The following lemma will be useful later.
\[prodpratique\]
1. For $\BL\subset_\ell\BQ$, defining $$f(Y_{\BQ\setminus \BL})=(-1)^{p(q-\ell)}\partial^{{[\BQ\setminus\BL]}}\left(V(Y_{\BQ\setminus\BL})\prod_{Y\in Y_{\BQ\setminus\BL}}P(Y)\right),$$ we have $$V[(\BQ \setminus \BL)\|\BP ]=V[\BP]f(\BQ\setminus\BL).$$
2. For $\BK\subset_k\BP$, defining $$g( X_{\BP\setminus\BK})=\partial^{{[\BP\setminus\BK]}}\left(V(X_{\BP\setminus\BK})\prod_{X \in X_{\BP\setminus\BK}}Q(X) \right),$$ we have $$V[\BQ\|(\BP\setminus\BK)]=V[\BQ]g{(\BP\setminus\BK)}$$
Defining $$\begin{array}{rcl}
h(X_\BP,Y_{\BQ\setminus\BL})&=&\partial^{[\BQ\setminus\BL]}V(Y_{\BQ\setminus\BL}\|X_\BP)
\\
&=&\partial^{[{\BQ\setminus\BL}]}\left(V(X_\BP)\Pi(X_\BP,Y_{\BQ\setminus\BL})V(Y_{\BQ\setminus\BL})\right)\\
&=&V(X_\BP)\partial^{[{\BQ\setminus\BL}]}\left(\Pi(X_\BP,Y_{\BQ\setminus\BL}) V(Y_{\BQ\setminus\BL})\right)
\end{array}$$ and applying Lemma \[exemple2\], we get $$\begin{array}{rcl}
\partial^{[\BP]}h(\BP,Y_{\BQ\setminus\BL})&=&V[\BP]\partial^{[{\BQ\setminus\BL}]}\left(V(Y_{\BQ\setminus\BL})\Pi(\BP,Y_{\BQ\setminus\BL})\right)\\
&=&V[\BP]\partial^{[{\BQ\setminus\BL}]}\left(V(Y_{\BQ\setminus\BL})\displaystyle{\prod_{Y\in Y_{\BQ\setminus\BL}}(-1)^pP(Y)}\right)\\
&=&V[\BP]f(Y_{\BQ\setminus \BL}) .
\end{array}$$ and finally $$V[(\BQ\setminus\BL) \|\BP]=\partial^{[\BP]}h(\BP,\BQ\setminus\BL)=V[\BP]f(\BQ\setminus\BL).$$ Which is Lemma \[prodpratique\].1.
The proof for Lemma \[prodpratique\].2 is similar.
Lemma \[prodpratique\] has the following corollary.
\[Regal0\] If $Q$ divides $P$, then $$\Sylv^{0,j}(P,Q)(U)=0$$
In this case, $f(\BQ\setminus\BL)=0$ as any root of $Q$ is a root of $P$ with at least the same multiplicity. So, applying Lemma \[prodpratique\].1, $V[(\BQ\setminus\BL)\|\BP]=0$. It follows $$\Sylv^{0,j}(P,Q)(U)=\sum_{\BL\subset_j\BQ}^{}s_\BL\frac{V[(\BQ\setminus\BL)\|\BP]V[\BL\|U)}{V[\BP]V[\BQ]}=0$$
Hermite Interpolation for multivariate symetric polynomials {#sec:DSandHermite}
===========================================================
\[symmetrichermite\]
We now introduce an Hermite interpolation for multivariate symmetric polynomials.
We consider an ordered set of $p-k$ variables $\BU$.
\[multih\] The set $$\mathcal{B}_{\BP,k}(\BU)=\ens{\frac{ V[\BK\|\BU)}{V[\BP]V(\BU)}\mid \BK\subset_k \BP }$$ is a basis of the vector-space of symmetric polynomials in $\BU$ of multidegree at most $k,\ldots,k$.
The proof of Proposition \[multih\] uses the following Lemma.
\[toutourien\]
1. $ V[\BK\|(\BP\setminus \BK)]=(-1)^{k(p-k)}s_\BK V[\BP]\not=0.$
2. If $\BK'\not= \BK$, $V[\BK'\|(\BP\setminus \BK) ]=0.$
<!-- -->
1. It is clear that $V[\BK\|(\BP\setminus \BK)]=(-1)^{k(p-k)}s_\BK V[\BP]\not=0,$ since $s_\BK$ is the signature of the permutation putting $\BP$ in the order $(\BP\setminus\BK)\| \BK$.
2. The fact that $V[\BK'\|(\BP\setminus \BK)]=0$ when $\BK'\not= \BK$ follows from the fact that the matrix $ {\cal V}[\BK'\|(\BP\setminus \BK)]$ has two equal columns.
\[Proof of Proposition \[multih\]\] Since the number of subsets of cardinality $k$ of $\BP$ is $\displaystyle{\binom{p}{k}}$ and that $\displaystyle{\binom{p}{k}}$ is also the dimension of the vector space of symmetric polynomials in $\BU$ of multidegree at most $k,\ldots,k$, it is enough to prove that $$\sum_{\BK' \subset_k \BP} c_{\BK'}\frac{ V[\BK'\|\BU)}{V[\BP]V(\BU)}=0$$ implies $c_{\BK}=0$ for all $\BK\subset_k \BP$.
Let us fix $\BK\subset_k \BP$. Since $$\sum_{\BK' \subset_k \BP} c_{\BK'} V[\BK'\|\BU)=0,$$ it follows by substitution and derivation that $$\sum_{\BK' \subset_k \BP} c_{\BK'} \partial^{[\BP\setminus \BK]} V[\BK'\|X_{\BP\setminus \BK})=0.$$ When replacing $X_{\BP\setminus \BK}$ by $\BP\setminus \BK$ we obtain $$\begin{aligned}
\sum_{\BK' \subset_k \BP} c_{\BK'} V[\BK'\|(\BP\setminus \BK)]&=0.\end{aligned}$$ Using Lemma \[toutourien\], we get $c_{\BK}=0$.
The following Proposition gives the connection between a symetric polynomial in $\BU$ of multidegree at most $k,\ldots,k$ and its coordinates in the basis $\mathcal{B}_{\BP,k}(\BU)$.
[**(Multivariate symmetric Hermite Interpolation)**]{} \[interpolationdata\] Let $g$ be a symetric polynomial in $\BU$ of multidegree at most $k,\ldots,k$. Writing $$g(\BU)=\sum_{\BK \subset_k \BP} g_{\BK}\frac{ V[\BK\|\BU)}{V[\BP]V(\BU)}$$ then $$g_\BK=(-1)^{k(p-k)}s_\BK h(\BP\setminus \BK)$$ with $$h(X_{\BP\setminus \BK})= \partial^{[\BP\setminus \BK]}(V(X_{\BP\setminus \BK})g(X_{\BP
\setminus \BK}) ).$$
We have $$\sum_{\BK \subset_k \BP} g_\BK V[\BK\|\BU)=V[\BP]g(\BU)V(\BU).$$ Derivating both sides by $\partial^{[\BP\setminus \BK']}$ and substituting $\BP\setminus \BK'$ for $\BU$ we get, using Lemma \[toutourien\] $$g_{\BK' } V[\BK'\|(\BP\setminus \BK')]=g_{\BK' }s_{\BK'} (-1)^{k(p-k)} V[\BP]= V[\BP]h(\BP\setminus \BK') ,$$ and finally $$g_{\BK'}=(-1)^{k(p-k)}s_{\BK'} h(\BP\setminus \BK').\qedhere$$
[Proposition \[multih\] generalizes a result in [@CL] given for Lagrange interpolation of symmetric multivariate polynomials.]{}
As a corollary of Proposition \[interpolationdata\], we recover the classical Hermite Interpolation
\[hermite\][**(Hermite Interpolation)**]{} Given an ordered list $${\bf q}=(q_{1,0},\ldots,q_{1,\mu_1-1},\ldots,q_{m,0},\ldots,q_{m,\mu_m-1})$$ of $p$ numbers, there is one and only one polynomial of degree at most $p-1$ satisfying the property $$\mbox{ for all }1\leq i\leq m, \mbox{ for all }0\leq j < \mu_i,
\begin{array}{lcl}
Q^{[j]}(x_i)&=&q_{i,j}.
\end{array}$$
If $k=p-1$ in Proposition \[multih\], then $$\mathcal{B}_{\BP,p-1}(U)= \ens{\frac{V[\BP\setminus \ens{x_{i,j}}\|U)}{V[\BP]}\mid x_{i,j}\in\BP}$$ is a basis of the vector space of univariate polynomials in $U$ of degree at most $\le p-1$.
Note that $(-1)^{p-1}s_{\BP\setminus \ens{x_{i,j}}}=(-1)^{\mu_{i}+\cdots+\mu_m-j-1}$. So, the family $$\ens{(-1)^{\mu_{i}+\cdots+\mu_m-j-1}q_{i,j}\mid i=1,\ldots,m, j=0,\ldots \mu_i-1}$$ is the coordinates in the basis $\mathcal{B}_{\BP,p-1}(U)$ of a polynomial $Q(U)$ (necessarily unique) of degree at most $p-1$ such that $Q^{[j]}(x_i)=q_{i,j}$, applying Proposition \[interpolationdata\].
Multi Sylvester double sums {#sec:fonda}
===========================
We introduce in subsection \[subsec:defmulti\] multi Sylvester double sums and study their properties, using the Hermite interpolation for symmetric multivariate polynomials. In subsection \[subsec:mult\] we compute the multi Sylvester double sums and Sylvester double sums for indices $(k,\ell)$ with $k+\ell\ge q$. In subsection \[subsec:fonda\] we prove the fundamental property of Sylvester double sums, i.e. that Sylvester double sums indexed by $k,\ell$, depend only (up to a constant) on $j=k+\ell<p$. This was already known in the simple roots case but even in this case our proof is new.
Definition of multi Sylvester double sums {#subsec:defmulti}
-----------------------------------------
The idea of replacing the variable $U$ by a block of indeterminates to define multi Sylvester double sums is directly inspired from [@KSV].
The [multi Sylvester double sum]{}, for $(k,\ell)$ a pair of natural numbers with $ k+\ell=j$, is the polynomial $\MSylv^{k,\ell}(P,Q)(\BU)$, where $\BU$ is a block of indeterminates of cardinality $p-j$, $$\begin{aligned}
\MSylv^{k,\ell}(P,Q)(\BU)=
\sum_{\genfrac{}{}{0pt}{}{\BK \subset_k \BP}
{\BL \subset_\ell \BQ}} s_{\BK} s_{\BL} \frac{ V[(\BQ \setminus \BL) \|
(\BP \setminus \BK)]V[\BL \| \BK \| \BU)}{ V[\BP] V[\BQ] V(\BU)}\end{aligned}$$
In particular $$\begin{aligned}
\MSylv^{j,0}(P,Q)(\BU)=
\sum_{\BK \subset_j \BP} s_{\BK} \frac{ V[\BQ\|(\BP \setminus \BK)]}{V[\BQ] }
\frac{ V[\BK \| \BU)}{V[\BP] V(\BU)}\end{aligned}$$
The following proposition gives the relationship between multi Sylvester double sums and Sylvester double sums.
\[lienMsylvSylv\] Denoting $\BU=U\|\BU'$ with $\BU'$ a block of $p-j-1$ indeterminates,
$\Sylv^{k,\ell}(P,Q)(U)$ is the coefficient of $\displaystyle{\prod_{U'\in \BU'}^{}U^{\prime j}}$ in $\MSylv^{k,\ell}(P,Q)(\BU)$.
The proof of Proposition \[lienMsylvSylv\] is based on the following Lemma.
\[coeff0\] $ V[\BK\|U)$ is the coefficient of $\displaystyle{\prod_{U'\in \BU'} U'^k}$ in $\displaystyle{\frac{ V[\BK\|U\|\BU')}{V(U\|\BU')}}$.
$$\begin{aligned}
\frac{ \partial^{[\BK]}V(X_\BK\|U\|\BU')}{V(U\|\BU')} &=\frac{ \partial^{[\BK]}(V(X_\BK\|U)\Pi(\BU',X_\BK)\Pi(\BU',U)V(\BU'))}{\Pi(\BU',U)V(\BU')}\\
&= \partial^{[\BK]}(V(X_\BK\|U)\Pi(\BU',X_\BK))\end{aligned}$$
Noting that $$\begin{gathered}
\partial^{[\BK]}
\left(V(X_\BK\|U)\Pi(\BU',X_\BK)\right)=
\partial^{[\BK]}V(X_\BK\|U)
\times\Pi(\BU',X_\BK)
+\sum_r V_r(X_\BK,U)\Pi_r(\BU',X_\BK)\end{gathered}$$ where each $\Pi_r(\BU',X_\BK)$ is obtained by partial derivation of $\Pi(\BU',X_\BK)$ with respect to at least one variable in $X_\BK$, it is clear that the degree of some $U'\in \BU'$ in $\Pi_r(\BU',X_\BK)$ is less than $k$. The claim follows, substituting $\BK$ to $X_\BK$.
The coefficient of $\displaystyle{\prod_{U'\in \BU'}^{}U^{\prime j}}$ in $\displaystyle{\frac{ V[\BL\|\BK\|U\|\BU')}{V(U\|\BU')}}$ is $V[\BL\|\BK \| U)$ by Lemma \[coeff0\]. The coefficient of $\displaystyle{\prod_{U'\in \BU'}^{}U^{\prime j}}$ in $\MSylv^{k,\ell}(P,Q)(\BU)$ is $$\sum_{\genfrac{}{}{0pt}{}{\BK \subset_k \BP}
{\BL \subset_\ell \BQ}} s_\BK s_\BL \frac{V[(\BQ\setminus\BL) \| (\BP\setminus\BK)] V[\BL\| \BK\| U)}{V[\BP] V[\BQ]}\
=\Sylv^{k,\ell}(P,Q)(U)$$ by Proposition \[lemme2bis\].
Computation of (multi) Sylvester double sums for ${j\ge q}$ {#subsec:mult}
-----------------------------------------------------------
\[enplus\] If $q\le j < p$ $$\MSylv^{j,0}(P,Q)(\BU)=(-1)^{j(p-j)}\prod_{U\in \BU}Q(U)$$
The polynomial $\displaystyle{\prod_{U\in \BU}Q(U)}$ is a symmetric polynomial in $\BU$ of multidegree $q,\ldots,q$, so at most $j,\ldots,j$. Its coordinates in the basis $\mathcal{B}_{\BP,j}(\BU)$ are, for $\BK\subset_j\BP$, $(-1)^{j(p-j)}s_\BK h(\BP\setminus\BK)$ where $$h(X_{\BP\setminus\BK})=\displaystyle{\partial^{[\BP\setminus\BK]}\left(V(X_{\BP\setminus\BK})\prod_{X\in X_{\BP\setminus\BK}}Q(X)\right)}$$ by Proposition \[interpolationdata\], and moreover $$h(\BP\setminus\BK)=\frac {V[\BQ\|(\BP \setminus \BK) ]}{V[\BQ]}$$ by Lemma \[prodpratique\].2.
So, the polynomials $\MSylv^{j,0}(P,Q)(\BU)$ and $\displaystyle{(-1)^{j(p-j)}\prod_{U\in \BU}Q(U)}$ have the same coordinates in the basis $\mathcal{B}_{\BP,k}(\BU)$ and are equal.
As a corollary
\[preouf\]
1. $\Sylv^{p-1,0}(P,Q)(U)=(-1)^{p-1} Q(U)$
2. For any $q<j<p-1$, $\Sylv^{j,0}(P,Q)(U)=0$
3. $\Sylv^{q,0}(P,Q)(U)=(-1)^{q (p-q)} Q(U)$
1. For $j=p-1$ Proposition \[enplus\] is exactly $$\Sylv^{p-1,0}(P,Q)(U)=(-1)^{p-1}Q(U).$$.
2. If $q<j<p-1$, denoting $\BU=U\|\BU'$ with $\BU'$ a block of $p-j-1$ indeterminates, the coefficient of $\displaystyle{\prod_{U'\in \BU'} U'^j}$ in $\displaystyle{\prod_{U'\in \BU}Q(U')}$ is equal to $0$, so $\Sylv^{j,0}(P,Q)(U)=0$ applying Proposition \[enplus\].
3. From Proposition \[enplus\] and Proposition \[lienMsylvSylv\], denoting $\BU=U\|\BU'$ with $\BU'$ a block of $p-q-1$ indeterminates, we know that $\Sylv^{q,0}(P,Q)(U)$ is equal to the coefficient of $\displaystyle{\prod_{U'\in \BU'} U'^q}$ in $(-1)^{q(p-q)}Q(U)\prod_{U'\in \BU'}Q(U')$. This coefficient is exactly $(-1)^{q(p-q)}Q(U)$.
\[jplusgrandqueq\] If $\ell \le q\leq k+\ell=j<p$ then $$\MSylv^{k,\ell}(P,Q,\BU)=(-1)_{}^{\ell(p-j)}\comb q \ell \MSylv^{j,0}(P,Q,\BU)$$
Let $\BL\subset_\ell\BQ$ and $\BU'=(U'_1,\ldots,U'_{p-k})$; the polynomial $\displaystyle{\frac{V[(\BQ\setminus\BL)\|\BU')}{V(\BU')}}$ is a symmetric polynomial in the indeterminates $\BU'$ of degree at most $q-\ell \leq k$ in each indeterminate $U'_i, 1\leq i\leq p-k$. So, we can write this polynomial in the basis $\mathcal{B}_{\BP,k}(\BU')$ $$\frac{V[(\BQ\setminus\BL)\|\BU')}{V(\BU')}=\sum_{\BK\subset_k\BP}g_\BK\frac{V[\BK\|\BU')}{V[\BP]V(\BU')}$$ where, by Proposition \[interpolationdata\] $$g_\BK=
(-1)^{k(p-k)}s_\BK V[(\BQ\setminus\BL)\|(\BP\setminus\BK)].$$ We deduce from this $${V[(\BQ\setminus\BL)\|\BU')}=\sum_{\BK\subset_k\BP}(-1)^{k(p-k)}s_\BK{V[(\BQ\setminus\BL)\|(\BP\setminus\BK)]}\frac{V[\BK\|\BU')}{V[\BP]}$$ We replace $\BU'$ by $\BU'=Y_\BL\|\BU$, where $\BU$ is a set of $p-j$ indeterminates, derivate with respect to $\partial^{[\BL]}$ and replace $Y_\BL$ by $\BL$; we obtain $$\begin{aligned}
\label{10}
{V[(\BQ\setminus\BL)\|\BL\|\BU)}&=\sum_{\BK\subset_k\BP} (-1)^{k(p-k})s_\BK{V[(\BQ\setminus\BL)\|(\BP\setminus\BK)]}\frac{V[\BK\|\BL\|\BU)}{V[\BP] }\\&=(-1)^{k\ell} (-1)^{k(p-k)}\sum_{\BK\subset_k\BP}s_\BK{V[(\BQ\setminus\BL)\|(\BP\setminus\BK)]}\frac{V[\BL\|\BK\|\BU)}{V[\BP] }\end{aligned}$$ As $$V[(\BQ\setminus\BL)\|\BL\|\BU)=
s_\BL
V[\BQ\|\BU),$$ we have $${V[\BQ\|\BU)}=\sum_{\BK\subset_k\BP}(-1)^{k(p-j)}s_\BK s_\BL{V[(\BQ\setminus\BL)\|(\BP\setminus\BK)]}\frac{V[\BL\|\BK\|\BU)}{V[\BP]}$$ and $$\frac{V[\BQ\|\BU)}{V(\BU)}=\sum_{\BK\subset_k\BP}(-1)_{}^{k(p-j)}s_\BK s_\BL V[(\BQ\setminus\BL)\|(\BP\setminus\BK)]\frac{V[\BL\|\BK\|\BU)}{V[\BP] V(\BU)}.$$
The polynomial $\displaystyle{\frac{V[\BQ\|\BU)}{V(\BU)}}$ is a symmetric polynomial in the indeterminates $\BU$ of degree at most $q\le j$ in each indeterminate $U_i, 1\leq i\leq p-j$. So, we can write it in the basis $\mathcal{B}_{\BP,j}(\BU)$ $$\frac{V[\BQ\|\BU)}{V(\BU)}=\sum_{\BW\subset_j\BP} g_\BW\frac{V[\BW\|\BU)}{V[\BP]V(\BU)}$$ where by Proposition \[interpolationdata\] $$g_\BW=
(-1)^{j(p-j)}
s_\BW V[\BQ\|(\BP\setminus\BW)].$$ So $$\sum_{\BW\subset_j\BP} s_\BW V[\BQ\|(\BP\setminus\BW)]V[\BW\|\BU)=\sum_{\BK\subset_k\BP}(-1)_{}^{\ell(p-j)}s_\BK s_\BL V[(\BQ\setminus\BL)\|(\BP\setminus\BK)]V[\BL\|\BK\|\BU).$$ It follows $$\begin{gathered}
\sum_{\genfrac{}{}{0pt}{}{\BW \subset_j \BP}
{\BL \subset_\ell \BQ}} s_\BW V[\BQ\|(\BP\setminus\BW)] V[\BW\|\BU)=(-1)_{}^{\ell(p-j)}\sum_{\genfrac{}{}{0pt}{}{\BK \subset_k \BP}
{\BL \subset_\ell \BQ}} s_\BK s_\BL V[(\BQ\setminus\BL)\|(\BP\setminus\BK)] V[\BL\|\BK\|\BU)\end{gathered}$$ $$\begin{gathered}
\comb q \ell \sum_{\BW\subset_j\BP} s_\BW V[\BQ\|(\BP\setminus\BW)] V[\BW\|\BU)=(-1)_{}^{\ell(p-j)}\sum_{\genfrac{}{}{0pt}{}{\BK \subset_k \BP}
{\BL \subset_\ell \BQ}}s_\BK s_\BL V[(\BQ\setminus\BL)\|(\BP\setminus\BK)] V[\BL\|\BK\|\BU)\end{gathered}$$ $$\begin{gathered}
\comb q \ell \frac{\sum_{\BW\subset_j\BP} s_\BW V[\BQ\|(\BP\setminus\BW)]V[\BW\|\BU)}{V[\BP]V[\BQ]V(\BU)}=(-1)_{}^{\ell(p-j)}\sum_{\genfrac{}{}{0pt}{}{\BK \subset_k \BP}
{\BL \subset_\ell \BQ}}s_\BK s_\BL\frac{V[(\BQ\setminus\BL)\|(\BP\setminus\BK)]V[\BL\|\BK\|\BU)}{V[\BP] V[\BQ] V(\BU)}\end{gathered}$$ and $$\begin{aligned}
\MSylv^{k,\ell}(P,Q,\BU)=(-1)_{}^{\ell(p-j)}\comb q \ell \MSylv^{j,0}(P,Q,\BU) .&\qedhere\end{aligned}$$
\[lienentreSylv\]For $q\leq j<p$, $$\Sylv^{k,\ell}(P,Q)=(-1)^{\ell(p-j)}\comb q \ell \Sylv _{}^{j,0}(P,Q)$$
Immediate using Proposition \[jplusgrandqueq\] and Proposition \[lienMsylvSylv\].
\[ouf\]
1. For any $(k,\ell)$ with $q=k+\ell$, $$\Sylv^{k,\ell}(P,Q)(U)=(-1)^{k(p-q)} \comb q k Q$$
2. For any $(k,\ell)$ with $\ell \le q,j=k+\ell$ with $q<j<p-1$, $$\Sylv^{k,\ell}(P,Q)(U)=0$$
3. For any $(k,\ell)$ with $\ell \le q,k+\ell=p-1$, $$\Sylv^{k,\ell}(P,Q)(U)=(-1)^{k}\comb q \ell Q(U)$$
Follows from Corollary \[lienentreSylv\] and Proposition \[preouf\].
Fundamental property of (multi) Sylvester double sums {#subsec:fonda}
-----------------------------------------------------
This section is essentially devoted to the proof of Theorem \[theo4\], which is a fundamental property of Sylvester double sums: up to a constant Sylvester double sums $\Sylv^{r,j-r}(P,Q)$ depend only on $j$. Such a result has been already given for $q\leq j<p$ by Corollary \[lienentreSylv\].
\[theo4\] If $k \in \mathbb{N}$, $\ell \in \mathbb{N}$, $k+\ell=j<q<p$ $$\Sylv^{k,\ell} (P,Q)(U) =( - 1 )^{\ell(p-j)}\comb j \ell\Sylv^{j,0}(P, Q)(U).$$
We, in fact, prove Theorem \[theo4\] as a corollary of a multivariate version (Theorem \[theo4mult\]). The proof of Theorem \[theo4mult\] uses in an essential way the Exchange Lemma coming from [@KSV].
\[theo4mult\] If $k \in \mathbb{N}$, $\ell \in \mathbb{N}$, $k+\ell=j<q<p$, and $\BU$ a set of $p-j$ indeterminates, $$\MSylv^{k,\ell} (P,Q)(\BU) =( - 1 )^{\ell(p-j)}\comb j \ell \MSylv^{j,0}(P, Q)(\BU).$$
To prove Theorem \[theo4mult\] for $j<q$, we need a lemma
\[coeff\] Let $\BK \subset_k \BP, \BL \subset_\ell \BQ$ and $\BU=U_1,\ldots,U_u$ an ordered set of variables. Then $(-1)^{u(u-1)/2}V[\BL\|\BK]$ is the coefficient of the leading monomial $$\prod_{i=1}^u U_i^{k+\ell+u-i}$$ of $V[\BL\|\BK\|\BU)$ with respect to the lexicographical ordering.
$$\begin{aligned}
V[\BL\|\BK\|\BU)&=\partial^{[\BK]}\partial^{[\BL]}(V(Y_\BL\|X_\BK\|\BU))(\BK,\BL,\BU)\\
&=V(\BU)\partial^{[\BK]}\partial^{[\BL]}(V(Y_\BL\|X_\BK)\Pi(\BU,Y_\BL\|X_\BK))(\BK,\BL)
.\end{aligned}$$
The coefficient of $\displaystyle{\prod_{i=1}^u U_i^{k+\ell+u-i}}$ in $V[\BL\|\BK\|\BU)$ is $(-1)^{u(u-1)/2}$ multiplied by the coefficient of $\displaystyle{\prod_{i=1}^u U_i^{k+\ell}}$ in $\partial^{[\BK]}\partial^{[\BL]}(V(Y_\BL\|X_\BK)\Pi(\BU,X_\BK\|Y_\BL)(\BK,\BL)$. This coefficient is $$\partial^{[\BK]}\partial^{[\BL]}V(Y_\BL\|X_\BK)(\BK,\BL)=V[\BL\|\BK];$$ indeed if any derivation is done on $\Pi(\BU,X_\BK\|Y_\BL)$, with respect to $\BK$ or $\BL$, the degree in at least one indeterminate $U_i\in\BU$ decreases strictly.
We have $j<q$. Let $\BU'$ be a block of $p-\ell$ indeterminates. On one hand, $$\begin{array}{rcl}
\displaystyle{\sum_{\BT\subset_\ell\BP }\Pi(X_{\BP\setminus \BT},Y_\BQ)\frac{\Pi(\BU',X_\BT)}{\Pi(X_{\BP\setminus \BT},X_\BT)}}&=&\displaystyle{\sum_{\BT\subset_\ell\BP }\frac{V(Y_\BQ\|X_{\BP\setminus \BT})V(X_\BT\|\BU')}{V(Y_\BQ)V(\BU')V(X_\BT\|X_{\BP\setminus \BT})}}\\
&=&\displaystyle{(-1)^{\ell(p-\ell)}\sum_{\BT\subset_\ell\BP }s_\BT\frac{V(Y_\BQ\|X_{\BP\setminus \BT} ) V(X_\BT\|\BU')}{V(Y_\BQ)V(\BU')V(X_\BP)}}.
\end{array}$$
On the other hand, $$\begin{array}{rcl}
\displaystyle{\sum_{\BL\subset_\ell\BQ} \Pi(X_\BP,Y_{\BQ\setminus \BL})\frac{\Pi(\BU',Y_\BL)}{\Pi(Y_\BL,Y_{\BQ\setminus \BL})}}
&=&
\displaystyle{\sum_{\BL\subset_\ell\BQ}\frac{V(Y_{\BQ\setminus \BL}\|X_\BP)V(Y_\BL\|\BU')}{V(X_\BP)V(\BU')V(Y_{\BQ\setminus \BL}\|Y_\BL)}}\\
&=&
\displaystyle{\sum_{\BL\subset_\ell\BQ}s_\BL\frac{V(Y_{\BQ\setminus \BL}\|X_\BP)V(Y_\BL\|\BU')}{V(X_\BP)V(\BU')V(Y_\BQ)}}
\end{array}$$ From the Exchange Lemma in [@KSV], we can write $$\begin{aligned}
\label{equation1}
\sum_{\BT\subset_\ell\BP }\Pi(X_{\BP\setminus \BT},Y_\BQ)\frac{\Pi(\BU',X_\BT)}{\Pi(X_{\BP\setminus \BT},X_\BT)}=\sum_{\BL\subset_\ell\BQ} \Pi(X_\BP,Y_{\BQ\setminus \BL})\frac{\Pi(\BU',Y_\BL)}{\Pi(Y_\BL,Y_{\BQ\setminus \BL})}\end{aligned}$$ So, we deduce from (\[equation1\]) $$\begin{aligned}
\sum_{\BT\subset_\ell\BP }s_\BT\frac{V(Y_\BQ\|X_{\BP\setminus \BT})V(X_\BT\|\BU')}{V(Y_\BQ)V(\BU')V(X_\BP)}=
(-1)^{\ell(p-\ell)}\sum_{\BL\subset_\ell\BQ}s_\BL\frac{V(Y_{\BQ\setminus \BL}\|X_\BP)V(Y_\BL\|\BU')}{V(X_\BP)V(\BU')V(Y_\BQ)}\end{aligned}$$ $$\begin{aligned}
\label{depart}
\sum_{\BT\subset_\ell \BP}s_\BT V(Y_\BQ\|X_{\BP\setminus \BT})
V(X_\BT \|\BU') =(-1)^{\ell(p-\ell)}\sum_{\BL\subset_\ell \BQ}s_\BL V( Y_{\BQ\setminus \BL}\|X_\BP) V(Y_\BL\|\BU')\end{aligned}$$ Hence, derivating with respect to $\BQ$ and substituting $\BQ$ to $Y_\BQ$, $$\begin{aligned}
\label{depart1}
\sum_{\BT\subset_\ell \BP}s_\BT V[\BQ\|X_{\BP\setminus \BT})
V(X_\BT \|\BU') =(-1)^{\ell(p-\ell)}\sum_{\BL\subset_\ell \BQ}s_\BL V[(\BQ\setminus \BL) \|X_\BP)V[\BL\|\BU')\end{aligned}$$
We fix $\BK\subset_k\BP$. The total degree with respect to $X_\BK$ of $V[\BQ\setminus \BL\|X_\BP ] V[\BL\|\BU')$ is $$d_1=\comb k 2+k (p+q-j).$$ Denoting, for any $\BT\subset_\ell\BP$, $c$ the cardinality of $\BK\cap \BT$, we note that the cardinality of $(\BP\setminus \BT)\cap \BK$ is $k-c$.
So, the total degree with respect to $X_\BK$ of $V[\BQ\|X_{\BP\setminus \BT})
V(X_\BT \|\BU')$ is $$d_{2,c}=\comb {k-c} 2 +(k-c)(p+q-j+c)+\comb c 2+c(p-c),$$ i.e. $$d_{2,c}=\comb k 2+k(p+q-j)-c(q-j+c)$$ and $$d_1-d_{2,c}^{}=c(q-j+c)$$ So $d_{2,c}<d_1$ if $c>0$ and $d_{2,c}=d_1$ if $c=0$ i.e. if $\BT \subset \BP \setminus \BK$. This implies that subsets $\BT$ which intersect $\BK$ don’t contribute to the homogeneous part of total degree $d_1$ in $X_\BK$ on the left side of (\[depart1\]).
Note that, if $\BT\subset_\ell\BP\setminus\BK$,
$$V[\BQ\|X_{\BP \setminus \BT})=r_{\BK,\BT} V[\BQ\|X_{\BP \setminus (\BK\cup \BT)}\| X_{\BK}),$$ where $r_{\BK,\BT}$ is the signature of the permutation $\rho_{\BK,\BT}$ taking the ordered set ${\BP}\setminus {\BT}$ to the ordered set $(\BP\setminus ({\BK}\cup {\BT}))\| {\BK}$.
We can also write $$V[(\BQ\setminus\BL)\|X_\BP)=s_\BK V[(\BQ\setminus\BL)\|X_{\BP\setminus \BK}\|X_\BK)$$
If $X_\BK=X_{u_1},\ldots, X_{u_k}$, taking the coefficient of $\prod_{i=1}^k X_{u_k}^{k-i+p+q-j}$ in both sides of (\[depart1\]) gives, by Lemma \[coeff\] $$\begin{gathered}
\label{suite}
\sum_{\BT\subset_\ell \BP\setminus \BK}r_{\BK,\BT}s_\BT
V[\BQ\|X_{\BP\setminus (\BK\cup \BT)} )
V(X_\BT \|\BU') =(-1)^{\ell(p-\ell)}\sum_{\BL\subset_\ell \BQ}s_\BK s_\BL V[(\BQ\setminus \BL)\|X_{\BP\setminus \BK} ) V[\BL\|\BU').\end{gathered}$$ Derivating both sides of (\[suite\]) with respect to $\partial^{[\BP \setminus \BK]}$ and replacing $X_{\BP\setminus \BK}$ by $\BP \setminus \BK$, followed by replacing $\BU'$ by $X_\BK\| \BU$, where $\BU$ is a set of $p-j$ indeterminates, derivating with respect to $\partial^{[\BK]}$ and replacing $X_\BK$ by $\BK$ gives $$\begin{gathered}
\label{suitebis}
\sum_{\BT\subset_\ell \BP\setminus \BK}r_{\BK,\BT}s_\BT V[\BQ\|(\BP\setminus (\BK\cup \BT))]
V[\BT \|\BK\|\BU) =(-1)^{\ell(p-\ell)}\sum_{\BL\subset_\ell \BQ}s_\BK s_\BL V[(\BQ\setminus \BL)\|(\BP\setminus \BK)] V[\BL\|\BK\|\BU).\end{gathered}$$ Summing with respect to $\BK$, ve get $$\sum_{\genfrac{}{}{0pt}{}{\BK \subset_k \BP}
{\BT\subset_\ell \BP\setminus \BK}}
r_{\BK,\BT}s_\BT V[\BQ\|(\BP\setminus (\BK\cup\BT))]
V[\BT \|\BK\|\BU) =(-1)^{\ell(p-\ell)}\MSylv^{k,\ell}(P,Q)(U)V[\BP] V[\BQ]V(\BU).$$ Denote $\BW$ the set $\BK\cup\BT$ ordered by the induced order on $\BP$. Let $\tau_{\BK,\BT}$ be the permutation sending the ordered set $\BP\setminus \BW \|\BW$ to the ordered set $\BP\setminus \BW \| \BT\|\BK$ and $t_{\BK,\BT}$ its signature. We deduce $$\sum_{\genfrac{}{}{0pt}{}{\BK \subset_k \BP}
{\BT\subset_\ell \BP\setminus \BK}}
t_{\BK,\BT}r_{\BK,\BT}s_\BT V[\BQ\|(\BP\setminus \BW) ]
V[\BW\|\BU)=
(-1)^{\ell(p-\ell)}\MSylv^{k,\ell}(P,Q)(U)V[\BP] V[\BQ]V(\BU).$$
We remark that $t_{\BK,\BT}r_{\BK,\BT}s_{\BT}=(-1)^{k\ell}s_\BW$. Indeed, denoting $\iota_{\BK,\BT}$ the permutation sending the ordered set $(\BP\setminus \BW) \| \BT\|\BK$ to the ordered set $(\BP\setminus \BW) \| \BK\|\BT$, with signature $(-1)^{k\ell}$, and by $\rho'_{\BK,\BT}$ the permutation sending the ordered set $(\BP\setminus\BT)\|\BT$ to the ordered set $(\BP\setminus(\BK\cup \BT))\|\BK\|\BT$, with signature $r_{\BK,\BT}$, we have the follwing sequence of permutations $$\begin{array}{rrcl}
\sigma_\BW:& \BP&\longleftrightarrow &(\BP\setminus\BW)\|\BW\\
\tau_{\BK,\BT} :& (\BP\setminus\BW)\|\BW&\longleftrightarrow &(\BP\setminus\BW)\|\BT\|\BK\\
\iota_{\BK,\BT} :& (\BP\setminus\BW)\|\BT\|\BK&\longleftrightarrow &(\BP\setminus\BW)\|\BK\|\BT\\
{\rho'}_{\BK,\BT}^{-1}:&(\BP\setminus\BW)\|\BK\|\BT&\longleftrightarrow&(\BP\setminus\BT)\|\BT\\
\sigma_\BT^{-1}:&(\BP\setminus\BT)\|\BT&\longleftrightarrow&\BP\quad\quad,
\end{array}$$ with $\sigma_\BT \circ {\rho'}_{\BK,\BT}^{-1}\circ \iota_{\BK,\BT} \circ \tau_{\BK,\BT} \circ \sigma_\BW^{-1}={\rm Id}$.
Noting that there are $\comb j \ell$ ways of decomposing $\BW\subset_j \BP$ as $\BW=\BK\cup \BT$, we get $$\MSylv^{k,\ell} (P,Q)(\BU) = ( - 1 )^{\ell(p-j)} \comb j \ell\MSylv^{j,0}(P, Q)(\BU).\qedhere$$
Theorem \[theo4\] is an immediate consequence of Theorem \[theo4mult\], by applying Proposition \[lienMsylvSylv\].
Sylvester double sums and remainders {#sec:remainders}
====================================
In Section \[sec:remainders\] we give a relationship between the Sylvester double sums of $P,Q$ and those of $Q,R$ where $R$ is the opposite of the remainder of $P$ by $Q$ in the Euclidean division.
We are now dealing with not necessarily monic polynomials.
\[defnonmonic\] Let $P$ be a polynomial of degree $p$ which leading coefficient is denoted $\lc(P)$. Let $Q$ be a polynomial of degree $q$ which leading coefficient is denoted $\lc(Q)$.
Let $(k,\ell)$ with $j=k+\ell\le p$ be a pair of natural numbers. We define $$\Sylv^{k,\ell}(P,Q)(U)=
{\lc(P)^{q-j}\lc(Q)^{p-j}}\Sylv^{k,\ell}\left(\frac P {\lc(P)},\frac Q{\lc(Q)}\right)(U)$$
\[proprietes\] [Note that if $k \in \mathbb{N}$, $\ell \in \mathbb{N}$,$\ell \le q$, $k+\ell=j<q$ $$\Sylv^{k,\ell} (P,Q)(U) = \left( - 1 \right)^{\ell(p-j)} \comb j \ell\Sylv^{j,0}(P, Q)(U)$$ follows immediately from Theorem \[theo4\] and Definition \[defnonmonic\]. ]{}
We use Notation \[notmultiset\] to define the ordered sets $\BP$ and $\BQ$ representing the multisets of roots of $P$ and $Q$.
Rewriting Lemma \[prodpratique\] in the non monic case, we get Lemma \[prodpratiquebis\].
\[prodpratiquebis\]
1. For $\BL\subset_\ell\BQ$, defining $$f(Y_{\BQ\setminus \BL})=(-1)^{p(q-\ell)}\partial^{{[\BQ\setminus\BL]}}\left(V(Y_{\BQ\setminus\BL})\prod_{Y \in Y_{\BQ\setminus\BL}}P(Y)\right),$$ we have $$f(\BQ\setminus\BL)= \lc(P)^{q-\ell}\frac{V[(\BQ\setminus\BL)\|\BP)]}{V[\BP]}$$
2. For $\BK\subset_k\BP$, defining $$g( X_{\BP\setminus\BK})=\partial^{{[\BP\setminus\BK]}}\left(V(X_{\BP\setminus\BK})\prod_{X\in X_{\BP\setminus\BK}}Q(X) \right),$$ we have $$g{(\BP\setminus\BK)}= \lc(Q)^{p-k}\frac{V[\BQ\|(\BP\setminus\BK)]}{V[\BQ]}$$
Similarly, reewriting Proposition \[preouf\] in the non monic case, we get Proposition \[preoufbis\].
\[preoufbis\]
1. $\Sylv^{p-1,0}(P,Q)(U)=(-1)^{p-1}\lc(P)^{q-p+1} Q(U)$
2. For any $q<j<p-1$, $\Sylv^{j,0}(P,Q)(U)=0$
3. $\Sylv^{q,0}(P,Q)(U)=(-1)^{q (p-q)} \lc(Q)^{p-q-1} Q(U)$
We proceed now to the proof of Proposition \[prorecurrence\] wich is the main result of Section \[sec:remainders\].
\[prorecurrence\] Let $R=-\rm{Rem}(P,Q).$ If $j\in \mathbb{N}$, $j< q$
- If $R=0$, $\Sylv^{j,0}(P,Q)(U) = 0$.
- If $R\not=0$, $\Sylv^{j,0}(P,Q)(U) = (-1)^{q(p-q)}\lc(Q)^{p-r}\Sylv^{j,0}(Q,R)(U)$
The following elementary lemma plays a key role in the proof of Proposition \[prorecurrence\].
\[remarque\] Let $R=-\rem(P,Q).$ For every $y_{i,j}\in \BQ$, $0\leq j'< j,$ $$P^{[j']}(y_i)=-R^{[j']}(y_i)$$
Write $P=CQ-R$, derivate $j'$ times and evaluate at $y_i$.
If $R=0$, $\displaystyle{\Sylv^{0,j}\left(\frac{P}{\lc(P)},\frac{Q}{\lc(Q)}\right)(U)=0}$ follows from Corollary \[Regal0\]. So, $$\Sylv^{j,0}(P,Q)(U)=(-1)^{j(p-j)}\Sylv^{0,j}(P,Q)(U)=0$$.
If $R\not=0$, let $r$ be the degree of $R$. Let $(z_1,\ldots,z_v)$ be an ordered set of the distinct roots of $R$ in an algebraic closure $\C$ of $\mathbb{K}$, with $z_i$ of multiplicity $\xi_i$, and, as in Notation \[notmultiset\], let $\BR$ be the multiset of roots of $R$, represented by the ordered set $$\BR=(z_{1,0},\ldots,z_{1,\xi_1-1},\ldots,z_{v,0},\ldots,z_{v,\xi_v-1}),$$ with $z_{i,j}=(z_i,j)$ for $0\leq j\leq \xi_i-1$, $\sum_{i=1}^v \xi_i=r.$
If $j\le q$, define for $\BL\subset_j \BQ$ $$\begin{array}{rcl}
f(Y_{\BQ\setminus\BL})&=&(-1)^{p(q-j)}\partial^{[\BQ\setminus \BL]}\left( V(Y_{\BQ\setminus\BL})\prod_{Y\in Y_{\BQ\setminus\BL}}P(Y) \right)\\
h(Y_{\BQ\setminus\BL})&=&\partial^{[\BQ\setminus \BL]}\left(V(Y_{\BQ\setminus\BL})\prod_{Y\in Y_{\BQ\setminus\BL}}R(Y) \right)
\end{array}$$ Note that $$f(\BQ\setminus\BL)=(-1)^{(p+1)(q-j)}h(\BQ\setminus\BL)$$ from Lemma \[remarque\].
So $$\begin{aligned}
\Sylv^{0,j}(P,Q)(U)&=
\displaystyle{\frac{\lc (P)^{q-j}\lc (Q)^{p-j}}{V[\BP]V[\BQ]}\sum_{\BL \subset_j\BQ}s_\BL V[(\BQ\setminus\BL)\|\BP]V[\BL\|U)}\\
&=\displaystyle{\frac{\lc (Q)^{p-j}}{V[\BQ]}\sum_{\BL \subset_j\BQ}s_\BL f(\BQ\setminus\BL)V[\BL\|U)}\hfill\mbox{ applying Lemma \ref{prodpratiquebis}.1}\\
&=\displaystyle{(-1)^{(p+1)(q-j)}\frac{\lc (Q)^{p-j}}{V[\BQ]}\sum_{\BL \subset_j\BQ}s_\BL h(\BQ\setminus\BL)V[\BL\|U)}\\
&=\displaystyle{(-1)^{(p+1)(q-j)}\frac{\lc (Q)^{p-j}}{V[\BQ]}\sum_{\BL \subset_j\BQ}s_\BL \frac{\lc (R)^{q-j}V[\BR\|(\BQ\setminus\BL)]}{V[\BR]}V[\BL\|U)\hfill\mbox{ applying Lemma \ref{prodpratiquebis}.2}}\\
&=\displaystyle{(-1)^{(p+1)(q-j)}\frac{\lc (Q)^{p-j}\lc (R)^{q-j}}{V[\BQ]V[\BR]}\sum_{\BL \subset_j\BQ}s_\BL V[\BR\|(\BQ\setminus\BL)]V[\BL\|U)}\\
&=\displaystyle{(-1)^{(p+1)(q-j)}\frac{\lc (Q)^{p-r}\lc (Q)^{r-j}\lc (R)^{q-j}}{V[\BQ]V[\BR]}\sum_{\BL\subset_j\BQ}s_\BL V[\BR\|(\BQ\setminus\BL)]V[\BL\|U)}\\
&=\displaystyle{(-1)^{(p+1)(q-j)}{\lc (Q)^{p-r}}\Sylv^{j,0}(Q,R)}\end{aligned}$$ The claim follows since $$\Sylv^{j,0}(P,Q)(U)=(-1)^{j(p-j)}\Sylv^{0,j}(P,Q)(U)$$ by Theorem \[theo4\] and $$(-1)^{j(p-j)}(-1)^{(p+1)(q-j)}=(-1)^{q(p-q)}.$$
Sylvester double sums and subresultants {#sec:subresdblsum}
=======================================
Finally we prove in this section that Sylvester double sums coincide (up to a constant) with subresultants, by an induction on the length of the remainder sequence of $P$ and $Q$.
This section is devoted to the proof of the link between double sums and subresultants which is known in the simple case (see [@ALPP; @AHKS; @RS]).
$\varepsilon_k=(-1)^{k(k-1)/2}$. The sign $\varepsilon_k$ is the signature of the permutation reversing the order i.e. sending $1,2,\ldots,k-1,k$ to $k, k-1,\ldots,2,1$. We have also $\varepsilon_k=1$ if $k\equiv 0,1 \mod 4$, $\varepsilon_k=-1$ if $k\equiv 2,3 \mod 4$. As a consequence $$\label{Eqvarepsilon}
\varepsilon_{i+1}=(-1)^{i}\varepsilon_{i}.$$ We have also $$\label{Eqvarepsilonbis}
\varepsilon_{i+j}=(-1)^{ij}\varepsilon_{i}\varepsilon_{j},$$ which follows from the fact that reversing $i+j$ numbers can be done in three steps: reversing the first $i$ ones, then the last $j$ one and placing the last $j$ numbers in front of the $i$ first.
The main theorem of this section is the following.
\[theoreme2\] Let $k \in \mathbb{N}$, $\ell \in \mathbb{N}$, $\ell \le q$, $k+\ell=j<p-1$ $$\Sylv^{k,\ell}(P,Q)(U)=(-1)^{k(p-j)}\varepsilon_{p-j}\comb j k\Sres_{j}(P,Q)(U).$$
[When $j=p-1$, $\Sres_{p-1}^{}(P,Q)(U)=Q(U)$ by convention; so, as $$\displaystyle{\Sylv^{k,\ell}(P,Q)(U)=(-1)^k\comb q \ell \lc(Q)^{q-p+1} Q(U)}\mbox{ for }k+\ell=p-1,\mbox{ we get }\hfill$$ $$\Sylv^{k,\ell}(P,Q)(U)=(-1)^k\comb q \ell \lc(Q)^{q-p+1} \Sres_{p-1}^{}(P,Q)(U).$$]{}
In order to prove Theorem \[theoreme2\], we use an induction on the length of the remainder sequence of $P$ and $Q$ based on Proposition \[prorecurrence\].
Before proving Theorem \[theoreme2\], we recall the following properties of subresultants.
\[rappel\] Let $R=-\rem (P,Q).$ $$\begin{array}{lll}
1.\,q<j<p-1&\quad\Sres_j(P,Q)(U)&=0\\
2.\, j=q&\quad\Sres_q(P,Q)(U)&=\varepsilon_{p-q}\lc(Q)^{p-q-1}Q(U)\\
3.\, j=q-1&\quad\Sres_{q-1}(P,Q)(U)&=\varepsilon_{p-q}\lc(Q)^{p-q+1}R(U)\\
4.\, j<q-1, R\not=0&\quad\Sres_j(P,Q)(U)&=\varepsilon_{p-q}\lc(Q)^{p-r}\Sres_j(Q,R)(U)\\
5.\, j<q-1, R=0 &\quad\Sres_j(P,Q)(U)&=0
\end{array}$$
All items follow from [@BPR] except the computation of $\Sres_{q-1}(P,Q)(U)$. $\Sres_{q-1}(P,Q)(U)$ is clearly equal to $\varepsilon_{p-q+2}\lc(Q)^{p-q+1}(-R(U))$ by replacing the row of $P$ by a row of $-R$ in the Sylvester-Habicht matrix, and reversing the order of its $p-q+2$ rows. Notice now that $\varepsilon_{p-q+2}=-\varepsilon_{p-q}$.
We also recall the following similar properties of Sylvester double sums.
\[rappelbis\] Let $R=-\rem(P,Q).$ Let $j\in \mathbb{N}$, $j<p-1$ $$\begin{array}{lll}
1.\,q<j<p-1&\Sylv^{j,0}(P,Q)(U)&=0,\\
2.\,j=q&\Sylv^{q,0}(P,Q)(U)&=(-1)^{q(p-q)} \lc(Q)^{p-q-1}Q(U)\\
3.\,j=q-1&\Sylv^{q-1,0} \left( P, Q \right)(U) &=(-1)^{(q-1)(p-q+1)+p-q} \lc(Q)^{p-q+1}R(U)\\
4.\,j<q-1,
R\not=0&\Sylv^{j,0}(P,Q)(U)&=(-1)^{q(p-q)} \lc(Q)^{p - r}
\Sylv^{j, 0} (Q,R)(U)
\\
5.\, j<q-1,
R=0&\Sylv^{j,0}(P,Q)(U)&=0,
\end{array}$$
All items follow from Proposition \[preoufbis\] except for the computation of $\Sylv^{q-1,0}(P,Q)(U)$. Using Proposition \[prorecurrence\] and Proposition \[preoufbis\] for $Q,R$, we obtain $$\Sylv^{q-1,0}(P,Q)(U)= (-1)^{q(p-q)}\lc (Q)^{p-r}\Sylv^{q-1,0}(Q,R)(U)=(-1)^{q-1+q(p-q)}\lc (Q)^{p-r+r-q+1}R(U)$$ It remains to remark that $(q-1)(p-q+1)+p-q=q(p-q)+q-1$.
The statement for $q\le j <p-1$ follows from Lemma \[rappel\] 1,2, Lemma \[rappelbis\] 1,2 and Theorem \[theo4\].
The statement for $j=q-1$ follows from Lemmma \[rappel\] 3, Lemma \[rappelbis\] 3, Theorem \[theo4\] and (\[Eqvarepsilon\]) since $\varepsilon_{p-q+1}=(-1)^{p-q}\varepsilon_{p-q}.$
For $j<q-1$ we first prove the special case $$\begin{aligned}
\label{star}
\Sylv^{j,0}(P,Q)=(-1)^{j(p-j)}\varepsilon_{p-j}\Sres_j(P,Q)\end{aligned}$$ The proof is by induction on the length of the remainder sequence of $P,Q$.
The basic case is when $Q$ divides $P$, i.e. $R=0$, and the claim is true by Lemma \[rappel\] 5, Lemma \[rappelbis\] 5.
Otherwise suppose, by induction hypothesis that $$\begin{aligned}
\Sylv^{j,0}(Q,R)=(-1)^{j(q-j)}\varepsilon_{q-j}\Sres_j(Q,R)\end{aligned}$$ Using Lemma \[rappel\] 5 and Lemma \[rappelbis\] 5 it remains to note that $$(-1)^{j(p-j)}(-1)^{j(q-j)}(-1)^{q(p-q)}=(-1)^{(q-j)(p-q)}$$ and conclude by (\[Eqvarepsilonbis\]) since $
\varepsilon_{p-j}=(-1)^{(q-j)(p-q)}\varepsilon_{p-q}\varepsilon_{q-j}.
$
The general case for $k,\ell$ now follows from Theorem \[theo4\].
The authors thank the referees for their relevant remarks. Special thanks to Daniel Perrucci for a very careful rereading.
|
---
abstract: 'The Status of BEPCII/BESIII upgrade is described, BEPCII is designed to reach a luminosity of $10^{33}/cm^{-2}s^{-1}$ at c. m. energy of 3.89GeV, and BESIII detector is fully rebuilt. The project is on track and is progressing well. The machine and detector are expected to commission together at the late half of 2007, and start to take some engineer run end of 2007 or beginning of 2008.'
author:
- Weiguo Li
title: 'BEPCII/BESIII Upgrade and the Prospective Physics'
---
Introduction
============
Beijing Electron Positron Collider (BEPC) started data taking in 1989, and they were upgraded in 1996, and the upgraded detector is called BESII [@besiidetector]. BES had taken data until the April of 2004, many results have been obtained using the data samples collected at J/$\psi$, $\psi(2S)$, $\psi(3770)$ and the scan data from 2.0 to 5.0 GeV.
After more 15 years data taking, the machine and detector are no more competitive to produce first grade physics at this energy region, especially after CESR reduced its energy to the $\tau$-charm energy region. After several years of preparation, the Chinese Government gave a green light to begin the BEPCII/BESIII upgrade at the end of 2003.
BEPCII is a two rings machine built in the existing tunnel, electrons and positrons are circulated in separate rings and only collider in the interaction point, multi-bunches and micro-beta are used to increase the machine luminosity. Super-conducting cavities and super-conducting quadruples are used. The Linac has to be upgraded also to increase its energy and currents, especially to meet the needs to inject the positions to the ring in a reasonable short time.
The detector has to be totally rebuilt, with a 43 layers small-cell main drift chamber (MDC), a time of flight (TOF) system, and an electro-magnetic calorimeter Made of CsI, a super-conducting magnet with a field of 1 tesla, and resistive plate chambers (RPC) are inserted in the magnet yoke to serve as the muon counters.
The whole project is scheduled to complete in 2008, with a goal that the luminosity of the machine should reach $3\times 10^{32}cm^{-2} s^{-1}$ at the end of 2008.
BEPCII
======
The main parameters of machine parameters are listed in Table I.
Energy Range 1-2.1 GeV
------------------ --------------------------------------------
Optimum energy 1.89 GeV
Luminosity $1\times 10^{33}cm^{-2}s^{-1}$ at 1.89 GeV
Injection Full energy inject.: 1.55-1.89 GeV
Position Injection $>$ 50 mA/min
Synchrotron mode 250 mA at 2.5 GeV
: BEPCII Design Goal.
Parameters Design Achieved BEPC
------------------------- ---------- ------------------------- -----------
Beam energy (GeV) 1.89 1.89($e^-$);1.89($e^+$) 1.55
$e^+$ Current 40 $>$ 63 4
$e^-$ Current 500 $>$ 500 50
Repetition rate (Hz) 50 25-50 25
$e^+$ emit. (mm-mrad) 1.60 $>$ 0.93 (1.89 GeV) 1.70
$e^-$ emit. (mm-mrad) 0.20 $>$ 0.30 (1.89 GeV) 0.58
$e^+$ energy spread (%) $\pm0.5$ $\pm$ 0.50 (1.89 GeV) $\pm$ 0.8
$e^-$ energy spread (%) $\pm0.5$ $\pm$ 0.55 (1.89 GeV) $\pm$ 0.8
: BEPCII Linac Achieved Performances.
To increase the injection energy and current to the rings, the Linac has been upgrades, the main systems are new acceleration tubes, new klystrons and modulators, new positron source, new electron gun and increase the beam repetition rate to 50 Hz, modified vacuum system, and other relevant modifications. The Linac has been debugged and tuned, some of the performances have already reached the designed values, and now the main efforts are to understand the operation of the Linac and to make the running condition to be stable and install the phase control system. The achieved performances of the Linac are listed in Table II.
---------------------------------------------------------------------------------------------------------------------
Parameters Unit BEPCII BEPC
--------------------------------------------------- ---------------- ------------------------ -----------------------
Operation energy GeV 1.0-2.1 1.0-2.5
Injection energy($E_{inj}$) GeV 1.55-1.89 1.3
Circumference(C) m 237.5 240.4
$\beta^*$-function at IP($\beta^*_x$/$\beta^*_y$) cm 100/1.5 120/5
Tunes($\nu_x/\nu_y/\nu_s$) 6.57/7.61/0.034 5.8/6.7/0.02
Hor. natural emittance($\theta_{x0}$) mm-mr 0.14 at 1.89 GeV 0.39 at 1.89 GeV
Dampling time ($\tau_{x}/\tau_{y}/\tau_{z}$) 25/25/12.5 at 1.89 GeV 28/28/14 at 1.89 GeV
RF frequency ($f_{rf}$) MHZ 499.8 199.533
RF voltage per ring ($V_{rf}$) MV 1.5 0.6-1.6
Number of bunches 93 2$\times$1
Bunch spacing m 2.4 240.4
Beam current colliding mA 910 1.89GeV 2$\times$35 1.89 GeV
Bunch length($\sigma_l$) cm 1.5 5
Impedance $|Z/n|_0$ $\Omega$ 0.2 4
Crossing angle mrad $\pm 11$ 0
Beam lifetime hrs. 2.7 6-8
Luminosity1.89 GeV $10^{31} 100 1
cm^{-2}s^{-1}$
---------------------------------------------------------------------------------------------------------------------
{width="135mm"}
The main parameters of the storage rings are listed in Table III. BEPCII storage rings have the following main systems: the super-conducting rf cavities and their power supplies and control system; the beam pipes; the magnets and their power supplies; the kickers; the beam instrumentations; the vacuum system; the control system. The dipole magnets of old ring are modified to be put in the outer ring. Up till now, most of the hardware devices are delivered, and the pieces not on site yet are scheduled to be delivered on time. The current plan is to assemble all the ring hardware pieces except the super-conducting quadruple magnets, for which more time is needed to get them tested and to measure the field with detector magnet together. The plan is to commission the whole rings without the quadruples first and provide some beam time for synchrotron use, then the quadruples are to be moved in the beam line to be tested together with other systems, after certain performance goals are met, especially certain amount of current should be reached so to verify the large currents can be run with certain luminosity and reasonable backgrounds to the detector, before the detector moves in to be debugged together with the machine. The machine and detector are expected to be commissioned together at the late half of 2007, and start to take some engineer run end of 2007 or beginning of 2008. The performance of BEPCII will be among the 2nd generation of $e^+e^-$ colliders, as shown in Figure 1.
BESIII Detector
===============
The BESIII detector [@besiiidetector] is a completely new detector. Figure 2 shows the schematic of BESIII detector.
{width="135mm"}
The beam pipe is made of two layers of Be pipes, with the thickness of 0.8 mm and 0.5 mm respectively, and a cooler is passed through between the pipes to take out the heat caused by the beams. MDC is the most inside detector component, it has 43 sense wires from inside to outside. To make room for the machine super-conducting magnet, the endplates are made from a cone shaped central section and a slightly inclined outer plate. The endplates are manufactured with the average hole accuracy better than 25 $\mu m$. Small cell structure is used with a half width of cell to be 6 mm for inner layers, and 8.1 mm for outer layers. The sense wire is made of 25 $\mu m$ tungsten, the field wire is made of 110 $\mu m$ gold plated Al. Totally there are about 28K wires. The gas chosen is $He(60)/C_3H_8(40)$ used by CLEOIII. All the wires have been strung and the tension and the leakage current of all wires are measured and satisfy the requirements. The electrons readout system adopts CERN HPTDC to measure the drift time, the prototype electron system is tested, and the performances meet the design requirements. A small chamber was tested in the KEK beam line, and the space resolution and dE/dx resolution are satisfactory. The expected chamber performances are, wire resolution to be better than 130 $\mu m$, and the momentum resolution is expected to better than 0.5% for 1 GeV tracks, two contributions from wire position resolution and multiple scattering are:$\sigma_{pt}/pt$ = $0.32\%/p$ and $ 0.37\%/\beta$ respectively. The outmost sense wire covers a polar angle of $cos(\theta)$ = 0.83, and the maximum acceptance is $cos(\theta)$ = 0.93, roughly reach 20th sense wire layer. The expected dE/dx resolution is about $6\%$.
Mounted on the MDC are two layers of TOF counters, each layer has 88 counters with a thickness of 5 cm and the two layers are staggered by half a counter to make a full coverage in the polar angle of $cos(\theta)$ = 0.82. There are photo-tubes at both ends of each counter to read the signals. And there are 48 single-layer counters in each side of the endcap region, and the signal is read out only at the inner end. The counters are mounted on the endcap EM calorimeter. BC408 and BC404 scintillators by Bicro are chosen for barrel and endcap respectively. The photo-tubes used are R2490-50 by Hamamatsu. From the test beam, 90 and 100 ps time resolutions are achieved for barrel and endcap respectively, as expected in the design.
Outside of TOF system is a CsI crystal EM calorimeter. The typical crystal has a dimension of $5\times 5 cm^2$ in the front face and $6.5\times 6.5 cm^2$ in the rear face, the length is 28 cm, corresponding to 15 radiation length. There are 5280 crystals in barrel and 960 in the endcap, 480 at each side. Two photo diodes (Hamamatsu S2744-08) with a photosensitive area of 10 mm $\times$ 20 mm are mounted on the rear face of each crystal to read the lights out. The prototype for readout electronics is tested to have a noise level of 1000 electrons. The expected energy resolution will be $2.5\%$ for 1 GeV photons. The crystals are made by Crismatec(France), Shanghai Institute of Ceramic and Beijing Hamamatsu. Most of the crystals are delivered and they met the specifications of dimensions, light yield and the radiation hardness. The mechanical structure of the calorimeter is designed such that there are no walls between crystals to reduce the dead material, the crystals are fixed to a support structure by 4 screws. $N_2$ will be flew in the crystal container to maintain the humidity inside, and the front-end electronics will be cooled by water. The EM calorimeter is scheduled to be assembled around the end of 2006.
Outside of EM calorimeter is the super-conducting magnet, its designed field is 1.0 tesla. The magnet uses the inner winding technique with the coil wound inside a Al support cylinder which in turn cooled by liquid He circled in the pipe welded on the outer surface of the support cylinder, the Al stabilized NbTi/Cu coil is made by Hitachi company, with the nominal operating current of about 3700 A, which has a stored energy of about 10 MJ. The field in the MDC volume has an uniformity of better than 5$\%$, the field will be measured with an accuracy of better than 0.25$\%$. The magnet is in the stage of testing the value box and after that to install the valve box with the magnet coil assembling, then the cryogenic system will be connected to the magnet and the whole magnet will be cooled and tested.
The return yoke is also served as the absorber of the muon detectors. The active muon detector is the Resistive Plate Chamber (RPC), there are 9 layers muon chambers in the barrel and 8 layers in the endcap, with two orthogonal strip readout alternatively layer by layer. The muon chambers are inserted in the steel slots in the yoke. The special feature of the RPC made in China is that there is no linseed oil used in the gap of RPC. The gas used is $Ar:C_2H_2F_4:Iso-Butane = 50:42:8$. All the RPC are installed in the yoke and the efficiencies and the dark currents measured are quite good, to the same level as those RPC with linseed oil used in other experiments.
The whole detector weights about 800 tons. And the detector hall will be air-conditioned, to control the temperature at $22\pm 2^oC$, and the humidity below $55\%$.
The electronics and trigger use pipelined arrangement, with a trigger latency of 6.4 $\mu$s. The trigger will use TOF, MDC and Muon information to make decisions. The simulation shows that good (almost 100$\%$) efficiency and a good background rejection can be achieved. The maximum trigger rate will be at J/$\psi$ energy with a total trigger rate of about 4000 Hz. At this energy, the good event rate will be about 2000 Hz. The DAQ bandwidth will be about 50 Mbytes per second. The readout system is based on VME. The offline analyzes package is under development, the preliminary version for event simulation and reconstruction is ready, the calibration codes and physics analyzes code are being worked on. The offline system will go through another two releases to get the package tested before the real data are taken.
The main designed detector performances are listed in Table IV, a comparison with CLEOc is made. Hopefully, some of the parameters may be better than designed.
Detector BESIII CLEOc
--------------- -------------------------------------- ------------------
$\sigma_{xy}$ = 130$\mu$m 90$\mu$m
MDC $\Delta p/p$(1 GeV) = 0.5$\%$ 0.5$\%$
$\sigma_{dE/dx}$ = 6-7$\%$ 6$\%$
EMC $\Delta E/\sqrt{E}$(1 GeV) = 2.5$\%$ 2.2$\%$
$\sigma_z$ = 0.6cm/$\sqrt{E}$ 0.5cm/$\sqrt{E}$
TOF $\sigma_T$ = 100-110ps Rich
$\mu$ counter 9 layers —
magnet 1.0 tesla 1.0 tesla
: BESIII Performances Compared with CLEOc.
Prospective Physics
===================
The rich physics topics at this energy region will require BESIII to take the data at J/$\psi$, $\psi(2S)$, $\psi(3770)$, and some energy point for $D_S$, also data at $\tau$ threshold and some scan data to measure hadronic cross-section in this energy region will be collected. The yearly yield of events are listed in Table V for different energy points the data are to be taken.
---------------- ------------- ------------------------ --------------- ------------------
Resonance Energy(GeV) Peak Lum. Physics Cross Nevents/yr
$10^{33}cm^{-2}s^{-1}$ Section(nb)
J/$\psi$ 3.097 0.6 3400 $10\times 10^9$
$\tau$ 3.670 1.0 2.4 $12\times 10^6$
$\psi(2S)$ 3.686 1.0 640 $3.2\times 10^9$
$D^0\bar{D^0}$ 3.770 1.0 3.6 $18\times 10^6$
$D^+D^-$ 3.770 1.0 2.8 $14\times 10^6$
$D_SD_S$ 4.030 0.6 0.32 $1.0\times 10^6$
$ D_SD_S$ 4.140 0.6 0.67 $2.0\times 10^6$
---------------- ------------- ------------------------ --------------- ------------------
The new X(1835) and other near threshold enhancements recently observed at BES will be studied with 100 times more data, X(3872) and states observed at 3940 MeV and 4260 MeV in other experiments may be studied in details. There are more scalars existed in this energy region to be accommodated in naive quark model, these states will be thoroughly studied.
The $\eta_C$, $\chi_{CJ}$ and $h_C$ can be studies with large statistics. The $\rho\pi$ puzzle will be studied with more decay channels and with better accuracy and different models can be tested and developed to explain the mechanism behind that.
A data sample at $\psi(3770)$ are to be taken, it will enable the measurements related with D decays to reach a new precision, for example, the $V_{cd}$, $V_{cs}$ can be measured to a statistic accuracy of about $1.6\%$, with a data sample of total accumulated luminosity of 20 $f_b^{-1}$. And the $D^0\bar{D^0}$ mixing and CP violation will be searched and studied. With huge data samples, the systematic errors should be well understood, and a lot of analyzes will adopt partial wave analyzes (PWA) to fully understand the decay dynamics. Efforts are being made and will be strengthened.
Right now, BES Collaboration has about 18 Chinese institutes and universities, and physicists from United State, Japan, Germany, Sweden and Russia have joined. BES welcomes new collaborators to join this exciting research project, which should last for at least 10 years. A conference called Charm2006 will be held in Beijing in June this year to discuss the physics potentials at BESIII and a US-CHINA workshop on the BESIII collaboration will be held right afterwards. All interested persons are welcome.
[2]{} J. Z. Bai *[et al.]{} (BES Collaboration), Nucl. Instrum. Methods Phys. Res. Sect. A **552**,344(2005). Preliminary Design Report of the BESIII Detector, **IHEP-BEPCII-SB-13**.*
|
---
abstract: 'We completely classify flows on approximately finite dimensional (AFD) factors with faithful Connes–Takesaki modules up to cocycle conjugacy. This is a generalization of the uniqueness of the trace-scaling flow on the AFD factor of type $\mathrm{II}_\infty $, which is equivalent to the uniqueness of the AFD factor of type $\mathrm{III}_1$. In order to achieve this, we show that a flow on any AFD factor with faithful Connes–Takesaki module has the Rohlin property, which is a kind of outerness for flows introduced by Kishimoto and Kawamuro.'
address: 'Department of Mathematical Sciences University of Tokyo, Komaba, Tokyo, 153-8914, Japan'
author:
- Koichi Shimada
title: 'A Classification of Flows on AFD Factors with Faithful Connes–Takesaki Modules'
---
Introduction {#intro}
============
In 1987, Haagerup [@H] showed the uniqueness of the approximately finite dimensional (AFD) factor of type $\mathrm{III}_1$. Although this is a great theorem which was the final step of the classification of AFD factors, here, we think of this theorem as a part of theory of flows on von Neumann algebras. In fact, by Takesaki’s duality theorem, the uniqueness of the AFD factor of type $\mathrm{III}_1$ is equivalent to the uniqueness of the trace-scaling flow on the AFD factor of type $\mathrm{II}_\infty $. In this paper, we will generalize this result as a part of theory of flows. More precisely, we will show the following theorem.
Let $M$ be any AFD factor. Then flows on $M$ which are not approximately inner at any non-trivial point or equivalently, have faithful Connes–Takesaki modules are classified by their Connes–Takesaki modules, up to strong cocycle conjugacy.
This result is not only a generalization of the uniqueness of the AFD factor of type $\mathrm{III}_1$. It is related to an important problem of classification of flows. The problem is about an “outerness” of flows, the Rohlin property.
Now, we explain the Rohlin property. First of all, it is important to note that classification of flows is difficult, compared with the complete classification of actions of discrete amenable groups on AFD factors (See Connes [@C3], Jones [@J], Ocneanu [@O], Sutherland–Takesaki [@ST2], Kawahigashi–Sutherland–Takesaki [@KwhST] and Katayama–Sutherland–Takesaki [@KtST]). The difficulty seems to come from the difference among various “outerness conditions” of flows. For example, one might consider that flows which are outer at any non-trivial point are outer. It seems to be reasonable to think of flows with full Connes spectra to be outer. However, the problem is that these outernesses do not coincide (See Example 2.3 of Kawamuro [@Kwm], which is based on Kawahigashi [@Kwh1], [@Kwh2]). Thus we need to clarify what appropriate outerness is. As a candidate for appropriate outerness, the Rohlin property was introduced by Kishimoto [@Ksm]. The Rohlin property is an analogue of the property in non-commutative Rohlin’s lemma in Connes’ classification of actions of $\mathbf{Z}$ (Theorem 1.2.5 of Connes [@C3]), which is derived from outerness of actions of $\mathbf{Z}$. Actually, Kishimoto’s definition is for flows on $\mathrm{C}^*$-algebras. After Kishimoto’s work, Kawamuro [@Kwm] introduced the Rohlin property for flows on von Neumann algebras. Recently, Masuda–Tomatsu [@MT] have presented a classification theorem for Rohlin flows. Thus the Rohlin property is now considered to be appropriate outerness. However, there is a problem. In general, it is not easy to see whether a given flow has the Rohlin property or not. Moreover, the Rohlin property is not written by “standard invariants” for flows. This can be an obstruction for the complete classification of flows on AFD factors. Hence it is important to characterize the Rohlin property in an appropriate way. At this point, it is conjectured that a flow on an AFD factor has the Rohlin property if and only if it has full Connes spectrum and is centrally free at each non-trivial point.
Now, we explain the relation between our main theorem and this characterization program. First of all, the uniqueness of the trace-scaling flow on the AFD factor of type $\mathrm{II}_\infty$ is deeply related to its having the Rohlin property. Indeed, by the results of Connes [@C2] and Haagerup [@H], it is possible to see that any trace-scaling flow on the AFD factor of type $\mathrm{II}_\infty$ has the Rohlin property (See Theorem 6.18 of Masuda–Tomatsu [@MT]). The uniqueness follows from the classification theorem of Rohlin flows. Thus it is expected that flows have the Rohlin property under our generalized assumption, that is, having faithful Connes–Takesaki modules (See Problem 8.5 of Masuda–Tomatsu [@MT]). In this paper, we actually show that flows on any AFD factor with faithful Connes–Takesaki modules have the Rohlin property, and obtain the main theorem by using Masuda–Tomatsu’s theorem. Hence it is possible to think of our main theorem as a partial answer to the characterization problem of the Rohlin property. The theorem means that if a flow is “very outer” at any non-trivial point, then it is globally “very outer”. Our main theorem provides interesting examples of Rohlin flows, and we believe that it is a useful observation for the characterization problem.
Hence the difficult point of the proof of the main theorem is to show the Rohlin property. In order to show the Rohlin property of a flow $\alpha$ on a factor $M$, we need to find good unitaries of $M$. To achieve this, we consider the continuous decomposition of $M$. The dual action $\theta $ of a modular flow of $M$ and the canonical extention $\tilde{\alpha }$ of $\alpha $ act on the continuous core $\tilde{M}$ of $M$. If the action $\theta \circ \tilde{\alpha }$ of $\mathbf{R}^2$ is faithful on the center of $\tilde{M}$, then the Rohlin property of $\alpha $ follows from ergodic theory. The problem is that even if $\alpha $ has faithful Connes–Takesaki module, the restriction of $\tilde{\alpha }\circ \theta $ on the center of $\tilde{M}$ may NOT be faithful. In order to overcome this problem, we consider a kind of decomposition of actions over the center of $\tilde {M}$ and reduce the problem to trace-scaling actions of $\mathbf{Z}$, $\mathbf{Z}^2$ or $\mathbf{R}$ on the AFD factor of type $\mathrm{II}_\infty$.
Besides our theorem, there is a similar result about actions of compact groups due to Izumi [@I]. He has shown that if an action of a compact group on a factor of type $\mathrm{III}$ has faithful Connes–Takesaki module, then it is minimal. It is also possible to consider that his result means that “pointwise outerness” implies “global outerness”. Although there is similarity between our main theorem and his theorem, however, there are some observations which show that our main theorem is essentially different from his theorem. For example, for actions of compact groups with faithful Connes–Takesaki modules, cocycle conjugacy always implies conjugacy. This is also true for trace-scaling flows on factors of type $\mathrm{II}_\infty$. However, for our flows, this does not hold.
Finally, we note that our main theorem depends on the uniqueness of the trace-scaling flow, which is based on the results of Connes [@C2] and Haagerup [@H]. Showing the uniqueness of the trace-scaling flow on the AFD factor of type $\mathrm{II}_\infty$ without using Connes and Haagerup’s theory is also an important problem (See Problem 8.8 of Masuda–Tomatsu [@MT]). However, our main theorem is different from that problem.
Preliminaries
=============
First of all, we explain things which are important to understand our main theorem and its proof, that is, Connes–Takesaki module and the Rohlin property.
Connes–Takesaki module
----------------------
First of all, we recall Connes–Takesaki module. Basic references are Connes–Takesaki [@CT] and Haagerup–Størmer [@HS].
Let $M$ be a properly infinite factor and let $\phi $ be a normal faithful semifinite weight on $M$. Set $N:=M\rtimes _{\sigma ^\phi}\mathbf{R}$. Then the von Neumann algebra $N$ is generated by $M$ and a one parameter unitary group $\{ \lambda _s\}_{s\in \mathbf{R}}$ satisfying $\lambda _s x \lambda _{-s}= \sigma _s^\phi (x)$ for $x\in M$, $s\in \mathbf{R}$. Let $\theta ^\phi$ be the dual action of $\sigma^\phi $ and let $C$ be the center of $N$. Then an automorphism $\alpha $ of $M$ extends to an automrphism $\tilde{\alpha }$ of $N$ by the following way (See Proposition 12.1 of Haagerup–Stømer [@HS]).
$$\tilde{\alpha }(x)=\alpha (x)\ \mathrm{for}\ x\in M,\ \tilde{\alpha }({\lambda _s})=[D\phi \circ \alpha ^{-1}: D\phi]_s\lambda _s\ \mathrm{for}\ s\in \mathbf{R}.$$
This $\tilde{\alpha }$ has the following properties (See Proposition 12.2 of Haagerup–Stømer [@HS]).
\(1) The automorphism $\tilde{\alpha }$ commutes with $\theta ^\phi$.
\(2) The automorphism $\tilde{\alpha }$ preserves the canonical trace on $N$.
\(3) The map $\alpha \mapsto \tilde {\alpha }$ is a continuous group homomorphism.
Set $\mathrm{mod}^\phi(\alpha ):=\tilde{\alpha }\mid _C$. This is said to be a Connes–Takesaki module of $\alpha$. Actually, this definition is different from the original definition of Connes–Takesaki [@CT]. However, in Proposition 13.1 of Haagerup–Stømer [@HS], it is shown that they are same. This Connes–Takesaki module does not depend on the choice of $\phi$, that is, if $\phi $ and $\psi $ are two normal faithful semifinite weights, then the action $\mathrm{mod}^\phi (\alpha ) \circ \theta ^\phi$ of $\mathbf{R}\times \mathbf{Z}$ on $C$ is conjugate to $\mathrm{mod}^\psi (\alpha ) \circ \theta ^\psi $. Hence, in the following, we will omit $\phi$, and write $\theta _t$ and $\mathrm{mod}(\alpha )$ if there is no danger of confusion. For an automorphism of any factor of type $\mathrm{II}_\infty$, considering its Connes–Takesaki module is equivalent to considering how it scales the trace. Hence flows with faithful Connes–Takesaki modules are natural generalization of trace-scaling flows.
We explain what property of automorphisms Connes–Takesaki module indicates. By Theorem 1 of Kawahigashi–Sutherland–Takesaki [@KwhST], an automorphism of any AFD factor is approximately inner if and only if its Connes–Takesaki module is trivial. Hence Connes–Takesaki module indicates “the degree of approximate innerness”.
Rohlin flows
------------
Next, we recall the Rohlin property, which is a kind of “outerness” for flows. A basic reference is Masuda–Tomatsu [@MT]. One of the typical forms of classification theorems of group actions is the following.
Two “very outer” actions whose difference are approximately inner are cocycle conjugate.
Hence, in order to classify flows on AFD factors, we need to clarify what is appropriate outerness. However, the problem is not so simple, compared with the problem for discrete group actions. As a candidate for appropriate outerness, the Rohlin property was introduced by Kishimoto [@Ksm] and Kawamuro [@Kwm]. In the following, we will explain the definition.
Let $\omega $ be a free ultrafilter on $\textbf{N} $. We denote by $l^{\infty }(M)$ the $\mathrm{C}^*$-algebra which consists of all bouded sequences in $M$. Set $$I_{\omega }:=\{ (x_n) \in l^{\infty }(M) \mid \text{strong*-lim}_{n\to \omega }(x_n)=0 \},$$ $$\begin{aligned}
N_{\omega }:=\{ (x_n) \in l^{\infty }(M) \mid &\text{for all } (y_n)\in I_{\omega }, \\
&\text{ we have } (x_ny_n)\in I_\omega \text{ and }(y_nx_n)\in I_\omega \} ,
\end{aligned}$$ $$C_{\omega }:=\{ (x_n) \in l^{\infty }(M) \mid \text{for all } \phi \in M_{*}, \lim _{n\to \omega }\| [\phi ,x_n] \| =0 \}.$$ Then we have $I_{\omega }\subset C_{\omega }\subset N_{\omega }$ and $I_{\omega }$ is a closed ideal of $N_{\omega}$. Hence we define the quotient $\mathrm{C}^{*}$-algebra $M^{\omega }:=N_{\omega }/I_{\omega }$. Denote the canonical quotient map $N_{\omega }\to M^{\omega }$ by $\pi $. Set $M_{\omega }:=\pi (C_{\omega })$. Then $M_{\omega }$ and $M^{\omega }$ are von Neumann algebras as in Proposition 5.1 of Ocneanu [@O]. Let $\alpha $ be an automorphism of $M$. We define an automorphism $\alpha ^{\omega }$ of $M^{\omega }$ by $\alpha^{\omega }((x_n))=(\alpha (x_n))$ for $(x_n)\in M^{\omega}$. Then we have $\alpha^{\omega }(M_{\omega })=M_{\omega }$. By restricting $\alpha ^{\omega }$ to $M_{\omega }$, we define an automorphism $\alpha _{\omega }$ of $M_{\omega }$. Hereafter we denote $\alpha ^{\omega }$ and $\alpha _{\omega }$ by $\alpha $ if there is no danger of confusion.
Choose a normal faithful state $\varphi $ on $M$. For a flow $\alpha $ on a von Neumann algebra $M$, set $$\begin{gathered}
M_{\omega ,\alpha }:=\{ (x_n)\in M_\omega \mid \text{for all } \epsilon >0,\text{ there exists } \delta >0 \text{ such that } \\
\{n\in \textbf{N}\mid \| \alpha _t(x_n)-x_n\|_{\varphi }^{\sharp } <\epsilon \ \mathrm{ for }\ |t|<\delta \}\in \omega \}. \end{gathered}$$
A flow $\alpha$ on a factor $M$ is said to have the Rohlin property if for each $p\in \mathbf{R}$, there exists a unitary $u$ of $M_{\omega, \alpha }$ satisfying $\alpha _t(u)=e^{-ipt} u$ for all $t\in \mathbf{R}$.
For flows with the Rohlin property, there is a classification theorem due to Masuda–Tomatsu [@MT].
Let $\alpha ^1$, $\alpha ^2$ be two Rohlin flows on a separable von Neumann algebra $M$. If $\alpha ^1_t\circ \alpha ^2_{-t}$ is approximated by inner automorphisms for each $t\in \mathbf{R}$, then they are mutually strongly cocycle conjugate For the definition of strong cocycle conjugacy, see Subsection 2.2 of Masuda–Tomatsu [@MT]. \[Masuda–Tomatsu\]
Hence, the Rohlin property is now considered to be appropriate outerness. However, one may feel that the definition of the Rohlin property is not so simple. Hence characterization of the Rohlin property is an important problem (See Conjecture 8.3 of Masuda–Tomatsu [@MT]).
Main Results
============
The main theorem of this paper is the following.
\[main\] Flows on any AFD factor with faithful Connes–Takesaki modules are completely classified by their Connes–Takesaki modules, up to strong cocycle conjugacy.
As a corollary, we obtain a classification theorem up to cocycle conjugacy. For a von Neumann algebra $C$ and a flow $\beta $ of $C$, set $\mathrm{Aut}_\beta (C):=\{ \sigma \in \mathrm{Aut}(C)\mid \sigma \circ \beta _t=\beta _t\circ \sigma, \ t\in \mathbf{R} \}$.
Let $\alpha ^1$ and $\alpha ^2$ be two flows on an AFD factor $M$ with faithful Connes–Takesaki modules. Then they are cocycle conjugate if and only if there exists an automorphism $\sigma \in \mathrm{Aut}_\theta (C)$ with $\mathrm{mod}(\alpha ^2_t)=\sigma \circ \mathrm{mod}(\alpha ^1_t) \circ \sigma ^{-1}$ for any $t\in \mathbf{R}$.
As an obvious application, we have the following example.
A flow on any AFD factor with faithful Connes–Takesaki module absorbs any flow on the AFD $\mathrm{II}_1$ factor, as a tensor product factor.
In order to show Theorem \[main\], by Theorem \[Masuda–Tomatsu\], and the characterization of approximate innerness of automorphisms of AFD factors (Theorem 1 of Kawahigashi–Sutherland–Takesaki [@KwhST]), it is enough to show the following theorem.
\[Rohlin\] A flow on any AFD factor with faithful Connes–Takesaki module has the Rohlin property.
From what we have explained in the previous section, this theorem means that a kind of “pointwise outerness” implies “global outerness”.
As we have explained in Section \[intro\] and the previous subsection, characterization of the Rohlin property is an important problem (Conjecture 8.3 of Masuda–Tomatsu [@MT]). Theorem \[Rohlin\] gives a partial answer to this problem. We will proceed further to this direction in Subsection \[generalization\].
The Proof of Main Results
=========================
In this section, we show Theorem \[Rohlin\]. In order to achieve this, we first note that we may assume that a flow has an invariant weight. This is seen in the following way. Let $\alpha $ be a flow on an AFD factor $M$. Then by the same argument as in Lemma 5.10 of Sutherland–Takesaki [@ST2] (or equivalently, by the combination of Lemma 5.11 and Lemma 5.12 of [@ST2]), there exists a flow $\beta $ and a dominant weight $\phi $ which satisfy the following conditions.
\(1) We have $\phi \circ \beta _t =\phi $ for all $t\in \mathbf{R}$.
\(2) The action $\beta $ is cocycle conjugate to $\alpha \otimes \mathrm{id}_{B(L^2\mathbf{R})}$.
By Lemma 2.11 of Connes [@C], $(M\otimes B(L^2\mathbf{R}))_\omega =M_\omega \otimes \mathbf{C}$. Hence, by replacing $\alpha $ by $\beta $, we may assume that the action $\alpha $ has an invariant dominant weight. In the rest of the paper, we denote the continuous core $M\rtimes _{\sigma ^\phi} \mathbf{R}$ by $N$ and the dual action of $\sigma ^\phi $ by $\theta$. Then by the same argument as in the proof of Proposition 13.1 of Haagerup–Størmer [@HS], the action $\tilde {\alpha }$ extends to a flow $\tilde{\tilde{\alpha }}$ of $N\rtimes _\theta \mathbf{R}$ so that if we identify $N\rtimes _\theta \mathbf{R}$ with $M\otimes B(L^2\mathbf{R})$ by Takesaki’s duality, $\tilde{\tilde{\alpha }}$ corresponds to $\alpha \otimes \mathrm{id}$. By Lemma 2.11 of Connes [@C] again, in order to show that $\alpha$ has the Rohlin property, it is enough to show that $\tilde{\tilde{\alpha }}$ has the Rohlin property. In order to achieve this, we need to choose $\{u_n \}\subset U(M\otimes B(L^2\mathbf{R}))_\omega $ which satisfies the conditions in the definition of the Rohlin property. Our strategy is to choose $\{u_n \}$ from $N$. Based on this strategy, it is sufficient to show the following lemma.
For each $p\in \mathbf{R}$, there exists a sequence $\{u_n\} \subset U(N)$ satisfying the following conditions.
We have $\| [u_n, \phi ]\| \to 0$ for any $\phi \in N_{*}$.
We have $\theta _s(u_n)-u_n \to 0$ compact uniformly for $s\in \mathbf{R}$ in the strong\* topology.
We have $\tilde{\alpha}_t (u_n)-e^{ipt}u_n \to 0$ compact uniformly for $t\in \mathbf{R}$ in the strong\* topology. \[key\]
By the first two conditions, this $\{ u_n\}$ asymptotically commutes with elements in a dense subspace of $M\otimes B(L^2\mathbf{R}) \cong M$. However, in general, this does not imply that $\{ u_n \}$ is centralizing (and this sometimes causes a serious problem). Hence, in order to assure that Lemma \[key\] implies Theorem \[Rohlin\], we need to show the following lemma.
\[7\] Let $M$ be an AFD factor of type $\mathrm{III}$ and let $M=N\rtimes _{\theta}\mathbf{R}$ be the continuous decomposition. Then a sequence $\{u_n\}\subset U(N)$ with conditions and of the above lemma is centralizing.
Let $H$ be the standard Hilbert space of $N$. Take $\xi \in H$ and $f\in L^2(\mathbf{R})$. Since $$x(\xi \otimes f)(s) = (\theta _{-s}(x)\xi )f(s),$$ $$\begin{aligned}
(\xi \otimes f)x(s)&=(J_Mx^*J_M(\xi \otimes f))(s) \\
&=(J_Nx^*J_N\xi )f(s) \\
&=(\xi x)f(s)\end{aligned}$$ for $s\in \mathbf{R}$, $x\in N$, we have $$\begin{aligned}
\| u_n(\xi \otimes f) - (\xi \otimes f)u_n \| ^2 &=\int _{\mathbf{R}}\| \theta _{-s}(u_n)\xi -\xi u_n\| ^2| f(s)| ^2 \ ds\\
&\leq \int _{\mathbf{R}} \| (\theta _{-s}(u_n) -u_n )\xi \| ^2 | f(s) | ^2\ ds \\
& + \int _{\mathbf{R}}\| u_n \xi -\xi u_n \| ^2| f(s)| ^2 \ ds \\
&\to 0\end{aligned}$$ by Lebesgue’s convergence theorem. Here, the convergence of the second term follows from Lemma 2.6 of Masuda–Tomatsu [@MT]. Any vector of $H\otimes L^2\mathbf{R}$ is approximated by finite sums of vectors of the form $\xi \otimes f$. Hence for any vector $\eta \in H\otimes L^2\mathbf{R}$, we have $\| u_n\eta -\eta u_n \| \to 0$. Hence $\{u_n \}$ is centralizing.
By this lemma, Lemma \[key\] implies Theorem \[Rohlin\]. In the following, we will show Lemma \[key\]. If $M$ is of type $\mathrm{II}_\infty$, Lemma \[key\] is shown in Theorem 6.18 of Masuda–Tomatsu [@MT], using Connes and Haagerup’s theory. If $M$ is of type $\mathrm{II}_1$ or is of type $\mathrm{III}_1$, then we need not do anything because Connes–Takesaki modules of automorphisms are always trivial. Hence we only need to consider the case when $M$ is of type $\mathrm{III}_0$ and the case when $M$ is of type $\mathrm{III}_\lambda$ ($0< \lambda <1$). Actually, as we will see in Remark \[lambda\], if $M$ is of type $\mathrm{III}_\lambda $ ($0<\lambda <1$), the Connes–Takesaki module of a flow cannot be faithful. Hence, the only problem is that how to handle the case when $M$ is of type $\mathrm{III}_0$.
Let $C$ be the center of $N$. First, we list up the form of the kernel of the action $\mathrm{mod}(\alpha ) \circ (\theta \mid_{C})$ of $\mathbf{R}^2$ on $C$. This is a closed subgroup of $\mathbf{R}^2$. Thus the kernel must be isomorphic to one of the following groups. $$0,\ \mathbf{Z},\ \mathbf{Z}^2,\ \mathbf{R},\ \mathbf{R}\times \mathbf{Z},\ \mathbf{R}^2.$$ However, since $\theta \mid _{C}$ is faithful, the kernel cannot be isomorphic to $\mathbf{R}\times \mathbf{Z}$ or $\mathbf{R}^2$. We handle the other four cases separately.
We first consider the case when $\mathrm{ker}(\mathrm{mod}(\alpha ) \circ (\theta \mid _{C}))=0$. In this case, by an argument similar to that of the proof of Theorem 3.3 of Shimada [@S], Lemma \[key\] follows from a Rohlin type theorem due to Feldman [@F]. In the following, we will explain this theorem.
**Settings.** A subset $Q$ of $\mathbf{R}^d$ is said to be a cube if $Q$ is of the form $$[-s_1,t_1]\times \cdots \times [-s_d,t_d]$$ for some $s_1, \cdots , s_d, t_1,\cdots , t_d >0$. Let $Q$ be a cube of $\mathbf{R}^d$ and $T$ be a non-singular action of $\mathbf{R}^d$ on a Lebesgue space $(X, \mu )$. Then a measurable subset $F$ of $X$ is said to be a $Q$-set if $F$ satisfies the following two conditions.
\(1) The map $Q\times F \ni (t,x) \mapsto T_t(x) \in X$ is injective.
\(2) The set $T_QF:=\{ T_t(x) \mid t\in Q, x\in F \}$ is measurable and non-null.
In this setting, the following theorem holds.
\[rp\] Let $T$ be a free non-singular action of $\mathbf{R}^d $ on the standard probability space $(X, \mu)$. Then for any $\epsilon >0$ and for any cube $P$ of $\mathbf{R}^d$, there exists a large cube $Q$ and a $Q$-set $F$ of $X$ with $$\mu (T_{\bigcap _{t\in P} (t+Q)}F) >1-\epsilon .$$
The proof is written in Feldman [@F2]. However, his paper is privately circulated. Hence we explain the outline of the proof in Appendix (Section \[appendix\]), which is based on Theorem 1 of Feldman–Lind [@FL] and Lind [@L]. As written in the proof of Theorem 1.1 (a) of Feldman [@F] (p.410 of Feldman [@F]), it is possible to introduce a measure $\nu $ on $F$ so that the map $Q\times F\ni (t,x) \mapsto T_t(x) \in T_QF$ is a non-singular isomorphism. The measure $\nu $ is defined in the following way. Set $$\mathcal{M}:= \{ A\subset F\mid T_Q(A) \ \mathrm{is} \ \mathrm{measurable} \ \mathrm{with} \ \mathrm{respect} \ \mathrm{to} \ \mu \}.$$ Then $\mathcal{M}$ is a $\sigma$-algebra of $F$ and it is possible to define a measure $\nu $ on $F$ by $$\nu (A):=\frac{\mu (T_QA)}{\mu(T_QF)}$$ for $A\in \mathcal{M}$. Then the map $(t,x) \mapsto T_t(x)$ is a non-singular isomorphism with respect to $\mathrm{Lebesgue}\otimes \nu$ and $\mu | _{T_QF}$. These things are written in p.410 of Feldman [@F] and the proof may be written in Feldman–Hahn–Moore [@FHM]. In this paper, for reader’s convenience, we present a proof of what we will use (Propositions \[A.4\] and \[A.5\] of Appendix).
When $\mathrm{ker}(\mathrm{mod}(\alpha )\circ (\theta | _C))$ is zero, Lemma \[key\] holds.
Think of $C$ as $L^\infty (X, \mu ) $ for some probability measured space $(X, \mu )$. Let $T$ be an action of $\mathbf{R}^2$ defined by the following way. $$f\circ T_{(s,t)}=\theta _{-s} \circ \tilde{\alpha}_{-t} (f)$$ for $f\in L^\infty (X, \mu )$, $(s,t) \in \mathbf{R}^2$. Fix a natural number $n\in \mathbf{N}$. Set $P:=[-n,n]^2$. Then by Theorem \[rp\], there exists a large cube $Q$ and a $Q$-set $F$ of $X$ with $$\mu (T_{\bigcap _{t\in P}(t+Q)}F)>1-\frac{1}{n}.$$ Define a function $u_n $ on $X$ by the following way. $$u_n = \begin{cases}
e^{-ipt} & (x=T_{(s,t)}(y), \ (s,t)\in Q, \ y\in F) \\
1 & (otherwise).
\end{cases}$$ Then by Proposition \[A.4\], the function $u_n$ is Borel measurable. Then for $x \in T_{\bigcap _{t\in P}(t+Q)}F$ and $(s,t)\in P$, we have $$\theta _s(u_n)(x) =u_n(x),$$ $$\tilde{\alpha }_t(u_n)(x)=e^{ipt}u_n(x).$$ Hence we have $$\begin{aligned}
\| \theta _s(u_n)-u_n \| _\mu^2 &\leq 4\mu (X\setminus T_{\bigcap _{t\in P}(t+Q)}F) \\
&\leq \frac{4}{n+1}\end{aligned}$$ for $s\in [-n,n]$. By the same computation, we have $$\| \tilde{\alpha }_t(u_n) -e^{ipt}u_n \| _\mu ^2\leq \frac{4}{n+1}$$ for $t\in [-n,n]$. Hence the sequence $\{ u_n \}$ of unitaries of $C$ satisfies the conditions in Lemma \[key\].
Next, we consider the following case.
\[z2\] When $\mathrm{ker}(\mathrm{mod}(\alpha )\circ (\theta \mid _{C} ))$ is isomorphic to $\mathbf{Z}^2$, Lemma \[key\] holds.
In this case, there exist two pairs $(p_1, q_1 )$, $(p_2,q_2)$ of non-zero real numbers with $\mathrm{ker}(\mathrm{mod}\circ \theta )=\mathbf{Z}(p_1,q_1)\oplus \mathbf{Z}(p_2,q_2)$. Here, we use our assumption that $\mathrm{mod}(\alpha ) $ is faithful for showing $q_i\not=0$. Set $\sigma _t:=\theta _{q_1t}\circ \tilde {\alpha }_{p_1t}$. In order to show Lemma \[z2\], it is enough to show the following lemma.
\[10\] For each $r\in \mathbf{R}$, there exists a sequence of unitaries $\{ u_n \}$ of $N$ which satisfies the following conditions.
We have $\| [u_n,\phi]\| \to 0$ for any $\phi \in N_*$.
We have $\theta _s(u_n)-u_n\to 0$ compact uniformly for $s\in \mathbf{R}$ in the strong\* topology.
We have $\sigma _t(u_n)-e^{irt}u_n \to 0$ compact uniformly for $t\in \mathbf{R}$ in the strong\* topology.
In order to show this lemma, we need to prepare some lemmas.
The action $\theta$ on $ C^\sigma $ is ergodic and has a period $p\in (0, \infty)$. \[11\]
Ergodicity follows from the ergodicity of $\theta: \mathbf{R}\curvearrowright C$. We show that the restriction of $\theta$ on $C^\sigma $ has a period. We first note that a Borel measurable map $T$ from $\mathbf{T}$ to itself which commutes with every translations of the torus must be a translation because we have $T(\gamma +t)-t=T(\gamma )$ for $t\in \mathbf{T}$ and for almost all $\gamma \in \mathbf{T}$. Now, we show that $C^\sigma \not=\mathbf{C}$. Assume that $C^\sigma $ were isomorphic to $\mathbf{C}$. Then since $\theta $ would commute with $\sigma$, which is a translation flow on the torus. Hence $\theta $ would be also a translation on the torus. Hence $\theta \circ \sigma $ would define a group homomorphism from $\mathbf{R}^2$ to the group of translations of the torus, which is isomorphic to $\mathbf{T}$. Hence the kernel of $\theta \circ \sigma $ would be isomorphic to $\mathbf{R}\times \mathbf{Z}$, which would contradict to the faithfulness of $\theta $. Combining this with the ergodicity of $\theta $, we have $\theta \mid _{C^\sigma}$ is non-trivial. Since $\mathrm{mod}(\alpha _{p_2})=\theta _{-q_2}\mid _{C}$, we have $(\sigma _{p_2/p_1}\circ \theta _{q_2-p_2q_1/p_1})\mid _C=\mathrm{id}_C$. Since $(p_1,q_1)$ and $(p_2,q_2)$ are independent, this $\theta \mid _{C^\sigma} $ has a non-trivial period.
By this lemma, we may assume the following.
\(1) We have $C^\sigma =L^\infty (\mathbf{T}_p)$, where $\mathbf{T}_p$ is the torus of length $p$, which is isomorphic to $[0,p)$ as a measured space.
\(2) We have $\theta _t(f) =f(\cdot -t)$ for $f\in L^\infty (\mathbf{T}_p)$, $t\in \mathbf{R}$.
Let $$N=\int _{[0,p)}^\oplus N_\gamma \ d\gamma$$ be the direct integral decomposition of $N$. For $\gamma _1, \gamma _2 \in \mathbf{R}$, $N_{\gamma_1}$ and $N_{\gamma _2}$ are mutually isomorphic by the following map. $$\theta _{\gamma _2-\gamma _1, \gamma _1}: N_{\gamma _1}\to N_{\gamma _2},$$ $$\theta _{\gamma _2-\gamma _1, \gamma _1}(x_{\gamma _1})=(\theta _{\gamma _2-\gamma _1}(x))_{\gamma _2}$$ for $x= \int _{[0,p) }^\oplus x_\gamma d\gamma \in N$. These $\theta_{\gamma_1, \gamma_2}$’s satisfy the following two conditions.
**Conditions.**
\(1) The equality $\theta _{0, \gamma}=\mathrm{id}_{N_\gamma }$ holds for each $\gamma \in [0,p)$.
\(2) The equality $\theta _{\gamma _3-\gamma _2, \gamma _2}\circ \theta_{\gamma _2-\gamma _1, \gamma _1}=\theta _{\gamma _3-\gamma_1,\gamma _1}$ holds for each $\gamma_1, \gamma _2, \gamma _3\in \mathbf{R}$.
By these $\theta _{\gamma _1, \gamma _2}$’s, all $N_\gamma$’s are mutually isomorphic. Thus it is possible to think of $N$ as $N_0\otimes L^\infty ([0,p))$.
Now, we need to consider the measurability of $\theta _{t, \gamma}$.
**Fact.**
If we identify $N$ with $N_0 \otimes L^\infty ([0,p))$, the map $[0, p)^2\ni(t, \gamma ) \mapsto \theta _{t, \gamma }\in \mathrm{Aut}(N_0)$ is Lebesgue measurable.
Although this fact is probably well-known for specialists, for the reader’s convenience, we present the proof in Appendix (Section \[appendix\]).
By measurability of $\theta _{t, \gamma }$, Lusin’s theorem and Fubini’s theorem, for almost all $\gamma \in [0,p)$, the map $t\mapsto \theta _{-t,t+\gamma}$ and $t\mapsto \theta _{t,\gamma}$ are also Lebesgue measurable. We may assume that $\gamma =0$ and we identify $N_{\gamma _1 }$ with $N_0$ by $\theta _{\gamma _1,0}$ for all $\gamma _1\in [0,p)$, that is, if we think of $N $ as the set of all essentially bounded weak \* Borel measurable maps from $[0,p)$ to $N_0$, then the set of constant functions is the following set. $$\{ \int_{[0,p)}^\oplus \theta _{\gamma , 0}(x_0)\ d\gamma \mid x_0\in N_0 \} .$$ Take a normal faithful state $\phi _0$ of $N_0$. Then $$\phi:=\frac{1}{p}\int _{[0,p)}^\oplus \phi _0 \circ \theta _{-\gamma , \gamma } \ d\gamma$$ is a normal faithful state on $N$. Choose $\phi _1, \cdots, \phi _n\in N_*$, $\epsilon >0$ and $T>0$. Then by the above identification of $N_*$ with $L^1_{(N_0)_*}([0, p))$, each $\phi _k$ is a Lebesgue measurable map from $[0, p)$ to $(N_0)_*$. Hence it is possible to approximate each $\phi _k$ by Borel simple step functions by the following way. $$\| \phi _k -\sum _{i=1}^{l_k}\phi _{k,i}\circ \theta _{-\gamma, \gamma}\chi _{I_i}(\gamma ) \| <\epsilon.$$ for each $k\in \{1, \cdots , n\}$. Here, $\phi _{k,i}\in{ N_0}_*$ for $i=1, \cdots , l_k$, $\{ I_i \} _{i=1}^{l_k}$ is a Borel partition of $[0, p)$. Next, we look at actions on $N_0$. Let $$\theta _p=\int _{[0,p)}^\oplus \theta _p^\gamma \ d\gamma ,$$ $$\sigma _t =\int _{[0,p)}^\oplus \sigma _t^\gamma \ d\gamma ,$$ $$\tau =\int _{[0,p)}^\oplus \tau _\gamma \ d\gamma$$ be the direct integral decompositions. Since $\theta $ is trace-scaling and $\tilde{\alpha }$ is trace-preserving, $\sigma $ is trace-scaling. Hence for almost all $\gamma \in [0,p)$, $\sigma ^\gamma $ is $\tau _\gamma$ -scaling. Thus we may assume that $\sigma ^0$ is $\tau_0$-scaling. In order to show Lemma \[10\], it is enough to show the following lemma.
In the above context, for real number $r\in \mathbf{R}$, there exists a unitary $u_0$ of $N_0$ which satisfies the following conditions. \[13\]
We have $\| [u_0, \phi _{k,i}] \| <\epsilon /(pl_k)$ for $k=1, \cdots , n$, $i=1, \cdots , l_k$.
We have $\| \theta _{mp}^0(u_0)-u_0\| _{\phi _0}^{\sharp }<\epsilon /p$ for $m\in \mathbf{Z}$, $| m| \leq p/T+2$.
We have $\| \sigma ^0_t(u_0)-e^{-irt}u_0 \| _{\phi _0}^{\sharp}<\epsilon /p$ for all $t\in [-T,T]$.
First, we show that Lemma \[13\] implies Lemma \[10\].
*Proof of Lemma \[13\] $\Rightarrow$ Lemma \[10\].* Assume that there exists a unitary $u_0$ in $N_0$ which satisfies the conditions in Lemma \[13\]. We set $$u_\gamma :=\theta _{\gamma ,0}(u_0),$$ $$u:=\int _{ [0,p) }^\oplus u_\gamma \ d\gamma .$$ Fix $t\in [-T, T]$ and $\gamma \in [0,p)$. For each $\gamma \in [0,p)$, choose $m_\gamma \in \mathbf{Z}$ so that $-t+m_\gamma p+\gamma \in [0,p)$. Then we have $$\begin{aligned}
(\theta _t(u))_\gamma &=\theta _{t,-t+\gamma}(u_{-t+\gamma}) \\
&=\theta _{t, -t+\gamma}(u_{-t+\gamma +m_\gamma p}) \\
&=\theta _{\gamma ,0}\circ \theta ^0_{m_\gamma p}\circ \theta _{t-m_\gamma p -\gamma ,-t+m_\gamma p+\gamma}(u_{-t+m_\gamma p+\gamma} )\\
&=\theta _{\gamma ,0}\circ \theta ^0_{m_\gamma p}(u_0).\end{aligned}$$ Hence we have $$\begin{aligned}
\| \theta _t(u)-u\| _{\phi }^\sharp &= \int _{[0,p)} \| (\theta _t(u))_\gamma -u_\gamma \| _{\phi _0 \circ \theta _{-\gamma ,\gamma }} ^\sharp \ d\gamma \\
&=\int _{[0,p) } \| \theta _{\gamma ,0}\circ \theta ^0_{m_\gamma p}(u_0 ) -\theta _{\gamma, 0} (u_0) \| _{\phi_0 \circ \theta _{-\gamma ,\gamma }} ^\sharp \ d\gamma \\
&=\int _{[0,p)} \| \theta ^0_{m_\gamma p }(u_0)-u_0 \| _{\phi _0}^\sharp \ d\gamma \\
&<\int _{[0,p)} \frac{\epsilon}{p} \ d\gamma \\
&=\epsilon .\end{aligned}$$ Here we use that $| m_\gamma | \leq T/p +2$ in the fourth inequality of the above estimation. By the same argument, we also have $$\| \sigma _t(u)-e^{-irt}u \| _\phi ^\sharp <\epsilon$$ for $t\in [-T,T]$. We also have $$\begin{aligned}
\| [u, \phi _k ]\| &\leq 2\| \phi _k -\sum _{i=1}^{l_k}\phi _{k,i} \otimes \chi _{I_i} \| +\|[u, \sum _{i=1}^{l_k}\phi _{k,i} \otimes \chi _{I_i} ]\| \\
&<2\epsilon +\sum _{i=1}^{l_k}\| [u, \phi _{k,i} \otimes \mathrm{id} ]\| \\
&=2\epsilon +\sum _{i=1}^{l_k}\int _{[0,p)} \| [\theta _{\gamma, 0}(u_0), \phi _{k,i}\circ \theta _{-\gamma, \gamma} ]\| \ d\gamma \\
&=2\epsilon +\sum _{i=1}^{l_k}\int _{[0,p)} \| [u_0, \phi _{k,i}] \| \ d\gamma \\
&<2\epsilon +\sum _{i=1}^{l_k}\int _{[0,p)} \frac{\epsilon }{pl_k} \ d\gamma \\
&=3\epsilon .\end{aligned}$$ Thus Lemma \[10\] holds.
In order to prove Lemma \[13\], we first rewrite the lemma in a simpler form. To do this, we show that there exists a number $s\in (0,1)$ with $(\theta ^0_p \circ \sigma ^0_s) \mid _{Z(N_0)}=\mathrm{id}$. Since the restriction of $\sigma^0$ on the center of $N_0$ has a period $1$ and is ergodic, we may assume that $Z(N_0)$ is isomorphic to $L^\infty ([0,1))$, which is canonically identified with $L^\infty(\mathbf{T})$, $\sigma ^0_s(f)=f(\cdot -s)$ for $s\in \mathbf{R}$, $f\in L^\infty (\mathbf{T})$. By this identification, $\theta ^0_p$ commutes with all $\sigma ^0_s$’s. Hence $\theta ^0$ is a translation on the torus. Thus there exists a unique $s\in (0,1)$ with $(\theta^0_p \circ \sigma ^0 _s) \mid _{Z(N_0)}=\mathrm{id}$. Set $\beta ^0:=\theta _p ^0 \circ \sigma ^0_s$. The proof of Lemma \[13\] reduces to that of the following lemma.
\[14\] The action $\{ \beta ^0_m \circ \sigma ^0_t\}_{(m,t)}$ of $\mathbf{Z}\times \mathbf{R}$ on $N_0$ has the Rohlin property.
*Proof of Lemma \[14\]$\Rightarrow$ Lemma \[13\].* Assume that the action $\{ \beta^0_m\circ \sigma^0_t\}_{m,t}$ has the Rohlin property. Then there exists a unitary element $u_0$ of $N_0$ with the following conditions.
\(1) We have $\|[u_0, \phi_{k,i}]\| <\epsilon/(pl_k)$ for $k=1, \cdots ,n$, $i=1, \cdots ,l_k$.
\(2) We have $\| \beta ^0_{m} (u_0)-e^{-irms}u_0\| _{\phi _0}^{\sharp} <\epsilon /(2p)$ for $m\in \mathbf{Z}$, $|m|\leq p/T+2$.
\(3) We have $\| \sigma ^0_t(u_0)-e^{-irt}u_0\| _{\phi _0\circ \theta _{mp}^0}^\sharp <\epsilon /(2p) $ for $t\in [-(1+s)(T+2p),(1+s)(T+2p)]$, $m\in \mathbf{Z}$, $|m|\leq p/T+2$.
Since $\beta^0_{m}=\theta ^0_{mp}\circ \sigma ^0_{ms}$, we have $$\begin{aligned}
\| \theta ^0_{mp}(u_0)-u_0 \| &=\|e^{-irms}\theta ^0_{mp}(u_0)-e^{-irms}u_0 \|_{\phi _0}^\sharp \\
&\leq \| \theta _{mp}^0 (e^{-irms}u_0-\sigma _{ms}^0(u_0))\|_{\phi_0}^\sharp +\| \beta _{m}^0(u_0)-e^{-irms}u_0\|^\sharp _{\phi _0} \\
&=\| e^{-irms}u_0-\sigma _{ms}^0(u_0)\|_{\phi _0\circ \theta _{mp}^0}^\sharp + \| \beta _{m}^0(u_0)-e^{-irms}u_0\|^\sharp _{\phi _0}\\
&<\frac{\epsilon}{2p}+\frac{\epsilon}{2p} \\
&=\frac{\epsilon}{p}\end{aligned}$$ for $m\in \mathbf{Z}$, $|m|\leq p/T+2$. Thus Lemma \[13\] holds.
In order to show Lemma \[14\], we need further to reduce the lemma to a simpler statement. Let $$N_0=\int _{[0,1)}^\oplus (N_0)_\gamma \ d\gamma$$ be the direct integral decomposition of $N_0$ over the center of $N_0$. For each $\gamma _1, \gamma _2\in [0,1)$, there exists an isomorphism from $(N_0)_{\gamma _1}$ to $(N_0)_{\gamma_2}$ defined by $$\sigma ^0 _{\gamma _2-\gamma _1, \gamma _1}((x_0)_{\gamma _1})=(\sigma ^0 _{\gamma _2-\gamma _1}(x_0))_{\gamma _2}$$ for $x_0=\int _{[0,1)}^\oplus (x_0)_\gamma d\gamma \in N_0$. These $\sigma ^0 _{\gamma _2-\gamma _1, \gamma _1}$’s satisfy the conditions similar to conditions (1) and (2) of $\theta _{t, \gamma}$ (See Conditions between Lemma \[11\] and Lemma \[13\]).
We identify $(N_0)_\gamma$’s with $(N_0)_0$ by $\sigma ^0_{\gamma, 0}$. Choose a normal faithful state $\psi_0 $ of $(N_0)_0$. Set $$\psi:=\int _{[0,1)} ^\oplus \psi _0\circ \sigma ^0_{-\gamma, \gamma} \ d\gamma .$$ This is a normal faithful state on $N_0$. Choose $\psi _1, \cdots \psi _n \in (N_0)_*$, $\epsilon >0$ and $T>0$. By the same argument as above, we may assume that $\psi _k$’s are simple step Borel functions. $$\psi _k =\sum _{i=1}^{l_k} \psi _{k,i} \circ \sigma ^0_{-\gamma , \gamma }\chi _{I_i}(\gamma )$$ for $k=1, \cdots , n$. Here, $\psi _{k,i}\in (N_0)_*$, $\{ I_i\}_{i=1}^{l_k}$ are partitions of $[0,1)$. Since $\beta ^0$ and $\sigma ^0$ fix the center of $N_0$, they are decomposed into the following form. $$\beta ^0 =\int _{[0,1)}^\oplus \beta^{0,\gamma } \ d\gamma ,$$ $$\sigma ^0_1=\int _{[0,1)}^\oplus \sigma ^{0,\gamma } \ d\gamma .$$ Then for each $\gamma \in [0,1)$, $ \{ \beta ^{0,\gamma }_n \circ \sigma ^{0,\gamma }_m \}_{(n,m)\in \mathbf{Z}^2} $ defines an action of $\mathbf{Z}^2$ on $(N_0)_\gamma$, which is isomorphic to the AFD factor of type $\mathrm{II}_\infty $. We show the following lemma, which is essentially important, that is, assumption that $\mathrm{mod}(\alpha )$ is faithful is essentially used for showing this lemma.
\[15\] For almost all $\gamma \in [0,1)$, the action $\{ \beta ^{0,\gamma }_n \circ \sigma ^{0,\gamma }_m \}$ is trace-scaling for $(n,m)\not=0$.
Take a pair $(n,m)\not=0$. By definition of $\beta^0$ and $\sigma ^0$, we have $$\begin{aligned}
\beta ^0_n \circ \sigma ^0_m &=(\theta _{np} \circ \sigma _{ns})^0\circ \sigma ^0_m \\
&=(\theta _{np} \circ \sigma _{ns+m})^0 \\
&=(\theta _{np} \circ \theta _{(ns+m)q_1} \circ \tilde{\alpha} _{(ns+m)p_1})^0 \\
&=(\theta _{np + (ns+m)q_1} \circ \tilde{\alpha} _{(ns+m)p_1})^0. \end{aligned}$$ If $n=0$, we need not show anything . Assume that $n\not=0$. Then since $\theta _{np}$ is not identity on the center of $N_0$, $\sigma _{ns+m}$ is not identity on $Z(N_0)$ by looking at the first equation. Hence $(ns+m)p_1\not=0$. Thus, by the faithfulness of $\mathrm{mod}(\alpha)$ and the last equation, we have $np +(ns+m)q_1\not =0$. Hence $\theta _{np +(ns+m)q_1}$ scales $\tau $. Besides, $\tilde{\alpha }$ preserves $\tau$. Hence we may assume that $\beta ^0 _n\circ \sigma ^0_m$ scales $\tau _0$. Hence if we decompose $\tau ^0$ by $$\tau^0 =\int _{[0,1)}^\oplus \tau ^{0,\gamma } \ d\gamma ,$$ $(\beta ^0 _n \circ \sigma ^0_m)^\gamma $ scales $\tau ^{0,\gamma}$ for almost all $\gamma \in [0,1)$.
From Lemma \[15\], we may assume that $\{ \beta ^{0,0}_n \circ \sigma ^{0,0}_m \}$ is trace-scaling. Now, we will return to prove Lemma \[14\], which completes the proof of Lemma \[10\].
*Proof of Lemma \[14\].* Let $\psi _0$, $\epsilon$ and $T$ be as explained after the statement of Lemma \[14\]. By Lemma \[15\], the action $\{ \beta ^{0,0}_n \circ \sigma ^{0,0} _m\}$ is centrally outer, and hence has the Rohlin property. Hence, for $A,B\in \mathbf{N}$ with $4(T+1)^2/\epsilon ^2 <B$ and $A>1/\epsilon ^2 $, there exists a family of projections $\{ e_{n,m}\}_{n=1,\cdots, B}^{m=1, \cdots A}$ of $N_{0,0}$ which satisfies the following conditions.
\(1) The projections are mutually orthogonal.
\(2) We have $$\begin{gathered}
\sum _{n,m} e_{n,m}=1, \\
\| \sum _{1\leq n\leq B, m=1,A} e_{n,m} \| _{\psi _0 +\psi _0 \circ \beta ^{0,0} }^\sharp \leq 2/\sqrt{A}, \\
\| \sum _{1\leq m\leq A,n\geq B-(T+1),n\leq T+1} e_{n,m}\| _{\psi _0+\sum _{l=-[T]-1}^{[T]+1} \psi _0\circ \sigma ^{0,0}_l }^\sharp \leq 2(T+1)/\sqrt{B}. \end{gathered}$$ Here, $[T]$ is the maximal natural number which is not larger than $T$.
\(3) We have $\| [e_{n,m}, \psi _{k,i}]\|<\epsilon/(ABl_k)$ for $n=1, \cdots B$, $m=1, \cdots , A$, $i=1, \cdots l_k$, $k=1, \cdots n$.
\(4) We have $\| \sigma ^{0,0}_l(e_{n,m})-e_{n+1,m} \| _{\psi _0}^\sharp <\epsilon /(AB)$ for $n,m$, $l \in \mathbf{Z}$ with $| l| \leq T+1$, $n\leq B-(T+1)$.
\(5) We have $\| \beta ^{0,0}(e_{n,m})-e_{n,m+1}\| _{\psi _0}^\sharp <\epsilon /(AB)$ for $n,m\in \mathbf{Z}$ with $m\not =A$.
Here, we define $e_{B+1,m}=e_{1,m}$ for $m=1, \cdots , A$, $e_{n,A+1}=e_{n,1}$ for $n=1, \cdots , B$. For $(s,t) \in \mathbf{T}\times \mathbf{R}$, we set $$u_\gamma :=e^{2\pi it\gamma }\sum _{n,m}e^{2\pi i (nt+ms)}\sigma ^0_{\gamma, 0 }(e_{n,m})$$ for $\gamma \in [0,1)$. We also set $$u:=\int _{[0,1)}^\oplus u_\gamma \ d\gamma \in U(N_0).$$ The above conditons (2) and (4) ensure that we can almost control $\sigma^{0,0}$, which is useful to show that $\sigma ^0_q(u)$ is close to $e^{-2\pi itq}u$. Conditions (2) and (5) is useful to show that $\beta ^0(u)$ is close to $e^{-2\pi is}u$. Condition (3) is useful to show that $[u, \psi _k]$ is small.
By condition (3) of the above, we have $$\begin{aligned}
\| [u, \psi _k ] \| &\leq \int _{[0,1)} \sum _{n,m}\sum _{i=1}^{l_k} \| [\sigma _{\gamma ,0}^0(e_{n,m}), \psi _{k,i} \circ \sigma _{-\gamma,\gamma}^0] \| \ d\gamma \\
&=\int _{[0,1)} \sum _{n,m}\sum _{i} \| [e_{n,m}, \psi _{k,i}] \| \ d\gamma \\
&<\int _{[0,1)} \sum _{n,m}\sum _{i} \frac{\epsilon}{ABl_k} \ d\gamma \\
&=\epsilon .\end{aligned}$$ By conditions (2) and (5), we have $$\begin{aligned}
&\| \beta ^0 (u)-e^{-2\pi is}u \| _\psi ^\sharp \\
&=\int _{[0,1)} \| \beta ^{0,\gamma } \circ \sigma ^0_{\gamma, 0}( \sum _{m,n} (e^{2\pi i ((n+\gamma )t+ms)}e_{n,m})) \\
& -\sum _{m,n} e^{2\pi i ((n+\gamma )t +(m-1)s)}\sigma ^0_{\gamma , 0}(e_{n,m}) \| _{\psi _0 \circ \sigma ^0_{-\gamma, \gamma}}^\sharp \ d\gamma \\
&=\int _{[0,1)} \| \sigma ^0 _{\gamma, 0}(\beta ^{0,0}(\sum _{m,n}e^{2\pi i((n+\gamma )t+ms)}e_{n,m}) \\
&-\sum _{m,n} e^{2\pi i ((n+\gamma )t+(m-1)s)}e_{n,m}) \| _{\psi _0\circ \sigma ^0_{-\gamma , \gamma}}^\sharp \ d\gamma \\
&\leq \sum _{m,n, m\not =A}\int _{[0,1)} \| e^{2\pi i ((n+\gamma )t+ms)}\sigma ^0_{\gamma ,0}(\beta ^{0,0}(e_{n,m})-e_{n,m+1}) \| _{\psi _0 \circ \sigma ^0_{-\gamma, \gamma}}^\sharp \ d\gamma \\
&+ \int _{[0,1)} \| \sigma ^0_{\gamma, 0} (\beta ^{0,0}(\sum _n e_{n,A})-\sum _n e_{n,A+1})\| _{\psi _0\circ \sigma ^0 _{-\gamma , \gamma}}^\sharp \ d\gamma \\
&\leq \sum _{m,n,m\not =A} \int _{[0,1)} \| \beta ^{0,0}(e_{n,m})-e_{n,m+1} \| _{\psi _0}^\sharp \ d\gamma +2/\sqrt{A} \\
&<AB(\frac{\epsilon}{AB})+2\epsilon \\
&=3\epsilon .\end{aligned}$$ Condition (2) is used in the fourth inequality and condition (5) is used in the fifth inequality. Next, we will compute $\| \sigma ^0 _q (u)-e^{2\pi itq}u \| _\psi ^\sharp$ for $q\in [-T,T]$. In order to do this, the following observation is useful. Let $\gamma \in [0,1)$ and let $q\in [-T,T]$. Choose $l_\gamma \in \mathbf{Z}$ so that $\gamma -q +l_\gamma \in [0,1)$. Then we have $$\begin{aligned}
(\sigma ^0_q(u))_\gamma &=\sigma ^0_{q, \gamma -q}(u_{\gamma -q+l_\gamma}) \\
&=\sigma ^0_{\gamma,0} \circ \sigma ^{0,0}_{l_\gamma} \circ \sigma ^0_{q-l_\gamma -\gamma, \gamma-q+l_\gamma}(u_{\gamma -q+l_\gamma}) \\
&=\sigma ^0_{\gamma , 0} \circ \sigma ^{0,0}_{l_\gamma} (e^{2\pi it(\gamma -q+l_\gamma )}\sum _{n,m}e^{2\pi i(nt+ms)}e_{n,m}) \\
&=e^{2\pi it(\gamma -q +l_\gamma )}\sigma ^0_{\gamma, 0} \circ \sigma ^{0,0}_{l_\gamma }(u_0).\end{aligned}$$ By conditions (2) and (4), we have $$\begin{aligned}
&\| \sigma ^0_q(u)-e^{-2\pi itq}u\| _{\psi}^\sharp \\
&= \int _{[0,1)} \| \sigma ^0 _{\gamma , 0} (e^{2\pi it(\gamma -q+l_\gamma )} \sigma ^{0,0}_{l_\gamma }(u_0) -e^{-2\pi itq}u_\gamma) \| _{\psi _0 \circ \sigma _{-\gamma, \gamma}^0}^\sharp \ d\gamma \\
&= \int _{[0,1)} \| \sum _{n,m}(e^{2\pi i((\gamma -q+l_\gamma +n)t+ms)}\sigma ^{0,0} _{l_\gamma }(e_{n,m})-e^{2\pi i((\gamma +n-q)t+ms)}e_{n,m}) \| _{\psi _0}^\sharp \ d\gamma \\
&\leq \int _{[0,1)} \sum _{m,l_\gamma<n\leq B-1} \| e^{2\pi i((\gamma -q+n)t+ms)}\sigma ^{0,0} _{l_\gamma }(e_{n-l_\gamma ,m}) \\
&-e^{2\pi i((\gamma +n-q)t+ms)}e_{n,m}) \| _{\psi _0}^\sharp \ d\gamma +2\epsilon \\
&<(\frac{\epsilon}{AB})AB +2\epsilon \\
&=3\epsilon .\end{aligned}$$ Condition (2) is used in the third inequality and condition (4) is used in the fourth inequality. Thus $\{ \sigma ^0_q \circ \beta ^0_m\}_{(q,m)}$ has the Rohlin property. Thus Lemma \[14\] holds.
\[r\] When $\mathrm{ker}(\mathrm{mod}(\alpha _s) \circ (\theta \mid_C)_t)\cong \mathbf{R}$, Lemma \[key\] holds.
There exists $(p,q)\in (\mathbf{R}\setminus \{0\})^2$ with $\mathrm{ker}(\mathrm{mod}(\alpha ) \circ \theta \mid _C)=(p,q)\mathbf{R}$. Set $\sigma _t:=\theta _{qt}\circ \tilde{\alpha} _{pt}$ for $t\in \mathbf{R}$. In order to show our lemma, it is enough to show that for each $r\in \mathbf{R}$, the action $\sigma $ admits a sequence of unitaries which satisfies the same conditions as in Lemma \[10\]. Take a normal faithful state $\phi _0$ of $N$, $\phi _1, \cdots, \phi _n\in N_*$ with $\|\phi _k \| =1$ ($k=1, \cdots,n$), $\epsilon >0$ and $T>0$. Think of $C$ as a standard probability measured space $L^\infty (\Gamma, \mu )$. Let $$N=\int _{\Gamma}^\oplus N_{\gamma} \ d\mu (\gamma ),$$ $$\phi _k =\int _\Gamma^\oplus \phi _{k,\gamma} \ d\mu (\gamma)$$ ($k=1, \cdots, n$) be the direct integral decompositions. Then by Theorem \[rp\] and Proposition \[A.5\], there exists a Borel subset $A$ of $\Gamma$ which satisfies the following three conditions.
\(1) There exists a large cube $Q:=[-T',T']$ and a $Q$-set $Y$ such that $A=T_QY$ and the map $Q\times Y \ni (t,x) \mapsto T_t(x)\in A$ is injective.
\(2) We have $$\mu (\bigcap _{t\in [-T,T]}T_tA) >1-\epsilon$$ and $$\int _{\bigcap _{t\in [-T,T]}T_tA}\| \phi _{k,\gamma}\| \ d\mu (\gamma )>1-\epsilon$$ for $k=1, \cdots ,n$.
\(3) There is a measure $\nu $ on $Y$ such that the map $Q\times Y\ni (t,x) \mapsto T_t(x) \in A$ is a non-singular isomorphism with respect to $\mathrm{Lebesgue} \otimes \nu $ and $\mu$ (Note that two measures $\mu +\sum _k \int _{\Gamma}\| \phi _{k,\gamma}\| \ d\mu (\gamma )$ and $\mu $ are mutually equivalent).
Here, we do not assume the existence of invariant probability measures for $\theta \mid _C$. Then $N$ is isomorphic to $$N_{\Gamma \setminus A}\oplus \int _{ [-T',T']} ^\oplus N_s \ ds.$$ Here, $$N_s=\int _{Y}^\oplus N_{(y,s)} \ d\nu (y) .$$ For $s,t\in [-T',T']$, $\theta $ defines an isomorphism $\theta _{s-t,t}$ from $N_t$ to $N_s$ by $\theta _{s-t,t}(x_t)=(\theta _{s-t}(x))_s$. As in Lemma \[10\], we identify $N_t$ with $N_0$ by this isomorphism. By this identification, we approximate $\phi _k$’s by simple step functions. $$\| \phi _k \chi _{A}-\sum _{i=1}^{l_k}\phi _{k,i}\circ \theta _{-t,t} \chi _{I_i}(t) \| <\epsilon$$ for $k=1,\cdots, n$, where $\phi _{k,i}\in (N_0)_*$ and $\{I_i\}_{i=1}^{l_k}$ are partitions of $[-T',T']$. Here, we note that it is possible to choose $\phi _{0,i}$’s so that they are positive. This is shown by the following way. Since $\phi _0: [-T',T'] \to (N_0)_*$ is measurable, by Lusin’s theorem, it is possible to choose a sufficiently large compact subset $K$ of $[-T',T']$ on which $\phi _0$ is continuous. Choose a finite partition $\{s_i\}_{i=1}^{l_0}$ of $K$ so that for every $ s\in K$, there exists a number $i$ such that $\phi _0(s) $ is close to $\phi (s_i)$. It is possible to choose a partition $\{ I_i\}$ of $K$ so that $\phi _0(s)$ is close to $\phi _0(s_i)$ on $I_i$. Then $\sum \phi _{0}(s_i)\chi _{I_i}$ well approximates $\phi _0$. Since $\sigma $ fixes the center of $N$, this is decomposed into the direct integral. $$\sigma =\sigma _{\Gamma \setminus A} \oplus \int _{[-T',T']}\sigma ^{t} \ dt.$$ Since $\sigma $ scales the canonical trace on $N$, for almost all $t\in \mathbf{R}$, the action $\sigma ^t$ is trace-scaling, and hence has the Rohlin property by Theorem 6.18 of Masuda–Tomatsu [@MT]. Hence, by the same argument as in the proof of Lemma \[14\], it is possible to choose a unitary element $u_0$ of $N_0$ satisfying the following conditions.
\(1) We have $\| [u_0, \phi _{k,i}]\| < \epsilon /(2l_kT')$ for $k=1, \cdots, n$, $i=1, \cdots, l_k$.
\(2) We have $\| \sigma ^{0}_t(u_0)-e^{-ipt}u_0 \| _{\phi _{0,i}}^\sharp <\epsilon /(2l_0T')$ for $t\in[-T,T]$, $i=1, \cdots, l_0$.
Set $u_t:=\theta _{t,0}(u_0)$ for $t\in [-T',T']$ and set $$u:=\chi _{X\setminus A} \oplus \int _{[-T',T']}^\oplus u_t \ dt.$$ Hence by the same aregument as in the proof of Lemma \[13\] $\Rightarrow $ Lemma \[10\], Lemma \[key\] holds.
\[z\] When $\mathrm{ker}(\mathrm{mod}(\alpha _s)\circ (\theta \mid _C) _t) \cong \mathbf{Z}$, Lemma \[key\] holds.
Let $(p,q )\in (\mathbf{R}\setminus \{0\})^2$ be a generator of $\mathrm{ker}(\mathrm{mod}(\alpha _s) \circ (\theta \mid_C)_t)$. Set $\sigma _t:=\theta _{qt}\circ \tilde{\alpha }_{pt}$ for $t\in \mathbf{R}$. Think of $C^\sigma $ as a standard probability space $L^\infty (\Gamma, \mu )$. We first show the following claim.
**Claim**. The action $\theta :\mathbf{R}\curvearrowright C^\sigma $ is faithful (and hence is free).
*Proof of Claim*. Assume that $\theta _t\mid _{C^\sigma}=\mathrm{id}_{C^\sigma}$. Then $\theta$ is decomposed into the direct integral over $\Gamma$. $$\theta _t=\int _{ \Gamma}^\oplus \theta _t^\gamma \ d\mu (\gamma ) ,$$ $$N=\int _{ \Gamma}^\oplus N_{ \gamma } \ d\mu (\gamma ).$$ We also decompose $\sigma $ by $$\sigma _s =\int _{\Gamma}^\oplus \sigma ^\gamma _s \ d\mu (\gamma ) .$$ Then for almost all $\gamma \in \Gamma$, $\{ \sigma ^\gamma _t\}_{t\in \mathbf{R}}$ defines a periodic ergodic action on the center of $N_\gamma$. Since the restriction of $\theta ^\gamma _t$ on the center of $N_\gamma $ commutes with that of $\sigma ^\gamma _s$’s, $\theta ^\gamma _t \mid _{Z(N_\gamma )}$ is of the form $\sigma ^\gamma _{s_\gamma}\mid _{Z(N_\gamma )}$. We show that there exists $s\in [0,1)$ such that $s_\gamma =s$ for almost all $\gamma $. Since we want to show the faithfulness of the action $\theta $, we may assume that $t\not =0$. We think of $C$ as a probability measured space $L^\infty (X, \mu_X)$. Then there exists a projection $p$ from $X$ to $\Gamma$ induced by the inclusion $L^\infty (\Gamma ) \to L^\infty (X)$. Let $T,S $ be two flows on $(X, \mu _X)$ defined by $f(T_tx)=\theta _{-t}(f)(x)$, $f(S_sx)=\sigma _{-s}(f)(x)$ for $x\in X$, $f\in L^\infty(X, \mu _X)$. We may assume that $X$ is a separable compact Hausdorff space and $T$ and $S$ are continuous. We show that the set $$A_s= \{ x\in X \mid T _t\circ S _{r}(x)=x \ \mathrm{for} \ \mathrm{some}\ 0\leq r\leq s\}$$ is Borel measurable. Let $f:\mathbf{R}\times X \to X^2$ is a map defined by $f:\mathbf{R}\times X\ni (s,x) \mapsto (T _t\circ S _s(x),x)\in X^2$. Then we have $A_s=\pi _X(f^{-1}(\Delta )\cap ([0,s] \times X))$, which is Borel measurable. Here, $\Delta $ is the diagonal set of $X\times X$ and $\pi _X :\mathbf{R} \times X \to X$ is the projection.
Next we show that there exists $s \in [0,1)$ such that $$B_s:=\{ x\in X \mid T_t\circ S_s(x)=x\}$$ has a positive measure. If not, the map $s\to \mu _X(A_s)$ would be continuous. By the first part of this proof, for each $\gamma \in \Gamma$, if $x\in X$ satisfies $p(x)=\gamma$, then we have $x\in A_{s_\gamma}$. Hence $\bigcup _{s>0}A_s$ is full measure. On the other hand, since $t\not =0$, we have $\mu (A_0)=0$. Thus there would exist $s \in [0,1)$ with $\mu _X(A_s)=1/2$. However, this would contradict to the ergodicity of $\theta$. Thus there exists $s\in [0,1)$ with $\mu _X(B_s)>0$.
By using the ergodicity of $\theta $ again, there exists $s\in [0,1)$ such that $B_s$ is full measure.
Hence there exists $s\in [0,1)$ such that $\sigma _s\mid _C =\theta _t\mid _C$. Since $\mathrm{ker}(\mathrm{mod}(\alpha )\circ (\theta \mid _C))=(p,q)\mathbf{Z}$, we have $s=t=0$, which is a contradiction. Hence Claim is shown.
Now, we return to the proof of Lemma \[z\]. For almost all $\gamma \in \Gamma$, the action $\sigma ^\gamma \mid _{Z(N_0)}$ is ergodic and has a period $1$, and $\sigma ^\gamma $ is trace-scaling. Hence this is the dual action of a modular automorphism of an AFD $\mathrm{III}_\lambda$ ($0<\lambda <1$) factor. Hence $\sigma ^\gamma $ has the Rohlin property. Hence by the same argument as in Lemma \[r\], our lemma is shown.
\[lambda\] When $M$ is of type $\mathrm{III}_\lambda$, $0<\lambda <1$, then Connes–Takesaki module of a flow on $M$ cannot be faithful. This is shown by the following way. Since $\mathrm{mod}(\alpha )$ commutes with $\theta$, as we have seen, this is a homomorphism from $\mathbf{R}$ to $\mathbf{T}$. Hence $\mathrm{mod}(\alpha ) $ cannot be faithful.
Remarks and Examples
====================
In this section, we present examples which have interesting properties.
Model Actions {#model}
-------------
In this subsection, we will construct model actions. If there were no flows with faithful Connes–Takesaki modules, then our main theorem would have no value. Hence, it is important to construct a flow which has a given flow as its Connes–Takesaki module.
Let $M$ be an AFD factor with its flow space $\{C, \theta \}$ and let $\sigma $ be a flow on $C$ which commutes with $\theta $ Here, we do not assume the faithfulness of $\sigma $. Then there exists a Rohlin flow $\alpha $ on $M$ with $\mathrm{mod}(\alpha )=\sigma $.
The proof is modeled after Masuda [@M]. As in Corollary 1.3 of Sutherland–Takesaki [@ST3], there exists an exact sequence $$1 \to\overline{ \mathrm{Int}}(M) \to \mathrm{Aut}(M) \to \mathrm{Aut}_\theta (C) \to 1,$$ and there exists a right inverse $s: \mathrm{Aut}_\theta (C) \to \mathrm{Aut}(M)$. The maps $p: \mathrm{Aut}(M)\to \mathrm{Aut}_\theta (C)$ and $s: \mathrm{Aut}_\theta (C) \to \mathrm{Aut}(M)$ are continuous.
Hence for a flow $\sigma $ on $C$ commuting with $\theta $, the homomorphism $\alpha :=s\circ \sigma :\mathbf{R}\to \mathrm{Aut}(M)$ gives an action with its Connes–Takesaki module $\sigma $. If $\sigma $ is faithful, by our main theorem, this has the Rohlin property. Assume that $\sigma $ is not faithful. Then $\mathrm{mod}(\alpha \otimes \beta )=\mathrm{mod}(\alpha )$ for a Rohlin flow $\beta $ on the AFD factor of type $\mathrm{II}_1$. Hence this $\alpha \otimes \beta$ does the job.
For actions on the AFD factor of type $\mathrm{II}_1$, strong cocycle conjugacy is equivalent to cocycle conjugacy because every automorphism of the AFD factor of type $\mathrm{II}_1$ is approximated by its inner automorphisms. However, for flows on some AFD factor of type $\mathrm{III}_0$, cocycle conjugacy does not always imply strong cocycle conjugacy.
Let $(X, \mu )$ be a probability measured space defined by $$(X, \mu ):=(\prod _{m\in \mathbf{Z}}(\prod _{n\in \mathbf{Z}}(\{ 0,1\},\{ \frac{1}{2}, \frac{1}{2} \} ))).$$ Let $S$, $T$ be two automorphisms of $X$ defined by the following way. $$S(m\mapsto (n\mapsto {x_n}^m \in \{0,1 \} )) =(m\mapsto (n\mapsto {x_{n+1}}^m)),$$ $$T(m\mapsto (n\mapsto {x_n}^m )) =(m\mapsto (n\mapsto {x_n}^{m+1})).$$ Then both $S$ and $T$ are ergodic and satisfy $S \circ T= T\circ S$. Let $\beta _1$, $\beta_2$ be two flows on $L^\infty(\mathbf{T})$ satisfying the following conditions.
Two flows are faithful.
The flow $\beta _1 $ is NOT conjugate to $\beta _2$.
Two flows preserve the Lebesgue measure.
As a probability measured space, we have $$\prod _{n\in \mathbf{Z}}\{ 0,1 \} ^n =(\prod _{n:\mathrm{odd}}\{ 0,1 \} ^n) \times (\prod _{n:\mathrm{even}}\{ 0,1 \} ^n )\cong \mathbf{T}^2.$$ By this identification, we set $$\beta :=( \bigotimes _{m\in \mathbf{Z}}(\beta _1\otimes \beta _2))\otimes \mathrm{id}: \mathbf{R}\curvearrowright L^\infty (X\times [0,1)).$$ Let $\theta $ be a flow on $L^\infty(X\times [0,1))$ defined by $T$ and the ceiling function $r=1$. Let $\rho $ be an automorphism of $L^\infty (X\times [0,1))$ defined by $S\times \mathrm{id}$. Then we have the following.
The flow $\theta $ commutes with both $\rho $ and $\beta $.
The action $\rho $ does not commute with $\beta $.
The flow $\theta $ is ergodic.
Now, we construct a pair of flows which are mutually cocycle conjugate but not strongly cocycle conjugate. Let $M$ be an AFD factor of type $\mathrm{III}_0$ with its flow of weights $\{\theta , X\times [0,1)\}$, $\alpha $ be a Rohlin action satisfying $\mathrm{mod}(\alpha )=\beta$ and let $\sigma $ be an an automorphism of $M$ satisfying $\mathrm{mod} (\sigma )=\rho $. Then we have $$\mathrm{mod}(\alpha )=\beta \not =\rho \circ \beta \circ \rho ^{-1}=\mathrm{mod}(\sigma \circ \alpha \circ \sigma ^{-1}).$$ Hence $\alpha $ is cocycle conjugate to $\sigma \circ \alpha \circ \sigma ^{-1}$ but they are not strongly cocycle conjugate.
On Stability {#extend}
------------
In Thoerem 5.9, Izumi [@I] has shown that an action of a compact group on any factor of type $\mathrm{III}$ with faithful Connes–Takesaki module is minimal. As well as our main theorem, this theorem means that actions which are “very outer” at any non-trivial point are “globally outer”. He has also shown that for these actions, cocycle conjugacy coincides with conjugacy. This phenomenon also occurs for trace-scaling flows on any factor of type $\mathrm{II}_\infty$. Hence it may be expected that this is true under our assumption. However, this is not the case.
\[non-stable\] Let $C$ be an abelian von Neumann algebra and $\theta $ be an ergodic flow on $C$. Let $M$ be an AFD factor with its flow of weights $(C, \theta )$. Let $\beta $ be a faithful flow on $C$ which commutes with $\theta$ and fixes a normal faithful semifinite weight $\mu $ of $C$. If the discrete spectrum of $\beta $ is NOT $\mathbf{R}$, then there are two flows $\alpha ^1$, $\alpha ^2$ which satisfies the following two conditions.
The Connes–Takesaki modules of $\alpha ^1 $ and $\alpha ^2 $ are $\beta$.
The flow $\alpha ^1 $ is NOT conjugate to $\alpha ^2$.
Before starting the proof, it is important to note that from the above theorem, cocycle conjugacy does not coincide with conjugacy at all. For example, if $\beta $ preserves a normal faithful state of $C$, then its discrete spectrum is countable. Hence it seems that our situation is different from that of actions of compact groups.
We return to the proof of the theorem. In the following, we actually construct these flows. In the following, we denote the AFD factor of type $\mathrm{II}_1$ by $R_0$ and denote the AFD factor of type $\mathrm{II}_\infty $ by $R_{0,1}$.
Let $\Lambda$ be the discrete spectrum of $\beta$ and $\mu $ be a $\beta$-invariant measure. In the rest of this subsection, we assume that $\Lambda $ is not $\mathbf{R}$. Then by the ergodicity of $\theta $ (Note that $\beta $ may not be ergodic), $\Lambda $ is a proper subgroup of $\mathbf{R}$. Hence there are at least two real numbers which do not belong to $\Lambda $. Let $\Gamma _j$ $(j=1,2)$ be two subgroups of $\mathbf{R}$ generated by two elements $\lambda_j, \mu _j$, respectively, satisfying the following conditions.
$$\Gamma _1 \cup \Lambda \not \subset \langle \Gamma _2, \Lambda \rangle ,$$
$$\Gamma _2 \cup \Lambda \not \subset \langle \Gamma _1, \Lambda \rangle .$$
Here, $\langle \Gamma _i, \Lambda \rangle $ is the subgroup of $\mathbf{R}$ generated by $\Gamma_i$ and $\Lambda $. Let $\gamma ^j$ ($j=1,2$) be two ergodic flows on $R_0$ with their discrete spectrum $\Gamma _j$, respectively. Namely, we think of $R_0$ as a weak closure of an irrational rotation algebra $A_s:=\mathrm{C}^*(u,v\mid u,v : \mathrm{unitaries} \ \mathrm{satisfying} \ vu=e^{2\pi is}uv)$ and define flows $\gamma ^j$, $j=1,2$, by the following way. This type of actions is considered by Kawahigashi [@Kwh1].
$$\gamma ^j_t(u)=e^{i\lambda _j t}u,\ \gamma ^j_t(v)=e^{i\mu _j t}v$$ for $t\in \mathbf{R}$.
Set $\tau :=\mu \otimes \tau _{R_{0,1}}\otimes \tau _{R_0}$. The flow $\theta $ is extended to a $\tau$-scaling flow on $N:=C\otimes R_{0,1}\otimes R_0$ as in equations (1.2) of Sutherland–Takesaki [@ST3]. Set $\overline{\alpha ^j}:=\beta \otimes \mathrm{id}_{R_{0,1}}\otimes \gamma ^j$. Then $\overline{\alpha ^j}$ commutes with $\theta$ (See the equation after equation (1.8) of Sutherland–Takesaki [@ST3]). Hence the flow $\overline{\alpha ^j}$ is extended to $M:=N\rtimes _\theta \mathbf{R}$ in the following way.
$$\alpha ^j_t(\lambda ^\theta _s)=\lambda ^\theta _s$$ for $s,t\in \mathbf{R}$. Note that the flow $\theta :\mathbf{R}\curvearrowright N$ is not so “easy”. However, the flow $\alpha ^j$ is very concrete. Here, we think of $M$ as a von Neumann algebra generated by $N$ and a one parameter unitary group $\{ \lambda _s \} _{s\in \mathbf{R}}$.
In order to show Theorem \[non-stable\], for these $\alpha ^j$’s, it is enough to show the following lemma.
\[two statements\] In the above context, we have the following two statements.
The Connes–Takesaki module of $\alpha ^j$ is $\beta $ for each $j=1,2$.
For the discrete spectrum of $\alpha ^j$, we have the following inclusion. $$\Gamma ^j \cup \Lambda \subset \mathrm{Sp}_d(\alpha ^j)\subset \langle \Gamma ^j , \Lambda \rangle .$$
From statement (1) of the lemma and our main theorem, it is shown that $ \alpha ^1$ and $\alpha^2$ are mutually cocycle conjugate. On the other hand, from statement (2) of the lemma, it is shown that the discrete spectrum of $\alpha ^1$ and that of $\alpha ^2 $ are different. Hence they are not conjugate.
In order to show this lemma, we first show the following lemma.
The weight $\hat{\tau}$ is invariant by $\alpha ^j$.
Set $$n_\tau:=\{a\in N \mid \tau (a^*a)<\infty\},$$ $$K(\mathbf{R},N):=\{ x:\mathbf{R}\to N\mid \mathrm{strongly}^* \ \mathrm{continuous} \ \mathrm{map} \ \mathrm{with} \ \mathrm{compact} \ \mathrm{support}\} ,$$ $$b_\tau :=\mathrm{span}\{ xa \mid x\in K(\mathbf{R},N), a\in n_\tau \}.$$ For $x\in b_\tau$, set $$\tilde{\pi}(x):=\int_{\mathbf{R}}x_t\lambda _t^\theta \ dt.$$
In order to show this lemma, it is enough to show the following two statements (For example, see Theorem X.1.17. of Takesaki [@T1]).
\(1) For $s,t\in \mathbf{R}$, we have $\sigma ^{\hat{\tau }}_t=\alpha ^j _{-s} \circ \sigma ^{\hat{\tau }}_t \circ \alpha ^j_s$.
\(2) For $x\in b_\tau$, $s\in \mathbf{R}$, we have $$\hat{\tau}\circ \alpha ^j_s (\tilde{\pi}(x)^*\tilde{\pi}(x))=\hat{\tau}(\tilde{\pi} (x)^*\tilde{\pi}(x)).$$
Statement (1) is trivial because $\alpha ^j$ commutes with $\sigma ^{\hat{\tau}}$. We show statement (2). Notice that $$\begin{aligned}
\alpha _s^j(\tilde{\pi}(x)) &=\alpha _s^j (\int _{\mathbf{R}}x_t\lambda _t \ dt) \\
&=\int _{\mathbf{R}}\overline{\alpha ^j}_s(x_t)\lambda _t \ dt \\
&=\tilde{\pi}(\overline{\alpha ^j}_s(x)).\end{aligned}$$
Since $\tau $ is invariant by $\overline{\alpha ^j}$, we have $$\begin{aligned}
\hat{\tau}\circ \alpha ^j _s(\tilde{\pi}(x)^*\tilde{\pi}(x)) &=\hat{\tau}(\tilde{\pi}(\overline{\alpha ^j}_s(x))^*\tilde{\pi}(\overline{\alpha ^j}_s(x))) \\
&=\tau (\int _{\mathbf{R}} \overline{\alpha ^j}_s (x_t^*x_t) \ dt) \\
&=\tau (\int _{\mathbf{R}}x_t^*x_t \ dt) \\
&=\hat{\tau}(\tilde{\pi}(x)^*\tilde{\pi}(x)).\end{aligned}$$ Thus statement (2) holds.
By this lemma, the canonical extention $\tilde{\alpha ^j}$ of $\alpha ^j$ is defined by $\tilde{\alpha ^j}_t(\lambda ^\sigma _s)=\lambda ^\sigma _s$ if we think of $\tilde {M}:=M\rtimes _{\sigma ^{\hat{\tau}}}\mathbf{R}$ as a von Neumann algebra generated by $M$ and a one parameter unitary group $\{\lambda ^\sigma _t\}$.
Hence by Lemma 13.3 of Haagerup–Størmer [@HS], if we identify $N\rtimes _\theta \mathbf{R} \rtimes _\sigma \mathbf{R}$ with $N \otimes B(H)$ by Takesaki’s duality theorem, we have $$\tilde{\alpha ^j} \cong \overline{\alpha ^j}\otimes \mathrm{id}.$$ Thus statement (1) of Lemma \[two statements\] holds.
In the following, we show statement (2) of Lemma \[two statements\]. We need to show the following lemma.
\[spect\] We have $\mathrm{Sp}_d\overline{(\alpha ^j})=\mathrm{Sp}_d(\alpha ^j)=\mathrm{Sp}_d(\tilde{\alpha ^j})$.
The action $\alpha ^j$ is an extension of the action $\overline{\alpha ^j}$, and the action $\tilde{\alpha ^j}$ is an extension of the action $\alpha ^j$. Hence we have $\mathrm{Sp}_d(\overline{\alpha ^j})\subset \mathrm{Sp}_d(\alpha ^j) \subset \mathrm{Sp}_d(\tilde {\alpha ^j})$. We show the implication $\mathrm{Sp}_d(\tilde{\alpha ^j})\subset \mathrm{Sp}_d(\overline{\alpha ^j})$. Note that if we identify $N\rtimes _\theta \mathbf{R} \rtimes _\sigma \mathbf{R}$ with $N\otimes B(H)$ by Takesaki’s duality theorem, we have $\tilde {\alpha ^j}=\overline{\alpha ^j} \otimes \mathrm{id}$. Choose $p\in \mathrm{Sp}_d(\tilde{\alpha ^j})$. Then there exists a non-zero element $x\in N\otimes B(H)$ with $\tilde{\alpha ^j}_t(x)=e^{ipt}x$ for $t\in \mathbf{R}$. If we write $x=(x_{kl})_{kl}\in N\otimes B(l^2(\mathbf{N}))$, then there exists $(k,l)$ with $x_{kl} \not =0$. Since we have $\overline{\alpha ^j}_t(x_{kl})=e^{ipt}x_{kl}$, we have $p\in \mathrm{Sp}_d(\overline{\alpha ^j})$.
Now, we return to the proof of statement (2) of Lemma \[two statements\], which completes the proof of Theorem \[non-stable\].
*Proof of Lemma \[two statements\].* The inclusion $\Gamma _j\cup \Lambda \subset \mathrm{Sp}_d(\overline{\alpha ^j})$ is trivial. We show the inclusion $\mathrm{Sp}_d(\overline{\alpha ^j})\subset \langle \Gamma _j, \Lambda \rangle$. If we think of $N=C\otimes R_{0,1}\otimes R_0$ as a subalgebra of $C\otimes B(H)\otimes R_0$, then $\overline{\alpha ^j} $ extends to $\beta \otimes \mathrm{id}_{B(H)}\otimes \gamma ^j$. Hence by the same argument as in Lemma \[spect\], we have $\mathrm{Sp}_d(\overline{\alpha ^j})=\mathrm{Sp}_d(\beta \otimes \gamma^j)$. Choose $p\in \mathrm{Sp}_d(\beta \otimes \gamma ^j)$. Let $x\in C\otimes R_0$ be a non-zero eigenvector for $p\in \mathrm{Sp}_d(\overline{\alpha^j})$. Then $x $ is expanded as $$x =\sum _{n,m}c_{n,m} u^nv^m$$ with $c_{n,m} \in C$ ($n,m\in \mathbf{Z}$). Hence we have $$\begin{aligned}
\sum _{n,m} e^{ipt}c_{n,m}u^nv^m &= e^{ipt}x \\
&=\beta _t \otimes \gamma ^j _t (x) \\
&=\sum _{n,m} \beta _t (c_{n,m}) e^{i(n\lambda _j + m\mu _j)t}u^nv^m. \end{aligned}$$ Since $x \not =0$, there exists $(n,m)$ with $c_{n,m}\not =0$. Hence by the uniqueness of the Fourier expansion, we have $$\beta _t(c_{n,m})=e^{i(p-n\lambda_j -m\mu _j )t} c_{n,m}.$$ Thus $p\in \langle \Gamma ^j, \Lambda \rangle$.
\(1) As shown in Corollary 8.2 of Yamanouchi [@Y], if we further assume that $\alpha ^1$, $\alpha ^2$ and $\beta $ are integrable, then $\alpha ^1$ is conjugate to $\alpha ^2$. In this case, $\beta $ contains the translation of $\mathbf{R}$ as a direct product component.
\(2) Another important difference between flows and actions of compact groups is about extended modular actions. The duals of extended modular flows are important examples of flows with faithful Connes–Takesaki modules (See Theorem 4.20 of Masuda–Tomatsu [@MT]). Actions of compact groups with faithful Connes–Takesaki modules are duals of skew products (See Definition 5.6 and Theorem 5.9 of Izumi [@I]). However, this is not true for flows by subsection \[model\] of this paper and Theorem 4.20 of Masuda–Tomatsu [@MT].
On a Characterization of the Rohlin Property {#generalization}
--------------------------------------------
One of the ultimate goals of the study of flows is to completely classify all flows on AFD von Neumann algebras. In order to achieve this, it is important to characterize the Rohlin property by using invariants for flows. A candidate for this characterization is the following conjecture.
\[conj\] Let $M$ be an AFD von Neumann algebra and let $\alpha $ be a flow on $M$. Let $\tilde{\alpha }:\mathbf{R} \curvearrowright \tilde{M}$ be a canonical extension of $\alpha$. Then the following three conditions are equivalent.
The action $\alpha $ has the Rohlin property.
We have $\pi _{\tilde{\alpha }}(\tilde{M})' \cap \tilde{M} \rtimes _{\tilde{\alpha }}\mathbf{R}=\pi _{\tilde{\alpha }}(Z(\tilde {M}))$.
The action $\alpha $ has full Connes spectrum and is centrally free.
We will give a partial answer for this conjecture by generalizing Theorem \[Rohlin\]. We start off by showing the following lemma.
Let $M$ be an AFD factor of type III. Let $\alpha $ be an automorphism of $M$ with trivial Connes–Takesaki module. Then $\alpha $ is centrally outer if and only if $\tilde{\alpha}^\gamma $ is outer for almost every $\gamma \in \Gamma$. Here, $C=L^\infty (\Gamma, \mu)$ is the center of $\tilde{M}$ and $\tilde{\alpha } =\int _\Gamma ^\oplus \tilde{\alpha }^\gamma d\mu (\gamma )$ is the direct integral decomposition. \[ct\]
This is shown by Proposition 5.4 of Haagerup–Størmer [@HS2] and Theorem 3.4 of Lance [@La].
In order to state our theorem, we define the following notion.
Let $C$ be an abelian von Neumann algebra and let $\beta $ be a flow on $C$. Then $\beta $ is said to be nowhere trivial if for any $e\in \mathrm{Proj}(C^\beta )$, $\beta \mid _{C_e}$ is not $\mathrm{id}_{C_e}$ as a flow.
The following theorem means that we need not consider Conjecture \[conj\] for flows on AFD von Neumann algebras of type $\mathrm{III}_0$ anymore.
\[characterization\]
Let $M$ be a von Neumann algebra of type $\mathrm{III}_0$ and $\alpha $ be a flow on $M$. Assume that $\mathrm{mod}(\alpha )$ is nowhere trivial, then conditions in the above conjecture are all equivalent to the following condition.
The action $\alpha $ is centrally free.
If conditions are equivalent for flows on the AFD factor of type $\mathrm{II}_\infty $, then these conditions are also equivalent for flows on AFD von Neumann algebras of type $\mathrm{III}_0$.
**Step 0.** The implications (1) $\Rightarrow $ (2) and (2) $\Rightarrow $ (3) are shown in Lemma 3.17 and Corollary 4.13 of Masuda–Tomatsu [@MT]. The implication (3) $\Rightarrow $ (4) is trivial.
**Step 1.** First, we show (a) and (b) when $M$ is a factor.
\(a) We show the implication (4) $\Rightarrow $ (1). If $\mathrm{mod}(\alpha ):\mathbf{R} \curvearrowright Z(M) $ is faithful, then $\alpha $ satisfies condition (1) by Theorem \[Rohlin\]. In the following, we assume that $\mathrm{mod}(\alpha ) $ is not faithful. By the ergodicity of $\theta $, $\mathrm{mod}(\alpha ) $ has a non-trivial period $p\in (0, \infty)$. Since $\theta $ is faithful and commutes with $\mathrm{mod}(\alpha )$, $C^{\mathrm{mod}(\alpha )}$ is not trivial. Hence, the restriction of $\theta $ to $C^{\mathrm{mod}(\alpha )}$ is either free or periodic.
When the restriction of $\theta $ to $C^{\mathrm{mod}(\alpha )}$ is free, then the proof goes parallel to Lemma \[z\], using Lemma \[ct\].
When the restriction of $\theta $ to $C^{\mathrm{mod}(\alpha ) }$ is periodic, then the proof goes parallel to Lemma \[z2\].
\(b) What remains to do is to reduce the case when $\mathrm{mod}(\alpha ) $ is trivial to Conjecture \[conj\] for flows on the AFD factor of type$\mathrm{II}_\infty $. This goes parallel to the proof of Lemma \[r\].
**Step 2.** Next, we consider the proof of this theorem for the case when $M$ is not a factor. Decomposing into a direct integral, we may assume that $\alpha $ is centrally ergodic. We need to consider the case when $\alpha \mid_{Z(M)}$ is faithful, the case when $\alpha \mid _{Z(M)}$ has a non-trivial period and the case when $\alpha \mid _{Z(M)}$ is trivial separately. When $\alpha |_{Z(M)}$ is faithful, the implication (4) $\Rightarrow $ (1) follows from Theorem \[rp\] and Proposition \[A.4\]. When $\alpha |_{Z(M)}$ has non-trivial period, then the proof is similar to that of Lemma \[z\]. When $\alpha |_{Z(M)}$ is trivial, then the implication follows from the case when $M$ is a factor.
By the same argument, it is possible to reduce Conjecture \[conj\] for flows on the AFD factor of type $\mathrm{III}_\lambda $ $0<\lambda <1$, $\mathrm{III}_1$ to Conjecture \[conj\] for actions of $\mathbf{R}\times \mathbf{Z}$, actions of $\mathbf{R}^2$ on the AFD factor of type $\mathrm{II}_\infty $, respectively.
Appendix
========
In this section, we explain the proof of two statements which are used in the proof of the main theorem.
Proof of Theorem \[rp\]
-----------------------
For readers who do not have any access to Feldman [@F2], we will explain the outline of the proof of Theorem \[rp\].
*Proof of Theorem \[rp\].*
The proof consists of two parts. The first is, for any cube $Q$ of $\mathbf{R}^d$, constructing a $Q$-set $F$ with $\mu (QF)>0$. This part is shown by the same argument as in the proof of Lemma of Lind [@L] (Note that Wiener’s ergodic theorem holds for actions without invariant measures). The second is to show this theorem by using the first part. This is achieved by the same argument as in the proof of Theorem 1 of Feldman–Lind [@FL]. In the proof, they show two key statements (Statements (i) and (ii) in p.341 of Feldman–Lind [@FL]). We need statements corresponding to them. Let $L$, $N$, $P$ be positive natural numbers. Assume that $P$ is a multiple of $L$. Set $$Q_P:=[0,P)^d,$$ $$S_L(Q_P):=\{ t=(t_1, \cdots , t_d)\in \mathbf{R}^d\mid \frac{P}{L}\leq t_j <P-\frac{P}{L} \ \mathrm{for} \ \mathrm{all} \ j\},$$ $$B_N(Q_P):=\{ t=(t_1, \cdots , t_d)\in \mathbf{R}^d\mid -N \leq t_j <P+N \ \mathrm{for} \ \mathrm{all} \ j\}\setminus Q_P,$$ $$C_{P/L}:=\{ n=(n_1, \cdots , n_d)\in \mathbf{Z}^d \mid 0\leq n_j <\frac{P}{L} \ \mathrm{for} \ \mathrm{all} \ j\}.$$ The corresponding statements are the following.
(i)’ Let $\eta >0$ be a positive number. Then for any sufficiently large even integer $M$, any integer $L$, any multiple $P=NLM$ of $LM$ and any $Q_P$-set $F$, we have $$\mu (B_{2N}(Q_P)(tF))<\eta$$ for over $9/10$ of the elements $t$ of $C_{P/L}$.
(ii)’ Let $\xi>0$ be a positive number. Then for any sufficiently large integer $L$, any integer $M$, any multiple $P=NLM$ of $LM$ by a multiple $N$ of $L$ and any $Q_P$-set $F$, we have $$\mu (S_L(Q_N)(NC_{P/L})(tF))>\mu (Q_P(tF))-\xi$$ for over $9/10$ of the elements $t$ of $C_{P/L}$.
The other parts of the proof is the same as that of Theorem 1 of Feldman–Lind [@FL].
We may assume that $X$ is a compact metric space and the map $T:\mathbf{R}^d\times X\to X$ is continuous.
\[A.3\] In the context of Theorem \[rp\], the set $F$ can be chosen to be a Borel subset of $X$.
This follows from the proof of Lemma of Lind [@L]. By removing a null set, we may assume that the set $D$ in p.181 of Lind [@L] is a Borel subset of $X$. Then the set $$\{ (t,x) \in \mathbf{R}^n \times X \mid T_t(x) \in D \}$$ is a Borel subset of $Q\times X$. Hence by Fubini’s theorem, the map $\psi _j^{\pm }(x)$ in p.181 of Lind is Borel measurable. Thus the set $F$ can be chosen to be a Borel subset.
\[A.4\] In the context of Theorem \[rp\], the map $$Q\times F \ni (t,x) \mapsto T_t(x) \in T_QF$$ is a Borel isomorphism.
By Lemma \[A.3\], if $C\subset T_QF $ is a Borel subset, then $C$ is also Borel in $X$. Hence the map $Q\times F \ni (t,x)\mapsto T_t(x) \in T_QF$ is a Borel bijection. Hence by Corollary A.10 of Takesaki [@T], this map is a Borel isomorphism.
\[A.5\] In the context of Theorem \[rp\], if $\mathbf{R}^d=\mathbf{R}$, then the map $$Q\times F \ni (t,x) \mapsto T_t(x) \in T_QF$$ is non-singular.
This is based on Lemma 3.1 of Kubo [@K]. The action $T$ of $\mathbf{R}$ on $X$ induces an action $\tilde{T}$ of $\mathbf{R}$ on $T_Q(F)$. Then $\tilde{T}$ defines an action $S$ of $\mathbf{Z}$ on $F$. Then $(F, \nu)$, $S$ and $(T_QF,\mu)$ satisfy the assumptions of Lemma 3.1 of Kubo [@K].
On a Measurability of a Certain Map
-----------------------------------
In the proof of Lemma \[10\], we use the fact that a map from a measured space to the automorphism group of a von Neumann algebra is measurable (See Fact between Lemma \[11\] and Lemma \[13\]). Probably it is well-known for specialists. However, we could not find appropriate references. Hence, we present the proof here.
If we identify $N$ with $N_0 \otimes L^\infty ([0,p))$, the map $[0, p)^2\ni(t, \gamma ) \mapsto \theta _{t, \gamma }\in \mathrm{Aut}(N_0)$ is Lebesgue measurable.
By Lusin’s theorem, it is enough to show that the map $[0,p)^2 \ni (t, \gamma ) \mapsto \phi _0 \circ \theta _{t, \gamma } \in (N_0)_*$ is Lebesgue measurable for $\phi _0 \in (N_0)_*$. We identify $N_*$ with $L^1 _{(N_0)_*}([0,p))$ and set $\phi :=\phi _0\otimes \mathrm{id}$. Since the map $s\mapsto \phi \circ \theta _s \in L^1_{(N_0)_*}([0,p))$ is continuous, for any $\epsilon >0$, there exists a positive number $\delta $ such that
$$\| \phi \circ \theta _s -\phi \| <\epsilon ^2$$
for $| s|<\delta$. Take a partition $0=s_0<s_1< \cdots <s_n=p$ so that $| s_i-s_{i+1}| <\delta$. For each $i=0, \cdots , n$, the map $[0,p)\ni \gamma \mapsto (\phi \circ \theta _{s_i})_\gamma$ is Lebesgue measurable and integrable. Hence it is possible to approximate $\phi \circ \theta _{s_i}$ by Borel simple step functions, that is, for each $i$, there exists a compact subset $K_i$ of $[0,p)$ which satisfies the following conditions.
\(2) We have $\mu (K_i)>p-\epsilon$.
\(3) There exist a Borel partition $\{ I_j\}$ of $K_i$ and $\phi _{i,j}\in( N_0)_*$ such that $$\| (\phi \circ \theta _{s_i})_\gamma -\sum _j \phi _{i,j}\chi _{I_j}(\gamma ) \| <\epsilon$$ for $ \gamma \in K_i$.
Set $$\psi _{t, \gamma }:=\sum _{i,j}\phi _{i,j}\chi _{[s_i,s_{i+1})}(t)\chi _{I_j}(\gamma ).$$ for each $(t, \gamma ) \in [0,p)^2$. For each $s\in [s_i,s_{i+1})$, set $$K_s:=\{ \gamma \in [0,p)\mid \| (\phi \circ \theta _s) _\gamma -(\phi \circ \theta _{s_i})_\gamma \| <\epsilon \} .$$ Then by the above inequality (1), we have $\mu (K_s)>p-\epsilon $. For $\gamma \in K_s\cap K_i$, we have $$\| (\phi \circ \theta _s)_\gamma -\psi _{s, \gamma } \| <2\epsilon .$$ Set $$K:=\{ (s, \gamma ) \in [0,p)^2 \mid \| (\phi \circ \theta _s)_\gamma -\psi _{s, \gamma } \| <2\epsilon \} .$$ Then we have $\mu (K) >p(p-2\epsilon )$. Hence $(s, \gamma ) \mapsto (\phi \circ \theta _s)_\gamma $ is well-approximated by simple step Borel functions in measure convergence. Hence this is Lebesgue measurable.
**Acknowledgements.** The author is thankful to Professor Toshihiko Masuda for pointing out a mistake in the proof of Lemma \[7\] in an early draft and giving him useful advice about Theorem \[rp\]. He also thanks to Professor Kawahigashi, who is his adviser, for encouragement on this work and for giving him useful advice about the presentation of the paper. He also thanks to Professor Izumi, Professor Tomatsu and Professor Ueda for giving him useful advice about subsection \[extend\]. He is supported by the Program for Leading Graduate Schools, MEXT, Japan.
[99]{} A. Connes, Almost periodic states and factors of type $\mathrm{III}_1$ . J. Funct. Anal. **16** (1974), 415–445. A. Connes, Factors of type $\mathrm{III}_1$, property $L'_\lambda$ and closure of inner automorphisms, J. Operator Theory **14**, no. 1 (1985), 189–211. A. Connes, Outer conjugacy classes of automorphisms of factors, Ann. Sci. É cole Norm. Sup. (4) **8** (1975), no. 3, 383–419. A. Connes and M. Takesaki, The flow of weights on factors of type III, Tohoku Math. J. (2) **29**, no. 4 (1977), 473–575. J. Feldman, Changing orbit equivalences of $\mathbf{R}^d$ actions, $d\geq 2$, to be $C^\infty $ on orbits, Internat. J. Math. **2**, no. 4 (1990), 409–427. J. Feldman, $\mathrm{C}^\infty$ Orbit Equivalence of Flows, xeroxed lecture notes, privately circulated. J. Feldman, P. Hahn and C. C. Moore, Orbit structure and countable sections for actions of continuous groups, Adv. Math. **28**, no. 3 (1978), 186–230. J. Feldman and D. Lind, Hyperfiniteness and the Halmos-Rohlin theorem for nonsingular Abelian actions. Proc. Amer. Math. Soc. **55**, no. 2 (1976), 339–344. M. Izumi, Canonical extension of endomorphisms of type III factors. Amer. J. Math. 125, no. 1 (2003), 1–56. V. F. R. Jones, Actions of finite groups on the hyperfinite type $\mathrm{II}_1$ factor, Mem. Amer. Math. Soc. **237** (1980). Y. Katayama, C. E. Sutherland and M. Takesaki, The characteristic square of a factor and the cocycle conjugacy of discrete group actions on factors, Invent. Math. **132**, no. 2, (1998), 331–380. Y. Kawahigashi, One-parameter automorphism groups of the injective $\mathrm{II}_1$ factor arising from the irrational rotation C$^*$-algebra, Amer. J. Math. **112** no. 4 (1990), 499–523. Y. Kawahigashi, One-parameter automorphism groups of the injective $\mathrm{II}_1$ factor with Connes spectrum zero, Canad. J. Math. **43** (1991), 108–118. Y. Kawahigashi, C. E. Sutherland, and M. Takesaki, The structure of the automorphism group of an injective factor and the cocycle conjugacy of discrete abelian group actions, Acta Math. **169** (1992), 105–130. K. Kawamuro, A Rohlin property for one-parameter automorphism groups of the hyperfinite $\mathrm{II}_1$ factor, Publ. Res. Inst. Math. Sci. **36** (2000), no. 5, 641–657. A. Kishimoto, A Rohlin property for one-parameter automorphism groups, Comm. Math. Phys. **179** (1996), no. 3, 599–622. I. Kubo, Quasi-flows, Nagoya Math. J. **35** (1969), 1–30. U. Haagerup, Connes bizentralizer problem and uniqueness of the injective factor of type $\mathrm{III}_1$, Acta Math. **158** (1987), 95–148. U. Haagerup and E. Størmer, Equivalence of normal states on von Neumann algebras and the flow of weights. Adv. Math. **83** (1990), no. 2, 180–262. U. Haagerup and E. Størmer, Pointwise inner automorphisms of von Neumann algebras. With an appendix by Colin Sutherland, J. Funct. Anal. **92** (1990), 177–201 C. Lance, Direct integrals of left Hilbert algebras, Math. Ann. **216** (1975), 11–28. D. Lind, Locally compact measure preserving flows, Adv. Math. **15** (1975), 175–193. T. Masuda, Unified approach to classification of actions of discrete amenable groups on injective factors, J. Reine Anguew. Math. **683** (2013), 1–47. T. Masuda and R. Tomatsu, Classification of actions of discrete Kac algebras on injective factors, Preprint (2013), arXiv:1306.5046. T. Masuda and R. Tomatsu, Rohlin flows on von Neumann algebras, Preprint (2012), arXiv:1206.0955. A. Ocneanu, Actions of discrete amenable groups on von Neumann algebras, Lecture Notes in Mathematics, **1138**. Springer-Verlag, Berlin, 1985. iv+115 pp. K. Shimada, Rohlin flows on Amalgamated Free Product Factors, to appear in Internat. Math. Res. Notices. C. E. Sutherland and M. Takesaki, Actions of discrete amenable groups on injective factors of type $\mathrm{III}_\lambda$, $\lambda \not=0$, Pacific J. Math. **137** (1989), 405–444. C. E. Sutherland and M. Takesaki, Right inverse of the module of approximately finite-dimensional factors of type III and approximately finite ergodic principal measured groupoids. Operator algebras and their applications, II (Waterloo, ON, 1994/1995), 149–159, Fields Inst. Commun., 20, Amer. Math. Soc. Providence, RI, 1998. M. Takesaki, Theory of operator algebras. I, Reprint of the first (1979) edition. Encyclopaedia of Mathematical Sciences, 124. Operator Algebras and Non-commutative Geometry, 5. Springer-Verlag, Berlin, 2002. xx+415 pp. M. Takesaki, Theory of operator algebras. II, Encyclopedia of Mathematical Sciences, 125. Operator Algebras and Non-commutative Geometry, 6. Springer-Verlag, Berlin, 2003. xxii+518 pp. T. Yamanouchi, One-cocycles on smooth flows of weights and extended modular coactions. Ergodic Theory Dynam. Systems **27**, no. 1 (2007), 285–318.
|
---
abstract: 'A scheme for separating the high- and low-frequency molecular dynamics modes in Hybrid Monte Carlo (HMC) simulations of gauge theories with dynamical fermions is presented. The algorithm is tested in the Schwinger model with Wilson fermions.'
address: |
The TrinLat Collaboration,\
School of Mathematics, Trinity College, Dublin 2, Ireland
author:
- 'Mike Peardon [^1] and James Sexton'
title: ' Multiple molecular dynamics time-scales in Hybrid Monte Carlo fermion simulations.'
---
At this conference, much discussion focussed on the dominant systematic error facing dynamical fermion simulations; the chiral extrapolation. It is proving to be a significant challenge to run Wilson fermion simulations of QCD using the HMC algorithm at light quark masses [@Irving:2002fx].
The standard implementation of HMC introduces pseudofermion fields to mimic the fermion action, and generates new proposals to a Metropolis test by integrating the molecular-dynamics (MD) equations of motion for a Hamiltonian in a fictitious simulation time. The maximum step-size for a useful Metropolis acceptance rate is set by a characteristic time-scale of the action used in the Hamiltonian [@Joo:2000dh]. Recent studies find that as the fermion is made lighter, the force induced by the pseudofermions generates increasingly violent high-frequency fluctuations. This effect is called “ultra-violet slowing-down” in Ref. [@Borici:2002zu]. In this report, we describe a modification to HMC that attempts to address this problem. A number of interesting algorithms have been devised recently that reduce the influence of the UV fermion modes on either the Monte Carlo update scheme or in the lattice discretisation [@Borici:2002zu; @Duncan:1998gq; @Hasenfratz:2002jn; @Orginos:1999cr; @Alexandrou:1999ii].
MULTIPLE MD TIME-SCALES
=======================
A scheme for integrating the equations of motion for the MD phase of the HMC algorithm by introducing different time-scales for different segments of the action was introduced in Ref. [@Sexton:1992nu].
The leap-frog integrator is constructed from the two simple time-evolution operators generated by the the kinetic and potential energy terms. Their effect on the system coordinates, $\{p,q\}$ are $$\begin{aligned}
V_T(\Delta\tau): \{p,q\} &\longrightarrow& \{p,q + \Delta \tau \;p\} \nonumber\\
V_S(\Delta\tau): \{p,q\} &\longrightarrow& \{p - \Delta \tau\;\partial S,q\}\end{aligned}$$ The simplest reversible leap-frog integrator is then $$V(\Delta \tau) =
V_S(\frac{\Delta \tau}{2})
V_T(\Delta \tau)
V_S(\frac{\Delta \tau}{2}).$$ If the action (and thus the Hamiltonian) is split into two parts, $$\begin{aligned}
{\cal H} = \underbrace{T(p) + S_1(q)}_{{\cal H}_1}
+ \underbrace{S_2(q)}_{{\cal H}_2}, \end{aligned}$$ then the two leap-frog integrators for these two pieces can be constructed as $$\begin{aligned}
V_1(\Delta \tau) &=&
V_{S_1}(\frac{\Delta \tau}{2})
\;V_T(\Delta \tau)\;
V_{S_1}(\frac{\Delta \tau}{2}) \nonumber \\
V_2(\Delta \tau) &=& V_{S_2}(\Delta \tau)\end{aligned}$$ and a reversible integrator for the full Hamiltonian can be constructed by combining the two schemes: $$V(\Delta \tau) =
V_2(\frac{\Delta \tau}{2})
\;\left[V_1(\frac{\Delta\tau}{m}) \right]^m\;
V_2(\frac{\Delta \tau}{2}),$$ with $m\in Z$. This compound integrator effectively introduces two evolution time-scales, $\Delta\tau$ and $\Delta\tau/m$.
We suggest that the multiple time-scale scheme is only helpful when two criteria are fulfilled simultaneously:
1. [the force term generated by $S_1$ is cheap to compute compared to that of $S_2$ and]{}
2. [the split captures the high-frequency modes of the system in $S_1$ and the low-frequency modes in $S_2$.]{}
The popular implementation of the scheme in dynamical fermion simulations with pseudofermions is to split the Hamiltonian into the Yang-Mills term and the pseudofermion action: $$\begin{aligned}
S_1 &=& S_G, \nonumber \\
S_2 &=& \phi^* [M^\dagger M]^{-1} \phi\end{aligned}$$ Unfortunately for light fermions, the highest frequency fluctuations are in the pseudofermion action, which also has the more computationally expensive force term, thus the criteria are not met. The key to using the method then is to implement a low-computational cost scheme that separates the high- and low-frequency fermion molecular-dynamics.
POLYNOMIAL FILTERING
====================
A very low-order polynomial approximation offers a cheap means of mimicking most of the short-distance physics of the fermion interations. Recent interest in polynomial approximations to a matrix inverse was inspired by the multi-boson algorithm of Ref. [@Luscher:1993xx]. This led to the development of the Polynomial HMC (PHMC) algorithm [@Frezzotti:1997ym]. Following this idea, we write an exact representation of the two-flavour probability measure $$\det M^\dagger M = \int\!\!
{\cal D}\phi
{\cal D}\phi^*
{\cal D}\chi
{\cal D}\chi^*
e^{\left\{-S_\phi-S_\chi\right\}}$$ with $$S_\phi = \left|[M {\cal P}(M)]^{-1} \phi \right|^2
\mbox{ and }
S_\chi = \left| {\cal P}(M) \chi \right|^2
\label{eqn:action}$$
The fields $\phi$ are modified pseudofermions and we term the new fields $\chi$, “guide” bosons. The algorithm exactly recovers the probability measure for the two-flavour theory for any choice of polynomial. This scheme is similar to the split-pseudofermion method of Ref. [@Hasenbusch:2001ne] with one distinction; using polynomials allows us to modify the fermion modes easily and cheaply. The success of the algorithm hinges on the empirical observation that the fermion modes that induce the high-frequency fluctuations are localised and can be handled by very low-order polynomials. This means the force term for the guide bosons, generated by $S_\chi$ can be computed cheaply and a multiple time-scale integrator can be built that simultaneously fulfills the two criteria discussed earlier. The time-scale split is then $$S =
\underbrace{S_G + S_\chi}_{UV,\;\; \partial S\mbox{ cheap}} +
\underbrace{S_\phi} _{IR,\;\; \partial S\mbox{ costly.}}$$
Note that in general, $p$ time-scales can be introduced straightforwardly by first adding $p-1$ sets of guide fields, then constructing a heirarchy of leapfrog integrators, with $m_i$ leap-frog steps at the $i^{\rm th}$ level. This introduces $p$ time-scales, $\Delta\tau >
\frac{\Delta\tau}{m_{p-1}} \dots
\frac{\Delta\tau}{m_{p-1} \dots m_{1}}$.
TESTING THE ALGORITHM
=====================
The performance of the method is under investigation in simulations of the two-flavour massive Schwinger model ($2d$ QED). HMC runs presented here are performed on $64\times64$ lattices, with $\beta=4.0, \kappa = 0.2618$. Non-hermitian Chebyshev polynomial approximations are used [@Borici:1995am]. Measurements of the dependence of the acceptance probability on the polynomial degree and tuning the MD parameters, $\Delta\tau$ and $m$ are being made.
Fig. \[fig:pacc\] shows the acceptance rate for the standard HMC algorithm (with even-odd preconditioned pseudofermions) along with the polynomial-filtered method. Two different polynomials were used; $N=4$ and $N=16$. The solid lines are fits to the expected behaviour of HMC as $\Delta\tau \rightarrow 0$, namely $$\langle P_{\rm acc}\rangle = \mbox{erfc}\left\{(\Delta\tau/\tau_0)^2\right\},$$ where the value of $\tau_0$ is determined by the best fit. $\tau_0$ is then a characteristic time-scale for the modes encapsulated in the pseudofermionic part of the action, $S_\phi$ in Eqn. \[eqn:action\]. Since evaluation of the force term $\partial S_\phi$ dominates in this parameter range, $\tau_0$ is a good indicator of algorithm performance.
TUNING THE ALGORITHM
====================
The algorithm has a number of free parameters, allowing a good deal of scope to optimise the performance. Fig. \[fig:tau0\] shows the characteristic time-scale for pseudofermion integration, $\tau_0$ as a function of Chebyshev polynomial degree. $\tau_0$ rises very rapidly initially as the degree of the filter algorithm is increased, and $\tau_0$ for $N=16$ is about four times larger than for the standard HMC algorithm. Inverting the pseudofermion matrix requires roughly the same computational cost, and thus the algorithm is four times more efficient than HMC (assuming autocorrelations are the same).
For low values of $N$ (the polynomial degree) the computational bottleneck is solver performance while for larger values, the evaluation of the force term arising from interactions between the gauge fields and the guide bosons begins to dominate. Effective and simple strategies for tuning the algorithm are still under investigation. We are also investigating alternative choices of polynomial filters beyond Chebyshev approximation.
MP is grateful to Enterprise-Ireland for support under grant SC/01/306.
[9]{}
A. C. Irving \[UKQCD\], hep-lat/0208065. B. Joo [*et.al.*]{} \[UKQCD\], hep-lat/0005023. A. Borici, hep-lat/0208048. A. Duncan, E. Eichten and H. Thacker, Phys. Rev. D [**59**]{} (1999) 014505. A. Hasenfratz and F. Knechtli, hep-lat/0203010. K. Orginos, D. Toussaint and R. L. Sugar \[MILC\], Phys. Rev. D [**60**]{} (1999) 054503. C. Alexandrou [*et.al.*]{} Phys. Rev. D [**61**]{} (2000) 074503. J. C. Sexton and D. H. Weingarten, Nucl. Phys. [**B380**]{}, 665 (1992). M. Hasenbusch, Phys. Lett. B [**519**]{} (2001) 177.
M. Luscher, Nucl. Phys. B [**418**]{} (1994) 637. R. Frezzotti and K. Jansen, Phys. Lett. B [**402**]{} (1997) 328. A. Borici and P. de Forcrand, Nucl. Phys. B [**454**]{} (1995) 645.
[^1]: Talk presented by Mike Peardon
|
---
author:
- |
Micha Livne\
University of Toronto\
Vector Institute\
`mlivne@cs.toronto.edu`\
Kevin Swersky\
Google Research\
`kswersky@google.com`\
David J. Fleet\
University of Toronto\
Vector Institute\
`fleet@cs.toronto.edu`\
bibliography:
- 'paper.bib'
title: High Mutual Information in Representation Learning with Symmetric Variational Inference
---
|
---
abstract: 'In the framework of an effective field theory of general relativity a model of scalar and vector bosons interacting with the metric field is considered. It is shown in the framework of a two-loop order calculation that for the cosmological constant term which is fixed by the condition of vanishing vacuum energy the graviton remains massless and there exists a self-consistent effective field theory of general relativity coupled to matter fields defined on a flat Minkowski background. This result is obtained under the assumption that the energy-momentum tensor of the gravitational field is given by the pseudotensor of Landau-Lifshitz’s classic textbook. Implications for the cosmological constant problem are also briefly discussed.'
author:
- 'J. Gegelia'
- 'Ulf-G. Mei[ß]{}ner'
title: Vacuum energy in the effective field theory of general relativity
---
Introduction
============
It is widely accepted that whatever the underlying fundamental theory of all interactions might be at low energies, the physics can be adequately described by an effective field theory (EFT) [@Weinberg:1995mt]. Gravitation can also included in the formalism of EFT by considering the most general effective Lagrangian of metric fields interacting with matter fields [@Donoghue:1994dn; @Donoghue:2015hwa] which is invariant under all underlying symmetries including the gauge symmetry of massless spin-two particles [@Veltman:1975vx]. This quantum field theoretical treatment of general relativity with the metric field presented as the Minkowski background plus the graviton field and the cosmological constant usually set equal to zero is considered as a well-defined approach in the modern sense, see, e.g., Ref. [@Donoghue:2017pgk]. It is well-known that for a non-vanishing cosmological constant term $\Lambda$ the graviton propagator has a pole corresponding to a massive ghost mode [@Veltman:1975vx]. Setting $\Lambda$ equal to zero as is usually done in the EFT of gravitation [@Donoghue:1994dn] does not solve the problem, as the radiative corrections re-generate the problem with the massive ghost [@Burns:2014bva]. This is because the cosmological constant term is not suppressed by any symmetry of the effective theory and therefore there is no protection against generating such a contribution to the effective action by radiative corrections. However, as it has been shown in Ref. [@Burns:2014bva], one can represent the cosmological constant as a power series in $\hbar$ and choose the coefficients of this series such that the graviton becomes a massless spin-two particle up to all orders in the loop expansion. Thus, within a perturbative EFT in flat Minkowski background, the cosmological constant, which is one of the parameters of the effective Lagrangian, is uniquely fixed. This does not solve the cosmological constant problem [@Weinberg:1988cp] (for a recent review of the cosmological constant problem see, e.g., Ref. [@Martin:2012bt]) but rather implies that taking into account a cosmological constant term other than obtained in Ref. [@Burns:2014bva] necessarily requires considering an EFT in a curved background field. In this case by imposing the equations of motion with respect to the background graviton field the mass term of the graviton is removed at tree level [@Gabadadze:2003jq], however, a systematic study of the issue at higher orders in loop expansion requires an EFT on a curved background metric which, to the best of our knowledge, is not available yet.
Experimental evidence of the accelerating expansion of the universe (see, e.g., Ref. [@Rubin:2016iqe] and references therein) leaves us with a very challenging problem, namely the huge discrepancy between the measured small value of the cosmological constant and its theoretical estimation [@Weinberg:1988cp]. An important question related to this problem is if there exists any condition that uniquely fixes the value of the cosmological constant. It seems natural to expect that the energy of the physical vacuum state of the theory describing the universe is exactly zero. This is the main assumption of this work and of importance for the later discussions. In the framework of the low-energy EFT of general relativity coupled to the fields of the Standard Model imposing such a condition uniquely fixes the cosmological constant term as a function of other parameters of the effective Lagrangian. In the current work we calculate the vacuum expectation value of the full four-momentum of the gravitational and matter fields at two-loop order in a simplified version of the Abelian model with spontaneous symmetry breaking considered in Ref. [@Burns:2014bva]. We obtain that as a result of a non-trivial cancellation between different diagrams the vacuum energy exactly vanishes for the value of the cosmological constant obtained in Ref. [@Burns:2014bva], i.e., for the value which guarantees the vanishing of the graviton mass and the vacuum expectation value of the graviton field at two-loop order. That is, provided that our result holds to all orders, the uniquely fixed value of the cosmological constant term, leading to a self-consistent perturbative EFT on Minkowki background is obtained as a consequence of imposing the condition of vanishing vacuum energy. Notice here that being aware of the lack of a commonly accepted expression of the energy-momentum tensor for the gravitational field (see, e.g., Refs. [@Babak:1999dc; @Butcher:2008rf; @Szabados:2009eka; @Butcher:2010ja; @Butcher:2012th]) in the current work we use the definition of the energy-momentum pseudotensor and the full four-momentum of the matter and gravitational fields given in the classic textbook by Landau and Lifshitz [@Landau:1982dva].
Our work is orginized as follows: In section \[three\] we specify the details of the considered EFT and calculate one- and two-loop order contributions to the vacuum energy. In section \[implications\] we briefly discuss the implications of the obtained results on the cosmological constant problem. We summarize in section \[summary\] and the appendix contains the Feynman rules and two-loop integrals required in our calculations.
Vacuum energy in an EFT of general relativity on a Minkowski background {#three}
=======================================================================
In the framework of EFT the action of matter interacting with gravity is given by the most general effective Lagrangian of gravitational and matter fields, which is invariant under general coordinate transformations and other symmetries of the Standard Model, $$S = \int d^4x \sqrt{-g}\, \left\{ {\cal L}_{\rm gr}(g)
+{\cal L}_{\rm m}(g,\psi)\right\} = \int d^4x \sqrt{-g}\, \left\{ \frac{2}{\kappa^2} (R-2\Lambda)+{\cal L}_{\rm gr,ho}(g)
+{\cal L}_{\rm m}(g,\psi)\right\} = S_{\rm gr}(g)+S_{\rm m}(g,\psi) ,
\label{action}$$ where $\kappa^2=32 \pi G$, with $G=6.70881\times 10^{-39}$ ${\rm GeV}^{-2}$ the gravitational (Newton’s) constant, $\psi$ and $g^{\mu\nu}$ denote the matter and metric fields, respectively, $g=\det g^{\mu\nu}$, $\Lambda$ is the cosmological constant and $R$ the scalar curvature. Further, ${\cal L}_{\rm gr,ho}(g)$ represents self-interaction terms of the gravitational field with higher orders of derivatives and ${\cal L}_{\rm matter}(g,\psi)$ is the effective Lagrangian of the matter fields interacting with gravity. Experimental evidence suggests that the contributions of ${\cal L}_{\rm gr,ho}(g)$ as well as the contributions of non-renormalizable interactions of ${\cal L}_{\rm matter}(g,\psi)$ in physical quantities are heavily suppressed. Vielbein tetrad fields have to be introduced for fermionic fields interacting with the gravitational field, however, we refrain from giving details on these as later we will perform calculations with bosonic degrees of freedom only.
The low-energy EFT of general relativity is obtained by representing the gravitational field as the sum of the Minkowskian background and the quantum fields [@tHooft:1974toh] $$\begin{aligned}
g_{\mu\nu} &=& \eta_{\mu\nu}+\kappa h_{\mu\nu},\nonumber \\
g^ {\mu\nu} &=& \eta^{\mu\nu}-\kappa h^{\mu\nu}+\kappa^2 h^\mu_\lambda h^{\lambda\nu}-\kappa^3 h^\mu_\lambda h^{\lambda}_{\sigma} h^{\sigma\nu}+\cdots ~,
\label{gexpanded}\end{aligned}$$ and calculating physical quantities perturbatively by applying standard QFT technique.
The energy-momentum tensor of the matter fields coupled to the gravitational field, $T^{\mu\nu}_{\rm m}$, and the pseudotensor of the gravitational field, $T^{\mu\nu}_{LL}$, are given by $$\begin{aligned}
T^{\mu\nu}_{\rm m} (g,\psi) & = & \frac{2}{\sqrt{-g}}\frac{\delta S_{\rm m} }{\delta g_{\mu\nu}}\,,
\label{EMTMatter}
\\
T^{\mu\nu}_{\rm gr} (g)&=&
\frac{4}{\kappa^2} \, \Lambda\,g^{\mu\nu} + T_{LL}^{\mu\nu}(g)\,,
\label{defTs}\end{aligned}$$ where $T_{LL}^{\mu\nu}(g)$ is defined via [@Landau:1982dva] $$\begin{aligned}
(-g)T^{\mu\nu}_{LL} (g) &=& \frac{2}{\kappa^2} \left(\frac{1}{8} \, g^{\lambda \sigma } g^{\mu \nu } g_{\alpha \gamma}
g_{\beta\delta} \, \mathfrak{g}^{\alpha \gamma},_{\sigma }
\, \mathfrak{g}^{\beta \delta},_\lambda -\frac{1}{4} \, g^{\mu \lambda } g^{\nu\sigma } g_{\alpha ,\gamma} g_{\beta \delta }
\, \mathfrak{g}^{\alpha\gamma},_\sigma \, \mathfrak{g}^{\beta\delta},_\lambda -\frac{1}{4} \, g^{\lambda \sigma } g^{\mu \nu }
g_{\beta \alpha} g_{\gamma \delta} \, \mathfrak{g}^{\alpha \gamma},_\sigma \, \mathfrak{g}^{\beta \delta},_\lambda \right.\nonumber\\
&+& \left. \frac{1}{2}\, g^{\mu \lambda } g^{\nu\sigma } g_{\beta \alpha} g_{\gamma \delta } \, \mathfrak{g}^{\alpha \gamma},_\sigma \, \mathfrak{g}^{\beta\delta},_\lambda
+g^{\beta \alpha } g_{\lambda \sigma }
\, \mathfrak{g}^{\nu \sigma},_\alpha \, \mathfrak{g}^{ \mu\lambda},_\beta +\frac{1}{2} \, g^{\mu \nu } g_{\lambda \sigma }
\, \mathfrak{g}^{\lambda \beta},_\alpha \, \mathfrak{g}^{\alpha\sigma},_\beta \right.\nonumber\\
&-& \left.g^{\mu \lambda } g_{\sigma \beta }
\, \mathfrak{g}^{\nu \beta},_\alpha \, \mathfrak{g}^{\sigma\alpha},_\lambda
-g^{\nu \lambda } g_{\sigma \beta} \, \mathfrak{g}^{\mu\beta},_\alpha \, \mathfrak{g}^{\sigma \alpha},_\lambda
+\, \mathfrak{g}^{\lambda \sigma},_\sigma \, \mathfrak{g}^{\mu\nu},_\lambda
- \, \mathfrak{g}^{\text{der}(\mu \lambda},_\lambda \, \mathfrak{g}^{\nu \sigma},_\sigma \right),
\label{LLEMT}\end{aligned}$$ with $\mathfrak{g}^{\mu\nu}=\sqrt{-g} \, g^{\mu\nu}$ and $\mathfrak{g}^{\mu\nu},_\lambda=\partial\mathfrak{g}^{\mu\nu}/\partial x^\lambda $.
The full energy-momentum tensor $T^{\mu\nu}=T^{\mu\nu}_{\rm m} (g,\psi)+T^{\mu\nu}_{\rm gr} (g)$ defines the conserved full four-momentum of the matter and the gravitational field as [@Landau:1982dva] $$P^\mu= \int (-g) \, T^{\mu\nu} d S_\nu\,,
\label{EMV}$$ where the integration is carried out over any hypersurface containing the whole three-dimensional space. Thus, the energy of the vacuum will be zero if the vacuum expectation value of the energy-momentum tensor times $(-g)$ vanishes. This quantity is given by the following path integral: $$\begin{aligned}
\langle 0| (-g) T^{\mu\nu} |0\rangle &=& \int {\cal D }g\, {\cal D}\psi \, (-g)\left[ T^{\mu\nu}_{\rm gr}(g)+ T_{\rm m}^{\mu\nu}(g,\psi) \right] \exp\left\{ i \int d^4 x \, \sqrt{-g}\,\left[ {\cal L}(g,\psi) +{\cal L}_{\rm GF}\right] \right\},
\label{VacuumE}\end{aligned}$$ where ${\cal L}_{\rm GF}$ is the gauge fixing term and the Faddeev-Popov determinant is included in the integration measure. The cosmological constant $\Lambda$ can be uniquely fixed by demanding that the right-hand-side of Eq. (\[VacuumE\]) vanishes. To demonstrate how one obtains a self-consistent EFT by imposing this condition, we consider a simple model of a massive scalar and a massive vector fields interacting with metric tensor field. It coincides with the bosonic part of the model with spontaneously broken Abelian gauge symmetry considered in Ref. [@Burns:2014bva] taken in unitary gauge for the Abelian gauge symmetry. The action of the matter part of the model is given by $$S_{\rm m} = \int d^4x \sqrt{-g}\, \left\{ -\frac{1}{4} g^{\mu\rho}g^{\nu\sigma} F_{\mu\nu}F_{\rho\sigma} +\frac{M^2}{2}\, g^{\mu\nu} A_\mu A_\nu +\frac{g^{\mu\nu}}{2}\,\partial_\mu H \partial_\nu H -\frac{m^2}{2}\,H^2+{\cal L}_{\rm MI}\right\},
\label{MAction}$$ where $F_{\mu\nu} = \partial_\mu A_\nu -\partial_\nu A_\mu$, $A_\mu$ is vector field, $H$ the scalar field and ${\cal L}_{\rm MI}$ denotes the interactions of matter fields, the specific form of which is not important for the current work as we will not include them in our calculations. The energy-momentum tensor corresponding to Eq. (\[MAction\]) has the form $$\begin{aligned}
T_m^{\mu\nu} & = & - g^{\mu\alpha}g^{\nu\rho}g^{\beta\sigma} F_{\alpha\beta}F_{\rho\sigma} + M^2\, g^{\mu\alpha}g^{\nu\beta} A_\alpha A_\beta +\partial_\mu H \partial_\nu H \nonumber\\
&-& g^{\mu\nu} \left\{ -\frac{1}{4} g^{\alpha\rho}g^{\beta\sigma} F_{\alpha\beta}F_{\rho\sigma} +\frac{M^2}{2}\, g^{\alpha\beta} A_\alpha A_\beta +\frac{g^{\alpha\beta}}{2}\,\partial_\alpha H \partial_\beta H -\frac{m^2}{2}\,H^2\right\} +T_{\rm MI}^{\mu\nu} ,
\label{MEMT}\end{aligned}$$ where $T_{\rm MI}^{\mu\nu}$ corresponds to ${\cal L}_{\rm MI}$.
By adding the following gauge fixing term to the effective Lagrangian $${\cal L}_{\rm GF}= \xi \left( \partial_\nu h^{\mu\nu}-\frac{1}{2} \partial^\mu h^\nu_\nu \right) \left( \partial^\beta h_{\mu\beta}-\frac{1}{2} \partial_\mu h^\alpha_\alpha \right),
\label{GFT}$$ where $\xi$ is the gauge parameter, we obtain the Feynman rules specified in the appendix.
![Diagrams contributing to the vacuum expectation value of the energy-momentum pseudotensor times $(-g)$ at tree order. Filled circles corresponds to the cosmological constant term. The cross stands for the energy-momentum pseudotensor times $(-g)$, and the wiggly line represents the graviton.[]{data-label="EMTTree"}](EMTTree.eps){height="3.00cm"}
For the vacuum expectation value of the full energy-momentum pseudotensor times $(-g)$ at tree order we obtain an infinite number of diagrams shown in Fig. \[EMTTree\]. All these contributions vanish if we take the cosmological constant vanishing at tree order. That is, we represent $\Lambda$ as $$\Lambda = \sum_{i=0}^\infty \hbar^i \Lambda _i \,,
\label{CCexpanded}$$ and take $\Lambda_0=0$. Notice that this also removes the graviton mass from the propagator at tree order.
![Diagrams contributing to the vacuum expectation value of the energy-momentum pseudotensor times $(-g)$. Filled circle corresponds to the cosmological constant term. The cross stands for the energy-momentum pseudotensor times $(-g)$, wiggly and solid lines represent the graviton and the scalar (vector), respectively.[]{data-label="EMTLoop"}](EMT.eps){height="2.0cm"}
Next, using the Feynman rules given in the appendix, we calculated the one-loop contributions to the vacuum expectation value of the full energy-momentum pseudotensor times $(-g)$ shown in Fig. \[EMTLoop\], and by demanding that $\Lambda_1$ cancels this contribution we obtain (in the calculations of the loop diagrams below we used the program FeynCalc [@Mertig:1990an; @Shtabovenko:2016sxi]) $$\Lambda_1 = -\frac{ \kappa ^2 \Gamma
\left(1-\frac{d}{2}\right) \left(m^d+(d-1)
M^d\right)}{ 2^{d+6} \pi ^{\frac{d}{2}+4} \, d}\,.$$ It is a trivial consequence of Eq. (\[EMTMatter\]) that the same value of $\Lambda_1$ cancels the one-loop contribution to the vacuum expectation value of the graviton field $h_{\mu\nu}$, shown in Fig. \[GrTP\], and consequently the graviton self-energy at zero momentum, i.e. graviton mass, as a result of a Ward identity [@Burns:2014bva]. The first non-trivial result is obtained at two-loop order by calculating the diagrams contributing to the vacuum expectation value of the full energy-momentum pseudotensor times $(-g)$ shown in Fig. \[EMTLoop\]. We also calculated the two-loop contributions to the vacuum expectation value of the gravitational field shown in Fig. \[GrTP\] and verified that the same value of $\Lambda_2$ cancels both quantities. The obtained result reads: $$\Lambda_2 = -\frac{ d (d+1) \kappa ^4 M^{2
d-2} \csc \left(\frac{\pi d}{2}\right) \Gamma
\left(1-\frac{d}{2}\right)}{2^{2 (d+3)} \pi ^{d-1}\Gamma \left(\frac{d}{2}\right)}.
\label{Lambda2}$$ While it is a trivial consequence of Eq. (\[EMTMatter\]) that the fourth diagrams in both figures \[EMTLoop\] and \[GrTP\] give equal contributions in $\Lambda$ it is only the sum of the corresponding second and third diagrams that lead to identical expressions. To check the obtained results we also calculated two-loop contributions to the graviton self-energy at zero momentum and verified that the same value of $\Lambda_2$ cancels the two-loop order contribution to the graviton mass in agreement with the Ward identity [@Burns:2014bva] (we do not give the expressions of the corresponding Feynman rules due to their huge size). While we expect that an analogous result holds to all orders we are not able to give a general argument supporting it.
![Diagrams contributing to the graviton tadpole. The filled circle corresponds to the cosmological constant term. Wiggly and solid lines represent the graviton and the scalar (vector) fields, respectively. []{data-label="GrTP"}](GR-Tadpole.eps){height="2.25cm"}
Implication on the cosmological constant problem {#implications}
================================================
It follows from the result of the previous section that unless the cosmological constant is chosen such that the energy of the vacuum is exactly zero, it cannot remove the graviton mass and the graviton tadpole order-by-order in perturbation theory and consequently a non-perturbative treatment of the cosmological constant term is mandatory. This is because for all physical processes there appear diagrams like ones shown in Fig. \[GrSE\] where the massless graviton propagator carries vanishing momentum and therefore $1/0$ singularities occur (this does not happen only if the tadpole vanishes order-by-order in the loop expansion).
The cosmological constant problem is often described as vacuum having tiny non-zero energy density. Due to this loose language one might think that the condition of vanishing vacuum energy [*a priory*]{} excludes the solution of the cosmological constant problem. A closer look reveals that exactly the opposite might be the case. Indeed, due to the condition imposed on the cosmological constant term of the effective Lagrangian the effective action calculated on the Minkowski background metric with vanishing background matter fields does not contain an effective cosmological constant term contributing to Einsten’s equations. However, for our universe the corresponding effective action has to be calculated in the presence of non-trivial background fields. The cosmological constant term of the effective Lagrangian exactly cancelling the loop contributions in a trivial background leads to a uniquely fixed effective cosmological constant contributing in the Einstein’s equation also in the presence of a non-trivial background. While for weak backgrounds we expect large cancellations leaving us with a tiny effective cosmological constant, a quantitative investigation of this estimation is a subject of a separate publication. To make this more precise, the background relevant for cosmology is not flat Minkowski, therefore the fixed cosmological constant term will not exactly cancel the quantum corrections to the effective cosmological constant, but rather only approximately, leaving a small finite piece. This remains to be calculated.
Summary and discussion {#summary}
======================
Consistency conditions of the perturbative EFT of general relativity in flat Minkowski background uniquely fix the cosmological constant term as a function of all other parameters of the theory [@Burns:2014bva]. This follows from the requirement of the presence of a massless graviton, instead of a massive spin-two ghost, in the spectrum of the theory. Notice that it is not possible to take into account perturbatively any other value of the cosmological constant term within an EFT on the flat Minkowski background. This is because of the $1/0$ singularities in the Feynman diagrams with tadpole contributions, see, e.g., Fig. \[GrSE\].
In our opinion if there is any fundamental reason for choosing a fixed value of the cosmological constant then it must be the condition of vanishing of the vacuum energy. It is often argued that vacuum has non-zero energy due to quantum fluctuations. A classical example is given by quantum oscillator. It is well-known that the ground state energy of a quantum oscillator is $\hbar \omega/2$, where $\omega$ is the angular frequency. A closer look reveals, however, that this expression is a result of an assumption. In particular, if we share the point of view that the real world is described by a quantum theory and the classical theory is only an approximation to it, then it is not possible to uniquely reproduce the quantum Hamiltonian of an oscillator by quantising the classical one. This non-uniqueness is of course well-known and is manifested in the problem of operator ordering. Indeed, by adding a vanishing term $\sim (p q-q p)$ to the classical Hamiltonian of the oscillator and quantizing it we obtain a quantum Hamiltonian with an arbitarary constant term and hence an arbitrary vacuum energy. Starting from the classical theory there is no way to tell which value of the vacuum energy is more “fundamental". Notice that the argument of the Casimir effect being a proof of the non-vanishing vacuum energy is not convincing either, see, e.g., Refs. [@Jaffe:2005vp; @Nikolic:2016kkp].
In the framework of low-energy EFT of general relativity coupled to the fields of the Standard Model imposing a condition of vanishing vacuum energy uniquely fixes the cosmological constant term as a function of other parameters of the effective Lagrangian. We expect that this leads to a self-consistent perturbative EFT defined on the Minkowsky background, i.e. to a massless graviton in the spectrum and the vanishing graviton tadpole. We were not able to give a general argument supporting our claim. Instead we calculated the vacuum expectation value of the full four-momentum of matter and gravitational fields at two-loop order in a simplified version of the Abelian model with spontaneous symmetry breaking considered in Ref. [@Burns:2014bva]. While at one-loop order the condition of vanishing vacuum energy automatically leads to the conditions of Ref. [@Burns:2014bva], at two-loop order the same agreement of two conditions appears as a result of a non-trivial cancellation between different diagrams. We notice here that there does not exist a commonly accepted expression of the energy-momentum tensor for the gravitational field (see, e.g., Refs. [@Babak:1999dc; @Butcher:2008rf; @Szabados:2009eka; @Butcher:2010ja; @Butcher:2012th]). In the current work we used the definition of the energy-momentum pseudotensor and the full four-momentum of the matter and gravitational fields given in the classic textbook by Landau and Lifshitz [@Landau:1982dva].
Within a self-consistent EFT all physical quantities should be finite after renormalizing (an infinite number of) parameters of the effective Lagrangian. Therefore it is mandatory that the uniquely fixed value of the cosmological constant term, which defines the perturbative EFT of the Standard Model coupled to gravitons on Minkowski flat background leads to a finite expression of the energy of the vacuum to all orders in loop expansion. Based on the two-loop order result of the current work we expect that this finite value is actually zero. Turning the argument around we expect that by demanding that the vacuum energy should be vanishing to all orders we obtain a self-consistent perturbative low-energy EFT of matter and gravitational fields on flat Minkowski background.
Relegating calculations and detailed discussion to a future work, we briefly comment on the implications of our results for the cosmological constant problem. In particular, we expect that the cosmological constant term of the effective Lagrangian exactly cancelling the loop contributions in flat background is very likely to cancel the bulk of such contributions also in the presence of a non-trivial background, relevant for our universe, thus leaving with a tiny effective cosmological constant contributing to Einstein’s equations.
We thank Dalibor Djukanovic for helpful comments on the manuscript. The work of JG was supported in part by BMBF (Grant No. 05P18PCFP1), and by the Georgian Shota Rustaveli National Science Foundation (Grant No. FR17-354). The work of UGM was supported in part by provided by Deutsche Forschungsgemeinschaft (DFG) through funds provided to the Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD" (Grant No. TRR110), by the Chinese Academy of Sciences (CAS) through a President’s International Fellowship Initiative (PIFI) (Grant No. 2018DM0034) and by the VolkswagenStiftung (Grant No. 93562).
Appendix {#appendix .unnumbered}
========
Below we give Feynman rules used in the calculation of the vacuum expectation values of the graviton field and the energy-momentum tensor.
Propagators:
- Scalar propagator with momentum $p$: $$\frac{ i }{ p^2-m^2+i
\epsilon }\,.
\label{sPr}$$
- Vector boson propagator with Lorentz indices $\mu$, $\nu$ and momentum $p$: $$-\frac{i \left( g^{\mu \nu} -p^\mu p^\nu/M^2\right)}{p^2-M^2+i \epsilon} \,.
\label{vPr}$$
- Graviton propagator in $D$ dimensions with Lorentz indices $(\mu,\nu)$, $(\alpha,\beta)$ and momentum $p$: $$\frac{i}{2} \, \frac{ g^{\lambda \nu } g^{\mu \sigma }+g^{\lambda \mu } g^{\nu
\sigma } - \frac{2 g^{\lambda \sigma } g^{\mu \nu
}}{D-2} }{p^2+i \epsilon } - \frac{i \, \xi}{2} \frac{
p^{\nu } \left(p^{\sigma } g^{\lambda \mu }+p^{\lambda }
g^{\mu \sigma }\right)+p^{\mu } \left(p^{\sigma } g^{\lambda \nu
}+p^{\lambda } g^{\nu \sigma }\right) }{ \left(p^2+i
\epsilon \right)^2} \,.
\label{gPr}$$
Vertices (all momenta in all vertices are incoming):
- Graviton with indices $(\mu,\nu)$: $$-\frac{2 i \Lambda g^{\mu \nu }}{\kappa } ;
\label{h}$$
- Graviton with indices $(\mu,\nu)$ and $(\alpha,\beta)$: $$i \Lambda \left(g^{\alpha \nu } g^{\beta \mu }+g^{\alpha \mu }
g^{\beta \nu }-g^{\alpha \beta } g^{\mu \nu }\right) ;
\label{h}$$
- Graviton with indices $(\mu,\nu)$ - scalars with momenta $p_1$ and $p_2$: $$\frac{1}{2} i \kappa \left(-g^{\mu \nu } \left(m^2+p_1\cdot
p_2\right)+p_2^{\mu } p_1^{\nu }+p_1^{\mu } p_2^{\nu }\right) ;
\label{hSS}$$
- Gravitons with indices $(\mu,\nu)$ and $(\alpha,\beta)$ - scalars with momenta $p_1$ and $p_2$: $$\begin{aligned}
&& -\frac{1}{4} i \kappa ^2 \left(-m^2 g^{\alpha \nu }
g^{\beta \mu }-m^2 g^{\alpha \mu } g^{\beta \nu }+m^2 g^{\alpha
\beta } g^{\mu \nu }+p_1^{\beta } p_2^{\nu } g^{\alpha \mu
}+p_1^{\beta } p_2^{\mu } g^{\alpha \nu }+p_1^{\alpha } p_2^{\nu
} g^{\beta \mu }
\right. \nonumber\\
&& \left.
+p_1^{\nu } \left(-p_2^{\mu } g^{\alpha \beta
}+p_2^{\beta } g^{\alpha \mu }+p_2^{\alpha } g^{\beta \mu
}\right)+p_1^{\alpha } p_2^{\mu } g^{\beta \nu }+p_1^{\mu }
\left(-p_2^{\nu } g^{\alpha \beta }+p_2^{\beta } g^{\alpha \nu
}+p_2^{\alpha } g^{\beta \nu }\right)-p_2^{\alpha } p_1^{\beta }
g^{\mu \nu }
\right. \nonumber\\
&& \left.
-p_1^{\alpha } p_2^{\beta } g^{\mu \nu }-p_1\cdot
p_2 \left(g^{\alpha \nu } g^{\beta \mu }+g^{\alpha \mu }
g^{\beta \nu }-g^{\alpha \beta } g^{\mu \nu }\right)\right) ;
\label{hhSS}\end{aligned}$$
- Graviton with indices $(\mu,\nu)$ - vector bosons with (Lorentz index, momentum) combinations $(\lambda,p_1)$ and $(\sigma, p_2)$: $$\begin{aligned}
&& -\frac{i}{2} \kappa \left( - M^2 g^{\lambda \sigma }
g^{\mu \nu }+M^2 g^{\lambda \nu } g^{\mu \sigma }+M^2 g^{\lambda
\mu } g^{\nu \sigma }+p_1^{\mu } p_2^{\nu } g^{\lambda \sigma
}-p_1^{\sigma } \left(p_2^{\nu } g^{\lambda \mu }+p_2^{\mu }
g^{\lambda \nu }-p_2^{\lambda } g^{\mu \nu }\right)
\right. \nonumber\\
&& \left.
+p_1^{\nu }
\left(p_2^{\mu } g^{\lambda \sigma }-p_2^{\lambda } g^{\mu
\sigma }\right)-p_2^{\lambda } p_1^{\mu } g^{\nu \sigma
}-p_1\cdot p_2 g^{\lambda \sigma } g^{\mu \nu }+p_1\cdot p_2
g^{\lambda \nu } g^{\mu \sigma }+p_1\cdot p_2 g^{\lambda \mu }
g^{\nu \sigma }\right);
\label{hVV}\end{aligned}$$
- Gravitons with indices $(\mu,\nu)$ and $(\alpha,\beta)$ - vector bosons with (Lorentz index, momentum) combinations $(\lambda,p_1)$ and $(\sigma, p_2)$: $$\begin{aligned}
&&
-\frac{i}{4} \kappa ^2 \left(-g^{\alpha \sigma } g^{\beta \nu }
g^{\lambda \mu } M^2-g^{\alpha \nu } g^{\beta \sigma }
g^{\lambda \mu } M^2-g^{\alpha \sigma } g^{\beta \mu }
g^{\lambda \nu } M^2-g^{\alpha \mu } g^{\beta \sigma }
g^{\lambda \nu } M^2+g^{\alpha \nu } g^{\beta \mu } g^{\lambda
\sigma } M^2
\right. \nonumber\\
&& \left.
+g^{\alpha \mu } g^{\beta \nu } g^{\lambda \sigma }
M^2+g^{\alpha \sigma } g^{\beta \lambda } g^{\mu \nu }
M^2+g^{\alpha \lambda } g^{\beta \sigma } g^{\mu \nu }
M^2-g^{\alpha \beta } g^{\lambda \sigma } g^{\mu \nu }
M^2-g^{\alpha \nu } g^{\beta \lambda } g^{\mu \sigma }
M^2-g^{\alpha \lambda } g^{\beta \nu } g^{\mu \sigma }
M^2
\right. \nonumber\\
&& \left.
+g^{\alpha \beta } g^{\lambda \nu } g^{\mu \sigma }
M^2-g^{\alpha \mu } g^{\beta \lambda } g^{\nu \sigma }
M^2-g^{\alpha \lambda } g^{\beta \mu } g^{\nu \sigma }
M^2+g^{\alpha \beta } g^{\lambda \mu } g^{\nu \sigma }
M^2-p_1^{\mu } p_2^{\nu } g^{\alpha \sigma } g^{\beta \lambda
}+p_1^{\mu } p_2^{\lambda } g^{\alpha \sigma } g^{\beta \nu
}
\right. \nonumber\\
&& \left.
-p_1^{\mu } p_2^{\nu } g^{\alpha \lambda } g^{\beta \sigma
}+p_1^{\mu } p_2^{\lambda } g^{\alpha \nu } g^{\beta \sigma
}+p_1^{\beta } p_2^{\nu } g^{\alpha \sigma } g^{\lambda \mu
}+p_1^{\alpha } p_2^{\nu } g^{\beta \sigma } g^{\lambda \mu
}+p_1^{\beta } p_2^{\mu } g^{\alpha \sigma } g^{\lambda \nu
}+p_1^{\alpha } p_2^{\mu } g^{\beta \sigma } g^{\lambda \nu
}+p_1^{\mu } p_2^{\nu } g^{\alpha \beta } g^{\lambda \sigma
}
\right. \nonumber\\
&& \left.
-p_1^{\beta } p_2^{\nu } g^{\alpha \mu } g^{\lambda \sigma
}-p_1^{\mu } p_2^{\beta } g^{\alpha \nu } g^{\lambda \sigma
}-p_1^{\beta } p_2^{\mu } g^{\alpha \nu } g^{\lambda \sigma
}-p_1^{\alpha } p_2^{\nu } g^{\beta \mu } g^{\lambda \sigma
}-p_1^{\mu } p_2^{\alpha } g^{\beta \nu } g^{\lambda \sigma
}-p_1^{\alpha } p_2^{\mu } g^{\beta \nu } g^{\lambda \sigma
}-p_1^{\beta } p_2^{\lambda } g^{\alpha \sigma } g^{\mu \nu
}
\right. \nonumber\\
&& \left.
-p_1^{\alpha } p_2^{\lambda } g^{\beta \sigma } g^{\mu \nu
}+p_1^{\beta } p_2^{\alpha } g^{\lambda \sigma } g^{\mu \nu
}+p_1^{\alpha } p_2^{\beta } g^{\lambda \sigma } g^{\mu \nu
}+p_1^{\sigma } \left(p_2^{\mu } g^{\alpha \nu } g^{\beta
\lambda }-p_2^{\alpha } g^{\mu \nu } g^{\beta \lambda
}
-p_2^{\lambda } g^{\alpha \nu } g^{\beta \mu }+p_2^{\mu }
g^{\alpha \lambda } g^{\beta \nu }
\right. \right. \nonumber\\
&& \left. \left.
-p_2^{\lambda } g^{\alpha \mu
} g^{\beta \nu }+p_2^{\alpha } g^{\beta \nu } g^{\lambda \mu
}+p_2^{\nu } \left(g^{\alpha \mu } g^{\beta \lambda }+g^{\alpha
\lambda } g^{\beta \mu }-g^{\alpha \beta } g^{\lambda \mu
}\right)-p_2^{\mu } g^{\alpha \beta } g^{\lambda \nu
}+p_2^{\alpha } g^{\beta \mu } g^{\lambda \nu }+p_2^{\lambda }
g^{\alpha \beta } g^{\mu \nu }
\right. \right. \nonumber\\
&& \left. \left.
+p_2^{\beta } \left(g^{\alpha \nu
} g^{\lambda \mu }+g^{\alpha \mu } g^{\lambda \nu }-g^{\alpha
\lambda } g^{\mu \nu }\right)\right)+p_1^{\beta } p_2^{\lambda }
g^{\alpha \nu } g^{\mu \sigma }+p_1^{\alpha } p_2^{\lambda }
g^{\beta \nu } g^{\mu \sigma }-p_1^{\beta } p_2^{\alpha }
g^{\lambda \nu } g^{\mu \sigma }-p_1^{\alpha } p_2^{\beta }
g^{\lambda \nu } g^{\mu \sigma }
\right. \nonumber\\
&& \left.
+p_1^{\nu } \left(-p_2^{\mu }
g^{\alpha \sigma } g^{\beta \lambda }+p_2^{\alpha } g^{\mu
\sigma } g^{\beta \lambda }-p_2^{\mu } g^{\alpha \lambda }
g^{\beta \sigma }+p_2^{\mu } g^{\alpha \beta } g^{\lambda \sigma
}-p_2^{\alpha } g^{\beta \mu } g^{\lambda \sigma }+p_2^{\lambda
} \left(g^{\alpha \sigma } g^{\beta \mu }+g^{\alpha \mu }
g^{\beta \sigma }-g^{\alpha \beta } g^{\mu \sigma
}\right)
\right. \right. \nonumber\\
&& \left. \left.
+p_2^{\beta } \left(g^{\alpha \lambda } g^{\mu \sigma
}-g^{\alpha \mu } g^{\lambda \sigma }\right)\right)-p_1^{\mu }
p_2^{\lambda } g^{\alpha \beta } g^{\nu \sigma }+p_1^{\mu }
p_2^{\beta } g^{\alpha \lambda } g^{\nu \sigma }+p_1^{\beta }
p_2^{\lambda } g^{\alpha \mu } g^{\nu \sigma }+p_1^{\mu }
p_2^{\alpha } g^{\beta \lambda } g^{\nu \sigma }+p_1^{\alpha }
p_2^{\lambda } g^{\beta \mu } g^{\nu \sigma }
\right. \nonumber\\
&& \left.
-p_1^{\beta }
p_2^{\alpha } g^{\lambda \mu } g^{\nu \sigma }-p_1^{\alpha }
p_2^{\beta } g^{\lambda \mu } g^{\nu \sigma }-g^{\alpha \sigma }
g^{\beta \nu } g^{\lambda \mu } p_1\cdot p_2-g^{\alpha \nu }
g^{\beta \sigma } g^{\lambda \mu } p_1\cdot p_2-g^{\alpha \sigma
} g^{\beta \mu } g^{\lambda \nu } p_1\cdot p_2
\right. \nonumber\\
&& \left.
-g^{\alpha \mu }
g^{\beta \sigma } g^{\lambda \nu } p_1\cdot p_2+g^{\alpha \nu }
g^{\beta \mu } g^{\lambda \sigma } p_1\cdot p_2+g^{\alpha \mu }
g^{\beta \nu } g^{\lambda \sigma } p_1\cdot p_2+g^{\alpha \sigma
} g^{\beta \lambda } g^{\mu \nu } p_1\cdot p_2+g^{\alpha \lambda
} g^{\beta \sigma } g^{\mu \nu } p_1\cdot p_2
\right. \nonumber\\
&& \left.
-g^{\alpha \beta }
g^{\lambda \sigma } g^{\mu \nu } p_1\cdot p_2-g^{\alpha \nu }
g^{\beta \lambda } g^{\mu \sigma } p_1\cdot p_2-g^{\alpha
\lambda } g^{\beta \nu } g^{\mu \sigma } p_1\cdot p_2+g^{\alpha
\beta } g^{\lambda \nu } g^{\mu \sigma } p_1\cdot p_2-g^{\alpha
\mu } g^{\beta \lambda } g^{\nu \sigma } p_1\cdot p_2
\right. \nonumber\\
&& \left.
-g^{\alpha
\lambda } g^{\beta \mu } g^{\nu \sigma } p_1\cdot p_2+g^{\alpha
\beta } g^{\lambda \mu } g^{\nu \sigma } p_1\cdot p_2\right);
\label{hhVV}\end{aligned}$$
- Energy-momentum tensor with indices $(\mu,\nu)$ - gravitons with (Lorentz indices, momentum) combinations $(\lambda,\sigma, p_1)$ and $(\alpha,\beta, p_2)$: $$\begin{aligned}
&&
\frac{1}{8} \left[\text{hhh}\left(\left\{\mu ,\nu
,p_1\right\},\left\{\alpha ,\beta ,p_2\right\},\left\{\lambda
,\sigma ,p_3\right\}\right)+\text{hhh}\left(\left\{\mu ,\nu
,p_1\right\},\left\{\alpha ,\beta ,p_2\right\},\left\{\sigma
,\lambda ,p_3\right\}\right) \right. \nonumber\\
&&
\left.
+\text{hhh}\left(\left\{\mu ,\nu
,p_1\right\},\left\{\beta ,\alpha ,p_2\right\},\left\{\lambda
,\sigma ,p_3\right\}\right)+\text{hhh}\left(\left\{\mu ,\nu
,p_1\right\},\left\{\beta ,\alpha ,p_2\right\},\left\{\sigma
,\lambda ,p_3\right\}\right)
\right. \nonumber\\
&&
\left.
+\text{hhh}\left(\left\{\nu ,\mu
,p_1\right\},\left\{\alpha ,\beta ,p_2\right\},\left\{\lambda
,\sigma ,p_3\right\}\right)+\text{hhh}\left(\left\{\nu ,\mu
,p_1\right\},\left\{\alpha ,\beta ,p_2\right\},\left\{\sigma
,\lambda ,p_3\right\}\right)
\right. \nonumber\\
&&
\left.
+\text{hhh}\left(\left\{\nu ,\mu
,p_1\right\},\left\{\beta ,\alpha ,p_2\right\},\left\{\lambda
,\sigma ,p_3\right\}\right)+\text{hhh}\left(\left\{\nu ,\mu
,p_1\right\},\left\{\beta ,\alpha ,p_2\right\},\left\{\sigma
,\lambda ,p_3\right\}\right)\right],\end{aligned}$$ where $$\begin{aligned}
&&
\text{hhh}\left(\left\{\mu ,\nu
,p_1\right\},\left\{\alpha ,\beta ,p_2\right\},\left\{\lambda
,\sigma ,p_3\right\}\right)=-\frac{1}{4} i \kappa \left(p_1^{\alpha } p_2^{\beta } g^{\lambda
\sigma } g^{\mu \nu }+p_2^{\beta } p_3^{\alpha } g^{\lambda
\sigma } g^{\mu \nu }+2 \Lambda g^{\alpha \beta } g^{\lambda
\sigma } g^{\mu \nu }
\right. \nonumber\\
&&
\left.
+2 g^{\alpha \beta } g^{\lambda \sigma }
\left(p_1\cdot p_2+p_1\cdot p_3+p_2\cdot p_3\right) g^{\mu \nu
}+2 g^{\alpha \beta } g^{\lambda \sigma }
\left(p_1^2+p_2^2+p_3^2\right) g^{\mu \nu }
\right. \nonumber\\
&&
\left.
+g^{\alpha \beta }
\left(p_1^{\nu } \left(p_2^{\mu }+p_3^{\mu }\right) g^{\lambda
\sigma }+\left(p_1^{\lambda }+p_2^{\lambda }\right) p_3^{\sigma
} g^{\mu \nu }\right)+4 \left(p_1^{\sigma } p_3^{\nu } g^{\alpha
\beta } g^{\lambda \mu }+p_1^{\beta } p_2^{\nu } g^{\alpha \mu }
g^{\lambda \sigma }+p_2^{\sigma } p_3^{\beta } g^{\alpha \lambda
} g^{\mu \nu }\right)
\right. \nonumber\\
&&
\left.
+4 \left(\left(p_1^{\sigma } p_2^{\nu
}+p_2^{\sigma } p_3^{\nu }\right) g^{\alpha \beta } g^{\lambda
\mu }+\left(p_2^{\nu } p_3^{\beta }+p_1^{\beta } p_3^{\nu
}\right) g^{\alpha \mu } g^{\lambda \sigma }+\left(p_1^{\beta }
p_2^{\sigma }+p_1^{\sigma } p_3^{\beta }\right) g^{\alpha
\lambda } g^{\mu \nu }\right)
\right. \nonumber\\
&&
\left.
-2 \left(\left(p_1^{\nu }
p_2^{\sigma }+p_2^{\nu } p_3^{\sigma }\right) g^{\alpha \beta }
g^{\lambda \mu }+\left(p_1^{\nu } p_3^{\beta }+p_2^{\beta }
p_3^{\nu }\right) g^{\alpha \mu } g^{\lambda \sigma
}+\left(p_1^{\sigma } p_2^{\beta }+p_1^{\beta } p_3^{\sigma
}\right) g^{\alpha \lambda } g^{\mu \nu }\right)
\right. \nonumber\\
&&
\left.
+2
\left(\left(p_1^{\nu } p_1^{\sigma }+p_3^{\nu } p_3^{\sigma
}\right) g^{\alpha \beta } g^{\lambda \mu }+\left(p_1^{\beta }
p_1^{\nu }+p_2^{\beta } p_2^{\nu }\right) g^{\alpha \mu }
g^{\lambda \sigma }+\left(p_2^{\beta } p_2^{\sigma }+p_3^{\beta
} p_3^{\sigma }\right) g^{\alpha \lambda } g^{\mu \nu }\right)
\right. \nonumber\\
&&
\left.
-2
\left(p_1^{\mu } p_1^{\nu } g^{\alpha \beta } g^{\lambda \sigma
}+\left(p_3^{\lambda } p_3^{\sigma } g^{\alpha \beta
}+p_2^{\alpha } p_2^{\beta } g^{\lambda \sigma }\right) g^{\mu
\nu }\right)-2 \left(p_1^{\beta } p_3^{\alpha } g^{\lambda
\sigma } g^{\mu \nu }+p_1^{\alpha } p_3^{\beta } g^{\lambda
\sigma } g^{\mu \nu }
\right. \right. \nonumber\\
&&
\left. \left.
+g^{\alpha \beta } \left(\left(p_2^{\nu }
p_3^{\mu }+p_2^{\mu } p_3^{\nu }\right) g^{\lambda \sigma
}+\left(p_1^{\sigma } p_2^{\lambda }+p_1^{\lambda } p_2^{\sigma
}\right) g^{\mu \nu }\right)\right)-4 \left(p_1^{\alpha }
p_1^{\beta } g^{\lambda \sigma } g^{\mu \nu }+p_3^{\alpha }
p_3^{\beta } g^{\lambda \sigma } g^{\mu \nu }
\right. \right. \nonumber\\
&&
\left. \left.
+g^{\alpha \beta }
\left(\left(p_2^{\mu } p_2^{\nu }+p_3^{\mu } p_3^{\nu }\right)
g^{\lambda \sigma }+\left(p_1^{\lambda } p_1^{\sigma
}+p_2^{\lambda } p_2^{\sigma }\right) g^{\mu \nu
}\right)\right)-5 \left(p_2^{\alpha } \left(p_1^{\beta
}+p_3^{\beta }\right) g^{\lambda \sigma } g^{\mu \nu }
+g^{\alpha
\beta } \left(p_1^{\mu } \left(p_2^{\nu }+p_3^{\nu }\right)
g^{\lambda \sigma }
\right. \right. \right. \nonumber\\
&&
\left. \left. \left.
+\left(p_1^{\sigma }+p_2^{\sigma }\right)
p_3^{\lambda } g^{\mu \nu }\right)\right)
+2 \left(p_3^{\mu }
p_3^{\sigma } g^{\alpha \beta } g^{\lambda \nu }+p_1^{\alpha }
p_1^{\nu } g^{\beta \mu } g^{\lambda \sigma }+p_3^{\alpha }
p_3^{\sigma } g^{\beta \lambda } g^{\mu \nu }+p_2^{\beta }
\left(p_2^{\mu } g^{\alpha \nu } g^{\lambda \sigma
}+p_2^{\lambda } g^{\alpha \sigma } g^{\mu \nu
}\right)
\right. \right. \nonumber\\
&&
\left. \left.
+p_1^{\lambda } p_1^{\nu } g^{\alpha \beta } g^{\mu
\sigma }\right)+2 \left(p_2^{\sigma } p_3^{\mu } g^{\alpha \beta
} g^{\lambda \nu }+p_2^{\mu } p_3^{\beta } g^{\alpha \nu }
g^{\lambda \sigma }+p_1^{\alpha } p_3^{\nu } g^{\beta \mu }
g^{\lambda \sigma }+p_1^{\beta } p_2^{\lambda } g^{\alpha \sigma
} g^{\mu \nu }+p_1^{\sigma } p_3^{\alpha } g^{\beta \lambda }
g^{\mu \nu }+p_1^{\lambda } p_2^{\nu } g^{\alpha \beta } g^{\mu
\sigma }\right)
\right. \nonumber\\
&&
\left.
+4 \left(p_1^{\sigma } p_3^{\mu } g^{\alpha \beta
} g^{\lambda \nu }+p_1^{\beta } p_2^{\mu } g^{\alpha \nu }
g^{\lambda \sigma }+p_1^{\alpha } p_2^{\nu } g^{\beta \mu }
g^{\lambda \sigma }+p_2^{\lambda } p_3^{\beta } g^{\alpha \sigma
} g^{\mu \nu }+p_2^{\sigma } p_3^{\alpha } g^{\beta \lambda }
g^{\mu \nu }+p_1^{\lambda } p_3^{\nu } g^{\alpha \beta } g^{\mu
\sigma }\right)
\right. \nonumber\\
&&
\left.
-4 \left(p_1^{\beta } p_3^{\nu } g^{\alpha \sigma
} g^{\lambda \mu }+p_1^{\sigma } \left(p_2^{\nu } g^{\alpha \mu
} g^{\beta \lambda }+p_3^{\beta } g^{\alpha \nu } g^{\lambda \mu
}\right)+p_2^{\sigma } \left(p_3^{\nu } g^{\alpha \lambda }
g^{\beta \mu }+p_1^{\beta } g^{\alpha \mu } g^{\lambda \nu
}\right)+p_2^{\nu } p_3^{\beta } g^{\alpha \lambda } g^{\mu
\sigma }\right)
\right. \nonumber\\
&&
\left.
-6 \left(p_2^{\nu } p_3^{\beta } g^{\alpha \sigma
} g^{\lambda \mu }+p_2^{\sigma } \left(p_3^{\nu } g^{\alpha \mu
} g^{\beta \lambda }+p_1^{\beta } g^{\alpha \nu } g^{\lambda \mu
}\right)+p_1^{\sigma } \left(p_2^{\nu } g^{\alpha \lambda }
g^{\beta \mu }+p_3^{\beta } g^{\alpha \mu } g^{\lambda \nu
}\right)+p_1^{\beta } p_3^{\nu } g^{\alpha \lambda } g^{\mu
\sigma }\right)
\right. \nonumber\\
&&
\left.
+8 \Lambda \left(g^{\alpha \sigma } g^{\beta \mu
} g^{\lambda \nu }+g^{\alpha \nu } g^{\beta \lambda } g^{\mu
\sigma }\right)-2 \left(p_2^{\nu } p_3^{\mu } g^{\alpha \sigma }
g^{\beta \lambda }+p_2^{\mu } p_3^{\nu } g^{\alpha \sigma }
g^{\beta \lambda }+p_1^{\sigma } p_2^{\lambda } g^{\alpha \nu }
g^{\beta \mu }+p_1^{\lambda } p_2^{\sigma } g^{\alpha \nu }
g^{\beta \mu }
\right. \right. \nonumber\\
&&
\left. \left.
+p_1^{\beta } p_3^{\alpha } g^{\lambda \nu }
g^{\mu \sigma }+p_1^{\alpha } p_3^{\beta } g^{\lambda \nu }
g^{\mu \sigma }\right)-2 \left(p_1^{\sigma } p_3^{\alpha }
g^{\beta \mu } g^{\lambda \nu }+g^{\alpha \sigma }
\left(p_1^{\lambda } p_2^{\nu } g^{\beta \mu }+p_2^{\mu }
p_3^{\beta } g^{\lambda \nu }\right)+p_1^{\alpha } p_3^{\nu }
g^{\beta \lambda } g^{\mu \sigma }
\right. \right. \nonumber\\
&&
\left. \left.
+g^{\alpha \nu }
\left(p_2^{\sigma } p_3^{\mu } g^{\beta \lambda }+p_1^{\beta }
p_2^{\lambda } g^{\mu \sigma }\right)\right)+2 \left(p_2^{\nu }
p_3^{\sigma } g^{\alpha \mu } g^{\beta \lambda }+p_1^{\sigma }
p_2^{\beta } g^{\alpha \nu } g^{\lambda \mu }+p_2^{\beta }
p_3^{\nu } g^{\alpha \sigma } g^{\lambda \mu }+p_1^{\beta }
p_3^{\sigma } g^{\alpha \mu } g^{\lambda \nu }
\right. \right. \nonumber\\
&&
\left. \left.
+p_1^{\nu }
g^{\alpha \lambda } \left(p_2^{\sigma } g^{\beta \mu
}+p_3^{\beta } g^{\mu \sigma }\right)\right)+2 \left(p_1^{\mu }
\left(p_2^{\sigma } g^{\alpha \beta } g^{\lambda \nu
}+p_3^{\beta } g^{\alpha \nu } g^{\lambda \sigma
}\right)+p_2^{\alpha } \left(p_3^{\nu } g^{\beta \mu }
g^{\lambda \sigma }+p_1^{\sigma } g^{\beta \lambda } g^{\mu \nu
}\right)
\right. \right. \nonumber\\
&&
\left. \left.
+p_3^{\lambda } \left(p_1^{\beta } g^{\alpha \sigma }
g^{\mu \nu }+p_2^{\nu } g^{\alpha \beta } g^{\mu \sigma
}\right)\right)-2 \left(p_1^{\mu } \left(p_3^{\sigma } g^{\alpha
\beta } g^{\lambda \nu }+p_2^{\beta } g^{\alpha \nu } g^{\lambda
\sigma }\right)+\left(p_2^{\beta } p_3^{\lambda } g^{\alpha
\sigma }+p_2^{\alpha } p_3^{\sigma } g^{\beta \lambda }\right)
g^{\mu \nu }
\right. \right. \nonumber\\
&&
\left. \left.
+p_1^{\nu } \left(p_2^{\alpha } g^{\beta \mu }
g^{\lambda \sigma }+p_3^{\lambda } g^{\alpha \beta } g^{\mu
\sigma }\right)\right)+2 \left(p_1^{\mu } \left(p_1^{\sigma }
g^{\alpha \beta } g^{\lambda \nu }+p_1^{\beta } g^{\alpha \nu }
g^{\lambda \sigma }\right)+p_2^{\alpha } \left(p_2^{\nu }
g^{\beta \mu } g^{\lambda \sigma }+p_2^{\sigma } g^{\beta
\lambda } g^{\mu \nu }\right)
\right. \right. \nonumber\\
&&
\left. \left.
+p_3^{\lambda } \left(p_3^{\beta }
g^{\alpha \sigma } g^{\mu \nu }+p_3^{\nu } g^{\alpha \beta }
g^{\mu \sigma }\right)\right)-4 \left(p_1^{\sigma } p_3^{\nu }
\left(g^{\alpha \mu } g^{\beta \lambda }+g^{\alpha \lambda }
g^{\beta \mu }\right)+p_2^{\sigma } p_3^{\beta } \left(g^{\alpha
\nu } g^{\lambda \mu }+g^{\alpha \mu } g^{\lambda \nu
}\right)
\right. \right. \nonumber\\
&&
\left. \left.
+p_1^{\beta } p_2^{\nu } \left(g^{\alpha \sigma }
g^{\lambda \mu }+g^{\alpha \lambda } g^{\mu \sigma
}\right)\right)-4 \left(p_2^{\nu } p_2^{\sigma } \left(g^{\alpha
\mu } g^{\beta \lambda }+g^{\alpha \lambda } g^{\beta \mu
}\right)+p_1^{\beta } p_1^{\sigma } \left(g^{\alpha \nu }
g^{\lambda \mu }+g^{\alpha \mu } g^{\lambda \nu
}\right)
\right. \right. \nonumber\\
&&
\left. \left.
+p_3^{\beta } p_3^{\nu } \left(g^{\alpha \sigma }
g^{\lambda \mu }+g^{\alpha \lambda } g^{\mu \sigma
}\right)\right)+2 \left(p_2^{\mu } p_3^{\sigma } g^{\alpha \nu }
g^{\beta \lambda }+p_2^{\beta } p_3^{\mu } g^{\alpha \sigma }
g^{\lambda \nu }+p_1^{\alpha } p_3^{\sigma } g^{\beta \mu }
g^{\lambda \nu }+p_1^{\lambda } p_2^{\beta } g^{\alpha \nu }
g^{\mu \sigma }
\right. \right. \nonumber\\
&&
\left. \left.
+p_1^{\nu } \left(p_2^{\lambda } g^{\alpha \sigma
} g^{\beta \mu }+p_3^{\alpha } g^{\beta \lambda } g^{\mu \sigma
}\right)\right)+16 \left(p_3^{\alpha } p_3^{\mu } g^{\beta \nu }
g^{\lambda \sigma }+p_1^{\alpha } p_1^{\lambda } g^{\beta \sigma
} g^{\mu \nu }+p_2^{\lambda } p_2^{\mu } g^{\alpha \beta }
g^{\nu \sigma }\right)-8 \left(p_1^{\alpha } p_2^{\mu } g^{\beta
\nu } g^{\lambda \sigma }
\right. \right. \nonumber\\
&&
\left. \left.
+p_2^{\lambda } p_3^{\alpha } g^{\beta
\sigma } g^{\mu \nu }+p_1^{\lambda } p_3^{\mu } g^{\alpha \beta
} g^{\nu \sigma }\right)+2 \left(p_1^{\alpha } p_1^{\mu }
g^{\beta \nu } g^{\lambda \sigma }+p_3^{\alpha } p_3^{\lambda }
g^{\beta \sigma } g^{\mu \nu }+p_2^{\alpha } \left(p_2^{\mu }
g^{\beta \nu } g^{\lambda \sigma }+p_2^{\lambda } g^{\beta
\sigma } g^{\mu \nu }\right)
\right. \right. \nonumber\\
&&
\left. \left.
+p_1^{\lambda } p_1^{\mu } g^{\alpha
\beta } g^{\nu \sigma }+p_3^{\lambda } p_3^{\mu } g^{\alpha
\beta } g^{\nu \sigma }\right)+2 \left(p_3^{\alpha }
\left(p_2^{\mu } g^{\beta \nu } g^{\lambda \sigma }+p_1^{\lambda
} g^{\beta \sigma } g^{\mu \nu }\right)+p_1^{\alpha }
\left(p_3^{\mu } g^{\beta \nu } g^{\lambda \sigma }+p_2^{\lambda
} g^{\beta \sigma } g^{\mu \nu }\right)
\right. \right. \nonumber\\
&&
\left. \left.
+\left(p_1^{\lambda }
p_2^{\mu }+p_2^{\lambda } p_3^{\mu }\right) g^{\alpha \beta }
g^{\nu \sigma }\right)-4 \left(p_2^{\mu } p_3^{\lambda }
g^{\alpha \sigma } g^{\beta \nu }+p_1^{\alpha } p_3^{\lambda }
g^{\mu \sigma } g^{\beta \nu }+p_1^{\mu } g^{\beta \sigma }
\left(p_2^{\lambda } g^{\alpha \nu }+p_3^{\alpha } g^{\lambda
\nu }\right)+p_2^{\alpha } p_3^{\mu } g^{\beta \lambda } g^{\nu
\sigma }
\right. \right. \nonumber\\
&&
\left. \left.
+p_1^{\lambda } p_2^{\alpha } g^{\beta \mu } g^{\nu
\sigma }\right)-8 \left(p_1^{\lambda } p_1^{\mu } g^{\alpha \nu
} g^{\beta \sigma }+p_1^{\alpha } p_1^{\mu } g^{\lambda \nu }
g^{\beta \sigma }+p_3^{\lambda } g^{\beta \nu } \left(p_3^{\mu }
g^{\alpha \sigma }+p_3^{\alpha } g^{\mu \sigma
}\right)+p_2^{\alpha } p_2^{\mu } g^{\beta \lambda } g^{\nu
\sigma }+p_2^{\alpha } p_2^{\lambda } g^{\beta \mu } g^{\nu
\sigma }\right)
\right. \nonumber\\
&&
\left.
+4 \left(p_3^{\lambda } p_3^{\sigma } g^{\alpha
\mu } g^{\beta \nu }+p_1^{\mu } p_1^{\nu } g^{\alpha \lambda }
g^{\beta \sigma }+p_2^{\alpha } p_2^{\beta } g^{\lambda \mu }
g^{\nu \sigma }\right)-2 \left(p_1^{\lambda } p_3^{\sigma }
g^{\alpha \mu } g^{\beta \nu }+p_2^{\lambda } p_3^{\sigma }
g^{\alpha \mu } g^{\beta \nu }+p_1^{\nu } p_2^{\mu } g^{\alpha
\lambda } g^{\beta \sigma }
\right. \right. \nonumber\\
&&
\left. \left.
+p_1^{\nu } p_3^{\mu } g^{\alpha
\lambda } g^{\beta \sigma }+p_1^{\alpha } p_2^{\beta }
g^{\lambda \mu } g^{\nu \sigma }+p_2^{\beta } p_3^{\alpha }
g^{\lambda \mu } g^{\nu \sigma }\right)+8 \left(p_1^{\sigma }
p_2^{\lambda } g^{\alpha \mu } g^{\beta \nu }+p_1^{\lambda }
p_2^{\sigma } g^{\alpha \mu } g^{\beta \nu }+p_2^{\nu } p_3^{\mu
} g^{\alpha \lambda } g^{\beta \sigma }
+p_2^{\mu } p_3^{\nu }
g^{\alpha \lambda } g^{\beta \sigma }
\right. \right. \nonumber\\
&&
\left. \left.
+p_1^{\beta } p_3^{\alpha }
g^{\lambda \mu } g^{\nu \sigma }+p_1^{\alpha } p_3^{\beta }
g^{\lambda \mu } g^{\nu \sigma }\right)+8 \left(p_1^{\lambda }
p_1^{\sigma } g^{\alpha \mu } g^{\beta \nu }+p_2^{\lambda }
p_2^{\sigma } g^{\alpha \mu } g^{\beta \nu }+p_2^{\mu } p_2^{\nu
} g^{\alpha \lambda } g^{\beta \sigma }+p_3^{\mu } p_3^{\nu }
g^{\alpha \lambda } g^{\beta \sigma }+p_1^{\alpha } p_1^{\beta }
g^{\lambda \mu } g^{\nu \sigma }
\right. \right. \nonumber\\
&&
\left. \left.
+p_3^{\alpha } p_3^{\beta }
g^{\lambda \mu } g^{\nu \sigma }\right)+10
\left(\left(p_1^{\sigma }+p_2^{\sigma }\right) p_3^{\lambda }
g^{\alpha \mu } g^{\beta \nu }+p_1^{\mu } \left(p_2^{\nu
}+p_3^{\nu }\right) g^{\alpha \lambda } g^{\beta \sigma
}+p_2^{\alpha } \left(p_1^{\beta }+p_3^{\beta }\right)
g^{\lambda \mu } g^{\nu \sigma }\right)
\right. \nonumber\\
&&
\left.
-4 \Lambda
\left(g^{\alpha \mu } g^{\beta \nu } g^{\lambda \sigma
}+g^{\alpha \lambda } g^{\beta \sigma } g^{\mu \nu }+g^{\alpha
\beta } g^{\lambda \mu } g^{\nu \sigma }\right)+4
\left(p_2^{\alpha } \left(p_3^{\sigma } g^{\beta \nu }+p_1^{\nu
} g^{\beta \sigma }\right) g^{\lambda \mu }+p_1^{\mu } g^{\alpha
\lambda } \left(p_3^{\sigma } g^{\beta \nu }+p_2^{\beta } g^{\nu
\sigma }\right)
\right. \right. \nonumber\\
&&
\left. \left.
+p_3^{\lambda } g^{\alpha \mu } \left(p_1^{\nu }
g^{\beta \sigma }+p_2^{\beta } g^{\nu \sigma }\right)\right)-8
\left(\left(p_3^{\alpha } p_3^{\sigma } g^{\beta \nu
}+p_1^{\alpha } p_1^{\nu } g^{\beta \sigma }\right) g^{\lambda
\mu }+g^{\alpha \mu } \left(p_1^{\lambda } p_1^{\nu } g^{\beta
\sigma }+p_2^{\beta } p_2^{\lambda } g^{\nu \sigma
}\right)
\right. \right. \nonumber\\
&&
\left. \left.
+g^{\alpha \lambda } \left(p_3^{\mu } p_3^{\sigma }
g^{\beta \nu }+p_2^{\beta } p_2^{\mu } g^{\nu \sigma
}\right)\right)-10 \left(p_2^{\alpha } \left(p_1^{\sigma }
g^{\beta \nu }+p_3^{\nu } g^{\beta \sigma }\right) g^{\lambda
\mu }+p_3^{\lambda } g^{\alpha \mu } \left(p_2^{\nu } g^{\beta
\sigma }+p_1^{\beta } g^{\nu \sigma }\right)
\right. \right. \nonumber\\
&&
\left. \left.
+p_1^{\mu }
g^{\alpha \lambda } \left(p_2^{\sigma } g^{\beta \nu
}+p_3^{\beta } g^{\nu \sigma }\right)\right)-4
\left(\left(p_2^{\sigma } p_3^{\alpha } g^{\beta \nu
}+p_1^{\alpha } p_2^{\nu } g^{\beta \sigma }\right) g^{\lambda
\mu }+g^{\alpha \lambda } \left(p_1^{\sigma } p_3^{\mu }
g^{\beta \nu }+p_1^{\beta } p_2^{\mu } g^{\nu \sigma
}\right)
\right. \right. \nonumber\\
&&
\left. \left.
+g^{\alpha \mu } \left(p_1^{\lambda } p_3^{\nu }
g^{\beta \sigma }+p_2^{\lambda } p_3^{\beta } g^{\nu \sigma
}\right)\right)-4 \left(\left(p_1^{\sigma } p_3^{\alpha }
g^{\beta \nu }+p_1^{\alpha } p_3^{\nu } g^{\beta \sigma }\right)
g^{\lambda \mu }+g^{\alpha \mu } \left(p_1^{\lambda } p_2^{\nu }
g^{\beta \sigma }+p_1^{\beta } p_2^{\lambda } g^{\nu \sigma
}\right)
\right. \right. \nonumber\\
&&
\left. \left.
+g^{\alpha \lambda } \left(p_2^{\sigma } p_3^{\mu }
g^{\beta \nu }+p_2^{\mu } p_3^{\beta } g^{\nu \sigma
}\right)\right)+10 \left(\left(p_1^{\alpha } p_2^{\sigma }
g^{\beta \nu }+p_2^{\nu } p_3^{\alpha } g^{\beta \sigma }\right)
g^{\lambda \mu }+g^{\alpha \mu } \left(p_2^{\lambda } p_3^{\nu }
g^{\beta \sigma }+p_1^{\lambda } p_3^{\beta } g^{\nu \sigma
}\right)
\right. \right. \nonumber\\
&&
\left. \left.
+g^{\alpha \lambda } \left(p_1^{\sigma } p_2^{\mu }
g^{\beta \nu }+p_1^{\beta } p_3^{\mu } g^{\nu \sigma
}\right)\right)+8 \left(p_2^{\alpha } \left(p_3^{\mu } g^{\beta
\nu } g^{\lambda \sigma }+p_1^{\lambda } g^{\beta \sigma }
g^{\mu \nu }\right)+p_1^{\mu } \left(p_3^{\alpha } g^{\beta \nu
} g^{\lambda \sigma }+p_2^{\lambda } g^{\alpha \beta } g^{\nu
\sigma }\right)
\right. \right. \nonumber\\
&&
\left. \left.
+p_3^{\lambda } \left(p_1^{\alpha } g^{\beta
\sigma } g^{\mu \nu }+p_2^{\mu } g^{\alpha \beta } g^{\nu \sigma
}\right)\right)+12 \left(p_2^{\alpha } p_3^{\lambda } g^{\beta
\sigma } g^{\mu \nu }+p_1^{\mu } \left(p_2^{\alpha } g^{\beta
\nu } g^{\lambda \sigma }+p_3^{\lambda } g^{\alpha \beta }
g^{\nu \sigma }\right)\right)-6 \left(p_2^{\mu } p_3^{\lambda }
g^{\alpha \nu } g^{\beta \sigma }
\right. \right. \nonumber\\
&&
\left. \left.
+p_2^{\alpha } p_3^{\mu }
g^{\lambda \nu } g^{\beta \sigma }+p_1^{\lambda } p_2^{\alpha }
g^{\beta \nu } g^{\mu \sigma }+p_1^{\alpha } p_3^{\lambda }
g^{\beta \mu } g^{\nu \sigma }+p_1^{\mu } \left(p_2^{\lambda }
g^{\alpha \sigma } g^{\beta \nu }+p_3^{\alpha } g^{\beta \lambda
} g^{\nu \sigma }\right)\right)-4 \left(p_2^{\lambda } p_2^{\mu
} \left(g^{\alpha \sigma } g^{\beta \nu }+g^{\alpha \nu }
g^{\beta \sigma }\right)
\right. \right. \nonumber\\
&&
\left. \left.
+p_3^{\alpha } p_3^{\mu } \left(g^{\beta
\sigma } g^{\lambda \nu }+g^{\beta \lambda } g^{\nu \sigma
}\right)+p_1^{\alpha } p_1^{\lambda } \left(g^{\beta \nu }
g^{\mu \sigma }+g^{\beta \mu } g^{\nu \sigma }\right)\right)+4
\left(p_1^{\lambda } p_3^{\mu } \left(g^{\alpha \sigma }
g^{\beta \nu }+g^{\alpha \nu } g^{\beta \sigma
}\right)
\right. \right. \nonumber\\
&&
\left. \left.
+p_1^{\alpha } p_2^{\mu } \left(g^{\beta \sigma }
g^{\lambda \nu }+g^{\beta \lambda } g^{\nu \sigma
}\right)+p_2^{\lambda } p_3^{\alpha } \left(g^{\beta \nu }
g^{\mu \sigma }+g^{\beta \mu } g^{\nu \sigma }\right)\right)-4
\left(p_1^{\alpha } p_3^{\mu } g^{\beta \sigma } g^{\lambda \nu
}+p_1^{\lambda } \left(p_2^{\mu } g^{\alpha \nu } g^{\beta
\sigma }+p_3^{\alpha } g^{\beta \nu } g^{\mu \sigma
}\right)
\right. \right. \nonumber\\
&&
\left. \left.
+p_2^{\mu } p_3^{\alpha } g^{\beta \lambda } g^{\nu
\sigma }+p_2^{\lambda } \left(p_3^{\mu } g^{\alpha \sigma }
g^{\beta \nu }+p_1^{\alpha } g^{\beta \mu } g^{\nu \sigma
}\right)\right)+2 \left(p_2^{\mu } p_3^{\alpha } g^{\beta \sigma
} g^{\lambda \nu }+p_2^{\lambda } \left(p_3^{\mu } g^{\alpha \nu
} g^{\beta \sigma }+p_1^{\alpha } g^{\beta \nu } g^{\mu \sigma
}\right)+p_1^{\alpha } p_3^{\mu } g^{\beta \lambda } g^{\nu
\sigma }
\right. \right. \nonumber\\
&&
\left. \left.
+p_1^{\lambda } \left(p_2^{\mu } g^{\alpha \sigma }
g^{\beta \nu }+p_3^{\alpha } g^{\beta \mu } g^{\nu \sigma
}\right)\right)-12 \left(p_2^{\alpha } p_3^{\lambda }
\left(g^{\beta \nu } g^{\mu \sigma }+g^{\beta \mu } g^{\nu
\sigma }\right)+p_1^{\mu } \left(p_3^{\lambda } \left(g^{\alpha
\sigma } g^{\beta \nu }+g^{\alpha \nu } g^{\beta \sigma
}\right)
\right. \right. \right. \nonumber\\
&&
\left. \left. \left.
+p_2^{\alpha } \left(g^{\beta \sigma } g^{\lambda \nu
}+g^{\beta \lambda } g^{\nu \sigma }\right)\right)\right)+4
\left(g^{\alpha \sigma } g^{\beta \nu } g^{\lambda \mu }
p_1\cdot p_2+g^{\alpha \mu } g^{\beta \lambda } g^{\nu \sigma }
p_1\cdot p_3+g^{\alpha \lambda } \left(g^{\beta \nu } g^{\mu
\sigma } p_1\cdot p_2+g^{\beta \mu } g^{\nu \sigma } p_1\cdot
p_3\right)
\right. \right. \nonumber\\
&&
\left. \left.
+g^{\alpha \nu } g^{\beta \sigma } g^{\lambda \mu }
p_2\cdot p_3+g^{\alpha \mu } g^{\beta \sigma } g^{\lambda \nu }
p_2\cdot p_3\right)+2 \left(g^{\alpha \nu } g^{\beta \mu }
g^{\lambda \sigma } p_1\cdot p_2+g^{\alpha \beta } g^{\lambda
\nu } g^{\mu \sigma } p_1\cdot p_3+g^{\alpha \sigma } g^{\beta
\lambda } g^{\mu \nu } p_2\cdot p_3\right)\right. \nonumber\\
&&
\left.
-8 \left(g^{\alpha \mu
} g^{\beta \nu } g^{\lambda \sigma } p_1\cdot p_2+g^{\alpha
\beta } g^{\lambda \mu } g^{\nu \sigma } p_1\cdot p_3+g^{\alpha
\lambda } g^{\beta \sigma } g^{\mu \nu } p_2\cdot p_3\right)+8
\left(g^{\alpha \mu } \left(g^{\beta \sigma } g^{\lambda \nu
}
\right. \right. \right. \nonumber\\
&&
\left. \left. \left.
+g^{\beta \lambda } g^{\nu \sigma }\right) p_1\cdot
p_2+g^{\alpha \sigma } g^{\beta \nu } g^{\lambda \mu } p_1\cdot
p_3+g^{\alpha \nu } g^{\beta \sigma } g^{\lambda \mu } p_1\cdot
p_3+g^{\alpha \lambda } g^{\beta \nu } g^{\mu \sigma } p_2\cdot
p_3+g^{\alpha \lambda } g^{\beta \mu } g^{\nu \sigma } p_2\cdot
p_3\right)
\right. \nonumber\\
&&
\left.
-4 \left(g^{\alpha \lambda } g^{\beta \sigma } g^{\mu
\nu } \left(p_1\cdot p_2+p_1\cdot p_3\right)+g^{\alpha \beta }
g^{\lambda \mu } g^{\nu \sigma } \left(p_1\cdot p_2+p_2\cdot
p_3\right)+g^{\alpha \mu } g^{\beta \nu } g^{\lambda \sigma }
\left(p_1\cdot p_3+p_2\cdot p_3\right)\right)
\right. \nonumber\\
&&
\left.
-4 \left(g^{\alpha
\lambda } g^{\beta \sigma } g^{\mu \nu } p_1^2+g^{\alpha \beta }
g^{\lambda \mu } g^{\nu \sigma } p_2^2+g^{\alpha \mu } g^{\beta
\nu } g^{\lambda \sigma } p_3^2\right)+8 \left(g^{\alpha \nu }
g^{\beta \sigma } g^{\lambda \mu } p_1^2+g^{\alpha \lambda }
g^{\beta \mu } g^{\nu \sigma } p_2^2+g^{\alpha \sigma } g^{\beta
\nu } g^{\lambda \mu } p_3^2
\right. \right. \nonumber\\
&&
\left. \left.
+g^{\alpha \lambda } g^{\beta \nu }
g^{\mu \sigma } p_3^2+g^{\alpha \mu } \left(g^{\beta \sigma }
g^{\lambda \nu } p_1^2+g^{\beta \lambda } g^{\nu \sigma }
p_2^2\right)\right)-4 \left(g^{\alpha \mu } g^{\beta \nu }
g^{\lambda \sigma } \left(p_1^2+p_2^2\right)
\right. \right. \nonumber\\
&&
\left. \left.+g^{\alpha \beta }
g^{\lambda \mu } g^{\nu \sigma }
\left(p_1^2+p_3^2\right)+g^{\alpha \lambda } g^{\beta \sigma }
g^{\mu \nu } \left(p_2^2+p_3^2\right)\right)\right)
;
\label{hhh}\end{aligned}$$
- Energy-momentum tensor with indices $(\mu,\nu)$ - scalars with momenta $p_1$ and $p_2$: $$g^{\mu \nu } \left(m^2+p_1\cdot p_2\right)-p_2^{\mu }
p_1^{\nu } -p_1^{\mu } p_2^{\nu } ;
\label{TSS}$$
- Energy-momentum tensor with indices $(\mu,\nu)$ - graviton with indices $(\alpha,\beta)$- scalars with momenta $p_1$ and $p_2$: $$\begin{aligned}
&& \frac{1}{2} \kappa \left(m^2 \left(-g^{\alpha \nu }\right)
g^{\beta \mu }-m^2 g^{\alpha \mu } g^{\beta \nu }+2 m^2
g^{\alpha \beta } g^{\mu \nu }+p_1^{\beta } p_2^{\nu } g^{\alpha
\mu }+p_1^{\beta } p_2^{\mu } g^{\alpha \nu }+p_1^{\alpha }
p_2^{\nu } g^{\beta \mu } \right. \nonumber\\
&& \left. +p_1^{\nu } \left(-2 p_2^{\mu }
g^{\alpha \beta }+p_2^{\beta } g^{\alpha \mu }+p_2^{\alpha }
g^{\beta \mu }\right)+p_1^{\alpha } p_2^{\mu } g^{\beta \nu
}+p_1^{\mu } \left(-2 p_2^{\nu } g^{\alpha \beta }+p_2^{\beta }
g^{\alpha \nu }+p_2^{\alpha } g^{\beta \nu }\right)-p_2^{\alpha
} p_1^{\beta } g^{\mu \nu }
\right. \nonumber\\
&& \left.
-p_1^{\alpha } p_2^{\beta } g^{\mu
\nu }-p_1\cdot p_2 g^{\alpha \nu } g^{\beta \mu }-p_1\cdot p_2
g^{\alpha \mu } g^{\beta \nu }+2 p_1\cdot p_2 g^{\alpha \beta }
g^{\mu \nu }\right);
\label{ThSS}\end{aligned}$$
- Energy-momentum tensor with indices $(\mu,\nu)$ - vector bosons with (Lorentz index, momentum) combinations $(\lambda,p_1)$ and $(\sigma, p_2)$: $$\begin{aligned}
&& - M^2 g^{\lambda \sigma } g^{\mu \nu }+M^2 g^{\lambda
\nu } g^{\mu \sigma }+M^2 g^{\lambda \mu } g^{\nu \sigma
}+p_1^{\mu } p_2^{\nu } g^{\lambda \sigma }-p_1^{\sigma }
\left(p_2^{\nu } g^{\lambda \mu }+p_2^{\mu } g^{\lambda \nu
}-p_2^{\lambda } g^{\mu \nu }\right)
\nonumber\\
&&
+p_1^{\nu } \left(p_2^{\mu }
g^{\lambda \sigma }-p_2^{\lambda } g^{\mu \sigma
}\right) -p_2^{\lambda } p_1^{\mu } g^{\nu \sigma }-p_1\cdot p_2
g^{\lambda \sigma } g^{\mu \nu }+p_1\cdot p_2 g^{\lambda \nu }
g^{\mu \sigma }+p_1\cdot p_2 g^{\lambda \mu } g^{\nu \sigma };
\label{TVV}\end{aligned}$$
- Energy-momentum tensor with indices $(\mu,\nu)$ - graviton with indices $(\alpha,\beta)$ - vector bosons with (Lorentz index, momentum) combinations $(\lambda,p_1)$ and $(\sigma, p_2)$: $$\begin{aligned}
&& -\frac{1}{2} \kappa \left(g^{\alpha \sigma } g^{\beta \nu }
g^{\lambda \mu } M^2+g^{\alpha \nu } g^{\beta \sigma }
g^{\lambda \mu } M^2+g^{\alpha \sigma } g^{\beta \mu }
g^{\lambda \nu } M^2+g^{\alpha \mu } g^{\beta \sigma }
g^{\lambda \nu } M^2-g^{\alpha \nu } g^{\beta \mu } g^{\lambda
\sigma } M^2-g^{\alpha \mu } g^{\beta \nu } g^{\lambda \sigma }
M^2
\right. \nonumber\\
&& \left.
-g^{\alpha \sigma } g^{\beta \lambda } g^{\mu \nu }
M^2-g^{\alpha \lambda } g^{\beta \sigma } g^{\mu \nu } M^2+2
g^{\alpha \beta } g^{\lambda \sigma } g^{\mu \nu } M^2+g^{\alpha
\nu } g^{\beta \lambda } g^{\mu \sigma } M^2+g^{\alpha \lambda }
g^{\beta \nu } g^{\mu \sigma } M^2-2 g^{\alpha \beta }
g^{\lambda \nu } g^{\mu \sigma } M^2
\right. \nonumber\\
&& \left.
+g^{\alpha \mu } g^{\beta
\lambda } g^{\nu \sigma } M^2+g^{\alpha \lambda } g^{\beta \mu }
g^{\nu \sigma } M^2-2 g^{\alpha \beta } g^{\lambda \mu } g^{\nu
\sigma } M^2+p_1^{\mu } p_2^{\nu } g^{\alpha \sigma } g^{\beta
\lambda }-p_1^{\mu } p_2^{\lambda } g^{\alpha \sigma } g^{\beta
\nu }+p_1^{\mu } p_2^{\nu } g^{\alpha \lambda } g^{\beta \sigma
}
\right. \nonumber\\
&& \left.
-p_1^{\mu } p_2^{\lambda } g^{\alpha \nu } g^{\beta \sigma
}-p_1^{\beta } p_2^{\nu } g^{\alpha \sigma } g^{\lambda \mu
}-p_1^{\alpha } p_2^{\nu } g^{\beta \sigma } g^{\lambda \mu
}-p_1^{\beta } p_2^{\mu } g^{\alpha \sigma } g^{\lambda \nu
}-p_1^{\alpha } p_2^{\mu } g^{\beta \sigma } g^{\lambda \nu }-2
p_1^{\mu } p_2^{\nu } g^{\alpha \beta } g^{\lambda \sigma
}+p_1^{\beta } p_2^{\nu } g^{\alpha \mu } g^{\lambda \sigma
}
\right. \nonumber\\
&& \left.
+p_1^{\mu } p_2^{\beta } g^{\alpha \nu } g^{\lambda \sigma
}+p_1^{\beta } p_2^{\mu } g^{\alpha \nu } g^{\lambda \sigma
}+p_1^{\alpha } p_2^{\nu } g^{\beta \mu } g^{\lambda \sigma
}+p_1^{\mu } p_2^{\alpha } g^{\beta \nu } g^{\lambda \sigma
}+p_1^{\alpha } p_2^{\mu } g^{\beta \nu } g^{\lambda \sigma
}+p_1^{\beta } p_2^{\lambda } g^{\alpha \sigma } g^{\mu \nu
}+p_1^{\alpha } p_2^{\lambda } g^{\beta \sigma } g^{\mu \nu
}
\right. \nonumber\\
&& \left.
-p_1^{\beta } p_2^{\alpha } g^{\lambda \sigma } g^{\mu \nu
}-p_1^{\alpha } p_2^{\beta } g^{\lambda \sigma } g^{\mu \nu
}-p_1^{\sigma } \left(-p_2^{\lambda } g^{\alpha \nu } g^{\beta
\mu }+p_2^{\alpha } g^{\lambda \nu } g^{\beta \mu }-p_2^{\lambda
} g^{\alpha \mu } g^{\beta \nu }+p_2^{\beta } g^{\alpha \nu }
g^{\lambda \mu }+p_2^{\alpha } g^{\beta \nu } g^{\lambda \mu
}
\right. \right. \nonumber\\
&& \left.\left.
+p_2^{\nu } \left(g^{\alpha \mu } g^{\beta \lambda }+g^{\alpha
\lambda } g^{\beta \mu }-2 g^{\alpha \beta } g^{\lambda \mu
}\right)+p_2^{\beta } g^{\alpha \mu } g^{\lambda \nu }+p_2^{\mu
} \left(g^{\alpha \nu } g^{\beta \lambda }+g^{\alpha \lambda }
g^{\beta \nu }-2 g^{\alpha \beta } g^{\lambda \nu }\right)+2
p_2^{\lambda } g^{\alpha \beta } g^{\mu \nu }
\right. \right. \nonumber\\
&& \left.\left.
-p_2^{\beta }
g^{\alpha \lambda } g^{\mu \nu }-p_2^{\alpha } g^{\beta \lambda
} g^{\mu \nu }\right)-p_1^{\beta } p_2^{\lambda } g^{\alpha \nu
} g^{\mu \sigma }-p_1^{\alpha } p_2^{\lambda } g^{\beta \nu }
g^{\mu \sigma }+p_1^{\beta } p_2^{\alpha } g^{\lambda \nu }
g^{\mu \sigma }+p_1^{\alpha } p_2^{\beta } g^{\lambda \nu }
g^{\mu \sigma }
\right. \nonumber\\
&& \left.
+p_1^{\nu } \left(p_2^{\beta } g^{\alpha \mu }
g^{\lambda \sigma }+p_2^{\alpha } g^{\beta \mu } g^{\lambda
\sigma }+p_2^{\mu } \left(g^{\alpha \sigma } g^{\beta \lambda
}+g^{\alpha \lambda } g^{\beta \sigma }-2 g^{\alpha \beta }
g^{\lambda \sigma }\right)-p_2^{\beta } g^{\alpha \lambda }
g^{\mu \sigma }-p_2^{\alpha } g^{\beta \lambda } g^{\mu \sigma
}
\right. \right. \nonumber\\
&& \left.\left.
+p_2^{\lambda } \left(-g^{\alpha \sigma } g^{\beta \mu
}-g^{\alpha \mu } g^{\beta \sigma }+2 g^{\alpha \beta } g^{\mu
\sigma }\right)\right)+2 p_1^{\mu } p_2^{\lambda } g^{\alpha
\beta } g^{\nu \sigma }-p_1^{\mu } p_2^{\beta } g^{\alpha
\lambda } g^{\nu \sigma }-p_1^{\beta } p_2^{\lambda } g^{\alpha
\mu } g^{\nu \sigma }-p_1^{\mu } p_2^{\alpha } g^{\beta \lambda
} g^{\nu \sigma }
\right. \nonumber\\
&& \left.
-p_1^{\alpha } p_2^{\lambda } g^{\beta \mu }
g^{\nu \sigma }+p_1^{\beta } p_2^{\alpha } g^{\lambda \mu }
g^{\nu \sigma }+p_1^{\alpha } p_2^{\beta } g^{\lambda \mu }
g^{\nu \sigma }+g^{\alpha \sigma } g^{\beta \nu } g^{\lambda \mu
} p_1\cdot p_2+g^{\alpha \nu } g^{\beta \sigma } g^{\lambda \mu
} p_1\cdot p_2+g^{\alpha \sigma } g^{\beta \mu } g^{\lambda \nu
} p_1\cdot p_2
\right. \nonumber\\
&& \left.
+g^{\alpha \mu } g^{\beta \sigma } g^{\lambda \nu
} p_1\cdot p_2-g^{\alpha \nu } g^{\beta \mu } g^{\lambda \sigma
} p_1\cdot p_2-g^{\alpha \mu } g^{\beta \nu } g^{\lambda \sigma
} p_1\cdot p_2-g^{\alpha \sigma } g^{\beta \lambda } g^{\mu \nu
} p_1\cdot p_2-g^{\alpha \lambda } g^{\beta \sigma } g^{\mu \nu
} p_1\cdot p_2
\right. \nonumber\\
&& \left.
+2 g^{\alpha \beta } g^{\lambda \sigma } g^{\mu
\nu } p_1\cdot p_2+g^{\alpha \nu } g^{\beta \lambda } g^{\mu
\sigma } p_1\cdot p_2+g^{\alpha \lambda } g^{\beta \nu } g^{\mu
\sigma } p_1\cdot p_2-2 g^{\alpha \beta } g^{\lambda \nu }
g^{\mu \sigma } p_1\cdot p_2
\right. \nonumber\\
&& \left.
+g^{\alpha \mu } g^{\beta \lambda }
g^{\nu \sigma } p_1\cdot p_2+g^{\alpha \lambda } g^{\beta \mu }
g^{\nu \sigma } p_1\cdot p_2-2 g^{\alpha \beta } g^{\lambda \mu
} g^{\nu \sigma } p_1\cdot p_2\right) ;
\label{ThVV}\end{aligned}$$
- Energy-momentum tensor with indices $(\mu,\nu)$ - gravitons with (Lorentz indices, momentum) combinations $(\lambda,\sigma, p_1)$ and $(\alpha,\beta, p_2)$: $$\begin{aligned}
&&
\frac{1}{8} \left(-4 p_1^{\sigma } p_2^{\lambda } g^{\alpha \nu }
g^{\beta \mu }-4 p_1^{\lambda } p_2^{\sigma } g^{\alpha \nu }
g^{\beta \mu }+4 p_1^{\sigma } p_2^{\alpha } g^{\lambda \nu }
g^{\beta \mu }-2 p_1^{\alpha } p_2^{\nu } g^{\lambda \sigma }
g^{\beta \mu }-2 p_1^{\alpha } p_2^{\nu } g^{\sigma \lambda }
g^{\beta \mu }+4 p_1^{\lambda } p_2^{\alpha } g^{\sigma \nu }
g^{\beta \mu }
\right. \nonumber\\
&& \left.
-4 g^{\alpha \sigma } g^{\lambda \nu } p_1\cdot
p_2 g^{\beta \mu }+6 g^{\alpha \nu } g^{\lambda \sigma }
p_1\cdot p_2 g^{\beta \mu }+6 g^{\alpha \nu } g^{\sigma \lambda
} p_1\cdot p_2 g^{\beta \mu }-4 g^{\alpha \lambda } g^{\sigma
\nu } p_1\cdot p_2 g^{\beta \mu }-4 p_1^{\sigma } p_2^{\lambda }
g^{\alpha \mu } g^{\beta \nu }
\right. \nonumber\\
&& \left.
-4 p_1^{\lambda } p_2^{\sigma }
g^{\alpha \mu } g^{\beta \nu }-4 p_1^{\sigma } p_2^{\nu }
g^{\alpha \beta } g^{\lambda \mu }+4 p_1^{\sigma } p_2^{\beta }
g^{\alpha \nu } g^{\lambda \mu }+4 p_1^{\beta } p_2^{\nu }
g^{\alpha \sigma } g^{\lambda \mu }-4 p_1^{\sigma } p_2^{\nu }
g^{\beta \alpha } g^{\lambda \mu }+4 p_1^{\sigma } p_2^{\alpha }
g^{\beta \nu } g^{\lambda \mu }
\right. \nonumber\\
&& \left.
+4 p_1^{\alpha } p_2^{\nu }
g^{\beta \sigma } g^{\lambda \mu }-4 p_1^{\sigma } p_2^{\mu }
g^{\alpha \beta } g^{\lambda \nu }+4 p_1^{\sigma } p_2^{\beta }
g^{\alpha \mu } g^{\lambda \nu }+4 p_1^{\beta } p_2^{\mu }
g^{\alpha \sigma } g^{\lambda \nu }-4 p_1^{\sigma } p_2^{\mu }
g^{\beta \alpha } g^{\lambda \nu }+4 p_1^{\alpha } p_2^{\mu }
g^{\beta \sigma } g^{\lambda \nu }
\right. \nonumber\\
&& \left.
-2 p_1^{\beta } p_2^{\nu }
g^{\alpha \mu } g^{\lambda \sigma }-2 p_1^{\beta } p_2^{\mu }
g^{\alpha \nu } g^{\lambda \sigma }-2 p_1^{\alpha } p_2^{\mu }
g^{\beta \nu } g^{\lambda \sigma }+2 p_1^{\sigma } p_2^{\lambda
} g^{\alpha \beta } g^{\mu \nu }+2 p_1^{\lambda } p_2^{\sigma }
g^{\alpha \beta } g^{\mu \nu }-2 p_1^{\beta } p_2^{\sigma }
g^{\alpha \lambda } g^{\mu \nu }
\right. \nonumber\\
&& \left.
-2 p_1^{\beta } p_2^{\lambda }
g^{\alpha \sigma } g^{\mu \nu }+2 p_1^{\sigma } p_2^{\lambda }
g^{\beta \alpha } g^{\mu \nu }+2 p_1^{\lambda } p_2^{\sigma }
g^{\beta \alpha } g^{\mu \nu }-2 p_1^{\alpha } p_2^{\sigma }
g^{\beta \lambda } g^{\mu \nu }-2 p_1^{\alpha } p_2^{\lambda }
g^{\beta \sigma } g^{\mu \nu }+2 p_1^{\beta } p_2^{\alpha }
g^{\lambda \sigma } g^{\mu \nu }
\right. \nonumber\\
&& \left.
+2 p_1^{\alpha } p_2^{\beta }
g^{\lambda \sigma } g^{\mu \nu }-2 p_1^{\beta } p_2^{\alpha }
g^{\lambda \nu } g^{\mu \sigma }-2 p_1^{\alpha } p_2^{\beta }
g^{\lambda \nu } g^{\mu \sigma }+2 p_1^{\sigma } p_2^{\lambda }
g^{\alpha \beta } g^{\nu \mu }+2 p_1^{\lambda } p_2^{\sigma }
g^{\alpha \beta } g^{\nu \mu }-2 p_1^{\beta } p_2^{\sigma }
g^{\alpha \lambda } g^{\nu \mu }
\right. \nonumber\\
&& \left.
-2 p_1^{\beta } p_2^{\lambda }
g^{\alpha \sigma } g^{\nu \mu }+2 p_1^{\sigma } p_2^{\lambda }
g^{\beta \alpha } g^{\nu \mu }+2 p_1^{\lambda } p_2^{\sigma }
g^{\beta \alpha } g^{\nu \mu }-2 p_1^{\alpha } p_2^{\sigma }
g^{\beta \lambda } g^{\nu \mu }-2 p_1^{\alpha } p_2^{\lambda }
g^{\beta \sigma } g^{\nu \mu }+2 p_1^{\beta } p_2^{\alpha }
g^{\lambda \sigma } g^{\nu \mu }
\right. \nonumber\\
&& \left.
+2 p_1^{\alpha } p_2^{\beta }
g^{\lambda \sigma } g^{\nu \mu }-2 p_1^{\beta } p_2^{\alpha }
g^{\lambda \mu } g^{\nu \sigma }-2 p_1^{\alpha } p_2^{\beta }
g^{\lambda \mu } g^{\nu \sigma }-2 p_1^{\beta } p_2^{\nu }
g^{\alpha \mu } g^{\sigma \lambda }-2 p_1^{\beta } p_2^{\mu }
g^{\alpha \nu } g^{\sigma \lambda }-2 p_1^{\alpha } p_2^{\mu }
g^{\beta \nu } g^{\sigma \lambda }
\right. \nonumber\\
&& \left.
+2 p_1^{\beta } p_2^{\alpha }
g^{\mu \nu } g^{\sigma \lambda }+2 p_1^{\alpha } p_2^{\beta }
g^{\mu \nu } g^{\sigma \lambda }+2 p_1^{\beta } p_2^{\alpha }
g^{\nu \mu } g^{\sigma \lambda }+2 p_1^{\alpha } p_2^{\beta }
g^{\nu \mu } g^{\sigma \lambda }-4 p_1^{\lambda } p_2^{\nu }
g^{\alpha \beta } g^{\sigma \mu }+4 p_1^{\beta } p_2^{\nu }
g^{\alpha \lambda } g^{\sigma \mu }
\right. \nonumber\\
&& \left.
+4 p_1^{\lambda } p_2^{\beta
} g^{\alpha \nu } g^{\sigma \mu }-4 p_1^{\lambda } p_2^{\nu }
g^{\beta \alpha } g^{\sigma \mu }+4 p_1^{\alpha } p_2^{\nu }
g^{\beta \lambda } g^{\sigma \mu }+4 p_1^{\lambda } p_2^{\alpha
} g^{\beta \nu } g^{\sigma \mu }-2 p_1^{\beta } p_2^{\alpha }
g^{\nu \lambda } g^{\sigma \mu }-2 p_1^{\alpha } p_2^{\beta }
g^{\nu \lambda } g^{\sigma \mu }\right. \nonumber\\
&& \left.
-2 p_1^{\nu } \left(-2
p_2^{\lambda } g^{\alpha \sigma } g^{\beta \mu }+2 p_2^{\alpha }
g^{\lambda \sigma } g^{\beta \mu }+2 p_2^{\alpha } g^{\sigma
\lambda } g^{\beta \mu }-2 p_2^{\lambda } g^{\alpha \mu }
g^{\beta \sigma }
\right. \right. \nonumber\\
&& \left. \left.
+p_2^{\sigma } \left(-2 g^{\alpha \mu }
g^{\beta \lambda }-2 g^{\alpha \lambda } g^{\beta \mu
}+\left(g^{\alpha \beta }+g^{\beta \alpha }\right) g^{\lambda
\mu }\right)+2 p_2^{\beta } g^{\alpha \mu } g^{\lambda \sigma
}+2 p_2^{\beta } g^{\alpha \mu } g^{\sigma \lambda }
\right. \right. \nonumber\\
&& \left. \left.
+2 p_2^{\mu
} \left(g^{\alpha \sigma } g^{\beta \lambda }+g^{\alpha \lambda
} g^{\beta \sigma }-\left(g^{\alpha \beta }+g^{\beta \alpha
}\right) \left(g^{\lambda \sigma }+g^{\sigma \lambda
}\right)\right)+p_2^{\lambda } g^{\alpha \beta } g^{\sigma \mu
}+p_2^{\lambda } g^{\beta \alpha } g^{\sigma \mu }\right)-4
p_1^{\lambda } p_2^{\mu } g^{\alpha \beta } g^{\sigma \nu }
\right. \nonumber\\
&& \left.
+4
p_1^{\beta } p_2^{\mu } g^{\alpha \lambda } g^{\sigma \nu }+4
p_1^{\lambda } p_2^{\beta } g^{\alpha \mu } g^{\sigma \nu }-4
p_1^{\lambda } p_2^{\mu } g^{\beta \alpha } g^{\sigma \nu }+4
p_1^{\alpha } p_2^{\mu } g^{\beta \lambda } g^{\sigma \nu }-2
p_1^{\beta } p_2^{\alpha } g^{\mu \lambda } g^{\sigma \nu }-2
p_1^{\alpha } p_2^{\beta } g^{\mu \lambda } g^{\sigma \nu }
\right. \nonumber\\
&& \left.
-2 p_1^{\mu } \left(-2 p_2^{\lambda } g^{\alpha \sigma } g^{\beta
\nu }+2 p_2^{\alpha } g^{\lambda \sigma } g^{\beta \nu }+2
p_2^{\alpha } g^{\sigma \lambda } g^{\beta \nu }-2 p_2^{\lambda
} g^{\alpha \nu } g^{\beta \sigma }
\right. \right. \nonumber\\
&& \left.\left.
+p_2^{\sigma } \left(-2
g^{\alpha \nu } g^{\beta \lambda }-2 g^{\alpha \lambda }
g^{\beta \nu }+\left(g^{\alpha \beta }+g^{\beta \alpha }\right)
g^{\lambda \nu }\right)+2 p_2^{\beta } g^{\alpha \nu }
g^{\lambda \sigma }+2 p_2^{\beta } g^{\alpha \nu } g^{\sigma
\lambda }
\right. \right. \nonumber\\
&& \left. \left.
+2 p_2^{\nu } \left(g^{\alpha \sigma } g^{\beta \lambda
}+g^{\alpha \lambda } g^{\beta \sigma }-\left(g^{\alpha \beta
}+g^{\beta \alpha }\right) \left(g^{\lambda \sigma }+g^{\sigma
\lambda }\right)\right)+p_2^{\lambda } g^{\alpha \beta }
g^{\sigma \nu }+p_2^{\lambda } g^{\beta \alpha } g^{\sigma \nu
}\right)-4 g^{\alpha \sigma } g^{\beta \nu } g^{\lambda \mu }
p_1\cdot p_2
\right. \nonumber\\
&& \left.
-4 g^{\alpha \nu } g^{\beta \sigma } g^{\lambda \mu
} p_1\cdot p_2
-4 g^{\alpha \mu } g^{\beta \sigma } g^{\lambda
\nu } p_1\cdot p_2+6 g^{\alpha \mu } g^{\beta \nu } g^{\lambda
\sigma } p_1\cdot p_2+2 g^{\alpha \sigma } g^{\beta \lambda }
g^{\mu \nu } p_1\cdot p_2+2 g^{\alpha \lambda } g^{\beta \sigma
} g^{\mu \nu } p_1\cdot p_2
\right. \nonumber\\
&& \left.
-3 g^{\alpha \beta } g^{\lambda
\sigma } g^{\mu \nu } p_1\cdot p_2-3 g^{\beta \alpha }
g^{\lambda \sigma } g^{\mu \nu } p_1\cdot p_2+3 g^{\alpha \beta
} g^{\lambda \nu } g^{\mu \sigma } p_1\cdot p_2+3 g^{\beta
\alpha } g^{\lambda \nu } g^{\mu \sigma } p_1\cdot p_2+2
g^{\alpha \sigma } g^{\beta \lambda } g^{\nu \mu } p_1\cdot
p_2
\right. \nonumber\\
&& \left.
+2 g^{\alpha \lambda } g^{\beta \sigma } g^{\nu \mu }
p_1\cdot p_2-3 g^{\alpha \beta } g^{\lambda \sigma } g^{\nu \mu
} p_1\cdot p_2-3 g^{\beta \alpha } g^{\lambda \sigma } g^{\nu
\mu } p_1\cdot p_2+3 g^{\alpha \beta } g^{\lambda \mu } g^{\nu
\sigma } p_1\cdot p_2+3 g^{\beta \alpha } g^{\lambda \mu }
g^{\nu \sigma } p_1\cdot p_2
\right. \nonumber\\
&& \left.
+6 g^{\alpha \mu } g^{\beta \nu }
g^{\sigma \lambda } p_1\cdot p_2-3 g^{\alpha \beta } g^{\mu \nu
} g^{\sigma \lambda } p_1\cdot p_2-3 g^{\beta \alpha } g^{\mu
\nu } g^{\sigma \lambda } p_1\cdot p_2-3 g^{\alpha \beta }
g^{\nu \mu } g^{\sigma \lambda } p_1\cdot p_2-3 g^{\beta \alpha
} g^{\nu \mu } g^{\sigma \lambda } p_1\cdot p_2
\right. \nonumber\\
&& \left.
-4 g^{\alpha \nu
} g^{\beta \lambda } g^{\sigma \mu } p_1\cdot p_2-4 g^{\alpha
\lambda } g^{\beta \nu } g^{\sigma \mu } p_1\cdot p_2+3
g^{\alpha \beta } g^{\nu \lambda } g^{\sigma \mu } p_1\cdot
p_2+3 g^{\beta \alpha } g^{\nu \lambda } g^{\sigma \mu }
p_1\cdot p_2-4 g^{\alpha \mu } g^{\beta \lambda } g^{\sigma \nu
} p_1\cdot p_2
\right. \nonumber\\
&& \left.
+3 g^{\alpha \beta } g^{\mu \lambda } g^{\sigma
\nu } p_1\cdot p_2+3 g^{\beta \alpha } g^{\mu \lambda }
g^{\sigma \nu } p_1\cdot p_2\right) ;
\label{Thh}\end{aligned}$$
Two-loop integral appearing in results of various two-loop diagrams: $$\begin{aligned}
&& \int\frac{d^n k_1 d^n k_2}{(2 \pi)^{2n}} \frac{1}{(k_1^2-M^2+i \epsilon)^\alpha
(k_2^2-M^2+i \epsilon)^\beta ((k_1-k_2)^2+i \epsilon)^\gamma} = \nonumber \\
&& \qquad \qquad \ \ \ \frac{ i^{2-2 \alpha -2 \beta -2
\gamma} M^{2 (n -\alpha -\beta
-\gamma)} \Gamma \left(\frac{n}{2}-\gamma \right) \Gamma
\left(-\frac{n}{2}+\alpha +\gamma \right) \Gamma
\left(-\frac{n}{2}+\beta +\gamma \right) \Gamma (-n+\alpha +\beta +\gamma )}{(4\pi)^n\Gamma (\alpha
) \Gamma (\beta ) \Gamma \left(\frac{n}{2}\right) \Gamma
(-n+\alpha +\beta +2 \gamma )} \,.
\label{int2loop}\end{aligned}$$
S. Weinberg, “The Quantum theory of fields. Vol. 1,2, Cambridge University Press (2005-06-02). J. F. Donoghue, Phys. Rev. D [**50**]{}, 3874 (1994).
J. F. Donoghue and B. R. Holstein, J. Phys. G [**42**]{}, no. 10, 103102 (2015).
M. J. G. Veltman, Conf. Proc. C [**7507281**]{}, 265 (1975). J. F. Donoghue, M. M. Ivanov and A. Shkerin, arXiv:1702.00319 \[hep-th\].
D. Burns and A. Pilaftsis, Phys. Rev. D [**91**]{}, no. 6, 064047 (2015).
S. Weinberg, Rev. Mod. Phys. [**61**]{}, 1 (1989).
J. Martin, Comptes Rendus Physique [**13**]{}, 566 (2012).
G. Gabadadze and A. Gruzinov, Phys. Rev. D [**72**]{}, 124007 (2005). D. Rubin and B. Hayden, Astrophys. J. [**833**]{}, no. 2, L30 (2016).
S. V. Babak and L. P. Grishchuk, Phys. Rev. D [**61**]{}, 024038 (2000). L. M. Butcher, A. Lasenby and M. Hobson, Phys. Rev. D [**78**]{}, 064034 (2008): L. B. Szabados, Living Rev. Rel. [**12**]{}, 4 (2009). L. M. Butcher, M. Hobson and A. Lasenby, Phys. Rev. D [**82**]{}, 104040 (2010). L. M. Butcher, M. Hobson and A. Lasenby, Phys. Rev. D [**86**]{}, 084013 (2012).
G. ’t Hooft and M. J. G. Veltman, Ann. Inst. H. Poincare Phys. Theor. A [**20**]{}, 69 (1974). L. D. Landau and E. M. Lifschits, “The Classical Theory of Fields,” Oxford: Pergamon Press (1975).
R. Mertig, M. Bohm and A. Denner, Comput. Phys. Commun. [**64**]{}, 345 (1991). V. Shtabovenko, R. Mertig and F. Orellana, Comput. Phys. Commun. [**207**]{} (2016) 432
R. L. Jaffe, Phys. Rev. D [**72**]{}, 021301 (2005). H. Nikolic, Phys. Lett. B [**761**]{}, 197 (2016).
|
---
abstract: 'Previous experiments have shown that spherical colloidal particles relax to equilibrium slowly after they adsorb to a liquid-liquid interface, despite the large interfacial energy gradient driving the adsorption. The slow relaxation has been explained in terms of transient pinning and depinning of the contact line on the surface of the particles. However, the nature of the pinning sites has not been investigated in detail. We use digital holographic microscopy to track a variety of colloidal spheres—inorganic and organic, charge-stabilized and sterically stabilized, aqueous and non-aqueous—as they breach liquid interfaces. We find that nearly all of these particles relax logarithmically in time over timescales much larger than those expected from viscous dissipation alone. By comparing our results to theoretical models of the pinning dynamics, we infer the area per defect to be on the order of a few square nanometers for each of the colloids we examine, whereas the energy per defect can vary from a few $kT$ for non-aqueous and inorganic spheres to tens of $kT$ for aqueous polymer particles. The results suggest that the likely pinning sites are topographical features inherent to colloidal particles—surface roughness in the case of silica particles and grafted polymer “hairs” in the case of polymer particles. We conclude that the slow relaxation must be taken into account in experiments and applications, such as Pickering emulsions, that involve colloids attaching to interfaces. The effect is particularly important for aqueous polymer particles, which pin the contact line strongly.'
address:
- 'School of Engineering & Applied Sciences, Harvard University, 29 Oxford Street, Cambridge, MA 02138, USA'
- 'Department of Physics, Harvard University, 17 Oxford Street, Cambridge, MA 02138, USA'
- 'Present address: Department of Physics and Biophysics, University of San Diego, San Diego, CA 92110 USA'
- 'Present address: Agilent Technologies, Santa Clara, CA 95051 USA'
author:
- Anna Wang
- Ryan McGorty
- 'David M. Kaz'
- 'Vinothan N. Manoharan'
bibliography:
- 'rsc.bib'
title: 'Contact-line pinning controls how quickly colloidal particles equilibrate with liquid interfaces'
---
contact-line pinning ,digital holography ,colloids ,interfaces
Introduction
============
The strong binding of colloidal particles to interfaces is exploited in a range of applications. Particles can stabilize oil-water interfaces in Pickering emulsions,[@pickering_spencer_umfreville_emulsions_1907] which are used in food,[@timgren_starch_2011; @destribats_emulsions_2014] oil recovery,[@zhang_nanoparticle_2010] pharmaceuticals, and cosmetics.[@marku_characterization_2012] Oil-water interfaces can also be used to scaffold the assembly of particles into colloidosomes,[@dinsmore_a.d._colloidosomes:_2002] Janus particles,[@paunov_novel_2003] monolayers,[@retsch_nanofab_2009] and photolithography masks.[@isa_particle_2010] Because the driving force for adsorption is large—the adsorption of a single particle reduces the interfacial energy of the system by many times the thermal energy $kT$—it is sometimes (and often tacitly) assumed that such particles reach their equilibrium contact angle rapidly once they breach the interface. Indeed, if viscous drag were the only force opposing the interfacial energy gradient, particles would relax to equilibrium exponentially with a time constant on the order of a microsecond.[@colosqui_colloidal_2013]
However, when Kaz, McGorty, and coworkers [@kaz_physical_2012] directly measured the adsorption dynamics of polystyrene microspheres at an interface between water/glycerol and oil, they found that the particles relaxed toward equilibrium logarithmically, not exponentially. Furthermore, the relaxation was so slow that the time projected for the particles to reach the equilibrium contact angle of 110$^\circ$ was months to years—far longer than typical experimental timescales. Later, Coertjens and coworkers [@coertjens_contact_2014] directly imaged polymer particles at vitrified interfaces and found that the average contact angle increased an hour after adsorption. Kaz *et al.* proposed that the slow relaxation is due to pinning and unpinning of the contact line on nanoscale heterogeneties (“defects”) on the particle surfaces. The pinning and unpinning events contribute to a larger dissipation of energy than viscosity alone. Using a model of contact-line hopping based on molecular-kinetic theory (described in the Background section below), they were able to infer the sizes of the defects.
More recent work has elucidated and expanded on how contact line pinning affects the dynamics of particles at interfaces. Colosqui and coworkers [@colosqui_colloidal_2013] developed a model based on Kramer’s theory for the full equilibrium dynamics of the particles, including not only the logarithmic regime, but also the dynamics shortly after the breach and close to equilibrium. As we describe below in the Background section, this model can be fit to experimental data to estimate the pinning energy per defect. Other work examines the effect of pinning on particle dynamics lateral to an interface. Recent experimental studies by Boniello *et al.* [@boniello_brownian_2015] indicate that the lateral diffusion of colloidal particles at a fluid interface is likely slowed by transient pinning events. Sharifi-Mood and coworkers [@sharifi-mood_curvature_2015] showed that strong pinning can locally distort the interface around a colloidal particle, affecting how particles migrate on a curved surface.
These studies highlight the importance of contact-line pinning for understanding the dynamics of colloids at interfaces. The observed slow relaxation has direct consequences for the applications we list above: in a collection of identical particles at an interface, such as the surface of a Pickering emulsion droplet, the particles can have different contact angles that change over time. Because the contact angle of a particle determines the length of the three-phase contact line and how much of the particle is exposed to the aqueous or oil phases, it affects the capillary [@kralchevsky_particles_2001] and electrostatic interactions between particles.[@mcgorty_colloidal_2010] Contact angles that change over time might help explain the heterogeneous pair-interactions [@park_heterogeneity_2010; @park_direct_2008] and long-ranged attractions observed between identically charged particles.[@nikolaides_m.g._electric-field-induced_2002] For the particular case of Pickering emulsions, the emulsion type (water-in-oil, or oil-in-water) also depends on the contact angle,[@binks_direct_2013] and so a changing contact angle might change the emulsion type and stability over time.
Here we focus on understanding how ubiquitous the pinning is and what causes it. To do this, we follow the approach of Kaz *et al.* [@kaz_physical_2012] and Wang *et al.* [@wang_relaxation_2013] and use digital holographic microscopy, a fast three-dimensional imaging technique, to measure the motion of spherical particles as they breach liquid interfaces. However, here we examine a much wider variety of particles and surface functionalities. We find that charge-stabilized polymer spheres (including a variety of emulsion-polymerized particles), surfactant-stabilized polymer spheres, and large (several micrometers in diameter) silica spheres all relax logarithmically to equilibrium, though some systems, including oil-dispersed PMMA particles and smaller silica spheres, reach equilibrium on experimental timescales. By fitting models to the data, we are able to extract details about the pinning sites. For example, we find that the heterogeneities on aqueous-dispersed polymer particles pin the contact line with an order of magnitude more energy than those on other particles, resulting in a longer logarithmic regime. We conclude that the likely pinning sites are nanoscale topographical features such as polymer “hairs.”
Background
==========
In this section we describe the theories that have been developed to explain the slow relaxation of colloidal particles at interfaces, and how fitting them to experimental data reveals details of the pinning dynamics. The logarithmic trajectories observed by Kaz *et al.* [@kaz_physical_2012] can be explained by using molecular kinetic theory (MKT) to model the motion of the contact line as a dynamic wetting process.[@blake_kinetics_1969] In this model, as the contact line moves across the surface of the particle, it encounters defects of area $A$ that each pin it with energy $\Delta U$. The contact line requires a thermal “kick” to keep it moving toward equilibrium: once it unpins from one defect, it can then move along the particle until it gets caught on another defect. The characteristic length the contact line traverses before reaching another defect is $\ell$ = $A$/$p$, [@colosqui_colloidal_2013] where $p$ is the perimeter of the contact line. This model explains why the particle motion appears continuous in the experiments: in practice, $\ell$ is on order of picometers [@colosqui_colloidal_2013], much smaller than displacements that we can measure. The model also explains why the particle slows as it progresses through the interface: the driving force decreases as the particle gets closer to equilibrium, while the pinning energy and the density of defects remain the same (Figure \[fcl\]).
The activated hopping of the contact line results in much more dissipation than that predicted from hydrodynamics. If hydrodynamics were the only relevant effect, we would expect the particle to follow an exponential path to equilibrium with a timescale $T_\mathrm{D} \approx
\eta r / \sigma_\mathrm{ow}$, where $\eta$ is a weighted average of the viscosities of the two fluids, $r$ the radius of the particle, and $\sigma_\mathrm{ow}$ the interfacial tension between oil and water.[@colosqui_colloidal_2013] For a 1-$\mu$m-radius particle at a water-alkane interface, $T_\mathrm{D}$ is approximately 0.1 $\mu$s, which is several orders of magnitude smaller than the times observed in experiments.[@kaz_physical_2012]
A model based on MKT, presented in the supplementary information section of Kaz *et al.*,[@kaz_physical_2012] captured the experimentally observed dynamics from 10$^{-2}$–10$^2$ s after the breach—the point where the particle first comes into contact with the interface and a three-phase contact line is formed. This model is not valid for shorter times, where the length of the contact line rapidly increases; instead, it is intended to model the behavior in the logarithmic regime, where the contact line perimeter changes slowly with time. By fitting the model to the data, the authors inferred that the area per pinning defect was on the order of a few square nanometers. This value is larger than the molecular scales the theory was derived for, but there are other successful applications of MKT to surfaces with defects larger than 1 nm$^2$.[@rolley_dynamics_2007; @snoeijer_moving_2013]
To show how the area per defect affects the dynamics, we present a brief derivation of the model from Kaz *et al.*[@kaz_physical_2012] We model the activated hopping process using an Arrhenius equation for the velocity of the contact line.[@blake_kinetics_1969] Far from equilibrium, we can neglect backward hops. In this case, the velocity of the contact line tangent to the particle is given by $$V = V_0 \exp\left(-\frac{\Delta U}{kT}+\frac{F_\mathrm{cl}(t)A}{2kT}\right)
\label{eqn:v}$$ where $V_0$ is a molecular velocity scale, and $kT$ is the thermal energy. The force per unit length on the contact line, $F_\mathrm{cl}$, is determined by the tangential component of the oil-water ($\sigma_\mathrm{ow}$), particle-oil ($\sigma_\mathrm{po}$), and particle-water ($\sigma_\mathrm{pw}$) interfacial tensions (Figure \[fcl\]): $$\begin{split}
F_\mathrm{cl} &= {\sigma}_\mathrm{ow}\cos{\theta}_\mathrm{D}(t)+\sigma_\mathrm{pw}-\sigma_\mathrm{po} \\
&= {\sigma}_\mathrm{ow}\left(\cos{\theta}_\mathrm{D}(t)-\cos{\theta}_\mathrm{E}\right)
\end{split}
\label{eqn:f}$$ where $\theta_\mathrm{D}$ is the dynamic contact angle.
Substituting Equation into Equation and rewriting the resulting equation of motion in terms of the observable axial coordinate $z$, we obtain $$\dot{z} = \nu r \sin(\theta_\mathrm{D}) \exp\left(\frac{A\sigma_\mathrm{ow}z}{2rkT}\right)
\newline = \nu \sqrt{z(2r-z)} \exp\left(\frac{A\sigma_\mathrm{ow}z}{2rkT}\right)
\label{eqn:zdot}$$ where $r$ is the radius of the particle and $$\nu =
(V_0/r)\exp\left(-\Delta U/kT+\left(1-\cos\theta_\mathrm{E}\right)A\sigma_\mathrm{ow}/2kT\right).$$ In deriving Equation we have assumed that the interface remains flat at all times. We have also let $z=0$ when the particle first touches the interface (at $\theta_\textnormal{D}$ = 0), from which we obtain $z = r\left(1-\cos\theta_\textnormal{D}\right)$ and $V =
r\dot{\theta}_\textnormal{D} = \dot{z}/\sin\theta_\textnormal{D}$.
When the particle is close to equilibrium, we can expand around the equilibrium contact angle and solve the resulting differential equation to obtain the equation of motion $$z \approx \frac{2rkT}{A\sigma_\mathrm{ow}} \log \left( \frac{A\sigma_\mathrm{ow}}{2rkT}\nu r \left(\sin\theta_\mathrm{E}\right) t \right)$$ which we can rewrite as $$\label{eqnsimple}
\frac{z}{r} \approx \frac{2kT}{A \sigma_\mathrm{ow}} \log \frac{t}{t_0} + C \ ; \quad C = \frac{2kT}{A \sigma_\mathrm{ow}} \log \left(\frac{A\sigma_\mathrm{ow}}{2rkT} \nu r \left(\sin\theta_\mathrm{E}\right) t_0 \right).$$ Equation shows that the trajectory of the particle is approximately logarithmic in time. We can infer the area per defect $A$ from the slope of a plot of $z$ as a function of $\log t$. We cannot determine the constant $C$—and hence the pinning energy per defect $\Delta U$, which is embedded in $\nu$—by fitting this model to the data. We therefore choose an arbitrary $t_0$ ($t_0$=1 s).
To determine the pinning energy per defect, $\Delta U$, we must observe where the logarithmic regime begins. Colosqui *et al.*,[@colosqui_colloidal_2013] using Kramer’s theory,[@kramers_brownian_1940] showed that particles having heterogeneous surface defects initially relax exponentially and then logarithmically. The models from Kaz *et al.* and Colosqui *et al.* are mathematically equivalent [@colosqui_colloidal_2013] when the dynamic contact angle $\theta_\mathrm{D}$ is approximately $\pi$/2. The area per defect $A$ from Kaz *et al.* is related to the length scale $l$ from Colosqui *et al.* by $A \sim 2{\pi}R^*l$, where $R^*$ is the radius of the contact line when the particle is at $z_\mathrm{C}$, and $z_\mathrm{C}$ is the height at which the relaxation changes from exponential to logarithmic. The crossover point between exponential and logarithmic regimes can be used to infer $\Delta U$, if the equilibrium height of the particle $z_\mathrm{E}$ is known or can be estimated: $$z_\mathrm{E}-z_\mathrm{C} = \frac{\Delta U \pi R^*}{2\sigma_\mathrm{ow}A}.
\label{crossover}$$
To analyze our experimental data we fit Equation to the logarithmic regime to obtain $A$ and then use Equation to determine the defect energy $\Delta U$. We note that these models capture only the gross features of the trajectories. A more recent model [@rahmani_colloidal_2016] expands on the model of Colosqui *et al.* to include extra dissipative effects. This model captures both the short- and long-time behavior of the experimental results from Kaz *et al.* well. Here, because we are interested primarily in the two parameters $A$ and $\Delta U$, we do not seek to capture the full time-dependence of the adsorption process, and we examine our results in the context of the simpler models from Kaz *et al.* and Colosqui *et al.*.
Materials and methods
=====================
Particles and interfaces
------------------------
To determine what kinds of surface features affect how a particle relaxes to equilibrium, we track particles with a variety of different surface properties as they breach an interface between an aqueous phase and oil. The types of particles we examine are listed in Table \[table:polymer\]. They include 1.9-$\mu$m-diameter charge-stabilized sulfate- and carboxyl-functionalized polystyrene (PS, Invitrogen), 2.48-$\mu$m-diameter sulfate-functionalized poly(methyl methacrylate) (PMMA, Bangs Laboratories, synthesized by emulsion polymerization), 1.7-$\mu$m-diameter polyvinylalcohol-stabilized PS (synthesized according to the procedure in Paine *et al.* [@paine_dispersion_1990]), 3.7-$\mu$m-diameter polyvinylpyrrolidone-stabilized PMMA (synthesized according to the procedure in Cao *et al.* [@cao_micron_2000]), and 1.0-$\mu$m-diameter bare silica microspheres with SiOH surface groups (Bangs Laboratories). We centrifuge and wash each suspension ten times in deionized water (EMD Millipore, resistivity = 18.2 M$\Omega \cdot$cm) to remove contaminants and surface-active compounds, then dilute them for use in experiments.
We also examine several different types of oil-dispersible particles: 1.0-$\mu$m-diameter (Bangs Laboratories) and 4.0-$\mu$m-diameter (AngstromSphere) silica microspheres, both with SiOH surface groups, 1.1-$\mu$m-diameter polydimethylsiloxane-stabilized PMMA (synthesized according to the procedure in Klein *et al.* [@klein_preparation_2003]), and 1.6-$\mu$m-diameter poly(12-hydroxystearic acid)-stabilized PMMA particles (synthesized according to the procedure in Elsesser *et al.* [@elsesser_revisiting_2010]). We wash the particles five times in decane ($\geq$99%, anhydrous, Sigma-Aldrich) to remove possible contaminants. We discard any macroscopic colloidal aggregates and keep the freely suspended particles for experiments.
[ p[2.5cm]{} p[5cm]{} p[2.5cm]{} m[1cm]{}]{} Name & Particle & Phase & Diameter ($\mu$m)\
PMMA & Sulfate-functionalized PMMA & aqueous & 2.48\
PVP-PMMA & Polyvinylpyrrolidone-stabilized PMMA & aqueous & 3.67\
sulfate-PS & Sulfate-functionalized PS & aqueous & 1.88\
carboxyl-PS & Carboxylate-functionalized PS & aqueous & 1.88\
PVA-PS & Polyvinylalcohol-stabilized PS & aqueous & 1.65\
PDMS-PMMA & Polydimethylsiloxane-stabilized PMMA & oil & 1.1\
PHSA-PMMA & Poly(12-hydroxystearic acid)-stabilized PMMA & oil & 1.6\
silica & Bare silica & aqueous or oil & 1.0\
large silica & Bare silica & oil & 4.0\
\[table:polymer\]
We prepare different aqueous phases from deionized water, anhydrous glycerol ($\geq$99%, Sigma-Aldrich), pure ethanol (100%, KOPTEC), and hydrochloric acid (Fluka). All of the aqueous solutions contain 100 mM NaCl (99.5%, EMD) to screen any electrostatic repulsion between the particle and the interface [@kaz_physical_2012]. For the oil phase we use decane ($\geq$99%, anhydrous, Sigma-Aldrich) which is first filtered through a PTFE membrane filter (Acrodisc). The different liquid-liquid interfaces we use in experiments are summarized in Table \[table:liquids\].
Name Aqueous phase (index) Oil (index)
---------------- ----------------------------------- ----------------
water/glycerol 59% w/w glycerol in water (1.411) decane (1.411)
water Water (1.333) decane (1.411)
water/ethanol 10% v/v ethanol in water (1.380) decane (1.411)
: Aqueous phase-decane interfaces used, along with the name we use to refer to them.
\[table:liquids\]
We measure the interfacial tension between water/glycerol and decane using the pendant drop method.[@rotenberg_determination_1983; @touhami_modified_1996] A 1 mL syringe (Sigma-Aldrich) with a blunt end syringe needle (18 gauge, Kimble) is filled with the aqueous phase. The needle is then submerged into a disposable cuvette (VWR) filled with decane. A droplet of the aqueous phase is slowly injected into decane while images are recorded. The profile of the droplet is analyzed from the images to determine the interfacial tension.
Sample preparation
------------------
Our custom-made polyether ether ketone (PEEK) sample cells are glued to a glass coverslip with UV-cured epoxy (Norland 60). A detailed description of their fabrication can be found in the supplemental information of Kaz *et al.*[@kaz_physical_2012] Using these cells, we create a stable oil-water interface consisting of a 30–80-$\mu$m-thick aqueous phase and a 2–3-mm-thick decane superphase. We use No. 1 coverslips (VWR) so that the interface is within the working distance of an oil-immersion objective (NA=1.4, Nikon CFI Plan Apo VC 100$\times$) or water-immersion objective (NA=1.2, Nikon CFI Plan Apo VC 60$\times$). We bake all glassware used to handle the colloidal particles and fluids in a pyrolysis oven (Pyro-Clean Tempyrox) to incinerate organics, then sonicate and wash the glassware with deionized water. This protocol is designed to eliminate interfacially-active contaminants.
We place the sample cell on a Nikon TE-2000 inverted microscope. We focus 5–15 $\mu$m below the interface to capture holograms of individual particles as they breach. If we start with particles that are suspended in the aqueous phase, we push them toward the interface using radiation pressure (force less than 1 pN) from out-of-focus optical tweezers, as shown in Figure \[dhm\]. If the particles are suspended in oil, we simply allow them to sediment toward the interface.
Tracking particles with Digital Holographic Microscopy
------------------------------------------------------
We use an in-line digital holographic microscope, based on a modified Nikon TE-2000 inverted microscope, to track the particles with high temporal and spatial resolution in all three dimensions (Figure \[dhm\]). We illuminate samples with a 660 nm imaging laser (Opnext HL6545MG) that is spatially filtered through a single-mode optical fiber (OzOptics SMJ-3U3U-633-4/125-3-5) and collimated. We use a counterpropagating 830 nm trap laser (Sanyo DL8142-201), which is spatially filtered through a single-mode optical fiber (OzOptics SMJ-3U3U-780-5/125-3-5), to push particles toward the interface.
![Experimental setup. The sample sits on an inverted microscope and is illuminated from above with a collimated 660 nm laser. The hologram formed by the interference of the scattered light from the sample with the undiffracted beam is then captured on a camera. We push spheres from the aqueous phase toward the interface with an 830 nm laser. To observe the breaching and relaxation of a particle from the aqueous phase, we push it gently toward the interface using optical tweezers, and measure its trajectory using holographic microscopy. To observe particles breaching from the oil phase, we simply let the particles fall to the interface.[]{data-label="dhm"}](optics_schematic.pdf)
The imaging beam (typically 50 mW power) scatters from the sample and interferes with undiffracted light to produce an interference pattern, or hologram. After passing through the objective, holograms are recorded on a monochrome CMOS camera (Photon Focus MVD-1024E-160-CL-12), captured with a frame grabber (EPIX PIXCI E4), and then saved to disk for further processing. We use a short camera exposure time, 20 $\mu$s, to minimize motion blur, and we capture holograms at up to 2000 frames per second, giving us sub-millisecond time resolution. A background image, taken in a part of the sample with no particles, is also recorded and divided from each time-series of holograms to remove artifacts arising from scattering from imperfections on the camera, lenses, and mirrors.
The background-divided holograms are analyzed using our open-source software package HoloPy (<http://manoharan.seas.harvard.edu/holopy>). To extract the particle trajectories, we fit the Lorenz-Mie scattering model to each background-divided hologram, as described in Fung *et al.*, [@fung_jqsrt_2012] following the work of Ovryn and Izen [@ovryn_imaging_2000] and of Lee and coworkers.[@lee_characterizing_2007]
The accuracy of the Lorenz-Mie model depends on the refractive index mismatch between the two liquid phases. The Lorenz-Mie scattering solution used to analyze the holograms is exact only for particles in an optically homogeneous medium. Exact light scattering solutions for particles straddling an optically discontinuous boundary do not exist. Therefore, to determine the position of bound particles with maximum accuracy, we index-match the aqueous phase to decane ($n$ = 1.41) by mixing anhydrous glycerol ($\geq$99%, Sigma-Aldrich) with water to make a solution of 59% w/w glycerol so that the system is optically continuous. This index-matching also eliminates reflections from the fluid-fluid interface, which would produce additional interference.
For some of the experiments, we cannot index match the aqueous medium to the oil phase. Because silica particles typically have a refractive index of 1.42 at our imaging wavelength (660 nm) and a high density compared to water, we cannot obtain sufficient radiation pressure to push them toward the interface if they are submerged in an aqueous medium with $n$ = 1.41. Instead, we disperse them in water ($n$ = 1.33) so that the refractive index contrast between the particles and medium is large enough for us to manipulate them with the trapping laser. In our analysis, we allow the refractive index of the particle relative to that of the medium to vary during the fit, which helps compensate for the change in medium index as the particle moves through the interface. In this way we are able to measure the approximate relaxation behavior of the silica spheres.
Because microscope objectives and their immersion fluids are designed to image objects in two dimensions, a difference in refractive index between the immersion oil and the medium leads to spherical aberration, which distorts distances in the axial direction [@egner_2006] and compromises the positioning accuracy. To mitigate this effect, we use an immersion oil with $n$=1.4140 (Series AA, Cargille) with our 100$\times$ oil-immersion objective for samples where we index-match the aqueous phase to decane ($n$=1.41). In experiments where pure water ($n$=1.33) is the aqueous phase, we use a water-immersion objective with water as the immersion fluid.
Results
=======
Slow relaxation is not particular to a water/glycerol-decane interface
----------------------------------------------------------------------
We begin by showing that the slow relaxation of particles at an interface is not particular to the decane-water/glycerol system of Kaz *et al.*[@kaz_physical_2012] In that work, the aqueous phase was designed to match the refractive index of decane yet retain an interfacial tension and Debye screening length similar to water. Here we track sulfate-PS particles as they approach interfaces from different aqueous solutions (Table \[table:liquids\]).
![Typical trajectories of sulfate-PS particles as they breach various aqueous phase-oil interfaces, as listed in Table \[table:liquids\]. The distance between the top surface of the particle and the interface is shown as a function of the time after the breach. All of the trajectories show logarithmic relaxation. We define $z$ = 0 $\mu$m as the height at which the particle and interface first touch.[]{data-label="nonmatched_log"}](log_liquids_decaneonly.pdf)
The motion of the polystyrene particles through the interface is approximately logarithmic with time in all of the systems, as shown in Figure \[nonmatched\_log\]. We note that the different starting times for the plots are an artifact of the logarithmic time-axis and the different frame rates used to acquire the data. The differences in slopes for the trajectories are due in part to the different refractive-index mismatches (and hence tracking errors) in the three systems, and in part to the different interfacial tensions and dielectric constants in the systems. However, we do not expect any of these effects to change the functional (logarithmic) relationship between height and time. Therefore, we conclude that the slow dynamics are not unique to the water/glycerol and decane system studied in Kaz *et al.* [@kaz_physical_2012] and are potentially relevant to a variety of other liquids.
Topographical features on polymer particles pin the contact line
----------------------------------------------------------------
Though colloidal particles may appear smooth under optical and even scanning electron microscopy, the particle surfaces contain nanoscale heterogeneities such as charges, asperities and, in the case of polymer particles, polymer “hairs.”[@rosen_saville_1990; @rosen_heatparticles_1992] To determine which of these features is responsible for the slow relaxation, we return to the index-matched water/glycerol and decane system and quantitatively measure how particles with different surface features breach the interface. From the trajectories we determine the area per defect, $A$, by fitting Equation to the logarithmic regime of the measured trajectories. Kaz and coworkers found that $A$ was on the order of the area per charge group for sulfate- ($A\approx$ 5 nm$^2$), carboxyl- ($A\approx$ 3 nm$^2$), amidine- ($A\approx$ 15 nm$^2$), and carboxylate-modified-latex ($A\approx$ 25 nm$^2$) spheres. These results suggest that the charges themselves, or some surface features associated with the charges, could be the pinning sites.
To understand if and how the charges influence the pinning, we do experiments on PS-carboxyl spheres suspended in a 59% glycerol in water solution containing 100 mM NaCl. We work with carboxyl-functionalized spheres because the pKa is higher than that of sulfate-functionalized spheres, so that the charge can be adjusted by changing the pH over a moderate range. We add acid to the suspensions and measure the zeta potentials of the particles using a Beckman Coulter DelsaNano C zeta potentiometer. The zeta potential decreases by a factor of about four over a range of acid concentrations from 0 to 10$^{-3}$ M (Table \[table:acidzeta\]). Measurements of the interfacial tension using the pendant drop method with a slight index mismatch, caused by increasing the water content in the aqueous phase by about 1% w/w, confirm that the interfacial tension does not vary with acid concentration.
Concentration of HCl (M) Zeta potential (mV)
-------------------------- ---------------------
0 -95 $\pm$ 10
10$^{-5}$ -68 $\pm$ 2
10$^{-4}$ -60 $\pm$ 2
10$^{-3}$ -24 $\pm$ 1
: Zeta potentials of PS-carboxyl latex at different acid concentrations.
\[table:acidzeta\]
![Trajectories of carboxyl-functionalized latex particles in solutions of varying acid concentration (HCl concentrations are marked above each curve). Lines are the average of five particle trajectories at each concentration. The gray shaded region shows the uncertainty in the zero-HCl-concentration measurement, as determined by the standard deviation on the five trajectories. It is representative of the uncertainties at the other concentrations.[]{data-label="acid"}](log_acid.pdf)
We find that at any given time after the particles breach, particles submerged in higher acid concentrations are at larger heights (Figure \[acid\]). There are two possible interpretations of this observation in the context of Kramer’s theory and Equation : either the energy of the defect decreases with acid concentration, or the equilibrium contact angle increases with acid concentration. These two quantities cannot be determined independently using either of the two dynamic models;[@colosqui_colloidal_2013; @kaz_physical_2012] however, it stands to reason that a smaller surface charge should increase the hydrophobicity of the particles and thus their equilibrium contact angle.
To better understand the nature of the pinning sites, we fit Equation to the the logarithmic regime in our data (Figure \[acid\]). The area per defect, which influences the slope of the trajectory, is between 4 nm$^2$ and 6 nm$^2$ for each of the four samples. The areas per defect measured here and in Kaz *et al.* [@kaz_physical_2012] differ by orders of magnitude from the roughly (100 nm)$^2$ chemical haterogeneities of polystyrene particles measured under atomic force microscopy [@chen_measured_2006]. This disparity suggests that the defects are not the chemical patches that are seen under surface characterization. Moreover, the fact that the areas per defect are nearly constant, despite the large variation in zeta potential (and hence area per charge) with acid concentration, suggests that the pinning sites are not the charges themselves but rather topographical features associated with the charged groups.
Prompted by a question from a reviewer of this manuscript, we also consider whether the slow relaxation might be related to swelling of the particles and subsequent deformation, as discussed by Park *et al.* [@park_fabrication_2010] and Tanaka *et al.* [@tanaka_novel_2010]. To determine whether the particles swell as they come into contact with decane, we measure the refractive index of our particles throughout the whole trajectory. If the polystyrene particles were swelling upon contact with decane, we would expect their refractive index to decrease, since $n_\mathrm{PS}$ is 1.59, while $n_\mathrm{decane}$ is 1.41. However, we find $n = 1.593 \pm
0.001$ before the breach and $n = 1.590 \pm 0.002$ several seconds after the breach (where the error is the standard error from fitting individual time points in a series). We conclude that there is no significant swelling during the breaching process.
![Typical trajectories of different polymer particles as they breach interfaces between water/glycerol and decane. The particle details are listed in Table \[table:polymer\]. The height of the particle above the interface as a function of the time after the breach is shown. All the trajectories show logarithmic relaxation.[]{data-label="polymerparticles"}](log_particles.pdf)
We probe the breaching behavior of a range of other polymer particles to gain further insights. We examine both charge-stabilized and sterically-stabilized particles. The charge-stabilized particles include sulfate-PS and PMMA, both of which are synthesized by emulsion polymerization, while the sterically-stabilized particles include PVA-PS and PVP-PMMA, both of which are synthesized by dispersion polymerization (Table \[table:polymer\]).
All of these polymer particles relax logarithmically after breaching, as shown in Figure \[polymerparticles\]. We fit Equation to the data to yield $A$ = 4.6–11 nm$^2$ for the particles. Using Equation , we calculate the pinning energies using $\sigma_\mathrm{ow}$ = 37 mN/m, $T$ = 295 K, and, for sulfate-PS, an equilibrium contact angle of 116$^\circ \pm
10^\circ$ [@isa_measuring_2011; @maestro_contact_2014] and $z_\mathrm{C}/r$ = 0.3; for PVA-PS, an equilibrium contact angle of 100$^\circ \pm 20^\circ$ [^1] and $z_\mathrm{C}/r$ = 0.2; and, for both types of water-dispersed PMMA particles, an equilibrium contact angle of 90$^\circ \pm
30^\circ$ [^2] and $z_\mathrm{C}/r$ = 0.2. We find that $\Delta U$ = 50–100 $kT$, as shown in Table \[table:polymerfits\].
Particle type $A$ (nm$^2$) $\Delta U$ ($kT$)
--------------- -------------- -------------------
PMMA 7.1 55 $\pm$ 35
PVP-PMMA 6.3 50 $\pm$ 30
sulfate-PS 4.6 55 $\pm$ 5
PVA-PS 11 100 $\pm$ 30
: Fitted $A$ and $\Delta U$ for various polymer particles. The uncertainties in $\Delta U$ account for uncertainties in the values of $z_\mathrm{C}$ and $z_\mathrm{E}$.
\[table:polymerfits\]
{width="\textwidth"}
We also examine the relaxation of PDMS-PMMA and PHSA-PMMA particles, both of which are sterically stabilized and dispersed in the oil phase. We can determine the contact angle of the particles (see Figure \[PMMAzoom\]) from the heights before and after they breach the interface. Both PHSA- and PDMS-stabilized particles reach a steady-state contact angle of 130–150$^\circ$ within a second of breaching. We find the steady-state contact angle is 135$^\circ \pm 10^\circ$ for the PDMS-stabilized particles, and 150$^\circ \pm 5^\circ$ for the PHSA-stabilized particles, where the uncertainty is determined from the standard error in the measurement of the height for five different particles. These contact angles are close to the those measured for PMMA particles at a water-decane interface using the freeze-fracture shadow-casting cryoSEM technique (130$^\circ \pm 12^\circ$) and using the gel-trapping technique (157$^\circ \pm 6^\circ$) [@leunissen_electrostatics_2007; @isa_measuring_2011; @maestro_contact_2014]
The relaxation of the the sterically stabilized PMMA particles is much faster than that of polymer spheres dispersed in the aqueous phase. To understand this difference, we use the two dynamical models to infer the area and pinning energy per defect. Fitting Equation to the first second of the breaching trajectory for the PDMS-PMMA particle (Figure \[PMMAzoom\]c) yields $A$ = 8 nm$^2$. From Equation , we calculate the pinning energy using $z_\mathrm{C}/r$ = 0.16, $\sigma_\mathrm{ow}$ = 37 mN/m, $T$ = 295 K and $R^*$ = 330 nm. We find $\Delta U$ = 4 $kT$. Thus the area per defect is comparable to that of the aqueous-dispersed particles, but the energy per defect is an order of magnitude smaller.
Because these particles likely have few charges, the area per defect is too small to be comparable to the area per charged group. So in this case, too, the evidence points to topographic features as the pinning sites. In the Discussion section we revisit the question of why the pinning energy is so much larger for the aqueous particles than the oil-dispersed ones. First, however, we examine the nature of the pinning sites on inorganic particles.
Inorganic particles can also pin contact lines
----------------------------------------------
We find that large, 4 $\mu$m bare silica spheres approaching a water/glycerol interface from the decane phase relax logarithmically after breaching but reach a steady-state height after less than 1 s (Figure \[silica\]). These silica particles are large enough that the slow evolution of the fringe pattern can be detected by eye, as shown in the insets in Figure \[silica\]. Fitting Equation to the logarithmic regime yields $A$ = 1 nm$^2$. We calculate $\Delta U$ using Equation with $z_\mathrm{C}/r$ between 0 and 0.84 and find that the pinning energy is 5–10 $kT$. This value is low compared to the pinning energies found for aqueous-dispersed polymer spheres. The result is consistent with the notion that particles that reach a steady-state contact angles on experimental timescales pin the contact line with smaller energies.
If we assume the surface asperities are roughly hemispherical caps, we can compare our fitted $A$ directly with measurements of the roughness of silica spheres from Ruiz and coworkers,[@ruiz_long_2015] who found the root-mean-squared roughness of 5.2-$\mu$m-diameter silica particles from Bangs Laboratories to be 1.4 nm using atomic force microscopy. The size we infer from our dynamic measurements is about 1 nm, in good agreement with the direct measurements.
We also examine 1-$\mu$m bare silica spheres approaching a water-decane interface from both phases. Most of the particles aggregate when we attempt to disperse them in decane. To obtain free particles, we discard the large aggregates that rapidly sediment and dilute the supernatant with more decane. We do not know whether the surface properties of silica in water and in decane are the same. However, we find that the smaller silica spheres reach a steady-state position within 20 ms when approaching from either phase, as shown in Figure \[silica\]. We do not observe a logarithmic regime, and the spheres reach a steady-state height within the time resolution of our experiment. Because the interface in these experiments is not index-matched, the measured height is only approximate, so we do not calculate a contact angle.
The fast relaxation and absence of any observable logarithmic relaxation means that we cannot determine if transient pinning or viscous dissipation sets the rate of relaxation of these spheres. We can, however, interpret the results in the context of the dynamic models if we assume that the relaxation is determined by pinning. In that case, the absence of a logarithmic regime suggests either that the crossover between the fast and logarithmic relaxation regimes is at timescales longer than what we can observe or that the difference between $z_\mathrm{E}$ and $z_\mathrm{C}$ is small. According to Equation , a small difference between $z_\mathrm{E}$ and $z_\mathrm{C}$ means that $\Delta U/A$ is small. Indeed, atomic-force-microscope measurements of similar-sized silica spheres (0.74 $\mu$m-diameter particles from Duke) by Chen and coworkers [@chen_attraction_2009] found the RMS roughness to be about 0.36 nm, which is smaller than the RMS roughness value for larger silica spheres (1.4 nm).[@ruiz_long_2015] Thus one interpretation of our results is that the small asperities do pin the contact line, but with a smaller energy than the larger asperities seen on the large silica spheres, leading to a faster relaxation to equilibrium.
Logarithmic relaxation may occur even in sheared emulsion formation
-------------------------------------------------------------------
Finally, we examine whether whether slow relaxation is an important effect to consider in the preparation of Pickering emulsion, which are usually made using vigorous mixing. For a 1.9-$\mu$m-diameter polystyrene sphere with an equilibrium contact angle of 110$^\circ$, Equation shows that the force on the particle integrated along the contact line is 10–100 nN for dynamic contact angles between 2$^\circ$ and 107$^\circ$. The force on a 1.9-$\mu$m-diameter particle in a suspension that is mixed at 11000 rpm in an Ultra Turrax homogenizer is about 1 nN,[@wang_image_2012] orders of magnitude smaller than the capillary driving force. Thus the relaxation of the particles is unlikely to be hastened by mixing, and long equilibration times may be important to take into account in the preparation of Pickering emulsions. One way to determine the equilibration time is to vitrify emulsions at different times after formation and image the interfaces using a method similar to that of Coertjens *et al.* [@coertjens_contact_2014]
Discussion
==========
We have shown that slow relaxation is common to many different kinds of particles, made of different materials and with different surface functionalities. Large silica particles and all of our polymer particles, whether stabilized in water or oil, relax to equilibrium at rates smaller than those expected from viscous dissipation alone. Thus we argue that the relaxation rate of colloidal particles at interfaces is likely controlled by transient pinning and unpinning of the three-phase contact line.
We have inferred certain features of the pinning sites by fitting dynamical models that account for pinning and depinning to our data. To gain further insight into the question of what surface features pin the contact line we now examine our results across the different types of systems. Our interpretation assumes that the dynamic models of Kaz *et al.* and Colosqui *et al.* capture the essential physics of the slow relaxation. Although there is little evidence that viscous dissipation controls the relaxation rate, we cannot—and do not attempt to—rule out the possibility that more complex wetting phenomena are responsible for the observed relaxation. Instead we focus on synthesizing a coherent explanation of the results in the context of the pinning models.
In all of the systems we observe, the area per defect is inferred to be on the order of a few square nanometers. This area is comparable to the area per charged group for aqueous charge-stabilized dispersions, as noted by Kaz *et al.*, but it is much smaller than the expected area per charged group for non-aqueous, sterically-stabilized polymer particles such as PHSA-PMMA. In the case of silica spheres, the area per defect is comparable to the measured surface roughness. We expect particles with more pronounced surface roughness to be affected more strongly by contact-line pinning. Taken together, these results suggest that the pinning sites are small-scale topographical features, perhaps associated with anchored charged groups in aqueous charge-stabilized colloids, but not the charges themselves.
In all of the aqueous polymer dispersions, whether charge- or sterically stabilized, the inferred pinning energy per defect is approximately 50 $kT$. This value contrasts markedly with that of the sterically stabilized non-aqueous particles and silica spheres, which is only a few $kT$. To explain this difference we must consider how the surface of the aqueous polymer spheres differs from that of the non-aqueous polymer spheres and the silica.
One feature of aqueous polymer spheres that is sometimes mentioned in the literature is polymer “hairs”; these are flexible polymer chains that are attached to the surface of the particles but extend out into solution and which may contain charged groups. The presence of such chains was originally inferred from electrophoretic mobility measurements: Rosen and Saville [@rosen_heatparticles_1992; @saville_electrokinetic_2000] found that both “hairy” polystyrene particles (with chains grafted onto their surface) and “bare” polystyrene particles had much lower electrophoretic mobilities than those predicted by classical electrokinetic theory. The discrepancy between experiment and theory was similar for both types of particles, suggesting that even “bare” particles have hairs. For both types of particles, the agreement between experiment and theory improved dramatically after the particles were heated past their glass transition temperature to allow the hairs to anneal to the surfaces of the particles. Further evidence for polymer hairs comes from optical measurements of the interaction between a polymer particle and a surface: Jensenius and Zocchi [@jensenius_measuring_1997] found that some polystyrene particles attached to surfaces, and, by measuring the displacement of the particle, they concluded that the attachment tether was a single polymer chain with a coil size of 50 nm. These experiments suggest that polymer hairs may be a common feature of polymer particles, whether there are chains deliberately grafted onto the surface or not.
We therefore hypothesize that polymer hairs are the pinning sites on aqueous-dispersed polymer particles. Furthermore we hypothesize that the pinning sites on the non-aqueous polymer particles are also polymer hairs, which are likely the polymer stabilizers grafted onto the particles. A possible explanation for why the hairs on the non-aqueous particles have a much lower pinning energy than the hairs on the aqueous particles is that the ones on the aqueous particles are polyelectrolytes. Moving a polyelectrolyte from the aqueous to the oil phase may involve a large energy barrier because all of the charges need to first be neutralized. This explanation is not inconsistent with our results for how the pH affects the relaxation in carboxyl-PS spheres. In those experiments we found that changing the pH to be closer to the isoelectric point did not change the area per defect; if the defects are indeed polyelectrolyte hairs, we expect that some, but not all of the charges would be neutralized, and so the area per defect (per hair) would not change. However, the pinning energy should change with the pH. Therefore this hypothesis can be tested by observing how the crossover point between exponential and logarithmic relaxation changes as a function of pH, while independently measuring how the equilibrium contact angle changes with pH. This is a point for future experiments to examine. Measurements closer to the isoelectric point could also help to better isolate the effect of charge on breaching behavior.
Conclusions
===========
The main message that emerges from our study is that slow, logarithmic relaxation is a common effect in colloidal particles bound to interfaces. By “slow” we mean slower than the rate expected from viscous dissipation alone. In many cases, however, the relaxation is slow even compared to experimental time scales. Our analysis of the forces involved suggests that the rate of relaxation will not be significantly altered by vigorous mixing; therefore experiments and applications (such as making Pickering emulsions) that involve attaching particles to interfaces and letting them assemble should account for the possibility that the particles are not in equilibrium on the timescale of assembly. We expect the out-of-equilibrium behavior to be most prominent in aqueous polymer particles a few hundred nanometers in diameter or larger; oil-dispersible polymer particles and silica spheres, even ones several micrometers in diameter, appear to equilibrate much more rapidly.
Based on the agreement between the observed logarithmic trajectories and the predictions of a model based on molecular kinetic theory, we have argued that the slow relaxation arises from surface heterogeneities that transiently pin the contact line. We ruled out the possiblity that the heterogeneities are charged groups directly attached to the surfaces of the particles. Instead, the likely culprits for the pinning are topographical features—nanoscale surface roughness in the case of silica particles and polymer “hairs” in the case of polymer particles. Beyond the implications described above for the assembly of particles at interfaces, these results also show that the adsorption trajectory is a sensitive probe of nanoscale surface features that are difficult to measure directly.
Acknowledgements
================
We acknowledge support from the National Science Foundation through grant number DMR-1306410 and by the Harvard MRSEC through grant number DMR-1420570. We thank W.B. Rogers for his careful reading of this manuscript and suggestions; C.E. Colosqui, M. Mani and M.P. Brenner for critical discussions regarding the interpretation of these results and the model; A. Hollingsworth for providing the PHSA-stabilized PMMA particles and for helpful discussions about hairy particles; J.G. Park for providing the PVP-stabilized PMMA and PVA-stabilized polystyrene particles, and G. Meng for providing the PDMS-stabilized PMMA particles. The hologram analysis computations were run on the Odyssey cluster, supported by the FAS Science Division Research Computing Group at Harvard University. The zeta potential characterization was performed at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Coordinated Infrustructure Network (NNCI), which is supported by the National Science Foundation under NSF award no. 1541959. CNS is part of Harvard University.
[^1]: Polystyrene with some PVA on the surface, angle taken from the measurement for “double-cleaned” polystyrene in Isa *et al.* [@coertjens_contact_2014]
[^2]: No measurements for the equilibrium contact angle of aqueous-dispersible PMMA particles could be found. The PMMA particles from Bangs Laboratories, Inc. are expected to be more hydrophilic than typical polystyrene particles.
|
---
author:
- 'Carter L. Johnson, Timothy J. Lewis, and Robert D. Guy'
bibliography:
- 'paper\_1.bib'
date: 'Draft Date: '
title: 'Neuromechanical Mechanisms of Gait Adaptation in *C. elegans*: Relative Roles of Neural and Mechanical Coupling'
---
**Abstract**
Understanding principles of neurolocomotion requires the synthesis of neural activity, sensory feedback, and biomechanics. The nematode *C. elegans* is an ideal model organism for studying locomotion in an integrated neuromechanical setting because its nervous system is well characterized and its forward swimming gait adapts to the surrounding fluid using sensory feedback. However, it is not understood how the gait emerges from mechanical forces, neuronal coupling, and sensory feedback mechanisms. Here, an integrated neuromechanical model of *C. elegans* forward locomotion is developed and analyzed. The model captures the experimentally observed gait adaptation over a wide range of parameters, provided that the muscle response to input from the nervous system is faster than the body response to changes in internal and external forces. The model is analyzed using the theory of weakly coupled oscillators to identify the relative roles of body mechanics, neural coupling, and proprioceptive coupling in coordinating the undulatory gait. The analysis shows that the wavelength of body undulations is set by the relative strengths of these three coupling forms. The model suggests that the experimentally observed decrease in wavelength in response to increasing fluid viscosity is the result of an increase in the relative strength of mechanical coupling, which promotes a short wavelength.
Introduction
============
The central goal of neuroethology is to understand how an organism’s body and nervous system interact with its environment to produce behaviors such as locomotion. Model organisms have been used to study the complex interactions between the nervous system, body mechanics, and environmental dynamics in generating and coordinating locomotion [@doi:10.1137/S0036144504445133; @Marder_1996]. Some studies of locomotion in model organisms highlight feedforward control of locomotion, where the nervous system drives motor activity and sensory feedback plays only a modulatory role; these include swimming behavior in lamprey, crayfish, and leeches [@Cohen_1992; @Mullins:2011aa; @Sigvardt:1996aa; @Skinner:1997aa; @Skinner:1998aa; @Tytell_2010; @Zhang_2014]. However, other organisms, such as cockroaches and stick insects, can only be understood as fully integrated neuromechanical systems because sensory feedback is essential to generate and coordinate movements [@Borgmann_2007; @Fuchs_2010; @doi:10.1137/S0036144504445133; @Ludwar_2005; @Pearson_2004]. This sensory feedback is necessary for navigating more complex environments and can often lead to gait adaptation. The nematode *C. elegans* is an ideal model organism for studying locomotion in an integrated neuromechanical setting because of its relatively simple and fully-described nervous system [@White:1986aa], limited stereotypical locomotive behavior [@Nigon:2017aa], dependence on sensory feedback for forward locomotion [@Schafer_2006; @Wen:2012aa], and undulatory gait that adapts to different fluid environments.
*C. elegans* locomote forward using alternating dorsal and ventral body bends that propogate from anterior to posterior. The properties of this undulatory gait adapt to fluid environments of different viscosities: higher external fluid viscosities result in slower undulations of shorter wavelengths [@Berri_2009; @Fang-Yen:2010aa; @Sznitman2010]. In water, *C. elegans* swim with a relatively long wavelength and relatively fast undulation frequency (roughly 1.5 bodylengths and 1.8 Hz) [@Fang-Yen:2010aa]. On agar, *C. elegans* crawl with a short wavelength and slow undulation frequency (0.65 bodylengths and 0.3 Hz) [@Fang-Yen:2010aa]. Previously, it was thought that these were two distinct gaits (swimming vs. crawling). However, recent experiments have shown that the wavelength and frequency of swimming in highly viscous fluids resemble crawling on agar surfaces [@Berri_2009; @Fang-Yen:2010aa], and instead of a sharp swim/crawl transition, there is a smooth transition between the two gaits as the fluid viscosity of the environment is varied [@Berri_2009; @Fang-Yen:2010aa; @Sznitman2010]. How this adaptation in gait emerges from the interactions between the external environment, mechanical forces, and internal sensory feedback mechanisms is not understood.
There are several hypotheses for how the undulatory gait is generated and coordinated [@Gjorgjieva_2014]; however, it is generally agreed that proprioception plays a key role [@Boyle:2012aa; @Niebur_1991; @Wen:2012aa]. One hypothesis is that there is a central pattern-generating (CPG) neural unit in the head that initiates the propagation of the bending wave — higher fluid viscosities slow the propagation and shorten the wavelength [@Karbowski_2007; @Wen:2012aa]. Another hypothesis is that the ventral nerve cord consists of a network of neural modules that are capable of either (i) intrinsic neural oscillations [@Olivares_2018] or (ii) intrinsic *neuromechanical* oscillations (i.e., involving an entire feedback loop from neural to muscular to body mechanics and back through proprioception) [@Boyle:2012aa; @Bryden_2008]. Recent experiments by Fouad et al. [@Fouad:2018aa] support the presence of multiple neural or neuromechanical oscillators, and gait adaptation has been demonstrated in computational models consisting of a chain of neuromechanical oscillators [@Boyle:2012aa; @cohen_ranner_2017; @Denham_2018]. However, it is still unclear how the interplay between neural, proprioceptive, and mechanical coupling gives rise to gait adaptation.
Here, we introduce a neuromechanical model of the *C. elegans* forward locomotion system. We use our model to systematically analyze the role of body mechanics, neural coupling, and proprioceptive coupling in gait adaptation. The model captures the experimentally observed gait adaptation over a wide range of parameters, provided that the muscle response to input from the nervous system is faster than the body response to changes in force. The modular structure of our model allows the use of the theory of weakly coupled oscillators to further dissect out the mechanisms underlying gait adaptation. Specifically, we assess the influence of each coupling modality (mechanical, neural, and proprioceptive). We find that proprioception leads to a posteriorly-directed traveling wave with a characteristic wavelength. Neural coupling promotes synchronous activity (long wavelength), and mechanical coupling promotes a high spatial frequency (short wavelength). The wavelength of the undulatory waveform is set by the relative strengths of these three coupling forms. As the external fluid viscosity increases, the mechanical coupling strength increases and the wavelength decreases, resulting in the observed wavelength trend of gait adaptation.
Neuromechanical Model {#chapter2}
=====================
The neuromechanical model developed here describes the motor circuit, body-wall muscles, and the resulting body shapes of *C. elegans*. The body description is derived from a continuous centerline-approximation of an active viscoelastic beam, whereas the muscles and neural subcircuits are discrete in nature. The model for the motor circuit uses the repeated motif of Haspel and O’Donovan [@Haspel14611]: 6 modules of roughly 12 motor neurons and 12 muscle cells, of these 12 repeated motor neurons roughly 6 (the dorsal/ventral B and D-class neurons) are responsible for forward locomotion. The model also includes proprioception: the B-class motor neurons respond to bending in the local and anterior regions of the body [@Wen:2012aa; @Zhen:2015aa].
A schematic of this model is shown in Figure \[wiring\_diag\], which highlights the modular structure of the neural circuit, body-wall muscles, and the corresponding body region. Within each module, the motor subcircuit drives the body-wall muscles, which in turn apply contractile forces to bend the corresponding body region. The body mechanics then feed back into the neural circuit through proprioceptive feedback, which translates body-wall length changes into neural signals. This structure allows each module to function, in isolation, as a neuromechanical oscillator, and it suggests that the full body functions as a system of coupled neuromechanical oscillators.
![The highlighted schematic here depicts the repeating neuromechanical module: a 4-motorneuron functional unit consisting of DB, VB, DD, and VD-class neurons, the post-synaptic muscles, and corresponding body wall region. The dorsal B-class (ventral B-class) neurons are excitatory and synapse onto the ipsilateral muscles and contralateral D-class neurons. The dorsal D-class (ventral D-class) neurons are inhibitory and synapse onto the dorsal (ventral) muscles. The B-class motorneurons also receive proprioceptive feedback from the local body segment (inhibitory) and anterior segments (excitatory). The interneuron AVB is connected to VB and DB via gap-junctions, and the VB (DB) neurons are also coupled via gap-junctions with their nearest neighbors of the same class. The body wall is modeled as a viscoelastic material connected to a contractile muscle. []{data-label="wiring_diag"}](wiring_diagram_body_only.png "fig:"){width=".9\textwidth"} ![The highlighted schematic here depicts the repeating neuromechanical module: a 4-motorneuron functional unit consisting of DB, VB, DD, and VD-class neurons, the post-synaptic muscles, and corresponding body wall region. The dorsal B-class (ventral B-class) neurons are excitatory and synapse onto the ipsilateral muscles and contralateral D-class neurons. The dorsal D-class (ventral D-class) neurons are inhibitory and synapse onto the dorsal (ventral) muscles. The B-class motorneurons also receive proprioceptive feedback from the local body segment (inhibitory) and anterior segments (excitatory). The interneuron AVB is connected to VB and DB via gap-junctions, and the VB (DB) neurons are also coupled via gap-junctions with their nearest neighbors of the same class. The body wall is modeled as a viscoelastic material connected to a contractile muscle. []{data-label="wiring_diag"}](wiring_diagram_module_only.png "fig:"){width=".9\textwidth"}
Model Development {#sect_model_dev}
-----------------
### Body Mechanics
The nematode body is modeled as an active viscoelastic beam for small amplitude displacements submerged in fluid. *C. elegans* usually operates in a regime where inertia plays a minor role (i.e., low Re), thus the equation of motion is a balance of internal elastic forces, internal viscous forces, and a fluid drag force described by a local drag coefficient $C_N$ [@Fang-Yen:2010aa; @Sznitman2010; @Wiggins:1998aa]: $$\begin{aligned}
C_N \dot{y} &= -k_b \partial_{xx} \qty(\kappa + \dfrac{\mu_b}{k_b}\dot{\kappa}+M(x,t)), \label{continuum_bodymechanics_PDE}\end{aligned}$$ where $x$ is the length-wise body coordinate, $t$ is time, $y(x,t)$ is the displacement in the ventral-dorsal plane, $\kappa(x,t)$ is the curvature, and $M(x,t)$ is the active moment that comes from internal muscle activity. The parameter $k_b$ is the bending modulus, $\mu_b$ is the effective internal viscosity, and the normal drag coefficient $C_N$ is proportional to the external fluid viscosity $\mu_f$ ($C_N = \alpha \mu_f$, see Appendix \[appendix\_deriv\_mech\_params\]). The values for these parameters are given in Table \[table\_1\], and a discussion of how they were selected is provided in Section \[sect\_param\_discuss\].
We consider small amplitude undulations, so that the curvature $\kappa(x,t)$ is approximately the second spatial derivative of the displacements $y(x,t)$: $$\kappa(x,t) \approx \partial_{xx}y(x,t). \label{kappa_is_yxx}$$ Taking two partial derivatives in $x$ of equation \[continuum\_bodymechanics\_PDE\] and applying force-free, moment-free boundary conditions, the curvature $\kappa(x,t)$ of the body satisfies $$\begin{aligned}
\alpha \mu_f \dot{\kappa} &= -k_b \partial_{xxxx} \qty(\kappa + \dfrac{\mu_b}{k_b}\dot{\kappa}+M(x,t)), \label{continuum_bodymechanics_PDE_v2}\\
\kappa(x,t) &+ \dfrac{\mu_b}{k_b}\dot{\kappa}(x,t) +M(x,t) = 0, \ &\text{ for } x = 0, x=L, \label{cont_pde_BC1}\\
\partial_{x} &\qty(\kappa + \dfrac{\mu_b}{k_b}\dot{\kappa}+M(x,t)) = 0,\ &\text{ for } x = 0, x=L, \label{cont_pde_BC2}\end{aligned}$$ where $x=0$ is the head and $x=L$ is the tail ($L$ is the body length). Note that in equations \[continuum\_bodymechanics\_PDE\_v2\]-\[cont\_pde\_BC2\], a positive curvature $\kappa(x,t)$ represents a bend towards the dorsal side. The active moment $M(x,t)$ comes from internal muscle activity, which will be defined below.
### Muscles
The body is driven by six modules of roughly 6 ventral and 6 dorsal muscle cells, that apply contractile forces to either the dorsal or ventral side [@Haspel14611; @Zhen:2015aa]. These muscle modules split the body into six distinct regions of length $\ell = L/6$. Each ventral/dorsal muscle group applies a contractile force as a function of its activity level $A(t)$. The ventral and dorsal ($k=V,D$) muscle activities $A_{k,j}$ in the $j^\text{th}$ module are given by $$\tau_m \dot{A}_{k,j} = - A_{k,j} + I_M(k,j), \label{muscle_module_single}\\$$ where $\tau_m$ is the timescale of muscle activation and $I_M(k,j)$ is the input from the $j^\text{th}$ neural module (described below). The tension $\sigma(A(t))$ generated by the muscle is only contractile ($\sigma\geq0$) and saturates at some peak force $c_{m}$: $$\begin{aligned}
\sigma(A(t)) &= \dfrac{c_m}{2}\qty(\tanh(c_{s}(A(t)-a_0)) + 1), \label{muscle_contract_force} \end{aligned}$$ where $c_s, a_0$ define the scale and shift of the nonlinear threshold. In the $j^{\text{th}}$ module, the dorsal and ventral muscles apply contractile forces to opposite sides of the body, which induces a moment $m_j(t)$ on the centerline from $x_{j-1}=(j-1)\ell$ to $x_j=j\ell$: $$m_j(t) = \sigma(A_{V,j}(t)) - \sigma(A_{D,j}(t)). \label{single_moment}$$
The active moment $M(x,t)$ as a function of the body coordinate $x$ is then given by $$\begin{aligned}
M(x,t) &= m_j(t) \text{ for } x \in [x_{j-1}, x_j).
\label{internal_moment_full}\end{aligned}$$
### Neural Module
The repeated neural module includes six motor neurons responsible for forward locomotion: DB (dorsal B-class), VB (ventral B-class), DD (dorsal D-class), and VD (ventral D-class) [@Boyle:2012aa; @Haspel14611; @Zhen:2015aa], as shown in Figure \[wiring\_diag\]. The neural modules are similar in structure to Boyle et al. [@Boyle:2012aa]. Each neural module is driven by constant input from the head interneuron AVB [@Haspel14611; @White:1986aa; @Zhen:2015aa]. The D-class neurons are assumed to invert excitation from the B-class neurons into inhibition of the contralateral muscles. The B-class neurons are modeled as bistable, non-spiking elements, in line with recordings of similar motor neurons involved in head-turns [@Mellem:2008aa]. The activities of the ventral and dorsal ($k=V,D$) B-class neurons in the $j^{\text{th}}$ neural module are given by $$\begin{aligned}
\tau_n \dot{V}_{k,j} &= F(V_{k,j}) + P(k,j) + I_{gj}(k,j), \label{nonlinear_bistable}\end{aligned}$$ where $$\begin{aligned}
F(V_k) &= V_k - V_k^3 + I. \label{non_fun_F}\end{aligned}$$ Here, $\tau_n$ is the timescale of neural activity, and $I$ is the offset from the constant “on” input from AVB. $P(k,j)$ is proprioceptive feedback into the neuron, and $I_{gj}(k,j)$ is gap-junctional (electrical) coupling between neurons, both of which will be described below.
The D-class neurons are excited by the ipsilateral B-class neurons and inhibit the contralateral body-wall muscles. This effect is captured by direct inhibition of the muscles by the B-class neurons. We model the B-class neurons as exciting the ipsilateral muscles and inhibiting the contralateral muscles. The input from the $j^\text{th}$ neural module to the ventral/dorsal muscles is given by $$I_M(k,j) = \begin{cases} V_{V,j} - V_{D,j}, & \text{ if } k=V\\
V_{D,j} - V_{V,j}, & \text{ if } k=D.
\end{cases} \label{input_to_muscles}$$
### Proprioceptive Feedback
To close the neuromechanical loop, the body segment curvatures feed back into the circuit via proprioceptive processes in the VB and DB neurons. There are two types of proprioception in this model: local (from the body region covered by the muscles of the module) and nonlocal (from neighboring anterior body regions).
Local proprioceptive feedback acts to reset the neural modules, i.e., switch between dorsal bend commands and ventral bend commands. Thus, *local* proprioception is modeled as an excitatory current to the ventral B-class neurons in response to positive average curvature over the local module of length $\ell = L/6$, and an inhibitory current in response to negative average local curvature. The input to the dorsal B-class neurons is the same but with the polarities reversed. This feedback acts to relax the contracted muscles and contract the relaxed muscles.
Nonlocal proprioception promotes a wave of activity that propogates from anterior to posterior. The anatomical structures underlying proprioception are unknown [@Zhen:2015aa], however, the evidence in Wen et al. [@Wen:2012aa] suggests that proprioceptive information affects the B-class motorneurons and is propagated posteriorly. In our model, positive nonlocal *anterior* segment curvature yields a weak inhibitory current to the ventral B-class neurons and a weak excitatory current to the dorsal B-class neurons. Negative nonlocal anterior segment curvature yields similar currents with the polarities reversed to each side. This is similar to the assumptions of Boyle et al. [@Boyle:2012aa], but diverges in the directionality and sign of nonlocal proprioception.
The proprioceptive feedback to the ventral and dorsal B-class neurons in the $j^{\text{th}}$ neural module ($j=1,\dots,6$) of length $\ell = L/6$ is modeled by $$\begin{aligned}
P(V,j) &= + c_p \dfrac{1}{\ell}\int_{(j-1)\ell}^{j \ell}\kappa(x,t) \dd x - {\varepsilon}_p \dfrac{1}{\ell} \int_{(j-2)\ell}^{(j-1) \ell}\kappa(x,t) \dd x, \label{prop_feedback_V}\\
P(D,j) &= - c_p \dfrac{1}{\ell} \int_{(j-1)\ell}^{j \ell} \kappa(x,t) \dd x + {\varepsilon}_p \dfrac{1}{\ell} \int_{(j-2)\ell}^{(j-1) \ell} \kappa(x,t) \dd x, \label{prop_feedback_D}\end{aligned}$$ where $c_p$ is the strength of *local* proprioception, ${\varepsilon}_p$ is the strength of *nonlocal* anterior proprioception, and $\kappa(x,t) = 0$ for $x\notin[0,L]$ for notational simplicity.
### Gap-Junctional Coupling
The B-class neurons are also connected via gap-junction synapses to neighboring B neurons of the same type (ventral/dorsal) [@Haspel14611; @White:1986aa; @Zhen:2015aa]. The gap-junctions are modeled as symmetric ohmic resistors with constant conductance, so that the gap-junctional coupling to the ventral and dorsal ($k=V,D$) B-class neurons in the $j^{\text{th}}$ neural module are described by
$$\begin{aligned}
I_{gj}(k,j) = {\varepsilon}_g (V_{k,j-1}-V_{k,j}) + {\varepsilon}_g(V_{k,j+1}-V_{k,j}), \label{gap_junction_coup}\end{aligned}$$
where ${\varepsilon}_g$ is the strength of gap-junction coupling and $V_{k,0} = V_{k,7} = 0$ for notational simplicity.
Model Discretization for Numerical Simulation {#sect_discret_and_sim}
---------------------------------------------
To simulate the model described in Section \[sect\_model\_dev\], the body is discretized into six modules in correspondence with the six neuromuscular modules, so that there are six discrete body segment curvatures. The 4th-order difference operator $D_4$ is used to approximate the 4th spatial derivative with zero-force, zero-torque boundary conditions: $$\dfrac{1}{\ell^4}D_4 = \dfrac{1}{\ell^4}\qty(\begin{matrix}
7 & -4 & 1 & & & \\
-4 & 6 & -4 & 1 & & \\
1 & -4 & 6 & -4 & 1 & \\
& 1 & -4 & 6 & -4 & 1\\
& & 1 & -4 & 6 & -4 \\
& & & 1 & -4 & 7 \\
\end{matrix}).\label{4th_diff_op}$$ Discretizing equations \[continuum\_bodymechanics\_PDE\_v2\]-\[cont\_pde\_BC2\] and using \[single\_moment\]-\[internal\_moment\_full\] yields a linear differential equation for the vector of body segment curvatures ${\underline{\mathbf{\kappa}}}$: $$\qty(\alpha \mu_f I_6 + \dfrac{\mu_b}{\ell^4} D_4) \dot{{\underline{\mathbf{\kappa}}}} = -\dfrac{k_b}{\ell^4} D_4 \qty({\underline{\mathbf{\kappa}}} + \sigma({\underline{\mathbf{A}}}_V) - \sigma({\underline{\mathbf{A}}}_D)), \label{full_kappa}$$ where $I_6$ is the $6\times6$ identity matrix.
In this discretization, the neural and muscle activity dynamics of all the modules are given by $$\begin{aligned}
\tau_m \dot{{\underline{\mathbf{A}}}}_V &= - {\underline{\mathbf{A}}}_V + {\underline{\mathbf{V}}}_V - {\underline{\mathbf{V}}}_D, \label{full_AV}\\
\tau_m \dot{{\underline{\mathbf{A}}}}_D &= - {\underline{\mathbf{A}}}_D + {\underline{\mathbf{V}}}_D - {\underline{\mathbf{V}}}_V, \label{full_AD} \\
\tau_n \dot{{\underline{\mathbf{V}}}}_V &= F({\underline{\mathbf{V}}}_V) + c_{p}{\underline{\mathbf{\kappa}}} - {\varepsilon}_p W_p {\underline{\mathbf{\kappa}}} + {\varepsilon}_g W_g {\underline{\mathbf{V}}}_V, \label{full_VV}\\
\tau_n \dot{{\underline{\mathbf{V}}}}_D &= F({\underline{\mathbf{V}}}_D) - c_{p}{\underline{\mathbf{\kappa}}} + {\varepsilon}_p W_p {\underline{\mathbf{\kappa}}} + {\varepsilon}_g W_g {\underline{\mathbf{V}}}_V, \label{full_VD}\end{aligned}$$ where each vector entry (e.g., $A_{V,j}$) is the corresponding activity of the $j^\text{th}$ module. In equations \[full\_VV\] and \[full\_VD\], $W_p$ is the nonlocal proprioceptive connectivity matrix (equation \[w\_p\_matrix\]), which comes from discretizing equation \[prop\_feedback\_V\], and $W_g$ is the gap-junction connectivity matrix (equation \[w\_g\_matrix\]), which comes from discretizing equation \[gap\_junction\_coup\]:
$$W_p = \qty(\begin{matrix}
0 & & & & \\
1 & 0 & & & \\
& & \ddots & \ddots & \\
& & & 1 & 0
\end{matrix})\label{w_p_matrix},$$
$$W_g = \qty(\begin{matrix}
-1 & 1 & & \\
1 & -2 & 1 & \\
& \ddots & \ddots & \ddots \\
& & 1 & -1 \\
\end{matrix}).\label{w_g_matrix}$$
\
A numerical solution to the system of differential equations \[full\_kappa\]-\[full\_VD\] is generated using the ode23 method in MATLAB.
Parameter Discussion {#sect_param_discuss}
--------------------
Some parameters in the model are well-constrained by experimental data, while others are not. Quantities that are directly measurable include the body length $L = 1$ mm, average body radius $R=40$ $\mu$m, cuticle width $r_{c} = 0.5$ $\mu$m, and wavelength $\lambda/L$ and frequency $f$ in fluids of various viscosities $\mu_f$. The timescales in the system are less certain. The range 50-200 ms is used for the muscle activation timescale $\tau_m$, which is the range of measurements of peak muscle force generation time in Milligar et al. (1997) [@Milligan2425]. As with previous models [@Boyle:2012aa; @Denham_2018; @Izquierdo_2018], the neural activity is chosen to be the fastest process in the model, but while Boyle et al. [@Boyle:2012aa] considered the B-neurons as instantaneous switches, here the neural activity timescale is set at $\tau_n = 10$ ms.
The internal viscosity $\mu_b$ and Young’s modulus $E$ have been estimated across several orders of magnitude [@Backholm_2013; @Fang-Yen:2010aa; @Sznitman2010], so caution is exercised in using one set of parameters from one source over another. Of more importance in the model is the mechanical timescale $$\tau_b = \dfrac{\mu_b}{k_b}, \label{mech_timescale}$$ which is the timescale of relaxation in an inviscid fluid. In equation \[mech\_timescale\], $k_b$ is the bending modulus, which is derived from the Young’s modulus $E$ and the geometry of the cuticle in Appendix \[appendix\_deriv\_mech\_params\] following previous modeling procedures [@Cohen_2014; @Sznitman2010]. Given the range of mechanical parameters reported in the literature, the mechanical timescale could be as small as $\tau_b = 1$ ms or as large as $\tau_b = 5$ s. The role of this timescale is explored in Section \[sect\_timescales\_param\_study\].
The electrophysiological details of the internal neural circui are largely unknown, thus all the feedback and coupling strengths $c_p, c_m, \varepsilon_p, \varepsilon_g$, the parameters of the nonlinear functions $F(V)$ and $\sigma(A)$ are not well constrained. The feedback strengths $c_m=10$, $c_p=1$ and parameters of the nonlinear functions $F(V)$ ($a=1, I = 0$) and $\sigma(A)$ ($c_s = 1, a_0 = 2$) were chosen so that the neuromechanical oscillator robustly gives the correct frequency ($\sim 1.76 Hz$) in a low-viscosity environment. The values for the coupling parameters $\varepsilon_p$ and $\varepsilon_g$, on the other hand, are explored in the next section.
Parameter Name Range of values References
---------------------- ------------------------------------------- ------------------------------------------------------------- ---------------------------------------------------
$L$ Body length 1 mm [@White:1986aa]
$R$ Average body radius 40 $\mu$m [@cohen_ranner_2017]
$r_{\text{cuticle}}$ Cuticle width 0.5 $\mu$m [@cohen_ranner_2017]
E Young’s modulus $3.77$ kPa - $1.3\times10^{4}$ kPa [@Backholm_2013; @Fang-Yen:2010aa; @Sznitman2010]
$I_c$ Second moment of area of cuticle $2.0 \times 10^{-7} (\text{mm})^4$ [@cohen_ranner_2017]
$k_b$ Bending modulus $7.53 \times 10^{-10} - 2.6\times 10^{-6}$ N$\cdot$(mm)$^2$ [@Backholm_2013; @Fang-Yen:2010aa; @Sznitman2010]
$\mu_b$ Body viscosity $2 \times 10^{-11} - 1.3\times10^{-7}$ N(mm)$^2$s [@Backholm_2013; @Fang-Yen:2010aa; @Sznitman2010]
$\mu_f$ Fluid viscosity $1 - 2.8 \times 10^4$ mPa$\cdot$s [@Fang-Yen:2010aa]
$C_N$ Normal drag coefficient $3.4 \mu_f$ [@cohen_ranner_2017; @Fang-Yen:2010aa]
$\tau_b$ Mechanical timescale $\tau_b = \mu_b/k_b$ 1 ms - 5 s [@Backholm_2013; @Fang-Yen:2010aa; @Sznitman2010]
$\tau_m$ Muscle activation timescale 50-200 ms [@Milligan2425]
: Range of parameters explored and sources. See Section \[sect\_param\_discuss\] for more details and Appendix \[appendix\_deriv\_mech\_params\] for derivations.
\[table\_1\]
Model Results {#sect_results}
=============
*C. elegans* locomote forward using alternating dorsal and ventral body bends that propogate in the form of a traveling wave from anterior to posterior. The spatial wavelength of this traveling wave changes in response to changes in the fluid viscosity [@Berri_2009; @Fang-Yen:2010aa; @Sznitman2010]. In this section, we show that our model captures this gait adaptation for a wide range of mechanical and neural parameters, provided that the muscle response to input from the nervous system is faster than the body response to changes in internal and external forces.
Model Captures Gait Adaptation {#sect_capt_gait_Adapt}
------------------------------
![The model captures the quantitative trend of gait modulation seen in experiments such as [@Fang-Yen:2010aa]. Here, $\tau_b = 0.5$ s, $\tau_m = 0.1$ s, and $\mu_b = 1.3\times 10^{-7}$ N(mm)$^2$s. In water ($\mu_f = 1$ mPa s) the wavelength is roughly 1.5 bodylengths, and increasing the fluid viscosity $\mu_f$ smoothly reduces the wavelength down to roughly 0.75 bodylengths in the most viscous case ($\mu_f = 28 $ Pa s). []{data-label="result1_lambdavgamma_fullonly"}](wvln_trends_tb_p5_tm_p1.png "fig:"){width=".7\textwidth"} ![The model captures the quantitative trend of gait modulation seen in experiments such as [@Fang-Yen:2010aa]. Here, $\tau_b = 0.5$ s, $\tau_m = 0.1$ s, and $\mu_b = 1.3\times 10^{-7}$ N(mm)$^2$s. In water ($\mu_f = 1$ mPa s) the wavelength is roughly 1.5 bodylengths, and increasing the fluid viscosity $\mu_f$ smoothly reduces the wavelength down to roughly 0.75 bodylengths in the most viscous case ($\mu_f = 28 $ Pa s). []{data-label="result1_lambdavgamma_fullonly"}](worm_shapes_tb_p5_tm_p1_version2.png "fig:"){width=".15\textwidth"}
We fit the model to match the wavelength and frequency in water, and then ran it in different fluid environments. Our model captures the quantitative effect of external fluid viscosity on the body wavelength seen in experiments and previous models. Figure \[result1\_lambdavgamma\_fullonly\] shows an example of the wavelength trend of the model for fixed body parameters $\tau_b = 500$ ms, $\tau_m = 50 $ms (the wavelengths were computed from the model output as described in Appendix \[appendix\_defining\_wvln\]). Figure \[result1\_lambdavgamma\_fullonly\] also shows that the model wavelengths are in close quantitative agreement with the experimentally-measured wavelengths of Fang-Yen et al. [@Fang-Yen:2010aa]. In water ($\mu_f = 1$ mPa$\cdot$s) the wavelength is roughly 1.5 bodylengths, and increasing the fluid viscosity $\mu_f$ smoothly reduces the wavelength down to roughly 0.75 bodylengths in the most viscous case ($\mu_f = 2.8\times 10^4$ mPa$\cdot$s). This wavelength trend is similar to what has been observed in other experiments [@Berri_2009; @Sznitman2010], and in Section \[sect\_timescales\_param\_study\], we show that our model captures this trend robustly over a wide range of parameters.
The undulation frequency also changes in response to changes in the fluid viscosity [@Berri_2009; @Fang-Yen:2010aa; @Sznitman2010]. In Fang-Yen et al. [@Fang-Yen:2010aa], the frequency decreases from 1.7 Hz to 0.30 Hz as fluid viscosity increases from 1 mPa s to $2.8 \times 10^4$ mPa s. Our model also exhibits a decrease in frequency as fluid viscosity $\mu_f$ increases, but not of the same magnitude (1.7 Hz - 1.6 Hz). Discussion of this discrepancy is given in Section \[sect\_discussion\].
Parameter Study Highlights Importance of Timescale Ordering in Capturing Gait Adaptation {#sect_timescales_param_study}
----------------------------------------------------------------------------------------
We performed a parameter study to show that the model robustly captures gait adaptation. For some parameter regimes, the body deformations were traveling waves for all fluid viscosities $\mu_f$, but this was not the case for other parameter regimes. Figure \[sample\_kymos\] shows kymographs of the body curvature that demonstrate two typical cases exhibited by the model.
![Sample model curvature kymographs (curvature vs. time) for various parameter regimes. For some parameter regimes, the gait adaptation trend generally held and there was a traveling wave at all $\mu_f$ values; (a) gives an example of this behavior for $\tau_b =$ 0.51 s, $\mu_b = 1.3 \times 10^{-7}$ N(mm$^2$) s, and $\mu_f$ = 28 Pa s. For other parameter regimes, high enough external fluid viscosity $\mu_f$ resulted in a loss of the traveling waveform; (b) gives an example of this behavior for $\tau_b =$ 0.51 s, $\mu_b = 1.5 \times 10^{-9}$ N(mm$^2$) s, and $\mu_f$ = 28 Pa s. []{data-label="sample_kymos"}](good_bad_kymos_withcolorbar.png){width="\textwidth"}
The model parameters were systematically varied to characterize the model behavior. For a given body timescale $\tau_b$ and body viscosity $\mu_b$, the muscle activity timescale $\tau_m$ was selected in the range 50-250 ms to match the undulation frequency (1.7 Hz) in water ($\mu_f = 1 $ mPa s). Next, the proprioceptive coupling strength $\varepsilon_p$ was selected to match the wavelength (1.5 bodylengths) in water, and the gap-junctional coupling strength was fixed at ${\varepsilon}_g = 0.0134$. The model was then run in different fluid viscosity $\mu_f$ environments and the emergent coordination trend is reported in Figure \[parameter\_Sweep\_wvln\_trends\]. The model behavior was classified as either: (1) not a traveling wave for all fluid environments, (2) incorrect wavelength trend, (3) qualitatively correct wavelength trend, or (4) incorrect frequency in water.
There is no traveling wave (red triangles) if, for any viscosity $\mu_f$, the difference between the minimum and maximum pairwise-phase difference is greater than or equal to 0.5, because this indicates that there is no consistent directionality to the phase differences in the body. A range of observed wavelength trends in various parameter regimes (the boxed markers in Figure \[parameter\_Sweep\_wvln\_trends\]) are illustrated in Figure \[sample\_runs\_vs\_gamma\]. Figure \[sample\_runs\_vs\_gamma\](a) and (b) show examples of the qualitatively correct wavelength trend (blue circles), while (c) shows the the incorrect trend, which was only obtained at a single parameter combination. The wavelength trend is incorrect because the wavelengths increased dramatically as the fluid viscosity increased, as opposed to generally decreasing.
![Classification of the model behavior for different mechanical parameters $\mu_b$ and $\tau_b$. For each parameter combination $(\mu_b, \tau_b)$, the muscle timescale $\tau_m$ was fit to match the undulation frequency in water ($\tau_m$ contours shown in black dashes). Boxed markers indicate parameter combinations which have the wavelength trend illustrated in Figure \[sample\_runs\_vs\_gamma\].[]{data-label="parameter_Sweep_wvln_trends"}](wvln_trends_tfs_vs_mus_redblue.png){width=".5\textwidth"}
(a)![Model wavelength vs. external fluid viscosity $\mu_f$ for various parameter regimes (the boxed markers in Figure \[parameter\_Sweep\_wvln\_trends\]). (a) and (b) show examples of the qualitatively correct wavelength trend, while (c) shows an incorrect trend. (a) has $\tau_b =$ 0.51 s , $\mu_b =1.4\times 10^{-8}$ N(mm$^2$)s, (b) has $\tau_b =$ 0.12 s , $\mu_b =1.5\times 10^{-9}$ N(mm$^2$)s, and (c) has $\tau_b =$ 0.01 s , $\mu_b =1.3\times 10^{-7}$ N(mm$^2$)s. []{data-label="sample_runs_vs_gamma"}](compare_wvln_trends_good1.png "fig:"){width="30.00000%"} (b)![Model wavelength vs. external fluid viscosity $\mu_f$ for various parameter regimes (the boxed markers in Figure \[parameter\_Sweep\_wvln\_trends\]). (a) and (b) show examples of the qualitatively correct wavelength trend, while (c) shows an incorrect trend. (a) has $\tau_b =$ 0.51 s , $\mu_b =1.4\times 10^{-8}$ N(mm$^2$)s, (b) has $\tau_b =$ 0.12 s , $\mu_b =1.5\times 10^{-9}$ N(mm$^2$)s, and (c) has $\tau_b =$ 0.01 s , $\mu_b =1.3\times 10^{-7}$ N(mm$^2$)s. []{data-label="sample_runs_vs_gamma"}](compare_wvln_trends_good2.png "fig:"){width="30.00000%"} (c)![Model wavelength vs. external fluid viscosity $\mu_f$ for various parameter regimes (the boxed markers in Figure \[parameter\_Sweep\_wvln\_trends\]). (a) and (b) show examples of the qualitatively correct wavelength trend, while (c) shows an incorrect trend. (a) has $\tau_b =$ 0.51 s , $\mu_b =1.4\times 10^{-8}$ N(mm$^2$)s, (b) has $\tau_b =$ 0.12 s , $\mu_b =1.5\times 10^{-9}$ N(mm$^2$)s, and (c) has $\tau_b =$ 0.01 s , $\mu_b =1.3\times 10^{-7}$ N(mm$^2$)s. []{data-label="sample_runs_vs_gamma"}](compare_wvln_trends_bad3.png "fig:"){width="30.00000%"}
A few key observations can be made from Figure \[parameter\_Sweep\_wvln\_trends\]. First, if the mechanical timescale $\tau_b$ is too large, then the frequency in water cannot be obtained (see the black squares in Figure \[parameter\_Sweep\_wvln\_trends\]). Second, if the mechanical timescale $\tau_b$ is too small, then there will not be a traveling wave for all fluid viscosities $\mu_f$. This suggests that while the body stiffness $k_b$ and body viscosity $\mu_b$ have been estimated across several orders of magnitude in various experiments and models, the effective mechanical body timescale $\tau_b = \mu_b/k_b$ lies within the relatively narrow range $0.07-1$ s.
In order to match the frequency, the muscle timescale $\tau_m$ must be inversely related to $\tau_b$. When the body timescale $\tau_b$ is increased, the muscle timescale $\tau_m$ must decrease to compensate. The frequency in water cannot be obtained for $\tau_b$ too large since it would require decreasing the muscle activity timescale $\tau_m$ below physiological limits. Similarly, when the body timescale $\tau_b$ is decreased, the muscle timescale $\tau_m$ must be increased to compensate for the frequency. For $\tau_b$ too small, there is not a traveling wave for all fluid viscosities $\mu_f$; this occurs soon after $\tau_b < \tau_m$. This suggests that the relative ordering of the timescales $\tau_b, \tau_m, \tau_n$ is key to the coordination. Generally, the mechanical timescale $\tau_b$ must be the largest, the muscle activity timescale $\tau_m$ intermediate, and the neural timescale $\tau_n$ the shortest. The mechanism by which this timescale ordering affects coordination is explained in Section \[sect\_two\_osc\_analysis\].
Remarkably, whenever there is a traveling wave in this systematic parameter search, it almost always has the qualitatively correct wavelength trend. This wavelength trend is consistent with gait adaptation across several orders of magnitude of the mechanical parameters.
The Neuromechanical Model as a Network of Coupled Oscillators: Insight Into Mechanisms Underlying Gait Adaptation
=================================================================================================================
The neuromechanical model is able to robustly capture the quantitative trend of gait adaptation across a wide range of parameters. In this section, the modular structure of the model will be exploited to uncover the fundamental mechanisms underlying gait adaptation. The isolated, uncoupled neuromechanical modules are intrinsic neuromechanical oscillators. These modules form a network of coupled oscillators with three forms of coupling: mechanical (through the body and external fluid), proprioceptive, and gap-junctional. Furthermore, this coupling is relatively weak, and thus the theory of weakly coupled oscillators [@Kopell_1986; @Schwemmer2012] can be applied to identify the coordinating effects of each coupling modality. We demonstrate that the competition between mechanical coupling and neural coupling provides an explicit mechanism for gait adaptation.
Isolated Neuromechanical Modules are Intrinsic Oscillators
----------------------------------------------------------
A single, isolated neuromechanical module is defined as a neural subcircuit, the corresponding muscles and body section, and local proprioceptive feedback (without coupling through the body or neural circuitry). The dynamics for this isolated module are governed by $$\begin{aligned}
\dot{\kappa} &= - \dfrac{1}{\tau_b}\qty(\kappa + \sigma(A_V)-\sigma(A_D)) ,\label{single_osc_kappa}\\
\dot{A}_V &= \dfrac{1}{\tau_m} \qty(- A_V + V_V - V_D) \label{single_osc_AV},\\
\dot{A}_D &= \dfrac{1}{\tau_m}\qty(- A_D + V_D - V_V) \label{single_osc_AD},\\
\dot{V}_V &= \dfrac{1}{\tau_n }\qty(F({V}_V) + c_{p}\kappa) \label{single_osc_VV}, \\
\dot{V}_D &= \dfrac{1}{\tau_n}\qty(F({V}_D)- c_{p}\kappa) \label{single_osc_VD}.\end{aligned}$$ Note that this is the model described in Section \[chapter2\], omitting the intermodular coupling. The isolated modules exhibit robust oscillations over a wide range of parameters, and a single period of the module is shown for each state-variable in Figure \[single\_oscillation\_3versions\](a). Thus, the neuromechanical modules are intrinsic oscillators, wherein each B-class neuron promotes either a dorsal or ventral bend and the local proprioceptive feedback acts to switch the bistable B neurons from one state to the other. The basic cycle of the oscillator is as follows: when activated, the ventral B-class neuron ($V_V$) excites the ventral muscles which build up activity ($A_V$) to induce a ventral bend (negative $\kappa$); when the curvature $\kappa$ is sufficiently large, the local proprioceptive feedback deactivates the ventral B-class neuron and activates the dorsal B-class neuron, and the cycle continues towards a dorsal bend.
The system of six identical, uncoupled neuromechanical oscillators is described by $$\dot{{\underline{\mathbf{X}}}}_j = S({\underline{\mathbf{X}}}_j), \ j = 1,\dots,6$$ where $${\underline{\mathbf{X}}}_j = \qty[
\kappa_j,
A_{V,j},
A_{D,j},
V_{V,j},
V_{D,j}]^T,$$ and $S({\underline{\mathbf{X}}})$ is given by equations \[single\_osc\_kappa\]-\[single\_osc\_VD\]. The oscillations correspond to a $T$-periodic limit cycle ${\underline{\mathbf{X}}}^{LC}(t)$ in $(\kappa, A_V, A_D, V_V, V_D)$-state-space. This limit cycle can be parametrized by phase $$\theta_j = \qty(\omega t + \theta_j^0) \text{ mod 1}$$ with the initial phase $\theta_j^0 \in [0,1)$. As $\theta_j$ increases at a constant rate $\omega = 1/T$, ${\underline{\mathbf{X}}}^{LC}(\theta_j)$ traces out the limit cycle through state-space and the state of each oscillator on the limit cycle is given by $${\underline{\mathbf{X}}}_j(t) = {\underline{\mathbf{X}}}^{LC}(\theta_j),$$ where the only distinguishing feature between the oscillators is their unique phase $\theta_j$. Figure \[single\_oscillation\_3versions\](a) shows the components of ${\underline{\mathbf{X}}}^{LC}(\theta)$.
(a)![The period and amplitude of the oscillations in $\kappa,A_V,A_D,V_D,V_V$ are all relatively similar for (a) the single, isolated neuromechanical module, (b) the single module in the full neuromechanical model at low viscosity ($\mu_f = 1$mPa s), and (c) the single module in the full neuromechanical model at high viscosity ($\mu_f = 2.8\times10^4$ mPa s). Ventral neural/muscle activities are given in green dashed lines, dorsal neural/muscle activities are given in red solid lines.[]{data-label="single_oscillation_3versions"}](single_isol_osc_cycle.png "fig:"){width="30.00000%"} (b)![The period and amplitude of the oscillations in $\kappa,A_V,A_D,V_D,V_V$ are all relatively similar for (a) the single, isolated neuromechanical module, (b) the single module in the full neuromechanical model at low viscosity ($\mu_f = 1$mPa s), and (c) the single module in the full neuromechanical model at high viscosity ($\mu_f = 2.8\times10^4$ mPa s). Ventral neural/muscle activities are given in green dashed lines, dorsal neural/muscle activities are given in red solid lines.[]{data-label="single_oscillation_3versions"}](single_osc_cycle_fullmodel_muf1.png "fig:"){width="30.00000%"} (c)![The period and amplitude of the oscillations in $\kappa,A_V,A_D,V_D,V_V$ are all relatively similar for (a) the single, isolated neuromechanical module, (b) the single module in the full neuromechanical model at low viscosity ($\mu_f = 1$mPa s), and (c) the single module in the full neuromechanical model at high viscosity ($\mu_f = 2.8\times10^4$ mPa s). Ventral neural/muscle activities are given in green dashed lines, dorsal neural/muscle activities are given in red solid lines.[]{data-label="single_oscillation_3versions"}](single_osc_cycle_fullmodel_muf2e4.png "fig:"){width="30.00000%"}
Network of Coupled Oscillators
------------------------------
Rearranging equations \[full\_kappa\]-\[w\_g\_matrix\], the neuromechanical model can be written as a network of coupled oscillators: $$\dot{{\underline{\mathbf{X}}}}_j = S({\underline{\mathbf{X}}}_j) + C_j({\underline{\mathbf{X}}}_1, \dots, {\underline{\mathbf{X}}}_6), \ \ j=1,\dots,6 \label{totally_separated_model}$$ where $C_j({\underline{\mathbf{X}}}_1, \dots, {\underline{\mathbf{X}}}_6)$ describes the coupling dynamics from all the modules to the $j^{th}$ module through gap-junctions, nonlocal proprioception, and body mechanics: $$C_j({\underline{\mathbf{X}}}_1,\dots,{\underline{\mathbf{X}}}_6) = \qty[\begin{array}{c}
{\varepsilon}_m \sum_{k=1}^6(D_4^{-1})_{jk}\ \dot{\kappa}_k,\\
0,\\
0,\\
\frac{1}{\tau_n} \sum_{k=1}^6 {\varepsilon}_p (W_p)_{jk}\ \kappa_k + {\varepsilon}_g(W_g)_{jk}\ V_{V,k},\\
\frac{1}{\tau_n} \sum_{k=1}^6 - {\varepsilon}_p (W_p)_{jk}\ \kappa_k + {\varepsilon}_g (W_g)_{jk}\ V_{D,k}\\
\end{array}].$$ The parameter ${\varepsilon}_m = \alpha \mu_f \ell^4/\mu_b$ is the effective mechanical coupling strength. The intrinsic oscillations of the isolated module (equations \[single\_osc\_kappa\]-\[single\_osc\_VD\]) in Figure \[single\_oscillation\_3versions\](a) are almost indistinguishable in both frequency and amplitude to the oscillations in Figure \[single\_oscillation\_3versions\](b,c) of a module within the fully-coupled network (equations \[full\_kappa\]-\[full\_VD\]) at both low and high external fluid visocisty $\mu_f$. Furthermore, Figure \[single\_oscillation\_3versions\](b,c) show that the change in oscillator period between the low and high viscosity cases is $\Delta T/T < 0.003$, so the change in frequency as external fluid viscosity $\mu_f$ is varied is small. This suggests that the coupling dynamics are “weak” relative to the intrinsic oscillatory dynamics.
Because the coupling is weak, the theory of weakly coupled oscillators can be applied (see [@Schwemmer2012] for details). The coupling only alters the phase of the oscillators on their respective limit cycles and the effect on amplitude is negligible, therefore the phase completely describes the state of a neuromechanical module. Equation \[totally\_separated\_model\] can be reduced to the so-called phase equations, a set of differential equations describing the evolution of the phases of each oscillator: $$\dot{\theta}_j = \omega_j + \sum_{k=1}^6 {\varepsilon}_m (D_4^{-1})_{jk} H_m (\theta_k-\theta_j) + {\varepsilon}_g (W_g)_{jk} H_g(\theta_k-\theta_j) + {\varepsilon}_p (W_p)_{jk} H_p(\theta_k-\theta_j) ,\label{weak_coupling_ThetaJ}$$ where $\theta_j$ is the phase of the $j^{th}$ oscillator, $\omega$ is the intrinsic frequency, and $H(\phi)$ are the interaction functions that describe the change in frequency (resulting from either mechanical, proprioceptive, or gap-junction coupling) as a function of the phase difference $\phi = \theta_k - \theta_j$ of a given pair of oscillators: $$\begin{aligned}
H_m(\phi) &= -\dfrac{1}{T} \int_0^T Z_\kappa(t) \dot{{\underline{\mathbf{\kappa}}}}^{LC}(t - \phi) \dd t, \label{mech_interact_fn}\\
H_p(\phi) &= \dfrac{1}{\tau_n}\dfrac{1}{T} \int_0^T Z_{V_V}(t) {\underline{\mathbf{\kappa}}}^{LC}(t - \phi) - Z_{V_D}(t) {\underline{\mathbf{\kappa}}}^{LC}(t - \phi) \dd t, \label{prop_interact_fn} \\
H_g(\phi) &= \dfrac{1}{\tau_n}\dfrac{1}{T} \int_0^T Z_{V_V}(t) \qty({\underline{\mathbf{V_V}}}^{LC}(t - \phi) - {\underline{\mathbf{V_V}}}^{LC}(t)) + Z_{V_D}(t) \qty({\underline{\mathbf{V_D}}}^{LC}(t - \phi) - {\underline{\mathbf{V_D}}}^{LC}(t))\dd t. \label{gap_interact_fn}\end{aligned}$$ Here, $Z_\kappa(t)$, $Z_{V_V}(t)$, $Z_{V_D}(t)$ are the $T-$periodic phase response functions to perturbations in the corresponding state variable.
The coupling modalities define the *structure* of the interaction functions, through the state variables that are coupled, as well as the coupling topology (the connectivity matrices $D_4^{-1}$, $W_g$, and $W_p$ in equation \[weak\_coupling\_ThetaJ\]). Note that there is a separate H-function for each of the three coupling modalities and these three coupling modalities add linearly to produce the full interaction of the modules. Therefore, the relative contributions of the various coupling types can be analyzed independently through varying the different coupling strengths: fluid viscosity $\mu_f$ (through ${\varepsilon}_m$) for mechanical, ${\varepsilon}_p$ for proprioceptive, and ${\varepsilon}_g$ for gap-junctional.
Two Oscillator Analysis Explains the Coordination Mechanism {#sect_two_osc_analysis}
-----------------------------------------------------------
Analyzing a pair of two coupled oscillators gives considerable insight into the coordination that each coupling modality produces separately and the mechanisms of coordination. With only two oscillators, the phase model reduces to $$\begin{aligned}
\dot{\theta}_1 &= \omega+ {\varepsilon}_m \sum_{j=1}^2 (D_4^{-1})_{1j} H_m (\theta_j-\theta_1) + {\varepsilon}_g H_g(\theta_2-\theta_1), \label{pair_osc_theta1}\\
\dot{\theta}_2 &= \omega + {\varepsilon}_m \sum_{j=1}^2 (D_4^{-1})_{2j} H_m (\theta_j-\theta_2) + {\varepsilon}_p H_p(\theta_{1}-\theta_2) + {\varepsilon}_g H_g(\theta_1-\theta_2). \label{pair_osc_theta2}\end{aligned}$$ In the two oscillator case, the matrix $D_4^{-1}$ is symmetric, so $(D_4^{-1})_{12} = (D_4^{-1})_{21} = d_{12}$. By defining $$\phi = \theta_2-\theta_1,$$ and subtracting equation \[pair\_osc\_theta1\] from equation \[pair\_osc\_theta2\], the dynamics of the two oscillator system can be described by a single differential equation for the phase difference between the two oscillators:\
$$\begin{aligned}
\dot{\phi} &= {\varepsilon}_m d_{12} G_m (\phi) + {\varepsilon}_p G_p(\phi) + {\varepsilon}_g G_g(\phi) = G(\phi), \label{pair_osc_phi}\end{aligned}$$ where $G_m(\phi) = H_m(-\phi) - H_m(\phi)$, $G_p(\phi) = H_p(-\phi)$, and $G_g(\phi) = H_g(-\phi) - H_g(\phi)$ are the pair-wise interaction functions, or *G-functions* of the pair. The stable phase-locked state of the system $\phi^*$ is given by $G(\phi^*)=0,$ $G'(\phi^*)<0$.
### Each Coupling Modality Promotes a Different Coordination Outcome
Figure \[gfns\_plot\] shows the G-functions and corresponding phase-locked states of the different coupling modalities. For mechanical coupling alone, i.e., ${\varepsilon}_p = {\varepsilon}_g =0$, the stable phase-locked state is anti-phase ($\phi^* = 0.5$), since $G(0.5)=0$ and $G'(0.5)<0$ (Figure \[gfns\_plot\](a)). Similarly, for proprioceptive coupling alone, the stable state is an intermediate phase-difference ($\phi^*\approx 0.75$, Figure \[gfns\_plot\](b)), so the first oscillator leads the second (front-to-back). For gap-junctional coupling alone, the stable state is synchrony ($\phi^* = 0$, Figure \[gfns\_plot\](c)).
The coordination outcome with all three coupling mechanisms present corresponds to the zero of the G-function (equation \[pair\_osc\_phi\]), which is a weighted sum of the three individual G-functions. Thus, coordination can be examined in the context of this weighted sum as the three coupling strengths are varied: external fluid viscosity $\mu_f$ for mechanical coupling, proprioceptive coupling strength ${\varepsilon}_p$, and gap-junction coupling strength ${\varepsilon}_g$.
(a)![Each coupling modality promotes a different coordination outcome in a pair of coupled neuromechanical oscillators based on the stable zero of the corresponding G-function: (a) mechanical coupling promotes antiphase since $G_m(0.5) = 0$ and $G_m'(0.5)<0$; (b) proprioceptive coupling promotes a phase-wave since $G_p(.75) = 0$ and $G_p'(.75)<0$; and (c) gap-junctional coupling promotes synchrony since $G_g(1) = 0$ and $G_g'(1)<0$.[]{data-label="gfns_plot"}](gfn_mech.png "fig:"){width=".3\textwidth"} (b)![Each coupling modality promotes a different coordination outcome in a pair of coupled neuromechanical oscillators based on the stable zero of the corresponding G-function: (a) mechanical coupling promotes antiphase since $G_m(0.5) = 0$ and $G_m'(0.5)<0$; (b) proprioceptive coupling promotes a phase-wave since $G_p(.75) = 0$ and $G_p'(.75)<0$; and (c) gap-junctional coupling promotes synchrony since $G_g(1) = 0$ and $G_g'(1)<0$.[]{data-label="gfns_plot"}](gfn_prop.png "fig:"){width=".3\textwidth"} (c)![Each coupling modality promotes a different coordination outcome in a pair of coupled neuromechanical oscillators based on the stable zero of the corresponding G-function: (a) mechanical coupling promotes antiphase since $G_m(0.5) = 0$ and $G_m'(0.5)<0$; (b) proprioceptive coupling promotes a phase-wave since $G_p(.75) = 0$ and $G_p'(.75)<0$; and (c) gap-junctional coupling promotes synchrony since $G_g(1) = 0$ and $G_g'(1)<0$.[]{data-label="gfns_plot"}](gfn_gap.png "fig:"){width=".3\textwidth"}
### Neural Coupling Sets the Low-Viscosity Wavelength
(a)![In the low-viscosity limit, the stable phase-locked states of the pair of neuromechanical oscillators is set by the competition between proprioceptive and gap-junctional coupling. (a) The linear combination of the G-functions given by equation \[pair\_osc\_phi\] for $\varepsilon_p = 0.05$, $\mu_f = 1$ mPa$\cdot$s, and various ${\varepsilon}_g$. Note that as the gap-junctional coupling strength ${\varepsilon}_g$ increases, the stable phase-locked phase difference $\phi^*$ moves from roughly $\phi^*=0.75$ towards $\phi^* = 1$. (b) The stable phase-locked states of the pair can be tuned by varying the two forms of neural coupling: proprioceptive and gap-junctional. When proprioceptive coupling dominates, the stable phase-locked state is a phase difference of roughly $\phi^* = 0.75$, and when gap-junctional coupling dominates, the stable phase-locked state is synchrony $\phi^* = 1$. The resulting wavelength in the body, if the pair-wise phase difference was constant in the six-oscillator model, can be tuned by varying the two forms of neural coupling: proprioceptive and gap-junctional. When proprioceptive coupling dominates, the wavelength is roughly 0.75 bodylengths, and when gap-junctional coupling dominated, the wavelength is infinite, since each oscillator pair is in perfect synchrony and thus the body is a standing wave. []{data-label="neural_coupling_plots"}](2osc_fullgfns_longwavelength.png "fig:"){width=".45\textwidth"} (b)![In the low-viscosity limit, the stable phase-locked states of the pair of neuromechanical oscillators is set by the competition between proprioceptive and gap-junctional coupling. (a) The linear combination of the G-functions given by equation \[pair\_osc\_phi\] for $\varepsilon_p = 0.05$, $\mu_f = 1$ mPa$\cdot$s, and various ${\varepsilon}_g$. Note that as the gap-junctional coupling strength ${\varepsilon}_g$ increases, the stable phase-locked phase difference $\phi^*$ moves from roughly $\phi^*=0.75$ towards $\phi^* = 1$. (b) The stable phase-locked states of the pair can be tuned by varying the two forms of neural coupling: proprioceptive and gap-junctional. When proprioceptive coupling dominates, the stable phase-locked state is a phase difference of roughly $\phi^* = 0.75$, and when gap-junctional coupling dominates, the stable phase-locked state is synchrony $\phi^* = 1$. The resulting wavelength in the body, if the pair-wise phase difference was constant in the six-oscillator model, can be tuned by varying the two forms of neural coupling: proprioceptive and gap-junctional. When proprioceptive coupling dominates, the wavelength is roughly 0.75 bodylengths, and when gap-junctional coupling dominated, the wavelength is infinite, since each oscillator pair is in perfect synchrony and thus the body is a standing wave. []{data-label="neural_coupling_plots"}](2osc_phases_neural_longwavelength.png "fig:"){width=".45\textwidth"}
The stable phase difference $\phi^*$ of the pair of the neuromechanical oscillators can be used to define a wavelength in the full body (for details see Appendix \[appendix\_defining\_wvln\]): $$\dfrac{\lambda}{L} = \dfrac{1}{6\qty(1-\phi^*)}.
\label{wvln_per_bodylength_defining_wvln}$$
In the low external fluid viscosity case ($\mu_f=1$ mPa$\cdot$s), setting ${\varepsilon}_p=0.05, {\varepsilon}_g = 0.01$ as in Section \[sect\_timescales\_param\_study\] provides a good approximation of the experimentally observed wavelength for the mechanical parameters $k_b = 2.6\times 10^{-7}$ N (mm)$^2$, $\mu_b = 1.3\times 10^{-7}$ N (mm)$^2$ s. For these parameters, the relative sizes of the G-functions in equation \[pair\_osc\_phi\] are $$\begin{aligned}
{\varepsilon}_m d_{12}\max|G_m (\phi)| &= 3.532 \times 10^{-5},\\
{\varepsilon}_p \max |G_p(\phi)| &= 2.016, \\
{\varepsilon}_g \max |G_g(\phi)| &= 1.259.\end{aligned}$$ Thus, at low viscosity, mechanical coupling is almost negligible compared to neural coupling, so the coordination is determined by proprioceptive and gap-junctional coupling.
How the wavelength is set in this low-viscosity case can be examined by varying the neural coupling strengths. Figure \[neural\_coupling\_plots\](a) shows that as the gap-junctional coupling strength ${\varepsilon}_g$ is increased relative to the proprioceptive coupling strength, the phase-locked states move from close to the zeros of $G_p(\phi)$ towards the zeros of $G_g(\phi)$. Figure \[neural\_coupling\_plots\](b) shows that when proprioceptive coupling dominates, the stable phase-locked state corresponds to a phase difference of roughly $\phi^* \approx 0.75$ and corresponds to a wavelength of 0.75 bodylengths according to equation \[wvln\_per\_bodylength\_defining\_wvln\]. When gap-junctional coupling dominates, the stable phase-locked state is close to synchrony $\phi^*\approx 1$, which corresponds to an infinite wavelength in the full body if this phase difference was constant. In this gap-junction-dominated case, each pair is in perfect synchrony and the body exhibits a standing wave.
To assess the predictive power of the two-oscillator phase model, a simulation of the neuromechanical model with only two modules was performed alongside the phase model. Figure \[neural\_coupling\_plots\](b) shows that the two-oscillator phase model is quantitatively accurate when compared to the phase differences and wavelengths derived from this two-module simulation. Thus, neural coupling sets the low-viscosity wavelength in the two-module neuromechanical model as well.
### Competition Between Mechanical and Neural Coupling Provides a Mechanism for Gait Adaptation
(a)![Gait adaptation is a result of the competition between mechanical and neural coupling in the pair of neuromechanical oscillators. (a) The linear combination of the G-functions given by equation \[pair\_osc\_phi\] for $\varepsilon_p = 0.05$, $\varepsilon_g = 0.0134$, and various $\mu_f$. Note that as $\mu_f$ increases, the strength of mechanical coupling increases and the stable phase-locked phase difference $\phi^*$ moves from roughly $\phi^*=0.8$ towards $\phi^* = 0.5$. (b) When neural coupling dominates, the stable phase-locked state is a phase difference of roughly $\phi^* = 0.88$, and when mechanical coupling dominates, the stable phase-locked state is antiphase, i.e., $\phi^* = 0.5$ phase difference. The resulting wavelength in the body, if the pair-wise phase difference was constant in the six-oscillator model, is set by the competition between the mechanical and neural coupling. When neural coupling dominates, the wavelength is roughly 1.5 bodylengths, and when mechanical coupling dominates, the wavelength is roughly 0.45 bodylengths. []{data-label="mech_coupling_plots"}](2osc_fullgfns_gaitadapt.png "fig:"){width=".45\textwidth"} (b)![Gait adaptation is a result of the competition between mechanical and neural coupling in the pair of neuromechanical oscillators. (a) The linear combination of the G-functions given by equation \[pair\_osc\_phi\] for $\varepsilon_p = 0.05$, $\varepsilon_g = 0.0134$, and various $\mu_f$. Note that as $\mu_f$ increases, the strength of mechanical coupling increases and the stable phase-locked phase difference $\phi^*$ moves from roughly $\phi^*=0.8$ towards $\phi^* = 0.5$. (b) When neural coupling dominates, the stable phase-locked state is a phase difference of roughly $\phi^* = 0.88$, and when mechanical coupling dominates, the stable phase-locked state is antiphase, i.e., $\phi^* = 0.5$ phase difference. The resulting wavelength in the body, if the pair-wise phase difference was constant in the six-oscillator model, is set by the competition between the mechanical and neural coupling. When neural coupling dominates, the wavelength is roughly 1.5 bodylengths, and when mechanical coupling dominates, the wavelength is roughly 0.45 bodylengths. []{data-label="mech_coupling_plots"}](2osc_phases_gait_adapt.png "fig:"){width=".45\textwidth"}
To examine the effect of mechanical coupling in the two-oscillator phase model, the neural coupling parameters are fixed to ${\varepsilon}_p = 0.05$ and ${\varepsilon}_g = 0.0134$ so that the wavelength in the low-viscosity case is roughly 1.5 bodylengths. The strength of mechanical coupling is increased in equation \[pair\_osc\_phi\] by increasing the external fluid viscosity $\mu_f$. Figure \[mech\_coupling\_plots\](a) shows that as the strength of mechanical coupling is increased, the phase-locked states move from close to the zeros set by ${\varepsilon}_p G_p(\phi) + {\varepsilon}_g G_g(\phi)$ towards the zeros of $G_m(\phi)$. Figure \[mech\_coupling\_plots\](b) shows how the stable phase-locked state changes as a function of the mechanical coupling strength $\mu_f$. When neural coupling dominates, the stable phase-locked state is a phase difference of roughly $\phi^*\approx 0.89$, and when mechanical coupling dominates, the stable phase-locked state is antiphase $\phi^* = 0.5$. Similarly, Figure \[mech\_coupling\_plots\](b) shows that when neural coupling dominates the resulting wavelength (according to equation \[wvln\_per\_bodylength\_defining\_wvln\]) is roughly 1.5 bodylengths, and when mechanical coupling dominates the wavelength is roughly 0.45 bodylengths.
This analysis shows that gait adaptation is a result of competition between mechanical and neural coupling. The decrease in wavelength as external viscosity $\mu_f$ increases is explained by the increased strength in mechanical coupling and its associated coordination outcome, antiphase. The two-oscillator phase model is quantitatively accurate when compared to phase differences derived from the neuromechanical model with two modules, as shown in Figure \[mech\_coupling\_plots\](b). Thus, this suggests that the mechanism underlying the behavior in the two-module neuromechanical model is the same as the mechanism of the phase model outlined here. However, note that the phase difference at the highest fluid viscosity ($\mu_f = 2.8\times 10^4$ mPa s) is different between the two-oscillator phase model and the full two-module neuromechanical model. This indicates the limit of weak coupling, as the phase reduction is not able to capture the transition to synchrony seen in the two-module neuromechanical model. However, weak coupling holds in the two-oscillator case for the rest of the viscosities $\mu_f$ considered. Furthermore, this transition to synchrony is not seen in the six-module neuromechanical model.
### Phase Reduction Gives Insight into Timescale Ordering
The phase reduction also explains why generally $\tau_b$ must be larger than $\tau_m$ in order to obtain the correct coordination trend (as described in Section \[sect\_timescales\_param\_study\]). The results in the previous subsection indicate that it is important for mechanical coupling to promote antiphase in order to get the correct wavelength trend as external viscosity $\mu_f$ is increased. Figure \[compare\_g\_fns\_tm\_tf\](a) shows that, when $\tau_b$ is sufficiently larger than $\tau_m$, the stable zero of $G_m$ is 0.5, i.e., the stable phase-locked state is antiphase. However, when $\tau_b$ is sufficiently smaller than $\tau_m$, the stable zero of $G_m$ is 0, i.e., the G-function is flipped and mechanical coupling promotes synchrony. In this case, the wavelength trend as external viscosity $\mu_f$ is increased is incorrect, since increasing the mechanical coupling strength would pull the oscillators towards synchrony, lengthening the wavelength instead of shortening it.
The shift in the stabilities of the phase-locked states from antiphase to synchrony is somewhat complicated, as Figure \[compare\_g\_fns\_tm\_tf\](c) shows that $\tau_b \approx \tau_m$ can yield tristable phase-locked states. A series of paired saddle-node bifurcations and paired super- and sub-critical pitchfork bifurcations (Figure \[compare\_g\_fns\_tm\_tf\]D), marks the transition from stable antiphase to tristability to stable synchrony as $\tau_b$ moves below $\tau_m$. The change in the stability of the antiphase state promoted by mechanical coupling is the cause of the rapid change in coordination in Figure \[parameter\_Sweep\_wvln\_trends\] as $\tau_b$ becomes sufficiently smaller than $\tau_m$.
(a)![(a) Mechanical G-function $G_m(\phi)$ for the pair of neuromechanical oscillators when $\tau_b$ is sufficiently larger than $\tau_m$ ($\tau_b = 0.5$ s, $\tau_m = 0.15$ s). Note the stable phase-locked state is antiphase since $G_m(0.5)=0$ and $G_m'(0.5)<0$. (b) Mechanical G-function $G_m(\phi)$ for the pair when $\tau_b$ is sufficiently smaller than $\tau_m$ ($\tau_b = 0.05$ s, $\tau_m = 0.15$ s). Note the stable phase-locked state is synchrony since $G_m(1)=0$ and $G_m'(1)<0$, while antiphase is unstable since $G_m'(0.5)>0$. (c) Mechanical G-function $G_m(\phi)$ for the pair when $\tau_b \approx \tau_m$ ($\tau_b = 0.14$ s, $\tau_m = 0.15$ s). (d) Bifurcation diagram for the phase-locked states $\phi^*$ of the mechanical G-function vs. $\tau_b$, for $\tau_m = .15$ s.[]{data-label="compare_g_fns_tm_tf"}](compare_gfns_tf_tm_1.png "fig:"){width=".45\textwidth"} (b)![(a) Mechanical G-function $G_m(\phi)$ for the pair of neuromechanical oscillators when $\tau_b$ is sufficiently larger than $\tau_m$ ($\tau_b = 0.5$ s, $\tau_m = 0.15$ s). Note the stable phase-locked state is antiphase since $G_m(0.5)=0$ and $G_m'(0.5)<0$. (b) Mechanical G-function $G_m(\phi)$ for the pair when $\tau_b$ is sufficiently smaller than $\tau_m$ ($\tau_b = 0.05$ s, $\tau_m = 0.15$ s). Note the stable phase-locked state is synchrony since $G_m(1)=0$ and $G_m'(1)<0$, while antiphase is unstable since $G_m'(0.5)>0$. (c) Mechanical G-function $G_m(\phi)$ for the pair when $\tau_b \approx \tau_m$ ($\tau_b = 0.14$ s, $\tau_m = 0.15$ s). (d) Bifurcation diagram for the phase-locked states $\phi^*$ of the mechanical G-function vs. $\tau_b$, for $\tau_m = .15$ s.[]{data-label="compare_g_fns_tm_tf"}](compare_gfns_tf_tm_2.png "fig:"){width=".45\textwidth"}\
(c)![(a) Mechanical G-function $G_m(\phi)$ for the pair of neuromechanical oscillators when $\tau_b$ is sufficiently larger than $\tau_m$ ($\tau_b = 0.5$ s, $\tau_m = 0.15$ s). Note the stable phase-locked state is antiphase since $G_m(0.5)=0$ and $G_m'(0.5)<0$. (b) Mechanical G-function $G_m(\phi)$ for the pair when $\tau_b$ is sufficiently smaller than $\tau_m$ ($\tau_b = 0.05$ s, $\tau_m = 0.15$ s). Note the stable phase-locked state is synchrony since $G_m(1)=0$ and $G_m'(1)<0$, while antiphase is unstable since $G_m'(0.5)>0$. (c) Mechanical G-function $G_m(\phi)$ for the pair when $\tau_b \approx \tau_m$ ($\tau_b = 0.14$ s, $\tau_m = 0.15$ s). (d) Bifurcation diagram for the phase-locked states $\phi^*$ of the mechanical G-function vs. $\tau_b$, for $\tau_m = .15$ s.[]{data-label="compare_g_fns_tm_tf"}](compare_gfns_tf_tm_3.png "fig:"){width=".45\textwidth"} (d)![(a) Mechanical G-function $G_m(\phi)$ for the pair of neuromechanical oscillators when $\tau_b$ is sufficiently larger than $\tau_m$ ($\tau_b = 0.5$ s, $\tau_m = 0.15$ s). Note the stable phase-locked state is antiphase since $G_m(0.5)=0$ and $G_m'(0.5)<0$. (b) Mechanical G-function $G_m(\phi)$ for the pair when $\tau_b$ is sufficiently smaller than $\tau_m$ ($\tau_b = 0.05$ s, $\tau_m = 0.15$ s). Note the stable phase-locked state is synchrony since $G_m(1)=0$ and $G_m'(1)<0$, while antiphase is unstable since $G_m'(0.5)>0$. (c) Mechanical G-function $G_m(\phi)$ for the pair when $\tau_b \approx \tau_m$ ($\tau_b = 0.14$ s, $\tau_m = 0.15$ s). (d) Bifurcation diagram for the phase-locked states $\phi^*$ of the mechanical G-function vs. $\tau_b$, for $\tau_m = .15$ s.[]{data-label="compare_g_fns_tm_tf"}](bif_diag_mech_coupling_phi_vs_tf_tm150ms.png "fig:"){width=".45\textwidth"}
Mechanism for Gait Adaptation Holds in Six-Oscillator Case
----------------------------------------------------------
We simulate the six-oscillator phase model in order to (i) assess the predictive power of the phase model by a quantitative comparison to the full six-module neuromechanical model and (ii) determine whether the mechanism of gait adaptation analyzed in the two-module case extends to the full six-module case.
Figure \[6box\_mech\_adapt\](a) shows the wavelengths for the six-oscillator phase model (line, circles) and neuromechanical model (crosses) as a function of external fluid viscosity $\mu_f$ for $\varepsilon_p = 0.05$ and $\varepsilon_g = 0.017$ (these coupling strengths were chosen so that the water-wavelength is approximately 1.5). The wavelengths were computed by equation \[wvln\_per\_bodylength\_defining\_wvln\_nonconstant\] in Appendix \[appendix\_defining\_wvln\]. The phase model and the neuromechanical model agree quantitatively even at high $\mu_f$, where the mechanical coupling strength is several orders of magnitude stronger. Figure \[6box\_mech\_adapt\](b) shows the stable phase differences between neighboring modules in the six-oscillator phase model (lines, circles) and neuromechanical model (crosses) as a function of external fluid viscosity $\mu_f$. Again, the phase model and the neuromechanical model are in quantitative agreement. Furthermore, Figure \[6box\_mech\_adapt\](b) shows that increasing fluid viscosity affects the phase-locked states in the six-oscillator case in a similar way as in the two-oscillator case. When neural coupling dominates at low viscosity, the stable phase differences are spread out near 0.9, and as fluid viscosity increases, the mechanical coupling strength increases and the stable phase differences decrease towards antiphase. The large variation between the phase differences across pairs of modules is due to the non-uniformity of coupling matrices $D_4^{-1}$, $W_g,$and $W_p$. The modules in the middle receive stronger mechanical coupling than the modules at the boundaries; the boundary modules receive less gap-junctional coupling because they have one fewer neighboring module; and the first module gets zero nonlocal proprioceptive feedback because it has no anterior neighboring module.
The general trend of each phase difference between neighboring modules (decreasing from near-synchrony towards antiphase) underlies the wavelength trend of gait adaptation in Figure \[6box\_mech\_adapt\](a) in both the six-oscillator phase model and the neuromechanical model. Thus, the results for the two-oscillator case in Section \[sect\_two\_osc\_analysis\] extend to the six-oscillator case: the decrease in wavelength in response to increasing fluid viscosity is the result of the corresponding increase in the relative strength of mechanical coupling, which decreases the phase differences between neighboring modules and yields shorter wavelengths.
(a)![(a) The wavelengths generated by the six-oscillator phase model (blue line with circles) and neuromechanical model (red crosses) as a function of external fluid viscosity $\mu_f$ for $\varepsilon_p = 0.05$ and $\varepsilon_g = 0.017$. The wavelength is set by the competition between the mechanical and neural coupling. (b) The phase differences between neighboring oscillator modules in the six-oscillator phase model (lines with circles) and neuromechanical model (crosses) as a function of external fluid viscosity $\mu_f$ for $\varepsilon_p = 0.05$ and $\varepsilon_g = 0.017$. Similar to the two-oscillator case, the stable phase differences here are set by the competition between mechanical and neural coupling. When neural coupling dominates, the stable phase differences are spread out around 0.9, and when mechanical coupling dominates, the stable phase differences move towards antiphase, i.e., closer to 0.5 phase difference, but with strong boundary effects. []{data-label="6box_mech_adapt"}](6box_wvlns_phase_Vs_nm.png "fig:"){width=".45\textwidth"} (b)![(a) The wavelengths generated by the six-oscillator phase model (blue line with circles) and neuromechanical model (red crosses) as a function of external fluid viscosity $\mu_f$ for $\varepsilon_p = 0.05$ and $\varepsilon_g = 0.017$. The wavelength is set by the competition between the mechanical and neural coupling. (b) The phase differences between neighboring oscillator modules in the six-oscillator phase model (lines with circles) and neuromechanical model (crosses) as a function of external fluid viscosity $\mu_f$ for $\varepsilon_p = 0.05$ and $\varepsilon_g = 0.017$. Similar to the two-oscillator case, the stable phase differences here are set by the competition between mechanical and neural coupling. When neural coupling dominates, the stable phase differences are spread out around 0.9, and when mechanical coupling dominates, the stable phase differences move towards antiphase, i.e., closer to 0.5 phase difference, but with strong boundary effects. []{data-label="6box_mech_adapt"}](6box_phasediffs_phase_Vs_NM.png "fig:"){width=".45\textwidth"}
Discussion {#sect_discussion}
==========
The analysis of the neuromechanical model presented here identifies a mechanism for gait adaptation to increasing fluid viscosity in *C. elegans* forward locomotion. We model the *C. elegans* forward locomotion system as a chain of neuromechanical oscillators coupled by body mechanics, proprioceptive coupling, and gap-junctional coupling. Using the theory of weakly coupled oscillators, we exploit the modular structure of the forward locomotion system to analyze the relative contributions of the various coupling modalities. We find that proprioceptive coupling between modules leads to a posteriorly-directed traveling wave with a characteristic wavelength. Gap-junction coupling between neural modules promotes synchronous activity (long wavelength), and mechanical coupling promotes a high spatial frequency (short wavelength). The wavelength of *C. elegans*’ undulatory waveform is set by the relative strengths of these three coupling forms. As the external fluid viscosity increases, the mechanical coupling strength increases and therefore wavelength decreases, as observed experimentally.
By tuning only a few coupling parameters, the model can robustly capture the gait adaptation seen in experiments [@Berri_2009; @Fang-Yen:2010aa; @Sznitman2010] over a wide range of mechanical parameters. The robustness of the model is of particular importance because the experimental measurements of mechanical body parameters vary widely. Our model suggests relationships between the parameters that need to hold in order to get the appropriate coordination and wavelength trend. In particular, the effective mechanical body timescale $\tau_b = \mu_b/k_b$ (the ratio of body viscosity to stiffness) plays a key role. Our model yields the correct coordination trend across the entire range of reported mechanical parameters, provided that $\tau_b$ is in the range . Furthermore, the muscle activity timescale $\tau_m$ must generally be shorter than the effective body mechanics timescale $\tau_b$. In other words, the system must generate contractile forces faster than the body responds, otherwise, there will not be a traveling wave of neuromechanical activity and therefore no effective locomotion for high external fluid viscosities. Our model is similar in structure to modeling work by Boyle et al. [@Boyle:2012aa]. In particular, the neural module is very similar to Boyle et al. [@Boyle:2012aa]. On the other hand, the description of the muscle dynamics and body mechanics are more complex in the Boyle et al. model [@Boyle:2012aa]. Boyle et al. [@Boyle:2012aa] also captures gait adaptation, and the large number of parameters and variables of the model allows it to more closely match the wavelengths, amplitudes, and undulation frequencies observed experimentally. However, the complexity of the model also limits the ability to systematically assess the relative roles of body mechanics and proprioception in coordination. Another difference between our model and Boyle et al. [@Boyle:2012aa] is in the sign, directionality, and extent of nonlocal proprioception. The directionality of proprioception in Boyle et al. [@Boyle:2012aa] is consistent with the directionality of undifferentiated processes extending posteriorly from the B-class neurons, which have postulated to be responsible for proprioception [@Zhen:2015aa]. We take the directionality of proprioception to be consistent with the functional directionality suggested by the experiments of Wen et al. [@Wen:2012aa]. Note that symmetry arguments can be made that reversing both the sign and direction of the nonlocal proprioception will not change the behavior of the models, as Denham et al. [@Denham_2018] points out. The extent of proprioception in Boyle et al. [@Boyle:2012aa] is over half a bodylength, and Denham et al. [@Denham_2018] showed that the larger the proprioceptive range, the longer the undulatory wavelength their model. We considered only nearest-neighbor proproception, which is sufficient to achieve the long-wavelength undulations in water because of our inclusion of gap-junctional coupling that promotes synchrony between the modules and thus long wavelengths. *C. elegans* gait adaptation is marked by a shortening of the wavelength and a decrease in undulation frequency with increasing fluid viscosity [@Berri_2009; @Fang-Yen:2010aa; @Sznitman2010]. Boyle et al. [@Boyle:2012aa] captures both wavelength and frequency adaptation as a function of external fluid viscosity. Our model captures the quantitative trend in wavelength and the qualitative trend in frequency. However, the model frequency range is only $1.7-1.6$ Hz as fluid visosity is increased as opposed to the range $1.7-0.3$ Hz given in Fang-Yen et al. [@Fang-Yen:2010aa]. Many differences between Boyle et al. [@Boyle:2012aa] and our model may account for this discrepancy in frequency adaptation; these differences include nonlinear and heterogeneous mechanical body parameters and a more sophisticated muscle model. Our model assumes that the undulatory gait emerges from a chain of neuromechanical oscillators coupled by both body mechanics and neural connectivity. However, there are several other hypotheses for how the undulatory gait is generated and coordinated [@Gjorgjieva_2014]: (1) a separate head circuit contains a CPG that drives the propogated bending wave along the body, and (2) a network of coupled CPGs generates and coordinates the bending wave in a feed-forward manner. Modeling work by Olivares et al. [@Olivares_2018] shows that the anatomical structure of the neural circuitry of *C. elegans* can be tuned to produce CPG-driven locomotion. However, there is no experimental evidence to date for such spontaneous isolated neural activity [@Cohen_2014; @Zhen:2015aa]. Furthermore, recent experiments by [@Fouad:2018aa] showed that *C. elegans* is capable of decoupled “two-frequency undulations”. By suppressing neural activity in the neck region, the head and body can undulate seemingly independent of one another at different frequencies (the head slower and the body faster). This evidence supports the presence of multiple neural or neuromechanical oscillators.
In the present study, the theory of weakly coupled oscillators is used to identify the roles of the various coupling modalities in generating coordination for forward locomotion in *C. elegans*. The phase models derived by the theory of weakly coupled oscillators capture the influence of one oscillating module on another through the interaction functions $H(\phi)$, which are convolution-like integrals of the coupling input and the corresponding phase response function $Z(t)$ of the individual modules. Therefore, our findings could be validated by experimentally measuring the phase-response curves of the neuromechanical circuit [@netoff_etal]. This could be achieved using a combination of optogenetic techniques and mechanical stimuli to perturb the system [@Fouad:2018aa; @hongfei_conf_abstract; @Wen:2012aa]. Note also that the structure of the phase equations could be exploited to further dissect out the biophysical mechanisms that underlie coordination of the undulatory motion of *C. elegans*. Because the shapes of the PRCs and the coupling signals combine to determine the interaction functions, a systematic analysis of how cellular and synaptic dynamics [@Zhang_2013], muscle properties, and body mechanics shape the PRCs and coupling signals would provide further insight into the integrated neuromechanical mechanisms underlying the generation and coordination of locomotion.
Appendix
========
Defining Wavelength {#appendix_defining_wvln}
-------------------
#### Constant Wavespeed
For a wavelength of undulation in the neuromechanical model traveling front-to-back at constant speed, the phase is defined as $$\theta(x,t) = \qty(\dfrac{t}{T} - \dfrac{x}{\lambda}) \text{ mod 1}$$ The phase corresponding to module $k$ ($k=1,\dots,6$) centered at body position $x = \ell(k - 1/2)$ is $$\theta_k = \qty(\dfrac{t}{T} - \dfrac{\ell}{\lambda}\qty(k - \frac{1}{2})) \text{ mod 1},
\label{theta_k_defining_wvln}$$ where $T$ is the oscillator period. Thus, the constant phase difference $\phi^*$ is $$\phi^* = \qty(\theta_{k+1} - \theta_k) \text{ mod 1} = \qty(- \dfrac{\ell}{\lambda}) \text{ mod 1} = 1-\dfrac{\ell}{\lambda}.
\label{phi_defining_wvln}$$ and the constant wavelength is $$\lambda = \dfrac{\ell}{1-\phi^*}.
\label{wvln_defining_wvln}$$ For the neuromechanical model, $\ell = L/6$, so the wavelength (normalized by bodylength) is $$\dfrac{\lambda}{L} = \dfrac{1}{6\qty(1-\phi^*)}.
\label{app_wvln_per_bodylength_defining_wvln}$$
#### Nonconstant Wavespeed
The non-uniform phase differences $\phi_k = \theta_{k+1}-\theta_{k}$ ($k=1,\dots,5$) between modules are used to define an effective wavelength of undulation when the wavespeed is nonconstant. The distance between the center of the first and center of the sixth module is $5/6L$, and the phase difference between them is $\sum_{k=1}^5(1-\phi_k)$. This gives an effective wavelength (normalized by 5/6 bodylengths) $$\dfrac{\lambda}{(5/6)L} = \dfrac{1}{\sum_{k=1}^{5}\qty(1-\phi_k)},
\label{wvln_per_bodylength_defining_wvln_nonconstant_v1}$$ so the wavelength (normalized by bodylength) is $$\dfrac{\lambda}{L} = \dfrac{1}{6\sum_{k=1}^{5}\qty(1-\phi_k)/5}.
\label{wvln_per_bodylength_defining_wvln_nonconstant}$$ Note that this is equivalent to the constant phase difference wavelength (equation \[app\_wvln\_per\_bodylength\_defining\_wvln\]) using the *average phase difference* between the modules as the constant phase difference $\phi^*$, i.e., with $$1 - \phi^* = \dfrac{\sum_{k=1}^{5}1-\phi_k}{5}.$$
For the neuromechanical model results, first the phase differences $\phi_k$ between the modules were computed, then the wavelength was computed according to equation \[wvln\_per\_bodylength\_defining\_wvln\_nonconstant\] above.
Derivation of Mechanical Parameters {#appendix_deriv_mech_params}
-----------------------------------
First, the bending modulus $k_b = EI_c$ of the cuticle of the worm was determined, where $E$ is the Young’s modulus and $I_c$ is the second moment of area of the cuticle. The nematode body can be thought of as a pressurized, fluid-filled tube or modeled as an annular cylinder as in Cohen and Ranner [@cohen_ranner_2017], so the only elasticity in the body is that of the cuticle. To approximate the second moment of area of the cuticle, $I_c$, note that the cuticle width $r_{\text{cuticle}} = 0.5$ $\mu$m is much smaller than the average worm radius $R=40$ $\mu$m. Following Cohen and Ranner [@cohen_ranner_2017],
$$I_c = 2\pi R^3 r_{\text{cuticle}} = 2.0 \times 10^{-7} \text{mm}^4. \label{moment_of_cuticle_area}$$
The Young’s modulus $E$ has been estimated to be as small as $E = 3.77 \pm 0.62$ kPa [@Sznitman2010] or as large as $E = 13$ MPa [@Fang-Yen:2010aa]. Backholm et al. [@Backholm_2013] gives a range of $110 \pm 30$ kPa $\leq E \leq 1.3 \pm 0.3$ MPa. Using these estimates, we explore the range of bending moduli $k_b = EI_c = 7.53 \times 10^{-10} - 2.6 \times 10^{-6}$ N(mm)$^2$.
The *cuticle* viscosity has been estimated as $5 \times 10^{-16}$ Nm$^2$s [@Fang-Yen:2010aa]. The internal *tissue* viscosity has been estimated to be constant and negative (energy-generating) as $c_d = -177.1 \pm 15.2$ Pa s so that $\mu_b = c_d I = - 1.7 \times 10^{-11}$ N(mm)$^2$s [@Sznitman2010] by a model fit, however this includes the active muscle components. Backholm et al. [@Backholm_2013] estimated the range $c_d \in \qty[1\times 10^{2}, 1\times 10^{4}]$ Pa s, so that the effective viscosity is $c_d I \in \qty[2\times 10^{-11}, 2\times 10^{-9}]$ N(mm)$^2$s. These experiments used different techniques and models for viscosity, so likely have different effects lumped into the viscosity parameter. In order to explore the range of effective body mechanics timescales $\tau_f = \mu_b / k_b = 0.001 - 5$ s, we use the range of body viscosities $\mu_b = 5 \times 10^{-10} - 1.3 \times 10^{-7}$ N(mm)$^2$s in our model.
Following previous modeling procedures [@Fang-Yen:2010aa; @cohen_ranner_2017], the normal drag coefficient $C_N$ of a slender body with length $L = 1$ mm and (average) radius $R = 40 \ \mu$m in a solution with viscosity $\mu_f$ is $$C_N = \dfrac{ 4 \pi \mu_f}{\ln(L/R)+0.5} = \alpha \mu_f \approx 3.4 \mu_f.\label{drag_coeff_Eqn}$$
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Netta Cohen for helpful discussions related to this work. The work of RDG was partially supported by NSF grant DMS-1664679.
|
---
abstract: 'We prove that the heat content determines planar triangles.'
address:
- 'Department of Mathematics, New College of Florida, Sarasota, Florida 34243'
- 'Department of Mathematics, New College of Florida, Sarasota, Florida 34243'
author:
- Reed Meyerson
- Patrick McDonald
date: 'June 1, 2016'
title: Heat Content Determines Planar Triangles
---
[^1]
[^2]
Introduction
============
Let $D$ be a bounded open subset of $\mathbb{R}^2.$ Suppose we heat $D$ to uniform initial temperature of 1, and then, holding the boundary of $D$ at temperature 0, we let the heat dissipate. We can describe the evolution of temperature via the solution of the heat equation on $D.$ To do so, let $\Delta$ be the Laplacian[^3] on $D$ and solve $$\begin{aligned}
\frac{\partial u}{\partial t} & = \Delta u \quad\text{on } D \times (0,\infty) \label{HE1.1} \\
\lim\limits_{x\rightarrow x_0} u(x,t) & = 0 \quad\forall\text{ } x_0 \in \partial D \times (0,\infty) \label{HE1.2} \\
u(x,0) & = 1 \quad\text{on } D\label{HE1.3} \end{aligned}$$ Using the temperature $u(x,t),$ we can associate to $D$ a measure of the heat in $D$ at time $t,$ the so-called heat content of $D:$ $$H_D(t) = \int\limits_{D}u(x,t) {\mathop{}\!\mathrm{d}}x \label{HC1.1}$$ We prove:
\[mainresult\] Heat content determines planar triangles.
Theorem (\[mainresult\]) should be viewed in the context of related results from spectral geometry. In particular, in her thesis C. Durso [@D] proved
Dirichlet spectrum determines planar triangles.
Durso used wave trace methods to establish her theorem. A recent paper of Grieser and Marrona [@GM] establshes the result using heat trace. We follow the argument of [@GM] to establish our result.
The relationship of Dirichlet spectrum to the geometry of the underlying domain is a well-studied topic with an extensive associated literature. The same is true for the relationship between heat content and geometry, but the associated geometric invariants are distinct. In particular, heat content is not spectral; that is, it is not determined by the Dirichlet spectrum of the domain $D.$ In fact, if we denote by $\lambda_n$ the $n$th Dirichlet eigenvalue enumerated in increasing order with multiplicity, and by $\phi_n$ a collection of associated orthonormal eigenfunctions, then $$H_D(t) = \sum A_n^2 e^{-\lambda_n t}$$ where the coefficient $A_n = \int_D \phi_n(x) {\mathop{}\!\mathrm{d}}x$ contributes off-diagonal information (for more on the relationship between Dirichlet spectrum and heat content, see [@G], [@MM]).
Heat content is closely related to Brownian motion and its associated exit times from a given domain. In more detail, suppose $X_t$ is Brownian motion in the plane, with ${\mathbb P}^x$ the associated collection of probability measures charging Brownian paths beginning at $x.$ Given a domain $D\subset {\mathbb R}^2,$ let $\tau$ be the first exit time from $D:$ $$\tau = \inf\{t\geq 0: X_t \notin D\}.$$ Let ${\mathbb E}^x$ be expectation with respect to the probability ${\mathbb P}^x$ and consider the moments of the exit time, $ {\mathbb E}^x[\tau^k],$ where $k$ varies over the natural numbers. Integrating over starting points in the domain $D$ results in a sequence of positive real numbers, the $L^1$-norms $ \|{\mathbb E}^x[\tau^k]\|_1,$ of the exit time moments of Brownian motion. It is easy to see that each element in the sequence is an invariant of the isometry class of $D.$ The extent to which this $L^1$-moment spectrum determines the geometry of $D$ is an active area of research, with roots in the nineteenth century (the first moment is also known as the torsional rigidity, a well-studied construct in the theory of elastica, and the objective function in the St. Venant Torsion Problem). For domains which are sufficiently regular, it is known that the $L^1$-moment spectrum determines heat content [@MM]. From this we have an immediate corollary:
The $L^1$-moment spectrum determines planar triangles.
In the remainder of the paper we prove our main result.
Proof of Theorem \[mainresult\]
===============================
As in the introduction, let $D$ be a bounded, open subset of $\mathbb{R}^2$ and let $u(x,t)$ denote the solution of the heat equation with uniform initial temperature distribution 1 and Dirichlet boundary conditions (i.e., $u(x,t)$ solves (\[HE1.1\])-(\[HE1.3\])). Let $H_D(t)$ be the heat content of $D:$ $$H_D(t) = \int\limits_{D}u(x,t) {\mathop{}\!\mathrm{d}}x.$$
As mentioned in the introduction, the relationship of heat content to the geometry of the underlying domain is a well-studied topic with an extensive associated literature. For our purposes, it is known that when the boundary of $D$ is sufficiently regular, there is a small time asymptotic expansion of $H_D(t).$ More precisely, when $D$ is smoothly bounded domain with compact closure in a Riemannian manifold, it is a theorem of Van den Berg and Gilkey [@BG] that $H_D(t)$ has an expansion of the form $$\label{asymptotics2.1}
H_D(t) \simeq \sum_{n=0}^\infty a_n t^{\frac{n}{2}}$$ where the coefficients $a_n$ are local invariants of the metric. A number of the coefficients have been computed; for example, it is known that $a_0= |D|$ and $a_1 = -\frac{2}{\sqrt{\pi}} |\partial D|$ where $|D|$ denotes the Riemannian volume of $D$ and $|\partial D|$ denotes the volume of the boundary of $D$ with respect to the induced surface measure. (For a more complete discussion of invariants appearing in the expansion (\[asymptotics2.1\]), and the relationship between heat content, heat trace and Dirichlet spectrum see [@G] and references therein).
When $D$ is a planar polygon, it is a theorem of Van den Berg and Srisatkunarajah [@BS] that $H_D$ has small time asymptotic expansion given by $$\label{asymptotics2.2}
H_D(t) = |D| - \frac{2|\partial D|}{\sqrt{\pi}} t^{\frac{1}{2}}
+ 4t\sum\limits_{i=1}^n\varphi(\theta_i) + O(e^{-q/t})$$ where $|D|$ is the area of $D,$ $|\partial D|$ is the perimeter of the boundary of $D,$ $\{\theta_i\}$ are the interior angles associated to $D,$ and $$\label{thirdterm}
\varphi(\theta) = \int\limits_0^\infty\frac{\sinh[(\pi-\theta)\xi]}{\sinh(\pi\xi)\cosh(\theta\xi)} {\mathop{}\!\mathrm{d}}\xi.$$
In the same paper, Van den Berg and Srisatkunarajah also establish a small time asymptotic expansion for the heat trace. More precisely, if $h_D(t)$ is the trace of the heat kernel, then, with notation as above, $$\label{asymptotics2.3}
h_D(t) = \frac{|D|}{4\pi} t^{-1} - \frac{|\partial D|}{8\sqrt{\pi}}t^{-\frac{1}{2}}
+ \frac{1}{24} \sum\limits_{i=1}^n \left(\frac{\pi}{\theta_i} - \frac{\theta_i}{\pi}\right) + O(e^{-r/t})$$
In [@GM], Grieser and Maronna use the heat trace asymptotics to show that a triangle’s Dirichlet spectrum is unique amongst triangles. To prove their theorem, they show that triangles are determined up to isometry by a triple given by area, perimeter, and the sum of the reciprocals of the interior angles (up to a constant, the third term appearing in the asymptotics of the heat trace for the domain). We employ a similar analysis to prove Theorem \[mainresult\]. More precisely, let $\Phi:(0,\pi)\times(0,\pi)\times(0,\pi)\rightarrow\mathbb{R}$ be defined by $$\label{G}
\Phi(x,y,z) = \varphi(x)+\varphi(y)+\varphi(z)$$ with $\varphi$ as in (\[thirdterm\]), the summands in the third term in the asymptotics of the heat content for the domain. We will prove that triangles are determined up to isometry by the first three coefficients of the heat content for the domain; i.e. the triple given by area, perimeter, and value of $\Phi$ as a function of the interior angles. This is equivalent to the following:
\[thm:main\] Let $T$ be the space of equivalence classes of isometric triangular domains in the plane. Define $H:T\rightarrow C^\infty(0,\infty)$ by $$H(D) = H_D(t), \quad\text{ for } D\in T$$ where $H_D(t)$ is the heat content associated to $D.$ Then $H$ is injective.
From equation (\[asymptotics2.2\]), heat content determines the area of our domain. Thus, we can restrict our attention to the space of triangles up to scaling, which we denote $T_s$.
We will use the following notation: Given a function $F:\mathbb{R}^3_+\rightarrow\mathbb{R}$ and a real number $r$, let $L_r(F)$ be the level set of $F$ with value $r:$ $$L_r(F) = \{(x,y,z)\in\mathbb{R}^3_+:F(x,y,z)=r\}$$
Where $\mathbb{R}^3_+$ denotes the positive octant of $\mathbb{R}^3$. Define a function $\Theta:\mathbb{R}^3_+\rightarrow\mathbb{R}$ by $$\label{theta}
\Theta(\alpha,\beta,\gamma) = \alpha + \beta + \gamma.$$
Let $\mathbb{T} = L_\pi(\Theta)$. Then $T_s$ may be identified with quotient $\mathbb{T}/\sim$ where the equivalence relation $\sim$ identifies permutations of a given triple. Because each ordering of the angles corresponds to one of the six sections in figure (\[fig:head\_on\]), there is a bijective correspondence between $T_s$ and any one of the six sections of $\mathbb{T}$. We will call a point in $\mathbb{T}$ an isosceles point if it represents an isosceles triangle. This occurs if and only if $p$ lies on one of the three lines in figure (\[fig:head\_on\]). The point at the intersection of these lines represents an equilateral triangle, thus we will call it the equilateral point.
[0.5]{} {width="\textwidth"}
[0.5]{} {width="\textwidth"}
\[fig:planes\]
Write $$\label{cot}
\psi(x) = \cot\left(\frac{x}{2}\right)$$ and let $\Psi:\mathbb{R}_3^+\rightarrow\mathbb{R}$ be defined by $$\label{psi}
\Psi(\alpha,\beta,\gamma) = \psi(\alpha) + \psi(\beta) + \psi(\gamma).$$ It can be shown using elementary geometry [@GM], that for any triangle with interior angles $(\alpha,\beta,\gamma),$ $$\Psi(\alpha,\beta,\gamma) = \frac{|\partial D|^2}{4|D|}.$$ Observe that $\Psi(p)\to \infty$ as $p$ approaches the boundary of $\mathbb{T}$. Thus, the sublevel sets of $\Psi$, given by $S_c(\Psi)=\{p\in\mathbb{T}:\Psi(p)\le c\}$, stay away from the boundary of $\mathbb{T}$. In addition, a straightforward computation of the Hessian of $\Psi$ indicates that $\Psi$ is strictly convex on the portion of the positive octant bounded by $\mathbb{T}.$ Thus $S_c$ is a convex set.
It follows from the definition of $\Psi$ that its level sets and sublevel sets will inherit the perumtation symmetry of $\mathbb{T}/\sim$. Using this symmetry and the convexity of $S_c$, it can be argued that $L_c(\Psi)\cap\mathbb{T}$ contains the equilateral point if and only if $L_c(\Psi)\cap\mathbb{T}$ is a singleton. Thus, if $F:T_s\rightarrow\mathbb{R}$ can be written as the sum of a strictly convex function of angles, then the equilateral triangle is determined by its area and the value of $F$.
Before proving Theorem \[mainresult\], we recall the method of Lagrange multipliers.
\[lagrangemultiplier\] Let $f,g,h:\mathbb{R}^3\rightarrow\mathbb{R}$ be $C^1$. Let $x_0\in\mathbb{R}$. Suppose $\nabla g(x_0)$ and $\nabla h(x_0)$ are linearly independent. If $f(x_0)$ is a local solution to the problem
- Maximize $f(x)$
- Subject to $g(x)=g(x_0)$, $h(x)=h(x_0)$
Then $\nabla f(x_0)\in\operatorname{span}[\nabla g(x_0),\nabla h(x_0)]$.
Theorem \[mainresult\] is a corollary of the following lemmas which we will prove in the sequel:
\[lin\_ind\] Let $f,g:(0,\pi)\rightarrow\mathbb{R}$ be $C^1$, monotone decreasing and convex. Suppose there exists a real $c>0$ such that $f'-c g'$ is increasing and convex. Suppose $F(\alpha,\beta,\gamma) = f(\alpha)+f(\beta)+f(\gamma),$ $G(\alpha,\beta,\gamma) = g(\alpha)+g(\beta)+g(\gamma)$ and $\Theta$ is as in (\[theta\]). Then $\nabla F, \nabla G, \nabla \Theta$ are linearly independent for all non-isosceles points of $\mathbb{T}.$
\[log2\_lemma\] Let $\varphi,\psi$ be as defined in (\[thirdterm\]) and (\[cot\]), respectively. Then $\varphi,\psi$ are decreasing and convex on $(0,\pi)$, and $\varphi'-\frac{\log 2}{2}\psi'$ is increasing and convex on $(0,\pi)$.
Let $t\in T_s$. Choose a representative $p\in\mathbb{T}$ of $t$. By the discussion preceding Lemma \[lagrangemultiplier\], we may assume $p$ is not the equilateral point. Then $L_{\Psi(p)}(\Psi)\cap\mathbb{T}$ is a closed curve around the equilateral point. Let $L$ be a segment of $L_{\Psi(p)}(\Psi)\cap\mathbb{T}$ such that $L$ contains $p$, $L$ is contained in one of the six sections of $\mathbb{T}$, and $L$ begins and ends at isosceles points. It follows that $L$ contains a representative of every triangle which agrees with $t$ on the value of $\Psi$. Thus, if $\Phi$ is monotone on $L$, then $\Phi$ differentiates between triangles with a fixed area and fixed value of $\Psi$.
Suppose $\Phi$ is not monotone on $L$. Then $\Phi\bigg|_L$ reaches a local extremum at a non-isosceles point. A contradiction follows from Lemma \[lagrangemultiplier\].
Suppose $\nabla F(x_0)+A\nabla G(x_0)-B\Theta(x_0)=0$. Then there are at least three solutions to the equation $f'(x_0)+Ag'(x_0)=B$. Suppose $\tilde f = f'-cg'$ is increasing and convex. Consider the equation $\tilde f(x) + c_1g(x) = c_2$ for fixed $c_1$ and $c_2$. If $c_1$ is positive, the left hand side is convex and there may be at most two solutions. If $c_1$ is negative, the left hand side is increasing and there may be at most one solution.
Write $$\begin{aligned}
\tilde \psi(x) &= \psi(\pi x)-\frac{2}{\pi x}\\
\tilde \varphi(x) &= \varphi(\pi x)-\frac{\log 2}{\pi x}.\end{aligned}$$ Then $$\varphi(\pi x)-\frac{\log 2}{2}\psi(\pi x) = \tilde \varphi(x)-\frac{\log 2}{2}\tilde \psi(x).$$
Thus, it will be sufficient to show that $\tilde \varphi'-\frac{\log 2}{2}\tilde \psi'$ is increasing and concave up on $(0,1)$. We change variables in the definition of $\varphi$ to obtain $$\varphi(\pi x) = \frac{1}{\pi}\int\limits_0^\infty\frac{\sinh(y-xy)}{\sinh(y)\cosh(xy)}dy.$$ Using the angle addition formula for $\sinh(x),$ we can write $$\varphi(\pi x) = \frac{1}{\pi}\int\limits_0^\infty\coth(y)[\tanh(y)-\tanh(xy)]dy.\\$$ It may be verified with elementary calculus that $\psi$ is decreasing and convex. Additionally, by observing the signs of $\frac{\partial}{\partial x}\tanh(xy)$ and $\frac{\partial^2}{\partial x^2}\tanh(xy)$, it follows that $\varphi$ is decreasing and convex.
We continue with our manipulation of $\varphi$ in an attempt to acquire a manageable form of $\tilde \varphi$. Integration by parts yields $$\begin{aligned}
\varphi(\pi x) =& \lim_{\stackrel{\epsilon \to 0}{\eta \to \infty}}\frac{1}{\pi}\coth(y)\left[\log(\cosh(y))-\frac{\log(\cosh(xy))}{x}\right]\bigg|_{y=\epsilon}^{y=\eta}\label{lims}\\
+&\frac{1}{\pi}\int\limits_0^\infty \left(\log(\cosh(y))-\frac{\log(\cosh(xy))}{x}\right)\operatorname{csch}^2(y)dy. \nonumber \end{aligned}$$ Applying L’Hopital’s rule we can evaluate (\[lims\]). $$\lim_{\stackrel{\epsilon \to 0}{\eta \to \infty}} \frac{1}{\pi}\coth(y)\left[\log(\cosh(y))-\frac{\log(\cosh(xy))}{x}\right]\bigg|_{y=\epsilon}^{y=\eta} =\frac{\log 2}{\pi x}-\frac{\log 2}{\pi}.$$ Noting that $$\int\limits_0^\infty \log(\cosh(y))\operatorname{csch}^2(y)dy = \log 2,$$ we obtain $$\label{varphi_rep}
\tilde \varphi(x) = -\frac{1}{\pi}\int\limits_0^\infty\frac{\log(\cosh(xy))}{x}\operatorname{csch}^2(y) dy.$$ Using this representation of $\tilde \varphi$, we compute relevant derivatives. Setting $$I(x) = \frac{\log(\cosh x)}{x}$$ we have $$\begin{aligned}
\tilde \varphi^{(k)}(x) &= -\frac{1}{\pi}\int\limits_0^\infty y^{k+1}I^{(k)}(xy)\operatorname{csch}^2(y)dy \nonumber \\
I''(x) &= \frac{\operatorname{sech}^2 x}{x}+2\frac{\log(\cosh x)-x\tanh{x}}{x^3}\label{second_deriv}\\
I'''(x) &= \frac{6(x\tanh x-\log(\cosh x))-3x^2\operatorname{sech}^2 x-2x^3\tanh x\operatorname{sech}^2 x}{x^4} \label{I3}\end{aligned}$$ The Laurent series for $\cot(x)$ is of the form $\frac{1}{x}-\sum\limits_{n=0}^\infty a_nx^{2n+1}$, where the $a_n$ are all positive, with convergence over the domain of interest. Thus, $\tilde \psi'(x)$ is decreasing and concave down on $(0,1)$. Thus, if $I''(x)$ is negative, it follows that $\tilde \varphi'(x)-\frac{\log 2}{2}\tilde \psi'(x)$ is increasing. By manipulating (\[second\_deriv\]), we see that $I''(x)$ is negative if and only if $$\frac{x^2}{2}\le x\sinh x \cosh x-\log(\cosh x)\cosh^2 x$$ This inequality is saturated at the origin. Thus, if the relation holds for the derivative of each side, we may recover the initial inequality by integration. Taking derivatives, it suffices to show $$x\le x\cosh(2x)-\sinh(2x)\log(\cosh x)$$
Again, this inequality is saturated at the origin. Thus, we differentiate again and would like to show $$1\le2x\sinh(2x)-2\cosh(2x)\log(\cosh x)+1$$ or equivalently $$0\le x\tanh(2x)-\log(\cosh x)$$ This inequality is also saturated at $x=0$. We differentiate one last time and seek to show $$0\le\tanh(2x)-\tanh(x)+2x\operatorname{sech}^2(2x)\label{final_increasing}$$ But $\tanh(x)$ is increasing, so $\tanh(2x)\ge\tanh(x)$. Thus, (\[final\_increasing\]) is valid for $x\ge 0$ and we have shown that $\tilde \varphi'(x)-\frac{\log 2}{2}\tilde \psi'(x)$ is increasing on $(0,1)$.
To verify convexity we must show $\tilde \varphi'''(x)-\frac{\log 2}{2}\tilde \psi'''(x)\ge0$. The representation in equation (\[varphi\_rep\]) demonstrates that $\tilde \varphi$ is a rescaled form of $\varphi$ with the pole at zero removed. Similarly, we now obtain a form of $\tilde \psi'$ which demonstrates that we have removed the pole of $\psi$ at zero. This process was carried out in [@GM] using the following representation of $\csc^2(x)$ $$\csc^2(x) = \sum\limits_{k=-\infty}^\infty\frac{1}{(k\pi+x)^2}$$ Thus, $$\tilde \psi'(x) = -\frac{2}{\pi}\sum\limits_{k\ne 0}\frac{1}{(2k+x)^2}$$ This series converges uniformly on $(0,\pi)$, so we may commute differentiation and summation to obtain $$\tilde \psi'''(x) = -\frac{12}{\pi}\sum\limits_{k\ne 0}\frac{1}{(2k+x)^4}$$ Thus, we must show that $$-\frac{1}{\pi}\int\limits_0^\infty y^4I^{(3)}(xy)\operatorname{csch}^2(y)dy\ge-\frac{6\log 2}{\pi} \sum\limits_{k\ne 0}\frac{1}{(2k+x)^4}\\$$ It will be sufficient to show that $$\int\limits_0^\infty y^4I^{(3)}(xy)\operatorname{csch}^2(y)dy\le\frac{6\log 2}{(x-2)^4}$$ We will use the following lemma:
\[xon6\] $I'''(x)\le\frac{x}{6}$ for $x>0$.
Assuming the lemma, we can finish the proof of Lemma \[log2\_lemma\].
In [@BM], Boyadzhiev and Moll demonstrated that $\int\limits_0^\infty y^5\operatorname{csch}^2(y)dy=\frac{15\zeta(5)}{2}$. Thus, $$\begin{aligned}
\int\limits_{0}^\infty y^4 I'''(xy)\operatorname{csch}^2(y)dy &\le \frac{1}{6}x\int\limits_0^{\infty}y^5\operatorname{csch}^2(y)dy\\
&=\frac{1}{6}\cdot\frac{15\zeta(5)}{2}x\\
&\le\frac{5\zeta(4)}{4}x\\
&=\frac{\pi^4}{72}x\\
&\le\frac{3}{2}x\\\end{aligned}$$ The line $y=\frac{3125}{2048}x$ is tangent to the graph of the convex function $\frac{4}{(x-2)^4}$. Thus, $\frac{3}{2}x\le\frac{3125}{2048}x\le\frac{4}{(x-2)^4}\le\frac{6\log 2}{(x-2)^4}$, which concludes the proof of Lemma \[log2\_lemma\].
All that remains to be done is the proof of Lemma \[xon6\].
Using (\[I3\]), we must establish $$6(x\tanh x-\log(\cosh x))-3x^2\operatorname{sech}^2 x-2x^3\tanh x\operatorname{sech}^2 x -\frac{x^5}{6} \le 0.$$
This inequality is saturated at the origin. Thus, taking derivatives and manipulating the result algebraically, it suffices to show $$\label{coshless2}
2(\cosh(2x)-2)-\frac{5}{6}x\cosh^4x\le 0.$$ If $\cosh(2x)<2$, inequality (\[coshless2\]) is clearly valid. Let $x_0=\frac{1}{2}\log(2+\sqrt{3})$ be the unique positive value where $\cosh(2x_0)=2$. Note that $x_0\in(\frac{1}{2},1)$. If $x\ge1$, we get $$\begin{aligned}
\frac{2(\cosh(2x)-2)}{x}-\frac{5}{6}\cosh^4x&\le 2(\cosh(2x)-2)-\frac{5}{6}\cosh^4x\\
&=2(2\cosh^2(x)-3)-\frac{5}{6}\cosh^4 x\end{aligned}$$ But the polynomial $2(2u-3)-\frac{5}{6}u^2$ has no real roots. Thus, (\[coshless2\]) is valid for $x\ge 1$. Now, suppose $x\in(\frac{1}{2},1)$. Then, we would like to show $$2(2\cosh^2x-3)-\frac{5}{6}x\cosh^4 x \le 0$$ Now consider the polynomial $2(2u-3)-\frac{5}{6}xu^2$. If $x>\frac{4}{5}$ this has no real roots. Otherwise, it has the solutions $u=\frac{12\pm6\sqrt{4-5x}}{5x}$. Thus, if we show that $$\label{ineq1}
\cosh^2(x) \le \frac{12-6\sqrt{4-5x}}{5x}$$ for $x\in(\frac{1}{2},\frac{4}{5})$, we will complete the verification of Lemma \[xon6\] for the entire range of $x>0$. By computing derivatives we can see that the the right-hand-side of (\[ineq1\]) is increasing. Thus, for $x\in(\frac{1}{2},\frac{4}{5})$, $$\frac{12-6\sqrt{4-5(\frac{1}{2})}}{5\left(\frac{1}{2}\right)}\le\frac{12-6\sqrt{4-5x}}{5x}$$ Additionally, for $x\in (\frac{1}{2},\frac{4}{5}).$ $$\cosh^2(x)\le\cosh^2\left(\frac{4}{5}\right)$$ It may be verified that $$\cosh^2\left(\frac{4}{5}\right)\le\frac{12-6\sqrt{4-5(\frac{1}{2})}}{5\left(\frac{1}{2}\right)}$$ from which the desired inequality follows immediately.
[10]{}
M. van den Berg, P. Gilkey *Heat content asymptotics of a Riemannian manifold with boundary*, J. Funct. Anal. [**120**]{}, 48-71, 1994.
M. van den Berg, S. Srisatkunarajah *Heat flow and Brownian motion for a region in $\mathbb{R}^2$ with a polygonal boundary*, Probab. Theory Related Fields, [**87**]{}, 41–52, 1990.
K. Boyadzhiev and V. Moll *The integrals in Gradshteyn and Ryzhik. Part 21: Hyperbolic functions*, SCIENTIA Series A: Mathematical Sciences, [**22**]{}, 109-127, 2011.
C. Durso *On the inverse spectral problem for polygonal domains*, Ph.D. thesis, MIT 1988.
P. Gilkey *Heat Content, Heat Trace, and Isospectrality*, Contemp. Math. [**491**]{}, 115–124, 2009.
D. Grieser, S. Maronna *Hearing the Shape of a Triangle*, Not. Amer. Math. Soc., [**60**]{}, 1440–1447, 2013.
P. McDonald, R. Meyers *Heat content and Dirichlet spectrum*, J. Funct. Anal. [**200**]{}, 150-159, 2003.
[^1]:
[^2]:
[^3]: We work with the convention that the Dirichlet Laplacian is positive.
|
---
abstract: |
We present distributions of countable models and correspondent structural characteristics of complete theories with continuum many types: for prime models over finite sets relative to Rudin–Keisler preorders, for limit models over types and over sequences of types, and for other countable models of theory.
[**Key words:**]{} countable model, theory with continuum many types, Rudin–Keisler preorder, prime model, limit model, premodel set.
author:
- 'Roman A. Popkov[^1] and Sergey V. Sudoplatov[^2]'
date: 'October 15, 2012'
title: |
Distributions of countable models\
of theories with continuum many types[^3]
---
Denote by $\mathcal{T}_c$ the class of all countable complete, non-small theories $T$, i. e., of theories with continual sets $S(T)$ of types. Below, unless otherwise stated, we shall assume that all theories, under consideration, belong to the class ${\cal T}_c$ and these theories will be called [*unsmall*]{} or [*theories with continuum many types*]{}.
In general case, for theories in ${\cal T}_c$, there is no correspondence between types and prime models over tuples that we observe for small theories (for given theory in ${\cal T}_c$, some prime models over realizations of types may not exist). Besides there are continuum many pairwise non-isomorphic countable models for each of these theories. However as we shall show, in this case, the structural links for types allow to distribute and to count the number of prime over finite sets, limit, and other countable models of a theory like small theories [@SuLP; @Su08] and arbitrary countable theories of unary predicates [@Pop].
**1. Examples**
Recall some basic examples of theories with continuum many types [@KeCh; @Spr]:
\(1) the theory ${\rm Th}(\langle\mathbb N;+,\cdot\rangle)$ of the standard model of arithmetic on naturals (for any subset $A$ of the set $P$ of all prime numbers, the set $\Phi(x)$ of formulas describing the divisibility of an element by a number in $A$ and its non-divisibility by each number in $P\setminus A$ is consistent);
\(2) the theory ${\rm Th}(\langle\mathbb Z;+,0\rangle)$ (there are continuum many $1$-types by the same reason as in the previous example);
\(3) the theory ${\rm Th}(\langle\mathbb Q;+,\cdot,\leq\rangle)$ of ordered fields (there are $2^\omega$ cuts for the set of rationals);
\(4) the theory $T_{\rm sdup}$ of a countable set of [*sequentially divisible unary predicates*]{} $S^{(1)}_\delta$, $\delta\in 2^{<\omega}$, with the following axioms:
$$\exists^{\geqslant \omega}x \left(S_{\overline{\delta}}(x)\wedge \neg S_{\overline{\delta} \textrm{\^{}}0}(x)
\wedge \neg S_{\overline{\delta} \textrm{\^{}}1}(x\right));$$ $$S_{\overline{\delta} \textrm{\^{}} {\varepsilon}}(x) \rightarrow
S_{\overline{\delta}}(x),\,\,\varepsilon\in\{0,1\};$$ $$\neg \exists x (S_{\overline{\delta} \textrm{\^{}} {0}}(x) \wedge S_{\overline{\delta} \textrm{\^{}} {1}}(x));$$
\(5) the theory $T_{\rm iup}$ of a countable set of [*independent unary predicates*]{} $P^{(1)}_k$, $k\in\omega$, axiomatizable by formulas: $$\exists x\,(P_{i_1}(x)\wedge\ldots\wedge P_{i_m}(x)\wedge\neg
P_{j_1}(x)\wedge\ldots\wedge\neg P_{j_n}(x)),$$ $\{i_1,\ldots,i_m\}\cap\{j_1,\ldots,j_n\}=\varnothing$ (one get continuum many $1$-types by consistency of any set of formulas $\{P^{\delta(k)}_k(x)\mid k\in\omega\}$, $\delta\in 2^\omega$);
(6)[^4] the theory $T_{\rm
ersiup}$ of a countable set of [*sequentially independent unary predicates $P^{(1)}_k$, $k\in\omega$, with an equivalence relation*]{} $E^{(2)}$, defined by the following axioms:
\(a) there are infinitely many $E$-classes and each $E$-class is infinite;
\(b) for any $k\in\omega$ there is unique $E$-class $X_k$ containing infinitely many solutions of each formula $P^{\delta_0}_{0}(x)\wedge\ldots\wedge P^{\delta_k}_{k}(x)$, $\delta_0,\ldots,\delta_k\in\{0,1\}$, and $X$ is disjoint with relations $P_i$, $i>k$; there is a prime model consisting of $E$-classes $X_k$, $k\in\omega$;
one get continuum many $1$-types in $E$-classes having nonempty intersections with each predicate $P_k$, $k\in\omega$;
\(7) the theory $T_{\rm sier}$ of a countable set of [*sequentially independent equivalence relations*]{} $E^{(2)}_n$, $n\in\omega$, with the following axioms:
\(a) $\vdash E_{n+1}(x,y)\to E_0(x,y)$, $n\in\omega$;
\(b) $\models\forall x,y(E_0(x,y)\to\exists z(E_m(x,z)\wedge
E_n(z,y)))$, $m\ne n$;
\(c) each $E_0$-class is infinite and each $E_{n+1}$-class is a singleton or infinite, $n\in\omega$;
\(d) if an $E_{n+1}$-class $X$ is contained in an $E_0$-class $Y$ then $Y$ consists of infinitely many $E_{n+1}$-classes, each of which is a singleton or infinite, $n\in\omega$;
\(e) if $X_{n+1}$ is an infinite $E_{n+1}$-class contained in an $E_0$-class $Y$ then $Y$ is represented as a union of infinite intersections $X_1\cap\ldots\cap X_n\cap X_{n+1}$ for $E_i$-classes $X_i$, $1\leq i\leq n$; moreover, for any $\delta_i\in\{0,1\}$ the sets $X^{\delta_1}_1\cap\ldots\cap
X^{\delta_n}_n\cap X^{\delta_{n+1}}_{n+1}\cap Y$ are infinite, $n\in\omega$;
\(f) for any $n\in\omega$ there is unique $E_0$-class containing infinite $E_1\mbox{-},\ldots,E_n$-class and one-element $E_m$-classes, $n<m$; there is a prime model consisting of these $E_0$-classes;
there are continuum many $2$-types in $E_0$-classes containing infinite $E_{n+1}$-classes, $n\in\omega$.
The structures $\langle\mathbb N;+,\cdot\rangle$ and $\langle\mathbb Q;+,\cdot,\leq\rangle$ are prime (since the universes of there structures equal to ${\rm dcl}(\varnothing)$), the structure $\langle\mathbb Z;+,0\rangle$ is prime over each its nonzero element (but it is not prime over $\varnothing$).
The theory $T_{\rm sdup}$ has a prime model and this model omits the type $p_\infty(x)$ deduced from the set of formulas describing the unbounded divisibility of $S_{\overline{\delta}}(x)$ by $S_{\overline{\delta}\textrm{\^{}} {\varepsilon}}(x)$. Moreover, the theory $T_{\rm sdup}$ has a prime model over every finite set, whence there are continuum many pairwise non-isomorphic prime models over tuples.
The theory $T_{\rm iup}$ does not have prime models over finite sets. The theories $T_{\rm ersiup}$ and $T_{\rm sier}$, having prime models over empty set, do not have prime models over non-principal types.
**2. Rudin–Keisler preorders**
Consider a theory $T\in\mathcal{T}_c$, a type $p\in S(T)$ and its realization $\bar{a}$. It is known that all prime models over realizations of $p$ are isomorphic. So if there is a [*prime model*]{} ${\cal M}(\bar{a})$ $\bar{a}$,, this model will be usually denoted by ${\cal M}_p$.
A consistent formula $\varphi(\bar{x})$ of $T$ belonging to an isolated type in $S(T)$ is called an [*$i$-formula*]{}, and if $\varphi(\bar{x})$ does not belong to isolated types in $S(T)$ then $\varphi(\bar{x})$ is a [*${\rm ni}$-formula*]{}.
Recall [@Va] that the prime model of $T$ exists if and only if every formula being consistent with $T$ is an $i$-formula.
Note that an expansion of any countable structure ${\cal M}$ by constants for each element transforms this structures to a prime one. Hence the property of absence of a prime model for a theory is not preserved under expansions of a theory. Clearly, this property is not also preserved under restrictions of a theory.
Let $p=p(\bar{x})$ and $q=q(\bar{y})$ be types in $S(T)$. Following [@SuLP; @Su041; @Su111] we say that $p$ *is dominated by a type* $q$, or $p$ *does not exceed $q$ under the Rudin–Keisler preorder* (written $p\leq_{\rm RK}
q$), if any model ${\cal M}\models T$ realizing $q$ realizes $p$ too.
By Omitting Types Theorem the condition $p\leq_{\rm RK}q$ can be syntactically characterized by the following: there is a [*$(q,p)$-formula*]{}, i. e., a formula $\varphi(\bar{x},\bar{y})$ such that the set $q(\bar{y})\cup\{\varphi(\bar{x},\bar{y})\}$ is consistent and $q(\bar{y})\cup\{\varphi(\bar{x},\bar{y})\}\vdash p(\bar{x})$. Herewith, in contrast small theories, a principal formula $\varphi(\bar{x},\bar{b})$ with the conditions specified, where $\models q(\bar{b})$, may not exist. If a principal formula $\varphi(\bar{x},\bar{b})$ of that form exists, the $(q,p)$-formula $\varphi(\bar{x},\bar{y})$ is called *$(q,p)$-principal*.
If $p\leq_{\rm RK}q$ and the models ${\cal M}_p$ and ${\cal M}_q$ exist, we say also that ${\cal M}_p$ *is dominated by* ${\cal M}_q$, or ${\cal M}_p$ *does not exceed ${\cal M}_q$ under the Rudin–Keisler preorder*, and write ${\cal M}_p\leq_{\rm RK}{\cal
M}_q$.
If the models ${\cal M}_p$ and ${\cal M}_q$ exist, the condition ${\cal M}_p\leq_{\rm RK}{\cal M}_q$ means that ${\cal M}_q\models
p$, i. e., some copy ${\cal M}'_p$ of ${\cal M}_p$ is an elementary submodel of ${\cal M}_q$: ${\cal M}'_p\preceq{\cal
M}_q$.
If the model ${\cal M}_q$ exists then the condition $p\leq_{\rm
RK} q$ implies an existence of $(q,p)$-principal formula, but not vice versa. Clearly, there is a theory $T$ with types $p$ and $q$ such that $p\leq_{\rm RK} q$, there is a $(q,p)$-principal formula, and the model ${\cal M}_q$ does not exists (it suffices to take the theory $T_{iup}$ and $1$-types $p$ and $q$ with $p=q$).
Obviously, no formula $\varphi(\bar{x},\bar{y})$ can not be both a $(q,p)$-formula and a $(q,p')$-formula for $p\ne p'$. At the same time, a fixed formula can be a $(q,p)$-formula even for continuum many types $q$.
A simplest example of that effect is given by an arbitrary principal formula $\varphi(\bar{x})$ forming a domination for a correspondent principal type by all types of given theory. Every (non)principal type $p(x)\in S(T)$ is dominated by an arbitrary type $q(\bar{y})\in S(T)$ containing the type $p(y_i)$ and it is witnessed by the formula $(x\approx y_i)$.
The following example illustrates the mechanism of the domination for a type by continuum many types in a situation different from the above.
Consider a disjunctive union of countable unary predicates $R_0$ and $R_1$ forming a universe of required structure. Define a coloring ${\rm Col}\mbox{\rm : }R_0\cup
R_1\to\omega\cup\{\infty\}$ with infinitely many elements for each color in each predicate $R_0, R_1$. Define a bipartite acyclic directed graph with a relation $Q$ linking parts $R_0$ and $R_1$ and satisfying the following conditions:
${\small\bullet}$ every element $a\in R_1$ of color $m\in\omega$ has infinitely many elements $b\in R_0$ of each color $n\geq m$ such that $(a,b)\in Q$ and there are no elements $c\in R_0$ with $(a,c)\in Q$ and ${\rm Col}(c)<m$;
${\small\bullet}$ every element $a\in R_0$ of color $m\in\omega$ has infinitely many elements $b\in R_1$ of each color $n\leq m$ such that $(b,a)\in Q$ and there are no elements $c\in R_1$ with $(c,a)\in Q$ and ${\rm Col}(c)>m$.
By the construction, for $1$-types $p_i$, isolated by sets $\{R_i(x)\}\cup\{\neg{\rm Col}_n(x)\mid n\in\omega\}$, $i=0,1$, we have $p_0\leq_{\rm RK}p_1$ (witnessed by the formula $Q(x,y)$) and $p_1\not\leq_{\rm RK}p_0$.
That structure is denoted by ${\cal M}_{01}$ and its theory by $T_{01}$. Expand the structure ${\cal M}_{01}$ by independent unary predicates $P_k$, $k\in\omega$, on each set defined by the formula $R_1(x)\wedge{\rm Col}_n(x)$, $n\in\omega$, such that the type $p_0$ preserves the completeness. Then the type $p_1(x)$ has continuum many completions $q(x)$, each of which dominates the type $p_0(x)$ by the formula $Q(x,y)$.
A modification of the example with the theory $T_{\rm sdup}$ instead of $T_{\rm uip}$ leads to the theory for which the formula $Q(x,y)$ produces the domination of the model ${\cal M}_{p_0}$ to continuum many models ${\cal M}_q$, where all types $q$ are completions of the type $p_0$ in $S^1(T_{01})$. $\Box$
Types $p$ and $q$ are said to be *domination-equivalent*, *realization-equivalent*, *Rudin–Keisler equivalent*, or *${\rm RK}$-equivalent* (written $p\sim_{\rm RK} q$) if $p\leq_{\rm RK} q$ and $q\leq_{\rm RK} p$. If $p\sim_{\rm RK} q$ and the models ${\cal M}_p$ and ${\cal M}_q$ exist then ${\cal M}_p$ and ${\cal M}_q$ are also said to be *domination-equivalent*, *Rudin–Keisler equivalent*, or *${\rm RK}$-equivalent* (written ${\cal M}_p\sim_{\rm RK}{\cal
M}_q$).
As in [@Ta4], types $p$ and $q$ are said to be *strongly domination-equivalent*, *strongly realization-equivalent*, *strongly Rudin–Keisler equivalent*, or *strongly ${\rm RK}$-equivalent* (written $p\equiv_{\rm
RK}q$) if for some realizations $\bar{a}$ and $\bar{b}$ of $p$ and $q$ accordingly both ${\rm
tp}(\bar{b}/\bar{a})$ and ${\rm tp}(\bar{a}/\bar{b})$ are principal. Moreover, If the models ${\cal M}_p$ and ${\cal M}_q$ exist, they are said to be *strongly domination-equivalent*, *strongly Rudin–Keisler equivalent*, or *strongly ${\rm RK}$-equivalent* (written ${\cal M}_p\equiv_{\rm RK}{\cal
M}_q$).
Clearly, domination relations form preorders, (strong) domination-equivalence relations are equivalence relations, and $p\equiv_{\rm RK}q$ implies $p\sim_{\rm RK}q$.
If ${\cal M}_p$ and ${\cal M}_q$ are not domination-equivalent then they are non-isomorphic. Moreover, non-isomorphic models may be found among domination-equivalent ones.
Repeating the proof [@Su111 Proposition 1] we get a syntactic characterization for an isomorphism of models ${\cal M}_p$ and ${\cal M}_q$. It asserts, as for small theories, that an existence of isomorphism between ${\cal M}_p$ and ${\cal M}_q$ is equivalent to the strong domination-equivalence of these models.
Denote by ${\rm RK}(T)$ the set ${\bf
P}$ of isomorphism types of models ${\cal M}_p$, $p\in S(T)$, on which the relation of domination is induced by $\leq_{\rm RK}$ for models ${\cal M}_p$: ${\rm RK}(T)=\langle{\bf
P};\leq_{\rm RK}\rangle$. We say that isomorphism types ${\bf
M}_1,{\bf M}_2\in{\bf P}$ are *domination-equivalent* (written ${\bf M}_1\sim_{\rm RK}{\bf
M}_2$) if so are their representatives.
We consider also the relation $\leq_{\rm RK}$, being defined on the set $S(T)$ of complete types of a theory $T$. Denote the structure $\langle S(T);\leq_{\rm RK}\rangle$ by ${\rm
RKT}(T)$.
Below we investigate links and properties of preordered sets ${\rm
RK}(T)$ and ${\rm RKT}(T)$ as well as links of arbitrary countable models of a theory with continuum many types.
The following assertion proposes criteria for the existence of the least element in ${\rm RK}(T)$.
PROOF. The equivalence $(1)\Leftrightarrow(2)$ forms a criterion for the existence of prime model of a theory [@Va]. The implications $(1)\Rightarrow(3)$ and $(3)\Rightarrow(2)$ are obvious. $\Box$
Since theories with continuum many types may not have prime models over tuples, the limits models may not exist too. Nevertheless the links between countable models can be observed by the following generalization of Rudin–Keisler preorder on isomorphism types of countable models that will be also denoted by $\leq_{\rm
RK}$. This generalization extends the preorder $\leq_{\rm RK}$ for isomorphism types of prime models over tuples and is based on the inclusion relation for finite diagrams ${\rm FD}(\mathcal{M})$.
Let ${\bf M}_1$ and ${\bf M}_2$ be isomorphism types of models $\mathcal{M}_1$ and $\mathcal{M}_2$ (of $T$) respectively. We say that ${\bf M}_1$ *is dominated by* ${\bf M}_2$ and write ${\bf M}_1\leq_{\rm
RK}{\bf M}_1$ if each type in $S^1(\varnothing)$, being realized in ${\bf M}_1$, is realized in ${\bf M}_2$: ${\rm FD}(\mathcal{M}_1)\subseteq{\rm
FD}(\mathcal{M}_2)$.
Since the relation $\leq_{\rm RK}$ does not depend on representatives $\mathcal{M}_1$ and $\mathcal{M}_2$ of isomorphism types ${\bf M}_1$ and ${\bf M}_2$, we shall also write $\mathcal{M}_1\leq_{\rm
RK}\mathcal{M}_2$ for the representatives $\mathcal{M}_1$ and $\mathcal{M}_2$ if ${\bf M}_1\leq_{\rm RK}{\bf M}_2$.
We denote by ${\rm CM}(T)$ the set ${\bf CM}$ of isomorphism types of countable models of $T$, equipped with the preorder $\leq_{\rm RK}$ of domination on this set: ${\rm
CM}(T)=\langle{\bf CM};\leq_{\rm RK}\rangle$.
Clearly, ${\rm RK}(T)\subseteq{\rm CM}(T)$. Since having non-principal types of a countable theory, there is a model of this theory being not represented in ${\rm RK}(T)$, the equality ${\rm RK}(T)={\rm CM}(T)$ is equivalent to the $\omega$-categoricity of $T$.
By the definition, a prime model over a type and a limit model over that type, being non-isomorphic, are domination-equivalent. Whence any limit models over a common type are also domination-equivalent.
The generalized relation of domination leads to a classification of countable models of an arbitrary theory of unary predicates [@Pop].
As we pointed out, a series of examples shows that, unlike small theories, for theories with continuum many types the relations of domination may not induce least elements (being isomorphism types of prime models). Besides, by the following example, isomorphism types of prime models over tuples can quite freely alternate with the other isomorphism types of countable models.
We consider a disjunctive union of countable unary predicates $R_0$ and $R_1$ forming the universe of required structure. We define a coloring ${\rm Col}\mbox{\rm :
}R_0\to\omega\cup\{\infty\}$ with infinitely many elements for each color. On the set $R_1$, we put a structure of independent unary predicates $P_k$, $k\in\omega$. We denote by $T_0$ the complete theory of the described structure.
Now we fix a dense (in the natural topology) set $X=\{q_m\mid
m\in\omega\}$ of $1$-types containing the formula $R_1(y)$. Using binary predicates $Q_m$, $m\in\omega$, the type $p_\infty(x)$, being isolated by the set $\{R_0(x)\wedge\neg{\rm Col}_n(x)\mid
n\in\omega\}$, and neighbourhoods $R_0(x)\wedge\bigwedge\limits_{i=0}^n\neg{\rm Col}_i(x)$ of $p_\infty(x)$, we get, in the expanded language, that all types in $X$ are approximated so that, if the type $p_\infty(x)$ is realized in a model ${\cal M}$ of expanded theory, then the type $q_m(y)$ is realized in ${\cal M}$ by the principal formula $Q_m(a,y)$, where $\models p_\infty(a)$ and $Q_m(a,y)\vdash
q_m(y)$, $m\in\omega$, and the realizability in a model of some types in $X$ does not imply the realizability of $p_\infty(x)$ in that model. Thus, a prime model over $p_\infty$ dominates a prime model over a set $A$, where $A$ consists of realizations of types in $X$ (one realization of each type).
In turn, the model ${\cal M}_{p_\infty}$ is dominated by a countable model (being not prime over tuples) which contains a realization of $p_\infty$ (with realizations of types in $X$) and a countable set of realizations of $1$-types consistent with $R_1(x)$ and not belonging to $X$. $\Box$
Using the notion of dense set of types for the theory $T_{\rm
iup}$ (without the predicate $R_1$) one describes (see [@Pop]) the preordered, with respect to $\leq_{\rm RK}$, set ${\bf M}$ of isomorphism types of countable models of $T_{\rm iup}$. Each countable model is defined by some countable set of realizations of a dense set. A model ${\cal M}_1$ is dominated by a model ${\cal M}_2$ if and only if each $1$-type $p$, realized in ${\cal
M}_1$, is realized in ${\cal M}_2$ and the number of realizations of $p$ in ${\cal M}_1$ does not exceed the number of realizations of $p$ in ${\cal M}_2$. Since the density of set of types is preserved under arbitrary removing or adding of a $1$-type, the set ${\bf M}$ does not have minimal and maximal elements.
Example 2.2 illustrates that the absence of prime model of a theory can be combined with the presence of a prime model over a tuple. At the same time, as the following proposition asserts, having a ${\rm ni}$-formula no prime model can not be dominated by all countable models of theory.
PROOF. By Omitting Type Theorem, there is a countable model ${\cal
M}$ of $T$ omitting the type $p(\bar{y})$. At the same time, by consistency of $\varphi(\bar{x})$ there is a tuple $\bar{a}$ such that ${\cal M}\models\varphi(\bar{a})$. The type $q(\bar{x})\rightleftharpoons{\rm tp}(\bar{a})$, contains the formula $\varphi(\bar{x})$ and, by the definition, does not dominate the type $p(\bar{y})$. $\Box$
Since each consistent conjunction of ${\rm ni}$-formula $\varphi(\bar{x})$ and a formula $\psi(\bar{x})$ is again a ${\rm
ni}$-formula, there are infinitely many types $q(\bar{x})\in S(T)$ containing the formula $\varphi(\bar{x})$ and do not dominating the type $p(\bar{y})$. Moreover, in a series of examples of $T$ like above, there are uncountably many these types since otherwise there is a countable expansion $T'$ of $T$ with new predicates $Q_n(\bar{x},\bar{y})$, $n\in\omega$, producing the isolation of each type $r(\bar{x})\in S(T')$, containing $\varphi(\bar{x})$, by its restriction to the language of $T$, and the domination of $p(\bar{y})$ by each type $q(\bar{x})$. Since the formula $\varphi(\bar{x})$ is again a ${\rm ni}$-formula, we get a contradiction by Proposition 2.4.
Note that if a type $p(\bar{y})$ is not dominated by a type $q(\bar{x})$ then, introducing new independent predicates $P_k(\bar{x})$, $k\in\omega$, transforming a neighbourhood of $q(\bar{x})$ to a ${\rm ni}$-formula and $q(\bar{x})$ to $2^\omega$ completions, we get a theory such that $p(\bar{x})$ is not dominated by continuum many types. By a similar way, as in Example 2.1, if a type $p(\bar{y})$ is dominated by a type $q(\bar{x})$ then, in an expansion, the type $p(\bar{y})$ is dominated by continuum many completions of $q(\bar{x})$.
Note also that a structure ${\rm RKT}(T)$ can have a minimal but not least $\sim_{\rm RK}$-class. Indeed, expanding the theory $T_{\rm iup}$ by binary predicates, one can obtain a dense set $S$ of $1$-types, each of which is domination-equivalent with the other, and the absence of prime model is preserved (it can be done by a countable set of new binary predicates, each of which is responsible for the domination-equivalence of two $1$-types in the given dense set, and this domination-equivalence is obtained by approximations for neighbourhoods of given types). The set $S$ and types, domination-equivalent to types in $S$, form a minimal $\sim_{\rm RK}$-class. By similar expansions, one get countably many minimal classes.
Together with Example 2.2 and Proposition 2.4, Example 2.1 illustrate a mechanism of domination of a non-principal type by all non-principal types of a theory with continuum many types and without ${\rm ni}$-formulas.
Having the features, in the following section, we propose a list of some basic properties of structures ${\rm RKT}(T)$ for theories $T$ in $\mathcal{T}_c$.[^5]
**3. Premodel sets**
A [*height*]{} ([*width*]{}) of preordered set $\langle
X;\leq\rangle$ is a supremum of cardinalities for its $\leq$-(anti)chains consisting of pairwise non-$\sim$-equivalent elements, where $\sim\,\,\rightleftharpoons(\leq\cap\geq)$. Recall [@SuLP], that if $a\in X$ then the set $\bigtriangleup(a)$ (respectively $\bigtriangledown(a)$) of all elements $x$ in $X$, for which $x\leq a$ ($a\leq x$), is a [*lower [(]{}upper[)]{} cone*]{} of $a$.
A continual preordered upward directed set $\langle
X;\leq$ $\rangle$ is called [*premodel*]{} if it has:
${\small\bullet}$ countably many elements under each element $a\in
X$ (i. e., $|\bigtriangleup(a)|=\omega$);
${\small\bullet}$ only countable $\sim$-classes (i. e., $|\bigtriangleup(a)\cap\bigtriangledown(a)|=\omega$ for any $a\in
X$);
${\small\bullet}$ countable, or continual and coinciding with $X$, co-countable, or co-continual set of common elements over any elements $a_1,\ldots,a_n\in X$ (i. e., $|\bigtriangledown(a_1)\cap\ldots\cap\bigtriangledown(a_n)|=\omega$, or $|\bigtriangledown(a_1)\cap\ldots\cap\bigtriangledown(a_n)|=2^\omega$ and $\bigtriangledown(a_1)\cap\ldots\cap\bigtriangledown(a_n)=X$, $|X\setminus(\bigtriangledown(a_1)\cap\ldots\cap\bigtriangledown(a_n))|=\omega$, or $|X\setminus(\bigtriangledown(a_1)\cap\ldots\cap\bigtriangledown(a_n))|=2^\omega$);
${\small\bullet}$ the countable height.
PROOF. The structure ${\rm RKT}(T)$ is upward directed since types $p(\bar{x})$, $q(\bar{y})\in S(T)$, where $\bar{x}$ and $\bar{y}$ are disjoint, are dominated by any type $r(\bar{x},\bar{y})\supset
p(\bar{x})\cup q(\bar{y})$ in $S(T)$.
As $T$ is countable, the set of formulas of $T$ is also countable and each type dominates at most countably many types. Having countably many types, being domination-equivalent with a given type (for instance, a type ${\rm tp}(\bar{a})$ is domination-equivalent with types ${\rm
tp}(\bar{a}\,\hat{\,}\,\bar{a})$, ${\rm
tp}(\bar{a}\,\hat{\,}\,\bar{a}\,\hat{\,}\,\bar{a}),\ldots$), we get that any type is domination-equivalent with countably many types of $T$.
Since each formula witnesses on domination of a type to at most countably many, or continuum and co-continuum many types, and there are countably many formulas of $T$, then any types $p_1,\ldots,p_n$ lay under countably many, or continuum many and coinciding with $S(T)$, co-countably many, or co-continuum many types.
As each type dominates countably many types, the height of ${\rm
RKT}(T)$ is at most countable. At the same time the height can not be finite since its finiteness, the upward direction of ${\rm
RKT}(T)$, and the countable domination imply that ${\rm RKT}(T)$ is countable in spite of $|S(T)|=2^\omega$. $\Box$
Since each $\sim_{\rm RK}$-class of a countable theory $T$ is countable and each type dominates countably many types, the ordered factor set ${\rm RKT}(T)/\!\!\sim_{\rm RK}$ can be linearly ordered only for small $T$. Moreover, as the height of ${\rm RKT}(T)$ is countable for $T\in\mathcal{T}_c$, this factor-set has continuum many incomparable elements, i. e., the width is continual:
PROOF. Assume the contrary that the width of a preordered set $\langle X;\leq$ $\rangle$ is not continual. Consider a maximal antichain $Y$. By the assumption, we have $|Y|=\lambda<2^\omega$. We link each element $y\in Y$ with a maximal chain $C_y$. Each chain $C_y$ is countable since the height is countable and each $\sim$-class is countable too. Now we note that $X=\bigcup\{\bigtriangleup(c)\mid c\in C_y,y\in Y\}$ since $\langle X;\leq$ $\rangle$ is upward directed. Then, as each lower cone $\bigtriangleup(c)$ is countable, we obtain $|X|\leq\lambda\cdot\omega\cdot\omega<2^\omega$ that contradicts the condition $|X|=2^\omega$. $\Box$
**4. Distributions for countable models of a theory**
**by $\leq_{\rm RK}$-sequences**
Recall that, by Tarski–Vaught criterion, a set $A$ in a structure ${\cal M}$ of language $\Sigma$ forms an elementary substructure if and only if for any formula $\varphi(x_0,x_1,\ldots,x_n)$ of the language $\Sigma$ and for any elements $a_1,\ldots,a_n\in A$ if ${\cal M}\models\exists
x_0\,\varphi(x_0,a_1,\ldots,a_n)$ then there is an element $a_0\in
A$ such that ${\cal M}\models\varphi(a_0,a_1,\ldots,a_n)$. It means that each formula $\varphi(\bar{x})$ over a finite set $A_0\subseteq A$ and belonging to a type over $A_0$ has a realization $\bar{a}\in A$.
Let ${\cal M}$ be a model of a countable theory $T$ and ${\bf
q}\rightleftharpoons(q_n)_{n\in\omega}$ be a [*$\leq_{\rm
RK}$-sequence*]{} of types of $T$, i. e., a sequence of non-principal types $q_n$ with $q_n\leq_{\rm
RK}q_{n+1}$, $n\in\omega$. We denote by $U({\cal M},{\bf
q})$ the set of all realizations in ${\cal M}$ of types of $T$, being dominated by some types in ${\bf
q}$. The $\leq_{\rm RK}$-sequence ${\bf q}$ is called [*elementary submodel*]{} if for any consistent formula $\varphi(\bar{y})$ of $T$ some type in ${\bf q}$ dominates a type $p(\bar{y})\in S(T)$ containing the formula $\varphi(\bar{y})$, and if the formula $\varphi(\bar{y})$ is equal to $\exists x\,\psi(x,\bar{y})$ then the type $p(\bar{y})$ is extensible to a type $p'(x,\bar{y})\in
S(T)$ dominated by a type in ${\bf q}$ and such that $\psi(x,\bar{y})\in p'$.
PROOF. $(1)\Rightarrow (2)$ is implied by Tarski–Vaught criterion.
$(2)\Rightarrow (1)$. Let ${\bf q}$ be an elementary submodel $\leq_{\rm RK}$-sequence. Using elements of $U({\cal M},{\bf q})$ we construct, by induction, a countable elementary submodel of ${\cal M}$. On initial step we enumerate, by natural numbers, all consistent, with $T$, formulas $\varphi(x,\bar{y})$ such that the enumeration $\nu$ starts with some formula $\varphi_0(x)$ and each formula has infinitely many numbers. We choose a realization $a_0\in U({\cal M},{\bf q})$ of the formula $\varphi_0(x)$ and put $A_0\rightleftharpoons\{a_0\}$. Assume that, on step $n$, a finite set $A_n\subset U({\cal M},{\bf q})$ is defined, the type of this set is dominated by some type in ${\bf q}$, and all possible tuples of elements in $A_n$ are substituted in initially enumerated formulas $\varphi(x,\bar{y})$ instead of tuples $\bar{y}$ such that there are infinitely many numbers for each formula, where tuples of elements in $A_n$ are not substituted. We assume also that the results $(\varphi(x,\bar{y}))^{\bar{y}}_{\bar{a}}$ of substitutions have the same numbers as before, a substitution is carried out for the formula with the number $n+1$, and this formula has the form $\varphi(x,\bar{a})$. If ${\cal M}\models\neg\exists
x\varphi(x,\bar{a})$, we put $A_{n+1}\rightleftharpoons A_n$. If ${\cal M}\models\exists x\varphi(x,\bar{a})$, we add fictitiously to the tuple $\bar{a}$ all missing elements of $A_n$ and choose an existing, by conjecture, type $p'(x,\bar{y})$ extending the type $p(\bar{y})={\rm tp}(A_n)$, where $\varphi(x,\bar{y})\in p'$ and the types $p$, $p'$ are dominated by some types in ${\bf q}$. We take for $a_{n+1}$ a realization in $U({\cal M},{\bf q})$ of the type $p'(x,A_n)$ (that exists since the model ${\cal M}$ is $\omega$-homogeneous) and put $A_{n+1}\rightleftharpoons
A_n\cup\{a_{n+1}\}$.
It is easy to see, using a mechanism of consistency [@ErPa], that $\bigcup\limits_{n\in\omega}A_n$ is a universe of required elementary submodel of ${\cal M}$. $\Box$
Since every $\omega$-saturated structure is $\omega$-homogeneous, Theorem 4.1 implies
Note that, in the proof of Theorem 4.1, we essentially use that the model $\mathcal{M}$ is $\omega$-homogeneous and all types of the sequence ${\bf q}$ are realized in $\mathcal{M}$. Possibly the types of a $\leq_{\rm RK}$-sequence ${\bf q}$ are not realized in an $\omega$-homogeneous model $\mathcal{M}$ but are realized in in some other $\omega$-homogeneous model $\mathcal{M}'$, where Theorem 4.1 can be applied.
Consider the theory $T_{\rm iup}$. By Theorem 4.1, each countable model of $T_{\rm iup}$ realizes a dense set $X$ of $1$-types (where $\bigcup X$ contains all formulas $$P_{i_1}(x)\wedge\ldots\wedge P_{i_m}(x)\wedge\neg
P_{j_1}(x)\wedge\ldots\wedge\neg P_{j_n}(x),$$ $\{i_1,\ldots,i_m\}\cap\{j_1,\ldots,j_n\}=\varnothing$) and vice versa, for each countable dense set $X$ of types, there is an ($\omega$-homogeneous) model of $T_{\rm iup}$ such that the set of types of elements equals to $X$.
Take two countable disjoint dense sets $P_0$ and $P_1$ of $1$-types, and $\omega$-homogeneous models $\mathcal{M}_0$ and $\mathcal{M}_1$ containing exactly one realization of each type in $P_0$ and $P_1$ respectively. Then there are $\leq_{\rm
RK}$-sequences ${\bf q}_i$ of types with realizations from given sets of realizations of types in $P_i$, $i=0,1$. Here, all types in ${\bf q}_i$ are realized $\mathcal{M}_i$ and are omitted in $\mathcal{M}_{1-i}$, $i=0,1$. $\Box$
By Theorem 4.1, each elementary submodel $\leq_{\rm RK}$-sequence ${\bf q}$ corresponds to some set of isomorphism types of countable models of a theory $T$, which can vary from $1$ to $2^\omega$. We denote this set by $I^{m}_{\bf
q}(T)$.
The sets $I^m_{\bf q}(T)$ can have nonempty intersections (for instance, having a prime model ${\cal M}_0$ its isomorphism type belongs to each set $I^m_{\bf q}(T)$) and can be disjoint (as in Example 4.1).
Distributing isomorphism types of countable model to pairwise disjoint sets, related to $\leq_{\rm RK}$-sequences ${\bf q}$ (and not related to the other $\leq_{\rm RK}$-sequences) and denoting the cardinalities of these sets by $I_{\bf q}$, we have the equality $$I(T,\omega)=\sum\limits_{\bf q}I_{\bf
q}=2^\omega.$$
**5. Three classes of countable models**
Recall [@SuLP; @Su08] that a model ${\cal M}$ of a theory $T$ is called [*limit*]{} if ${\cal M}$ is not prime over tuples and ${\cal M}=\bigcup\limits_{n\in\omega}{\cal
M}_n$ for some elementary chain $({\cal M}_n)_{n\in\omega}$ of prime models of $T$ over tuples. In this case the model ${\cal M}$ is said to be [*limit over a sequence ${\bf q}$ of types*]{}, where ${\bf
q}=(q_n)_{n\in\omega}$, ${\cal M}_n={\cal M}_{q_n}$, $n\in\omega$. If a cofinite subset of the set of types $q_n$ is a singleton containing a type $p$ then the limit model over ${\bf q}$ is said to be [*limit over the type*]{} $p$.
Consider a countable complete theory $T$. Denote by ${\bf
P}\rightleftharpoons{\bf P}(T)$, ${\bf L}\rightleftharpoons{\bf L}(T)$, and ${\bf NPL}\rightleftharpoons{\bf
NPL}(T)$ respectively the set of prime over tuples, limit, and other countable models of $T$, and by $P(T)$, $L(T)$, and ${\rm
NPL}(T)$ the cardinalities of these sets.
By the definition, each value $P(T)$, $L(T)$, and ${\rm NPL}(T)$ may vary from $0$ to $2^\omega$ and the following equality holds: $$I(T,\omega)=P(T)+L(T)+{\rm NPL}(T).$$ Since $I(T,\omega)=2^\omega$ for theories $T$ in $\mathcal{T}_c$, some value $P(T)$, $L(T)$, or ${\rm NPL}(T)$ is equal to $2^\omega$.
The tuple $(P(T),L(T),{\rm NPL}(T))$ is called a [*triple of distribution of countable models of $T$*]{} and is denoted by ${\rm cm}_3(T)$.
A theory $T$ is called [*$p$-zero*]{} (respectively [*$l$-zero*]{}, [*${\rm
npl}$-zero*]{}) if $P(T)=0$ (respectively $L(T)=0$, ${\rm NPL}(T)=0$).
A theory $T$ is called [*$p$-categorical*]{} (respectively [*$l$-categorical*]{}, [*${\rm
npl}$-categorical*]{}) if $P(T)=1$ (respectively $L(T)=1$, ${\rm NPL}(T)=1$).
A theory $T$ is called [*$p$-Ehrenfeucht*]{} (respectively [*$l$-Ehrenfeucht*]{}, [*${\rm
npl}$-Ehrenfeucht*]{}) if $1<P(T)<\omega$ (respectively $1<L(T)<\omega$, $1<{\rm
NPL}(T)<\omega$).
A theory $T$ is called [*$p$-countable*]{} (respectively [*$l$-countable*]{}, [*${\rm
npl}$-countable*]{}) if $P(T)=\omega$ (respectively $L(T)=\omega$, ${\rm NPL}(T)=\omega$).
A theory $T$ is called [*$p$-continual*]{} (respectively [*$l$-continual*]{}, [*${\rm
npl}$-continual*]{}) if $P(T)=2^\omega$ (respectively $L(T)=2^\omega$, ${\rm
NPL}(T)=2^\omega$).
By the definition, each $p$-zero theory is $l$-zero.
Recall [@SuLP; @Su08] that the $p$-categoricity of a small theory $T$ is equivalent to its countable categoricity as well as to the absence of limit models. The $p$-Ehrenfeuchtness of $T$ means that the structure ${\rm RK}(T)$ is finite and has at least two elements. The theory $T$ is Ehrenfeucht if and only if $T$ is $p$-Ehrenfeucht and $L(T)<\omega$. Besides every small theory is ${\rm npl}$-zero, i. e., each its countable model is prime over a tuple or is limit. Since by Vaught’s and Morley’s theorems [@Va; @Mo2], $I(T,\omega)\in(\omega\setminus\{0,2\})\cup\{\omega,\omega_1,2^\omega\}$ and for small theories $T$ the inequalities $1\leq P(T)\leq\omega$ hold, we have the following
As shown in [@SuLP; @Su08], all values, pointed out in Theorem 5.1 (for $\lambda_2\ne\omega_1$) have realizations in the class of small theories.
Similarly Theorem 5.1, for the classification of theories in the class $\mathcal{T}_c$, the problem arises for the description of all possible triples $(\lambda_1,\lambda_2,\lambda_3)$ realized by ${\rm cm}_3(T)$ for theories $T\in\mathcal{T}_c$.
Examples in Section 1 confirm the realizability of triples $(0,0,2^\omega)$ and $(2^\omega,2^\omega,0)$ in the class $\mathcal{T}_c$ (by the $p$-zero, ${\rm npl}$-continual theory $T_{\rm iup}$ and the $p$-continual, ${\rm npl}$-zero theory $T_{\rm sdup}$ respectively). Some fusion of theories $T_{\rm
iup}$ and $T_{\rm sdup}$ substantiates the realizability of triple $(2^\omega,2^\omega,2^\omega)$. E. A. Palyutin noted that the theory $T_{\rm ersiup}$ realizes the triple $(1,0,2^\omega)$. This triple is also realized by the theory $T_{\rm sier}$.
The following theorem produces a characterization for the class of ${\rm npl}$-zero theories.
PROOF. If for a tuple $\bar{b}\in M$ every consistent formula $\varphi(\bar{x},\bar{b})$ is an $i$-formula then there is a model ${\cal M}(\bar{b})\preccurlyeq {\cal M}$. Repeating the proof of Proposition 1.1.7 in [@SuLP] or of Proposition 4.1 in [@Su08] in respect that any tuple $\bar{a}$ is extensible to a tuple $\bar{b}$ of described form, we get a representation of ${\cal M}$ as a union of elementary chain of prime models over finite sets. Thus, ${\cal M}$ is prime over a finite set or limit.
If a tuple $\bar{a}\in M$ is not extensible to a tuple $\bar{b}\in
M$ such that each consistent formula $\varphi(\bar{x},\bar{b})$ is $i$-formula, then $\bar{a}$ is not contained in prime models over tuples, being elementary submodels of ${\cal M}$, whence the model ${\cal M}$ is neither prime over a tuple nor limit. $\Box$
Theorem 5.2 implies
Below we describe some families of triples $(\lambda_1,\lambda_2,\lambda_3)$ that cannot be realized by ${\rm
cm}_3(T)$, where $T\in\mathcal{T}_c$.
PROOF. (1) If $P(T)<2^\omega$ and ${\rm NPL}(T)<2^\omega$ then there are less than continuum many types that realized in models representing isomorphism types in the classes ${\bf P}(T)$ and ${\bf NPL}(T)$. Since each type, realized in a limit model, is also realized in a prime model over a tuple, there are continuum many types, being not realized in countable models of $T$, that is impossible.
\(2) Assume that ${\rm NPL}(T)<2^\omega$. Then there are $<2^\omega$ types in $S(T)$, over which prime models do not exist. Therefore, for any type $p\in S(T)$ there are continuum many types $q\in S(T)$ extending $p$ and having models $\mathcal{M}_q$. Since there are continuum many types $q$ and the model $\mathcal{M}_p$ is countable, then there are continuum many these non-domination-equivalent types $q$ dominating $p$ and not dominated by $p$. Whence, for any model $\mathcal{M}_p$ there are continuum many possibilities for elementary extensions by pairwise non-isomorphic models $\mathcal{M}_q$, being non-isomorphic to $\mathcal{M}_p$. Since the process of extension of models $\mathcal{M}_p$ by continuum many models $\mathcal{M}_q$ can be continued unboundedly many times, there are continuum many pairwise non-isomorphic limit models, i. e., $L(T)=2^\omega$. $\Box$
The following proposition gives a sufficient condition for the existence of continuum many prime models over finite sets in the assumption of uncountably many these models.
PROOF. Since there are uncountably many types $p(\bar{x})$, we have neighbourhoods $\chi_\delta(\bar{x})$ of these types, $\delta\in 2^{<\omega}$, each of which belongs to uncountably many given types $p(\bar{x})$ and satisfies the following conditions:
${\small\bullet}$ $\chi_\delta(\bar{x})\equiv(\chi_{\delta\,\hat{\,}\,0}(\bar{x})\vee\chi_{\delta\,\hat{\,}\,1}(\bar{x}))$;
${\small\bullet}$ $\models\neg\exists\bar{x}(\chi_{\delta\,\hat{\,}\,0}(\bar{x})\wedge\chi_{\delta\,\hat{\,}\,1}(\bar{x}))$.
For each sequence $\delta\in 2^\omega$, the local consistency implies the consistency of the set $\Phi_\delta(\bar{x})$ of formulas $\chi_{\delta\upharpoonright n}(\bar{x})$, $n\in\omega$. Whence there are continuum many types in $S^{l(\bar{x})}(\varnothing)$. Moreover, since the formulas $\psi$ can be chosen by $\varphi$ independently of realizations of types $p$, by compactness each set $\Phi_\delta(\bar{x})$ has a completion $q(\bar{x})\in S(\varnothing)$ such that for any consistent formula $\varphi(\bar{a},\bar{y})$, $\models
q(\bar{a})$, there is a principal formula $\psi(\bar{a},\bar{y})$ with $\psi(\bar{a},\bar{y})\vdash\varphi(\bar{a},\bar{y})$ and this formula does not depend of $q$ as well as it was independent of $p$. Thus, there is a model $\mathcal{M}_q$ and there are continuum many these models, i. e., $P(T)=2^\omega$. $\Box$
By Proposition 5.5, we have a partial solution of a variant of the Vaught problem, being formulated by E. A. Palyutin as the implication $P(T)>\omega\Rightarrow P(T)=2^\omega$. Namely, this implication is true for prime models over realizations of types $p$ having the specified, in the proposition, [*uniform choice property*]{} of formulas $\psi$ by formulas $\varphi$.
**6. Operators acting on a class of structures**
Consider a non-principal $1$-type $p_{\infty}(x)$ and formulas $\varphi_n(x) ~\in~ p_{\infty}(x)$, $n \in \omega$, such that $\varphi_0(x) = (x \approx x)$, $\vdash \varphi_{n+1}(x)
\rightarrow \varphi_{n}(x)$, $\{ \varphi_n(x) \mid n \in \omega \}
\vdash p_{\infty}(x)$. The formula ${\rm
Col}_n(x)\rightleftharpoons\varphi_n(x) \wedge \neg
\varphi_{n+1}(x)$ is the $n$-*th approximation* of $p_{\infty}(x)$ or the $n$-*th color*. Then the type $p_{\infty}(x)$ is isolated by the set $\{\neg {\rm Col}_n \mid n
\in \omega \}$ of formulas.
The *operator of continual partition* ${\rm icp}(\mathcal{A}, \mathcal{A}_0, Y,
\{R_i^{(2)}\}_{i \in \omega})$ takes for input:
\(1) a predicate structure $\mathcal{A}$;
\(2) a substructure $\mathcal{A}_0 \subset \mathcal{A}$, where its universe equals to an infinite set for solutions of a formula $\psi(x)$ in $\mathcal{A}$, the substructure generates unique non-principal 1-type $p_{\infty}(x)\in S(\varnothing)$ and $p_{\infty}(x)$ is realized in $\mathcal{A}_0$;
\(3) an infinite set $Y$ with $Y \cap A = \varnothing$;
\(4) a sequence $(R_i^{(2)})_{i\in\omega}$ of binary predicate symbols.
We assume that $A_0$ is the domain of predicates $R_i$, $Y$ is the range of, $\vdash R_i(x,y) \rightarrow R_0(x,y)$, $i
> 0$. The work of the operator is defined by the following schemes of formulas:
\(1) $\forall x \, \exists^{\infty}y({\rm Col}_0(x) \rightarrow
R_0(x,y))$;
\(2) $\forall x, x' \, (\neg (x \approx x') \rightarrow \neg
\exists y (R_0(x,y) \wedge R_0(x',y)))$, i. e., $R_0$-images of distinct element satisfying $\psi(x)$ are disjoint and an equivalence relation on $Y$ with infinitely many infinite classes is refined by the formula $R_0(x,y)$;
\(3) $\forall x ({\rm Col}_n(x) \rightarrow \exists^{\infty}y
(R_0(x,y) \wedge \bigwedge\limits_{i = 1}^{n} R^{\delta_i}_i(x,y))
\wedge \neg \exists z \bigvee\limits_{i > n} R_i(x,z))$ for all possible binary tuples $(\delta_1,\ldots, \delta_n)$, i. e., for any element $a \in A_0$ of color $n$, the set of solutions for the formula $R_0(a,y)$ is divided, by $R_n(x,y)$, into $2^n$ disjoint sets, each of which is infinite.
Thus, the set of solutions for the formula $R_0(a,y)$, where $a
\models p_{\infty}(x)$, is divided, by $R_n(x,y)$, into continuum many disjoint sets similar Example 5. For output of the operator, we obtain a structure $\mathcal{B}$ with continuum many non-principal types $\{R^{\delta_i}_i(a,y) \mid i \in \omega
\setminus \{0\}\}$, and there are no prime models over the type $p_{\infty}(x)$.
The *operator of allocation for a countable subset* ${\rm
css} (\mathcal{A}, \mathbf{q}_{\omega}, \mathcal{A}_0,
\{R^{(2)}_{j}\}_{j \in \omega})$ takes for input:
\(1) a predicate structure $\mathcal{A}$ with a continual set $\mathbf{q}$ of non-principal $1$-types;
\(2) a countable subset $\mathbf{q}_{\omega} \subset \mathbf{q}$;
\(3) a substructure $\mathcal{A}_0 \subset \mathcal{A}$ with unique non-principal 1-type $p_{\infty}(x)\in S(\varnothing)$ and such that $p_{\infty}(x)$ is realized in $\mathcal{A}_0$;
\(4) a sequence $(R_j^{(2)})_{j\in\omega}$ of binary predicate symbols.
Denote by ${\rm Col}_{ij}(x)$ approximations of types $q_j(x) \in
\mathbf{q}_{\omega}$, $j \in \omega$. Then the type $q_j$ is isolated by the set of formulas $\{\neg {\rm Col}_{ij}(x) \mid i
\in \omega\}$. At the operator’s work, we assume that $A_0$ is the domain of predicates $R_{ij}$ and their range contains the set of realizations for types in $\mathbf{q}_{\omega}$. The work of the operator is defined by the following schemes of formulas:
\(1) $\forall x ({\rm Col}_i(x) \rightarrow \bigwedge\limits_{k
\geq i} \exists^{\infty} y (R_j(x,y) \wedge {\rm Col}_{kj}(y))
\wedge \bigwedge\limits_{k < i} \neg \exists y (R_j(x,y) \wedge
{\rm Col}_{kj}(y)))$, i. e., for any element $a \in A_0$ of $i$-th color, there are infinitely many images of each color $k$, $k \geq
i$, and there are no images of colors $k$, $k < i$;
\(2) $\forall x, x'(\neg (x \approx x') \rightarrow \neg \exists y
(R_{j}(x,y) \wedge R_{j}(x', y)))$, i. e., images of distinct elements belonging to $A_0$ are disjoint.
If the continual set $\mathbf{q}$ of non-principal types is obtained by the operator ${\rm icp}$ (and there are no prime models over each type in $\mathbf{q}$) then after passing all colors ${\rm Col}$ by all predicates $R_j$, the countable subset $\mathbf{q}_{\omega}$ is selected and, using a generic construction for a structure with required properties, there exists a prime model $\mathcal{M}_{p_{\infty}}$ over a realization of $p_{\infty}$ and realizing exactly all types in $\mathbf{q}_{\omega}$. If $\mathbf{q}_{\omega}$ is dense in $\mathbf{q}$ with respect to natural topology then, assuming that types in $\mathbf{q}_{\omega}$ are free (are not linked with $a
\in A_0$), we can remove elements in $\mathbf{q}_{\omega}$ and obtain new prime model $\mathcal{M}_{p_{\infty},\tilde{\mathbf{q}}_{\omega}}$, $\tilde{\mathbf{q}}_{\omega} \subset \mathbf{q}_{\omega}$, being an elementary submodel of $\mathcal{M}_{p_{\infty},\mathbf{q}_{\omega}}$. But having links of the dense set $\mathbf{q}_{\omega}$ with the type $p_{\infty}$ by predicates, the removing of a type in $\mathbf{q}_{\omega}$ leads to the removing of $p_{\infty}$. Whence applying the operator ${\rm css}$ with input parameters, satisfying the conditions above, there are no other (non-isomorphic) prime models being an elementary submodel of $\mathcal{M}_{p_{\infty},\mathbf{q}_{\omega}}$. Thus if we focus on this property, the given operator is called the *operator of ban for downward movement* and it is denoted by ${\rm bd}$ with the same input parameters.
The *operator of ban for upward movement* ${\rm bu} (\mathcal{A}, \mathcal{A}_1,
\mathcal{A}_2, Z, \{R^{(3)}_n\}_{n\in\omega})$ takes for input:
\(1) a predicate structure $\mathcal{A}$;
\(2) two disjoint substructures $\mathcal{A}_1$ and $\mathcal{A}_2$ of $\mathcal{A}$ with unique non-principal 1-types $p_1$ and $p_2$, being realized in $\mathcal{A}_1$ and $\mathcal{A}_2$ respectively;
\(3) an infinite set $Z$ such that $A_1 \cap Z = \varnothing$ and $A_2 \cap Z = \varnothing$;
\(4) a sequence $(R_n^{(3)})_{n\in\omega}$ of ternary predicate symbols.
We denote approximations of $p_1$ and $p_2$ by ${\rm Col}_{i1}$ and ${\rm Col}_{i2}$, $i \in \omega$, respectively. The set $A_1
\times A_2$ is the domain of predicates $R_{n}$, and $Z$ is their range, $\vdash R_n(x,y,z) \rightarrow R_0(x,y,z)$, $i
> 0$. The work of the operator is defined by the following schemes of formulas:
\(1) $\forall x, y({\rm Col}_{01}(x) \wedge {\rm Col}_{02}(y)
\rightarrow \exists^{\infty} z R_0(x, y, z))$;
\(2) $\forall x, y, x', y'(\neg( x \approx x') \wedge \neg (y
\approx y') \rightarrow \neg \exists z (R_{n}(x,y,z) \wedge
R_{n}(x',y',z)))$, i. e., $R_{n}$-images of distinct pairs $(a_1,a_2) \in A_1 \times A_2$ are disjoint and the set $Z$ is divided into infinitely many infinite equivalence classes;
\(3) $\forall x, y ({\rm Col}_{k1}(x) \wedge {\rm Col}_{n2}(y)
\rightarrow \exists^{\infty}z (R_0(x,y,z) \wedge
\bigwedge\limits_{1 \leq i \leq \min(k,n)} R^{\delta_i}_i(x,y,z))
\wedge \neg \exists z \bigvee\limits_{i > \min(k,n)} R_i(x,y,z))$ for all possible binary tuples $(\delta_1,\ldots,
\delta_{\min(k,n)})$.
Hence, if a pair $(a_1,a_2)$ has the $(\infty,\infty)$-color, the set of solutions for the formula $R_0(a_1,a_2,z)$ is divided on continuum many parts. Thus, there is a prime model over each realization of $p_1(x)$ and of $p_2(y)$, but there are no prime models over types $q(x,y) \supset p_1(x) \cup p_2(y)$.
The *operator for construction of limit models over a type*, ${\rm lmt}(p, \lambda,
\{R^{(2)}_i\}_{i\in\omega})$ takes for input:
\(1) a non-principal 1-type $p(x)$;
\(2) a number $\lambda \in \omega+1$ of limit models over $p(x)$;
\(3) a sequence $(R_i^{(2)})_{i\in\omega}$ of binary predicate symbols.
We assume that predicates $R_i$ act on a set of realizations of $p(x)$ such that $R_i(a, y) \vdash p(y)$ and $\models\exists y
R_i(a, y)$ and realizations $R_i(a, y)$ do not semi-isolate $a$, where $a \models p(x)$. We construct a tree of $R_i$-extensions over a realization $a$ of $p$. Consider sequences $i_0, \ldots,
i_n, \ldots \in 2^\omega$ correspondent to pathes $R_{i_0}(a, a_1)
\wedge \ldots \wedge R_{i_n}(a_n, a_{n+1}) \wedge \ldots$. There are $2^{\omega}$ extensions. As shown in [@SuLP; @Su072], given number $\lambda \in \omega + 1$ of limit models can be obtained by some family of identities.
For $n\in\omega\setminus\{0\}$ limit models, we use the following identities:
\(1) $n - 1 \approx m$, $m \geq n$,
\(2) $mm \approx m$, $m < n$,
\(3) $n_1 n_2 \ldots n_s \approx n_s$, $\min \{n_1, n_2, \ldots,
n_{s-1}\} > n_s$.
For countably many limit models, we introduce identities:
\(1) $nn \approx n$, $n \in \omega$,
\(2) $n_1 n_2 \ldots n_s \approx n_s$, $\min \{n_1, n_2, \ldots,
n_{s-1}\} > n_s$,
\(3) $n_1 n_2 \approx n_1 (n_1+1) (n_2 +2) \ldots (n_2 - 1)n_2$, $n_1 < n_2$.
The *operator for construction of limit models over a $\leq_{RK}$-sequence* $${\rm
lms}((q_n)_{n \in \omega}, \lambda, \{R^{(2)}_i\}_{i \in
\omega})$$ takes for input:
\(1) a $\leq_{RK}$-sequence $(q_n)_{n \in \omega}$;
\(2) a number $\lambda \in \omega+1$ of limit models over the sequence $(q_n)_{n \in \omega}$;
\(3) a sequence $(R_i^{(2)})_{i\in\omega}$ of binary predicate symbols.
Consider types $q_n$ and $q_{n+1}$. Since they belong to the $\leq_{RK}$-sequence, there is a formula $\varphi(x,y)$ such that $q_{n+1}(y) \cup \{\varphi(x,y)\}$ is consistent and $q_{n+1}(y)
\cup \{\varphi(x,y)\} \vdash q_{n}(x)$. We assume that predicates $R_i$ act so that $R_i(x,y) \vdash \varphi(x,y)$ and for every $a
\models q_{n+1}(y)$, $R_i(x,a) \vdash q_{n}(x)$. Below we consider numbers $i$ instead of predicates $R_i$. Then for the $\leq_{RK}$-sequence, there are $\omega^{\omega}$ sequences $i_l,
\ldots, i_k, \ldots$ correspondent to $R_l(a_n, a_{n-1}) \wedge
\ldots \wedge R_k(a_j, a_{j-1}) \wedge \ldots $, where $a_n
\models q_{n}(x),$ $\ldots,$ $a_1 \models q_1(x)$.
By the sequence $(q_n)_{n\in\omega}$, we construct sequences of prime models $\mathcal{M}_{q_n}$ over realizations of $q_n$, where $(n+1)$-th model is an elementary extension of $n$-th one. Any limit model is a union of countable chain of a sequence of prime models over tuples. Predicates $R_i$, $i \in \omega$, link types in $(q_n)$ leading to required number of limit models. As shown in [@SuLP; @Su08], the problem of extension of a theory producing a given number of limit models over $(q_n)$ is reduced to a factorization of the set $\omega^{\omega}$ by an identification of some words such that the result of this factorization contains as many classes as there are limit models.
For $n\in\omega\setminus\{0\}$ limit models, we use the following identities:
\(1) $n - 1 \approx m$, $m \geq n$;
\(2) $n_0 n_1 \ldots n_s \approx \underbrace{n_s \ldots n_s}_{s+1
\, {\textrm{times}}}$, $\max\{n_0, n_1, \ldots, n_{s-1}\} < n_s$.
For countably many limit models, we take identities:
\(1) $n_0 n_1 \ldots n_s \approx \underbrace{n_s \ldots n_s}_{s+1
\, {\textrm{times}}}$, $\max\{n_0, n_1, \ldots, n_{s-1}\} < n_s$;
\(2) $n_0 n_1 \ldots n_s \approx n_0(n_0+1)\ldots(n_0+s)$, $n_0 + s
\leq n_s$;
\(3) $n_0 n_1 \ldots n_s \approx
n_0(n_0+1)\ldots(n_0+t)\underbrace{(n_0+t)\ldots(n_0+t)}_{s - t \,
\textrm{times}}$, $n_0 + s$, $n_0 + t = n_s$, $t > 0$, $s > t$.
**7. Distributions of prime and limit models**
**for finite Rudin–Keisler preorders**
If $\widetilde{{\bf M}}$ is a $\sim_{RK}$-class containing an isomorphism type ${\bf M}$ of a prime model over a tuple, then as usual we denote by ${\rm IL}(\widetilde{{\bf M}})$ the number of limit models, being unions of elementary chains of models, whose isomorphism types belong to the class $\widetilde{{\bf M}}$.
Clearly, for theories $T$ with finite structures ${\rm RK}(T)$, any limit model is limit over a type.
The following two theorems show that for $p$-Ehrenfeucht small theories, the number of countable models is defined by the number of prime models over tuples and by the distribution function ${\rm
IL}$ of numbers of limit models over types. Assuming Continuum Hypothesis, all possible basic characteristics are realized.
[@SuLP; @Su072].
*Any small theory $T$ with a finite Rudin–Keisler preorder satisfies the following conditions:*
[(a)]{} ${\rm RK}(T)$ contains the least element ${\bf M}_0$ [(]{}the isomorphism type of a prime model[)]{}, and ${\rm
IL}(\widetilde{{\bf M}_0})=0$;
[(b)]{} ${\rm RK}(T)$ contains the greatest $\sim_{\rm
RK}$-class $\widetilde{{\bf M}_1}$ [(]{}the class of isomorphism types of all prime models over realizations of powerful types[)]{}, and $|{\rm RK}(T)|>1$ implies ${\rm IL}(\widetilde{{\bf
M}_1})\geq 1$;
[(c)]{} if $|\widetilde{\bf M}|>1$, then ${\rm
IL}(\widetilde{\bf M})\geq 1$.
Moreover, we have the following [*decomposition formula*]{}: $$I(T,\omega)=|{\rm RK}(T)|+\sum_{i=0}^{|{\rm RK}(T)/\sim_{\rm
RK}|-1} {\rm IL}(\widetilde{{\bf M}_i}),$$ where $\widetilde{{\bf M}_0},\ldots, \widetilde{{\bf M}_{|{\rm
RK}(T)/\sim_{\rm RK}|-1}}$ are all elements of the partially ordered set ${\rm RK}(T)/\!\!\sim_{\rm RK}$ and ${\rm
IL}(\widetilde{{\bf
M}_i})\in\omega\cup\{\omega,\omega_1,2^\omega\}$ for each $i$.
[@SuLP; @Su072]. [*For any finite preordered set $\langle X;\leq\rangle$ with the least element $x_0$ and the greatest class $\widetilde{x_1}$ in the ordered factor set $\langle X;\leq\rangle/\!\!\sim$ with respect to $\sim$ [(]{}where $x\sim y\Leftrightarrow x\leq y\mbox{ and }y\leq
x$[)]{}, and for any function $f\mbox{\rm :
}X/\!\!\sim\:\to\omega\cup\{\omega,2^\omega\}$, satisfying the conditions $f(\widetilde{x_0})=0$, $f(\widetilde{x_1})>0$ for $|X|>1$, and $f(\widetilde{y})>0$ for $|\widetilde{y}|>1$, there exist a small theory $T$ and an isomorphism $g\mbox{\rm : }\langle
X;\leq\rangle\:\widetilde{\to}\:{\rm RK}(T)$ such that ${\rm
IL}(g(\widetilde{y}))=f(\widetilde{y})$ for any $\widetilde{y}\in
X/\!\!\sim$.*]{}
Note that, by criterion of existence of prime model, an unsmall theory $T$ is $p$-categorical if and only if there is a unique $\equiv_{\rm RK}$-class $S\subset S(T)$ such that for any realization $\bar{a}$ of some (any) type in $S$ every consistent formula $\varphi(\bar{x},\bar{a})$ is an ${\rm i}$-formula.
Similarly, an unsmall theory $T$ is $p$-Ehrenfeucht if and only if there are finitely many pairwise non-$\equiv_{\rm RK}$-equivalent types $p_j$, $j<n$, $1<n<\omega$, such that for any $j$ and for some (any) realization $\bar{a}_j$ of $p_j$ every consistent formula $\varphi(\bar{x},\bar{a}_j)$ is an ${\rm i}$-formula.
The proofs of the following assertions repeat according proofs for the class of small theories [@SuLP; @Su041; @BSV].
By Proposition 7.3, we have the following analogue of Theorem 7.1 for the class ${\cal T}_c$.
The following theorem is an analogue of Theorem 7.2 for the class ${\cal T}_c$.
PROOF. Denote the cardinality of $X$ by $m$ and consider the theory $T_0$ of unary predicates $P_i$, $i < m$, forming a partition of a set $A$ on $m$ disjoint infinite sets with a coloring ${\rm Col}$: $A \rightarrow \omega\cup\{\infty\}$ such that for any $i < m$, $j \in \omega$, there are infinitely many realizations for each type $\{ {\rm Col}_j(x) \wedge P_i(x)\}$, $\{\neg {\rm Col}_j(x) \mid j \in \omega \} \cup \{P_i(x)\} =
p_i(x)$. In this case, each set of formulas isolates a complete type.
Let $X_1, \ldots, X_n$ be connected components of the preordered set $\langle X;\leq\rangle$, consisting of $m_1, \ldots, m_n$ elements respectively, $m_1+\ldots +m_n=m$. Now we assume that each element in $X$ corresponds to a predicate $P_i$, $i < m$.
We expand the theory $T_0$ to a theory $T_1$ by binary predicates $Q_{kl}$, whose domain coincides with the set of solutions for the formula $P_k(x)$ and the range is the set of solutions for the formula $P_l(x)$; we link types $p_k$ and $p_l$ if correspondent elements $x_k$ and $x_l$ in $X$ belong to a common connected component and $x_l$ covers $x_k$. Moreover, the coloring ${\rm
Col}$ will be $1$-inessential and $Q_{kl}$-ordered [@SuLP]:
\(1) for any $i \geq j$, there are elements $x,y \in M$ such that $$\models {\rm Col}_i(x) \wedge {\rm Col}_j(y) \wedge Q_{kl}(x,y) \wedge P_{k}(x) \wedge P_{l}(y);$$
\(2) if $i < j$ then there are no elements $u,v \in M$ such that $$\models {\rm Col}_i(u) \wedge {\rm Col}_j(v) \wedge Q_{kl}(u,v) \wedge P_{k}(u) \wedge P_{l}(v).$$
Applying a generic construction we get that if $a \models
p_{l}(y)$ then the formula $Q_{kl}(x,a)$ is isolating and $p_{l}(y) \cup Q_{kl}(x,y) \vdash p_{k}(x)$, moreover, realizations of $p_k$ do not semi-isolate realizations of $p_l$. Thus the set of non-principal 1-types $p_i(x)$ has a preorder correspondent to the preorder $\leq$.
We construct, by induction, an expansion of theory $T_1$ to a required theory $T$.
On initial step, we expand the theory $T_1$ by binary predicates $\{R_i^{(2)}\}_{i \in \omega}$ and apply the operator of continual partition ${\rm icp}(\mathcal{A}, \mathcal{A} \upharpoonright P_0,
Y, \{R_i^{(2)}\}_{i \in \omega}) = \mathcal{B}$, where $\mathcal{A}$ is a model of $T_1$. We consider an arbitrary connected component $X_i$ and enumerate its elements so that if $x_k
> x_l$ then $k > l$. On further $m_i$ steps, we apply the operator of allocation for a countable subset ${\rm css} (\mathcal{B},
\mathbf{q}_{\omega}, \mathcal{A} \upharpoonright P_{l_i},
\{R^{(2)}_{j}\}_{j \in \omega})$, where $l_1, \ldots, l_i$ are numbers of elements forming the connected component $X_i$, $\mathbf{q}_{\omega}$ is a countable dense subset of set $\mathbf{q}$ of 1-types for the structure $\mathcal{B}$. We organize a similar process for all connected components in $X$. Now for all types corresponding to elements in distinct connected components and to maximal elements in a common component, we apply the operator of ban for upward movement ${\rm bu} (\mathcal{A},
\mathcal{A} \upharpoonright P_{i}, \mathcal{A} \upharpoonright
P_{j},\{R^{(3)}_{\Delta}\})$, expanding the theory by disjoint families ternary predicates $R^{(3)}_n$, $n\in\omega$.
The required number of limit models can be done by application, for each $g(\widetilde{x})$, of the operator ${\rm
lmt}(g(\widetilde{x}), f(\widetilde{x}),
\{R_i^{g(\widetilde{x})}\}_{i \in \omega})$ expanding the theory by predicates $R_i^{g(\widetilde{x})}$ for each $g(\widetilde{x})$. $\Box$
By the proof of Theorem 7.7, positive values $P(T)$ for the class ${\cal T}_c$ can be defined by prime models, being not prime over $\varnothing$. Modifying the proof, one can realize an arbitrary finite preordered set $\langle X;\leq\rangle$ with the least element by ${\rm RK}(T)$ for a theory $T\in{\cal T}_c$ with a prime model over $\varnothing$.
By the construction for the proof of Theorem 7.7, we get
**8. Distributions of prime and limit models**
**for countable Rudin–Keisler preorders**
We say [@SuLP; @Su08] that a family ${\bf Q}$ of $\leq_{\rm
RK}$-sequences ${\bf q}$ of types [*represents*]{} a $\leq_{\rm RK}$-sequence ${\bf q}'$ of types if any limit model over ${\bf q}'$ is limit over some ${\bf q}\in
{\bf Q}$.
[@SuLP; @Su08].
*Any small theory $T$ satisfies the following conditions:*
[(a)]{} the structure ${\rm RK}(T)$ is upward directed and has the least element ${\bf M}_0$ [(]{}the isomorphism type of prime model of $T$[)]{}, ${\rm IL}(\widetilde{{\bf M}_0})=$ $0$;
[(b)]{} if ${\bf q}$ is a $\leq_{\rm RK}$-sequence of non-principal types $q_n$, $n\in\omega$, such that each type $q$ of $T$ is related by $q\leq_{\rm RK} q_n$ for some $n$, then there exists a limit model over ${\bf q}$; in particular, $I_l(T)\geq 1$ and the countable saturated model is limit over ${\bf q}$, if ${\bf q}$ exists;
[(c)]{} if ${\bf q}$ is a $\leq_{\rm RK}$-sequence of types $q_n$, $n\in\omega$, and $({\cal M}_{q_n})_{n\in\omega}$ is an elementary chain such that any co-finite subchain does not consist of pairwise isomorphic models, then there exists a limit model over ${\bf q}$;
[(d)]{} if ${\bf q}'=(q'_n)_{n\in\omega}$ is a subsequence of $\leq_{\rm RK}$-sequence ${\bf q}$, then any limit model over ${\bf q}$ is limit over ${\bf q}'$;
[(e)]{} if ${\bf q}=(q_n)_{n\in\omega}$ and ${\bf
q}'=(q'_n)_{n\in\omega}$ are $\leq_{\rm RK}$-sequences of types such that for some $k,m\in\omega$, since some $n$, any types $q_{k+n}$ and $q'_{m+n}$ are related by ${\cal
M}_{q_{k+n}}\simeq{\cal M}_{q'_{m+n}}$, then any model ${\cal M}$ is limit over ${\bf q}$ if and only if ${\cal M}$ is limit over ${\bf q}'$.
Moreover, the following [*decomposition formula*]{} holds: $$I(T,\omega)=|{\rm RK}(T)|+\sum\limits_{{\bf q}\in {\bf Q}}{\rm
IL}_{\bf q},$$ where ${\rm IL}_{\bf q}\in\omega\cup\{\omega,\omega_1,2^\omega\}$ is the number of limit models related to the $\leq_{\rm
RK}$-sequence ${\bf q}$ and not related to extensions and to restrictions of ${\bf q}$ that used for the counting of all limit models of $T$, and the family ${\bf Q}$ of $\leq_{\rm
RK}$-sequences of types represents all $\leq_{\rm RK}$-sequences, over which limit models exist.
[@SuLP; @Su08].
*Let $\langle
X,\leq\rangle$ be at most countable upward directed preordered set with a least element $x_0$, $f\mbox{\rm :
}Y\to\omega\cup\{\omega,2^\omega\}$ be a function with at most countable set $Y$ of $\leq_0$-sequences, i. e., of sequences in $X\setminus\{x_0\}$ forming $\leq$-chains, and satisfying the following conditions:*
[(a)]{} $f(y)\geq 1$ if for any $x\in X$ there exists some $x'$ in the sequence $y$ such that $x\leq x'$;
[(b)]{} $f(y)\geq 1$ if any co-finite subsequence of $y$ does not contain pairwise equal elements;
[(c)]{} $f(y)\leq f(y')$ if $y'$ is a subsequence of $y$;
[(d)]{} $f(y)=f(y')$ if $y=(y_n)_{n\in\omega}$ and $y'=(y'_n)_{n\in\omega}$ are sequences such that there exist some $k,m\in\omega$ for which $y_{k+n}=y'_{m+n}$ since some $n$.
Then there exists a small theory $T$ and an isomorphism $$g\mbox{\rm : }\langle X,\leq\rangle\:\widetilde{\to}\:{\rm RK}(T)$$ such that any value $f(y)$ is equal to the number of limit models over $\leq_{\rm RK}$-sequence $(q_n)_{n\in\omega}$, correspondent to the $\leq_0$-sequence $y=(y_n)_{n\in\omega}$, where $g(y_n)$ is the isomorphism type of the model ${\cal M}_{q_n}$, $n\in\omega$.
Repeating the proof of Theorem 8.1, we obtain
Similarly Theorem 7.2, Theorem 8.2 has a generalization for the class $\mathcal{T}_c$:
PROOF. We assume that $X$ is countable since for finite $X$, the proof repeats the construction for the proof of Theorem 7.7. Now we consider the theory $T_0$ of unary predicates $P_i$, $i \in
\omega$, forming, with the type $p_{\infty}(x) = \{\neg P_i(x)
\mid i \in \omega \}$, a partition of a set $A$ by disjoint infinite classes with a coloring ${\rm Col}$: $A \rightarrow
\omega\cup\{\infty\}$ such that for any $i,j \in \omega$, there are infinitely many realizations for each of types $\{ {\rm
Col}_j(x) \wedge P_i(x)\}$, $\{\neg {\rm Col}_j(x) \mid i \in
\omega \} \cup \{P_i(x)\} = p_i(x)$, $\{ {\rm Col}_j(x) \} \cup
p_{\infty}(x)$, $\{\neg {\rm Col}_j(x) \mid j \in \omega \}\cup
p_{\infty}(x)$. Here, each set of formulas isolates a complete type. We link the type $\{\neg {\rm Col}_j(x) \mid j \in
\omega\}\cup p_{\infty}(x)$ with the type $p_0(x)$ by an extension of $T_0$ to a theory $T_1$ with a binary predicate $Q_0$ such that for all $j \in \omega$, we have:
\(1) $\forall x, y \left( {\rm Col}_j(x) \wedge P_0(x) \wedge
Q_0(x,y) \rightarrow {\rm Col}_j(y) \wedge P_j(y)\right);$
\(2) $\forall x, y \left( {\rm Col}_j(y) \wedge P_j(y) \wedge
Q_0(x,y) \rightarrow {\rm Col}_j(x) \wedge P_0(x)\right);$
\(3) $Q_0$ is a bijection between sets of solutions for the formulas ${\rm Col}_j(x) \wedge P_0(x)$ and ${\rm Col}_j(y) \wedge
P_j(y)$.
These conditions allow not to care about the type $p_{\infty}(x)$ with respect to the existence of prime model over it, since $p_0(x)$ and $p_{\infty}(x)$ are strongly ${\rm RK}$-equivalent.
Let $X_1, \ldots, X_n,\ldots$ be connected components in the preordered set $\langle X,\leq\rangle$. We consider a one-to-one correspondence between $X$ and the set of predicates $P_i(x)$, $i
\in \omega$.
Similar the proof of Theorem 7.7, we expand the theory $T_1$ to a theory $T_2$ by binary predicates $Q_{kl}$ with domains $P_k(x)$ and ranges $P_l$, and link types $p_k$ and $p_n$ if correspondent elements in $X$ lay in common connected component and an element $x_l$ corresponding to $p_l$ covers an element $x_k$ corresponding to $p_k$. Moreover, using a generic construction, the coloring ${\rm Col}$ should be $1$-inessential and $Q_{kl}$-ordered.
The further proof repeats arguments for the proof of Theorem 7.7, where the operator ${\rm css}$ of allocation for a countable set is applied countably many times, for non-principal types corresponding to elements in $X$. In this case, if non-principal types are not exhausted, we apply the operator ${\rm icp}$ of continual partition for remaining types.
For the required number of limit models with respect to a sequence $(q_n)_{n \in \omega}$, we expand the theory by predicates $R^{(q_n)}_i$, $i \in \omega$, and apply the operator ${\rm
lms}((q_n)_{n \in \omega}, f(y), \{R^{(q_n)}_i\}_{i \in \omega}),$ where $y$ is a sequence in $Y$ correspondent to the sequence $(q_n)_{n \in \omega}$. $\Box$
By the construction for the proof of Theorem 8.4, we obtain
**9. Interrelation of classes ${\bf P}$, ${\bf L}$, and ${\bf NPL}$**
**in theories with continuum many types.**
**Distributions of triples ${\rm cm}_3(T)$ in the class $\mathcal{T}_c$**
PROOF. The construction of preordered set of types, isomorphic to the structure $\langle X,\leq\rangle$ and without prime models over the type $p_0$ is similar the proof of Theorem 8.4. Then for each non-principal type $p_i$, correspondent to an element in $P$, we apply the operator of allocation for a countable subset ${\rm
dss}(\mathcal{A}, \textbf{q}_{\omega}, \mathcal{A} \upharpoonright
P_i, \{R_n\}_{n \in \omega})$. If there are types $p_i$, correspondent to elements in $NPL$, we apply, for these types, the operator of continual partition ${\rm icp}(\mathcal{A},
\mathcal{A} \upharpoonright P_i, Z, \{R_n\}_{n \in \omega})$. For all types, corresponding to elements in distinct connected components in $\langle X,\leq\rangle$ as well as to maximal elements in a common component, we apply the operator of ban for upward movement. For the removing of prime models over remaining continuum many types, we apply, for $n$-tuples of elements the operator of continual partition, using $(n+1)$-ary predicates. The required number of limit models is obtained by the operator for construction of limit models over a sequence of types. $\Box$
PROOF is similar the proof of Theorem 9.1 with the only difference that before we use the operator of continual partition and then, if non-principal types $p_i$ are not exhausted, we apply the operator of allocation for a countable set. For getting prime models over remaining continuum many types, we apply, for $n$-tuples of elements the operator of allocation for a countable set, using $(n+1)$-ary predicates. The required number of limit models is obtained by the operator for construction of limit models over a sequence of types. $\Box$
By the construction for the proof of Theorem 9.2, we obtain
Proposition 5.4 and Corollaries 7.8, 8.5, 9.3 imply the following analogue of Theorem 5.1 for the class $\mathcal{T}_c$.
In conclusion, the authors thank Evgeniy A. Palyutin for helpful remarks.
[99]{} The Lachlan problem / S. V. Sudoplatov. — Novosibirsk : NSTU, 2009. — 336 p. Hypergraphs of prime models and distributions of countable models of small theories / S. V. Sudoplatov // J. Math. Sciences. — 2010. — Vol. 169, No. 5. — P. 680–695. Classification of countable models for complete theories of unary predicates with a substitution of a bounded order / R. A. Popkov // Algebra and Model Theory 8. Collection of papers / eds.: A. G. Pinus, K. N. Ponomaryov, S. V. Sudoplatov, and E. I. Timoshenko. — Novosibirsk : NSTU, 2011. — P. 73–82. Model theory / C. C. Chang, H. J. Keisler. — Amsterdam : Elsevier, 1990. — Studies in Logic and the Foundations of Mathematics. — Vol. 73. — 650 p. Handbook of mathematical logic / ed. J. Barwise. — Moscow : Nauka, 1982. — Vol. 1. Model Theory. — 392 p. \[in Russian\] Mathematical logic / Yu. L. Ershov, E. A. Palyutin. — Moscow : FIZMATLIT, 2011. — 356 p. \[in Russian\] Denumerable models of complete theories / R. Vaught // Infinistic Methods. — London : Pergamon, 1961. — P. 303–321. Complete theories with finitely many countable models. I / S. V. Sudoplatov // Algebra and Logic. — 2004. — Vol. 43, No. 1. — P. 62–69. On Rudin-Keisler preorders in small theories / S. V. Sudoplatov // Algebra and Model Theory 8. Collection of papers / eds.: A. G. Pinus, K. N. Ponomaryov, S. V. Sudoplatov, and E. I. Timoshenko. — Novosibirsk : NSTU, 2011. — P. 94–102. Theories with constants and three countable models / P. Tanović // Archive for Math. Logic. — 2007. — Vol. 46, No. 5–6. — P. 517–527. The number of countable models / M. Morley // J. Symbolic Logic. — 1970. — Vol. 35, No. 1. — P. 14–18. On the number of countable models of complete theories with finite Rudin–Keisler preorders / S. V. Sudoplatov // Siberian Math. J. — 2007. — Vol. 48, No. 2. — P. 334–338. Conditions for non-symmetric relations of semi-isolation / B. S. Baizhanov, S. V. Sudoplatov, V. V. Verbovskiy // Siberian Electronic Mathematical Reports. — 2012. — Vol. 9. — P. 161–184.
[^1]: r-popkov@yandex.ru
[^2]: sudoplat@math.nsc.ru
[^3]: [*Mathematics Subject Classification:*]{} 03C15. The work is supported by RFBR (grant 12-01-00460-a).
[^4]: The example is proposed by E. A. Palyutin.
[^5]: Recall that for countable structures ${\rm RKT}(T)$ the basic properties (the countable cardinality, the upward direction, the countability of $\sim_{\rm
RK}$-classes, the presence of the least $\sim_{\rm RK}$-classes) are presented in [@Su111].
|
---
abstract: 'The influence of a Lorentz-violating fixed background on fermions is considered by means of a torsion-free non-minimal coupling. The non-relativistic regime is assessed and the Lorentz-violating Hamiltonian is determined. The effect of this Hamiltonian on the hydrogen spectrum is determined to first-order evaluation (in the absence of external magnetic field), revealing that there appear some energy shifts that modify the fine structure of the spectrum. In the case the non-minimal coupling is torsion-like, no first order correction shows up in the absence of an external field; in the presence of an external field, a secondary Zeeman effect is implied. Such effects are then used to set up stringent bounds on the parameters of the model.'
author:
- 'H. Belich $^{b,e}$, T. Costa-Soares$^{c,d,e}$, M.M. Ferreira Jr.$^{a,e}$, J.A. Helayël-Neto$^{c,e}$ and F. M. O. Mouchereck$^{a}$'
title: '** Lorentz-violating corrections on the hydrogen spectrum induced by a non-minimal coupling**'
---
Introduction
============
Since the pioneering work by Carroll-Field-Jackiw [@Jackiw], Lorentz-violating theories have been extensively studied and used as an effective probe to test the limits of Lorentz covariance.** ** Nowadays, these theories are encompassed in the framework of the Extended Standard Model (SME), conceived by Colladay and Kosteletcký [Colladay]{} as a possible extension of the minimal Standard Model of the fundamental interactions. The SME admits Lorentz and CPT violation in all sectors of interactions by incorporating tensor terms (generated possibly as vacuum expectation values of a more fundamental theory) that account for such a breaking. Actually, the SME model sets out as an effective model that keeps unaffected the $SU(3)\times SU(2)\times U(1)$ gauge structure of the underlying fundamental theory while it breaks Lorentz symmetry at the particle frame.
Concerning the gauge sector of the SME, many studies have been developed that focus on many different respects [@Colladay]-[@Belich]. The fermion sector has been investigated as well, initially by considering general features (dispersion relations, plane-wave solutions, and energy eigenvalues) [@Colladay], and later by scrutinizing CTP-violating probing experiments conceived to find out in which extent the Lorentz violation may turn out manifest and to set up upper bounds on the breaking parameters. The CPT theorem, valid for any local Quantum Field Theory, predicts the equality of some quantities (life-time, mass, gyromagnetic ratio, charge-to-mass ratio) for particle and anti-particle. Thus, in the context of quantum electrodynamics, the most precise and sensitive tests of Lorentz and CPT invariance refer to comparative measurement of these quantities. A well-known example of this kind of test involves high-precision measurement of the gyromagnetic ratio [@Gyro] and cyclotron frequencies [@Cyclotron] for electron and positron confined in a Penning trap for a long time. The unsuitability of the usual figure of merit adopted in these works, based on the difference of the g-factor for electron and positron, was shown in refs. [@Penning], in which an alternative figure of merit was proposed, able to constrain the Lorentz-violating coefficients (in the electron-positron sector) to $1$part in $10^{20}$. Other interesting and precise experiments, also devised to establish stringent bounds on Lorentz violation, proposed new figures of merit involving the analysis of the hyperfine structure of the muonium ground state [@Muon], clock-comparison experiments [@Clock], hyperfine spectroscopy of hydrogen and anti-hydrogen [@Hydrog], and experiments with macroscopic samples of spin-polarized matter [@Spin].
The influence of Lorentz-violating and CPT-odd terms specifically on the Dirac equation has been studied in refs. [@Halmi], with the evaluation of the nonrelativistic Hamiltonian and the associated energy-level shifts. A similar investigation searching for deviations on the spectrum of hydrogen has been recently performed in ref. [@Fernando], where the nonrelativistic Hamiltonian was derived directly from the modified Pauli equation. Some interesting energy-level shifts, such as a Zeeman-like splitting, were then reported. These results may also be used to set up bounds on the Lorentz-violating parameters.
In another paper involving the fermion sector [@ACNminimo], the influence of a non-minimally coupled Lorentz-violating background on the Dirac equation has been investigated. It was then shown that such a coupling, given at the form $\epsilon _{\mu \nu \alpha \beta }\gamma ^{\mu
}v^{\nu }F^{\alpha \beta }$, is able to induce topological phases (Aharonov-Bohm and Aharonov-Casher [@Casher]) at the wave function of an electron (interacting with the gauge field and in the presence of the fixed background). Lately, in connection with this particular effect, it has been shown that (non-minimally coupled) particles and antiparticles develop opposite A-Casher phases. This fact, in the context of a suitable experiment, may be used to constrain the Lorentz-violating parameter [Thales]{}. In these papers, however, it was not addressed the issue concerning the nonrelativistic corrections induced by this kind of coupling in an atomic system.
The present work has as its main goal to examine the effects of the Lorentz-violating background, whenever non-minimal coupled as in ref. [ACNminimo]{}, on the Dirac equation, with special attention to its nonrelativistic regime and possible implications on the hydrogen spectrum. The starting point is the Dirac Lagrangian supplemented by Lorentz and CPT-violating terms. The investigation of the nonrelativistic limit is performed and the Lorentz-violating Hamiltonian is written down. The effect of the background on the spectrum of hydrogen atom is then evaluated by considering a first-order perturbation. In the absence of external magnetic field, it is verified that three different corrections are attained, able to modify the fine structure of the spectrum. For the case of the torsion-like non-minimal coupling, $g_{a}\epsilon _{\mu \nu \alpha \beta }\gamma
_{5}v^{\nu }F^{\alpha \beta }$, no correction is found out. In the presence of an external field, this term yields a Zeeman splitting proportional to the background magnitude. The theoretical modifications here obtained are used to set up stringent bounds on the magnitude of the corresponding Lorentz-violating coefficient.
This paper is outlined as follows. In Sec. II, it is analyzed the influence of the non-minimal coupling on the nonrelativistic limit of the Dirac equation, focusing on the possible corrections induced on the spectrum of the hydrogen atom. This is done both for a torsion-free and torsion-like non-minimal coupling. In Sec. III, we present our Final Remarks.
Non-minimal coupling to the gauge field and background
======================================================
The non-minimal coupling of the particle to** **the Lorentz-violating background is here considered in two versions**:** a torsion-free and a torsion-like coupling. We begin by analyzing the torsion-free case, which is implemented by defining a covariant derivative with non-minimal coupling, as below:
$$D_{\mu }=\partial _{\mu }+ieA_{\mu }+igv^{\nu }F_{\mu \nu }^{\ast },
\label{covader}$$
where $F_{\mu \nu }^{\ast }$ is the dual electromagnetic tensor $(F_{\mu \nu
}^{\ast }=\frac{1}{2}\epsilon _{\mu \nu \alpha \beta }F^{\alpha \beta }).$ In this situation, the additional term sets a non-minimal coupling of the fermion sector to a fixed background $v^{\mu }$, responsible for the breaking of Lorentz symmetry [@Jackiw] at the particle frame. The mass dimensions of the gauge field and the coupling constant are: $\left[ A_{\mu }\right] =1,\left[ g\right] =-2.$ The Dirac equation with such a coupling,
$$(i\hbar \gamma ^{\mu }D_{\mu }-m_{e}c)\Psi =0, \label{Dirac1}$$
is the starting point to investigate the influence of this background on the dynamics of the fermionic particle. Working with the Dirac representation[^1] of the $\gamma $-matrices, and writing $\Psi $ in terms of two-component spinors, $\Psi =\left(
\begin{tabular}{l}
$$ \\
$$\end{tabular}\right) ,$ there follow two coupled equations for $\phi $ and $\chi $ in momentum space: $$\begin{aligned}
\left( E/c-m_{e}c-eA_{0}/c+g\overrightarrow{v}\cdot \overrightarrow{B}\right) \phi -\overrightarrow{\sigma }\cdot (\overrightarrow{p}-e\overrightarrow{A}/c+gv^{0}\overrightarrow{B}-g\overrightarrow{v}\times
\overrightarrow{E})\chi &=&0, \label{phi1} \\
-\left( E/c+m_{e}c-eA_{0}/c+g\overrightarrow{v}\cdot \overrightarrow{B}\right) \chi +\overrightarrow{\sigma }\cdot (\overrightarrow{p}-e\overrightarrow{A}+gv^{0}\overrightarrow{B}-g\overrightarrow{v}\times
\overrightarrow{E})\phi &=&0. \label{phi2}\end{aligned}$$To investigate the low-energy behavior of this system, the natural option is to search for its nonrelativistic limit, where the energy is given as $E=m_{e}c^{2}+H,$ with $H$ being the nonrelativistic Hamiltonian. Writing the weak component ($\chi )$ in terms of the strong one ($\phi )$, the following equation for $\phi $ holds: $\left( H/c-eA_{0}/c+g\overrightarrow{v}\cdot
\overrightarrow{B}\right) \phi =\frac{1}{2m_{e}c}\left( \overrightarrow{\sigma }\cdot \overrightarrow{\Pi }\right) \left( \overrightarrow{\sigma }\cdot \overrightarrow{\Pi }\right) \phi ,$ where the generalized canonical moment is defined as $\overrightarrow{\Pi }=\left( \overrightarrow{p}-e\overrightarrow{A}/c+gv^{0}\overrightarrow{B}-g\overrightarrow{v}\times
\overrightarrow{E}\right) $. After some algebra, the nonrelativistic Hamiltonian for the particle comes out: $$\begin{aligned}
H &=&\biggl\{\left[ \frac{1}{2m_{e}}(\overrightarrow{p}-\frac{e}{c}\overrightarrow{A})^{2}+eA_{0}-\frac{e\hbar }{2m_{e}c}(\overrightarrow{\sigma }\cdot \overrightarrow{B})\right] +\frac{g^{2}}{2m_{e}}(\overrightarrow{v}\times \overrightarrow{E})^{2}+\frac{\hbar }{2m_{e}}gv_{0}\overrightarrow{\sigma }\cdot (\overrightarrow{\nabla }\times
\overrightarrow{B}) \notag \\
&&-\frac{g\hbar }{2m_{e}}\overrightarrow{\sigma }\cdot \lbrack
\overrightarrow{\nabla }\times (\overrightarrow{v}\times \overrightarrow{E})]+\frac{gv_{0}}{m_{e}}(\overrightarrow{p}-\frac{e}{c}\overrightarrow{A})\cdot \overrightarrow{B}-\frac{g}{m_{e}}(\overrightarrow{p}-\frac{e}{c}\overrightarrow{A})\cdot (\overrightarrow{v}\times \overrightarrow{E})-\frac{g^{2}v_{0}}{m_{e}}\overrightarrow{B}\cdot (\overrightarrow{v}\times
\overrightarrow{E})\biggr\}, \label{H1}\end{aligned}$$In the expression above, there appears the Pauli Hamiltonian (between brackets) corrected by the terms that compose the Lorentz-violating Hamiltonian, $H_{LV\text{ }},$ which truly constitutes our object of interest. The purpose is to investigate the contribution of the Hamiltonian with Lorentz-violation $(H_{LV})$ in the states of the hydrogen atom. Such a calculation will be initially performed for the case of a free hydrogen atom (without external field, $\overrightarrow{A}=0$), for which only three terms contribute. For all the terms that do not involve the spin operator, we shall use the hydrogen 1-particle wave functions $\left( \Psi \right) $ labeled in terms of the quantum numbers $n,l,m$, $\Psi _{nlm}(r,\theta ,\phi
)=R_{nl}(r)\Theta _{lm}(\theta )\Phi _{m}(\phi ),$ whereas the evaluation of the terms involving $\overrightarrow{\sigma }$ requires the use of the wave function $\Psi _{nljm_{j}m_{s}},$ with $j,m_{j\text{ }}$being the quantum numbers suitable to deal with the addition of angular momentum. Here, $r,\theta ,\phi $ are spherical coordinates.
As our initial evaluation[^2], we consider the first-order correction induced by the term $g^{2}(\overrightarrow{v}\times
\overrightarrow{E})^{2}/2m,$ namely:
$$\Delta E_{1}=\frac{g^{2}}{2m_{e}}\int \Psi _{nlm}^{\ast }(\overrightarrow{v}\times \overrightarrow{E})^{2}\Psi _{nlm}d^{3}r. \label{Energ}$$
To solve it, we write $(\overrightarrow{v}\times \overrightarrow{E})^{2}=v^{2}E^{2}-(\overrightarrow{v}\cdot \overrightarrow{E})^{2}$ and take the Coulombian electric field given by $\overrightarrow{E}=-e\widehat{r}/r^{2},$ so that the result is: $$\Delta E_{1}=\frac{g^{2}e^{2}}{2m_{e}}[v^{2}\left\langle
nlm|1/r^{4}|nlm\right\rangle -\left\langle nlm|(\overrightarrow{v}\cdot
\widehat{r})^{2}/r^{4}|nlm\right\rangle ].$$
In spherical coordinates, $\overrightarrow{v}\cdot \widehat{r}=v_{x}\sin
\theta \cos \phi +v_{y}\sin \theta \sin \phi +v_{z}\cos \theta ,$ which leads us to:
$$\Delta E_{1}=\frac{g^{2}e^{2}}{4m_{e}}\left[ \overline{\left( \frac{1}{r^{4}}\right) }(v_{x}^{2}+v_{y}^{2}+2v_{z}^{2})+(v_{x}^{2}+v_{y}^{2}-2v_{z}^{2})\left\langle nlm|\frac{\cos ^{2}\theta }{r^{4}}|nlm\right\rangle \right] .$$
Considering the intermediate result, $$\left\langle nlm|\frac{\cos ^{2}\theta }{r^{4}}|nlm\right\rangle =\overline{\left( \frac{1}{r^{4}}\right) }\left[ \frac{(l^{2}-m^{2})}{(2l-1)(2l+1)}+\frac{(l^{2}-m^{2}+2l+1)}{\left( 2l+3\right) (2l+1)}\right] ,$$the following energy correction is obtained for the case the background is aligned along the z-axis$\left( \overrightarrow{v}=v_{z}\overset{\symbol{94}}{z}\right) $:
$$\Delta E_{1}=\frac{g^{2}e^{2}v_{z}^{2}}{4m_{e}}\overline{\left( \frac{1}{r^{4}}\right) }\left[ 1-\left( \frac{(l^{2}-m^{2})}{(2l-1)(2l+1)}+\frac{(l^{2}-m^{2}+2l+1)}{\left( 2l+3\right) (2l+1)}\right) \right] . \label{CE1}$$
where $\overline{\left( 1/r^{4}\right) }=\left\langle
nlm|1/r^{4}|nlm\right\rangle
=3[1-l(l+1)/3n^{2}]/[n^{3}a_{0}^{4}(l+3/2)(l+1)(l+1/2)l(l-1/2)]$ is a well-known result for the hydrogen system. Here, $a_{0}=\hbar
^{2}/e^{2}m_{e} $ is the Bohr radius ($a_{0}=0.0529nm$). This result shows that the non-minimal coupling is able to remove the accidental degeneracy, regardless the spin-orbit interaction. This effect, therefore, implies a modification on the fine structure of the spectrum. The order of magnitude of this correction is given by the ratio $g^{2}v_{z}^{2}e^{2}/(m_{e}a_{0}^{4}),$ which is numerically $2\times
10^{53}\left( gv_{z}\right) ^{2}eV.$ Considering that spectroscopic experiments are able to detect effects of one part in $10^{10}$ in the spectrum$,$ the correction (\[CE1\]) may not be larger than $10^{-10}eV,$ which implies an upper bound for the product $gv_{z}$, namely: $gv_{z}\leq
10^{-32}.$
In the absence of an external magnetic field, the next term to be taken into account is $\frac{g}{m_{e}}(\overrightarrow{p}-e\overrightarrow{A})\cdot (\overrightarrow{v}\times \overrightarrow{E}),$ whose non-trivial part is $\frac{g}{m_{e}}\overrightarrow{p}\cdot (\overrightarrow{v}\times
\overrightarrow{E}).$ Hence, the first-order energy correction is: $$\Delta E_{2}=-i\hbar m_{e}\int \Psi _{nlm}^{\ast }\nabla \cdot (\overrightarrow{v}\times \overrightarrow{E})\Psi _{nlm}d^{3}r=-i\hbar \frac{g}{m_{e}}\int \Psi ^{\ast }[\nabla \cdot (\overrightarrow{v}\times
\overrightarrow{E})]\Psi d^{3}r-i\hbar \frac{g}{m_{e}}\int \Psi ^{\ast }(\overrightarrow{v}\times \overrightarrow{E})\cdot \nabla \Psi d^{3}r.
\label{E2}$$Taking the gradient of $\Psi $ in spherical coordinates, and the scalar product with $(\overrightarrow{v}\times \overrightarrow{E}),$ many terms are obtained that depend linearly on $\sin \phi $, $\cos \phi $ or $\sin 2\phi ,
$ except for two of them. These are the ones that survive after the angular integration is performed. The remaining expression is: $$\Delta E_{2}=\frac{egv_{z}m\hbar }{m_{e}}\int R_{nl}^{\ast }(r)\Theta
_{lm}^{\ast }(\theta )\frac{1}{r^{3}}R_{nl}(r)\Theta _{lm}(\theta
)r^{2}dr\sin \theta d\theta =\frac{egv_{z}m\hbar }{m_{e}}\overline{\left(
\frac{1}{r^{3}}\right) }.$$which can be explicitly written as: $$\Delta E_{2}=\frac{egv_{z}\hbar }{m_{e}}\frac{m}{a_{0}^{3}n^{3}l(l+1/2)(l+1)}, \label{CE2}$$where the well-known result $\overline{\left( 1/r^{3}\right) }=[a_{0}^{3}n^{3}l(l+1/2)(l+1)]^{-1}$ has been used$.$ A previous superficial examination of eq. (\[E2\]) could lead to the misleading expectation of a vanishing result, once it consists of the average of a linear function of the momentum $\left( p\right) $ on the state $\Psi $. Yet, in the development of this expression, there arise the angular momentum, $\overrightarrow{L}=\overrightarrow{r}\times \overrightarrow{p},$ whose expectation value in a bound state is generally non-vanishing, justifying the result of eq. (\[CE2\]). The order of magnitude of this correction is $e\hbar gv_{z}/(m_{e}a_{0}^{3}),$ whose numerical value is $2\times
10^{27}\left( gv_{z}\right) eV.$ Taking into account the possibility of detection of one part in $10^{10},$ we arrive at the following bound for the parameters: $gv_{z}\leq 10^{-19}.$
In order to evaluate the correction associated with the terms involving the spin operator, it is necessary to work with the wave functions $\Psi
_{nljm_{j}m_{s}}=\psi _{nljm_{j}}(r,\theta ,\phi )\chi _{sm_{s}},$ suitable to treat the situations where there occurs addition of angular momenta ($J=L+ $ $S$), with $n,l,j,m_{j}$ being the associated quantum numbers. Considering the free hydrogen atom, the first non-null spin term is $\overrightarrow{\sigma }\cdot \lbrack \overrightarrow{\nabla }\times (\overrightarrow{v}\times \overrightarrow{E})],$ which implies the following first-order correction:
$$\Delta E_{3}=\frac{g\hbar }{2m_{e}}\langle nljm_{j}m_{s}|\overrightarrow{\sigma }\cdot (\overrightarrow{\nabla }\times (\overrightarrow{v}\times
\overrightarrow{E}))|nljm_{j}m_{s}\rangle . \label{EB}$$
For the case of the Coulombian electric field, $\overrightarrow{\sigma }\cdot \lbrack \overrightarrow{\nabla }\times (\overrightarrow{v}\times
\overrightarrow{E})]=2e(\overrightarrow{\sigma }\cdot \overrightarrow{v})/r^{3}-e(\overrightarrow{v}\cdot \overrightarrow{\nabla })(\overrightarrow{\sigma }\cdot \overrightarrow{r})/r^{3}.$ After some algebraic manipulations, one obtains:
$$\Delta E_{3}=\frac{ge\hbar }{m_{e}}\langle nljm_{j}m_{s}|(\overrightarrow{\sigma }{}\cdot \overrightarrow{v})/r^{3}-(\overrightarrow{f}\cdot
\overrightarrow{\sigma })|nljm_{j}m_{s}\rangle , \label{EB2}$$
with: $f_{x}=e(-v_{x}+3v_{x}\sin ^{2}\theta \cos ^{2}\phi +3v_{y}\sin
^{2}\theta \cos \phi \sin \phi +v_{z}\cos \theta \sin \theta \cos \phi
)/r^{3};$ $f_{y}=e(-v_{y}+3v_{y}\sin ^{2}\theta \sin ^{2}\phi +3v_{x}\sin
^{2}\theta \cos \phi \sin \phi +v_{z}\cos \theta \sin \theta \cos \phi
)/r^{3};$ $f_{z}=e(-v_{z}+3v_{z}\cos ^{2}\theta +3v_{x}\sin \theta \cos
\theta \cos \phi +v_{y}\cos \theta \sin \theta \cos \phi )/r^{3}.$
To complete this calculation, it is necessary to write the $|jm_{j}\rangle $ kets in terms of the spin eigenstates $|mm_{s}\rangle ,$ which is done by means of the general expression: $|jm_{j}\rangle
=\dsum\limits_{m,m_{s}}\langle mm_{s}|jm_{j}\rangle $ $|mm_{s}\rangle ,$ where $\langle mm_{s}|jm_{j}\rangle $ are the Clebsch-Gordon coefficients. Evaluating such coefficients for the case $j=l+1/2,m_{j}=m+1/2,$ one has: $\
|jm_{j}\rangle =\alpha _{1}|m\uparrow \rangle +\alpha _{2}|m+1\downarrow
\rangle ;$ one the other hand, for $j=l-1/2,m_{j}=m+1/2,$ one obtains: $|jm_{j}\rangle =\alpha _{2}|m\uparrow \rangle -\alpha _{1}|m+1\downarrow
\rangle ,$ with: $\alpha _{1}=\sqrt{(l+m+1)/(2l+1)},\alpha _{2}=\sqrt{(l-m)/(2l+1)}.$ Taking now into account the orthonormalization relation $\langle m^{\prime }m_{s}^{\prime }|mm_{s}\rangle =\delta _{m^{\prime
}m}\delta _{m_{s}^{\prime }m_{s}},$ it is possible to show that eq. ([EB]{}) leads to:
$$\Delta E_{3}=\pm \frac{3e\hbar gv_{z}}{2m_{e}}\frac{m_{j}}{a_{0}^{3}n^{3}l(l+1/2)(l+1)\left( 2l+1\right) }\left\{ 1-\left( \frac{(l^{2}-m^{2})}{(2l-1)(2l+1)}+\frac{(l^{2}-m^{2}+2l+1)}{\left( 2l+3\right)
(2l+1)}\right) \right\} ,$$
where the positive and negative signs correspond to $j=l+1/2$ and $j=l-1/2,$ respectively; it was also used: $\langle nljm_{j}m_{s}|\sigma
_{z}|nljm_{j}m_{s}\rangle =\pm m_{j}/(2l+1)$, $\langle nljm_{j}m_{s}|\sigma
_{x}|nljm_{j}m_{s}\rangle =\langle nljm_{j}m_{s}|\sigma
_{y}|nljm_{j}m_{s}\rangle =0,$ and the expression for $\overline{\left(
1/r^{3}\right) }.$ The order of magnitude of this correction is $gv_{z}e\hbar /(m_{e}a_{0}^{3}),$ the same of the correction $\Delta E_{2}.$
Next, we still consider an external fixed field and we evaluate the corrections induced by it. In principle, three terms of the Hamiltonian ([H1]{}) might yield non-zero contributions in the presence of a magnetic field, namely: $\Delta E_{1B}=\frac{gv_{0}}{m_{e}}\left\langle nlm|(\overrightarrow{p}-e\overrightarrow{A})\cdot \overrightarrow{B}|nlm\right\rangle ,$ $\Delta E_{2B}=-\frac{eg}{m_{e}c}\left\langle nlm|\overrightarrow{A}\cdot (\overrightarrow{v}\times \overrightarrow{E})|nlm\right\rangle ,$ $\Delta E_{3B}=-\frac{g^{2}v_{0}}{m_{e}}\left\langle
nlm|\overrightarrow{B}\cdot (\overrightarrow{v}\times \overrightarrow{E})|nlm\right\rangle .$ For a fixed magnetic field along the z-axis, $\overrightarrow{B}=B_{0}\widehat{z}$, the vector potential in the symmetric gauge reads: $\overrightarrow{A}=-B_{0}(y/2,-x/2,0).$ Concerning the first term, only the product $\overrightarrow{A}\cdot \overrightarrow{B}$ could provide a non-trivial contribution, once the evaluation of the product $\overrightarrow{p}\cdot \overrightarrow{B}$ on the wave function obviouslyvanish. After a simple inspection, one gets $\Delta E_{1B}=\frac{gv_{0}}{m_{e}}\left\langle nlm|\overrightarrow{A}\cdot \overrightarrow{B}|nlm\right\rangle =0.$
In order to solve the second term, we should write $(\overrightarrow{v}\times \overrightarrow{E})=$ $-\frac{e}{r^{2}}[(v_{y}\cos \theta -v_{z}\sin
\theta \sin \phi )\widehat{i}+(v_{z}\sin \theta \cos \phi -v_{x}\cos \theta )\widehat{j}+(v_{z}\sin \theta \sin \phi -v_{y}\sin \theta \cos \phi )\widehat{k}.$ The explicit calculation of this term yields a trivial result. Finally, it remains to evaluate the third term, which turns out to be also vanishing. We thus verify that the magnetic field does not yield any correction associated with the background; it only leads to the well-known Zeeman effect. This is the situation for the torsion free coupling.
Another possible way to couple the Lorentz-violating background ($v^{\mu }$) to the fermion field is by proposing a torsion-like non-minimal coupling, $$D_{\mu }=\partial _{\mu }+eA_{\mu }+ig_{a}\gamma _{5}v^{\nu }F_{\mu \nu
}^{\ast },$$which has a chiral character, and has been examined in ref. [@ACNminimo] as well.
Writing the spinor $\Psi $ in terms of the so-called small and large components in much the same way as it was done in the previous case, there follow two coupled equations for the 2-component spinors $\phi ,\chi ,$ $$\begin{aligned}
\left[ \left( E/c-mc-eA_{0}/c\right) -g_{a}\overrightarrow{\sigma }\cdot
\left( v^{0}\overrightarrow{B}-\overrightarrow{v}\times \overrightarrow{E}\right) \right] -\phi \lbrack \overrightarrow{\sigma }\cdot \left(
\overrightarrow{p}-e\overrightarrow{A}/c\right) -g_{a}\overrightarrow{v}\cdot \overrightarrow{B}]\chi =0, && \label{phi4} \\
\lbrack \overrightarrow{\sigma }\cdot \left( \overrightarrow{p}-e\overrightarrow{A}/c\right) +g_{a}\overrightarrow{v}\cdot \overrightarrow{B}]\phi -\left[ \left( E/c+mc-eA_{0}/c\right) -g_{a}\overrightarrow{\sigma }\cdot \left( v^{0}\overrightarrow{B}-g_{a}\overrightarrow{v}\times
\overrightarrow{E}\right) \right] \chi &=&0,\end{aligned}$$from which we can read the weak component in terms of the strong one, $\chi =\frac{1}{2m_{e}}\left[ \overrightarrow{\sigma }\cdot \left( \overrightarrow{p}-\frac{e}{c}\overrightarrow{A}\right) +g_{a}\overrightarrow{v}\cdot
\overrightarrow{B}\right] \phi .$ It is then possible to write the Pauli equation, $$\left( H/c-eA_{0}/c-g_{a}\overrightarrow{\sigma }\cdot \left( v^{0}\overrightarrow{B}-\overrightarrow{v}\times \overrightarrow{E}\right)
\right) \phi =\left[ \overrightarrow{\sigma }\cdot \left( \overrightarrow{p}-\frac{e}{c}\overrightarrow{A}\right) -g_{a}\overrightarrow{v}\cdot
\overrightarrow{B}\right] \frac{1}{2m_{e}}\left[ \overrightarrow{\sigma }\cdot \left( \overrightarrow{p}-\frac{e}{c}\overrightarrow{A}\right) +g_{a}\overrightarrow{v}\cdot \overrightarrow{B}\right] \phi ,$$whose structure reveals as canonical generalized moment the usual relation,$\overrightarrow{\text{ }\Pi }=(\overrightarrow{p}-\frac{e}{c}\overrightarrow{A}).$ Simplifying the equation above, the nonrelativistic Hamiltonian takes the form: $$H=\left[ \frac{(\overrightarrow{p}-e\overrightarrow{A}/c)^{2}}{2m_{e}}+eA_{0}-\frac{e\hbar }{2m_{e}c}(\overrightarrow{\sigma }\cdot
\overrightarrow{B})\right] +g_{a}v_{0}c\overrightarrow{\sigma }\cdot
\overrightarrow{B}-g_{a}c\overrightarrow{\sigma }\cdot (\overrightarrow{v}\times \overrightarrow{E})-\frac{g_{a}}{2m_{e}}(\overrightarrow{v}\cdot
\overrightarrow{B})^{2}. \label{H2}$$This Hamiltonian has yet two additional terms, $(\overrightarrow{\sigma }\cdot \overrightarrow{p})(\overrightarrow{v}\cdot \overrightarrow{B})-(\overrightarrow{v}\cdot \overrightarrow{B})(\overrightarrow{\sigma }\cdot
\overrightarrow{p}),$ which are equal (canceling each other) for the case of a uniform magnetic field. They will not be considered here.
In the absence of magnetic field, only the term $\overrightarrow{\sigma }\cdot (\overrightarrow{v}\times \overrightarrow{E})$ contributes for the energy, implying the following correction:
$$\Delta E_{\sigma }=g_{a}\langle nljm_{j}m_{s}|\overrightarrow{\sigma }\cdot (\overrightarrow{v}\times \overrightarrow{E})|nljm_{j}m_{s}\rangle .$$
Considering that $\overrightarrow{\sigma }\cdot (\overrightarrow{v}\times
\overrightarrow{E})=$ $-\frac{e}{r^{2}}[(v_{y}\cos \theta -v_{z}\sin \theta
\sin \phi )\sigma _{x}+(v_{z}\sin \theta \cos \phi -v_{x}\cos \theta )\sigma
_{y}+(v_{z}\sin \theta \sin \phi -v_{y}\sin \theta \cos \phi )\sigma _{z},$ and the action of the spin operators on the kets $|nljm_{j}m_{s}\rangle ,$ it is easy to note that: $\Delta E_{\sigma }=0.$ Hence, the non-minimal pseudoscalar coupling yields no background contribution for the energy levels.
Now, the presence of an external magnetic field shall be taken into account. In this case, there appears a non-zero new contribution associated with the term $cg_{a}v_{0}\overrightarrow{\sigma }\cdot \overrightarrow{B},$ which generates a Zeeman splitting of the levels, whose separation is linear on the product $cg_{a}v_{0}.$ For the case the magnetic field is aligned with the z-axis, the implied energy correction is $\Delta
E_{1B}=cg_{a}v_{0}B_{0}\langle nljm_{j}m_{s}|\sigma
_{z}|nljm_{j}m_{s}\rangle ,$ which yields:
$$\Delta E_{1B}=\pm g_{a}v_{0}cB_{0}\frac{m_{j}}{2l+1},$$
where the positive and negative signs correspond to $j=l+1/2$ and $j=l-1/2,$ respectively. This is exactly the same pattern of splitting of the Zeeman effect, here with amplitude given as $g_{a}v_{0}B_{0}.$ Hence, besides the usual Zeeman effect, there occurs this secondary Zeeman splitting that implies a correction to the effective splitting. The last term of eq. ([H2]{}) only implies a constant correction on all levels, which does not lead to any change in the spectrum. The magnitude of this correction is proportional to $g_{a}v_{0}cB_{0}.$ If such an effect is not detectable for a magnetic strength of $1$ $G$, it should not imply a correction larger than $10^{-10}eV,$ so that the bound $g_{a}v_{0}\leq 10^{-18}$ is attained.
Final Remarks
=============
In this work, we have studied low-energy effects of a Lorentz-violating background (non-minimally coupled to the fermion and gauge fields) on a nonrelativistic system. Indeed, the nonrelativistic limit has been worked out and the Lorentz-violating Hamiltonian (derived from the non-minimal coupling) evaluated. The first-order corrections induced on the energy levels of the hydrogen atom have been determined. As a result, we have observed effective shifts on the hydrogen spectrum, both in the presence and absence of an external magnetic field. In the absence of the external magnetic field, the term $\epsilon _{\mu \nu \alpha \beta }\gamma ^{\mu
}v^{\nu }F^{\alpha \beta }$ induces three different corrections, all of them implying modifications on the fine structure of the spectrum. This result indicates the breakdown of the accidental degeneracy, with the energy depending on $l,m$ quantum numbers. Stipulating $10^{-10}eV$ as the magnitude of a maximally undetectable change in the spectrum, we have set up an upper bound on the product of parameters: $gv_{z}\leq 10^{-32}.$
In the case of the torsion-like non-minimal coupling, no correction is implied in the absence of external magnetic field; on the other hand, in the presence of such a fixed field, a secondary Zeeman effect is obtained. Considering that such a correction should be smaller than $10^{-10}eV,$ an upper bound is set up for the product, namely: $g_{a}v_{0}\leq 10^{-18}$. These results show that Lorentz violation in the context of the non-minimal coupling regarded here turns out as a real negligible effect.
MMFJr and JAHN express their gratitude to CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) and FMOM acknowledges FAPEMA (Fundação de Amparo à Pesquisa do Estado do Maranhão) for financial support.
[99]{} S.M. Carroll, G.B. Field and R. Jackiw, *Phys. Rev.* D **41**, 1231 (1990).
D. Colladay and V. A. Kostelecký, *Phys. Rev.* D **55**,6760 (1997); D. Colladay and V. A. Kostelecký, *Phys. Rev.* D **58**, 116002 (1998).
S.R. Coleman and S.L. Glashow, *Phys. Rev*. D** 59**, 116008 (1999); V. A. Kostelecký and M. Mewes, *Phys. Rev. Lett*. **87**, 251304 (2001); *Phys. Rev*. D**66**, 056005 (2002).
C. Adam and F. R. Klinkhamer, *Nucl. Phys.* B **607**, 247 (2001); C. Adam and F.R. Klinkhamer, Phys. Lett. B** 513**, 245 (2001); V.A. Kostelecky and R. Lehnert, Phys. Rev. D** 63,** 065008 (2001); R. Lehnert, *Phys. Rev.* D **68**, 085003 (2003); R. Lehnert, *J. Math. Phys.* **45**, 3399 (2004); R. Montemayor and L.F. Urrutia, *Phys. Rev.* D **72**, 045018 (2005); Q. G. Bailey and V. Alan Kostelecky, *Phys. Rev.* D **70**, 076006 (2004); B. Altschul, *Phys. Rev.* D **72**, 085003 (2005); -ibid, *Phys. Rev.* D** 70** (2004) 056005; F.A. Brito, T. Mariz, J.R. Nascimento, E. Passos, R.F. Ribeiro, JHEP 0510 (2005) 019; O. Gagnon and G. D. Moore, *Phys. Rev. D* **70**, 065002 (2004); L.P. Colatto, A.L.A. Penna, W.C. Santos, *Eur. Phys. J*. **C** 36, 79 (2004); M. Lubo, *Phys. Rev.* D **71**, 047701 (2005); P. A. Bolokhov, S. G. Nibbelink, M.Pospelov, *Phys. Rev.* D**72,** 015013 (2005); M. M. Ferreira Jr, P*hys. Rev.* D **70,** 045013 (2004); -ibid, P*hys. Rev.* D **71,** 045003 (2005); A.A. Andrianov and R. Soldati, *Phys. Rev.* D** 51**, 5961 (1995); -ibid, *Phys. Lett.* B** 435**, 449 (1998); A.A. Andrianov, R. Soldati and L. Sorbo, *Phys. Rev.* D** 59**, 025002 (1999); R. Lehnert and R. Potting, *Phys. Rev. Lett*. **93**, 110402 (2004); -ibid, Phys. Rev. D **70**, 125010 (2004).
A. P. Baêta Scarpelli, H. Belich, J. L. Boldo and J. A. Helayël-Neto, *Phys. Rev.* D **67**, 085021 (2003); H. Belich, M.M. Ferreira Jr., J. A. Helayël-Neto and M.T.D. Orlando, Phys. Rev. D **67**, 125011 (2003); -ibid, Erratum to Phys. Rev. D **67**, 125011 (2003); H. Belich, M.M. Ferreira Jr., J. A. Helayël-Neto and M.T.D. Orlando, Phys. Rev. D **68**, 025005 (2003); H. Belich *et al*., *Phys.Rev*. D** 68,** 065030 (2003); Nucl. Phys. B** ** Suppl. **127**, 105 (2004); H. Belich, M. M. Ferreira Jr and J. A. Helayel-Neto, *Eur. Phys. J. C* **38**, 511 (2005); H. Belich, T. Costa-Soares, M.M. Ferreira Jr., J. A. Helayël-Neto, *Eur. Phys. J. C* **42,** 127 (2005).
R. S. Van Dyck Jr., P. B. Schwinberg and H. G. Dehmelt, Phys. Rev*.* Lett. **59**, 26 (1987); -ibid, Phys. Rev*.* D **34**, 722 (1986); L.S. Brown and G. Gabrielse, *Rev. Mod. Phys.* **58**, 233 (1986).
P. B. Schwinberg, R. S. Van Dyck Jr.and H. G. Dehmelt, Phys. Lett. **81**A, 119 (1981).
R. Bluhm, V.A. Kostelecky, and N. Russell, *Phys. Rev. Lett*. **79**, 1432 (1997); R. Bluhm, V.A. Kostelecky, and N. Russell, *Phys. Rev.* D **57**, 3932 (1998)**.**
R. Bluhm, V.A. Kostelecky, and C. Lane, *Phys. Rev. Lett*. **84**, 1098 (2000).
V.A. Kostelecky and C. Lane, *Phys. Rev.* D **60**, 116010 (1999); R. Bluhm, V.A. Kostelecky, C. Lane, and N. Russell, *Phys. Rev. Lett*. **88**, 090801 (2002).
R. Bluhm, V.A. Kostelecky, and N. Russell, *Phys. Rev. Lett*. **82**, 2254 (1999).
R. Bluhm and V.A. Kostelecky, *Phys. Rev. Lett*. **84**, 1381 (2000).
V.A. Kostelecky and C. D. Lane, J. Math. Phys. 40, 6245 (1999); R. Lehnert, *J. Math. Phys.* **8**, 3399 (2004).
M. M. Ferreira Jr and F. M. O. Mouchereck, hep-th/0601018.
H. Belich, T. Costa-Soares, M.M. Ferreira Jr. and J. A. Helayël-Neto, Eur. Phys. J. C **42**, 127 [2005]{}; H. Belich, T. Costa-Soares, M.M. Ferreira Jr., J. A. Helayël-Neto and M.T.D. Orlando, hep-th/0407260, to appear in Int. J. Mod. Phys. A (2006).
Y. Aharanov and A. Casher, Phys. Rev. Lett. **53**, 319 (1984); J. Anandan, Phys. Lett. B** 138**, 347, (1989); Erik Sjöqvist, Phys. Rev. Lett. **89**, 210401 (2002).
H. Belich, T. Costa-Soares, M.M. Ferreira Jr., J. A. Helayël-Neto, and M. T. D. Orlando, hep-th/0603091.
[^1]: In the such a representation, the Dirac matrices are written as: $\gamma
^{0}=\left(
\begin{tabular}{ll}
$1$ & $0$ \\
$0$ & $-1$\end{tabular}\right) ,$ $\overset{\rightarrow }{\gamma }=\left(
\begin{tabular}{ll}
$0$ & $$ \\
$-$ & $0$\end{tabular}\right) ,$ $\gamma _{5}=\left(
\begin{tabular}{ll}
$0$ & $1$ \\
$1$ & $0$\end{tabular}\right) ,$where $\overrightarrow{\sigma }=(\sigma _{x},\sigma _{y},\sigma
_{z})$ are the Pauli matrices.
[^2]: It is worthwhile to mention here that all calculations have been carried out in the Gaussian unit system, adopted through this work.
|
---
abstract: 'Due to advances in computer hardware and new algorithms, it is now possible to perform highly accurate many-body simulations of realistic materials with all their intrinsic complications. The success of these simulations leaves us with a conundrum: how do we extract useful physical models and insight from these simulations? In this article, we present a formal theory of downfolding–extracting an effective Hamiltonian from first-principles calculations. The theory maps the downfolding problem into fitting information derived from wave functions sampled from a low-energy subspace of the full Hilbert space. Since this fitting process most commonly uses reduced density matrices, we term it density matrix downfolding (DMD).'
author:
- |
Huihuo Zheng$^{1}$, Hitesh J. Changlani$^{2}$,\
Kiel T. Williams$^{3}$, Brian Busemeyer$^{3}$, and Lucas K. Wagner$^{3}$
bibliography:
- 'refs.bib'
title: From real materials to model Hamiltonians with density matrix downfolding
---
Introduction to downfolding the many electron problem
=====================================================
In multiscale modeling of many-particle systems, the effective Hamiltonian (or Lagrangian) is one of the most core concepts. The effective Hamiltonian dictates the behavior of the system on a coarse-grained level, where ‘sub-grid’ effects are folded into the parameters and form of the effective Hamiltonian. Many concepts in condensed matter physics can be viewed as statements about the behavior of the effective Hamiltonian. In particular, identification of ‘strongly correlated’ materials as materials where band theory is not an accurate representation of the systems is a statement about effective Hamiltonians. Effective Hamiltonians at different length scales also form the basis of the renormalization group [@Wilson]. A major goal in condensed matter physics is to determine what effective Hamiltonians apply to different physical situations, in particular quantum effective Hamiltonians, which lead to large-scale emergent quantum phenomena.
The dominant effective model for quantum particles in condensed systems is band structure, and for metals, Fermi liquid theory. However, a major challenge is how this paradigm should be altered when it is no longer a good description of the physical system. Examples of these include the high-T$_c$ cuprates and other transition metal oxides, which do not appear to be well-described by these simple effective Hamiltonians. For these systems, many models have been proposed, such as the Hubbard [@Hubbard1963], Kanamori [@Kanamori1963], $t$-$J$ [@tJSpalek] and Heisenberg models. While these models have been extensively studied analytically and numerically, and have significantly enhanced our understanding of the physics of correlated electrons, their effectiveness for describing a real complex system of interest is often unclear. At the same time, more complex effective models can be commensurately more difficult to solve, so one would like to also find an accurate effective model that is computationally tractable.
To address the need for a link between [*ab initio*]{} electron-level models and larger scale models, downfolding has most commonly been carried out using approaches based on density functional theory (DFT). The one particle part is obtained from a standard DFT calculation which is projected onto localized Wannier functions and gives an estimate of the effective hoppings of the lattice model based on Kohn-Sham band structure calculations [@Pavirini]. Then, to estimate the interactions, one assumes a model of screening of the Coulomb interactions based on constrained DFT, RPA, or some other methods. Since effects of interactions between the orbitals of interest have already been accounted for by DFT, a double counting correction is required to obtain the final downfolded Hamiltonian. The approach has been developed and widely applied [@Pavirini; @Dasgupta; @Aryasetiawan2004; @Jeschke2013]; but remains an active area of research [@Haule_doublecounting]. There are other downfolding approaches that include the traditional Löwdin method, coupled to a stochastic approach [@Tenno; @Zhou_Ceperley] and the related method of canonical transformations [@White_CT; @Yanai_CT]. While they have many advantages, it is typically not possible to know if a given model [*ansatz*]{} was a good guess or not, and it is very rare for a technique to provide an estimate of the quality of the resultant model.
The situation described above stands in contrast to the derivation of effective classical models. For concreteness, let us discuss classical force fields computed from [*ab initio*]{} electronic structure calculations. Typically, a data set is generated using an [*ab initio*]{} calculation in which the positions of the atoms and molecules are varied, creating a set of positions and energies. The parameters in the force field [*ansatz*]{} are varied to obtain a best-fit classical model. Then, using standard statistical tools, it is possible to assess how well the fit reproduces the [*ab initio*]{} data within the calculation, without appealing to experiment. While translating that error to error in properties is not a trivial task, this approach has the important advantage that in the limit of a high quality fit and high quality [*ab initio*]{} results, the resultant model is predictive.
Naïvely, one might think to reconcile the fitting approach used in classical force fields with quantum models by matching eigenstates between a quantum model and [*ab initio*]{} systems, varying the model parameters until the eigenstates match [@Wagner2013]. However, this strategy does not work well in practice because it is often not possible to obtain exact eigenstates for either the model or the [*ab initio*]{} system. To resolve this, we develop a general theory for generating effective quantum models that is exact when the wave functions are sampled from the manifold of low-energy states. Because this method is based on fitting the energy functional, we will show the practical application of this theory using both exact solutions and [*ab initio*]{} quantum Monte Carlo (QMC) to derive several different quantum models.
The endeavor we pursue here is to develop a multi-scale approach in which the effective interactions between quasiparticles (such as dressed electrons) are determined after an *ab initio* simulation (but not necessarily exact solution) of the continuum Schroedinger equation involving all the electrons. The method uses reduced density matrices (RDMs), of low-energy states, not necessarily eigenstates, to cast downfolding as a fitting problem. We thus call it density matrix downfolding (DMD). In this paper, our application of DMD to physical problems employ one body (1-RDM) and two body (2-RDM) density matrices. The many-body states used in DMD will typically be generated using QMC techniques \[either variational Monte Carlo (VMC) or diffusion Monte Carlo (DMC)\] to come close to the low energy manifold.
The remainder of the paper is organized as follows:
- In Section \[sec:theory\], we clarify and make precise what it means to downfold a many-electron problem to a few-electron problem. We recast the problem into minimization of a cost function that needs to be optimized to connect the many and few body problems. We further these notions both in terms of physical as well as information science descriptions, which allows us to connect to compression algorithms in the machine learning literature.
- Section \[sec:examples\] discusses several representative examples where we consider multiband lattice models and [*ab initio*]{} systems to downfold to a simpler lattice model.
- In Section \[sec:conclusion\], we discuss future prospects of applications of the DMD method, ongoing challenges and clear avenues for methodological improvements.
Downfolding as a compression of the energy functional {#sec:theory}
=====================================================
Theory
------
### Energy functional
Suppose we start with a quantum system with Hamiltonian $H$ and Hilbert space ${\mathcal H}$.
Let the energy functional be $E[\Psi] = \frac{\bra{\Psi}H\ket{\Psi}}{\braket{\Psi|\Psi}}$ for a wavefunction $\ket{\Psi} \in {\mathcal H}$.
\[theorem:criticalpoint\] $E[\Psi]$ has a critical point only where $\Psi$ is an eigenstate of $H$.
$$\begin{aligned}
\frac{\delta }{\delta \Psi^*} E[\Psi] = \frac{\delta}{\delta \Psi^*}\frac{\langle \Psi |H|\Psi\rangle}{\langle \Psi |\Psi\rangle} = \frac{H|\Psi\rangle}{\langle \Psi |\Psi\rangle} - \langle \Psi |H|\Psi\rangle \frac{|\Psi \rangle}{|\langle \Psi | \Psi\rangle|^2} =\frac{ (H-E[\Psi])|\Psi\rangle }{\langle\Psi|\Psi\rangle}\,.\end{aligned}$$
Therefore, $\frac{\delta }{\delta \Psi^*} E[\Psi] = 0$ if and only if $(H-E[\Psi])|\Psi\rangle =0$, i.e., $\Psi$ is an eigenvector of $H$ corresponding to eigenvalue $E[\Psi]$.
### Low energy space
Let $\mathcal{LE}(H,N)$ be a subset of ${\mathcal H}$ spanned by $N$ vectors given by the lowest energy solutions to $H\ket{\Psi_i}=E_i{\Psi_i}$.
$H_{eff}$ is an operator on the Hilbert space ${\mathcal {LE}(H,N)}$.
The effective model $E_{eff}[\Psi]=\frac{\bra{\Psi}H_{eff}\ket{\Psi}}{\braket{\Psi|\Psi}}$ is a functional from $\mathcal{LE} \rightarrow \mathbb{R}$.
If $\ket{\Psi}\in \mathcal{LE}$ and $\ket{\Phi}\in {\mathcal H} \setminus \mathcal{LE}$, then $\ket{\Psi} \oplus \ket{\Phi} \in {\mathcal H}$. In the following, we will use the direct sum operator $\oplus 0$ to translate between the larger ${\mathcal H}$ and the smaller $\mathcal{LE}$.
\[lemma:zeroderiv\] Suppose that $\ket{\Psi}\in \mathcal{LE}$ and $\ket{\Phi} \in {\mathcal H} \setminus \mathcal{LE}$. Then $\left.\frac{\delta E[\Psi \oplus \Phi]}{\delta \Phi}\right|_{\Phi=0}=0$.
$\langle \Psi\oplus 0 | H | 0\oplus \Phi \rangle=0$ because the two states have non-overlapping expansions in the eigenstates of $H$. Using that fact, we can evaluate $$\begin{aligned}
\left.\frac{\delta E[\Psi \oplus \Phi]}{\delta \Phi}\right|_{\Phi=0} &= \left.\frac{\left(H-E[\Psi\oplus\Phi]\right) \ket{\Phi} }{\braket{\Psi|\Psi} + \braket{\Phi|\Phi}} \right|_{\Phi=0} = 0.\end{aligned}$$
This is equivalent to noting that $H$ is block diagonal in the partitioning of ${\mathcal H}$ into $\mathcal{LE}$ and ${\mathcal H} \setminus \mathcal{LE}$. Importantly, if $\ket{\Psi} \in \mathcal{LE}$, then $\frac{\delta E[\Psi\oplus 0] }{\delta (\Psi\oplus 0)^*} = \ket{\Psi'} \oplus 0$, where $\ket{\Psi'} \in \mathcal{LE} $.
\[theorem:matching\] Assume $ E[\Psi\oplus 0] = E_{eff}[\Psi]+C$ for any $\ket{\Psi } \in \mathcal{LE}$, where $C$ is a constant. Then $(H_{eff}+C)|\Psi\rangle\oplus 0 = H (|\Psi\rangle \oplus 0)$.
Note that $$\begin{aligned}
\frac{\delta E[\Psi\oplus 0]}{\delta (\Psi\oplus 0)^*}=\frac{(H-E[\Psi\oplus 0])\ket{\Psi\oplus 0} }{\braket{\Psi|\Psi}}
\label{eqn:psider}\end{aligned}$$ and $$\begin{aligned}
\frac{\delta E_{eff}[\Psi]}{\delta \Psi^*}=\frac{(H_{eff}-E_{eff}[\Psi])\ket{\Psi} }{\braket{\Psi|\Psi}}.
\label{eqn:psieffder}\end{aligned}$$ Since the derivatives are equal, setting Eq. equal to Eq. , $$\begin{aligned}
H\ket{\Psi\oplus 0}= (H_{eff}+E[\Psi\oplus 0]-E_{eff}[\Psi])\ket{\Psi}\oplus 0 =(H_{eff}+C)\ket{\Psi}\oplus 0.\end{aligned}$$
Theorem \[theorem:matching\] combined with Lemma \[lemma:zeroderiv\] means that the eigenstates of $H_{eff}$ are be the same as the eigenstates of $H$ if its derivatives match $H$. Such $H_{eff}$ always exists. Let $H_{eff} = \sum_{i}^N E_i |\Psi_i\rangle \langle \Psi_i|$ where $|\Psi_i\rangle$’s are eigenstates belong to $\mathcal{LE}(H,N)$. This satisfies $E[\Psi] = E_{eff}[\Psi]$ and $H_{eff}|\Psi\rangle = H |\Psi \rangle$ for any $\Psi$ in $\mathcal{LE}(H,N)$.
We have thus reduced the problem of finding an effective Hamiltonian $H_{eff}$ that reproduces the low-energy spectrum of $H$ to matching the corresponding energy functionals $E[\Psi]$ and $E_{eff}[\Psi]$. This involves sampling the low-energy space, choosing the form of $H_{eff}$, and optimizing the parameters. An important implication of this is that it is not necessary to diagonalize either of the Hamiltonians; one must only be able to select wave functions from the low-energy space $\mathcal{LE}$. As we shall see, this can be substantially easier than attaining eigenstates.
Some further notes about this derivation:
- Fitting $\Psi$’s must come from $\mathcal{LE}$. It is not enough that the energy functional $E[\Psi]$ is less than some cutoff.
- In the case of sampling an approximate $\mathcal{LE}$, the error comes from non-parallelity of $E[\Psi]$ with the correct low energy manifold, up to a constant offset.
- While $H_{eff}$ is unique, it has many potential representations and approximations.
- Our method can be applied to any manifold spanned by eigenstates
- Model fitting is finding a compact approximation to $E_{eff}[\Psi]$. This is a high-dimensional space, so we use descriptors to do this.
- For operators that are not the Hamiltonian, it is possible to fit $\mathcal{O}_{eff}[\Psi] \simeq {\mathcal O}[\Psi]$ in a similar way. However, the eigenstates of ${\mathcal O}$ and ${\mathcal O}_{eff}$ will not coincide in general unless $\mathcal{O}$ commutes with the Hamiltonian.
The theory presented above maps coarse-graining into a functional approximation problem. This is still rather intimidating, since even supposing one can generate wave functions in the low-energy space, they are still complicated objects in a very large space. An effective way to accomplish this is through the use of descriptors, $d_i[\Psi]$, which map from ${\mathcal H} \rightarrow \mathbb{R}$. Then we can approximate the energy functional as follows $$E_{eff}[\Psi] \simeq \sum_i f_i(d_i[\Psi]),$$ where $f_i$ are some parameterized functions. This will allow us to use techniques from statistical learning to efficiently describe $E_{eff}$.
Practical protocol
------------------
= \[diamond, draw, fill=blue!10, text width=4.5em, text badly centered, node distance=3cm, inner sep=0pt\] = \[rectangle, draw, fill=blue!10, text width=5em, text centered, rounded corners, minimum height=4em\] = \[rectangle, draw, fill=red!10, text width=5em, text centered, rounded corners, minimum height=4em\] = \[draw,-latex’,very thick\] = \[draw, ellipse,fill=red!20, node distance=3cm, minimum height=2em\]
(wfs) [Generate $\ket{\Psi_i} \in \mathcal{LE}$]{}; (descriptors) [Generate $d_j[\Psi_i]$,$E[\Psi_i]$]{}; (assess) [Assess descriptors]{}; (ansatz) [Ansatz: $E_i \simeq \sum_j p_j d_{ij}$]{}; (fit) [Fit optimal model]{}; (model) [Effective model]{}; (wfs) – (descriptors); (descriptors) – (assess); (assess) – (ansatz); (ansatz) – (fit); (fit) – (model);
(assess.south) – ($ (assess.south) + (0,-0.2)$) – node \[below\] [Incomplete sampling]{} ($ (wfs.south) + (0,-0.2)$) – (wfs.south); (assess.north) – ($ (assess.north) + (0,0.2)$) – node \[above\] [Incomplete descriptor space]{} ($ (descriptors.north) + (0,0.2)$) – (descriptors.north);
A practical protocol is presented in Figure \[fig:protocol\]. In this section we go through this procedure step by step.
#### Generating $\ket{\Psi_i}\in \mathcal{LE}$
Ideally one would be able to sample the entire low-energy space. Typically, however, the space will be too large and it will need to be sampled. The optimal wave functions to use depend on the models one expects to fit, which we will discuss in detail in later steps. Simple strategies that we will use in the examples below include excitations with respect to a determinant and varying spin states.
#### Generate $d_j[\Psi_i]$ and $E[\Psi_i]$
The choice of descriptor is fundamental to the success of the downfolding. In the case of a second-quantized Hamiltonian $$H_{eff} = E_0 + \sum_{ij} t_{ij} (c_i^\dagger c_j + h.c.) + \sum_{ijkl} V_{ijkl} c_i^\dagger c_j^\dagger c_k c_l,$$ a set of linear descriptors by simply taking the expectation value of both sides of the equation. Then for example, the occupation descriptor for orbital $k$ is $d_{occ(k)}[\Psi_i] = \braket{\Psi_i | c_k^\dagger c_k | \Psi_i}$; the double occupation descriptor for orbital $k$ is $d_{double(k)}[\Psi_i] = \braket{\Psi_i | n_{k\uparrow}n_{k\downarrow} | \Psi_i}$. The orbital that $c_k$ represents is part of the descriptor, and in the examples below we will discuss this choice as well. One is not limited to static orbital descriptors; they may have a more complex functional dependence on the trial function to include orbital relaxation.
#### Assess descriptors
At this point, one has collected the data $E_i$ and $d_{ij}$. If two descriptors have a large correlation coefficient, then they are redundant in the data set. This could either mean that the sampling of the low-energy Hilbert space $\mathcal{LE}$ was insufficient, or that they are both proxies for the same differences in states. If two data points have the same or very similar descriptor sets, but different energies, then either the descriptor set is not enough to describe the variations in the low-energy space, or the sampling has generated states that are not in the low-energy space. To resolve these possibilities, one should analyze the difference between the two wave functions.
In either case, when the model is accurate, the fits will be accurate. If descriptors values available in the reduced Hilbert space are not represented in the sampled wave functions, then intruder states can appear upon solution of the effective model. In that case, the model fitting is an extrapolation instead of an interpolation. For this reason it is desirable to have eigenstates or near-eigenstates in the sample set if possible; they are guaranteed to be on the corners of the descriptor space if the model is accurate.
#### Ansatz: $E_i \simeq \sum_i d_{ij} p_j$
If the descriptors are chosen well, then the model can be written in linear form: $$E[\Psi_i] = \sum_j p_j d_j[\Psi_i],$$ which we shorten to $${\bf E} = D{\bf p} .
\label{eqn:EdP}$$ If this can be done, the fitting problem is reduced to a linear regression optimization. More complex functions of the descriptors are also possible, although at the cost of making the effective model more difficult to solve and complicating the fitting procedure.
#### Fit optimal model
Finally, one wishes to find a set of parameters such that Eq. is satisfied as closely as possible. There are many choices to make in this step, which will often depend on the desired properties of the final model. One can imagine choosing different cost functions to minimize, which can also include a penalty for complicated models. In our tests, we have successfully used LASSO [@Lasso] and matching pursuit techniques [@MP_Zhang1993] to select high quality and compact model parameters. A detailed example of using the latter technique is presented in Section \[subsection:fese\].
Representative Examples {#sec:examples}
=======================
Given the theoretical framework for downfolding a many orbital (or many-electron) problem to a few orbital (or few-electron) problem, we now discuss examples which elucidate the DMD method. The examples are as follows:
- Section \[subsection:3band\]: Three-band Hubbard $\rightarrow$ one-band Hubbard at half filling. Demonstrates finding a basis set for the second quantized operators and uses a set of eigenstates directly sampled from the low-energy space to find a one-band model.
- Section \[subsection:1dhydrogen\]: Hydrogen chain $\rightarrow$ one-band Hubbard model at half filling. Demonstrates basis sets for [*ab initio*]{} systems and the possibility to use this technique to determine the quality of a model to a given physical situation.
- Section \[subsection:graphene\]: Graphene $\rightarrow$ one-band Hubbard model with and without $\sigma$ electrons. Demonstrates using the downfolding procedure to examine the effects of screening due to core electrons.
- Section \[subsection:fese\]: FeSe molecule $\rightarrow$ $3d,4p,4s$ system. Demonstrates the use of matching pursuit to assess the importance of terms in an effective model and to select compact effective models.
In all examples we will highlight the important ingredients associated with DMD. First and foremost is the choice of low energy space or energy window i.e. how our database of wave functions was generated. Associated with this is the choice of the one body space in terms of which the effective Hamiltonian is expressed. Finally, we discuss aspects of the functional forms or parameterizations that are expected to describe our physical problem. An important effective Hamiltonian that enters three out of our four representative examples is the one-band or single-band Hubbard model: $$H = E_0 -t \;\sum_{\langle i,j \rangle, \eta} \tilde{d}_{i,\eta}^{\dagger} \tilde{d}_{j,\eta} + U \;\sum_{i} \tilde{n}^{i}_{\uparrow} \tilde{n}^{i}_{\downarrow}\,,
\label{eq:oneband}$$ where $t$ and $U$ are downfolded (renormalized) parameters, $\eta$ is a spin index, $\tilde{d}_{i,\eta}$ is the effective one-particle operator associated with spatial orbital (or site) $i$ and $n_{i,\eta}=\tilde{d}_{i,\eta}^{\dagger} \tilde{d}_{i,\eta}$ is the corresponding number operator. $\langle i,j \rangle$ is used to denote nearest neighbor pairs. We will sometimes drop the constant energy shift $E_0$ when we write equations like Eq. .
Three-band Hubbard model to one-band Hubbard model at half filling {#subsection:3band}
------------------------------------------------------------------
Our first example is motivated by the high $T_c$ superconducting cuprates [@Bednorz1986] that have parent Mott insulators with rich phase diagrams on electron or hole doping [@Dagotto_RevModPhys; @LeeWen_RevModPhys]. Many works have been devoted to their model Hamiltonians and corresponding parameter values [@tJSpalek; @Pavirini; @Emery; @ZhangRice; @Hybertsen_PRB1989; @Hybertsen_PRB1990; @Kent_Hubbard]. A minimal model involving both the copper and oxygen degrees of freedom is the three-orbital or three-band Hubbard model, $$\begin{aligned}
H &=& \epsilon_p \sum_{j\in p,\eta} n_{j,\eta} + \epsilon_{d} \sum_{i \in d,\eta} n_{i,\eta}
+ t_{pd} \sum_{\langle i\in d ,j \in p \rangle, \eta} \text{sgn}(p_i,d_j) \Big( c_{i,\eta}^{\dagger} c_{j,\eta} + \text{h.c.} \Big) \nonumber \\
& & + U_p \sum_{j\in p} n_{j,\uparrow} n_{j,\downarrow} + U_d \sum_{i\in d} n_{i,\uparrow} n_{i,\downarrow} + V_{pd} \sum_{\langle i \in p ,j \in d \rangle} n_j n_i\,,\end{aligned}$$ where $d_i,p_j$ refer to the $d_{x^2 - y^2}$ orbitals of copper at site $i$ and $p_x$ or $p_y$ oxygen at site $j$, respectively. $\text{sgn}(p_i,d_j)$ is the sign of the hopping $t_{pd}$ between nearest neighbors, shown schematically in Figure \[fig:threeband\]. $\epsilon_d$ and $\epsilon_p$ are orbital energies, $U_d$ and $U_p$ are strengths of onsite Hubbard interactions, and $V_{pd}$ is the strength of the density-density interactions between a neighboring $p$ and $d$ orbital. To simplify we consider only the case where $\epsilon_p$, $U_d$ and $t_{pd}$ are non zero; $t_{pd}$ is chosen throughout this section to be the typical value of $1.3$ eV to give the reader a sense of overall energy scales. Since we work with fixed number of particles we set our reference zero energy to be $\epsilon_d = 0$, thus the charge transfer energy $\Delta \equiv \epsilon_p - \epsilon_d$ equals $\epsilon_p$ in our notation. We work in the hole notation; half filling corresponds to two spin-up and two spin-down holes on the $2\times2$ cell.
![Schematic for downfolding the three-band Hubbard model on the $2\times2$ cell to the one-band Hubbard model. The oxygen orbitals are eliminated to give “dressed" $d$-like orbitals of the one-band model, with modified hopping and interaction parameters. The relationship between the $\tilde{d}$ and the copper and oxygen orbitals is encoded by a linear transformation which is parameterized by $\alpha_1$, $\alpha_2$, $\alpha_3$, $\alpha_4$ and $F$ (see Appendix for more details). Here the diameter of the circles has been shown to be proportionate to the magnitude of $F$ or $\alpha_i$.[]{data-label="fig:threeband"}](three_band_figure.eps){width="0.9\linewidth"}
It is our objective to determine what one-band Hubbard model \[Eq. \] “best" describes the three-band data. The effective *d-like* orbitals $\tilde{d}_{i,\eta}$, that enter the low energy description are mixtures of copper and oxygen orbitals; this optimal transformation also remains an unknown. Thus the model determination involves two aspects (1) what are the composite objects that give a compact description of the low energy physics? and (2) given this choice what are the effective interactions between them? (A similar problem was posed and solved by one of us in the context of spin systems [@Changlani_percolation].) In addition, the best effective Hamiltonian description depends on the energy scale of interest. All these issues will be addressed in the remainder of the section.
We begin by encoding the relationship between the bare and effective operators as a linear transformation ${\bf T}$, $$\tilde{d}_{i,\eta} = \sum_{j} T_{ij} c_{j,\eta}
\label{eq:dc}$$ where $c_{j,\eta}$ is the hole (destruction) operator and refers to either the bare $d$ or $p$ orbitals. Further generalizations of this relationship (for example, including higher body terms) are also possible, but have not been considered here. For the $2\times2$ unit cell [**T**]{} is a $4 \times 12 $ matrix, which we parameterize by four distinct parameters. These correspond to mixing of a copper orbital with nearest neighbor oxygens ($\alpha_1$), nearest neighbor coppers ($\alpha_2$), next-nearest neighbor oxygens ($\alpha_3$) and next-nearest neighbor coppers ($\alpha_4$) as shown schematically in Figure \[fig:threeband\]. The explicit form of ${\bf T}$ after accounting for the symmetries of the lattice has been written out in the Appendix. These parameters are optimized to minimize a certain cost function, which will be explained shortly.
All RDMs in the three-band and one-band descriptions are also related via ${\bf T}$; the ones that we focus on are evaluated in eigenstate $s$ and are given by,
$$\begin{aligned}
\langle {\tilde{d}_{i,\eta}}^{\dagger} \tilde{d}_{j,\eta} \rangle_{s} &=& \sum_{mn} T^{*}_{im} \langle {c_{m,\eta}}^{\dagger} c_{n,\eta} \rangle_{s} T_{jn} \label{eq:dmstransformations1} \,,\\
\langle \tilde{n}_{i,\uparrow} \tilde{n}_{i,\downarrow} \rangle_{s} &=& \sum_{jkmn} T^{*}_{ij} T^{*}_{im} \langle {c_{j,\uparrow}}^{\dagger} {c_{m,\downarrow}}^{\dagger} c_{n,\downarrow} c_{k,\uparrow} \rangle_{s} T_{in} T_{ik}\,.
\label{eq:dmstransformations2}\end{aligned}$$
We optimize ${\bf T}$ by demanding two conditions be satisfied, (1) the effective orbitals ($\tilde{d}_{i,\eta}$) are orthogonal to each other i.e. $\Big({\bf T} {\bf T}^{\dagger}\Big)_{mn} = \delta_{mn}$ and (2) the sum of all diagonal entries (trace) of the 1-RDM of the effective orbitals for all low energy eigenstates equals the number of electrons of a given spin i.e. $\sum_{i} \sum_{\eta} \langle {\tilde{d}_{i,\eta}}^{\dagger} \tilde{d}_{i,\eta} \rangle_{s} = N_{\eta}$. These conditions are enforced by minimizing a cost function, $$C = \sum_{s} \sum_{\eta} \Big( \sum_{i} \langle \tilde{d}_{i,\eta}^{\dagger} \tilde{d}_{i,\eta} \rangle_{s} - N_{\eta} \Big)^{2} + \sum_{mn} ( \Big({\bf T} {\bf T}^{\dagger}\Big)_{mn} -\delta_{mn})^{2}\,.
\label{eq:C}$$ For the $2\times2$ cell, $N_{\uparrow}=N_{\downarrow}=2$ and $i=1,2,3,4$. The number of states $s$ was varied from three to six, depending on the energy window of interest.
Figure \[fig:varyUdep\] shows regimes of the three-band model where the lowest six eigenstates are separated from the higher energy manifold; the fourth and fifth eigenstates are degenerate. In the large $U_d$ limit, charge fluctuations are suppressed and these six states correspond to the Hilbert space of $4 \choose 2$ states of the effective spin model in its $S_z=0$ sector. These states have primarily *d-like* character, an aspect we will verify in this section. The eigenstates outside of this manifold involve *p-like* excitations which the one-band model is not designed to capture.
We chose the lowest three eigenstates of the three-band model for minimizing the cost in Eq. . The four dimensional space of parameters of ${\bf T}$ was scanned for this purpose. The corresponding trace and orthogonality conditions are simultaneously satisfied with only small deviations, confirming the validity of Eq. . Importantly, the 1-RDM elements in the transformed basis corresponding to nearest neighbors $\langle \tilde{d}_1^{\dagger} \tilde{d}_2 \rangle_s$ already provide estimates for $U/t$ of the effective model. Since the exact knowledge of the corresponding eigenstates of the one-band Hubbard model is available for arbitrary $U/t$ by exact diagonalization, we directly look up the $U/t$ with the same 1-RDM value. These estimates complement the one obtained by DMD which was carried out with the same three low-energy eigenstates, using their energies and the computed values of $\langle \tilde{d}_1^{\dagger} \tilde{d}_2 \rangle_s$ and $\langle \tilde{n}_{i,\uparrow} \tilde{n}_{i,\downarrow} \rangle_{s}$ from Eqs. and . [^1] A representative example of our results for $U_{d}/t_{pd}=8$ and $\Delta/t_{pd}=3$ has been discussed in the Appendix.
Some trends in the one-band description are explored in Figure \[fig:varyUdep\] by monitoring the downfolded parameters as a function of varying $\Delta/t_{pd}$ and $U_d/t_{pd}$. For example, when $U_d/t_{pd}=8$ is fixed and $\Delta/t_{pd}$ is increased, we find that the effective hopping $t$ decreases and $U/t$ increases. This is physically reasonable since an increasing difference in the single particle energies of the copper and oxygen orbitals makes it energetically unfavorable for holes to hop between the two orbitals. When $\Delta/t_{pd}=3$ is fixed and $U_d/t_{pd}$ is increased, $U/t$ increases. As one mechanism of avoiding the large $U_d$, the copper orbitals are forced to hybridize more with the oxygen ones; on the other hand, hole delocalization is suppressed in a bid to maintain mostly one hole per $\tilde{d}$ due to the larger $U/t$. The net result of these effects is that the $t$ also increases. An important check for the one-band model is its ability to reproduce the low energy gaps of the three-band model; these have been compared in Figure \[fig:energyfit\]. For the case of $\Delta/t_{pd}=3$, we observe that for all $U_d/t_{pd}$ the lowest three eigenstates were reproduced well. This model also reproduces the states outside of the DMD energy window, although with slightly larger errors. Similar trends are seen for the case of $\Delta/t_{pd}=5$, with the noticeable difference being that the energy error of the highest state has reduced. This also reflects that the parameters obtained from DMD are, in general, dependent on the energy window of interest, a point which we will highlight shortly by investigating it systematically.
------------------- --
[ 1= ]{} [ 1= ]{}
------------------- --
A promise of downfolding is the reduction of the size of the effective Hilbert space; allowing simulations of bigger unit cells to be carried out. To show that this actually works well in practice for the three-band case, we consider the $2\sqrt{2} \times 2 \sqrt{2}$ square unit cell, comprising of 8 copper and 16 oxygen orbitals. For representative test cases, we performed exact diagonalization calculations at half filling; the Hilbert space comprises of 112,911,876 basis states. Roughly 200 Lanczos iterations were carried out, enabling convergence of the lowest four energies. We compared the lowest gaps with the corresponding calculation on the one-band model on the same square geometry, with a Hilbert space size of only 4,900, using the downfolded parameters obtained from the smaller $2 \times 2$ cell.
Our results are summarized in Figure \[fig:predictivity\]. Panel (A) shows the six representative parameter sets of the three-band model and the corresponding downfolded one-band parameters. Panels (B) and (C) show the lowest three energy gaps for representative values of $U_d/t_{pd}=4,8,12$ for $\Delta/t_{pd}=3$ and $\Delta/t_{pd}=5$ respectively. In all cases, the agreement between the three-band and one-band models is remarkably good. The energy gap error of the lowest gap is within $0.0004$ eV (1% relative error). The largest error in the third gap is of the order of $0.005$ eV (3% relative error). These results indicate the reliability of the downfolding procedure and highlight its predictive power.
[ 1= ]{} [ 1= ]{}
Until this point, all our results focused on downfolding using only the lowest three eigenstates of the $2\times2$ cell. We now explore the effect of increasing the energy window, by including higher eigenstates, using our test example of $U_d/t_{pd}=8$ and $\Delta/t_{pd}=3$. To do so, we now use all six low energy eigenstates for optimizing the cost function in Eq. . We find similar (but not exactly the same) values of $\alpha_i$ compared to the case when only the three lowest states were used. The fact that a solution with small cost can be attained confirms our expectation that the entire low energy space of six states is consistently described by a set of $\tilde{d}_i$ operators.
------------------- --
[ 1= ]{} [ 1= ]{}
------------------- --
However, as Figure \[fig:windows\](A) shows, the estimates of $U/t$ and $t$ depend on how many eigenstates are used in the DMD procedure. This is because the DMD aims to provide the one-band description that best describes *all* states in a given window. If the model is not perfect within a given energy window, an energy dependent model is expected, consistent with the renormalization group perspective. For our test example, increasing the number of eigenstates from three to six changed $U/t$ from $13.8$ to $9.44$ and $t$ from $0.3045$ to $0.2750$ eV. [^2]
The features associated with the energy dependence are further confirmed in Figure \[fig:windows\](B). which shows a comparison of energy gaps of the three-band and downfolded one-band model on the $2\times2$ cell. When only three states are used, the one-band (nearest neighbor) Hubbard model is insufficient for *accurately* describing states outside the window. When all six states are used, the DMD tries to minimize the error of the largest energy gap at the cost of errors in the smaller energy gaps. One could of course choose a different parameterization, say with additional next nearest neighbor $t'$, for which is may be possible to reduce this energy dependence significantly and thus have a model that describes the smaller and larger energy scales equally well.
One dimensional hydrogen chain {#subsection:1dhydrogen}
------------------------------
We now move on to one of the simplest extended *ab initio* systems, a hydrogen chain in one dimension with periodic boundary conditions. The one-dimensional hydrogen chain has been used as a model for validating a variety of modern *ab initio* many-body methods [@H10_Simons]. We consider the case of $10$ atoms with periodic boundary conditions and work in a regime where the inter-atomic distance $r$ is in the range $1.5 - 3.0$ Å, such that the system is well described in terms of primarily $s$-like orbitals.
For a given $r$, we first obtain single-particle Kohn-Sham orbitals from a set of spin-unrestricted and spin-restricted DFT-PBE calculations. The localized orbital basis upon which the RDMs (descriptors) are evaluated is obtained by generating intrinsic atomic orbitals (IAO) [@knizia_intrinsic_2013] from the Kohn-Sham orbitals orthogonalized using the Löwdin procedure (see Figure \[fig:fit\_quality\]). These are the orbitals that enter the one-band Hubbard Hamiltonian. Then, to generate a database of wavefunctions needed for the DMD, we produce a set of Slater-Jastrow wavefunctions consisting of singles and doubles excitations to the Slater determinant:
$$\begin{aligned}
| s \rangle = & e^J \Big[a^\dagger_{i \eta} a_{k \eta} | KS \rangle \Big] \,,\\
| d \rangle = & \: e^J \Big[a^\dagger_{i \eta} a^\dagger_{j \eta'} a_{k \eta'} a_{l \eta} | KS \rangle\Big] ,\end{aligned}$$
where $|KS\rangle$ is the Slater determinant of occupied Kohn-Sham orbitals, $\eta \neq \eta'$ are spin indices, and $a_{i}^\dagger$ ($a_{i}$) is a single-electron creation (destruction) operator corresponding to a particular Kohn-Sham orbital. The $k,l$ indices label occupied orbitals in the original Slater determinant, while $i,j$ are virtual orbitals. $e^J$ is a Jastrow factor optimized by minimizing the variance of the local energy.
We compute the energies (expectation values of the Hamiltonian) and the RDMs for each wave function within DMC. By computing the trace of the resulting 1-RDMs, we verify that all the electrons present in the system are represented within the localized basis of $s$-like orbitals. If the trace of the 1-RDM deviates from the nominal number of electrons for a particular state by more than some chosen threshold - 2% in this example - it indicates that some orbitals are occupied ($2s$- or $2p$-like orbitals for hydrogen) that are not represented within the localized IAO basis used for computing the descriptors. Hence, these states do not exist within the $\mathcal{LE}$ space, and cannot be described by a one-band $s$-orbital model. We exclude such states from the wave function set. The acquired data is then used in DMD to downfold to a one-band Hubbard Hamiltonian.
------------------- --
[ 1= ]{} [ 1= ]{}
------------------- --
Figure \[fig:fit\_quality\] shows the fitting results of the energy functional $E[\Psi]$ within the sampled $\mathcal{LE}$ for two representative distances (1.5 and 2.25Å). As we can see, the model $E_{eff}[\Psi]$ reproduces the *ab initio* $E[\Psi]$ up to certain error that decreases with atomic separation. That is, the fitted Hubbard model provides a more accurate description as separation distance increases, and the system becomes more atomic-like.
Figure \[fig:Parameters-vs-Bond-t\] shows the fitted values of the downfolding parameters $t$ and $U/t$ at various distances. $t$ decreases as the interatomic distance increases, and the value of $U/t$ increases. The single-band Hubbard model qualitatively captures how the system approaches the atomic limit, in which $t$ becomes zero.
---------------------------- --
[ 1= ]{} [ 1= ]{} [ 1= ]{}
---------------------------- --
The R$^2$ values obtained from fitting the descriptors to the *ab initio* energy \[see Figure \[fig:Parameters-vs-Bond-t\](C)\] also show that the single-band Hubbard model is a good description of the system at large distances, but not at small distances. This is primarily because the dynamics of other degrees of freedom (e.g. $2s$ and $2p$ orbitals) become important to the low energy spectrum at small distances. Other interaction terms beyond the on-site Hubbard $U$, such as nearest-neighbor Coulomb interactions and Heisenberg coupling, can also become significant. Without including higher orbitals or additional many-body interaction terms, the model gives rise to an incorrect insulator state at small distances. Conversely, at larger separations ($r>1.8$Å), where the system is in an insulator phase [@Stella2011], the model provides a better description.
Graphene and hydrogen honeycomb lattice {#subsection:graphene}
---------------------------------------
Our third example highlights the role of the high energy degrees of freedom not present in the low energy description but which are instrumental in renormalizing the effective interactions. We demonstrate this by considering the case of graphene, and by comparing it to artificially constructed counterparts without the high energy electrons. Although many electronic properties of graphene can be adequately described by a noninteracting tight-binding model of $\pi$ electrons [@Castro2009], electron-electron interactions are crucial for explaining a wide range of phenomena observed in experiments [@Kotov2012]. In particular, electron screening from $\sigma$ bonding renormalizes the low energy plasmon frequency of the $\pi$ electrons [@Zheng2016]. In fact a system of $\pi$ electrons with bare Coulomb interactions has been shown to be an insulator instead of a semimetal [@DrutPRL2009; @DrutPRB2009; @Smith2014; @Zheng2016]. Using DMD, we demonstrate how the screening effect of $\sigma$ electrons is manifested in the low energy effective model of graphene.
In order to disentangle the screening effect of $\sigma$ electrons from the bare interactions between $\pi$ electrons, we apply DMD to three different systems, graphene, $\pi$-only graphene, and a honeycomb lattice of hydrogen atoms. In the $\pi$-only graphene, the $\sigma$ electrons are replaced with a static constant negative charge background. The role of $\sigma$ electrons is then clarified by comparing the effective model Hamiltonians of these two systems. The hydrogen system we study has the same lattice constant $a=2.46$ Å as graphene, which has a similar Dirac cone dispersion as graphene [@Zheng2016].
By constructing the one-body space by Wannier localizing Kohn-Sham orbitals obtained from DFT calculations (see Figure \[fig:honeycomb\_wan\]), we verify that the low energy degrees of freedom correspond to the $\pi$ orbitals in graphene and its $\pi$-only system and $s$ orbitals in hydrogen; these enter the effective one-band Hubbard model description in Eq. . Due to the vanishing density of states at the Fermi level, the Coulomb interaction remains long-ranged, in contrast to usual metals where the formation of electron-hole pairs screens the interactions strongly [@Zheng2016]. However, for certain aspects, the long ranged part can be considered as renormalizing the onsite Coulomb interaction $U$ at low energy [@Schuler2013; @Changlani2015].
------------------- --
[ 1= ]{} [ 1= ]{}
------------------- --
To estimate the one-band Hubbard parameters, we used the DMD method using a set of 50 Slater-Jastrow wave functions that correspond to the electron-hole excitations within the $\pi$ channel for the graphene systems or $s$ channel for the hydrogen system. In particular, for graphene, the Slater-Jastrow wave functions are constructed from occupied $\sigma$ bands and occupied $\pi$ bands, whereas for $\pi$-only graphene, Slater-Jastrow wave functions constructed from occupied $\pi$ Kohn-Sham orbitals of graphene. The *ab initio* simulations were performed on a $3\times3$ cell (32 carbons or hydrogens) and the energy and RDMs of these wave functions were evaluated with VMC. The error bars on our downfolded parameters are estimated using the jackknife method [@Jackknife1981]. The results from our calculations are summarized in Figure \[fig:ne\_aidmd\_gh\].
---------------------------- --
[ 1= ]{} [ 1= ]{} [ 1= ]{}
---------------------------- --
We find that the one-band Hubbard model describes graphene and hydrogen very well, as is seen from the fact that $R^2$ is closed to 1 for the fits. Our fits are shown in Figure \[fig:ne\_aidmd\_gh\]. For both graphene and hydrogen, $U/t$ is smaller than the critical value of the semimetal-insulator transition $(U/t)_c \approx 3.8$ for the honeycomb lattice [@Sorella2012], which is consistent with both systems being semimetals. The two systems indeed have similar hopping constant $t$, consistent with the fact that they have similar Fermi velocities at the Dirac point. However, the difference in their high energy structure manifests itself as differently renormalized electron-electrons interactions, explaining the difference in $U$. Most prominently, the $\pi$-only system has much larger $U/t$ ($\sim4.9$) compared to graphene, which is large enough to push it into the insulating (antiferromagnetic) phase. Thus, downfolding shows the clear significance of $\sigma$ electrons in renormalizing the effective onsite interactions of the $\pi$ orbitals,making graphene a weakly interacting semimetal instead of an insulator.
FeSe diatomic molecule {#subsection:fese}
----------------------
Transition metal systems are often difficult to model due to the many orbital and possibly magnetic descriptors introduced by $d$ electrons. This is seen in the proliferation of models for transition metals, which include terms like spin-spin coupling, spin-orbital coupling, hopping, Hund’s like coupling, and so on. Models containing all possible descriptors are unwieldy, and it is difficult to determine which degrees of freedom are needed for a minimal model to reproduce an interesting effect. Transition metal systems are challenging to describe using most electronic structure methods because of the strong electron correlations and multiple oxidation states possible in these systems. Fixed-node DMC has been shown to be a highly accurate method on transition metal materials in improving the description of the ground state properties and energy gaps [@Foyevtsova2014; @Wagner_Abbamonte; @Zheng2015; @Wagner2016]. In this section, we apply DMD using fixed-node DMC to quantify the importance of various interactions in a FeSe diatomic molecule with a bond length equal to that of the iron based superconductor, FeSe [@kumar_crystal_2010], in order to help identifying the descriptors that may be relevant in the bulk material.
{width="80.00000%"}
We considered a low-energy space spanned by the Se $4p$, Fe $3d$, and Fe $4s$ orbitals. We sampled singles and doubles excitations from a reference Slater determinant of Kohn-Sham orbitals taken from DFT calculations with PBE0 functional with total spin 0, 2, and 4, which were then multiplied by a Jastrow factor and further optimized using fixed-node DMC. After this procedure, 241 states were within a low energy window of 8 eV. Of these, eight states had a significant iron $4p$ component, which excludes them from the low-energy subspace. This leaves us with 233 states in the low-energy subspace.
We consider a set of 21 possible descriptors consisting of local operators on the iron $4s$, iron $3d$ states, and selenium $4p$ states, which is a total of 9 single-particle orbitals. We use the same IAO construction as Section \[subsection:1dhydrogen\] to generate the basis for these operators. At the one-body level, we consider orbital energy descriptors:
$$\begin{aligned}
&\epsilon_{s} n_s,&
&\epsilon_{\pi,\mathrm{Se}} (n_{p_x} + n_{p_y}), &
&\epsilon_{z} n_{p_z},&
\nonumber \\
&\epsilon_{z^2} n_{d_{z^2}},&
&\epsilon_{\pi,\mathrm{Fe}} (n_{d_{xz}} + n_{d_{yz}}).&
&\epsilon_{\delta} (n_{d_{xy}} + n_{d_{x^2-y^2}}),&\end{aligned}$$
and the symmetry-allowed hopping terms:
$$\begin{aligned}
&t_{\sigma,d} \sum_{\eta} \left( c_{d_{z^2},\eta}^{\dagger} c_{p_z,\eta} + \text{h.c.} \right),&
&t_{\sigma,s} \sum_{\eta} \left(c_{s,\eta}^{\dagger} c_{p_z,\eta} + \text{h.c.} \right),&
&t_{\pi} \sum_{\eta} \left(c_{d_{xz},\eta}^{\dagger} c_{p_x,\eta} + c_{d_{yz},\eta}^{\dagger} c_{p_y,\eta} + \text{h.c.} \right).&\end{aligned}$$
As before, $\eta$ represents the spin index. At the two-body level, we consider Hubbard interactions:
$$\begin{aligned}
&U_p \sum_{i \in p} n_{i,\uparrow} n_{i,\downarrow},&
&U_{d,\delta} \sum_{i\in \{d_{xy},d_{x^2-y^2}\}} n_{i,\uparrow} n_{i,\downarrow},&
\nonumber \\
&U_d \sum_{i \in d} n_{i,\uparrow} n_{i,\downarrow},&
&U_{d,\pi} \sum_{i\in \{d_{xz},d_{yz}\}} n_{i,\uparrow} n_{i,\downarrow},&
&U_{d_{z^2}} n_{d_{z^2},\uparrow} n_{d_{z^2},\downarrow},&\end{aligned}$$
where $p$ refers to the Se-$4p$ orbitals and $d$ refers to the Fe-$3d$ orbitals. Importantly, we also account for the Hund’s coupling terms for the iron atom:
$$\begin{aligned}
&J \sum_{\substack{i\ne j \\i,j \in d}} S_i \cdot S_j,&
&J_{\delta} S_{d_{xy}} \cdot S_{d_{x^2-y^2}},&
&J_{\delta,d_{z^2}} (S_{d_{xy}} + S_{d_{x^2-y^2}}) \cdot S_{d_{z^2}},& \label{eqn:hund1}
\nonumber \\
&J_{\pi} S_{d_{xz}} \cdot S_{d_{yz}},&
&J_{\pi,d_{z^2}} (S_{d_{xz}} + S_{d_{yz}}) \cdot S_{d_{z^2}}.&
&J_{\pi,\delta} (S_{d_{xz}} + S_{d_{yz}}) \cdot (S_{d_{xy}} + S_{d_{x^2-y^2}}),&\end{aligned}$$
Finally, we also add a nearest neighbor Hubbard interaction: $V \sum_{i\in p, j\in d} n_{i} n_j$.
To generate a minimal description of the system, we employed a matching pursuit (MP) method [@MP_Zhang1993]. MP chooses to add descriptors based on their correlation with the residual of the linear fit. We started with a model that only consists of $E_0$. The Hund’s coupling descriptor \[first term in Eq. \] has the largest correlation coefficient with the residual fit, so it is added first. The fact that the Hund’s coupling is chosen first in MP is consistent with the several studies in the literature, which find a prominent Hund’s coupling can explain some of the properties of bulk FeSe. [@demedici_hunds_2011; @de_medici_janus-faced_2011; @georges_strong_2013; @busemeyer_competing_2016]. Next, MP includes the descriptor that correlates most strongly with the residuals of this first minimal model, in this case the hopping between $d$ and $p$ $\sigma$-symmetry orbitals. We repeated this procedure until the RMS error did not improve more than 0.05 eV upon adding a new parameter. This criterion was chosen to strike a balance between the complexity of the model and the accuracy in reproducing the sample set.
The following model was produced: $$\begin{aligned}
H_{eff} &=& \epsilon_{\delta,\mathrm{Fe}} (n_{d_{xy}} + n_{d_{x^2-y^2}}) + \epsilon_s n_{s}+\epsilon_{z} n_{p_z} \nonumber \\
&&+ t_{\sigma,d} \sum_{\eta} \left( c_{d_{z^2},\eta}^{\dagger} c_{p_z,\eta} + \text{h.c.} \right)+t_{\sigma,s} \sum_{\eta} \left(c_{s,\eta}^{\dagger} c_{p_z,\eta} + \text{h.c.} \right)
\nonumber \\
&&+ U_d \sum_{i \in d} n_{i,\uparrow} n_{i,\downarrow} + J \sum_{\substack{i\ne j \\i,j \in d}} S_i \cdot S_j + E_0. \label{eq:fesemodel}\end{aligned}$$ As before, $\eta$ is the spin index and $i$ is the orbital index, and $d$ is the set of iron $3d$ orbitals, as above. $E_0$ is an overall energy shift, also included as a fit parameter. The parameter values and corresponding error of each model produced by MP are shown in Figure \[fig:fese\]. Note that all parameters may change at each step because the entire model is refitted when an addition parameter is included in each iteration. The parameters are smoothly varying with the inclusion of new parameters, and they take the correct signs based on symmetry (where applicable). The RMS error decreases with each additional parameter, but less so as the algorithm appends additional parameters. Eventually the diminishing improvements do not merit the additional complexity of more parameters.
Conclusion and Future prospects {#sec:conclusion}
===============================
The density matrix downfolding (DMD) technique uses data derived from low-energy approximate solutions to a high energy Hamiltonian to systematically determine an effective Hamiltonian that describes the low-energy behavior of the system. It is based on several rather simple proofs which occupy a role similar to the variational principle; they allow us to know which effective models are closer to the correct solution than others. The method is very general and does not require a quasiparticle picture to apply, and neither does it have double-counting issues. It treats all interactions on an equal footing, so hopping parameters are naturally modified by interaction parameters and so on. While most of the applications have used the first principles quantum Monte Carlo method to obtain the low-energy solutions, the method is completely general and can be used with any solution method that can produce high quality energy and reduced density matrices. We have discussed several examples to present the conceptual and algorithmic aspects of DMD.
The resultant lattice model can be efficiently and accurately solved for large system sizes [@LeBlanc_PRX] using techniques designed and suited for small local Hilbert spaces; these include exact or selected diagonalization [@DeRaedt; @Tubman_selci; @Holmes_Tubman_Umrigar], density matrix renormalization group (DMRG) [@White1992], tensor networks [@PEPS; @Changlani_CPS; @NeuscammanCPS], dynamical mean field theory (DMFT) [@Kotliar2006], density matrix embedding (DMET) [@DMET_2012] and lattice QMC methods [@Scalapino; @Trivedi_Ceperley; @Zhang_AFQMC; @Sandvik_loops; @Prokofiev; @Booth2009; @SQMC; @Holmes_Changlani_Umrigar; @Booth2013]. These methods have also been used to obtain excited states, dynamical correlation functions and thermal properties, that have been difficult to obtain in *ab initio* approaches.
DMD, though conceptually simple, is still a method in its development stages, with room for algorithmic improvements and new applications. Advances in the field of inverse problems [@Berg2017] could be incorporated into DMD to mitigate the problems associated with optimization and over-fitting. Here we briefly outline some aspects that need further research:
1. The wave function database ($\ket{\Psi} \in \mathcal{LE}$): The DMD method relies crucially on the availability of a low energy space of *ab initio* wave functions. While these wave functions do not have to be eigenstates, automating their construction remains challenging and realistically requires knowledge of the physics to be described.
2. Optimal choice of basis functions. The second-quantized operators in the effective Hamiltonian correspond to a basis in the continuum. The quality of the model depends on the basis describing the changes between low-energy wave functions accurately.
3. Form of the low energy model Hamiltonian. While the exact effective Hamiltonian is unique, there may be many ways of approximating it with varying levels of compactness and accuracy.
The advantage of the DMD framework is that all these can be resolved internally. Given a good sampling of $\mathcal{LE}$, (2) and (3) can be resolved using regression. Given that (2) and (3) are correct or near correct, then (1) can be resolved by finding binding planes, as noted in Section \[sec:theory\]. The method thus has a degree of self consistency; it will return low errors only when 1-3 are correct.
We have shown applications to strongly correlated models (3-band), [*ab initio*]{} bulk systems hydrogen chain and graphene, and a transition metal molecule FeSe. The technique is on the verge of being applied to transition metal bulk systems; there are no major barriers to this other than a polynomially scaling computational cost and the substantial amount of work involved in parameterizing and fitting models to these systems. Looking into the future, we anticipate that this technique can help with the definition of a correlated materials genome–what effective Hamiltonian best describes a given material is highly relevant to its behavior.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank David Ceperley, Richard Martin, Cyrus Umrigar, Garnet Chan, Shiwei Zhang, Steven White, Lubos Mitas, So Hirata, Bryan Clark, Norm Tubman, Miles Stoudenmire and Victor Chua for extremely useful and insightful discussions. This work was funded by the grant DOE FG02-12ER46875 (SciDAC). HZ acknowledges support from Argonne Leadership Computing Facility, a U.S. Department of Energy, Office of Science User Facility under Contract DE-AC02-06CH11357. HJC acknowledges support from the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-FG02-08ER46544 for his work at the Institute for Quantum Matter (IQM). KTW acknowledges support from the National Science Foundation under the Graduate Research Fellowship Program, Fellowship No. DGE-1144245. This research is part of the Blue Waters sustained-petascale computing project, which is supported by the National Science Foundation (awards OCI-0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications.
Author Contributions {#author-contributions .unnumbered}
====================
HJC, HZ and LKW conceived the initial DMD ideas and designed the project and organization of the paper. All authors contributed to the theoretical developments and various representative *ab initio* and lattice examples. All authors contributed to the analysis of the data, discussions and writing of the manuscript. LKW oversaw the project. HJC and HZ contributed equally to this work.
Appendix {#appendix .unnumbered}
========
In Section \[subsection:3band\] we discussed parameterizing the transformation ${\bf T}$ as a $4\times12$ matrix for the $2 \times 2$ unit cell, in terms of $\alpha_1$, $\alpha_2$, $\alpha_3$ and $\alpha_4$. Using the numbering of the orbitals corresponding to Figure \[fig:threeband\], the explicit form of ${\bf T}$ is, $$\begin{aligned}
{\bf T} =
\left(
\begin{array}{cccccccccccc}
F & \alpha_2 & \alpha_2 & \alpha_4 & \alpha_1 & \alpha_1 & -\alpha_1 & -\alpha_1 & \alpha_3 & -\alpha_3 & \alpha_3 & -\alpha_3 \\
\alpha_2 & F & \alpha_4 & \alpha_2 & \alpha_3 & -\alpha_1 & \alpha_1 & -\alpha_3 & -\alpha_3 & \alpha_3 & \alpha_1 & -\alpha_1 \\
\alpha_2 & \alpha_4 & F & \alpha_2 & -\alpha_1 & \alpha_3 & -\alpha_3 & \alpha_1 & \alpha_1 & -\alpha_1 & -\alpha_3 & \alpha_3 \\
\alpha_4 & \alpha_2 & \alpha_2 & F & -\alpha_3 & -\alpha_3 & \alpha_3 & \alpha_3 & -\alpha_1 & \alpha_1 & -\alpha_1 & \alpha_1 \\
\end{array}
\right)\,,\end{aligned}$$ where we have defined $F \equiv \sqrt{1-4{\alpha_1}^2 - 2{\alpha_2}^2 - 4 {\alpha_3}^2 -{\alpha_4}^2}$.
A concrete and representative example of our results, shown in Section \[subsection:3band\] for the $2\times2$ cell, is explained for the case of $U_d/t_{pd}=8$ and $\Delta/t_{pd}=3$. The first task was to obtain the optimal transformation ${\bf T}$ for which the lowest three eigenstates ($s=1,2,3$) of the three-band model were used for computing the cost in Eq. . The minimum of the cost was determined by a brute force scan in the four dimensional space of $\alpha$’s and using a linear grid spacing of $0.002$ found $\alpha_1=0.216$, $\alpha_2=0.042$, $\alpha_3=0.018$ and $\alpha_4=0.016$. The two terms in the cost i.e. the trace and orthogonality conditions are individually satisfied to a relative error of less than 0.5 percent.
$\langle c_i^{\dagger} c_j \rangle_s$ and $\langle {c_{j,\uparrow}}^{\dagger} {c_{m,\downarrow}}^{\dagger} c_{n,\downarrow} c_{k,\uparrow} \rangle_{s}$ were computed from the exact knowledge of the three-band model eigenstates and hence $\langle {\tilde{d}_{i,\eta}}^{\dagger} \tilde{d}_{j,\eta} \rangle_{s}$ and $\langle \tilde{n}_{i,\uparrow} \tilde{n}_{i,\downarrow} \rangle_{s}$ are obtained once the optimal ${\bf T}$ has been determined. As mentioned in the main text, the value of $\langle \tilde{d}_{1,\eta}^{\dagger} \tilde{d}_{2,\eta} \rangle_s$ provides estimates for $U/t$ of the effective model by direct comparison of its value to that in the corresponding eigenstate in the one-band model. For our test example, the absolute values of $\langle \tilde{d}_{1,\uparrow}^{\dagger} \tilde{d}_{2,\uparrow} \rangle_s$ in states $s=1,2,3$ are approximately $0.159$, $0.142$ and $0.084$ respectively which correspond to $(U/t)_1 \approx 14.1 $, $(U/t)_2 \approx 13.2 $, $(U/t)_3 \approx 12.7 $. Performing DMD with the three eigenenergies and their calculated RDMs gave $t=0.3025$ eV and $U/t = 13.45 $; the latter in the correct range of the other estimates.
[^1]: We also mimicked the situation characteristic of *ab initio* examples where no eigenstates are generally available. Several non eigenstates were generated as random linear combinations of the lowest three eigenstates and input into the DMD procedure, with similar outcomes.
[^2]: When three states were used for optimizing the orbitals and for the DMD, we found $U/t\approx 13.45$ and $t=0.3025$ eV. This is because $\tilde{d}$ are slightly different in the two cases.
|
---
abstract: |
Obtaining semantic labels on a large scale radiology image database (215,786 key images from 61,845 unique patients) is a prerequisite yet bottleneck to train highly effective deep convolutional neural network (CNN) models for image recognition. Nevertheless, conventional methods for collecting image labels (e.g., Google search followed by crowd-sourcing) are not applicable due to the formidable difficulties of medical annotation tasks for those who are not clinically trained. This type of image labeling task remains non-trivial even for radiologists due to uncertainty and possible drastic inter-observer variation or inconsistency.
In this paper, we present a looped deep pseudo-task optimization procedure for automatic category discovery of visually coherent and clinically semantic (concept) clusters. Our system can be initialized by domain-specific (CNN trained on radiology images and text report derived labels) or generic (ImageNet based) CNN models. Afterwards, a sequence of pseudo-tasks are exploited by the looped deep image feature clustering (to refine image labels) and deep CNN training/classification using new labels (to obtain more task representative deep features). Our method is conceptually simple and based on the hypothesized “convergence” of better labels leading to better trained CNN models which in turn feed more effective deep image features to facilitate more meaningful clustering/labels. We have empirically validated the convergence and demonstrated promising quantitative and qualitative results. Category labels of significantly higher quality than those in previous work are discovered. This allows for further investigation of the hierarchical semantic nature of the given large-scale radiology image database.
author:
- |
Xiaosong Wang Le Lu Hoo-chang Shin Lauren Kim Isabella Nogues Jianhua Yao\
Ronald Summers\
Imaging Biomarkers and Computer-aided Detection Laboratory\
Department of Radiology and Imaging Sciences\
National Institutes of Health Clinical Center\
10 Center Drive, Bethesda, MD 20892\
[xiaosong.wang,le.lu,hoo-chang.shin,lauren.kim,isabella.nogues,rms@nih.gov, jyao@cc.nih.gov]{}
bibliography:
- 'LDPO\_eccv2016.bib'
title: |
Unsupervised Category Discovery via Looped Deep Pseudo-Task Optimization\
Using a Large Scale Radiology Image Database
---
Introduction
============
The rapid and tremendous success of applying deep convolutional neural networks (CNNs) [@krizhevsky2012imagenet; @simonyan2014very; @Szegedy2014Going] to many challenging computer vision tasks derives from the accessibility of the well-annotated ImageNet [@deng2009imagenet; @russakovsky2014imagenet] and PASCAL VOC [@Everingham2015Pascal] datasets. Deep CNNs perform significantly better than previous shallow learning methods and hand-crafted image features, however, at the cost of requiring greater amounts of training data. ImageNet pre-trained deep CNN models [@jia2014caffe; @krizhevsky2012imagenet; @Lin2015Nin] serve an indispensable role to be bootstrapped upon for all externally-sourced data exploitation tasks [@Liang2015Baby; @Chen2015Webly]. In the medical domain, however, no comparable labeled large-scale image dataset is available except the recent [@Shin2015]. Vast amounts of radiology images/reports are stored in many hospitals’ Picture Archiving and Communication Systems (PACS), but the main challenge lies in how to obtain ImageNet-level semantic labels on a large collection of medical images [@Shin2015].
Nevertheless, conventional means of collecting image labels (e.g. Google image search using the terms from WordNet ontology hierarchy [@Miller1995], SUN/PLACE databases [@Xiao2010Sun; @Zhou2014Place] or NEIL knowledge base [@Chen2013Neil]; followed by crowd-sourcing [@deng2009imagenet]) are not applicable due to 1) the formidable difficulties of medical annotation tasks for clinically untrained annotators, 2) unavailability of a high quality or large capacity medical image search engine. On the other hand, even for well-trained radiologists, this type of “assigning labels to images” task is not aligned with their regular diagnostic routine work so that drastic inter-observer variations or inconsistency may be demonstrated. The protocols of defining image labels based on visible anatomic structures (often multiple), or pathological findings (possibly multiple) or using both cues have a lot of ambiguity.
Shin et al. [@Shin2015] first extract the sentences depicting disease reference key images (similar concept to “key frames in videos”) using natural language processing (NLP) out of $\sim780$K patients’ radiology reports, and find 215,786 key images of 61,845 unique patients from PACS. Then, image categorization labels are mined via unsupervised hierarchical Bayesian document clustering, i.e. generative latent Dirichlet allocation (LDA) topic modeling [@blei2003latent], to form 80 classes at the first level of hierarchy. The purely text-computed category information offers some coarse level of radiology semantics but is limited in two aspects: 1) The classes are [*highly unbalanced*]{}, in which one dominating category contains 113,037 images while other classes contain a few dozens. 2) The classes are not [*visually*]{} coherent. As a result, transfer learning from the CNN models trained in [@Shin2015] to other medical computer-aided detection (CAD) problems performs less compellingly than those transferred directly from ImageNet CNNs [@Shin2015Deep; @krizhevsky2012imagenet; @Szegedy2014Going]. In this paper, we present a [**L**]{}ooped [**D**]{}eep [**P**]{}seudo-task [**O**]{}ptimization (LDPO) approach for automatic category discovery of visually coherent and clinically semantic (concept) clusters. The true semantic category information is assumed to be latent and not directly observable. The main idea is to learn and train CNN models using pseudo-task labels (when human annotated labels are unavailable) and iterate this process with the expectation that pseudo-task labels will eventually resemble latent true image categories. Our work is partly related to the recent progress of semi-supervised learning or self-taught image classification, which has advanced both image classification and clustering processes [@Singh2012DiscPat; @Raina2007Self; @Li2011Towards; @Juneja2013Blocks; @Dai2015EnProDeepFets; @Dai2013EnPro]. The iterative optimization in [@Singh2012DiscPat; @Juneja2013Blocks] seeks to identify discriminative local visual patterns and reject others, whereas our goal is to set better labels for all images during iterations towards auto-annotation. Our contributions are in several fold. **1)**, We propose a new “iteratively updated” deep CNN representation based on the LDPO technique. Thus it requires no hand-crafted image feature engineering [@Singh2012DiscPat; @Raina2007Self; @Li2011Towards; @Juneja2013Blocks] which may be challenging for a large scale medical image database. Our method is conceptually simple and based on the hypothesized “convergence” of better labels lead to better trained CNN models which in turn, offer more effective deep image features to facilitate more meaningful clustering/labels. [*This looped property is unique to deep CNN classification-clustering models since other types of classifiers do not learn better image features simultaneously.*]{} We use the database from [@Shin2015] to conduct experiments with the proposed method in different LDPO settings. Specifically, different pseudo-task initialization strategies, two CNN architectures of varying depths (i.e., AlexNet [@krizhevsky2012imagenet] and GoogLeNet [@Szegedy2014Going]), different deep feature encoding schemes [@Cimpoi2015Filter; @Cimpoi2015Deep] and clustering via K-means only or over-fragmented K-means followed by Regularized Information Maximization (RIM [@Gomes2010Discriminative] as an effective model selection method), are extensively explored and empirically evaluated. **2)**, We consider the deep feature clustering followed by supervised CNN training as the outer loop and the deep feature clustering as the inner loop. Model selection on the number of clusters is critical and we carefully employ over-fragmented K-means followed by RIM model pruning/tuning to implement this criterion. This helps prevent cluster labeling amongst similar images, which can consequently compromise the CNN model training in the outer loop iteration. **3)**, The convergence of our LDPO framework can be observed and measured in both the cluster-similarity score plots and the CNN training classification accuracies. **4)**, Given the deep CNN LDPO models, hierarchical category relationships in a tree-like structure can be naturally formulated and computed from the final pairwise CNN classification confusion measures, as described in \[sec:hcl\]. We will make our discovered image annotations (after reviewed and verified by board-certified radiologists in a with-humans-in-the-loop fashion [@Yu2015Construction]) together with trained CNN models publicly available upon publication.
To the best of our knowledge, this is the first work exploiting to integrate unsupervised deep feature clustering and supervised deep label classification for self-annotating a large scale radiology image database where the conventional means of image annotation are not feasible. The measurable LDPO “convergence” makes this ill-posed problem well constrained, at no human labeling costs. Our proposed LDPO method is also quantitatively validated using Texture-25 dataset [@Dai2015EnProDeepFets; @Lazebnik2005Sparse] where the “unsupervised” classification accuracy improves over LDPO iterations. The ground truth labels of texture images [@Dai2015EnProDeepFets; @Lazebnik2005Sparse] are known and used to measure the accuracy scores against LDPO clustering labels. Our results may grant the possibility of 1), investigating the hierarchical semantic nature (object/organ, pathology, scene, modality, etc.) of categories [@Rematas2015Dataset; @Johnson2015Image]; 2), finer level image mining for tag-constrained object instance discovery and detection [@Wu2015Harvesting; @Bazzani2015Self], given the large-scale radiology image database.
Related Work
============
[**Unsupervised and Semi-supervised Learning:**]{} Dai *et al.* [@Dai2015EnProDeepFets; @Dai2013EnPro] study the semi-supervised image classification/clustering problem on texture [@Lazebnik2005Sparse], small to middle-scale object classes (e.g., Caltech-101 [@FeiFei2004101]) and scene recognition datasets [@Quattoni2009indoor]. By exploiting the data distribution patterns that are encoded by so called ensemble projection (EP) on a rich set of visual prototypes, the new image representation derived from clustering is learned for recognition. Graph based approaches [@Liu2010Large; @Kingma2014SSL] are used to link the unlabeled image instances to labeled ones as anchors and propagate labels by exploiting the graph topology and connectiveness weights. In an unsupervised manner, Coates *et al.* [@Coates2011] employ k-means to mine image patch filters and then utilize the resulted filters for feature computation. Surrogate classes are obtained by augmenting each image patch with its geometrically transformed versions and a CNN is trained on top of these surrogate classes to generate features [@Dosovitskiy2014]. Wang *et al.* [@Wang2015Unsupervised] design a Siamese-triplet CNN network, leveraging object tracking information in $100$K unlabeled videos to provide the supervision for visual representation learning. Our work initializes an unlabeled image collection with labels from a pseudo-task (e.g., text topic modeling generated labels [@Shin2015]) and update the labels through an iterative looped optimization of deep CNN feature clustering and CNN model training (towards better deep image features).
[**Text and Image:**]{} [@berg2013babytalk] is a seminal work that models the semantic connections between image contents and the text sentences. Those texts describe cues of detecting objects of interest, attributes and prepositions and can be applied as contextual regularizations. [@Karpathy2015Deep] proposes a structured objective to align the CNN based image region descriptors and bidirectional Recurrent Neural Networks (RNN) over sentences through the multimodal embedding. [@Vinyals2015Show] presents a deep recurrent architecture from “Sequence to Sequence” machine translation [@Sutskever2014Sequence] to generate image description in natural sentences, via maximizing the likelihood of the target description sentence given the training image. [@Sun2015Automatic] applies extensive NLP parsing techniques (e.g., unigram terms and grammatical relations) to extract concepts that are consequently filtered by the discriminative power of visual cues and grouped by joint visual and semantic similarities. [@Chen2015Sense] further investigates an image/text co-clustering framework to disambiguate the multiple semantic senses of some Polysemy words. The NLP parsing in radiology reports is arguably much harder than processing those public datasets of image captions [@Karpathy2015Deep; @Vinyals2015Show; @berg2013babytalk] where most plain text descriptions are provided. Radiologists often rule out or indicate pathology/disease terms, [*not existing in the corresponding key images*]{}, but based on patient priors and other long-range contexts or abstractions. In [@Shin2015Interleaved], only $\sim8$% key images (18K out of 216K) can be tagged from NLP with the moderate confidence levels. We exploit the interactions from the text-derived image labels, to the proposed LDPO (mainly operating in the image modality) and the final term extraction from image groups.
[**Domain Transfer and Auto-annotation:**]{} Deep CNN representation has made transfer learning or domain adaption among different image datasets practical, via straightforward fine-tuning [@Girshick2015RCNN; @Razavian2014CNN]. Using pre-trained deep CNNs allows for the cross-domain transfer between weakly supervised video labels and noisy image labels. It can further output localized action frames by mutually filtering out low CNN-confidence instances [@Sun2015Temporal]. A novel CNN architecture is exploited for deep domain transfer to handle unlabeled and sparsely labeled target domain data [@Tzeng2015Simultaneous]. An image label auto-annotation approach is addressed via multiple instance learning [@Wu2015Deep] but the target domain is restricted to a small subset (25 out of 1000 classes) of ImageNet [@deng2009imagenet] and SUN [@Xiao2010Sun]. [@Wigness2015] introduces a method to identify a hierarchical set of unlabeled data clusters (spanning a spectrum of visual concept granularities) that are efficiently labeled to produce high performing classifiers (thus less label noise at instance level). By learning visually coherent and class balanced labels through LDPO, we expect that the studied large-scale radiology image database can markedly improve its feasibility in domain transfer to specific CAD problems where very limited training data are available per task.
Looped Deep Pseudo-Task Optimization {#sec-method}
====================================
Traditional detection and classification problems in medical imaging, e.g. Computer Aided Detection (CAD) [@roth2015improving], require precise labels of lesions or diseases as the training/testing ground-truth. This usually requires a large amount of annotation from well-trained medical professionals (especially at the era of “deep learning”). Employing and converting the medical records stored in the PACS into labels or tags is very challenging [@Shin2015Interleaved]. Our approach performs the category discovery in an empirical manner and returns accurate key-word category labels for all images, through an iterative framework of deep feature extraction, clustering, and deep CNN model fine-tuning.
As illustrated in Fig. \[fig:flowchart:png\], the iterative process begins by extracting the deep CNN feature based on either a fine-tuned (with high-uncertainty radiological topic labels [@Shin2015]) or generic (from ImageNet labels [@krizhevsky2012imagenet]) CNN model. Next, the deep feature clustering with $k$-means or $k$-means followed by RIM is exploited. By evaluating the purity and mutual information between discovered clusters, the system either terminates the current iteration (which leads to an optimized clustering output) or takes the refined cluster labels as the input to fine-tune the CNN model for the following iteration. Once the visually coherent image clusters are obtained, the system further extracts semantically meaningful text words for each cluster. All corresponding patient reports per category cluster are finally adopted for the NLP. Furthermore, the hierarchical category relationship is built using the class confusion measures of the latest converged CNN classification models.
{width="1.0\linewidth"}
Convolution Neural Networks
---------------------------
The proposed LDPO framework is applicable to a variety of CNN models. We analyze the CNN activations from layers of different depths in AlexNet [@krizhevsky2012imagenet] and GoogLeNet [@Szegedy2014Going]. Pre-trained models on the ImageNet ILSVRC data are obtained from Caffe Model Zoo [@jia2014caffe]. We also employ the Caffe CNN implementation [@jia2014caffe] to perform fine-tuning on pre-trained CNNs using the key image database (from [@Shin2015]). Both CNN models with/without fine-tuning are used to initialize the looped optimization. AlexNet is a common CNN architecture with 7 layers and the extracted features from its convolutional or fully-connected layers have been broadly investigated [@Girshick2015RCNN; @Razavian2014CNN; @Karpathy2015Deep]. The encoded convolutional features for image retrieval tasks are introduced in [@Ng15], which verifies the image representation power of convolutional features. In our experiments we adopt feature activations of both the 5th convolutional layer $Conv5$ and 7th fully-connected (FC) layer $FC7$ as suggested in [@Cimpoi2015Deep; @Chatfield14]. GoogLeNet is a much deeper CNN architecture compared to AlexNet, which comprises 9 inception modules and an average pooling layer. Each inception modules is truly a set of convolutional layers with multiple window sizes, i.e. $1\times1, 3\times3, 5\times5$. Similarly, we explore the deep image features from the last inception layer $Inception5b$ and final pooling layer $Pool5$. Table \[tab:model\] illustrates the detailed model layers and their activation dimensions.
Encoding Images using Deep CNN Features
---------------------------------------
While the features extracted from fully-connected layer are able to capture the overall layout of objects inside the image, features computed at the last convolution layer preserve the local activations of images. Different from the standard max-pooling before feeding the fully-connected layer, we adopt the same setting ([@Cimpoi2015Filter]) to encode the convolutional layer outputs in a form of dense pooling via Fisher Vector (FV) [@Perronnin2010FV] and Vector Locally Aggregated Descriptor (VLAD) [@Jegou2012VLAD]. Nevertheless, the dimensions of encoded features are much higher than those of the FC feature. Since there is redundant information from the encoded features and we intend to make the results comparable between different encoding schemes, Principal Component Analysis (PCA) is performed to reduce the dimensionality to 4096, equivalent to the FC features’ dimension.
Given a pre-trained (generic or domain-specific) CNN model (i.e., Alexnet or GoogLeNet), an input image $I$ is resized to fit the model definition and feed into the CNN model to extract features $\{f^{L}_{i,j}\}$ ($1\leqslant i,j\leqslant s^{L}$) from the $L$-th convolutional layer with dimensions $s^{L}\times s^{L}\times d^{L}$, e.g., $13\times13\times256$ of $Conv5$ in AlexNet and $7\times7\times1024$ of $Pool5$ in GoogLeNet. For Fisher Vector implementation, we use the settings as suggested in [@Cimpoi2015Deep]: 64 Gaussian components are adopted to train the Gaussian mixture Model(GMM). The dimension of resulted FV features is significantly higher than $FC7$’s, i.e. $32768 (2\times64\times256) \ vs \ 4096$. After PCA, the FV representation per image is reduced to a $4096$-component vector. A list of deep image features, the encoding methods and output dimensions are provided in Table \[tab:model\]. To be consistent with the settings of FV representation, we initialize the VLAD encoding of convolutional image features by $k$-means clustering with $k=64$. Thus the dimensions of VLAD descriptors are $16384(64\times256)$ of $Conv5$ in AlexNet and $65536(64\times1024)$ of $Inception5b$ in GoogLeNet. PCA further reduces the dimensions of both to $4096$.
Image Clustering
----------------
Image clustering plays an indispensable role in our LDPO framework. We hypothesize that the newly generated clusters driven by looped pseudo-task optimization are better than the previous ones in the following terms: 1) Images in each cluster are visually more coherent and discriminative from instances in other clusters; 2) The numbers of images per cluster are approximately equivalent to achieve class balance; 3) The number of clusters is self-adaptive according to the statistical properties of a large collection of image data. Two clustering methods are employed here, i.e. $k$-means alone and an over-segmented $k$-means (where $K$ is much larger than the first setting, e.g., 1000) followed by Regularized Information Maximization (RIM) [@Gomes2010Discriminative] for model selection and optimization.
$k$-means is an efficient clustering algorithm provided that the number of clusters is known. We explore $k$-means clustering here for two reasons: 1) To set up the baseline performance of clustering on deep CNN image features by fixing the number of clusters $k$ at each iteration; 2) To initialize the RIM clustering since $k$-means is only capable of fulfilling our first two hypotheses, and RIM will help satisfy the third. Unlike $k$-means, RIM works with fewer assumptions on the data and categories, e.g. the number of clusters. It is designed for discriminative clustering by maximizing the mutual information between data and the resulted categories via a complexity regularization term. The objective function is defined as $$f(\mathbf{W};\mathbf{F},\lambda)=I_{\mathbf{W}}\{c;\mathbf{f}\}-R(\mathbf{W};\lambda),
\label{eq:RIM:objective}$$ where $c\in\{1,...,K\}$ is a category label, $\mathbf{F}$ is the set of image features $\mathbf{f_{i}}=(f_{i1},...,f_{iD})^{T}\in\mathbb{R}^{D}$. $I_{\mathbf{W}}\{c;\mathbf{f}\}$ is an estimation of the mutual information between the feature vector $\mathbf{f}$ and the label $c$ under the conditional model $p(c|\mathbf{f},\mathbf{W})$. $R(\mathbf{W};\lambda)$ is the complexity penalty and specified according to $p(c|\mathbf{f},\mathbf{W})$. As demonstrated in [@Gomes2010Discriminative], we adopt the unsupervised multilogit regression cost. The conditional model and the regularization term are consequently defined as $$\begin{aligned}
p(c=k|\mathbf{f},\mathbf{W})&\propto& exp(w^{T}_{k}\mathbf{f}+b_{k}) \\ R(\mathbf{W};\lambda)&=&\lambda\sum_{k}w^{T}_{k}w_{k},
\label{eq:RIM:multilogit}\end{aligned}$$ where $\mathbf{W}=\{\mathbf{w}_{1},...,\mathbf{w}_{K},b_{1},...,b_{K}\}$ is the set of parameters and $\mathbf{w}_{k}\in\mathbb{R}^{D},b_{k}\in\mathbb{R}$. Maximizing the objective function is now equivalent to solving a logistic regression problem. $R$ is the $L_{2}$ regulator of weight $\{w_{k}\}$ and its power is controlled by $\lambda$. Large $\lambda$ values will enforce to reduce the total number of categories considering that no penalty is given for unpopulated categories [@Gomes2010Discriminative]. This characteristic enables RIM to attain the optimal number of categories coherent to the data. $\lambda$ is fixed to $1$ in all our experiment.
Convergence in Clustering and Classification {#sec:cl}
--------------------------------------------
Before exporting the newly generated cluster labels to fine-tune the CNN model of the next iteration, the LDPO framework will evaluate the quality of clustering to decide if convergence has been achieved. Two convergence measurements have been adopted [@Tuytelaars09], i.e., Purity and Normalized Mutual Information (NMI). We take these two criteria as forms of empirical similarity examination between two clustering results from adjacent iterations. If the similarity is above a certain threshold, we believe the optimal clustering-based categorization of the data is reached. We indeed find that the final number of categories from the RIM process in later LDPO iterations stabilize around a constant number. The convergence on classification is directly observable through the increasing top-1, top-5 classification accuracy levels in the initial few LDPO rounds which eventually fluctuate slightly at higher values. Convergence in clustering is achieved by adopting the underlying classification capability stored in those deep CNN features through the looped optimization, which accents the visual coherence amongst images inside each cluster. Nevertheless, the category discovery of medical images will further entail clinically semantic labeling of the images. From the optimized clusters, we collect the associated text reports for each image and assemble each cluster’s text reports together as a unit. Then NLP is performed on each report unit to find highly recurring words to serve as key word labels for each cluster by simply counting and ranking the frequency of each word. Common words to all clusters are removed from the list. The resultant key words and randomly sampled exemplary images are ultimately compiled for review by board-certified radiologists. This process shares some analogy to the human-machine collaborated image database construction [@Yu2015Construction; @Wigness2015]. In future work, NLP parsing (especially term negation/assertion) and clustering can be integrated into LDPO framework.
Hierarchical Category Relationship {#sec:hcl}
----------------------------------
ImageNet [@deng2009imagenet] are constructed according to WordNet ontology hierarchy [@Miller1995]. Recently, a new formalism so-called Hierarchy and Exclusion (HEX) graphs has been introduced [@Deng2014Large] to perform object classification by exploiting the rich structure of real world labels [@deng2009imagenet; @krizhevsky2012imagenet]. In this work, our converged CNN classification model can be further extended to explore the hierarchical class relationship in a tree representation. First, the pairwise class similarity or affinity score $A_{i,j}$ between class (i,j) is modeled via an adapted measurement from CNN classification confusion [@Chen2015Webly].
A\_[i,j]{} &= (Prob(i|j) + Prob(j|i) ) &\
&= ( + ) \[eq:HCL:rij\]
where $C_i$, $C_j$ are the image sets for class $i$,$j$ respectively, $|\cdot|$ is the cardinality function, $CNN(I_m|j)$ is the CNN classification score of image $I_m$ from class $C_i$ at class $j$ obtained directly by the N-way CNN flat-softmax. Here $A_{i,j} = A_{j,i}$ is symmetric by averaging $Prob(i|j)$ and $Prob(j|i)$.
Affinity Propagation algorithm [@frey07affinitypropagation] (AP) is invoked to perform “tuning parameter-free” clustering on this pairwise affinity matrix $\{A_{i,j}\} \in \mathbb{R}^{K\times K}$. This process can be executed recursively to generate a hierarchically merged category tree. Without loss of generality, we assume that at level L, classes $i^L$,$j^L$ are formed by merging classes at level L-1 through AP clustering. The new affinity score $A_{i^L,j^L}$ is computed as follows. [ $$\begin{aligned}
A_{i^L,j^L} = \frac{1}{2} \Big(Prob(i^L|j^L) + Prob(j^L|i^L) \Big)
\\ Prob(i^L|j^L) = \frac{\sum_{I_m\in C_{i^L}}\sum_{k \in {j^L}}CNN(I_m|k)}{|C_{i^L}|}
\label{eq:HCL:rijH}
\end{aligned}$$ ]{} where L-th level class label ${j^L}$ include all merged original classes (i.e., 0-th level before AP is called) $k \in {j^L}$ so far. From the above, the N-way CNN classification scores (Sec. \[sec:cl\]) only need to be evaluated once. $A_{i^L,j^L}$ at any level can be computed by summing over these original scores. The discovered category hierarchy can help alleviate the highly uneven visual separability between different object categories in image classification [@Yan2015hd] from which the category-embedded hierarchical deep CNN could be beneficial.
Experimental Results & Discussion {#sec-Exp}
=================================
### Dataset:
We experiment on the same dataset used in [@Shin2015]. The image database contains totally 216K $2D$ key-images which are associated with $\sim62$K unique patients’ radiology reports. Key-images are directly extracted from the Dicom file and resized as $256\times256$ bitmap images. Their intensity ranges are rescaled using the default window settings stored in the Dicom header files (this intensity rescaling factor improves the CNN classification accuracies by $\sim2\%$ to [@Shin2015]). Linked radiology reports are also collected as separate text files with patient-sensitive information removed for privacy reasons. At each LDPO iteration, the image clustering is first applied on the entire image dataset so that each image will receive a cluster label. Then the whole dataset is randomly reshuffled into three subgroups for CNN fine-tuning via Stochastic Gradient Descent (SGD): i.e. training ($70\%$), validation ($10\%$) and testing ($20\%$). In this way, the convergence is not only achieved on a particular data-split configuration but generalized to the entire database.
In order to quantitatively validate our proposed LDPO framework, we also apply category discovery on the texture-25 dataset [@Dai2015EnProDeepFets; @Lazebnik2005Sparse]: 25 texture classes, with 40 samples per class. The images from Texture-25 appear drastically different from those natural images in ImageNet, similar to our domain adaptation task from natural to radiology images. The ground truth labels are first hidden from the unsupervised LDPO learning procedure and then revealed to produce the quantitative measures (where purity becomes accuracy) against the resulted clusters. The cluster number is assumed to be known to LDPO and thus the model selection module of RIM in clustering is dropped.
### CNN Fine-tuning:
The Caffe [@jia2014caffe] implementation of CNN models are used in the experiment. During the looped optimization process, the CNN is fine-tuned for each iteration once a new set of image labels is generated from the clustering stage. Only the last softmax classification layer of the models (i.e. ’FC8’ in AlexNet and ’loss3/classifier’ in GoogLeNet) is significantly modulated by 1) setting a higher learning rate than all other layers and 2) updating the (varying but converging) number of category classes from the newly computed results of clustering.
LDPO Convergence Analysis
-------------------------
We first study how the different settings of proposed LDPO framework will affect convergence as follows:
### Clustering Method:
We perform $k$-means based image clustering with $k \in$ $\{80,100,$ $200,300,500,800\}$. Fig. \[fig:kmeans:png\] shows the changes of top-1 accuracy, cluster purity and NMI with different $k$ across iterations. The classification accuracies quickly plateau after 2 or 3 iterations. Smaller $k$ values naturally trigger higher accuracies ($>86.0$% for $k=80$) as less categories make the classification task easier. Levels of Purity and NMI between clusters from two consecutive iterations increase quickly and fluctuate close to $0.7$, thus indicating the convergence of clustering labels (and CNN models). The minor fluctuation are rather due to the randomly re-sorting of the dataset in each iteration. RIM clustering takes an over-segmented $k$-means results as initialization, e.g., $k=1000$ in our experiments. As shown in Fig. \[fig:RIM\_Encode:png\] Top-left, RIM can estimate the category capacities or numbers consistently under different image representations (deep CNN feature + encoding approaches). $k$-means clustering enables LDPO to approach the convergence quickly with high classification accuracies; whereas, the added RIM based model selection delivers more balanced and semantically meaningful clustering results (see more in Sec. \[sec-label-results\]). This is due to RIM’s two unique characteristics: 1), less restricted geometric assumptions in the clustering feature space; 2), the capacity to attain the optimal number of clusters by maximizing the mutual information of input data and the induced clusters via a regularized term.
{width="0.325\linewidth"} {width="0.325\linewidth"} {width="0.325\linewidth"}
{width="0.47\linewidth"} {width="0.47\linewidth"}\
{width="0.47\linewidth"} {width="0.47\linewidth"}
{width="0.31\linewidth"} {width="0.61\linewidth"}
### Pseudo-Task Initialization:
Both ImageNet and domain-specific [@Shin2015] CNN models have been employed to initialize the LDPO framework. In Fig. \[fig:RIM\_Encode:png\], two CNNs of AlexNet-FC7-ImageNet and AlexNet-FC7-Topic demonstrate their LDPO performances. LDPO initialized by ImageNet CNN reach the steady state noticeably slower than its counterpart, as AlexNet-FC7-Topic already contains the domain information from this radiology image database. However, similar clustering outputs are produced after convergence. Letting LDPO reach $\sim10$ iterations, two different initializations end up with very close clustering results (i.e., Cluster number, purity and NMI) and similar classification accuracies (shown in Table \[tab:CNN-Acc\]).
### CNN Deep Feature and Image Encoding:
Different image representations can vary the performance of proposed LDPO framework as shown in Fig. \[fig:RIM\_Encode:png\]. As mentioned in Sec. \[sec-method\], deep CNN images features extracted from different layers of CNN models (AlexNet and GoogLeNet) contain the level-specific visual information. Convolutional layer features retain the spatial activation layouts of images while FC layer features do not. Different encoding approaches further lead to various outcomes of our LDPO framework. The numbers of clusters range from 270 ([**AlexNet-FC7-Topic**]{} with no deep feature encoding) to 931 (the more sophisticated [**GoogLeNet-Inc.5b-VLAD**]{} with VLAD encoding). The numbers of clusters discovered by RIM reflect the amount of information complexity stored in the radiology database.
### Computational Cost:
LDPO runs on a node of Linux computer cluster with 16 CPU cores (x2650), 128G memory and Nvidia K20 GPUs. The Computational costs of different LDPO configurations are shown in Table \[tab:CNN-Acc\] per looped iteration. The more sophisticated and feature rich settings, e.g., [**AlexNet-Conv5-FV**]{}, [**GoogLeNet-Pool5**]{} and [**GoogLeNet-Inc.5b-VLAD**]{}, require more time to converge.
--------------------- -------------------- --------------- ---------------
[**CNN setting**]{} [**Cluster \#**]{} [**Top-1**]{} [**Top-5**]{}
\[0.3ex\] 270 0.8109 0.9412
\[0.3ex\] 275 0.8099 0.9547
\[0.3ex\] 712 0.4115 0.4789
\[0.3ex\] 624 0.4333 0.5232
\[0.3ex\] 462 0.4109 0.5609
\[0.3ex\] 929 0.3265 0.4001
\[0.3ex\]
--------------------- -------------------- --------------- ---------------
: Classification Accuracy of Converged CNN Models
\[tab:CNN-Acc\]
--------------------- -------------------------------
[**CNN setting**]{} [**Time per iter.(HH:MM)**]{}
\[0.3ex\] 14:35
\[0.3ex\] 14:40
\[0.3ex\] 17:40
\[0.3ex\] 15:44
\[0.3ex\] 21:12
\[0.3ex\] 23:35
\[0.3ex\]
--------------------- -------------------------------
: Computational Cost of LDPO
\[tab:CNN-Time\]
{width="0.165\linewidth"} {width="0.20\linewidth"} {width="0.20\linewidth"} {width="0.20\linewidth"} {width="0.20\linewidth"} {width="0.165\linewidth"} {width="0.20\linewidth"} {width="0.20\linewidth"} {width="0.20\linewidth"} {width="0.20\linewidth"} {width="0.165\linewidth"} {width="0.20\linewidth"} {width="0.20\linewidth"} {width="0.20\linewidth"} {width="0.20\linewidth"} {width="0.165\linewidth"} {width="0.20\linewidth"} {width="0.20\linewidth"} {width="0.20\linewidth"} {width="0.20\linewidth"}
{width="0.96\linewidth"} {width="0.96\linewidth"}
LDPO Categorization and Auto-annotation Results {#sec-label-results}
-----------------------------------------------
The category discovery clusters employing our LDPO method are found to be more visually coherent and cluster-wise balanced in comparison to the results in [@Shin2015] where clusters are formed only from text information ($\sim780K$ radiology reports). Fig. \[fig:Clusters:png\] [**Left**]{} shows the image numbers for each cluster from the AlexNet-FC7-Topic setting. The numbers are uniformly distributed with a mean of 778 and standard deviation of 52. Fig. \[fig:Clusters:png\] [**Right**]{} illustrates the relation of clustering results derived from image cues or text reports [@Shin2015]. Note that there is no instance-balance-per-cluster constraints in the LDPO clustering. The clusters in [@Shin2015] are highly uneven: 3 clusters inhabit the majority of images. Fig. \[fig:Cluster\_Sample:png\] shows sample images and top-10 associated key words from 4 randomly selected clusters (more results in the supplementary material). The LDPO clusters are found to be semantically or clinically related to the corresponding key words, containing the information of (likely appeared) anatomies, pathologies (e.g., adenopathy, mass), their attributes (e.g., bulky, frontal) and imaging protocols or properties.
Next, from the best performed LDPO models in Table \[tab:CNN-Acc\], [**AlexNet-FC7-Topic**]{} has [**Top-1**]{} classification accuracy of 0.8109 and [**Top-5**]{} accuracy 0.9412 with 270 formed image categories; [**AlexNet-FC7-ImageNet**]{} achieves accuracies of 0.8099 and 0.9547, respectively, from 275 discovered classes. In contrast, [@Shin2015] reports [**Top-1**]{} accuracies of 0.6072, 0.6582 and [**Top-5**]{} as 0.9294, 0.9460 on 80 text only computed classes using AlexNet [@krizhevsky2012imagenet] or VGGNet-19 [@simonyan2014very], respectively. Markedly better accuracies (especially on [**Top-1**]{}) on classifying higher numbers of classes (being generally more difficult) highlight advantageous quality of the LDPO discovered image clusters or labels. This means that the LDPO results have rendered significantly better performance on automatic image labeling than the most related previous work [@Shin2015], under the same radiology database. [*After the subjective evaluation by two board-certified radiologists, [**AlexNet-FC7-Topic**]{} of 270 categories and [**AlexNet-FC7-ImageNet**]{} of 275 classes are preferred, out of total six model-encoding setups. Interestingly, both CNN models have no deep feature encoding built-in and preserve the gloss image layouts (capturing somewhat global visual scenes without unordered FV or VLAD encoding schemes [@Cimpoi2015Deep; @Cimpoi2015Filter; @Jegou2012VLAD].)*]{}.
For the quantitative validation, LDPO is also evaluated on the Texture-25 dataset as an unsupervised texture classification problem. The purity and NMI are computed between the resulted LDPO clusters per iteration and the ground truth clusters (of 25 texture image classes [@Dai2015EnProDeepFets; @Lazebnik2005Sparse]) where purity becomes classification accuracy. [**AlexNet-FC7-ImageNet**]{} is employed and the quantitative results are plotted in Fig. \[fig:texture:png\]. Using the same clustering method of k-means, the purity or accuracy measurements improve from 53.9% (0-th) to 66.1% at the 6-th iteration, indicating that LDPO indeed learns better deep image features and labels in the looped process. Similar trend is found for another texture dataset [@Cimpoi2015Filter]. Exploiting LDPO for other domain transfer based auto-annotation tasks will be left as future work.
![Purity (Accuracy) and NMI plots between the ground truth classes and LDPO discovered clusters versus the iteration numbers.[]{data-label="fig:texture:png"}](purity_NMI_texture.pdf){width="0.90\linewidth"}
The final trained CNN classification models allow to compute the pairwise category similarities or affinity scores using the CNN classification confusion values between any pair of classes (Sec. \[sec:hcl\]). Affinity Propagation algorithm is called recursively to form a hierarchical category tree. The resulted category tree has (270, 64, 15, 4, 1) different class labels from bottom (leaf) to top (root). The random color coded category tree is shown in Fig. \[fig:CL\_Tree:png\]. The high majority of images in the clusters of this branch are verified as CT Chest scans by radiologists. [*Enabling to construct a semantic and meaningful hierarchy of classes offers another indicator to validate the proposed LDPO category discovery method and results.*]{} Refer to the supplementary material for more results. We will make our trained CNN models, computed deep image features and labels publicly available upon publication.
Conclusion & Future Work
========================
In this paper, we present a new Looped Deep Pseudo-task Optimization framework to extract visually more coherent and semantically more meaningful categories from a large scale medical image database. We systematically and extensively conduct experiments under different settings of the LDPO framework to validate and evaluate its quantitative and qualitative performance. The measurable LDPO “convergence” makes the ill-posed auto-annotation problem well constrained without the burden of human labeling costs. For future work, we intend to explore the feasibility/performance on implementing our current LDPO clustering component by deep generative density models [@Bengio-et-al-2015-Book; @Salakhutdinov2015Learning; @Kingma2014SSL]. It may therefore be possible that both classification and clustering objectives can be built into a multi-task CNN learning architecture which is “end-to-end” trainable by alternating two task/cost layers during SGD optimization [@Tzeng2015Simultaneous].
|
---
author:
- 'Yoh Yamamoto$^*$'
- 'Alan Salcedo$^*$'
- 'Carlos M. Diaz$^{*\S}$'
- 'Md Shamsul Alam$^{*\S}$'
- 'Tunna Baruah$^{*\S}$'
- 'Rajendra R. Zope$^{*\S}$'
bibliography:
- 'bibtex\_references.bib'
title: 'Supplementary information for: comparison of regularized SCAN functional with SCAN functional with and without self-interaction for a wide-array of properties'
---
polynomial used in SCAN {#polynomial-used-in-scan .unnumbered}
=======================
The polynomial function used in the rSCAN implementation is defined as $$f(\alpha)=c_1 + c_2 \alpha + c_3 \alpha^2 + c_4 \alpha^3 + c_5 \alpha^4 + c_6 \alpha^5 + c_7 \alpha^6 + c_8 \alpha^7$$ for $\alpha \in [0,2.5]$ where the coefficients $c$’s are as shown in Table \[tab:rscancoef\]. Those are the same as in Ref. [@doi:10.1063/1.5094646]. The constraints used are $f^{(0,1,2)}(0)$ and $f^{(0,1,2,3)}(2.5)$ to be identical values as the $f(\alpha)$ in SCAN at these two points. In addition, $f(1)=0$ was also used as a constraint. The plots of $f_x(\alpha)$ and $f_x'(\alpha)$ are shown in Fig. \[fig:switchingfunction\]. The plot of $\partial\epsilon_{XC}/\partial\rho$ for an Ar atom is shown in Fig. \[fig:scan\_v\_rscan\]. We have also tested different choices of polynomials with the same constraints and found essentially the same results.
------- ------------ -------------
Coef. Exchange Correlation
$c_1$ $ 1.000$ $ 1.000$
$c_2$ $-0.677$ $-0.640$
$c_3$ $-0.44456$ $-0.4352$
$c_4$ $-0.62109$ $-1.53568$
$c_5$ $ 1.39690$ $ 3.06156$
$c_6$ $-0.85920$ $-1.91571$
$c_7$ $ 0.22746$ $ 0.51688$
$c_8$ $-0.02252$ $-0.05185$
------- ------------ -------------
![\[fig:switchingfunction\] The switching function $f_x(\alpha)$ and $f_x'(\alpha)$ for SCAN and rSCAN.](scan_rscan1.eps){width="0.8\columnwidth"}
![\[fig:scan\_v\_rscan\] $\partial\epsilon_{XC}/\partial\rho$ as a function of $r$ for Ar atom with SCAN and rSCAN.](Ar_localxcpot.eps){width="0.8\columnwidth"}
---------------- ------ ------- -------------
System SCAN rSCAN CCSD(T)[^1]
AlF 1.31 1.31 1.47
AlH$_2$ 0.40 0.40 0.40
BeH 0.58 0.58 0.23
BF 1.05 1.05 0.82
BH 1.58 1.59 1.41
BH$_2$ 0.48 0.49 0.50
BH$_2$Cl 0.55 0.55 0.68
BH$_2$F 0.68 0.68 0.83
BHCl$_2$ 0.56 0.56 0.67
BHF$_2$ 0.83 0.83 0.96
BN 2.12 2.06 2.04
BO 2.35 2.33 2.32
BS 0.89 0.85 0.78
C$_2$H 0.76 0.74 0.76
C$_2$H$_3$ 0.71 0.70 0.69
C$_2$H$_5$ 0.34 0.34 0.31
CF 0.89 0.90 0.68
CF$_2$ 0.72 0.72 0.54
CH 1.48 1.48 1.43
CH$_2$BH 0.56 0.57 0.62
CH$_2$BOH 2.31 2.31 2.26
CH$_2$F 1.27 1.27 1.38
CH$_2$NH 2.02 2.00 2.07
CH$_2$PH 1.02 0.99 0.87
CH$_2$-singlet 1.83 1.82 1.49
CH$_2$-triplet 0.59 0.59 0.59
CH$_3$BH$_2$ 0.69 0.70 0.58
CH$_3$BO 3.77 3.78 3.68
CH$_3$Cl 1.94 1.91 1.90
CH$_3$F 1.72 1.71 1.81
CH$_3$Li 5.77 5.76 5.83
CH$_3$NH$_2$ 1.36 1.35 1.39
CH$_3$O 2.11 2.09 2.04
CH$_3$OH 1.66 1.65 1.71
CH$_3$SH 1.65 1.66 1.59
ClCN 3.00 3.02 2.85
ClF 0.82 0.75 0.88
ClO$_2$ 1.76 1.75 1.86
CN 1.41 1.41 1.43
CO 0.13 0.17 0.12
CS 1.92 1.96 1.97
---------------- ------ ------- -------------
--------------- ------ ------ ------
CSO 0.79 0.79 0.73
FCN 2.33 2.35 2.18
FCO 0.85 0.84 0.77
FH-BH$_2$ 3.03 3.03 2.97
FH-NH$_2$ 4.67 4.67 4.63
FH-OH 3.40 3.40 3.38
FNO 1.54 1.51 1.70
H$_2$CN 2.52 2.50 2.49
H$_2$O 1.86 1.87 1.86
H$_2$O-Al 4.53 4.54 4.36
H$_2$O-Cl 3.04 3.05 2.24
H$_2$O-F 2.58 2.64 2.19
H$_2$O-H$_2$O 2.78 2.78 2.73
H$_2$O-Li 2.98 2.98 3.62
H$_2$O-NH$_3$ 3.57 3.58 3.50
H$_2$S-H$_2$S 1.05 1.06 0.92
H$_2$S-HCl 2.36 2.36 2.13
HBH$_2$BH 0.85 0.88 0.84
HBO 2.72 2.71 2.73
HBS 1.43 1.40 1.38
HCCCl 0.31 0.28 0.50
HCCF 0.56 0.52 0.75
HCHO 2.37 2.32 2.39
HCHS 1.86 1.80 1.76
HCl 1.16 1.15 1.11
HCl-HCl 1.91 1.90 1.78
HCN 3.03 3.02 3.01
HCNO 2.60 2.57 2.96
HCO 1.67 1.65 1.69
HCOF 2.09 2.07 2.12
HCONH$_2$ 3.96 3.95 3.92
HCOOH 1.48 1.47 1.38
HCP 0.48 0.45 0.35
HF 1.80 1.80 1.81
HF-HF 3.42 3.42 3.40
HN$_3$ 1.77 1.79 1.66
HNC 3.05 3.07 3.08
HNCO 2.05 2.04 2.06
HNO 1.57 1.56 1.65
HNO$_2$ 1.96 1.95 1.93
HNS 1.39 1.39 1.41
HO$_2$ 2.17 2.21 2.17
HOCl 1.55 1.56 1.52
HOCN 3.97 3.99 3.80
HOF 1.92 1.89 1.92
--------------- ------ ------ ------
--------------- ------ ------ ------
HOOH 1.57 1.57 1.57
HPO 2.34 2.32 2.63
LiBH$_4$ 6.11 6.11 6.13
LiCl 7.10 7.10 7.10
LiCN 6.99 7.00 6.99
LiF 6.28 6.28 6.29
LiH 5.82 5.82 5.83
LiN 6.84 6.83 7.06
LiOH 4.53 4.53 4.57
N$_2$H$_2$ 2.83 2.83 2.88
N$_2$H$_4$ 2.71 2.71 2.72
NaCl 8.85 8.85 9.01
NaCN 8.81 8.82 8.89
NaF 7.99 7.99 8.13
NaH 6.33 6.33 6.40
NaLi 0.23 0.25 0.48
NaOH 6.63 6.63 6.77
NCl 1.14 1.16 1.13
NCO 0.83 0.84 0.79
NF 0.15 0.18 0.07
NF$_2$ 0.13 0.11 0.19
NH 1.54 1.54 1.54
NH$_2$ 1.80 1.80 1.79
NH$_2$Cl 2.03 2.01 1.95
NH$_2$F 2.28 2.24 2.27
NH$_2$OH 0.70 0.67 0.70
NH$_3$ 1.55 1.55 1.53
NH$_3$-BH$_3$ 5.33 5.33 5.28
NH$_3$-NH$_3$ 2.18 2.19 2.13
NH$_3$O 5.21 5.21 5.39
NO 0.17 0.19 0.13
NO$_2$ 0.29 0.29 0.34
NOCl 1.86 1.83 2.08
NP 2.74 2.75 2.87
NS 1.77 1.79 1.82
O$_3$ 0.63 0.64 0.57
OCl 1.43 1.43 1.28
OCl$_2$ 0.50 0.48 0.56
OF 0.16 0.19 0.02
OF$_2$ 0.33 0.31 0.33
OH 1.65 1.65 1.66
P$_2$H$_4$ 1.06 1.08 1.00
PCl 0.42 0.38 0.57
PF 0.65 0.66 0.81
PH 0.50 0.49 0.44
--------------- ------ ------ ------
------------ ------ ------ ------
PH$_2$ 0.62 0.62 0.55
PH$_2$OH 1.89 1.92 0.68
PH$_3$ 0.67 0.69 0.61
PH$_3$O 3.63 3.63 3.77
PO 1.89 1.85 1.96
PO$_2$ 1.36 1.33 1.44
PPO 1.74 1.69 1.88
PS 0.55 0.52 0.68
S$_2$H$_2$ 1.19 1.19 1.14
SCl 0.19 0.22 0.07
SCl$_2$ 0.34 0.31 0.39
SF 0.63 0.59 0.81
SF$_2$ 0.90 0.87 1.06
SH 0.83 0.83 0.77
SH$_2$ 1.06 1.07 0.99
SiH 0.19 0.20 0.11
SiH$_3$Cl 1.28 1.28 1.36
SiH$_3$F 1.23 1.23 1.31
SiO 2.99 2.95 3.11
SO$_2$ 1.54 1.52 1.63
SO-triplet 1.40 1.40 1.56
------------ ------ ------ ------
[crrr]{} System & SCAN & rSCAN & CCSD(T)[^2]\
2-pyridoxine–2-aminopyridine & 17.0 & 17.1 & 17\
Adenine–thymine stack & 8.7 & 8.5 & 11.66\
Adenine–thymine WC & 16.0 & 16.2 & 16.74\
Ammonia dimer & 3.2 & 3.2 & 3.17\
Benzene–ammonia & 2.1 & 2.0 & 2.32\
Benzene dimer C2h & 1.0 & 0.9 & 2.62\
Benzene dimer C2v & 1.5 & 1.5 & 2.71\
Benzene–HCN & 4.2 & 4.1 & 4.55\
Benzene–methane & 0.9 & 0.9 & 1.45\
Benzene–water & 3.4 & 3.3 & 3.29\
Ethene dimer & 1.2 & 1.1 & 1.5\
Ethene–ethyne & 1.4 & 1.4 & 1.51\
Formamide dimer & 16.6 & 16.7 & 16.12\
Formic acid dimer & 21.0 & 20.9 & 18.8\
Indole–benzene stack & 2.1 & 1.9 & 4.59\
Indole–benzene T-shape & 4.2 & 4.1 & 5.62\
Methane dimer & 0.4 & 0.4 & 0.53\
Phenol dimer & 6.0 & 5.9 & 7.09\
Pyrazine dimer & 2.7 & 2.5 & 4.2\
Uracil dimer HB & 20.5 & 20.6 & 20.69\
Uracil dimer stack & 8.1 & 7.9 & 9.74\
Water dimer & 5.5 & 5.5 & 5.02\
------------------------------------------------------------- ----------- ------ ------- ----------
Reaction Direction SCAN rSCAN Ref.[^3]
H + HCl $\rightarrow$ H$_2$ + Cl Forward -1.4 -0.1 5.7
Reverse 0.1 -0.3 8.7
OH + H$_2$ $\rightarrow$ H$_2$O + H Forward -2.1 -2.6 5.1
Reverse 11.1 13.2 21.2
CH$_3$ + H$_2$ $\rightarrow$ CH$_4$ + H Forward 7.2 6.9 12.1
Reverse 7.0 8.0 15.3
OH + CH$_4$ $\rightarrow$ H$_2$O + CH$_3$ Forward -1.6 -1.9 6.7
Reverse 11.8 12.7 19.6
H + H$_2$ $\rightarrow$ H$_2$ + H Forward 2.4 2.3 9.6
Reverse 2.4 2.3 9.6
OH + NH$_3$ $\rightarrow$ H$_2$O + NH$_2$ Forward -7.4 -7.9 3.2
Reverse 3.2 3.3 12.7
HCl + CH$_3$ $\rightarrow$ CH$_4$ + Cl Forward -3.1 -3.3 1.7
Reverse -1.7 -2.3 7.9
OH + C$_2$H$_6$ $\rightarrow$ H$_2$O + C$_2$H$_5$ Forward -4.8 -5.3 3.4
Reverse 13.0 14.0 19.9
F + H$_2$ $\rightarrow$ HF + H Forward -7.7 -8.2 1.8
Reverse 22.2 24.8 33.4
O + CH$_4$ $\rightarrow$ OH + CH$_3$ Forward 2.2 2.1 13.7
Reverse 3.3 2.9 8.1
H + PH$_3$ $\rightarrow$ H$_2$ + PH$_2$ Forward -3.2 -3.4 3.1
Reverse 19.5 19.3 23.2
H + HO $\rightarrow$ H$_2$ + O Forward 3.2 3.0 10.7
Reverse 2.1 1.1 13.1
H + H$_2$S $\rightarrow$ H$_2$ + HS Forward -2.7 -2.7 3.5
Reverse 11.1 10.1 17.3
O + HCl $\rightarrow$ OH + Cl Forward -4.0 -5.0 9.8
Reverse -1.5 -3.2 10.4
CH$_3$ + NH$_2$ $\rightarrow$ CH$_4$ + NH Forward 4.5 3.9 8
Reverse 12.5 12.8 22.4
C$_2$H$_5$ + NH$_2$ $\rightarrow$ C$_2$H$_6$ + NH Forward 6.0 5.5 7.5
Reverse 9.5 9.6 18.3
NH$_2$ + C$_2$H$_6$ $\rightarrow$ NH$_3$ + C$_2$H$_5$ Forward 4.8 4.5 10.4
Reverse 12.0 12.6 17.4
NH$_2$ + CH$_4$ $\rightarrow$ NH$_3$ + CH$_3$ Forward 7.7 7.6 14.5
Reverse 10.4 11.0 17.8
s-trans cis-C$_5$H$_8$ $\rightarrow$ s-trans cis-C$_5$H$_8$ Forward 33.6 32.5 38.4
Reverse 33.6 32.5 38.4
------------------------------------------------------------- ----------- ------ ------- ----------
------------------------------------------------------ --------- ------- ------- --------
H + N$_2$O $\rightarrow$ OH + N$_2$ Forward 18.7 19.2 18.14
Reverse 66.1 62.6 83.22
H + FH $\rightarrow$ HF + H Forward 38.3 38.2 42.18
Reverse 38.3 38.2 42.18
H + ClH $\rightarrow$ HCl + H Forward 19.5 19.3 18
Reverse 19.5 19.3 18
H + FCH$_3$ $\rightarrow$ HF + CH$_3$ Forward 29.2 28.3 30.38
Reverse 46.3 46.4 57.02
H + F$_2$ $\rightarrow$ HF + F Forward -1.6 -0.5 2.27
Reverse 88.6 89.2 106.18
CH$_3$ + FCl $\rightarrow$ CH$_3$F + Cl Forward -5.1 -4.7 7.43
Reverse 45.9 45.4 60.17
F$^-$ + CH$_3$F $\rightarrow$ FCH$_3$ + F$^-$ Forward -8.3 -7.8 -0.34
Reverse -8.3 -7.8 -0.34
F$^-$...CH$_3$F $\rightarrow$ FCH$_3$...F$^-$ Forward 7.5 7.9 13.38
Reverse 7.5 7.9 13.38
Cl$^-$ + CH$_3$Cl $\rightarrow$ ClCH$_3$ + Cl$^-$ Forward -6.1 -4.6 3.1
Reverse -6.1 -4.6 3.1
Cl$^-$...CH$_3$Cl $\rightarrow$ ClCH$_3$...Cl$^-$ Forward 6.1 7.2 13.61
Reverse 6.1 7.2 13.61
F$^-$ + CH$_3$Cl $\rightarrow$ FCH$_3$ + Cl$^-$ Forward -21.7 -20.5 -12.54
Reverse 14.3 14.2 20.11
F$^-$...CH$_3$Cl $\rightarrow$ FCH$_3$...Cl$^-$ Forward -2.1 -1.2 2.89
Reverse 24.4 24.2 29.62
OH$^-$ + CH$_3$F $\rightarrow$ HOCH$_3$ + F$^-$ Forward -11.2 -10.7 -2.78
Reverse 9.8 10.8 17.33
OH$^-$...CH$_3$F $\rightarrow$ HOCH$_3$...F$^-$ Forward 3.9 4.3 10.96
Reverse 43.9 45.1 47.2
H + N$_2$ $\rightarrow$ HN$_2$ Forward 13.9 13.7 14.69
Reverse 9.8 9.3 10.72
H + CO $\rightarrow$ HCO Forward 5.9 6.1 3.17
Reverse 24.2 24.2 22.68
H + C$_2$H$_4$ $\rightarrow$ CH$_3$CH$_2$ Forward 5.2 6.4 1.72
Reverse 43.3 43.2 41.75
CH$_3$ + C$_2$H$_4$ $\rightarrow$ CH$_3$CH$_2$CH$_2$ Forward 0.7 2.1 6.85
Reverse 31.0 31.9 32.97
HCN $\rightarrow$ HNC Forward 46.1 46.1 48.16
Reverse 32.0 31.6 33.11
------------------------------------------------------ --------- ------- ------- --------
Infrared and Raman spectra of water cluster
===========================================
--------------------------------------------------------------------------------------------------------- ------- ------------- -------- ------- ------- -------- -------- --------
CCSD(T)[^4]
(l)[1-4]{} (l)[5-8]{} (l)[9-9]{} Freq. IR Raman Depol. Freq. IR Raman Depol. Freq.
(l)[1-1]{} (l)[2-2]{} (l)[3-3]{} (l)[4-4]{} (l)[5-5]{} (l)[6-6]{} (l)[7-7]{} (l)[8-8]{} (l)[9-9]{} 1640 70.77 0.66 0.72 1636 71.72 0.66 0.71 1638.1
3802 2.96 100.86 0.05 3803 3.36 100.12 0.05 3786.8
3909 54.50 25.46 0.75 3911 55.89 25.03 0.75 3904.5
--------------------------------------------------------------------------------------------------------- ------- ------------- -------- ------- ------- -------- -------- --------
------------------------------------------------------------------------------------------------------- -------- ------------- -------- ------- -------- -------- -------- --------
CCSD(T)[^5]
(l)[1-4]{} (l)[5-8]{} (l)[9-9]{} Freq. IR Raman Depol. Freq. IR Raman Depol. Freq.
(l)[1-1]{} (l)[2-2]{} (l)[3-3]{} (l)[4-4]{} (l)[5-5]{} (l)[6-6]{} (l)[7-7]{} (l)[8-8]{} (l)[9-9]{} 94 124.45 0.04 0.73 105 154.49 0.05 0.74 132.6
147 47.70 0.05 0.74 163 22.31 0.03 0.71 145.3
171 118.93 0.14 0.66 164 125.78 0.15 0.66 146.1
203 173.20 0.06 0.54 201 161.80 0.04 0.58 183.6
388 43.82 0.15 0.68 382 46.30 0.12 0.71 355.2
649 89.10 0.34 0.74 653 88.74 0.32 0.74 629.7
1639 85.86 0.92 0.75 1635 87.59 0.88 0.74 1639.6
1664 36.35 0.72 0.30 1660 37.28 0.69 0.31 1658.8
3654 386.50 151.68 0.14 3659 397.39 170.27 0.15 3712.1
3790 8.85 83.27 0.05 3799 8.71 80.66 0.05 3782.5
3876 74.56 51.35 0.26 3885 75.17 50.81 0.26 3875.5
3894 78.12 23.97 0.75 3903 79.83 23.69 0.75 3896.1
------------------------------------------------------------------------------------------------------- -------- ------------- -------- ------- -------- -------- -------- --------
-------------------------------------------------------------------------------------------------------- -------- ------------- -------- ------- -------- -------- -------- --------
CCSD(T)[^6]
(l)[1-4]{} (l)[5-8]{} (l)[9-9]{} Freq. IR Raman Depol. Freq. IR Raman Depol. Freq.
(l)[1-1]{} (l)[2-2]{} (l)[3-3]{} (l)[4-4]{} (l)[5-5]{} (l)[6-6]{} (l)[7-7]{} (l)[8-8]{} (l)[9-9]{} 190 102.95 0.04 0.71 200 37.87 0.03 0.71 157
197 44.45 0.10 0.62 204 25.98 0.05 0.56 170
213 57.00 0.19 0.74 220 71.17 0.30 0.70 183
214 79.27 0.21 0.73 224 148.95 0.15 0.75 190.3
240 7.45 0.27 0.18 239 5.47 0.26 0.18 216.7
250 42.44 0.18 0.63 264 43.85 0.19 0.66 234.1
375 62.91 0.93 0.34 385 76.09 0.72 0.41 334.5
388 51.73 0.49 0.62 396 30.54 0.63 0.49 343.5
481 122.23 0.49 0.28 488 124.93 0.53 0.25 434.1
636 186.88 0.76 0.56 638 186.37 0.59 0.63 558.7
719 290.05 0.30 0.46 724 291.11 0.27 0.57 650.1
953 8.74 0.52 0.74 958 9.02 0.49 0.73 850.8
1649 57.25 1.28 0.65 1647 52.22 1.29 0.63 1647.8
1651 81.78 0.90 0.53 1649 89.09 0.88 0.52 1650.1
1674 14.87 0.98 0.75 1672 14.12 0.98 0.75 1671.7
3429 29.13 304.92 0.06 3427 26.38 300.24 0.06 3596.5
3515 754.03 37.68 0.74 3514 759.62 37.20 0.74 3647.8
3537 683.55 47.93 0.54 3533 687.80 38.67 0.59 3655
3864 78.86 53.18 0.22 3865 87.49 46.96 0.25 3865.7
3865 77.77 51.03 0.43 3866 70.63 62.53 0.32 3869.9
3867 42.63 114.89 0.08 3869 45.91 109.97 0.09 3871.4
-------------------------------------------------------------------------------------------------------- -------- ------------- -------- ------- -------- -------- -------- --------
------------------------------------------------------------------------------------------------------- --------- ------------- -------- ------- --------- -------- -------- --------
CCSD(T)[^7]
(l)[1-4]{} (l)[5-8]{} (l)[9-9]{} Freq. IR Raman Depol. Freq. IR Raman Depol. Freq.
(l)[1-1]{} (l)[2-2]{} (l)[3-3]{} (l)[4-4]{} (l)[5-5]{} (l)[6-6]{} (l)[7-7]{} (l)[8-8]{} (l)[9-9]{} 63 0.15 0.17 0.67 54 0.01 0.12 0.73 48.8
77 2.91 0.30 0.74 85 1.90 0.39 0.75 76.1
222 0.16 0.24 0.11 223 0.10 0.22 0.12 192.1
236 38.03 0.96 0.75 234 37.55 0.81 0.75 208.2
251 130.83 0.04 0.75 263 37.78 0.05 0.75 230.8
260 47.33 0.06 0.74 265 20.32 0.05 0.75 230.8
277 146.05 0.03 0.66 283 234.89 0.00 0.75 248.7
282 216.38 0.00 0.34 286 248.59 0.00 0.73 248.7
287 9.42 0.03 0.60 289 3.94 0.05 0.75 254.3
316 1.50 0.22 0.36 327 0.31 0.20 0.34 283.5
433 0.20 0.19 0.29 442 0.02 0.21 0.29 391
461 19.63 0.78 0.74 478 15.73 0.80 0.75 421
481 38.75 0.56 0.75 493 37.35 0.52 0.75 437.9
484 37.65 0.62 0.75 494 38.02 0.59 0.75 437.9
807 132.49 0.62 0.73 811 136.70 0.69 0.75 730.5
891 159.02 0.41 0.75 898 163.54 0.40 0.75 800.2
900 161.73 0.37 0.74 900 161.87 0.41 0.75 800.2
1074 0.02 0.68 0.01 1078 0.00 0.42 0.08 971
1649 83.66 0.66 0.75 1647 84.91 0.64 0.75 1653
1664 42.80 0.54 0.75 1663 42.73 0.53 0.75 1666.6
1665 42.21 0.54 0.74 1664 42.67 0.55 0.75 1666.6
1697 0.05 1.75 0.20 1696 0.00 1.91 0.19 1693.3
3161 0.21 360.22 0.08 3160 0.23 357.14 0.08 3446.4
3299 1886.40 2.13 0.73 3299 1893.09 2.10 0.74 3526.6
3303 1917.26 2.27 0.72 3300 1891.83 2.38 0.73 3526.6
3354 26.31 105.31 0.75 3353 26.84 106.52 0.75 3559.1
3860 67.22 48.06 0.35 3861 69.75 44.94 0.34 3860.9
3861 45.97 95.66 0.08 3862 73.59 39.86 0.34 3861.4
3862 71.11 43.90 0.45 3863 45.34 97.98 0.11 3861.4
3864 45.77 95.13 0.07 3865 46.19 95.84 0.09 3861.5
------------------------------------------------------------------------------------------------------- --------- ------------- -------- ------- --------- -------- -------- --------
------------------------------------------------------------------------------------------------------- --------- ------------- -------- ------- --------- -------- -------- --------
CCSD(T)[^8]
(l)[1-4]{} (l)[5-8]{} (l)[9-9]{} Freq. IR Raman Depol. Freq. IR Raman Depol. Freq.
(l)[1-1]{} (l)[2-2]{} (l)[3-3]{} (l)[4-4]{} (l)[5-5]{} (l)[6-6]{} (l)[7-7]{} (l)[8-8]{} (l)[9-9]{} 30 4.38 0.02 0.75 30 4.57 0.02 0.74 22.5
44 0.06 0.18 0.62 43 0.03 0.20 0.61 41.4
68 0.10 0.22 0.74 67 0.15 0.25 0.75 60.6
78 1.84 0.29 0.66 71 1.45 0.26 0.75 63.5
189 2.21 0.21 0.22 191 0.91 0.22 0.15 179.2
203 31.05 0.59 0.74 206 31.14 0.60 0.75 190.1
213 31.08 0.62 0.74 219 39.08 0.59 0.75 196
238 115.26 0.17 0.70 244 95.34 0.17 0.73 226.6
255 4.33 0.01 0.73 258 6.82 0.01 0.71 232.3
267 146.94 0.03 0.74 266 148.96 0.04 0.71 236.9
286 270.53 0.09 0.29 288 276.54 0.05 0.59 265.1
321 82.37 0.18 0.74 324 76.91 0.17 0.75 291.8
330 11.90 0.06 0.75 331 19.75 0.04 0.73 295
336 1.11 0.04 0.70 338 1.52 0.02 0.72 297.7
449 48.52 0.84 0.39 453 50.97 0.62 0.46 400.5
463 14.93 0.31 0.22 467 10.20 0.30 0.16 420.7
490 15.64 0.81 0.74 495 15.78 0.75 0.74 443.9
500 17.78 0.79 0.73 507 19.11 0.79 0.73 456.7
559 62.28 0.99 0.35 561 62.38 0.89 0.40 503.1
771 23.74 1.31 0.65 778 24.16 1.32 0.70 702.5
848 128.11 1.18 0.69 852 126.67 1.07 0.74 766.2
934 130.17 0.62 0.69 941 130.08 0.59 0.75 843.9
955 124.66 0.26 0.73 960 123.33 0.22 0.73 860
1063 8.15 0.07 0.19 1068 8.54 0.15 0.11 963.7
1649 83.44 0.17 0.69 1647 83.94 0.18 0.74 1657.6
1662 20.27 0.72 0.58 1660 19.34 0.72 0.59 1667.9
1672 58.45 0.55 0.37 1671 61.06 0.49 0.42 1675.2
1696 34.66 0.48 0.05 1695 36.19 0.44 0.04 1694.4
1705 4.22 0.62 0.63 1703 3.96 0.73 0.49 1701.4
3095 50.75 495.91 0.08 3097 48.68 493.31 0.08 3413
3223 2923.51 4.41 0.74 3224 2857.86 4.04 0.74 3482.9
3235 2624.40 13.29 0.38 3237 2630.15 13.05 0.38 3490.2
3297 93.02 74.00 0.75 3298 79.76 73.89 0.74 3529.6
3317 113.31 79.85 0.72 3317 120.20 75.18 0.73 3535.5
3861 50.36 53.55 0.28 3867 56.87 60.48 0.19 3859.1
3862 67.18 61.22 0.15 3870 66.32 49.40 0.24 3861.1
3862 50.73 57.29 0.18 3871 49.89 88.56 0.12 3862.9
3864 52.53 87.50 0.12 3873 40.97 23.24 0.71 3862.9
3867 57.65 108.78 0.08 3875 67.76 142.57 0.05 3865.7
------------------------------------------------------------------------------------------------------- --------- ------------- -------- ------- --------- -------- -------- --------
FLOSIC-rSCAN calculations
=========================
---- ----------- ----------- -----------------------------
Z rSCAN SIC-rSCAN E$_\text{Accu}$ (Ref. \[\])
1 -0.500 -0.500 -0.5
2 -2.905 -2.900 -2.90
3 -7.480 -7.474 -7.48
4 -14.650 -14.643 -14.67
5 -24.641 -24.628 -24.65
6 -37.841 -37.813 -37.85
7 -54.594 -54.538 -54.59
8 -75.076 -74.997 -75.07
9 -99.752 -99.640 -99.73
10 -128.963 -128.799 -128.94
11 -162.286 -162.100 -162.25
12 -200.082 -199.874 -200.05
13 -242.383 -242.147 -242.35
14 -289.404 -289.131 -289.36
15 -341.311 -340.993 -341.26
16 -398.164 -397.807 -398.11
17 -460.208 -459.802 -460.15
18 -527.606 -527.141 -527.54
19 -599.981 -599.467
20 -677.627 -677.061
21 -760.692 -760.077
22 -849.446 -848.791
23 -944.011 -943.254
24 -1044.596 -1043.686
25 -1151.143 -1150.196
26 -1263.849 -1262.832
27 -1382.936 -1381.869
28 -1508.506 -1507.310
29 -1640.748 -1639.332
30 -1779.669 -1778.237
31 -1925.102 -1923.629
32 -2077.232 -2075.698
33 -2236.145 -2234.541
34 -2401.837 -2400.160
35 -2574.471 -2572.710
36 -2754.143 -2752.290
---- ----------- ----------- -----------------------------
---- -------- ----------- -----------
Z rSCAN SIC-rSCAN Expt.[^9]
2 24.624 24.483 24.587
3 5.400 5.374 5.392
4 8.802 8.820 9.323
5 8.788 8.732 8.298
6 11.716 11.437 11.26
7 14.889 14.308 14.534
8 13.707 13.431 13.618
9 17.609 17.005 17.423
10 21.633 20.589 21.565
11 5.180 5.140 5.139
12 7.393 7.395 7.646
13 6.181 6.180 5.986
14 8.341 8.215 8.152
15 10.670 10.441 10.487
16 10.345 10.419 10.36
17 13.045 12.962 12.968
18 15.879 15.646 15.76
19 4.282 4.472 4.341
20 5.839 6.089 6.113
21 6.241 7.054 6.561
22 6.984 8.324 6.828
23 7.126 7.070 6.746
24 7.312 7.015 6.767
25 6.908 7.133 7.434
26 7.795 8.068 7.902
27 8.396 8.279 7.881
28 8.789 8.443 7.64
29 8.090 7.488 7.726
30 9.238 9.519 9.394
31 6.156 6.560 5.999
32 8.093 8.324 7.899
33 10.134 10.423 9.789
34 9.685 10.280 9.752
35 11.913 12.379 11.814
36 14.250 14.743 14
---- -------- ----------- -----------
---- ------- ----------- ------------
Z rSCAN SIC-rSCAN Expt.[^10]
1 0.725 0.510 0.754
3 0.446 0.465 0.618
5 0.608 0.167 0.280
6 1.547 0.875 1.262
8 1.573 0.637 1.462
9 3.428 2.123 3.401
11 0.457 0.480 0.548
13 0.626 0.383 0.434
14 1.570 1.277 1.390
15 0.765 0.602 0.747
16 2.157 1.843 2.077
17 3.714 3.277 3.613
19 0.409 0.424 0.501
22 1.008 -1.211 0.087
29 1.178 1.061 1.236
31 0.544 0.187 0.43
32 1.540 1.274 1.233
33 0.830 0.659 0.814
34 2.135 1.806 2.021
35 3.584 3.232 3.364
---- ------- ----------- ------------
----------------- -------- ----------- -----------
System rSCAN SIC-rSCAN Ref.[^11]
C$_2$O$_2$H$_2$ 643.0 589.5 634.0
CH$_3$CCH 710.2 678.3 705.1
C$_4$H$_8$ 1160.8 1134.0 1149.4
S$_2$ 109.6 98.7 104.3
SiH$_4$ 322.4 326.9 325.0
SiO 188.8 157.8 193.1
----------------- -------- ----------- -----------
--------------------------------------- ----------- ------- ----------- -----------
Reaction Direction rSCAN SIC-rSCAN Ref.[^12]
OH + CH$_4\rightarrow$ CH$_3$+ H$_2$O Forward -14.6 11.6 6.7
Reverse 12.6 14.2 19.6
H +OH$\rightarrow$ H$_2$ + O Forward 2.1 10.6 10.7
Reverse 12.8 14.0 13.1
H + H$_2$S$\rightarrow$ H$_2$+ HS Forward -2.7 1.6 3.6
Reverse 4.3 14.3 17.3
--------------------------------------- ----------- ------- ----------- -----------
----------------------------------------------------------- ------- ----------- -----------
Reaction rSCAN SIC-rSCAN Ref.[^13]
H$_2^+$ $\rightarrow$ H $+$ H$^+$
R/R$_e$ = 1.0 67.8 64.4 64.4
R/R$_e$ = 1.25 64.9 58.9 58.9
R/R$_e$ = 1.5 57.8 48.7 48.7
R/R$_e$ = 1.75 50.8 38.2 38.3
He$_2^+$ $\rightarrow$ He $+$ He$^+$
R/R$_e$ = 1.0 74.4 56.5 56.9
R/R$_e$ = 1.25 71.5 44.6 46.9
R/R$_e$ = 1.5 63.4 27.5 31.3
R/R$_e$ = 1.75 58.5 14.3 19.1
(NH$_3$)$_2^+$ $\rightarrow$ NH$_3$ $+$ NH$_3^+$
R/R$_e$ = 1.0 43.4 36.3 35.9
R/R$_e$ = 1.25 38.3 25.4 25.9
R/R$_e$ = 1.5 30.9 11.6 13.4
R/R$_e$ = 1.75 27.2 4.1 4.9
(H$_2$O)$_2^+$ $\rightarrow$ H$_2$O $+$ H$_2$O$^+$
R/R$_e$ = 1.0 52.9 36.3 39.7
R/R$_e$ = 1.25 48.8 22.8 29.1
R/R$_e$ = 1.5 42.7 11.9 16.9
R/R$_e$ = 1.75 40.1 6.2 9.3
C$_4$H$_{10}^+$ $\rightarrow$ C$_2$H$_5$ $+$ C$_2$H$_5^+$ 42.0 34.3 35.28
(CH$_3$)$_2$CO$^+$ $\rightarrow$ CH$_3$ $+$ CH$_3$CO$^+$ 30.1 40.5 22.57
ClFCl $\rightarrow$ ClClF -22.3 -2.9 -1.01
C$_2$H$_4$...F$_2$ $\rightarrow$ C$_2$H$_4$ $+$ F$_2$ 2.5 0.5 1.08
C$_6$H$_6$...Li $\rightarrow$ Li $+$ C$_6$H$_6$ 7.7 12.1 9.5
NH$_3$...ClF $\rightarrow$ NH$_3$ $+$ ClF 17.1 12.0 10.5
NaOMg $\rightarrow$ MgO $+$ Na 75.7 95.3 69.56
FLiF $\rightarrow$ Li $+$ F$_2$ 120.4 92.4 94.36
----------------------------------------------------------- ------- ----------- -----------
[^1]: Reference \[\]
[^2]: Reference \[\]
[^3]: Reference \[\]
[^4]: Reference [@doi:10.1063/1.4820448]
[^5]: Reference [@doi:10.1063/1.4820448]
[^6]: Reference [@doi:10.1063/1.4820448]
[^7]: Reference [@doi:10.1063/1.4820448]
[^8]: Reference [@doi:10.1063/1.4820448]
[^9]: Reference [@NIST_ASD]
[^10]: Reference [@NIST_CCCBD]
[^11]: Reference [@doi:10.1021/jp035287b]
[^12]: Reference [@doi:10.1021/jp035287b]
[^13]: Reference [@doi:10.1021/ct900489g]
|
---
abstract: 'We discuss the recent claim that hadron multiplicities measured at RHIC energies are [*directly*]{} described in terms of gluon degrees of freedom fixed from the initial conditions of central heavy ion collisions. The argument is based on the parton saturation scenario expected to be valid at high parton densities and on the assumption of conserved gluon number. Alternatively we conjecture that “bottom-up” equilibration before hadronization modifies this picture, due to nonconservation of the number of gluons.'
address: |
$^a$ Fakultät für Physik, Universität Bielefeld, D-33501 Bielefeld, Germany\
$^b$ Department of Physics, Columbia University, New York, NY 10027, USA\
$^c$ LPT, Université Paris-Sud, Bâtiment 210, F-91405 Orsay, France\
$^d$ Institute for Nuclear Theory, University of Washington, Seattle, WA 98195-1550, USA
author:
- 'R. Baier$^a$, A.H. Mueller$^b$, D. Schiff$^c$, and D.T. Son$^d$'
title: 'Does parton saturation at high density explain hadron multiplicities at RHIC ?'
---
At RHIC and LHC energies in the central rapidity region of heavy ion collisions a high density of energy is deposited mainly in the form of gluons. In recent papers by Kharzeev et al. [@Nardi; @Levin; @Kharzeev; @Kreview] (hereafter referred to as KLN) it is assumed that the initial gluon density determines the number of produced (charged) particles, i.e. that there is a correspondence between the number of partons in the initial state and the number of particles in the final state. The argument is formulated in the framework of the saturation scenario [@Gribov; @Qiu; @Blaizot; @MV; @Krasnitz], which determines the initial gluon distribution inside the colliding nuclei, and where the characteristic momentum scale is the hard scale $Q_s >> \Lambda_{QCD}$.
Based on this scenario the initial gluon multiplicity immediately after the high energy nuclear collisions can be calculated in the McLerran-Venugopalan model [@MV; @Krasnitz; @McLerran; @Mueller]. At $\tau_0 \simeq 1/Q_s $ this initial gluon density is [@Mueller], $$n_{hard}(\tau_0) = c~ \frac{(N_c^2-1) Q_s^3}{4 \pi^2 N_c \alpha_s~ (Q_s \tau_0)} \, .
\label{dens}$$
In [@Nardi; @Levin; @Kharzeev] this relation is used to obtain the hadron multiplicities in A-A collisions, which are then compared with the corresponding RHIC data [@Nagle; @Back; @Adcox; @Bearden; @Adler].
In this note we propose a possibly more realistic description, which departs from taking the initial condition as the only one determining ingredient for the final hadronic state. This description is based on the “bottom-up” scenario [@Baier1; @Son; @Mueller3], which leads to thermalization of the gluons produced after the collision. In contrast to KLN, this scenario stresses the importance of branching processes of gluons which may allow for a short enough equilibration time, at least at high energies. As a consequence the number of gluons is increasing between initial and equilibration times. In this context let us note that already in [@KrasnitzV] a phenomenological factor $\kappa_{inel} > 1$ is introduced accounting for gluon number changing processes which may occur at late times beyond when the classical approach is applicable.
In the following we investigate in the “bottom-up” picture the resulting hadron multiplicities and compare with the predictions of the KLN model. We discuss in some detail the conceptual differences between these two interesting possibilities.
The arguments of KLN [@Nardi; @Levin; @Kharzeev] go through a number of steps, which we first critically review:
i\) The number of gluons in the initial state, and at the time when gluons transform into hadrons, is assumed to be equal, which can be true when only $2 \leftrightarrow 2$ processes are taken into account. Only in this circumstance the measured hadron multiplicities are reflecting [*directly*]{} the initial conditions.
ii\) In (\[dens\]) the parameter $c$ is a constant linking the number of gluons in the nucleus wave function to the number of gluons which are freed during the collision. It is expected to be of $O(1)$ [@Mueller]. An (approximate) analytical calculation gives $c = 2~\ln{2} \simeq 1.39$ [@Kovchegov]. However, extracting $c$ from the numerical simulation in [@Krasnitz; @KrasnitzV] leads to the estimate [[^1]]{} $c \simeq 0.5$ for $Q_s = 1~GeV$.
iii\) The hard saturation scale $Q_s^2$ in the case of one nucleus has been determined to be [@Mueller], $$Q_s^2 (\vec{s}) =
\frac{4 \pi^2 N_c}{N_c^2 -1}~ \alpha_s (Q_s^2)~ x G(x, Q_s^2)~
\rho_{nucleon}(\vec{s}) \, ,
\label{nscale}$$ where $\vec{s}$ is the impact parameter, $x G(x, Q^2)$ is the gluon structure function in the nucleon, and $\rho_{nucleon}$ is the transverse density of nucleons in the nucleus. Notice that in the above equation an additional numerical multiplicative factor may come as a consequence of the inherent uncertainty in the precise determination of the saturation momentum outside of the McLerran-Venugopalan model [@MV]. We shall call this factor $K$ and introduce it a little later on. KLN generalize (\[nscale\]) to the case of two colliding nuclei. Let us first write [[^2]]{}
$$Q_s^2 (\vec{s},\vec{b}) =
\frac{4 \pi^2 N_c}{N_c^2 -1}~ \alpha_s (Q_s^2)~ x G(x, Q_s^2)
~\frac{ \rho_{part}(\vec{s},\vec{b})}{2} \, .
\label{nuscale}$$
where $\rho_{part}(\vec{s},\vec{b})$ is the density of participating nucleons in the transverse plane as a function of the impact parameter $\vec{b}$ of the A-A collision, and of $\vec{s}$ the transverse coordinate of the produced gluon. The factor $\frac{1}{2}$ is required by the proper definition of $Q_s^2$, relative to one nucleus [@Mueller]. Integrating the density with respect to $\vec{s}$ leads to $$\int~ d^2s ~\rho_{part}(\vec{s},\vec{b}) = N_{part} (\vec{b}) ~,
\label{npart}$$ with $N_{part} (\vec{b}) $ the number of participants in the A-A collision at fixed $\vec{b}$.
iv\) Focusing on the distribution of freed gluons $dN/{d \eta}$ for ${\eta=0}$ at given $\vec{b}$, we can write $$\begin{aligned}
\label{spect}
\frac{d N}{d \eta} (\vec{b})
&& = c \frac{N_c^2 -1 }{4 \pi^2 N_c} \,
\int~ d^2s ~ \frac{1}{\alpha_s} \,
Q_s^2(\vec{s},\vec{b}) \nonumber \\
&& \simeq c ~ xG(x, {\bar Q_s^2}) ~\frac{ N_{part} (\vec{b})}{2}
\, .\end{aligned}$$ The corresponding charged hadron multiplicity for the most central collisions is then obtained as $$\label{Nard0}
\langle \frac{2}{N_{part}} \frac{d N_{ch}}{d\eta} \rangle
%\langle \frac{2}{N_{part}} \frac{d N_{ch}}{d\eta} \rangle \Bigg|_{\eta<1}
\simeq \frac{1}{3} ~c~\left[ \ln \frac{\bar{Q^2_s} }{\Lambda^2_{QCD}} \right] \, ,$$ using the gluon structure function (\[glu\]) given below. As reference energy we take $\sqrt{s} = 130~GeV$. One should remark that in (\[spect\]) and (\[Nard0\]), ${\bar Q_s^2}$ shows up as an effective average over the variable $\vec{s}$ in (\[nuscale\]). As a first approximation the following is used: $$\label{aver}
{\bar{Q_s^2}} (\vec{b}) =
K~\frac{4 \pi^2 N_c}{N_c^2 -1}~ \alpha_s ({\bar{Q_s^2}})~ x G(x,{\bar{ Q_s^2}})
~\frac{ \rho_{part}(\vec{b})}{2} \, .$$
v\) Information on the gluon structure function is necessary. In the small $x$ regime, the main feature is that it increases with $Q^2$ at fixed $x$. Following KLN [@Nardi; @Levin; @Kharzeev], it is reasonable to take $$x G(x, Q^2) = 0.5 ~\ln (\frac{Q^2}{\Lambda_{QCD}^2}) \, ,
\label{glu}$$ with $\Lambda_{QCD} = 200~ MeV$, such that $ x G(x, Q^2) \simeq 2 $ at $Q^2 = 2 ~GeV^2$ (at $x=0.02$).
vi\) The strong coupling constant is $$\alpha_s(Q^2) \simeq \frac{1}{\beta_0~\ln (\frac{Q^2}{\Lambda_{QCD}^2})} \, ,
\label{coupl}$$ with $\beta_0 = (11 - 2 n_f/3)/4 \pi$ for $N_c =3$; we take $n_f = 3$.
vii\) Using (\[aver\]) together with the above choices for $\alpha_s$ and $xG(x,Q^2)$, one finds for central $Au-Au$ collisions at RHIC (at $\sqrt{s} = 130~ GeV$), taking for the moment $K = 1$:
${\bar{Q_s^2}} (\vec{b} =0) \simeq 0.63 ~GeV^2$, using $\rho_{part} (\vec{b} =0) = 3.06 ~fm^{-2}$,
as quoted in Table 2 by [@Nardi]. For larger values of $\vec{b}$, $\bar{Q_s^2}$ becomes even smaller, e.g. for ${b} = 10~ fm$, $\bar{Q_s^2}
\simeq 0.32~ GeV^2$ ! This casts some doubts on the applicability of this model, even for central collisions at RHIC energies.
As already noticed, the missing factor $\frac{1}{2}$ in KLN and a larger value used for $\alpha_s$, namely $\alpha_s (Q^2) = 0.6$ at $Q^2 = 2~ GeV^2$, allows them to get larger values of $Q_s^2$, i.e. $Q_s^2 \ge 2~ GeV^2$ for the central collisions.
Taking in the following an optimistic point of view, we use (\[aver\]) with a multiplicative $K $ factor of order $O(1)$, explicitely $K \simeq 1.6$, such that for $\vec{b} = 0$, ${\bar{Q_s^2}} (\vec{b} =0) = 1 ~GeV^2$. Our use of $K$ is to a large extent cosmetic. This factor only appears in calculating $Q_s^2$, but does not change (\[Nard0\]) or (\[Nard1\]). Such a factor will affect the average transverse momentum per produced gluon but not the total number of produced gluons. The difference between ${\bar{Q_s}} (\vec{b} =0) = 1 ~GeV$ and ${\bar{Q_s}} (\vec{b} =0) = 0.8 ~GeV$, corresponding to $K = 1.6 $ and $K = 1$, respectively, has little effect on the equilibration temperature $T_{eq}$ and time $\tau_{eq}$ quoted later in this note.
viii\) Finally, there is an equality assumed between the number of partons in the final state and the number of observed hadrons (“parton-hadron duality” [@duality]).
Let us now turn to the “bottom-up” scenario [@Baier1; @Son; @Mueller3], which in contrast to the KLN [@Nardi; @Levin; @Kharzeev] prescription, does not relate the multiplicities to the initial condition only, but also to the way gluons are thermalized.
In the framework of perturbative QCD the time evolution of the gluonic system, when described by a non-linear Boltzmann equation based on $2 \leftrightarrow 2$ processes, a relatively long time ($\sim Q_s^{-1}~\exp(\alpha_s^{-1/2} )$) is required for the approach to kinetic equilibration [@Mueller; @Serreau]. Fast and efficient thermalization occurs when inelastic processes, namely gluon splittings $2 \leftrightarrow 3$ are taken into account, and kinetic equilibration occurs much faster at times $\sim \alpha_s^{-13/5} Q_s^{-1}$ [@Baier1; @Son; @Mueller3]. In this “bottom-up” scenario the time evolution of the system proceeds through several regimes (Fig. \[fig:pict\]), with $Q_s\tau\sim\alpha_s^{-3/2},\alpha_s^{-5/2} $ and $\alpha_s^{-13/5}$.
The difference between the “bottom-up” and the KLN picture is schematically illustrated in this Fig. \[fig:pict\]. Under the KLN assumption the hard gluons (on the momentum scale $Q_s$) are conserved in number, i.e. $n_{hard} \tau = const$, and they finally hadronize, after passing through a hydrodynamical stage. The “bottom-up” scenario is characterized by the fact that hard gluons are degrading, soft ones are formed and start to dominate the system. As a result the interactions of gluons in this kinetic scenario modify strongly the initial gluon spectrum. The gluons are redistributed and thermalizing, such that a quark gluon plasma is formed. The number of gluons, together with the entropy [@Mueller3], is increasing with proper time $\tt$, such that the ratio is, $$R = [n_{soft}(\tt) (Q_s \tt)] \vert_{ \tt_{eq}}~
/ ~ [n_{hard}(\tt) (Q_s \tt)] \vert_{\tt_0} \sim \alpha_s^{-2/5} \gg 1 \, .
\label{ratio}$$ The following processes are expected to be present (at RHIC and higher energies in the central region of pseudorapidity $\eta \le 1$): at $Q_s \tau \ge 1$ saturated hard gluons $\longrightarrow$ elastic scatterings and branching/production of soft gluons $\longrightarrow$ at $Q_s \tau |_{eq} \sim \alpha_s^{-13/5}$ thermalization of these soft gluons with temperature $T_{eq} \sim \alpha_s^{2/5} Q_s \longrightarrow$ hydrodynamic expansion $\longrightarrow$ hadronization (Fig. \[fig:pict\]).
In more detail the parametrically estimated time scales are determined as follows [@Baier1; @Son; @Mueller3].
[ Early times $1 \ll Q_s\tau \ll \alpha_s^{-3/2}$: ]{}
At the earliest time, $\tau\sim Q_s^{-1}$, gluons, i.e. hard gluons, have typical large transverse momentum of order $Q_s$ and occupation number of order $1/\alpha_s$. Later on gluons with smaller momenta, but still larger than $\Lambda_{QCD}$ will be produced (nevertheless, they are denoted as soft gluons). The density of hard gluons $n_{hard}$ decreases with time due to the one-dimensional expansion. Gluons interact by elastic scatterings at small angle, with exchange momentum $\ll Q_s$. The typical occupation number is large until $Q_s\tau\sim\alpha_s^{-3/2}$, when it becomes of $O(1)$.
This regime is the transition region from the non-linear classical gluon field to the one where the transport description by Boltzmann equations should become applicable.
[ Times $\alpha_s^{-3/2} \ll Q_s\tau \ll \alpha_s^{-5/2}$: ]{}
Inelastic scatterings produce (soft) gluons $n_{soft}$ with characteristic momentum estimated to be of order $\alpha_s^{1/2} Q_s > \Lambda_{QCD}$ (Fig. \[fig:pict\]), namely via $hard + hard \rightarrow hard + hard + soft$. The number of these soft gluons becomes comparable to that of hard ones at $Q_s\tau \sim \alpha_s^{-5/2}$, namely $n_{soft} \sim n_{hard}$.
[Times $Q_s \tt \gg \alpha_s^{-5/2}$: ]{}
After $Q_s\tau\sim\alpha_s^{-5/2}$ most gluons are soft, $n_{soft}\gg n_{hard}$; they achieve thermal equilibration amongst themselves. Although the whole system is still not in thermal equilibrium, the soft part is characterized by the temperature $T$, $n_{soft} \sim T^3$. The few hard gluons collide with the soft ones of the thermal bath and constantly loose energy to the latter. A hard gluon emits one with a softer energy, which splits into gluons with comparable momenta. The products of this branching quickly cascade further, giving all their energy to the thermal bath. This increases the temperature in the thermal bath found to be, $$T = c_T~ \alpha_s^3 Q_s^2 \tau \, ,$$ i.e. it increases linearly with time, even when the system is expanding, due to the hard gluons which serve as an energy source. $c_T$ is a numerical constant to be discussed later. One can verify a posteriori that soft gluons are indeed equilibrated due to many interactions: the size of the expanding system of $O(\tau)$ is indeed much larger than the mean free path of the soft gluons, i.e. $$\tau/\lambda_{soft} \sim \tau n_{soft}(\tau) \sigma
\sim c_T~ \alpha_s^5 (Q_s \tau )^2 \gg 1 \, ,$$ where the cross section is estimated as $\sigma \sim \alpha_s^2 /T^2$.
In the following, in order to provide predictions for particle multiplicities in this scenario we go as much as possible beyond the parametric estimates, ending up with consistency arguments which specify the allowed range of parameters.
First we start by considering the constant $c_T$ which may be written in terms of the parameter $c$ as [@Baier1], $$c_T \simeq \frac{15}{8 \pi^5} c N_c^3 \, \simeq 0.16~c \, .
\label{cT}$$ The linear growth of $T$ as shown in Fig. \[fig:pict\] terminates, when the hard gluons loose all of their energy, i.e. when
$$\tau = \tt_{eq}
= c_{eq}~ \alpha_s^{-13/5} Q_s^{-1} \,
\label{final} ,$$
where the parameter $c_{eq}$ is unknown for the moment, although in principle it could be calculated in the “bottom-up” framework. The temperature achieves a maximal value, i.e. the equilibration temperature, which is expressed by $$T_{max} = T_{eq} =0.16~ c~c_{eq}~ \alpha_s^{2/5}(Q_s^2)~Q_s \, .$$ Subsequently the temperature decreases as $\tau^{-1/3}$ [@Bjorken], such that $\tau n_{soft} (\tau) = const$.
We now derive the charged hadron multiplicity from the number density of the equilibrated soft gluons, $$n_{soft} (\tau_{eq}) = 2 (N_c^2 -1) \frac{\zeta(3)}{\pi^2} T_{eq}^3 \, ,$$ which is much larger than $n_{hard}$ in (\[dens\]) used by KLN. One finds for the ratio $R$ defined in (\[ratio\]), $$\begin{aligned}
R = && 8 \zeta(3) N_c~ \frac{c_T^3}{c} ~\alpha_s^{10} (Q_s^2)~ (Q_s \tau)^4
\Bigg|_{\tau_{eq}} \nonumber \\
\simeq && 0.13~ c^2~ c_{eq}^4 \alpha_s^{-2/5}(Q_s^2) \, .
\label{rat}\end{aligned}$$
For the charged hadron multiplicity the result for $\sqrt{s} = 130 ~GeV$ and the most central collisions, is $$\begin{aligned}
\langle \frac{2}{N_{part}} \frac{d N_{ch}}{d\eta} \rangle
%\langle \frac{2}{N_{part}} \frac{d N_{ch}}{d\eta} \rangle \Bigg|_{\eta<1}
&& \simeq \frac{R~c}{3}~ \ln \frac{\bar{Q^2_s} }{\Lambda^2_{QCD}}
\nonumber \\
&& \simeq 0.04~ c^3 ~ c_{eq}^4
\left[ \ln \frac{\bar{Q^2_s} }{\Lambda^2_{QCD}} \right]^{7/5} \, ,
\label{Nard1}\end{aligned}$$ which replaces (\[Nard0\]).
In order to compare the predicted charged hadron multiplicity, (\[Nard0\]) and (\[Nard1\]), with RHIC data we use as a reference the result by the PHOBOS Collaboration [@Back], namely $3.24 \pm 0.1 (stat) \pm 0.25 (syst)$, which is in good agreement within errors with the experimental measurements of the other collaborations at RHIC. This is best seen by comparing the values for ${d N_{ch}}/{d\eta}$ at midrapidity at $\sqrt{s} = 130~GeV$: ${d N_{ch}}/{d\eta} = 555 \pm 12 (stat) \pm 35 (syst)$ [@Back], $609 \pm 1 (stat) \pm 37 (syst)$ [@Adcox], $549 \pm 1 (stat) \pm 35 (syst)$ [@Bearden], $567 \pm 1 (stat) \pm 38 (syst)$ [@Adler], respectively.
As reference value we take $$\label{norm}
\langle \frac{2}{N_{part}} \frac{d N_{ch}}{d\eta} \rangle
= 3.24 \, .$$
First we note that in the KLN approach agreement with experimental data can only be achieved with the value $c \simeq 3$, which is different from the current numerical estimate quoted before, $c \simeq 0.5$.
Turning to the “bottom-up” approach, the experimental value (\[norm\]) meets the theoretical expectation (\[Nard1\]) for $R~c \simeq 3$, or equivalently $$\label{ceq}
c_{eq} \simeq \frac{2.0}{c^{3/4}} \, .$$ This relation is to be confronted with the consistency requirement for the “bottom-up” scenario , that the ratio $R$ (\[rat\]) be larger than $2$, implying $$\label{cc}
c^2~c_{eq}^4 \ge 10, \, \, \, \, i.e.\, \, \, c_{eq} \ge \frac{1.8}{\sqrt{c}} \, .$$
It is not too difficult to see that (\[ceq\], \[cc\]) constrain the two parameters: $$\label{constr}
c~ \le~ 1.5 \, \, \, and \, \, \, c_{eq} ~\ge~ 1.5 \, .$$
This in turn allows us to infer the actual properties of the medium, especially to answer tentatively the question of the formation of the equilibrated plasma. We do this via discussing the temperature $T_{eq}$ and the equilibration time $\tau_{eq}$. In order that the quark gluon plasma is produced, $T_{eq}$ should be bigger than the phase transition temperature $T_{deconf}$, which is of order $T_{deconf} = 173 \pm 8 ~MeV$ and $154 \pm 8 ~MeV$ for 2 and 3 flavour QCD, respectively [@Karsch], i.e. $$T_{eq} \ge T_{deconf} \, .
\label{cond3}$$ This constrains $$c~ c_{eq} > 1.3, \, \, \, or \, \, c > 0.2 \, .
\label{contemp}$$
Finally, we may correlatively discuss $\tau_{eq}$. Under the condition (\[constr\]) one finds (for central collisions) $$\tau_{eq} ~ \ge 2.6 ~ fm \sim 1/2 ~R_{Au} \, ,
\label{eqtime}$$ which is much bigger than the current estimate of $\simeq 0.7 ~ fm$ [@McLerran] [[^3]]{}. Within the uncertainties inherent to these estimates the formation of the equilibrated plasma may indeed be realized.
One may consider the energy dependence at RHIC energies of the charged multiplicities. Following [@Kharzeev] it is controlled by the energy dependence of the saturation scale, i.e. $Q^2_s (s)/Q^2_s (s_0) = (s/s_0)^{\lambda /2}$ with $\lambda = 0.25$ [@Golec]. As discussed we choose as the reference value $\bar{Q^2_s} (s_0) = 1~GeV^2$ at $\sqrt{s_0} = 130~GeV$. For the most central collisions the expression for the energy dependence of the pseudorapidity density of charged particles at midrapidity reads in the “bottom-up” scenario, $$\label{energyb}
\langle \frac{2}{N_{part}} \frac{d N_{ch}}{d\eta} \rangle
\simeq 0.64 \left( \frac{\sqrt s }{\sqrt s_0}\right)^{\lambda} \,
\left[\ln \frac{\bar{Q^2_s} (s)}{\Lambda_{QCD}^2} \right]^{7/5}
\, .$$
Based on (\[energyb\]) the multiplicity at $\sqrt{s} = 200~GeV$ is obtained: $$\label{edep}
\langle \frac{2}{N_{part}} \frac{dN_{ch}}{d\eta} \rangle
%\langle \frac{2}{N_{part}} \frac{dN_{ch}}{d\eta} \rangle_{\eta < 1}
= 3.84 \, ,$$ compared to the value given by the PHOBOS Collaboration [@Back], $$\label{PHdat}
\langle \frac{2}{N_{part}} \frac{dN_{ch}}{d\eta} \rangle
= 3.78 \pm 0.25 ~(syst) \, .$$
Finally we consider the centrality dependence, i.e. the dependence of $d N_{ch}/ d\eta$ as a function of $N_{part}$. In Fig. \[fig:phob\] the comparison of the “bottom-up” expectation as derived from (\[energyb\]) with data from the PHOBOS Collaboration [@Back] at $\sqrt{s} = 130~GeV$ and at $\sqrt{s} = 200~GeV$ at RHIC is shown. For this comparison $\bar{Q^2_s} (\vec{b}) $ is calculated from (\[aver\]) as a function of $N_{part}$, using $\rho_{part}(\vec{b})$ given in Table 2 of [@Nardi], together with the scaling relation for the energy dependence of $Q_s$ quoted above [@Golec]. One has to note, that for $N_{part} < 100$ the values of $Q_s^2$ are becoming smaller than $0.6~GeV^2$. The structure of the shape of $d N_{ch}/ d\eta$ as a function of $N_{part}$ seen in the data (Fig. \[fig:phob\]) could be attributed, cf. with (\[spect\]), to details of $x G(x, Q_s^2)$ at small $x$ as a function of $Q_s^2$.
In summary, the description provided by the “bottom-up” scenario is in agreement with RHIC data, provided the parameters $c$ and $c_{eq}$, which are not determined in the picture, lie in a given, limited range. In particular for $c \simeq 1$ and $c_{eq} \simeq 2$ the picture looks both reasonable and attractive. For these values the results (for the most central collisions) for $R, T_{eq}$ and $\tau_{eq}$ are $3,~ 230~MeV$ and $3.6~fm$, respectively, for $Q_s = 1~GeV$; for $Q_s = 0.8~GeV$, i.e. for $K=1$, the values are $R=3, T_{eq} = 210 ~MeV$ and $\tau_{eq} = 3.2 ~fm$. However, the picture probably does not make much sense for a small value of $c \simeq 1/2$, which is currently favoured by numerical calculations of classical field equations, because the ratio $R$ turns out to become $R \simeq 6 $, which is too much inelasticity when $Q_s = 1~GeV$. This discrepancy deserves further examination.
On the other hand, in order to accomodate the present RHIC data, the KLN description requires $c \simeq 3$. For the moment, due to a number of ambiguities, it is not yet clear whether this large value is compatible with the various constraints of the saturation picture.
The “bottom-up” scenario is likely to provide a more convincing agreement with data. However, this analysis will have to be supplemented by further ingredients, e.g. the calculation of $c_{eq}$, a more precise estimate of $c$, before one can finally claim a significant agreement with data.
Discussions with D. Kharzeev, M. Nardi, K. Redlich and R. Venugopalan are kindly acknowledged. RB acknowledges support, in part, by DFG, project FOR 339/2-1. The work of AHM is supported, in part, by a DOE Grant. The work of DTS is supported, in part, by a DOE Grant No. DOE-ER-41132 and by the Alfred P. Sloan Foundation. This research was supported in part by the National Science Foundation under Grant No. PHY99-07949; RB and DS thank ITP, UCSB, for kind hospitality, and for the facilities during the completion of this work.
[99]{}
D. Kharzeev and M. Nardi, Phys. Lett. B 507 (2001) 121.
D. Kharzeev and E. Levin, Phys. Lett. B 523 (2001) 79.
D. Kharzeev, E. Levin and M. Nardi, hep-ph/0111315 (2001).
For a recent review, see D. Kharzeev, hep-ph/0204014 (2001).
L. V. Gribov, E. M. Levin and M. G. Ryskin, Phys. Rep. 100 (1983) 1.
A. H. Mueller and J. W. Qiu, Nucl. Phys. B 268 (1986) 427.
J.-P. Blaizot and A. H. Mueller, Nucl. Phys. B 289 (1987) 847.
L. McLerran and R. Venugopalan, Phys. Rev. D 49 (1994) 2233; Phys. Rev. D 49 (1994) 3352;\
Phys. Rev. D 50 (1994) 2225.
A. Krasnitz and R. Venugopalan, Nucl. Phys. A 698 (2002) 209c, Plenary Talk by R. Venugopalan at QM2001,\
and references therein.
For a recent review, e.g. L. McLerran, hep-ph/0202025 (2002).
A. H. Mueller, Nucl. Phys. B 572 (2000) 227; Phys. Lett. B 475 (2000) 220.
For a review on RHIC experiments , e.g. see J. L. Nagle, nucl-ex/0109016 (2001).
B. B. Back et al. (PHOBOS Collaboration), Phys. Rev. Lett. 85 (2000) 3100; Phys. Rev. Lett. 88 (2002) 022302;\
nucl-ex/0201005 (2002).
K. Adcox et al. (PHENIX Collaboration), Phys. Rev. Lett. 86 (2001) 3500; Phys. Rev. Lett. 87 (2001) 052301.
I. G. Bearden et al. (BRAHMS Collaboration), Phys. Lett. B523 (2001) 227; nucl-ex/0112001 (2001).
C. Adler et al. (STAR Collaboration), Phys. Rev. Lett. 87 (2001) 0112303.
R. Baier, A.H. Mueller, D. Schiff and D. T. Son, Phys. Lett. B 502 (2001) 51.
R. Baier, A.H. Mueller, D. Schiff and D. T. Son, Nucl. Phys. A 698 (2002) 217c,\
Talk given by D. T. Son at QM2001.
A. H. Mueller, Talk given at “Statistical QCD”, Bielefeld, August 2001,\
to appear in the Proceedings (Nucl. Phys. A).
A. Krasnitz and R. Venugopalan, Phys. Rev. Lett. 86 (2001) 1717;\
A. Krasnitz, Y. Nara and R. Venugopalan, Phys. Rev. Lett. 87 (2001) 192302.
Yu. V. Kovchegov, Nucl. Phys. A 692 (2001) 557.
We acknowledge discussions with R. Venugopalan on the numerical evaluation of the “parton-liberation” coefficient $c$.
Yu. L. Dokshitzer, V. A. Khoze, A. H. Mueller and S. I. Troyan, [*Basics of perturbative QCD*]{},\
Editions Frontiérs, Gif-sur-Yvette, 1991, and references therein.
J. Serreau and D. Schiff, JHEP 0111 (2001) 039.
J. D. Bjorken, Phys. Rev. D 27 (1983) 140.
F. Karsch, E. Laermann and A. Peikert, Nucl. Phys. B605 (2001) 579.
D. Teaney, J. Lauret and E. V. Shuryak, Phys. Rev. Lett. 86 (2001) 4783;\
U. Heinz and P. F. Kolb, hep-ph/0204061;\
U. Heinz, Talk given at “Statistical QCD”, Bielefeld, August 2001,\
to appear in the Proceedings (Nucl. Phys. A).
K. Golec-Biernat and M. Wüsthof, Phys. Rev. D59 (1999) 014017; Phys. Rev. D60 (1999) 114023.
[^1]: This estimate is based on the relation $c= 4 \pi^2 f_N / [(N_c^2 - 1)~\ln{Q_s^2/\Lambda^2_{QCD}} ]$, where $f_N = 0.3$ [@Raju].
[^2]: Notice in (\[nuscale\]), consistently with (\[nscale\]), the factor $\frac{1}{2}$ compared to eq. (14) in [@Nardi]. This has the consequence of lower values of $Q_s^2$ appearing in our discussion of the KLN approach.
[^3]: This short equilibration time is required in order to describe the hadron spectra for different particles as measured at RHIC by hydrodynamic calculations, assuming the existence of the quark gluon plasma [@Heinz] .
|
---
abstract: 'We give a complete algorithm and source code for constructing what we refer to as heterotic risk models (for equities), which combine: i) granularity of an industry classification; ii) diagonality of the principal component factor covariance matrix for any sub-cluster of stocks; and iii) dramatic reduction of the factor covariance matrix size in the Russian-doll risk model construction. This appears to prove a powerful approach for constructing out-of-sample stable short-lookback risk models. Thus, for intraday mean-reversion alphas based on overnight returns, Sharpe ratio optimization using our heterotic risk models sizably improves the performance characteristics compared to weighted regressions based on principal components or industry classification. We also give source code for: a) building statistical risk models; and ii) Sharpe ratio optimization with homogeneous linear constraints and position bounds.'
---
**Heterotic Risk Models**
Zura Kakushadze$^\S$$^\dag$[^1]
*$^\S$ Quantigic$^\circledR$ Solutions LLC*
*1127 High Ridge Road \#135, Stamford, CT 06905[^2]*
*$^\dag$ Free University of Tbilisi, Business School & School of Physics*
*240, David Agmashenebeli Alley, Tbilisi, 0159, Georgia*
(April 30, 2015)
Introduction
============
When the number of stocks in a portfolio is large and the number of available (relevant) observations in the historical time series of returns is limited – which is essentially a given for short-horizon quantitative trading strategies – the sample covariance matrix (SCM) is badly singular. This makes portfolio optimization – [*e.g.*]{}, Sharpe ratio maximization – challenging as it requires the covariance matrix to be invertible. A standard method for circumventing this difficulty is to employ factor models,[^3] which, instead of computing SCM for a large number of stocks, allow to compute a factor covariance matrix (FCM) for many fewer risk factors. However, the number of relevant risk factors itself can be rather large. [*E.g.*]{}, in a (desirably) granular industry classification (IC), the number of industries[^4] can be in 3 digits for a typical (liquid) trading universe. Then, even the sample FCM can be singular.
In (Kakushadze, 2015c) a simple idea is set forth: model FCM itself via a factor model, and repeat this process until the remaining FCM is small enough and can be computed. In fact, at the end of this process we may even end up with a single factor, for which “FCM" is simply its variance.[^5] This construction – termed as “Russian-doll" risk models (Kakushadze, 2015c) – dramatically reduces the number of or altogether eliminates the factors for which (off-diagonal) FCM must be computed. The “catch" is that at each successive step we must: i) identify the risk factors; and ii) compute the specific (idiosyncratic) risk (ISR) and FCM consistently.
Identifying the risk factors in the Russian-doll construction is facilitated by using a binary industry classification:[^6] using BICS as an illustrative example, industries serve as the risk factors for sub-industries; sectors – there are only 10 of them – serve as the risk factors for industries; and – if need be – the “market" serves as the sole risk factor for sectors. Correctly computing ISR and FCM is more nontrivial: the algorithms for this are generally deemed proprietary. One method in the “lore" is to use a cross-sectional linear regression, where the returns are regressed over a factor loadings matrix (FLM), and FCM is identified with the (serial) covariance matrix of the regression coefficients, whereas ISR squared is identified with the (serial) variance of the regression residuals. However, as discussed in (Kakushadze, 2015c), generally this does not satisfy a nontrivial requirement (which is often overlooked in practice) that the factor model reproduce the historical in-sample total variance.
In this paper we share a complete algorithm and source code for building what we refer to as “heterotic" risk models. It is based on a simple observation that, if we use principal components (PCs) as FLM, the aforementioned total variance condition is automatically satisfied. Unfortunately, the number of useful PCs is few as it is limited by the number of observations, and they also tend to be unstable out-of-sample (as they are based on off-diagonal covariances), with the first PC being most stable. We circumvent this by building FLM from the first PCs of the blocks (sub-matrices) of the sample correlation matrix[^7] corresponding to – in the BICS language – the sub-industries. [*I.e.*]{}, if there are $N$ stocks and $K$ sub-industries, FLM is $N\times K$, and in each column the elements corresponding to the tickers in that sub-industry are proportional to the first PC of the corresponding block, with all other elements vanishing.[^8] The total variance condition is automatically satisfied. Then, applying the Russian-doll construction yields a nonsingular factor model covariance matrix, which, considering it sizably adds value in Sharpe ratio optimization for certain intraday mean-reversion alphas we backtest, appears to be stable out-of-sample.
Heterotic risk models are based on our proprietary know-how. We hope sharing it with the investment community encourages organic custom risk models building.
This paper is organized as follows. In Section \[sec2\] we briefly review some generalities of factor models and discuss in detail the total variance condition. In Section \[sub.pc\] we discuss the PC approach and an algorithm for fixing the number of PC factors, with the R source code in Appendix \[app.A\]. We discuss heterotic risk models in detail in Section \[sec.het\], with the complete Russian-doll embedding in Section \[sec.RD\] and the R source code in Appendix \[app.B\]. In Section \[sec.horse\] we run a horse race of intraday mean-reversion alphas via i) weighted regressions and ii) optimization using heterotic risk models. For optimization with homogeneous linear constrains and (liquidity/position) bounds we use the R source code in Appendix \[app.C\].[^9] We briefly conclude in Section \[sec.conc\].
Multi-factor Risk Models {#sec2}
========================
Generalities
------------
In a multi-factor risk model, a sample covariance matrix $C_{ij}$ for $N$ stocks, $i,j = 1,\dots,N$, which is computed based on a time series of stock returns $R_i$ ([*e.g.*]{}, daily close-to-close returns), is modeled via a constructed covariance matrix $\Gamma_{ij}$ given by $$\begin{aligned}
\label{Gamma}
&&\Gamma \equiv \Xi + \Omega~\Phi~\Omega^T\\
&& \Xi_{ij} \equiv \xi_i^2 ~\delta_{ij}\end{aligned}$$ where $\delta_{ij}$ is the Kronecker delta; $\Gamma_{ij}$ is an $N\times N$ matrix; $\xi_i$ is the specific risk (a.k.a. idiosyncratic risk) for each stock; $\Omega_{iA}$ is an $N\times K$ factor loadings matrix; and $\Phi_{AB}$ is a $K\times K$ factor covariance matrix, $A,B=1,\dots,K$, where $K\ll N$. [*I.e.*]{}, the random processes $\Upsilon_i$ corresponding to $N$ stock returns are modeled via $N$ random processes $\chi_i$ (specific risk) together with $K$ random processes $f_A$ (factor risk): $$\begin{aligned}
\label{Upsilon}
&&\Upsilon_i = \chi_i + \sum_{A=1}^K \Omega_{iA}~f_A\\
&&\mbox{Cov}(\chi_i, \chi_j) = \Xi_{ij}\\
&&\mbox{Cov}(\chi_i, f_A) = 0\\
&&\mbox{Cov}(f_A, f_B) = \Phi_{AB}\\
&&\mbox{Cov}(\Upsilon_i, \Upsilon_j) = \Gamma_{ij}\end{aligned}$$ When $M<N$, where $M+1$ is the number of observations in each time series, the sample covariance matrix $C_{ij}$ is singular with $M$ nonzero eigenvalues. In contrast, assuming all $\xi_i>0$ and $\Phi_{AB}$ is positive-definite, then $\Gamma_{ij}$ is automatically positive-definite (and invertible). Furthermore, the off-diagonal elements of $C_{ij}$ typically are not expected to be too stable out-of-sample. On the contrary, the factor model covariance matrix $\Gamma_{ij}$ is expected to be much more stable as the number of risk factors, for which the factor covariance matrix $\Phi_{AB}$ needs to be computed, is $K\ll N$.
Conditions on Total Variances
-----------------------------
The prime aim of a risk model is to predict the covariance matrix out-of-sample as precisely as possible, including the out-of-sample total variances. However, albeit this requirement is often overlooked in practical applications, a well-built factor model had better reproduce the in-sample total variances. That is, we require that the factor model total variance $\Gamma_{ii}$ coincide with the in-sample total variance $C_{ii}$: $$\label{tot.risk}
\Gamma_{ii} = \xi_i^2 + \sum_{A,B=1}^K \Omega_{iA}~\Phi_{AB}~\Omega_{iB} = C_{ii}$$ [*A priori*]{} this gives $N$ conditions[^10] for $N + K(K+1)/2$ unknowns $\xi_i$ and $\Phi_{AB}$, so we need additional assumptions[^11] to compute $\xi_i$ and $\Phi_{AB}$.
Linear Regression
-----------------
One such assumption – [*intuitively*]{} – is that the total risk should be attributed to the factor risk to the greatest extent possible, [*i.e.*]{}, the part of the total risk attributed to the specific risk should be minimized. One way to formulate this requirement mathematically is via least squares. First, mimicking (\[Upsilon\]), we decompose the stock returns $R_i$ via a linear model $$R_i = \epsilon_i + \sum_{A=1}^K \Omega_{iA}~f_A$$ Here the residuals $\epsilon_i$ are [*not*]{} the same as $\chi_i$ in (\[Upsilon\]); in particular, generally the covariance matrix $\mbox{Cov}(\epsilon_i, \epsilon_j)$ is not diagonal (see below). We can require that $$\label{lin.w}
\sum_{i=1}^N z_i~\epsilon_i^2 \rightarrow \mbox{min}$$ where $z_i > 0$, and the minimization is w.r.t. $f_A$. This produces a weighted linear regression[^12] with the regression weights $z_i$. So, what should these weights be?
Correlations, Not Covariances {#cor.not.cov}
-----------------------------
While choosing unit weights $z_i\equiv 1$ might appear as the simplest thing to do, this suffers from a shortcoming. Intuitively it is clear that – [*on average*]{} – the residuals $\epsilon_i$ are larger for more volatile stocks, so the regression with unit weights would produce skewed results.[^13] This can be readily rectified using nontrivial regression weights. A “natural" choice is $z_i = 1/C_{ii}$. In fact, we have a regression with unit weights: $$\begin{aligned}
&&{\widetilde R}_i = {\widetilde \epsilon}_i + \sum_{A=1}^K {\widetilde \Omega}_{iA}~f_A\\
&&\sum_{i=1}^N {\widetilde \epsilon}_i^2 \rightarrow \mbox{min}\label{lin.unit}\end{aligned}$$ where ${\widetilde R}_i \equiv R_i /\sqrt{C_{ii}}$, ${\widetilde \Omega}_{iA} \equiv \Omega_{iA} /\sqrt{C_{ii}}$, and ${\widetilde \epsilon}_i \equiv \epsilon_i /\sqrt{C_{ii}}$ on average are expected to be much more evenly distributed compared with $\epsilon_i$ – we have scaled away the volatility skewness via rescaling the returns, factor loadings and residuals by $\sqrt{C_{ii}}$.
So, we are now modeling the sample [*correlation*]{} matrix $\Psi_{ij} \equiv C_{ij}/ \sqrt{C_{ii}}\sqrt{C_{jj}}$ (note that $\Psi_{ii}=1$, while $|\Psi_{ij}|\leq 1$ for $i\neq j$)[^14] via another factor model matrix $${\widetilde\Gamma}_{ij} = {\widetilde \xi}^2_i~\delta_{ij} + \sum_{A,B=1}^K {\widetilde \Omega}_{iA}~\Phi_{AB}~{\widetilde \Omega}_{jB}$$ where ${\widetilde\Gamma}_{ij} \equiv \Gamma_{ij}/\sqrt{C_{ii}}\sqrt{C_{jj}}$, and ${\widetilde \xi}^2_i\equiv \xi_i^2/C_{ii}$. The solution to (\[lin.unit\]) is given by (in matrix notation) $$\begin{aligned}
\label{fac.ret}
&&f = \left({\widetilde \Omega}^T~{\widetilde\Omega}\right)^{-1}{\widetilde\Omega}^T~{\widetilde R}\\
&&{\widetilde \epsilon} = \left[1 - Q\right] {\widetilde R}\\
&&Q \equiv {\widetilde \Omega}\left({\widetilde \Omega}^T~{\widetilde\Omega}\right)^{-1}{\widetilde\Omega}^T\end{aligned}$$ where $Q$ is a projection operator: $Q^2 = Q$. Consequently, we have: $$\begin{aligned}
&&{\widehat \Xi} \equiv \mbox{Cov}\left({\widetilde\epsilon},{\widetilde \epsilon}^T\right) = \left[1 - Q\right] \Psi \left[1 - Q^T\right]\\
&&{\widetilde \Omega}~\Phi~{\widetilde \Omega}^T = {\widetilde \Omega}~\mbox{Cov}\left(f, f^T\right){\widetilde \Omega}^T = Q~\Psi~Q^T\end{aligned}$$ Note that the matrix ${\widehat \Xi}$ is not diagonal. However, the idea here is to identify ${\widetilde\xi}_i^2$ with the diagonal part of ${\widehat \Xi}$: $$\label{xi}
{\widetilde \xi}_i^2 \equiv {\widehat \Xi}_{ii} = \left(\left[1 - Q\right] \Psi \left[1 - Q^T\right]\right)_{ii}$$ and we have $${\widetilde \Gamma}_{ij} = {\widetilde \xi}_i^2~\delta_{ij} + \left(Q~\Psi~Q^T\right)_{ij}$$ Note that ${\widetilde \xi}_i^2$ defined via (\[xi\]) are automatically positive (nonnegative, to be precise – see below). However, we must satisfy the conditions (\[tot.risk\]), which reduce to $${\widetilde\Gamma}_{ii} = {\widetilde\xi}_i^2 + \sum_{A,B=1}^K {\widetilde\Omega}_{iA}~\Phi_{AB}~{\widetilde\Omega}_{iB} = \Psi_{ii} = 1$$ and imply $$\begin{aligned}
\label{T}
&&T_{ii} = 0\\
&&T \equiv 2~Q~\Psi~Q^T - Q~\Psi - \Psi~Q^T\end{aligned}$$ The $N$ conditions (\[T\]) are not all independent. Thus, we have $\mbox{Tr}(T) = 0$.
Principal Components {#sub.pc}
====================
The conditions (\[T\]) are nontrivial. They are not satisfied for an arbitrary factor loadings matrix ${\widetilde \Omega}_{iA}$. However, there is a simple way of satisfying these conditions, to wit, by building ${\widetilde \Omega}_{iA}$ from the principal components of the correlation matrix $\Psi_{ij}$.
Let $V_i^{(a)}$, $a=1,\dots,N$ be the $N$ principal components of $\Psi_{ij}$ forming an orthonormal basis $$\begin{aligned}
&&\sum_{j=1}^N \Psi_{ij}~V_j^{(a)} = \lambda^{(a)}~V_i^{(a)}\\
&&\sum_{i=1}^N V_i^{(a)}~V_i^{(b)} = \delta_{ab}\end{aligned}$$ such that the eigenvalues $\lambda^{(a)}$ are ordered decreasingly: $\lambda^{(1)} > \lambda^{(2)} >\dots$. More precisely, some eigenvalues may be degenerate. For simplicity – and this is not critical here – we will assume that all positive eigenvalues are non-degenerate. However, we can have multiple null eigenvalues. Typically, the number of nonvanishing eigenvalues[^15] is $M$, where, as above, $M+1$ is the number of observations in the stock return time series. We can readily construct a factor model with $K\leq M$: $$\label{FLM.PC}
{\widetilde \Omega}_{iA} = \sqrt{\lambda^{(A)}}~V_i^{(A)}$$ Then the factor covariance matrix $\Phi_{AB} = \delta_{AB}$ and we have $$\begin{aligned}
\label{PC}
&& {\widetilde \Gamma}_{ij} = {\widetilde \xi}_i^2~\delta_{ij} + \sum_{A=1}^K \lambda^{(A)}~V_i^{(A)}~V_j^{(A)}\\
&&{\widetilde \xi}_i^2 = 1 - \sum_{A=1}^K \lambda^{(A)}\left(V_i^{(A)}\right)^2\end{aligned}$$ so ${\widetilde \Gamma}_{ii} = \Psi_{ii} = 1$. See Appendix \[app.B\] for the R code including the following algorithm.
Fixing $K$ {#sub.fix.K}
----------
When $K = M$ we have ${\widetilde \Gamma} = \Psi$, which is singular.[^16] Therefore, we must have $K \leq K_{max} < M$. So, how do we determine $K_{max}$? And is there $K_{min}$ other than the evident answer $K_{min} = 1$? Here we can do a lot of complicated, even convoluted things. Or we can take a pragmatic approach and come up with a simple heuristic. Here is one simple algorithm that does a very decent job at fixing $K$.
The idea here is simple. It is based on the observation that, as $K$ approaches $M$, $\mbox{min}({\widetilde \xi}^2_i)$ goes to 0 ([*i.e.*]{}, less and less of the total risk is attributed to the specific risk, and more and more of it is attributed to the factor risk), while as $K$ approaches 0, $\mbox{max}({\widetilde \xi}^2_i)$ goes to 1 ([*i.e.*]{}, less and less of the total risk is attributed to the factor risk, and more and more of it is attributed to the specific risk). So, as a rough cut, we can think of $K_{max}$ and $K_{min}$ as the maximum/minimum values of $K$ such that $\mbox{min}({\widetilde \xi}^2_i) \geq \zeta^2_{min}$ and $\mbox{max}({\widetilde \xi}^2_i)\leq \zeta^2_{max}$, where $\zeta_{min}$ and $\zeta_{max}$ are some desired bounds on the fraction of the contribution of the specific risk into the total risk. [*E.g.*]{}, we can set $\zeta_{min} = 10\%$ and $\zeta_{max}=90\%$. In practice, we actually need to fix the value of $K$, not $K_{max}$ and $K_{min}$, especially that for some preset values of $\zeta_{min}$ and $\zeta_{max}$ we may end up with $K_{max} < K_{min}$. However, the above discussion aids us in coming up with a simple heuristic definition for what $K$ should be. Here is one: $$\begin{aligned}
\label{K}
&&|g(K) - 1| \rightarrow \mbox{min}\\
\label{g}
&&g(K) \equiv \sqrt{\mbox{min}({\widetilde \xi}^2_i)} + \sqrt{\mbox{max}({\widetilde \xi}^2_i)}\end{aligned}$$ [*i.e.*]{}, we take $K$ for which $g(K)$ (which monotonically decreases with increasing $K$) is closest to 1. This simple algorithm works pretty well in practical applications.[^17]
Limitations
-----------
An evident limitation of the principal component approach is that the number of risk factors is limited by $M$. If long lookbacks are unavailabe/undesirable, as, [*e.g.*]{}, in short-holding quantitative trading strategies, then typically $M \ll N$. Yet, the number of the actually relevant underlying risk factors can be substantially greater than $M$, and most of these risk factors are missed by the principal component approach. In this regard, we can ask: can we use other than the first $M$ principal components to build a factor model? The answer, prosaically, is that, without some additional information, it is unclear what to do with the principal components with null eigenvalues. They simply do not contribute to any sample factor covariance matrix. However, not all is lost. There is a way around this difficulty.
Heterotic Construction {#sec.het}
======================
Industry Risk Factors
---------------------
Without long lookbacks, the number of risk factors based on principal components is limited.[^18] However, risk factors based on a granular enough industry classification can be plentiful. Furthermore, they are independent of the pricing data and, in this regard, are insensitive to the lookback. In fact, typically they tend to be rather stable out-of-sample as companies seldom jump industries, let alone sectors.
For terminological definiteness, here we will use the BICS nomenclature for the levels in the industry classification, albeit this is not critical here. Also, BICS has three levels “sector $\rightarrow$ industry $\rightarrow$ sub-industry" (where “sub-industry" is the most granular level). The number of levels in the industry hierarchy is not critical here either. So, we have: $N$ stocks labeled by $i=1,\dots,N$; $K$ sub-industries labeled by $A=1,\dots,K$; $F$ industries labeled by $a=1,\dots,F$; and $L$ sectors labeled by $\alpha=1,\dots,L$. More generally, we can think of such groupings as “clusters". Sometimes, loosely, we will refer to such cluster based factors as “industry" factors.[^19]
“Binary" Property
-----------------
The binary property implies that each stock belongs to one and only one sub-industry, industry and sector (or, more generally, cluster). Let $G$ be the map between stocks and sub-industries, $S$ be the map between sub-industries and industries, and $W$ be the map between industries and sectors: $$\begin{aligned}
\label{G.map}
&&G:\{1,\dots,N\}\mapsto\{1,\dots,K\}\\
&&S:\{1,\dots,K\}\mapsto\{1,\dots,F\}\label{S.map}\\
&&W:\{1,\dots,F\}\mapsto\{1,\dots,L\}\label{W.map}\end{aligned}$$ The nice thing about the binary property is that the clusters (sub-industries, industries and sectors) can be used to identify blocks (sub-matrices) in the correlation matrix $\Psi_{ij}$. [*E.g.*]{}, at the most granular level, for sub-industries, the binary matrix $\delta_{G(i), A}$ defines such blocks. Thus, the sum $B_A\equiv \sum_{i=1}^N \delta_{G(i), A}~X_i$, where $X_i$ is an arbitrary $N$-vector, is the same as $\sum_{i\in J(A)} X_i$, where $J(A)$ is the set of tickers in the sub-industry $A$. These blocks are the backbone of the following construction.
Heterotic Models
----------------
Consider the following factor loadings matrix: $$\begin{aligned}
\label{ind.pc}
&&{\widetilde \Omega}_{iA} = \delta_{G(i), A}~U_i\\
&&U_i \equiv [U(A)]_i,~~~i\in J(A),~~~A=1,\dots K\end{aligned}$$ where $J(A)\equiv \{i|G(i)=A\}$ is the set of tickers (whose number $N(A)\equiv |J(A)|$) in the sub-industry labeled by $A$. Then the $N(A)$-vector $U(A)$ is the [*first*]{} principal component of the $N(A) \times N(A)$ matrix $\Psi(A)$ defined via $[\Psi(A)]_{ij} \equiv \Psi_{ij}$, $i,j\in J(A)$. (Note that $\sum_{i\in J(A)} [U(A)]_i^2 =1$; also, let the corresponding (largest) eigenvalue of $\Psi(A)$ be $\lambda(A)$.)[^20] With this factor loadings matrix we can compute the factor covariance matrix and specific risk via a linear regression as above, and we get: $$\begin{aligned}
&&{\widetilde \xi}_i^2 = 1 - \lambda(G(i))~ U_i^2\\
&&\left({\widetilde \Omega}~\Phi~{\widetilde \Omega}^T\right)_{ij} = U_i~U_j \sum_{k\in J(G(i))} ~\sum_{l\in J(G(j))} U_k ~\Psi_{kl}~ U_l\end{aligned}$$ so we have[^21] $${\widetilde \Gamma}_{ij} = \left[1 - \lambda(G(i))~ U_i^2\right]\delta_{ij} + U_i ~U_j \sum_{k\in J(G(i))}~ \sum_{l\in J(G(j))} U_k ~\Psi_{kl}~ U_l$$ and automatically ${\widetilde \Gamma}_{ii} = 1$. This simplicity is due to the use of the (first) principal components corresponding to the blocks $\Psi(A)$ of the sample correlation matrix.
### Multiple Principal Components
For the sake of completeness, let us discuss an evident generalization. Above in (\[ind.pc\]) we took the binary map between the tickers and sub-industries and augmented it with the first principal components of the corresponding blocks in the sample correlation matrix. Instead of taking only the first principal component, we can take the first $P(A)\geq 1$ principal components for each block labeled by the sub-industry $A$ ($A=1,\dots, K$). Then we have ${\widehat K} = \sum_{A=1}^K P(A)$ risk factors labeled by pairs ${\widehat A} \equiv (A, I)$, where for a given value of $A$ we have $I\in D(A)$ (with $|D(A)| = P(A)$). The factor loadings matrix reads: $${\widetilde \Omega}_{i{\widehat A}} = \delta_{G(i), A}~[U(A)]_i^{(I)}$$ where $U(A)$ is the $N(A)\times P(A)$ matrix whose columns are the first $P(A)$ principal components (with eigenvalues $[\lambda(A)]^{(I)}$) of the $N(A) \times N(A)$ matrix $\Psi(A)$ (as above, $[\Psi(A)]_{ij} \equiv \Psi_{ij}$, $i,j\in J(A)$, and $\sum_{i\in J(A)} [U(A)]_i^{(I)}~[U(A)]_i^{(J)} = \delta_{IJ}$, $I,J\in D(A)$.) In order to have nonvanishing specific risks, it is necessary that we take $P(A) < M$ ($M+1$ is the number of observations in the time series). We then have $$\begin{aligned}
&&\Gamma_{ij} = \left[1 - \sum_{I\in D(G(i))} [\lambda(G(i))]^{(I)}~ \left([U(G(i))]^{(I)}_i\right)^2\right]\delta_{ij} +\nonumber\\
&&\,\,\,\,\,\,\,+\sum_{I\in D(G(i))}~\sum_{J\in D(G(j))} [U(G(i))]_i^{(I)}~[U(G(j))]_j^{(J)}\times\nonumber\\
&&\,\,\,\,\,\,\,\times\sum_{k\in J(G(i))}~ \sum_{l\in J(G(j))} [U(G(i))]_k^{(I)} ~\Psi_{kl}~ [U(G(j))]_l^{(J)}\end{aligned}$$ and (as in the case above with all $P(A)\equiv 1$) automatically ${\widetilde \Gamma}_{ii} = 1$.
### Caveats
The above construction might look like a free lunch, but it is not. Let us start with the $P(A)\equiv 1$ case (first principal components only). For short lookbacks, the number of risk factors typically is too large: $K$ can easily be greater than $M$, so[^22] $$\Phi_{AB} = \sum_{i\in J(A)}~\sum_{j\in J(B)} U_i~\Psi_{ij}~U_j$$ is singular. In general, the sample factor covariance matrix is singular if $K > M$. We will deal with this issue below via the nested Russian-doll risk model construction.
This issue is further exacerbated in the multiple principal component construction (with at least some $P(A)>1$) as the number of risk factors ${\widehat K} > K$ is even larger. This too can be dealt with via the Russian-doll construction. However, there is yet another caveat pertinent to using multiple principal components, irrespective of whether the factor covariance matrix is singular or not. The principal components are based on off-diagonal elements of $\Psi_{ij}$ and tend to be unstable out-of-sample, the first principal component typically being the most stable. So, for the sake of simplicity, below we will focus on the case with only first principal components.
Russian-Doll Construction {#sec.RD}
=========================
General Idea
------------
As discussed above, the sample factor covariance matrix $\Phi_{AB}$ is singular if the number of factors $K$ is greater than $M$. The simple idea behind the Russian-doll construction is to model such $\Phi_{AB}$ itself via yet another factor model matrix $\Gamma^\prime_{AB}$ (as opposed to computing it as a sample covariance matrix of the risk factors $f_A$):[^23] $$\begin{aligned}
&&\Gamma^\prime_{AB} = \sqrt{\Phi_{AA}}\sqrt{\Phi_{BB}}~{\widetilde \Gamma}^\prime_{AB}\\
&&{\widetilde \Gamma}^\prime_{AB} = ({\widetilde \xi}^{\prime}_A)^2~\delta_{AB} + \sum_{a,b=1}^F {\widetilde\Omega}^\prime_{Aa}~\Phi^\prime_{ab}~{\widetilde\Omega}^\prime_{Bb}\end{aligned}$$ where ${\widetilde \xi}^\prime_A$ is the specific risk for the “normalized" factor return ${\widetilde f}_A \equiv f_A/\sqrt{\Phi_{AA}}$; ${\widetilde \Omega}^\prime_{Aa}$, $A=1,\dots,K$, $a=1,\dots,F$ is the corresponding factor loadings matrix; and $\Phi^\prime_{ab}$ is the factor covariance matrix for the underlying risk factors $f^\prime_a$, $a=1,\dots,F$, where we assume that $F\ll K$. If the smaller factor covariance matrix $\Phi^\prime_{ab}$ is still singular, we model it via yet another factor model with fewer risk factors, and so on – until the resulting factor covariance matrix is nonsingular. If, at the final stage, we are left with a single factor, then the resulting $1\times 1$ factor covariance matrix is automatically nonsingular – it is simply the sample variance of the remaining factor.
Complete Heterotic Russian-Doll Embedding {#sub.het.rd}
-----------------------------------------
For concreteness we will use the BICS terminology for the levels in the industry classification, albeit this is not critical here. Also, BICS has three levels “sector $\rightarrow$ industry $\rightarrow$ sub-industry" (where “sub-industry" is the most granular level). For definiteness, we will assume three levels here, and the generalization to more levels is straightforward. So, we have: $N$ stocks labeled by $i=1,\dots,N$; $K$ sub-industries labeled by $A=1,\dots,K$; $F$ industries labeled by $a=1,\dots,F$; and $L$ sectors labeled by $\alpha=1,\dots,L$. A nested Russian-doll risk model then is constructed as follows: $$\begin{aligned}
\label{Gamma.RD}
&&\Gamma_{ij} = \sqrt{C_{ii}}\sqrt{C_{jj}}~{\widetilde \Gamma}_{ij}\\
&&{\widetilde \Gamma}_{ij} = {\widetilde \xi}_i^2~\delta_{ij} + U_i~U_j~\Gamma^\prime_{G(i), G(j)}\\
&&\Gamma^\prime_{AB} = \sqrt{\Phi_{AA}}\sqrt{\Phi_{BB}}~{\widetilde\Gamma}^\prime_{AB}\label{Gamma.prime.RD}\\
&&{\widetilde\Gamma}^\prime_{AB} = ({\widetilde \xi}^\prime_A)^2~\delta_{AB} + U^\prime_A~U^\prime_B~\Gamma^{\prime\prime}_{S(A), S(B)}\\
&&\Gamma^{\prime\prime}_{ab} = \sqrt{\Phi^\prime_{aa}}\sqrt{\Phi^\prime_{bb}}~{\widetilde\Gamma}^{\prime\prime}_{ab}\\
&&{\widetilde\Gamma}^{\prime\prime}_{ab} = ({\widetilde \xi}^{\prime\prime}_a)^2~\delta_{ab} + U^{\prime\prime}_a~U^{\prime\prime}_b~\Gamma^{\prime\prime\prime}_{W(a), W(b)}\\
&&\Gamma^{\prime\prime\prime}_{\alpha\beta} = \sqrt{\Phi^{\prime\prime}_{\alpha\alpha}}\sqrt{\Phi^{\prime\prime}_{\beta\beta}}~{\widetilde \Gamma}^{\prime\prime\prime}_{\alpha\beta}\\
&&{\widetilde \Gamma}^{\prime\prime\prime}_{\alpha\beta} = ({\widetilde \xi}^{\prime\prime\prime}_\alpha)^2~\delta_{\alpha\beta} + U^{\prime\prime\prime}_\alpha~U^{\prime\prime\prime}_\beta~\Phi^{\prime\prime\prime}\end{aligned}$$ where $$\begin{aligned}
\label{xi.RD}
&&{\widetilde \xi}_i^2 = 1 - \lambda(G(i))~U_i^2\\
&&({\widetilde \xi}^\prime_A)^2 = 1 - \lambda^\prime(S(A))~(U^\prime_A)^2\\
&&({\widetilde \xi}^{\prime\prime}_a)^2 = 1 - \lambda^{\prime\prime}(W(a))~(U^{\prime\prime}_a)^2\\
&&({\widetilde \xi}^{\prime\prime\prime}_\alpha)^2 = 1 - \lambda^{\prime\prime\prime}~(U^{\prime\prime\prime}_\alpha)^2\end{aligned}$$ and $$\begin{aligned}
&&\Phi_{AB} = \sum_{i\in J(A)}~\sum_{j\in J(B)} U_i~\Psi_{ij}~U_j\\
&&\Phi^\prime_{ab} = \sum_{A\in J^\prime(a)} ~\sum_{B\in J^\prime(b)} U^\prime_A~\Psi^\prime_{AB}~U^\prime_B\\
&&\Phi^{\prime\prime}_{\alpha\beta} = \sum_{a\in J^{\prime\prime}(\alpha)} ~\sum_{b\in J^{\prime\prime}(\beta)} U^{\prime\prime}_a~ \Psi^{\prime\prime}_{ab}~U^{\prime\prime}_b\label{Theta.1}\\
&&\Phi^{\prime\prime\prime}= \sum_{\alpha,\beta = 1}^L U^{\prime\prime\prime}_\alpha~\Psi^{\prime\prime\prime}_{\alpha\beta}~U^{\prime\prime\prime}_\beta\end{aligned}$$ so we have $\Phi_{AA} = \lambda(A)$, $\Phi^\prime_{aa} = \lambda^\prime(a)$, $\Phi^{\prime\prime}_{\alpha\alpha} = \lambda^{\prime\prime}(\alpha)$, and $\Phi^{\prime\prime\prime} = \lambda^{\prime\prime\prime}$ (see below). Also, $J(A) = \{i| G(i)=A\}$ ($N_A \equiv |J(A)|$ tickers in sub-industry $A$), $J^\prime(a) = \{A|S(A) = a\}$ ($N^\prime(a) \equiv |J^\prime(a)|$ sub-industries in industry $a$), $J^{\prime\prime}(\alpha) = \{a|W(a) = \alpha\}$ ($N^{\prime\prime}(\alpha)\equiv |J^{\prime\prime}(\alpha)|$ industries in sector $\alpha$), and the maps $G$ (tickers to sub-industries), $S$ (sub-industries to industries) and $W$ (industries to sectors) are defined in (\[G.map\]), (\[S.map\]) and (\[W.map\]). Furthermore, $$\begin{aligned}
&&\Psi_{ij} = C_{ij} / \sqrt{C_{ii}}\sqrt{C_{jj}}\\
&&\Psi^\prime_{AB} = \Phi_{AB} / \sqrt{\Phi_{AA}}\sqrt{\Phi_{BB}}\\
&&\Psi^{\prime\prime}_{ab} = \Phi^\prime_{ab} / \sqrt{\Phi^\prime_{aa}}\sqrt{\Phi^\prime_{bb}}\\
&&\Psi^{\prime\prime\prime}_{\alpha\beta} = \Phi^{\prime\prime}_{\alpha\beta} / \sqrt{\Phi^{\prime\prime}_{\alpha\alpha}}\sqrt{\Phi^{\prime\prime}_{\beta\beta}}\end{aligned}$$ and $$\begin{aligned}
\label{U.RD}
&&U_i \equiv [U(A)]_i,~~~i\in J(A)\\
&&U^\prime_A \equiv [U^\prime(a)]_A,~~~A\in J^\prime(a)\\
&&U^{\prime\prime}_a \equiv [U^{\prime\prime}(\alpha)]_a,~~~a\in J^{\prime\prime}(\alpha)\end{aligned}$$ The $N(A)$-vector $U(A)$ is the first principal component of $\Psi(A)$ with the eigenvalue $\lambda(A)$ ($[\Psi(A)]_{ij} \equiv \Psi_{ij}$, $i,j\in J(A)$), the $N^\prime(a)$-vector $U^\prime(a)$ is the first principal component of $\Psi^\prime(a)$ with the eigenvalue $\lambda^\prime(a)$ ($[\Psi^\prime(a)]_{AB} \equiv \Psi^\prime_{AB}$, $A,B\in J^\prime(a)$), the $N^{\prime\prime}(\alpha)$-vector $U^{\prime\prime}(\alpha)$ is the first principal component of $\Psi^{\prime\prime}(\alpha)$ with the eigenvalue $\lambda^{\prime\prime}(\alpha)$ ($[\Psi^{\prime\prime}(\alpha)]_{ab} \equiv \Psi^{\prime\prime}_{ab}$, $a,b\in J^{\prime\prime}(\alpha)$), while $U^{\prime\prime\prime}_\alpha$ is the first principal component of $\Psi^{\prime\prime\prime}_{\alpha\beta}$ with the eigenvalue $\lambda^{\prime\prime\prime}$. The vectors $U(A)$, $U^\prime(a)$ and $U^{\prime\prime}(\alpha)$ are normalized, so $\sum_{i\in J(A)} U_i^2 = 1$, $\sum_{A\in J^\prime(a)} (U^\prime_A)^2 = 1$, $\sum_{a\in J^{\prime\prime}(\alpha)} (U^{\prime\prime}_a)^2 = 1$, and also $\sum_{\alpha=1}^L (U^{\prime\prime\prime}_\alpha)^2 = 1$.
For the sake of completeness, above we included the step where the sample factor covariance matrix $\Phi^{\prime\prime}_{\alpha\beta}$ for the sectors is further approximated via a 1-factor model ${\widetilde \Gamma}^{\prime\prime\prime}_{\alpha\beta}$. If $\Phi^{\prime\prime}_{\alpha\beta}$ computed via (\[Theta.1\]) is nonsingular, then this last step can be omitted,[^24] so at the last stage we have $L$ factors (as opposed to a single factor).[^25] Similarly, if we have enough observations to compute the sample covariance matrix $\Phi^\prime_{ab}$ for the industries, we can stop at that stage. Finally, note that in the above construction we are guaranteed to have $({\widetilde \xi}^{\prime\prime\prime}_\alpha)^2 > 0$, $({\widetilde \xi}^{\prime\prime}_a)^2 >0$, $({\widetilde \xi}^\prime_A)^2> 0$ and ${\widetilde \xi}_i^2\geq 0$ (with the last equality occurring only for single-ticker sub-industries and not posing a problem – see below).[^26] In Appendix \[app.B\] we give the R code for building heterotic risk models.
Model Covariance Matrix and Its Inverse {#sub.inv}
---------------------------------------
The model covariance matrix is given by $\Gamma_{ij}$ defined in (\[Gamma.RD\]). For completeness, let us present it in the “canonical" form: $$\Gamma_{ij} = \xi_i^2~\delta_{ij} + \sum_{A,B=1}^K \Omega_{iA}~\Phi^*_{AB}~\Omega_{jB}$$ where $$\begin{aligned}
&&\xi_i^2 \equiv C_{ii}~{\widetilde \xi}_i^2\\
&&\Omega_{iA} \equiv \sqrt{C_{ii}}~U_i~\delta_{G(i),A}\\
&&\Phi^*_{AB} \equiv \Gamma^\prime_{AB}\end{aligned}$$ where ${\widetilde \xi}_i^2$ is defined in (\[xi.RD\]), $U_i$ is defined in (\[U.RD\]), $\Gamma^\prime_{AB}$ is defined in (\[Gamma.prime.RD\]), and we use the star superscript in the our factor covariance matrix $\Phi^*_{AB}$ (which is nonsingular) to distinguish it from the sample factor covariance matrix $\Phi_{AB}$ (which is singular).
In many applications, such as portfolio optimization, one needs the inverse of the matrix $\Gamma$. When we have no single-ticker sub-industries, the inverse is given by (in matrix notation) $$\begin{aligned}
\label{Gamma.inv}
&&\Gamma^{-1} = \Xi^{-1} - \Xi^{-1}~\Omega~\Delta^{-1}~\Omega^T~\Xi^{-1}\\
&&\Delta \equiv (\Phi^*)^{-1} + \Omega^T~\Xi^{-1}~\Omega\\
&&\Xi\equiv \mbox{diag}(\xi^2_i)\end{aligned}$$ However, when there are some single-ticker sub-industries, the corresponding $\xi_i^2=0$, $i\in H$ ($H\equiv \{i|N(G(i)) = 1\}$), so (\[Gamma.inv\]) “breaks". Happily, there is an easy “fix". This is because for such tickers the specific risk and factor risk are [*indistinguishable*]{}. Recall that $U_i = 1$, $i\in H$, and $\Phi^*_{AA} = 1$, $A\in E$ ($E\equiv\{A|N(A)=1\}$). We can rewrite $\Gamma_{ij}$ via $$\Gamma_{ij} = {\widehat \xi}_i^2~\delta_{ij} + \sum_{A,B=1}^K \Omega_{iA}~{\widehat \Phi}^*_{AB}~\Omega_{jB}$$ where: ${\widehat \xi}_i^2 = \xi_i^2$ for $i\not\in H$; ${\widehat \xi}_i^2 = C_{ii}~\zeta_i$ for $i\in H$ with arbitrary $\zeta_i$, $0 < \zeta_i \leq 1$; ${\widehat \Phi}^*_{AB} = \Phi^*_{AB}$ if $A\not\in E$ or $B\not\in E$ or $A\neq B$; and ${\widehat \Phi}^*_{AA} = 1 - \delta_{A, G(i)}~\zeta_i$ for $A\in E$. (Here we have taken into account that $U_i=1$, $i\in H$.) Now we can invert $\Gamma$ via $$\begin{aligned}
\label{Gamma.inv.1}
&&\Gamma^{-1} = {\widehat \Xi}^{-1} - {\widehat \Xi}^{-1}~\Omega~{\widehat \Delta}^{-1}~\Omega^T~{\widehat \Xi}^{-1}\\
&&{\widehat \Delta} \equiv ({\widehat \Phi}^*)^{-1} + \Omega^T~{\widehat \Xi}^{-1}~\Omega\\
&&{\widehat \Xi}\equiv \mbox{diag}({\widehat \xi}^2_i)\end{aligned}$$ Note that, due to the factor model structure, to invert the $N\times N$ matrix $\Gamma$, we only need to invert two $K\times K$ matrices ${\widehat \Phi}^*$ and ${\widehat \Delta}$. If there are no single-ticker sub-industries, then $\Phi^*$ itself has a factor model structure and involves inverting two $F\times F$ matrices, one of which has a factor model structure, and so on.
Horse Race {#sec.horse}
==========
So, suppose we have built a complete heterotic risk model. How do we know it adds value? [*I.e.*]{}, how do we know that the off-diagonal elements of the factor model covariance matrix $\Gamma_{ij}$ are stable out-of-sample to the extent that they add value. We can run a horse race. There are many ways of doing this. Here is one. For a given trading universe we compute some expected returns, [*e.g.*]{}, based on overnight mean-reversion. We can construct a trading portfolio by using our heterotic risk model covariance matrix in the optimization whereby we maximize the Sharpe ratio (subject to the dollar neutrality constraint). On the other hand, we can run the same optimization with a diagonal sample covariance matrix $\mbox{diag}(C_{ii})$ subject to neutrality (via linear homogeneous constraints) w.r.t. the underlying heterotic risk factors (plus dollar neutrality).[^27] In fact, optimization with such diagonal covariance matrix and subject to such linear homogeneous constraints is equivalent to a weighted cross-sectional regression with the loadings matrix (over which the expected returns are regressed) identified with the factor loadings matrix (augmented by the intercept, [*i.e.*]{}, the unit vector, for dollar neutrality) and the regression weights identified with the inverse sample variances $1/C_{ii}$ (see (Kakushadze, 2015a) for details). So, we will refer to the horse race as between optimization (using the heterotic risk model) and weighted regression (with the aforementioned linear homogeneous constraints).[^28]
Notations
---------
Let $P_{is}$ be the time series of stock prices, where $i=1,\dots,N$ labels the stocks, and $s=0,1,\dots,M$ labels the trading dates, with $s=0$ corresponding to the most recent date in the time series. The superscripts $O$ and $C$ (unadjusted open and close prices) and $AO$ and $AC$ (open and close prices fully adjusted for splits and dividends) will distinguish the corresponding prices, so, [*e.g.*]{}, $P^C_{is}$ is the unadjusted close price. $V_{is}$ is the unadjusted daily volume (in shares). Also, for each date $s$ we define the overnight return as the previous-close-to-open return: $$\label{c2o.ret}
E_{is} \equiv \ln\left({P^{AO}_{is} / P^{AC}_{i,s+1}}\right)$$ This return will be used in the definition of the expected return in our mean-reversion alpha. We will also need the close-to-close return $$\label{c2c.ret}
R_{is} \equiv \ln\left({P^{AC}_{is} / P^{AC}_{i,s+1}}\right)$$ An out-of-sample (see below) time series of these returns will be used in constructing the heterotic risk model and computing, among other things, the sample variances $C_{ii}$. Also note that all prices in the definitions of $E_{is}$ and $R_{is}$ are fully adjusted.
We assume that: i) the portfolio is established at the open[^29] with fills at the open prices $P^O_{is}$; ii) it is liquidated at the close on the same day – so this is a purely intraday alpha – with fills at the close prices $P^C_{is}$; and iii) there are no transaction costs or slippage – our aim here is not to build a realistic trading strategy, but to test that our heterotic risk model adds value to the alpha. The P&L for each stock $$\Pi_{is} = H_{is}\left[{P^C_{is}\over P^O_{is}}-1\right]$$ where $H_{is}$ are the [*dollar*]{} holdings. The shares bought plus sold (establishing plus liquidating trades) for each stock on each day are computed via $Q_{is} = 2 |H_{is}| / P^O_{is}$.
Universe Selection
------------------
For the sake of simplicity,[^30] we select our universe based on the average daily dollar volume (ADDV) defined via (note that $A_{is}$ is out-of-sample for each date $s$): $$\label{ADDV}
A_{is}\equiv {1\over d} \sum_{r=1}^d V_{i, s+r}~P^C_{i, s+r}$$ We take $d=21$ ([*i.e.*]{}, one month), and then take our universe to be the top 2000 tickers by ADDV. To ensure that we do not inadvertently introduce a universe selection bias, we rebalance monthly (every 21 trading days, to be precise). [*I.e.*]{}, we break our 5-year backtest period (see below) into 21-day intervals, we compute the universe using ADDV (which, in turn, is computed based on the 21-day period immediately preceding such interval), and use this universe during the entire such interval. We do have the survivorship bias as we take the data for the universe of tickers as of 9/6/2014 that have historical pricing data on http://finance.yahoo.com (accessed on 9/6/2014) for the period 8/1/2008 through 9/5/2014. We restrict this universe to include only U.S. listed common stocks and class shares (no OTCs, preferred shares, [*etc.*]{}) with BICS sector, industry and sub-industry assignments as of 9/6/2014.[^31] However, as discussed in detail in Section 7 of (Kakushadze, 2015a), the survivorship bias is not a leading effect in such backtests.[^32]
Backtesting
-----------
We run our simulations over a period of 5 years (more precisely, 1260 trading days going back from 9/5/2014, inclusive). The annualized return-on-capital (ROC) is computed as the average daily P&L divided by the intraday investment level $I$ (with no leverage) and multiplied by 252. The annualized Sharpe Ratio (SR) is computed as the daily Sharpe ratio multiplied by $\sqrt{252}$. Cents-per-share (CPS) is computed as the total P&L divided by the total shares traded.[^33]
Weighted Regression Alphas {#sub.reg}
--------------------------
We will always require that our portfolio be dollar neutral: $$\label{DN}
\sum_{i=1}^N H_{is}= 0$$ We will further require neutrality $$\sum_{i=1}^N H_{is}~\Lambda_{iAs}= 0$$ with the three different incarnations for the loadings matrix $\Lambda_{iA}$ (for each trading day $s$, so we omit the index $s$)[^34] defined via: $$\begin{aligned}
\label{load.pc}
&&\mbox{principal components:}~~~\Lambda_{iA} = \sqrt{C_{ii}}~\sqrt{\lambda^{(A)}}~V^{(A)}_i,~~~A=1,\dots,K_{PC}\\
&&\mbox{sub-industries:}~~~\Lambda_{iA} = \delta_{G(i),A},~~~A=1,\dots,K\label{load.sub}\\
&&\mbox{heterotic risk factors:}~~~\Lambda_{iA} = \sqrt{C_{ii}}~U_i~\delta_{G(i),A},~~~A=1,\dots,K\label{load.het}\end{aligned}$$ Here $V^{(A)}_i$ are the first $K_{PC}$ principal components (with the eigenvalues $\lambda^{(A)}$)[^35] of the sample correlation matrix $\Psi_{ij}$. For each date $s$ we take $M+1=d=21$ trading days as our lookback ([*i.e.*]{}, the number of observations) in the out-of-sample time series of close-to-close (see (\[c2c.ret\])) returns $(R_{i,(s+1)}, R_{i,(s+2)}, \dots, R_{i,(s+d)})$ (based on which we compute the sample covariance (correlation) matrix $C_{ijs}$ ($\Psi_{ijs}$) for each $s$), so the number of the nonvanishing eigenvalues $\lambda^{(A)}>0$ is $M=20$, and we take $K_{PC}=M$. Further, the map $G$ between tickers and sub-industries is defined in (\[G.map\]), and $K$ is the number of sub-industries.[^36] Finally, the vector $U_i$ in (\[load.het\]) is defined in (\[U.RD\]).
For each date labeled by $s$, we run cross-sectional regressions of the overnight (see (\[c2o.ret\])) returns $E_{is}$ over the corresponding loadings matrix, call it $Y$ (with indices suppressed), which has 3 different incarnations: i) for principal components, $Y$ is an $N\times (K_{PC}+1)$ matrix, whose first column in the intercept (unit $N$-vector), and the remaining columns are populated by $\Lambda_{iA}$ defined in (\[load.pc\]); ii) for sub-industries, the elements of $Y$ are the same as $\Lambda_{iA}$ defined in (\[load.sub\]); and iii) for heterotic risk factors, $Y$ is an $N\times (K+1)$ matrix, whose first column in the intercept (unit $N$-vector), and the remaining columns are populated by $\Lambda_{iA}$ defined in (\[load.het\]). We take the regression weights to be $z_i \equiv 1/C_{ii}$. More precisely, to avoid unnecessary variations in the weights $z_i$ (as such variations could result in unnecessary overtrading), we do not recompute $z_i$ daily but every 21 trading days, same as with the trading universe.
In the cases i)-iii) above, we compute the residuals $\varepsilon_{is}$ of the weighted regression and the dollar holdings $H_{is}$ via (we use matrix notation and suppress indices): $$\begin{aligned}
&&{\widetilde E} \equiv Z~\varepsilon = Z \left[E - Y~(Y^T~Z~Y)^{-1}~Y^T~Z~E\right]\\
&&H_{is} \equiv -{\widetilde E}_{is} ~ {I\over\sum_{j=1}^N \left|{\widetilde E}_{js}\right|}\end{aligned}$$ where $Z\equiv\mbox{diag}(z_i)$, we have dollar neutrality (\[DN\]),[^37] and $\sum_{i=1}^N \left|H_{is}\right| = I$ (the total [*intraday*]{} dollar investment level (long plus short), which is the same for all dates $s$).
The simulation results are given in Table \[table2\] and P&Ls for the 3 cases i)-iii) are plotted in Figure 2. For comparison purposes – and to alley any potential concerns that the results in Table \[table2\] may not hold for realistic position bounds, in Table \[table3\] we give the simulation results for the same cases i)-iii) above with the strict bounds $$\label{liq}
|H_{is}| \leq 0.01~A_{is}$$ so not more than 1% of each stock’s ADDV is bought or sold. We use the bounded regression algorithm and the R source code of (Kakushadze, 2015b) to run these simulations. Expectedly, the liquidity bounds (\[liq\]) lower ROC and CPS while improving SR, but in the same fashion for all 3 weighted regression alphas i)-iii). The results in Tables \[table2\] and \[table3\] confirm our prior intuitive argument that the sub-industries outperform the principal components simply because they are more numerous.[^38]
If we compute $\Lambda_{iA}$ in (\[load.pc\]) and (\[load.het\]) every 21 trading days (instead of daily – see fn. \[fn.load\]), the difference is very slight. [*E.g.*]{}, for the heterotic risk factors computed every 21 days (with no bounds) we get: ROC = 51.66%, SR = 13.42, CPS = 2.26.
Finally, let us also mention that, in the weighted regressions ii) and iii), the dollar holdings for the tickers in the single-ticker sub-industries are automatically null. This is [*not*]{} the case for optimized alphas (see below). Generally, if single-ticker (or small) sub-industries are undesirable, one can “prune" the industry hierarchy tree by merging (single-ticker and/or small) sub-industries at the industry level.
Optimized Alphas {#sub.opt}
----------------
As mentioned above, our goal is to determine whether the heterotic risk model construction adds value by comparing the simulated performance of the weighted regression alphas above to the simulated performance of the optimized alphas (via maximizing the Sharpe ratio) based on the same expected returns $E_{is}$. In maximizing the Sharpe ratio, we use the heterotic risk model covariance matrix $\Gamma_{ij}$ given by (\[Gamma.RD\]), which we compute every 21 trading days (same as for the universe). For each date (we omit the index $s$) we maximize the Sharpe ratio subject to the dollar neutrality constraint: $$\begin{aligned}
&&{\cal S} \equiv {\sum_{i=1}^N H_i~E_i\over{\sqrt{\sum_{i,j=1}^N \Gamma_{ij}~H_i~H_j}}} \rightarrow \mbox{max}\\
&&\sum_{i=1}^N H_i = 0\label{d.n.opt}\end{aligned}$$ The solution is given by $$\label{H.opt}
H_i = -\gamma \left[\sum_{j = 1}^N \Gamma^{-1}_{ij}~E_j - \sum_{j=1}^N \Gamma^{-1}_{ij}~{{\sum_{k,l=1}^N \Gamma^{-1}_{kl}~E_l}\over{\sum_{k,l = 1}^N \Gamma^{-1}_{kl}}}\right]$$ where $\Gamma^{-1}$ is the inverse of $\Gamma$ (see Subsection \[sub.inv\]), and the overall normalization constant $\gamma > 0$ (this is a mean-reversion alpha) is fixed via the requirement that $$\sum_{i=1}^N \left|H_i\right| = I$$ Note that (\[H.opt\]) satisfies the dollar neutrality constraint (\[d.n.opt\]).
The simulation results are given in Table \[table2\] in the bottom row. The P&L plot for this optimized alpha is included in Figure 2. For the same reasons as in the case of weighted regression alphas, in the bottom row of Table \[table3\] we give the simulation results for the same optimized alpha with the strict liquidity bounds (\[liq\]).[^39] We use the optimization algorithm for maximizing the Sharpe ratio subject to linear homogeneous constraints and bounds discussed in (Kakushadze, 2015a).[^40] Also, in the second rows in Tables \[table2\] and \[table3\] we have included the simulation results for the optimized alpha where in the optimization we use the risk factor model covariance matrix $\Gamma_{ij}$ based on the principal components discussed in Section \[sub.pc\].[^41] From our simulation results in Tables \[table2\] and \[table3\] it is evident that the heterotic risk model predicts off-diagonal elements of the covariance matrix (that is, correlations) out-of-sample rather well. Indeed, using it in the optimization sizably improves ROC, SR and CPS compared with the weighted regressions with all three loadings i)-iii) above.[^42]
Concluding Remarks {#sec.conc}
==================
The heterotic risk model construction we discuss in this paper is based on a “heterosis" of: i) granularity of an industry classification; ii) diagonality of the principal component factor covariance matrix for any sub-cluster of stocks; and iii) dramatic reduction of the size of the factor covariance matrix in the Russian-doll construction. This is a powerful approach, as is evident from the horse race we ran above.
Naturally, one may wonder if we can extend our construction to risk models which do not include any statistical risk factors ([*i.e.*]{}, principal components) or include other non-binary factors such as style factors. A key simplifying feature in the heterotic construction is that the industry classification, which is used as the backbone (and is augmented with the principal components to satisfy the conditions (\[tot.risk\])), is binary. Once non-binary risk factors are included, it is more nontrivial to compute the specific risk and the factor covariance matrix (such that (\[tot.risk\]) are satisfied). However, there exist proprietary algorithms for dealing with this, which are outside of the scope of this paper. We hope to make these algorithms a public knowledge elsewhere.
One final remark concerns purely statistical risk models based on principal components. Albeit their market share is rather limited, it is unclear why a portfolio manager would be willing to pay for such models considering that they are straightforward to build in-house, especially now that we have provided the source code for constructing them. One argument is that using option implied volatility (which is available only for optionable stocks) to model stock volatility should work better,[^43] and if a portfolio manager does not possess the implied volatility data or the know-how for incorporating it into a statistical risk model, he or she would be better off simply buying one from a provider. However, this argument appears to be thin, at best. Nowadays, with ever-shortening lookbacks, it is unclear if the implied volatility indeed adds any value when the risk model is used in actual portfolio optimization for actual alphas. In this regard a new study would appear to be warranted. In any event, as we saw above, heterotic risk models outperform principal component risk models by a significant margin, so one can build heterotic risk models in-house (instead of buying less powerful statistical models) now that this know-how is in the public domain. The only data needed to construct a heterotic risk model is: i) adjusted close prices; and ii) a granular enough binary industry classification, such as GICS, BICS, ICB, [*etc.*]{} Most quantitative traders already have this data in-house. So, we hope this paper further encourages/aids organic custom risk model building.
R Code: Principal Component Risk Model {#app.A}
======================================
In this appendix we give the R (R Package for Statistical Computing, http://www.r-project.org) source code for building a purely statistical risk model (principal components) based on the algorithm we discuss in Section \[sub.pc\], including the algorithm for fixing the number of factors $K$ in Section \[sub.fix.K\]. The code below is essentially self-explanatory and straightforward. It consists of a single function . The input is: i) , an $N\times d$ matrix of returns ([*e.g.*]{}, daily close-to-close returns), where $N$ is the number of tickers, $d$ is the number of observations in the time series ([*e.g.*]{}, the number of trading days), and the ordering of the dates is immaterial; and ii) , where for (default) the risk factors are computed based on the principal components of the sample correlation matrix $\Psi_{ij}$, whereas for they are computed based on the sample covariance matrix $C_{ij}$. The output is a list: is the specific risk $\xi_i$ (not the specific variance $\xi_i^2$), is the factor loadings matrix $\Omega_{iA} = \sqrt{C_{ii}}~{\widetilde \Omega}_{iA}$, is the factor covariance matrix $\Phi_{AB}$ (with the normalization (\[FLM.PC\]) for the factor loadings matrix, $\Phi_{AB} = \delta_{AB}$), is the factor model covariance matrix $\Gamma_{ij} = \sqrt{C_{ii}}\sqrt{C_{jj}}~{\widetilde \Gamma}_{ij}$, and is the matrix $\Gamma^{-1}_{ij}$ inverse to $\Gamma_{ij}$.\
\
R Code: Heterotic Risk Model {#app.B}
============================
In this appendix we give the R source code for building the heterotic risk model based on the algorithm we discuss in Section \[sub.het.rd\]. The code below is essentially self-explanatory and straightforward as it simply follows the formulas in Section \[sub.het.rd\]. It consists of a single function . The input is: i) , an $N\times d$ matrix of returns ([*e.g.*]{}, daily close-to-close returns), where $N$ is the number of tickers, $d$ is the number of observations in the time series ([*e.g.*]{}, the number of trading days), and the ordering of the dates is immaterial; ii) is a list whose length [*a priori*]{} is arbitrary, and its elements are populated by the binary matrices (with rows corresponding to tickers, so is $N$) corresponding to the levels in the input binary industry classification hierarchy in the order of decreasing granularity (so, in the BICS case is the $N\times K$ matrix $\delta_{G(i), A}$ (sub-industries), is the $N\times F$ matrix $\delta_{G^\prime(i), a}$ (industries), and is the $N\times L$ matrix $\delta_{G^{\prime\prime}(i), \alpha}$ (sectors), where the map $G$ is defined in (\[G.map\]) (tickers to sub-industries), $G^\prime \equiv GS$ (tickers to industries), and $G^{\prime\prime} \equiv GSW$ (tickers to sectors), with the map $S$ (sub-industries to industries) defined in (\[S.map\]), and the map $W$ (industries to sectors) defined in (\[W.map\])); iii) , where for at the final step we have a single factor (“market"), while for (default) the factors correspond to the least granular level in the industry classification hierarchy; and iv) , where for the tickers corresponding to the single-ticker clusters at the most granular level in the industry classification hierarchy (in the BICS case this would be the sub-industry level) are dropped altogether, while for (default) the output universe is the same as the input universe. The output is a list: is the specific risk $\xi_i$ (not the specific variance $\xi_i^2$), is the factor loadings matrix $\Omega_{iA} = \sqrt{C_{ii}}~{\widetilde \Omega}_{iA}$, is the factor covariance matrix $\Phi_{AB}$, is the factor model covariance matrix $\Gamma_{ij} = \sqrt{C_{ii}}\sqrt{C_{jj}}~{\widetilde \Gamma}_{ij}$, and is the matrix $\Gamma^{-1}_{ij}$ inverse to $\Gamma_{ij}$.\
\
R Code: Optimizer with Constraints & Bounds {#app.C}
===========================================
In this appendix we give the R source code for the optimization algorithm with linear homogeneous constraints and position bounds we use in Section \[sub.opt\]. This code is similar to the code for the bounded regression algorithm discussed in detail in (Kakushadze, 2015b) with one important difference, so our discussion here will be brief. The entry function is . The of are: , which is the $N$-vector of stock returns (for a given date); , a matrix whose columns are the coefficients of the homogeneous constraints, so is $N$ ([*e.g.*]{}, if the sole constraint is the dollar neutrality constraint, then is an $N\times 1$ matrix with unit elements); , which is the $N\times N$ inverse factor model covariance matrix $\Gamma^{-1}_{ij}$; , which is the $N$-vector of the upper bounds $w_i^+$ on the weights $w_i$ (see below); , which is the $N$-vector of the lower bounds $w_i^-$ on the weights $w_i$; and , which is the desired precision with which the output weights $w_i$, the $N$-vector of which returns, must satisfy the normalization condition $\sum_{i=1}^N |w_i| = 1$. Here the weights are defined as $w_i\equiv H_i/I$ (the dollar holdings over the total investment level). See (Kakushadze, 2015b) for more detail.[^44]\
\
DISCLAIMERS {#app.D}
===========
Wherever the context so requires, the masculine gender includes the feminine and/or neuter, and the singular form includes the plural and [*vice versa*]{}. The author of this paper (“Author") and his affiliates including without limitation Quantigic$^\circledR$ Solutions LLC (“Author’s Affiliates" or “his Affiliates") make no implied or express warranties or any other representations whatsoever, including without limitation implied warranties of merchantability and fitness for a particular purpose, in connection with or with regard to the content of this paper including without limitation any code or algorithms contained herein (“Content").
The reader may use the Content solely at his/her/its own risk and the reader shall have no claims whatsoever against the Author or his Affiliates and the Author and his Affiliates shall have no liability whatsoever to the reader or any third party whatsoever for any loss, expense, opportunity cost, damages or any other adverse effects whatsoever relating to or arising from the use of the Content by the reader including without any limitation whatsoever: any direct, indirect, incidental, special, consequential or any other damages incurred by the reader, however caused and under any theory of liability; any loss of profit (whether incurred directly or indirectly), any loss of goodwill or reputation, any loss of data suffered, cost of procurement of substitute goods or services, or any other tangible or intangible loss; any reliance placed by the reader on the completeness, accuracy or existence of the Content or any other effect of using the Content; and any and all other adversities or negative effects the reader might encounter in using the Content irrespective of whether the Author or his Affiliates is or are or should have been aware of such adversities or negative effects.
The R code included in Appendix \[app.A\], Appendix \[app.B\] and Appendix \[app.C\] hereof is part of the copyrighted R code of Quantigic$^\circledR$ Solutions LLC and is provided herein with the express permission of Quantigic$^\circledR$ Solutions LLC. The copyright owner retains all rights, title and interest in and to its copyrighted source code included in Appendix \[app.A\], Appendix \[app.B\] and Appendix \[app.C\] hereof and any and all copyrights therefor.
[99]{}
biblabel\#1
Acharya, V.V. and Pedersen, L.H. (2005) Asset pricing with liquidity risk. [*Journal of Financial Economics*]{} 77(2): 375-410.
Ang, A., Hodrick, R., Xing, Y. and Zhang, X. (2006) The Cross-Section of Volatility and Expected Returns. [*Journal of Finance*]{} 61(1): 259-299.
Anson, M. (2013/14) Performance Measurement in Private Equity: Another Look at the Lagged Beta Effect. [*The Journal of Private Equity*]{} 17(1): 29-44.
Asness, C.S. (1995) The Power of Past Stock Returns to Explain Future Stock Returns. Goldman Sachs Asset Management. Working paper.
Asness, C. and Stevens, R. (1995) Intra- and Inter-Industry Variation in the Cross-Section of Expected Stock Returns. Goldman Sachs Asset Management. Working paper.
Asness, C., Krail, R.J. and Liew, J.M. (2001) Do Hedge Funds Hedge? [*The Journal of Portfolio Management*]{} 28(1): 6-19.
Bai, J. (2003) Inferential theory for factor models of large dimension. [*Econometrica*]{} 71(11): 135-171.
Bai, J. and Li, K. (2012) Statistical Analysis of Factor Models of High Dimension. [*The Annals of Statistics*]{} 40(1): 436-465.
Bai, J. and Ng, S. (2002) Determining the number of factors in approximate factor models. [*Econometrica*]{} 70(1): 191-221.
Bansal, R. and Viswanathan, S. (1993) No Arbitrage and Arbitrage Pricing: A New Approach. [*The Journal of Finance*]{} 48(4): 1231-1262.
Banz, R. (1981) The relationship between return and market value of common stocks. [*Journal of Financial Economics*]{} 9(1): 3-18.
Basu, S. (1977) The investment performance of common stocks in relation to their price to earnings ratios: A test of the efficient market hypothesis. [*Journal of Finance*]{} 32(3): 663-682.
Black, F. (1972) Capital market equilibrium with restricted borrowing. [*Journal of Business*]{} 45(3): 444-455.
Black, F., Jensen, M. and Scholes, M. (1972) The capital asset pricing model: Some empirical tests. In: Jensen, M. (ed.) [*Studies in the Theory of Capital Markets*]{}. New York, NY: Praeger Publishers, pp. 79-121.
Blume, O. and Friend, L. (1973) A new look at the capital asset pricing model. [*Journal of Finance*]{} 28(1): 19-33.
Brandt, M.W., Brav, A., Graham, J.R. and Kumar, A. (2010) The idiosyncratic volatility puzzle: Time trend or speculative episodes? [*Review of Financial Studies*]{} 23(2): 863-899.
Briner, B. and Connor, G. (2008) How much structure is best? A comparison of market model, factor model and unstructured equity covariance matrices. [*Journal of Risk*]{} 10(4), 3-30.
Burmeister, E. and Wall, K.D. (1986) The arbitrage pricing theory and macroeconomic factor measures. [*Financial Review*]{} 21(1): 1-20.
Campbell, J. (1987) Stock returns and the term structure. [*Journal of Financial Economics*]{} 18(2): 373-399.
Campbell, J.Y., Lettau, M., Malkiel, B.G. and Xu, Y. (2001) Have individual stocks become more volatile? An empirical exploration of idiosyncratic risk. [*Journal of Finance*]{} 56(1): 1-43.
Campbell, J. and Shiller, R. (1988) The dividend-price ratio and expectations of future dividends and discount factors. [*Review of Financial Studies*]{} 1(3): 195-227.
Carhart, M.M. (1997) Persistence in mutual fund performance. [*Journal of Finance*]{} 52(1): 57-82.
Chamberlain, G. and Rothschild, M. (1983) Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets. [*Econometrica*]{} 51(5): 1281-1304.
Chan, K.C., Chen, N. and Hsieh, D. (1985) An Exploratory Investigation of the Firm Size Effect. [*Journal of Financial Economics*]{} 14(3): 451-471.
Chen, N., Grundy, B. and Stambaugh, R.F. (1990) Changing Risk, Changing Risk Premiums, and Dividend Yield Effects. [*The Journal of Business*]{} 63(1): 51-70.
Chen, N., Roll, R. and Ross, S. (1986) Economic forces and the stock market. [*Journal of Business*]{} 59(3): 383-403.
Chicheportiche, R. and Bouchaud, J.-P. (2014) A nested factor model for non-linear dependencies in stock returns. [*Quantitative Finance*]{} (forthcoming), DOI:10.1080/14697688.2014.994668.
Cochrane, J.H. (2001) [*Asset Pricing.*]{} Princeton, NJ: Princeton University Press.
Connor, G. (1984) A unified beta pricing theory. [*Journal of Economic Theory*]{} 34(1): 13-31.
Connor, G. (1995) The Three Types of Factor Models: A Comparison of Their Explanatory Power. [*Financial Analysts Journal*]{} 51(3): 42-46.
Connor, G. and Korajczyk, R. (1988) Risk and return in an equilibrium APT: Application of a new test methodology. [*Journal of Financial Economics*]{} 21(2): 255-289.
Connor, G. and Korajczyk, R. (1989) An intertemporal beta pricing model. [*Review of Financial Studies*]{} 2(3): 373-392.
Connor, G. and Korajczyk, R. (2010) Factor Models in Portfolio and Asset Pricing Theory. In: Guerard Jr, J.B. (ed.) [*Handbook of Portfolio Construction: Contemporary Applications of Markowitz Techniques*]{}. New York, NY: Springer, pp. 401-418.
Daniel, K. and Titman, S. (1997) Evidence on the Characteristics of Cross Sectional Variation in Stock Returns. [*Journal of Finance*]{} 52(1): 1-33.
DeBondt, W. and Thaler, R. (1985) Does the stock market overreact? [*Journal of Finance*]{} 40(3): 739-805.
Dhrymes, P.J., Friend, I. and Gultekin, N.B. (1984) A Critical Reexamination of the Empirical Evidence on the Arbitrage Pricing Theory. [*The Journal of Finance*]{} 39(2): 323-346.
Ederington, L. and Guan, W. (2002) Is implied volatility an informationally efficient and effective predictor of future volatility? [*The Journal of Risk*]{} 4(3): 29-46.
Fama, E. and French, K. (1992) The cross-section of expected stock returns. [*Journal of Finance*]{} 47(2): 427-465.
Fama, E.F. and French, K.R. (1993) Common risk factors in the returns on stocks and bonds. [*J. Financ. Econ.*]{} 33(1): 3-56.
Fama, E. and French, K. (1996) Multifactor explanations for asset pricing anomalies. [*Journal of Finance*]{} 51(1): 55-94.
Fama, E. and French, K. (2015) A Five-Factor Asset Pricing Model. [*Journal of Financial Economics*]{} (forthcoming), DOI: 10.1016/j.jfineco.2014.10.010.
Fama, E.F. and MacBeth, J.D. (1973) Risk, Return and Equilibrium: Empirical Tests. [*Journal of Political Economy*]{} 81(3): 607-636.
Ferson, W. and Harvey, C. (1991) The variation in economic risk premiums. [*Journal of Political Economy*]{} 99(2): 385-415.
Ferson, W. and Harvey, C. (1999) Conditioning variables and the cross section of stock returns. [*Journal of Finance*]{} 54(4): 1325-1360.
Forni, M., Hallin, M., Lippi, M. and Reichlin, L. (2000) The generalized dynamic factor model: Identification and estimation. [*The Review of Economics and Statistics*]{} 82(4): 540-554.
Forni, M., Hallin, M., Lippi, M. and Reichlin, L. (2005) The generalized dynamic factor model: One-sided estimation and forecasting. [*Journal of the American Statistical Association*]{} 100(471): 830-840.
Forni, M. and Lippi, M. (2001). The generalized dynamic factor model: Representation theory. [*Econometric Theory*]{} 17(6): 1113-1141.
Goyal, A., Perignon, C. and Villa, C. (2008) How common are common return factors across the NYSE and Nasdaq? [*Journal of Financial Economics*]{} 90(3): 252-271.
Goyal, A. and Santa-Clara, P. (2003) Idiosyncratic risk matters! [*Journal of Finance*]{} 58(3): 975-1007.
Grinold, R.C. and Kahn, R.N. (2000) [*Active Portfolio Management.*]{} New York, NY: McGraw-Hill.
Hall, A.D., Hwang, S. and Satchell, S.E. (2002) Using bayesian variable selection methods to choose style factors in global stock return models. [*Journal of Banking and Finance*]{} 26(12): 2301-2325.
Haugen, R.A. (1995) [*The New Finance: The Case Against Efficient Markets.*]{} Upper Saddle River, NJ: Prentice Hall.
Heaton, J. and Lucas, D.J. (1999) Stock Prices and Fundamentals. [*NBER Macroeconomics Annual*]{} 14(1): 213-242.
Heston, S.L. and Rouwenhorst, K.G. (1994) Does Industrial Structure Explain the Benefits of International Diversification? [*Journal of Financial Economics*]{} 36(1): 3-27.
Jagannathan, R. and Wang, Z. (1996) The conditional CAPM and the cross-section of expected returns. [*Journal of Finance*]{} 51(1): 3-53.
Jegadeesh, N. and Titman, S. (1993) Returns to buying winners and selling losers: Implications for stock market efficiency. [*Journal of Finance*]{} 48(1): 65-91.
Jegadeesh, N. and Titman, S. (2001) Profitability of Momentum Strategies: An Evaluation of Alternative Explanations. [*Journal of Finance*]{} 56(2): 699-720.
Kakushadze, Z. (2014) 4-Factor Model for Overnight Returns. [*Wilmott Magazine*]{} (forthcoming); http://ssrn.com/abstract=2511874 (October 19, 2014).
Kakushadze, Z. (2015a) Mean-Reversion and Optimization. [*Journal of Asset Management*]{} 16(1): 14-40.
Kakushadze, Z. (2015b) Combining Alphas via Bounded Regression. SSRN Working Papers Series, http://ssrn.com/abstract=2550335 (January 15, 2015).
Kakushadze, Z. (2015c) Russian-Doll Risk Models. [*Journal of Asset Management*]{} 16(3): 170-185.
Kakushadze, Z. and Liew, J.K.-S. (2015) Custom v. Standardized Risk Models. [*Risks*]{} 3(2): 112-138.
King, B.F. (1966) Market and Industry Factors in Stock Price Behavior. [*Journal of Business*]{} 39(1): 139-190.
Korajczyk, R.A. and Sadka, R. (2008) Pricing the Commonality Across Alternative Measures of Liquidity. [*Journal of Financial Economics*]{} 87(1): 45-72.
Kothari, S. and Shanken, J. (1997) Book-to-market, dividend yield and expected market returns: A time series analysis. [*Journal of Financial Economics*]{} 44(2): 169-203.
Lakonishok, J., Shleifer, A. and Vishny, R.W. (1994) Contrarian Investment, Extrapolation, and Risk. [*The Journal of Finance*]{} 49(5): 1541-1578.
Lee, J.-H. and Stefek, D. (2008) Do Risk Factors Eat Alphas? [*The Journal of Portfolio Management*]{} 34(4): 12-24.
Lehmann, B. and Modest, D. (1988) The empirical foundations of the arbitrage pricing theory. [*Journal of Financial Economics*]{} 21(2): 213-254.
Liew, J. and Vassalou, M. (2000) Can Book-to-Market, Size and Momentum be Risk Factors that Predict Economic Growth? [*Journal of Financial Economics*]{} 57(2): 221-245.
Lintner, J. (1965) The valuation of risky assets and the selection of risky investments in stock portfolios and capital budgets. [*The Review of Economics and Statistics*]{} 47(1): 13-37.
Lo, A.W. (2010) [*Hedge Funds: An Analytic Perspective.*]{} Princeton, NJ: Princeton University Press.
Lo, A.W. and MacKinlay, A.C. (1990) Data-snooping biases in tests of financial asset pricing models. [*Review of Financial Studies*]{} 3(3): 431-468.
MacKinlay, A.C. (1995) Multifactor models do not explain deviations from the CAPM. [*Journal of Financial Economics*]{} 38(1): 3-28.
MacQueen, J. (2003) The structure of multifactor equity risk models. [*Journal of Asset Management*]{} 3(4) 313-322.
Markowitz, H.M. (1952) Portfolio Selection. [*Journal of Finance*]{} 7(1): 77-91.
Markowitz, H.M. (1984) The Two-Beta Trap. [*Journal of Portfolio Management*]{} 11(1): 12-19.
Menchero, J. and Mitra, I. (2008) The Structure of Hybrid Factor Models. [*Journal of Investment Management*]{} 6(3): 35-47.
Merton, R. (1973) An intertemporal capital asset pricing model. [*Econometrica*]{} 41(5): 867-887.
Miller, G. (2006) Needles, Haystacks, and Hidden Factors. [*The Journal of Portfolio Management*]{} 32(2): 25-32.
Motta, G., Hafner, C. and von Sachs, R. (2011) Locally stationary factor models: identification and nonparametric estimation. [*Econometric Theory*]{} 27(6): 1279-1319.
Mukherjee, D. and Mishra, A.K. (2005) Multifactor Capital Asset Pricing Model Under Alternative Distributional Specification. SSRN Working Papers Series, http://ssrn.com/abstract=871398 (December 29, 2005).
Ng, V., Engle, R.F. and Rothschild, M. (1992) A multi-dynamic-factor model for stock returns. [*Journal of Econometrics*]{} 52(1-2): 245-266.
Pastor, L. and Stambaugh, R.F. (2003) Liquidity Risk and Expected Stock Returns. [*The Journal of Political Economy*]{} 111(3): 642-685.
Roll, R. and Ross, S.A. (1980) An Empirical Investigation of the Arbitrage Pricing Theory. [*Journal of Finance*]{} 35(5): 1073-1103.
Rosenberg, B. (1974) Extra-Market Components of Covariance in Security Returns. [*Journal of Financial and Quantitative Analysis*]{} 9(2): 263-274.
Ross, S.A. (1976) The arbitrage theory of capital asset pricing. [*Journal of Economic Theory*]{} 13(3): 341-360.
Ross, S.A. (1978a) A Simple Approach to the Valuation of Risky Streams. [*Journal of Business*]{} 51(3): 453-475.
Ross, S.A. (1978b) Mutual Fund Separation in Financial Theory – The Separating Distributions. [*Journal of Economic Theory*]{} 17(2): 254-286.
Scholes, M. and Williams, J. (1977) Estimating Betas from Nonsynchronous Data. [*Journal of Financial Economics*]{} 5(3): 309-327.
Schwert, G. (1990) Stock returns and real activity: A century of evidence. [*Journal of Finance*]{} 45(4): 1237-1257.
Shanken, J. (1987) Nonsynchronous data and the covariance-factor structure of returns. [*Journal of Finance*]{} 42(2): 221-231.
Shanken, J. (1990) Intertemporal Asset Pricing: An Empirical Investigation. [*Journal of Econometrics*]{} 45(1-2): 99-120.
Shanken, J. and Weinstein, M.I. (2006) Economic Forces and the Stock Market Revisited. [*Journal of Empirical Finance*]{} 13(2): 129-144.
Sharpe, W.F. (1963) A simplified model for portfolio analysis. [*Management Science*]{} 9(2): 277-293.
Sharpe, W. (1964) Capital asset prices: A theory of market equilibrium under conditions of risk. [*Journal of Finance*]{} 19(3): 425-442.
Stock, J.H. and Watson, M.W. (2002a) Macroeconomic forecasting using diffusion indexes. [*Journal of Business and Economic Statistics*]{} 20(2): 147-162.
Stock, J.H. and Watson, M.W. (2002b) Forecasting using principal components from a large number of predictors. [*Journal of the American Statistical Association*]{} 97(460): 1167-1179.
Stroyny, A.L. (2005) Estimating a Combined Linear Factor Model. In: Knight, J. and Satchell, S.E. (eds.) [*Linear Factor Models in Finance.*]{} Oxford: Elevier Butterworth-Heinemann.
Treynor, J.L. (1999) Towards a Theory of Market Value of Risky Assets. In: Korajczyk, R. (ed.) [*Asset Pricing and Portfolio Performance: Models, Strategy, and Performance Metrics.*]{} London: Risk Publications.
Vassalou, M. (2003) News Related to Future GDP Growth as a Risk Factor in Equity Returns. [*Journal of Financial Economics.*]{} 68(1): 47-73.
Whitelaw, R. (1997) Time variations and covariations in the expectation and volatility of stock market returns. [*Journal of Finance*]{} 49(2): 515-541.
Zangari, P. (2003) Equity factor risk models. In: Litterman, B. (ed.) [*Modern Investment Management: An Equilibrium Approach.*]{} New York, NY: John Wiley & Sons, Inc., pp. 334-395.
Zhang, C. (2010) A Re-examination of the Causes of Time-varying Stock Return Volatilities. [*Journal of Financial and Quantitative Analysis*]{} 45(3): 663-684.
------------- ------- -------------- -------- ------- -------------- ------- --------
$K$ Min 1st Quartile Median Mean 3rd Quartile Max $g(K)$
\[0.5ex\] 1 0.16 0.62 0.824 0.771 0.953 1 0.4
2 0.137 0.525 0.693 0.682 0.867 1 0.37
3 0.084 0.453 0.629 0.618 0.8 0.999 0.29
4 0.075 0.405 0.562 0.56 0.718 0.992 0.27
5 0.06 0.355 0.501 0.51 0.668 0.981 0.235
6 0.06 0.312 0.449 0.462 0.606 0.977 0.233
7 0.057 0.272 0.396 0.417 0.552 0.931 0.203
8 0.033 0.233 0.347 0.375 0.503 0.916 0.139
9 0.029 0.203 0.306 0.334 0.446 0.884 0.111
10 0.019 0.172 0.264 0.294 0.39 0.84 0.056
11 0.01 0.144 0.227 0.256 0.339 0.84 0.018
12 0.009 0.118 0.194 0.22 0.294 0.84 0.013
13 0.008 0.093 0.157 0.186 0.25 0.728 0.056
14 0.002 0.07 0.122 0.152 0.204 0.696 0.12
15 0.002 0.05 0.089 0.119 0.162 0.686 0.128
16 0 0.029 0.062 0.088 0.122 0.608 0.211
17 0 0.014 0.035 0.057 0.077 0.606 0.221
18 0 0.003 0.011 0.028 0.034 0.592 0.231
\[1ex\]
------------- ------- -------------- -------- ------- -------------- ------- --------
: First column: the number of principal components $K$; last column: $g(K)$ defined in (\[g\]); Min, 1st Quartile, Median, Mean, 3rd Quartile and Max refer to the corresponding quantities for the ratio ${\widetilde \xi}_i^2 = \xi^2_i/C_{ii}$ (specific variance over total variance). The number of observations (days) in the time series is $M+1 = 20$. The number of (randomly selected) stocks is $N=2316$. All quantities are rounded to 3 digits. The value of $K$ fixed via (\[K\]) is $K=12$. See Figure 1 for a density plot.
\[table.prin.comp\]
-------------------------------------------- -------- ------- ------
Alpha ROC SR CPS
\[0.5ex\] Regression: Principal Components 46.80% 11.50 2.05
Optimization: Principal Components 47.74% 11.88 2.26
Regression: BICS Sub-Industries 49.36% 12.89 2.16
Regression: Heterotic Risk Factors 51.89% 13.63 2.27
Optimization: Heterotic Risk Model 55.90% 15.41 2.67
\[1ex\]
-------------------------------------------- -------- ------- ------
: Simulation results for the weighted regression alphas discussed in Section \[sub.reg\] and the optimized alphas discussed in Section \[sub.opt\], without any bounds on the dollar holdings. All quantities are rounded to 2 digits. See Figure 2 for P&L plots.
\[table2\]
-------------------------------------------- -------- ------- ------
Alpha ROC SR CPS
\[0.5ex\] Regression: Principal Components 41.27% 14.24 1.84
Optimization: Principal Components 40.92% 14.33 1.96
Regression: BICS Sub-Industries 44.56% 16.51 1.97
Regression: Heterotic Risk Factors 46.86% 18.30 2.08
Optimization: Heterotic Risk Model 49.00% 19.23 2.36
\[1ex\]
-------------------------------------------- -------- ------- ------
: Simulation results for the weighted regression alphas discussed in Section \[sub.reg\] and the optimized alphas discussed in Section \[sub.opt\], with the liquidity bounds (\[liq\]) on the dollar holdings. All quantities are rounded to 2 digits. See Figure 3 for P&L plots.
\[table3\]
4.truein 4.truein
4.truein 4.truein
4.truein 4.truein
[^1]: Zura Kakushadze, Ph.D., is the President of Quantigic$^\circledR$ Solutions LLC, and a Full Professor at Free University of Tbilisi. Email: zura@quantigic.com
[^2]: DISCLAIMER: This address is used by the corresponding author for no purpose other than to indicate his professional affiliation as is customary in publications. In particular, the contents of this paper are not intended as an investment, legal, tax or any other such advice, and in no way represent views of Quantigic$^\circledR$ Solutions LLC, the website or any of their other affiliates. The title of this paper is inspired by the heterotic string theory, with no substantive connection therewith.
[^3]: For a partial list of literature related to factor risk models, see, [*e.g.*]{}, [@Q1], [@Q2], [@Q3], [@Q4], [@Q5], [@Q6], [@Q7], [@Q8], [@Q9], [@Q10], [@Q11], [@Q12], [@Q13], [@Q14], [@Q15], [@Q16], [@Q17], [@Q18], [@Q19], [@Q20], [@Q21], [@Q22], [@Q23], [@Q24], [@Q26; @Q25], [@Q27], [@Q28], [@Q29; @Q30], [@Q31; @Q32; @Q33], [@Q34], [@Q35], [@Q36], [@Q37; @Q38; @Q39; @Q40], [@Q41], [@Q42; @Q43], [@Q44; @Q45], [@Q46], [@Q47], [@Q48], [@Q49], [@Q50], [@Q51], [@Q52], [@Q53], [@Q54], [@Q55; @Q56], [@4F; @MeanRev; @RusDoll], [@CustomRM], [@Q62], [@Q63], [@Q64], [@Q65], [@Q66], [@Q67], [@Q68], [@Q69], [@Q70], [@Q71], [@Q72], [@Q73], [@Q74; @Q75], [@Q76], [@Q77], [@Q78], [@Q79], [@Q80], [@Q81], [@Q82], [@Q83], [@Q84], [@Q85; @Q86; @Q87], [@Q88], [@Q89], [@Q90; @Q91], [@Q92], [@Q93; @Q94], [@Q95; @Q96], [@Q97], [@Q98], [@Q99], [@Q100], [@Q101], [@Q102], and references therein.
[^4]: By this we mean the stock clusters at the most granular level in the IC hierarchy. [*E.g.*]{}, in BICS these would be sub-industries, whereas other ICs have different naming conventions.
[^5]: Generally, off-diagonal elements of a sample (stock or factor) covariance matrix tend to be unstable out-of-sample, whereas its diagonal elements (variances) typically are much more stable.
[^6]: The number of non-binary style factors is at most of order 10 and does not pose a difficultly for computing the factor covariance matrix. It is the ubiquitous industry factors that are problematic.
[^7]: And not SCM – this is an important technical detail, see the discussion in Section \[cor.not.cov\].
[^8]: Note that this is not the same as a“hybrid" (mixture) of industry and statistical risk factors.
[^9]: The source code given in the appendices is not written to be “fancy" or optimized for speed or in any other way. Its sole purpose is to illustrate the algorithms described in the main text in a simple-to-understand fashion. Some legalese relating to this code is given in Appendix \[app.D\].
[^10]: With additional assumptions not all of these conditions are nontrivial (see below).
[^11]: There are no “natural" $K(K+1)/2$ conditions we can impose on $\Gamma_{ij}$, $i\neq j$ in terms of out-of-sample unstable $C_{ij}$, $i\neq j$. Note that the variances $C_{ii}$ typically are much more stable.
[^12]: Without the intercept, that is, unless the intercept is already subsumed in $\Omega_{iA}$.
[^13]: Cross-sectionally, stock volatility typically has a roughly log-normal distribution.
[^14]: If $R_i(t_s)$ ($t_s$ labels the observations in the time series and in the above notations the index $s$ takes $M+1$ values) are the time series of the stock returns based on which the sample covariance matrix $C_{ij}$ is computed (so $C_{ii} = \mbox{Var}(R_i(t_s))$, where the variance is serial), then $\Psi_{ij}$ is the sample covariance matrix for the “normalized" returns ${\widetilde R}_i(t_s) \equiv R_i(t_s)/\sqrt{C_{ii}}$, [*i.e.*]{}, $\Psi_{ij} = \mbox{Cov}({\widetilde R}_i(t_s), {\widetilde R}_j(t_s)) = \mbox{Cor}(R_i(t_s), R_j(t_s))$, where $\mbox{Cov}(\cdot, \cdot)$ and $\mbox{Cor}(\cdot, \cdot)$ are serial.
[^15]: This number can be smaller if some stock returns are 100% correlated or anti-correlated. For the sake of simplicity – and this not critical here – we will assume that there are no such returns.
[^16]: Note that ${\widetilde \xi}_i^2 = \sum_{a=K+1}^M \lambda^{(a)}\left(V_i^{(a)}\right)^2 \geq 0$. We are assuming $\lambda^{(a)}\geq 0$, which (up to computational precision) is the case if there are no N/As in the stock return times series.
[^17]: The distribution of ${\widetilde \xi}^2_i$ is skewed; typically, ${\widetilde \xi}^2_i$ has a tail at higher values, while $\ln({\widetilde \xi}^2_i)$ has a tail at lower values, and the distribution is only roughly log-normal. So $K$ is not (the floor/cap of) $M/2$, but somewhat higher, albeit close to it. See Table \[table.prin.comp\] and Figure 1 for an illustrative example.
[^18]: The number of style factors is also limited (especially for short horizons), of order 10 or fewer.
[^19]: Albeit in the BICS context we may be referring to, [*e.g.*]{}, sub-industries, while in other classification schemes the actual naming may be altogether different.
[^20]: If $N(A)=1$, [*i.e.*]{}, we have only one ticker in the sub-industry labeled by $A$, then $[U(A)]_i =1$ and $\lambda(A) = \Psi_{ii} = 1$, $i\in J(A)$.
[^21]: For single-ticker sub-industries ($N(A) = 1$) the specific risk vanishes: ${\widetilde\xi}^2_i=0$; however, this does not pose a problem as this does not cause the matrix ${\widetilde\Gamma}_{ij}$ to be singular (see below).
[^22]: Note that $\Phi_{AA} = \lambda(A)$.
[^23]: We use a prime on $\Gamma^\prime_{AB}$, ${\widetilde \xi}^{\prime}_A$, $\Phi_{ab}^\prime$, [*etc.*]{} to avoid confusion with $\Gamma_{ij}$, ${\widetilde \xi}_i$, $\Phi_{AB}$, [*etc.*]{}
[^24]: That is, assuming there are enough observations in the time series for out-of-sample stability.
[^25]: This last factor can be interpreted as the “market" risk factor. For the sake of completeness, the definitions of the factors at each stage are as follows: (i) for the sub-industries $f_A = \sum_{i\in J(A)} U_i~{\widetilde R}_i$, where ${\widetilde R}_i = R_i/\sqrt{C_{ii}}$; (ii) for the industries $f^\prime_a = \sum_{A\in J^\prime(a)} U^\prime_A~{\widetilde f}_A$, where ${\widetilde f}_A = f_A/\sqrt{\Phi_{AA}}$; (iii) for the sectors $f^{\prime\prime}_\alpha = \sum_{a\in J^{\prime\prime}(\alpha)} U^{\prime\prime}_a~{\widetilde f}^\prime_a$, where ${\widetilde f}^\prime_a = f^\prime_a/\sqrt{\Phi^\prime_{aa}}$; and (iv) for the “market" $f^{\prime\prime\prime} = \sum_{\alpha=1}^L U^{\prime\prime\prime}_\alpha~{\widetilde f}^{\prime\prime}_\alpha$, where ${\widetilde f}^{\prime\prime}_\alpha = f^{\prime\prime}_\alpha/\sqrt{\Phi^{\prime\prime}_{\alpha\alpha}}$.
[^26]: For a typical, large trading universe, industries and sectors usually contain more than one ticker; however, there can be cases of single-ticker sub-industries. Nonetheless, ${\widetilde\Gamma}_{ij}$ is nonsingular. Indeed, for an arbitrary $N$-vector $X_i$ we have $X^T~{\widetilde \Gamma}~X > 0$ unless $X_i \equiv 0$, $i\not\in H$, where $H\equiv\{i|N(G(i))=1\}$. For such $X_i$ we have $X^T~{\widetilde \Gamma}~X = \sum_{A,B\in E} Y_A~\Gamma^\prime_{AB}~Y_B > 0$, where $E\equiv\{A|N(A)=1\}$, $Y_A \equiv\delta_{A, G(i)} X_i$, $A\in E$, and we have taken into account that by construction $\Gamma^\prime_{AB}$ (and its sub-matrix with $A,B\in E$) is positive-definite, and also that $U_i = 1$, $i\in H$. More on this below.
[^27]: For comparative purposes, we will also run separate backtests where we require neutrality w.r.t. BICS sub-industries and principal components.
[^28]: The remainder of this section somewhat overlaps with Section 7 of (Kakushadze, 2015a) as the backtesting models are similar, albeit not identical.
[^29]: This is a so-called “delay-0" alpha: the same price, $P^O_{is}$ (or adjusted $P^{AO}_{is}$), is used in computing the expected return (via $E_{is}$) and as the establishing fill price.
[^30]: In practical applications, the trading universe of liquid stocks typically is selected based on market cap, liquidity (ADDV), price and other (proprietary) criteria.
[^31]: The choice of the backtesting window is based on what data was readily available.
[^32]: Here we are after the [*relative outperformance*]{}, and it is reasonable to assume that, to the leading order, individual performances are affected by the survivorship bias approximately equally.
[^33]: As mentioned above, we assume no transaction costs, which are expected to reduce the ROC of the optimization and weighted regression alphas by the same amount as the two strategies trade the exact same amount by design. Therefore, including the transaction costs would have no effect on the actual [*relative outperformance*]{} in the horse race, which is what we are after here.
[^34]: The loadings $\Lambda_{iA}$ in (\[load.pc\]) and (\[load.het\]) are computed for each trading date $s$ (as opposed to, say, every 21 days – see below); in (\[load.sub\]) they change only with the universe (every 21 days).\[fn.load\]
[^35]: The factor $\sqrt{\lambda^{(A)}}$ in (\[load.pc\]) does not affect the regression residuals below.
[^36]: In (\[load.sub\]) we deliberately take $\Lambda_{iA} = \delta_{G(i),A}$ as opposed to $\Lambda_{iA} = \sqrt{C_{ii}}~\delta_{G(i),A}$ (see below). Note that with (\[load.sub\]) the intercept is subsumed in $\Lambda_{iA}$ as we have $\sum_{A=1}^K \Lambda_{iA} = 1$, so (\[DN\]) is automatic.
[^37]: Due to ${\widetilde E}_{is}$ having 0 cross-sectional means, which in turn is due to the intercept either being included (the cases i) and iii)), or being subsumed in the loadings matrix $Y$ (the case ii)).
[^38]: In Table \[table2\] the heterotic risk factors outperform the sub-industries. However, this is largely an artifact of defining $\Lambda_{iA}$ as in (\[load.sub\]). If we take $\Lambda_{iA} = \sqrt{C_{ii}}~\delta_{G(i),A}$ instead (and augment the regression loadings matrix $Y$ with the intercept for dollar neutrality), we will get (without the bounds (\[liq\]) – the results with the bounds are similar): ROC = 51.62%, SR = 13.45, CPS = 2.26.
[^39]: In Tables \[table2\] and \[table3\] at the final stage the heterotic risk factors are the (10) BICS sectors: there are enough (20) observations in the time series. The 1-factor model gives almost the same results.
[^40]: The source code for this algorithm is not included in (Kakushadze, 2015a), so we include it in Appendix \[app.C\]. It is similar to the source code of (Kakushadze, 2015b) for the bounded regression.
[^41]: This matrix is given by $\Gamma_{ij} =\sqrt{C_{ii}}\sqrt{C_{jj}}~{\widetilde\Gamma}_{ij}$, where ${\widetilde\Gamma}_{ij}$ is defined in (\[PC\]), and $K$ is determined via the algorithm of Section \[sub.fix.K\]. For the $d=21$ trading day lookback in our backtests, the value of $K$ fixed by this algorithm turns out to be $K=13$.
[^42]: There exist further (proprietary) performance improvements using the heterotic risk model.
[^43]: In this context, the paper [@ImpliedVol] sometimes is referred to.
[^44]: The analog of the line below, reads in (Kakushadze, 2015b), where the analog of is diagonal and both lines give the same result; however, for non-diagonal here they do not.
|
---
abstract: 'In wireless sensor networks, where energy is scarce, it is inefficient to have all nodes active because they consume a non-negligible amount of battery. In this paper we consider the problem of jointly selecting sensors, relays and links in a wireless sensor network where the active sensors need to communicate their measurements to one or multiple access points. Information messages are routed stochastically in order to capture the inherent reliability of the broadcast links via multiple hops, where the nodes may be acting as sensors or as relays. We aim at finding optimal sparse solutions where both, the consistency between the selected subset of sensors, relays and links, and the graph connectivity in the selected subnetwork are guaranteed. Furthermore, active nodes should ensure a network performance in a parameter estimation scenario. Two problems are studied: sensor and link selection; and sensor, relay and link selection. To solve such problems, we present tractable optimization formulations and propose two algorithms that satisfy the previous network requirements. We also explore an extension scenario: only link selection. Simulation results show the performance of the algorithms and illustrate how they provide a sparse solution, which not only saves energy but also guarantees the network requirements.'
author:
- 'Rocío Arroyo-Valles[^1], Andrea Simonetto[^2], and Geert Leus[^3]'
bibliography:
- '../../TeX/PaperCollection2.bib'
- 'bib.bib'
title: |
Consistent Sensor, Relay, and Link Selection\
in Wireless Sensor Networks[^4]
---
[^1]: R. Arroyo-Valles was with the Dept. of Teoría de la Señal y Comunicaciones, Universidad Carlos III de Madrid, Avda. de la Universidad 30, Leganés, 28911,Madrid, Spain. She is now with the European Patent Office, 2280 HV Rijswijk, The Netherlands. E-mail: marrval@tsc.uc3m.es.
[^2]: Andrea Simonetto was with the applied mathematics department, Université catholique de Louvain, Louvain-la-Neuve, Belgium. He is now with the optimization and control group of IBM Research Ireland. E-mail: andrea.simonetto@ibm.com
[^3]: Geert Leus is with the Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, 2628CD Delft, The Netherlands. E-mail: g.j.t.leus@tudelft.nl
[^4]: The work in this paper was supported by Spanish Government grant TEC2014-52289.
|
---
author:
- |
Hai-Quan Li$^{1}$, Naihuan Jing$^{1,2}$$^\ast$ & Xi-Lin Tang$^{1}$\
\
\
---
[**Abstract**]{} Recently, the method that using an entanglement as a resource to distinguish orthogonal product states by local operations and classical communication (LOCC) has brought into focus. Zhang et al. presented protocols which use an entanglement to distinguish some classes of orthogonal product states in $\mathbb{C}^m\otimes \mathbb{C}^n$[@Zhang016]. In this paper, we mainly study the local distinguishability of multipartite product states. For the class of locally indistinguishable multipartite product states constructed by Wang et al. in [@Wang17], we present a protocol that distinguishes perfectly these quantum states by LOCC using an entangled state as a resource for implementing quantum measurements.
[**Keywords**]{} Entangled states $\cdot$ Multipartite product states $\cdot$ Distinguishability
Introduction
============
In quantum information theory, the relationship between quantum nonlocality and quantum entanglement has received considerable attention in the last several decades due to their deep connections [@Ghosh01; @Ghosh04; @Hayashi06]. However, Bennett et al. found nine states without entanglement in $\mathbb{C}^3\otimes\mathbb{C}^3$, which cannot be distinguished perfectly by local operations and classical communication (LOCC) [@Ben99]. This latter interesting phenomenon is called “nonlocality without entanglement”, i.e., the local indistinguishability of mutually orthogonal product states by LOCC, and it has since attracted much attention [@Walgate02; @Horodecki03; @De04; @Nathanson05; @Niset06; @Feng09; @Duan10; @Childs13; @Zhang14; @Wang15; @Zhang15; @Zhang16; @Xu15; @Xu16; @Wang17; @ZhangX17; @Zhang17]. These developments provided a better understanding of nonlocality without entanglement.
In $2008$, using an entanglement, Cohen perfectly distinguished certain classes of unextendible product bases (UPB) by LOCC in $\mathbb{C}^m\otimes\mathbb{C}^n$ [@Cohen07]. His method using entanglement as a resource to distinguish the quantum states showed that entanglement is a valuable resource for quantum information. This is parallel to well-known theoretical applications of entanglement such as in quantum information processing, quantum cryptography[@Ekert91; @Gisin02], quantum teleportation[@Karlsson98; @Kim01], and quantum secure direct communication [@Wang05; @Tian08; @Yang11]. It is thus interesting to ask what entanglement resources are necessary and sufficient for distinguishing indistinguishable quantum states with LOCC.
In $2016$, Zhang et al. have shown that a $\mathbb{C}^2\otimes\mathbb{C}^2$ maximally entangled states is sufficient to perfectly distinguish certain classes by LOCC in $\mathbb{C}^m\otimes\mathbb{C}^n$ [@Zhang016] and have raised the question for a multipartite system. We believe that any class of locally indistinguishable orthogonal product states can be perfectly distinguished by LOCC with enough entanglements as resources. On the other hand, Bandyopadhyay et al. have proved that there is no entangled state as a universal resource for local state discrimination in multipartite systems [@Bandyopadhyay16]. This result says that distinguishability of multipartite orthogonal product states has to be dealt individually according to the system. The phenomenon of “nonlocality without entanglement” in multipartite quantum systems has also been studied in [@Niset06; @Xu16; @Wang17; @Zhang17]. Niset et al. constructed a set of locally indistinguishable multipartite orthogonal product bases in $\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2}\otimes\cdots\otimes\mathbb{C}^{d_{n-1}}\otimes\mathbb{C}^{d_{n}} (d_{i}\geq n-1)$[@Niset06]. Xu et al. constructed two different classes of locally indistinguishable multipartite orthogonal product states in any multipartite quantum system [@Xu16]. In addition, Wang et al. constructed a set of LOCC indistinguishable multipartite orthogonal product states by using a set of locally indistinguishable bipartite orthogonal product states [@Wang17]. Recently, Zhang et al.[@Zhang17] gave a general construction of locally indistinguishable multipartite orthogonal product states. It is interesting to see how to use entanglement as a resource to distinguish multipartite orthogonal product states by LOCC. In this paper, we consider the set of LOCC indistinguishable multipartite product states constructed in [@Wang17] and show that they can be indeed distinguished by LOCC with entanglement as a resource. It is shown by separating the problem into the cases of even and odd partite states, and each case is solved by considering the lower rank cases. For the even case, the local distinguishability of a class of orthogonal product states is considered in the bipartite system. For the odd case, this is dealt with by first establishing the result in tripartite systems.
**Local distinguishability of multipartite product states**
===========================================================
Let $\{|i\rangle\}_{i=1}^{d}$ be the standard orthonormal basis in $\mathbb{C}^d$. A pure state $|\psi\rangle$ of $\mathbb{C}^{d}\otimes\mathbb{C}^{d'}(d' > d)$ is said to be maximally entangled if for any orthogonal basis ${|i_{A}\rangle}$ of the subsystem $A$, there exists an orthogonal basis ${|i_{B}\rangle}$ of the subsystem $B$ such that $|\psi\rangle$ can be written as $|\psi\rangle=\frac{1}{\sqrt{d}}\sum\limits_{i=1}^{d}|i_{A}i_{B}\rangle$[@Li12].
In the following, we shall study local distinguishability of multipartite product states in even-partite and odd-partite cases respectively.
Local distinguishability for even-partite case
----------------------------------------------
In this subsection, we first consider local distinguishability of orthogonal product states in a bipartite system. We claim that any class of locally indistinguishable orthogonal product states can be perfectly distinguished by LOCC with enough entanglements as resources.
In $\mathbb{C}^m\otimes\mathbb{C}^n(4\leq m\leq n)$, the following $2n-1$ orthogonal product states are LOCC indistinguishable[@Wang17]. $$\begin{split}
&|\alpha\pm \beta\rangle=\frac{1}{\sqrt{2}}(|\alpha\rangle\pm|\beta\rangle),0\leq\alpha\leq\beta,\\
&|\phi_1\rangle=(|1\rangle_A+|2\rangle_A+\cdots+|m\rangle_A)(|1\rangle_B+|2\rangle_B+\cdots+|n\rangle_B), \\
&|\phi_i\rangle=|i\rangle_A|1-i\rangle_B, \ \ i=2, 3, \ldots, m, \\
&|\phi_{m+1}\rangle=|1-m\rangle_A|2\rangle_B, \\
&|\phi_{m+j-1}\rangle=|1-(j-1)\rangle_A|j\rangle_B, \ \ j=3, 4, \ldots, m, \\
&|\phi_{m+l-1}\rangle=|1-2\rangle_A|l\rangle_B, \ \ l=m+1, m+2, \ldots, n, \\
&|\phi_{m+n}\rangle=|m\rangle_A|3-(m+1)\rangle_B, \\
&|\phi_{n+s}\rangle=|m-1\rangle_A|s-(s+1)\rangle_B, \\
&|\phi_{n+t}\rangle=|m\rangle_A|t-(t+1)\rangle_B, \\
&s=m+2k-1, \, t=m+2k, \, k=1, 2, \ldots, \lfloor\frac{n-m}{2}\rfloor.
\end{split}$$
Using the method presented by Cohen[@Cohen07], we first need to add two ancillary systems $a$ and $b$ by sharing an entangle state $|\psi\rangle_{ab}$. Now the system $a$ and the system $A$ are both held by Alice, the system $b$ and the system $B$ are both held by Bob, i.e., Alice and Bob control $aA$ system and $bB$ system respectively. Then, Bob measures $\{\emph{B}_i\}$ and performs one of the projectors on $bB$. The result is denoted by an operation $\emph{B}_1(|\omega_i\rangle_{AB}\otimes|\psi\rangle_{ab})$, where $|\omega_i\rangle_{AB}$ is the considered state. Finally, Alice and Bob can proceed from here to distinguish the states using only LOCC.
[**Theorem 1**]{} In $\mathbb{C}^m\otimes\mathbb{C}^n(4\leq m\leq n)$, a $\mathbb{C}^n\otimes\mathbb{C}^n$ maximally entangled state is sufficient to perfectly distinguish the $2n-1$ orthogonal product states $(1)$ by LOCC.
[ *Proof:*]{} Let $|\psi\rangle_{ab}$ be a $\mathbb{C}^n\otimes\mathbb{C}^n$ maximally entangled state, $$|\psi\rangle_{ab}=\frac{1}{\sqrt{n}}\sum_{i=1}^n|ii\rangle_{ab}.$$ Alice and Bob share $|\psi\rangle_{ab}$, and then Bob performs the projector $\emph{B}_1=\sum_{i=1}^n|ii\rangle_{bB}\langle ii|$ on $bB$.
From $|\varphi_i\rangle=\emph{B}_1(|\phi_i\rangle_{AB}\otimes|\psi\rangle_{ab})$, we have
$$\begin{split}
&|\varphi_1\rangle=(|1\rangle_A+|2\rangle_A+\cdots+|m\rangle_A)(|1\rangle_B|11\rangle_{ab}+|2\rangle_B|22\rangle_{ab}+\cdots+|n\rangle_B|nn\rangle_{ab}), \\
&|\varphi_i\rangle=|i\rangle_A(|1\rangle_B|11\rangle_{ab}-|i\rangle_B|ii\rangle_{ab}), \ \ i=2, 3, \ldots, m, \\
&|\varphi_{m+1}\rangle=|1-m\rangle_A|2\rangle_B|22\rangle_{ab}, \\
&|\varphi_{m+j-1}\rangle=|1-(j-1)\rangle_A|j\rangle_B|jj\rangle_{ab}, \ \ j=3, 4, \ldots, m, \\
&|\varphi_{m+l-1}\rangle=|1-2\rangle_A|l\rangle_B|ll\rangle_{ab}, \ \ l=m+1, m+2, \ldots, n, \\
&|\varphi_{m+n}\rangle=|m\rangle_A(|3\rangle_B|33\rangle_{ab}-|m+1\rangle_B|(m+1)(m+1)\rangle_{ab}), \\
&|\varphi_{n+s}\rangle=|m-1\rangle_A(|s\rangle_B|ss\rangle_{ab}-|s+1\rangle_B|(s+1)(s+1)\rangle_{ab}), \\
&|\varphi_{n+t}\rangle=|m\rangle_A(|t\rangle_B|tt\rangle_{ab}-|t+1\rangle_B|(t+1)(t+1)\rangle_{ab}), \\
&s=m+2k-1, \, t=m+2k, \, k=1, 2, \ldots, \lfloor\frac{n-m}{2}\rfloor.
\end{split}$$
To distinguish these states using LOCC, Alice makes a projective measurement with $2n-2$ outcomes. Considering the outcome $\emph{A}_{i-1}=|1\rangle_a\langle1|\otimes|i\rangle_A\langle i|+|i\rangle_a\langle i|\otimes|i\rangle_A\langle i|$, it leaves $|\varphi_{i}\rangle$ invariant and transforms $|\varphi_{1_{i}}\rangle$ to $|i\rangle_A(|1\rangle_{B}|11\rangle_{ab}+|i\rangle_{B}|ii\rangle_{ab})$ for $i=2, 3, \ldots, m$. Since $|\varphi_{i}\rangle$ and $|\varphi_{1_{i}}\rangle$ are two mutually orthogonal states, Bob can easily distinguish the two states using the projector $\emph{B}_{(i-1)1, (i-1)2}=|1\rangle_b\langle1|\otimes|1\rangle_B\langle1|\pm|i\rangle_b\langle i|\otimes|i\rangle_B\langle i|$ for $i=2, 3, \ldots, m$.
Take the outcome $\emph{A}_m=|2\rangle_a\langle2|\otimes|1-m\rangle_A\langle1-m|$, the only remaining possibility is $|\varphi_{m+1}\rangle$, which has thus been successfully identified. In the same way, Alice can identify $|\varphi_{m+j-1}\rangle$ and $|\varphi_{m+l-1}\rangle$ by projector $\emph{A}_{m+j-2}=|j\rangle_a\langle j|\otimes|1-(j-1)\rangle_A\langle1-(j-1)|$ and $\emph{A}_{m+l-2}=|l\rangle_a\langle l|\otimes|1-2\rangle_A\langle1-2|$ ($j=3, 4, \ldots, m$, $l=m+1, m+2,\ldots, n$) respectively.
For the outcome $\emph{A}_{m+n-1}=|3\rangle_a\langle3|\otimes|m\rangle_A\langle m|+|(m+1)\rangle_a\langle(m+1)|\otimes|m\rangle_A\langle m|$, it leaves $|\varphi_{m+n}\rangle$ invariant and transforms $|\varphi_{1_{m+n}}\rangle$ to $|m\rangle_A(|3\rangle_{B}|33\rangle_{ab}+|m+1\rangle_{B}|(m+1)(m+1)\rangle_{ab})$. Bob can easily identify the two states, since any two orthogonal states can be distinguished. In the same way, $|\varphi_{n+s}\rangle$ and $|\varphi_{m+t}\rangle$ can be distinguished for $s=m+2k-1$ and $t=m+2k,$ where $k=1, 2, \ldots, \lfloor\frac{n-m}{2}\rfloor$.
Therefore, we have succeeded in distinguishing the states (1) by LOCC using a $\mathbb{C}^n\otimes\mathbb{C}^n$ maximally entangled state as a resource.
[**Example 1** ]{} In $\mathbb{C}^4\otimes\mathbb{C}^5$, a $\mathbb{C}^5\otimes\mathbb{C}^5$ maximally entangled state is sufficient to perfectly distinguish the following $9$ LOCC indistinguishable states by LOCC.
$$\begin{split}
&|\alpha\pm \beta\rangle=\frac{1}{\sqrt{2}}(|\alpha\rangle\pm|\beta\rangle),0\leq\alpha\leq\beta,\\
&|\phi_1\rangle=(|1\rangle_A+|2\rangle_A+\cdots+|4\rangle_A)(|1\rangle_B+|2\rangle_B+\cdots+|5\rangle_B), \\
&|\phi_2\rangle=|2\rangle_A|1-2\rangle_B,\ \ \ \ \ \ \ \ |\phi_3\rangle=|3\rangle_A|1-3\rangle_B, \\
&|\phi_4\rangle=|4\rangle_A|1-4\rangle_B, \ \ \ \ \ \ \ \ |\phi_5\rangle=|1-4\rangle_A|2\rangle_B, \\
&|\phi_6\rangle=|1-2\rangle_A|3\rangle_B, \ \ \ \ \ \ \ \ |\phi_7\rangle=|1-3\rangle_A|4\rangle_B, \\
&|\phi_8\rangle=|1-2\rangle_A|5\rangle_B, \ \ \ \ \ \ \ \ |\phi_9\rangle=|4\rangle_A|3-5\rangle_B. \\
\end{split}$$
These states can be described more specifically by a box-diagram (Fig. 1), where all states are shown except $|\phi_1\rangle$, as the latter would have covered the whole picture. In fact, it follows from the proof of Theorem 1 that one uses the states $$|\psi\rangle_{ab}=\frac{1}{\sqrt{5}}(|11\rangle_{ab}+|22\rangle_{ab}+|33\rangle_{ab}+|44\rangle_{ab}+|55\rangle_{ab}),$$ $$\emph{B}_1=|11\rangle_{bB}\langle11|+|22\rangle_{bB}\langle22|+|33\rangle_{bB}\langle33|+|44\rangle_{bB}\langle44|+|55\rangle_{bB}\langle55|$$ to perform $\emph{B}_1(|\phi_i\rangle_{AB}\otimes|\psi\rangle_{ab})$ and the results are the following states $$\begin{split}
&|\varphi_1\rangle=(|1\rangle_A+|2\rangle_A+|3\rangle_A+|4\rangle_A)(|1\rangle_B|11\rangle_{ab}+|2\rangle_B|22\rangle_{ab}+|3\rangle_B|33\rangle_{ab}+|4\rangle_B|44\rangle_{ab} \\
&~~~~~~~~~+|5\rangle_B|55\rangle_{ab}),\\
&|\varphi_2\rangle=|2\rangle_A(|1\rangle_B|11\rangle_{ab}-|2\rangle_B|22\rangle_{ab}), \\
&|\varphi_3\rangle=|3\rangle_A(|1\rangle_B|11\rangle_{ab}-|3\rangle_B|33\rangle_{ab}), \\
&|\varphi_4\rangle=|4\rangle_A(|1\rangle_B|11\rangle_{ab}-|4\rangle_B|44\rangle_{ab}), \\
&|\varphi_5\rangle=|1-4\rangle_A|2\rangle_B|22\rangle_{ab}, \\
&|\varphi_6\rangle=|1-2\rangle_A|3\rangle_B|33\rangle_{ab}, \\
&|\varphi_7\rangle=|1-3\rangle_A|4\rangle_B|44\rangle_{ab}, \\
&|\varphi_8\rangle=|1-2\rangle_A|5\rangle_B|55\rangle_{ab}, \\
&|\varphi_9\rangle=|4\rangle_A(|3\rangle_B|33\rangle_{ab}-|5\rangle_B|55\rangle_{ab}),
\end{split}$$
These states are depicted on Fig. 2, where the stopper state $|\varphi_1\rangle$ is not shown due to the same reason as above.
As an example, the state $|\phi_2\rangle$ is obtained as follows. $$\begin{split}
|\varphi_2\rangle&=\emph{B}(|\phi_2\rangle_{AB}\otimes|\psi\rangle_{ab})\\
&=(|11\rangle_{bB}\langle11|+\cdots+|55\rangle_{bB}\langle55|)(|2\rangle_A|1-2\rangle_B
(|11\rangle_{ab}+|22\rangle_{ab}+\cdots+|55\rangle_{ab}))\\
&=(|11\rangle_{bB}\langle11|+\cdots+|55\rangle_{bB}\langle55|)(|2\rangle_A(|111\rangle_{abB}+|221\rangle_{abB}+\cdots+|551\rangle_{abB})\\
&~~~ -|2\rangle_A(|112\rangle_{abB}+|222\rangle_{abB}+\cdots+|552\rangle_{abB}))\\
&=|2\rangle_A(|1\rangle_B|11\rangle_{ab}-|2\rangle_B|22\rangle_{ab})
\end{split}$$
Next we can distinguish the states easily as follows. Alice makes an eight-outcome projective measurement, and one begins by considering the first outcome, $\emph{A}_1=|1\rangle_a\langle1|\otimes|2\rangle_A\langle2|+|2\rangle_a\langle2|\otimes|2\rangle_A\langle2|$. This leaves the state $|\varphi_{2}\rangle$ invariant and transforms $|\varphi_{1}\rangle$ to $|2\rangle_A(|1\rangle_B|11\rangle_{ab}+|2\rangle_B|22\rangle_{ab})$. Then Bob uses the projectors $\emph{B}_{11, 12}=|1\rangle_b\langle1|\otimes|1\rangle_B\langle1|\pm|2\rangle_b\langle2|\otimes|2\rangle_B\langle2|$, which can be easily identified to be $|\varphi_{1}\rangle$ and $|\varphi_{2}\rangle$ respectively. In the same way, for outcomes $\emph{A}_2=|1\rangle_a\langle1|\otimes|3\rangle_A\langle3|+|3\rangle_a\langle3|\otimes|3\rangle_A\langle3|$ and $\emph{A}_3=|1\rangle_a\langle1|\otimes|4\rangle_A\langle4|+|4\rangle_a\langle4|\otimes|4\rangle_A\langle4|$, Bob can identify $|\varphi_{3, 4}\rangle$ by projectors $\emph{B}_{21, 22}=|1\rangle_b\langle1|\otimes|1\rangle_B\langle1|\pm|3\rangle_b\langle3|\otimes|3\rangle_B\langle3|$ and $\emph{B}_{31, 32}=|1\rangle_b\langle1|\otimes|1\rangle_B\langle1|\pm|4\rangle_b\langle4|\otimes|4\rangle_B\langle4|$, respectively.
Consider the outcome $\emph{A}_4=|2\rangle_a\langle2|\otimes|1-4\rangle_A\langle1-4|$, the only remaining possibility is $|\varphi_5\rangle$, which has thus been successfully identified. In the same way, Alice can identify $|\varphi_{6, 7, 8}\rangle$ by three projectors $\emph{A}_5=|3\rangle_a\langle3|\otimes|1-2\rangle_A\langle1-2|$, $\emph{A}_6=|4\rangle_a\langle4|\otimes|1-3\rangle_A\langle1-3|$, $\emph{A}_7=|5\rangle_a\langle5|\otimes|1-2\rangle_A\langle1-2|$, respectively.
For the last outcome $\emph{A}_8=|3\rangle_a\langle3|\otimes|4\rangle_A\langle4|+|5\rangle_a\langle5|\otimes|4\rangle_A\langle4|$, it leaves $|\varphi_{9}\rangle$ invariant and transforms $|\varphi_{1}\rangle$ to $|4\rangle_A(|3\rangle_B|33\rangle_{ab}+|5\rangle_B|55\rangle_{ab})$. Since $|\varphi_{1}\rangle$ and $|\varphi_{9}\rangle$ are two mutually orthogonal states, Bob can easily distinguish them by using the projectors $\emph{B}_{81, 82}=|3\rangle_b\langle3|\otimes|3\rangle_B\langle3|\pm|5\rangle_b\langle5|\otimes|5\rangle_B\langle5|$.
Thus the $9$ quantum states (3) have been perfectly distinguished using solely LOCC with a $\mathbb{C}^5\otimes\mathbb{C}^5$ maximally entangled state.
Next we consider local distinguishability of orthogonal product states for the general even-partite case. Surprisingly, a set of locally indistinguishable multipartite product states can be constructed by using a set of locally indistinguishable bipartite orthogonal product states. Recall that Wang et al.[@Wang17] have constructed $2(n_2+n_4+\cdots+n_{2k}-k)+1$ LOCC indistinguishable orthogonal product states in $\mathbb{C}^{n_1}\otimes\mathbb{C}^{n_2}\otimes\cdots\otimes\mathbb{C}^{n_{2k-1}}\otimes\mathbb{C}^{n_{2k}}~(4\leq n_1\leq n_2\leq\cdots\leq n_{2k-1}\leq n_{2k}, k\geq 2)$.
The set is given explicitly by [$$\begin{array}{rcl}
\mathcal{S}_1 & = & \{|\phi_{i_1}\rangle_1|11\rangle_2\cdots|11\rangle_k\ \big| \ i_1=1,2,\ldots,2n_2-2\}, \\
\mathcal{S}_2 & = & \{|111\rangle_1|\phi_{i_2}\rangle_2|11\rangle_3\cdots|11\rangle_k \big| \ i_2=1,2,\ldots,2n_4-2\}, \\
& \vdots & \\
\mathcal{S}_k & = & \{|111\rangle_1\cdots|11\rangle_{k-1}|\phi_{i_k}\rangle_k \big| \ i_k=1,2,\ldots,2n_{2k}-2\}.
\end{array}$$ ]{}
Here the stopper state $|\phi\rangle=|\phi_1\rangle|\phi_2\rangle\ldots|\phi_{k}\rangle$, where $|\phi_s\rangle$ is the stopper state of the bipartite system $\mathbb{C}^{n_{2s-1}}\otimes\mathbb{C}^{n_{2s}}$ ($s = 1,2, \ldots, k$).
[**Theorem 2.**]{} In $\mathbb{C}^{n_1}\otimes\mathbb{C}^{n_2}\otimes\cdots\otimes\mathbb{C}^{n_{2k-1}}\otimes\mathbb{C}^{n_{2k}}$, it is sufficient to perfectly distinguish the set $\mathcal{S}$ of orthogonal product states by LOCC using entanglement as a resource.
[ *Proof:*]{} The set $\mathcal{S}$ of the $2(n_2+n_4+\cdots+n_{2k}-k)+1$ multipartite product states consists of the sets $\mathcal{S}_{i} ~(i=1,2,\ldots,k)$ and a stopper state. If the states in each set $\mathcal{S}_i$ can be distinguished, then all the states in $\mathcal{S}$ can be distinguished successfully.
Let $\mathcal{H}$ = $\mathbb{C}^{n_1}\otimes\mathbb{C}^{n_2}\otimes\cdots\otimes\mathbb{C}^{n_{2k-1}}\otimes\mathbb{C}^{n_{2k}}(4\leq n_1\leq n_2\leq\cdots\leq n_{2k-1}\leq n_{2k})$. Suppose that each spatially separated observer $\mathcal{O}_i$ ($i=1,2,\ldots,2k$) controls a subsystem of $\mathcal{H}$. For example, Alice controls the subsystem $\mathcal{H^{A}}$ and Bob controls the subsystem $\mathcal{H^{B}}$ in the bipartite system $\mathcal{H^{A}}\otimes\mathcal{H^{B}}$. We call the observers $\mathcal{O}_{2s-1}$ and $\mathcal{O}_{2s}$ the $s$-th Alice and the $s$-th Bob respectively. Here we can see that $\mathcal{H}$ consists of $k$ local subsystems $\mathbb{C}^{n_{2s-1}}\otimes\mathbb{C}^{n_{2s}}, s = 1,2, \ldots, k$. Let $|\psi\rangle$ = $\otimes_{s=1}^k|\psi\rangle_{s}$ be an entangle state in $\mathcal{H}$, where $|\psi\rangle_{s}$ is a $\mathbb{C}^{n_{2s}}\otimes\mathbb{C}^{n_{2s}}$ maximally entangled state for $s = 1,2, \ldots, k.$ The parties share the entangle state $|\psi\rangle$. That is, the $s$-th Alice and the $s$-th Bob share the maximally entangled state $|\psi\rangle_{s}$.
We denote the states $|\psi_{i_s}\rangle=|\phi_{i_s}\rangle_1|11\rangle_2\cdots|11\rangle_k$ ($i_s=1, 2, \ldots, 2n_{2s}-2$) and $|\psi_{2n_{2s}-1}\rangle=|\phi\rangle$. Because of the particularity of multipartite product states, the local distinguishability of the bipartite states $|\phi_{i_s}\rangle$ ($i_s=1, 2, \ldots, 2n_{2s}-2$) and $|\phi_{s}\rangle$ is corresponding to the local distinguishability of the multipartite states $|\psi_{i_s}\rangle$ ($i_s=1, 2, \ldots, 2n_{2s}-1$). By Theorem 1, since the $s$-th Alice and the $s$-th Bob share the maximally entangled state $|\psi\rangle_{s}$, the bipartite states $|\phi_{i_s}\rangle$, $i_s=1, 2, \ldots, 2n_{2s}-2$ and $|\phi_{s}\rangle$ can be distinguished with LOCC. Thus, the multipartite states $|\psi_{i_s}\rangle$, $i_s=1, 2, \ldots, 2n_{2s}-1$, also can be distinguished with LOCC.
Therefore, the $2(n_2+n_4+\cdots+n_{2k}-k)+1$ indistinguishable quantum states can be LOCC distinguished using the maximally entangled state $|\psi\rangle$ as a resource.
Local distinguishability for odd-partite case
---------------------------------------------
In this subsection, we study local distinguishability of orthogonal product states for odd-partite systems. We use a concrete example to illustrate the idea, and then present the general method for the tripartite system. Based on tripartite systems, we then establish the general result for local distinguishability in any odd multipartite system.
In $\mathbb{C}^4\otimes\mathbb{C}^5\otimes\mathbb{C}^6$, Wang. et al.[@Wang17] have given $17$ LOCC indistinguishable orthogonal product states as follows
$$\begin{split}
&|\phi_1\rangle=|1+2+3+4\rangle_C|1+2+3+4+5\rangle_B|1+2+3+4+5+6\rangle_A,\\
&|\phi_2\rangle=|4\rangle_C|2\rangle_B|1-2\rangle_A,~~~~~~~~|\phi_{10}\rangle=|4\rangle_C|5\rangle_B|3-6\rangle_A,\\
&|\phi_3\rangle=|4\rangle_C|3\rangle_B|1-3\rangle_A,~~~~~~~~|\phi_{11}\rangle=|4\rangle_C|1-2\rangle_B|6\rangle_A,\\
&|\phi_4\rangle=|4\rangle_C|4\rangle_B|1-4\rangle_A,~~~~~~~~|\phi_{12}\rangle=|3\rangle_C|1-2\rangle_B|6\rangle_A,\\
&|\phi_5\rangle=|4\rangle_C|5\rangle_B|1-5\rangle_A,~~~~~~~~|\phi_{13}\rangle=|2\rangle_C|1-2\rangle_B|6\rangle_A,\\
&|\phi_6\rangle=|4\rangle_C|1-5\rangle_B|2\rangle_A,~~~~~~~~|\phi_{14}\rangle=|1\rangle_C|1-2\rangle_B|6\rangle_A,\\
&|\phi_7\rangle=|4\rangle_C|1-2\rangle_B|3\rangle_A,~~~~~~~~|\phi_{15}\rangle=|1-2\rangle_C|4\rangle_B|6\rangle_A,\\
&|\phi_8\rangle=|4\rangle_C|1-3\rangle_B|4\rangle_A,~~~~~~~~|\phi_{16}\rangle=|2-3\rangle_C|5\rangle_B|6\rangle_A,\\
&|\phi_9\rangle=|4\rangle_C|1-4\rangle_B|5\rangle_A,~~~~~~~~|\phi_{17}\rangle=|3-4\rangle_C|4\rangle_B|6\rangle_A.\\
\end{split}$$
The LOCC indistinguishability of the $17$ states is derived from the fact that no party can go first, i.e., Alice and Bob cannot apply any nontrivial measurements, and the third party Charlie can only apply the trivial measurement. By introducing a maximally entangled state shared by Alice and Bob, Alice use projectors such that the three parties can perform measurements to distinguish the results by LOCC. The procedure is carried in two steps. First Alice and Bob share the $\mathbb{C}^6\otimes\mathbb{C}^6$ maximally entangled state $$|\psi\rangle_{ab}=|11\rangle_{ab}+|22\rangle_{ab}+|33\rangle_{ab}+|44\rangle_{ab}+|55\rangle_{ab}+|66\rangle_{ab}.$$
Then Alice performs the projector $$\emph{A}_1=|11\rangle_{aA}\langle11|+|22\rangle_{aA}\langle22|+|33\rangle_{aA}\langle33|+|44\rangle_{aA}\langle44|
+|55\rangle_{aA}\langle55|+|66\rangle_{aA}\langle66|$$ to get the following states: $$\begin{split}
&|\phi'_1\rangle=|1+2+3+4\rangle_C|1+2+3+4+5\rangle_B\sum_{i=1}^6|iii\rangle_{Aab},\\
&|\phi'_2\rangle=|4\rangle_C|2\rangle_B(|111\rangle_{Aab}-|222\rangle_{Aab}),\\
&|\phi'_3\rangle=|4\rangle_C|3\rangle_B(|111\rangle_{Aab}-|333\rangle_{Aab}),\\
&|\phi'_4\rangle=|4\rangle_C|4\rangle_B(|111\rangle_{Aab}-|444\rangle_{Aab}),\\
&|\phi'_5\rangle=|4\rangle_C|5\rangle_B(|111\rangle_{Aab}-|555\rangle_{Aab}),\\
&|\phi'_6\rangle=|4\rangle_C|1-5\rangle_B|222\rangle_{Aab},\\
&|\phi'_7\rangle=|4\rangle_C|1-2\rangle_B|333\rangle_{Aab},\\
&|\phi'_8\rangle=|4\rangle_C|1-3\rangle_B|444\rangle_{Aab},\\
&|\phi'_9\rangle=|4\rangle_C|1-4\rangle_B|555\rangle_{Aab},\\
&|\phi'_{10}\rangle=|4\rangle_C|5\rangle_B(|333\rangle_{Aab}+|666\rangle_{Aab}),\\
&|\phi'_{i}\rangle=|\phi_{i}\rangle|66\rangle_{ab}, i=11,12, \ldots, 17.\\
\end{split}$$
Then Bob makes a 11-outcome projective measurement. For outcomes $\emph{B}_1=|2\rangle_b\langle2|\otimes|1-5\rangle_B\langle1-5|$, $\emph{B}_2=|3\rangle_b\langle3|\otimes|1-2\rangle_B\langle1-2|$, $\emph{B}_3=|4\rangle_b\langle4|\otimes|1-3\rangle_B\langle1-3|$ and $\emph{B}_4=|5\rangle_b\langle5|\otimes|1-4\rangle_B\langle1-4|$, the states $|\phi'_6\rangle, |\phi'_7\rangle, |\phi'_8\rangle$ and $|\phi'_9\rangle$ can be identified directly.
For the outcome $\emph{B}_{i+3}=|1\rangle_b\langle1|\otimes|i\rangle_B\langle i|+|i\rangle_b\langle i|\otimes|i\rangle_B\langle i|, i=2,3,4,5$, it leaves $|\phi'_{i}\rangle$ invariant and transforms $|\phi'_{1_{i}}\rangle$ to $|1+2+3+4\rangle_C|i\rangle_B(|111\rangle_{Aab}+|iii\rangle_{Aab})~(i=2,3,4,5).$ Alice can easily identify these states, since any two orthogonal states can be distinguished.
For the outcome $\emph{B}_9=|6\rangle_b\langle6|\otimes|1-2\rangle_B\langle1-2|$, it leaves $|\phi'_{i}\rangle~(i=11,12,13,14)$ invariant. Then, Charles can identify these states by four projectors $\emph{C}_{9i}=|i\rangle_C\langle i|, i=1,2,3,4$.
For the outcome $\emph{B}_{10}=|3\rangle_b\langle3|\otimes|6\rangle_B\langle 6|+|6\rangle_b\langle 6|\otimes|5\rangle_B\langle 5|$, it leaves $|\phi'_{10}\rangle$, $|\phi'_{16}\rangle$ invariant and transforms $|\phi'_{1}\rangle$ to $|1+2+3+4\rangle_C|5\rangle_B(|333\rangle_{Aab}+|666\rangle_{Aab}).$ Then, Charles uses the projector $\emph{C}_{10,1}=|2-3\rangle_C\langle 2-3|$ to identify $|\phi'_{16}\rangle$. When Charles uses the projector $\emph{C}_{10,2}=|4\rangle_C\langle 4|,$ it leaves $|\phi'_{10}\rangle$ invariant and transforms $|\phi'_{1}\rangle$ to $|4\rangle_C|5\rangle_B(|333\rangle_{Aab}+|666\rangle_{Aab}).$ Alice also can distinguish them, since they are two orthogonal states in Alice’s Hilbert space.
For the last outcome $\emph{B}_{11}=|6\rangle_b\langle 6|\otimes|4\rangle_B\langle 4|$, it leaves $|\phi'_{15}\rangle$, $|\phi'_{17}\rangle$ invariant and transforms $|\phi'_{1}\rangle$ to $|1+2+3+4\rangle_C|4\rangle_B|666\rangle_{Aab}.$ Then, Charles can distinguish these states by the projectors $\emph{C}_{11,1}=|1-2\rangle_C\langle 1-2|,$ $\emph{C}_{11,2}=|1+2\rangle_C\langle 1+2|,$ $\emph{C}_{11,3}=|3-4\rangle_C\langle 3-4|,$ and $\emph{C}_{11,4}=|3+4\rangle_C\langle 3+4|.$
Therefore, we have succeeded in distinguishing the states (6) by LOCC using entanglement as a resource.
In the space $\mathbb{C}^{n_1}\otimes\mathbb{C}^{n_2}\otimes\mathbb{C}^{n_3}~(4\leq n_1\leq n_2\leq n_{3}),$ there exist $2(n_1+n_3)-3$ orthogonal product states which are LOCC indistinguishable [@Wang17] and collectively denoted by $\mathcal{G}$.
The set $\mathcal{G}$ can be divided into three parts: $\mathcal{G}$={$|\phi\rangle\}\cup \mathcal{T}\cup\mathcal{R}$. The state $|\phi\rangle=|1+2+\cdots+n_1\rangle_C|1+2+\cdots+n_2\rangle_B|1+2+\cdots+n_3\rangle_A$. The subset $\mathcal{T}$ consists of $|\phi_i\rangle=|n_1\rangle\otimes|\psi_i\rangle$, $i=2,3,\ldots ,2n_3-1$, where $\{|\psi_i\rangle\}$ is the set of the states constructed in $\mathbb{C}^{n_2}\otimes\mathbb{C}^{n_3}$ using [@Wang17 Theorem 1] (except the stopper state).
To describe the subset $\mathcal{R}$, we consider three cases: (a) $n_2<n_3$ and $n_3-n_2$ is odd; (b) $n_2<n_3$ and $n_3-n_2$ is even; (c) $n_2=n_3$. Let $\{|i'\rangle\}_{i'=1}^{n_2}$ be another base in the second factor (Bob’s). By relabeling, one can such that $i'=i$ for all $i'$ in case (a); $(n_2-1)'=n_2,$ $n_2'=n_2-1$, $i'=i$ for the other $i'$ in case (b); and for case (c), $2'=n_2-1,$ $(n_2-1)'=2$, $i'=i$ for the other $i'$. Let $\mathcal{R}=H\cup V$, here $H=\{|i\rangle|1'-2'\rangle|n_3\rangle \ \big| \ 1 \leq i\leq n_1\},$ and $V=\{|i-(i+1)\rangle|(n_2-\delta_{(n_1-i)})'\rangle|n_3\rangle \ \big| 1 \ \leq i \leq n_1-1\},$ where $\delta_i=\frac{1}{2}(1+(-1)^i)$.
[**Theorem 3.**]{} Over the space $\mathbb{C}^{n_1}\otimes\mathbb{C}^{n_2}\otimes\mathbb{C}^{n_3}~(4\leq n_1\leq n_2\leq n_{3}),$ it is sufficient to perfectly distinguish the set $\mathcal{G}$ of the orthogonal product states by LOCC using a maximally entangled state.
[ *Proof:*]{} We only consider the case (a) to show the idea, as the other cases can be dealt with similarly. We assume the first, second and third system belong to Charles, Bob, and Alice respectively. Similar to the previous example of the tripartite system, Alice and Bob share the $\mathbb{C}^{n_3}\otimes\mathbb{C}^{n_3}$ maximally entangled state $$|\psi\rangle_{ab}=|11\rangle_{ab}+|22\rangle_{ab}+\cdots+|{n_3}{n_3}\rangle_{ab}.$$ And then Alice performs a projector $$\emph{A}_1=|11\rangle_{aA}\langle11|+|22\rangle_{aA}\langle22|+\cdots+|{n_3}{n_3}\rangle_{aA}\langle{n_3}{n_3}|.$$ The parties can proceed from here to distinguish the states using LOCC. As the proof of Theorem 1, Alice and Bob can distinguish the states in the set $\mathcal{T}$ except the state $|n_1\rangle_C\otimes |n_2\rangle_B\otimes |3-(n_2+1)\rangle_A$ and the state $|n_1\rangle_C\otimes |1-2\rangle_B\otimes |n_3\rangle_A(\in\mathcal{R})$. In addition, the states in the set $\mathcal{R}$ can be distinguished by Bob and Charles. Finally, the last state $|n_1\rangle_C\otimes |n_2\rangle_B\otimes |3-(n_2+1)\rangle_A$ also can be distinguished by the three parties with LOCC.
Therefore, for a general tripartite system, using a $\mathbb{C}^{n_3}\otimes\mathbb{C}^{n_3}$ maximally entangled state is sufficient to perfectly distinguish the set $\mathcal{G}$ of orthogonal product states by LOCC.
We now generalize the method to the general odd partite quantum systems. It is known [@Wang17] that there are $2(n_1+n_3+\cdots+n_{2k+1}-k)+1$ LOCC indistinguishable product states in the space $\mathbb{C}^{n_1}\otimes\mathbb{C}^{n_2}\otimes\cdots\otimes\mathbb{C}^{n_{2k}}\otimes\mathbb{C}^{n_{2k+1}}$ $(4\leq n_1\leq n_2\leq\cdots\leq n_{2k+1})$. They are explicitly given as follows.
Let $|\psi_{i_1}\rangle_1$ $(i_1=1,2,\ldots,2(n_1+n_3)-4)$ be the product states (except the stopper state) constructed in [@Wang17 Theorem 3] for the general tripartite system. The corresponding systems are assigned to 1st Charles, Bob and Alice-system respectively. Let $|\psi_{i_s}\rangle_s$, $i_s=1,2,\ldots,2n_{2s+1}-2$ be the product states (except the stopper state) given in [@Wang17 Theorem 1] for the general bipartite system. The corresponding systems are designated as the s-th Alice and s-th Bob system for each integer $2\leq s\leq k$.
The set $\mathcal{S'}$ of product states in a multipartite quantum system is then the union of a stopper state $|\psi\rangle$ and the following sets. [$$\begin{array}{rcl}
\mathcal{S'}_1 & = & \{|\psi_{i_1}\rangle_1|11\rangle_2\cdots|11\rangle_k\ \big| \ i_1=1,2,\ldots,2(n_1+n_3)-4\}, \\
\mathcal{S'}_2 & = & \{|111\rangle_1|\psi_{i_2}\rangle_2|11\rangle_3\cdots|11\rangle_k \big| \ i_2=1,2,\ldots,2(n_5)-2\}, \\
& \vdots & \\
\mathcal{S'}_k & = & \{|111\rangle_1\cdots|11\rangle_{k-1}|\psi_{i_k}\rangle_k \big| \ i_k=1,2,\ldots,2(n_{2k+1})-2\},
\end{array}$$ ]{} where $|\psi\rangle=|\psi_1\rangle|\psi_2\rangle\ldots|\psi_{2k+1}\rangle$ and $|\psi_i\rangle=|1\rangle+|2\rangle+\cdots+|n_i\rangle$ for $2\leq i\leq 2k+1$. As the general odd partite quantum systems are composed of the tripartite subsystem and even-partite subsystem, the local distinguishability for odd-partite case is transformed into those for the tripartite system and even-partite system. Let $|\psi\rangle$ = $\otimes_{s=1}^k|\psi\rangle_{s}$ be an entangled state in $\mathbb{C}^{n_1}\otimes\mathbb{C}^{n_2}\otimes\cdots\otimes\mathbb{C}^{n_{2k}}\otimes\mathbb{C}^{n_{2k+1}}$, where $|\psi\rangle_{s}$ is a $\mathbb{C}^{n_{2s+1}}\otimes\mathbb{C}^{n_{2s+1}}$ maximally entangled state for $s = 1,2, \ldots, k.$ It follows from Theorem 2 and Theorem 3 that if the parties share the entangled state $|\psi\rangle$ we have the following result.
[**Theorem 4.**]{} Over the space $\mathbb{C}^{n_1}\otimes\mathbb{C}^{n_2}\otimes\cdots\otimes\mathbb{C}^{n_{2k}}\otimes\mathbb{C}^{n_{2k+1}}$, it is sufficient to perfectly distinguish the set $\mathcal{S'}$ of the orthogonal product states by LOCC using entanglement as a resource.
Conclusions
===========
Recently, much attention has been given to the problem of distinguishing bipartite orthogonal product states by LOCC with entanglement. In this paper, we have studied the case of multipartite systems and show that entanglement can also be used as a resource to locally distinguish the sets of multipartite product states. We have demonstrated how to come up with the entangled state to distinguish the sets of multipartite product states in both the even and odd partite systems. Our results provided are expected to help us understand the distinguishability of multipartite product states by LOCC.
[99]{} Ghosh, S., Kar, G., Roy, A., Sen, A. & Sen, U. Distinguishability of Bell States. *Phys. Rev. Lett*. [**87**]{}, 277902 (2001). Ghosh, S., Kar, G., Roy, A., Sarkar, D.: Distinguishability of maximally entangled states. Phys. Rev. A. **70**, 022304 (2004)
Hayashi, M., Markham, D., Murao, M., Owari, M., Virmani, S.: Bounds on multipartite entangled orthogonal state discrimination using local operations and classical communication.Phys. Rev. Lett. **96**, 040501 (2006).
Bennett, C. H., DiVincenzo, D. P., Fuchs, C. A., Mor, T., Rains, E., Shor, P. W., Smolin ,J. A., Wootters, W. K.: Quantum nonlocality without entanglement. Phys. Rev. A. **59**, 1070 (1999).
Walgate, J., Hardy, L.: Nonlocality, asymmetry, and distinguishing bipartite states. Phys. Rev. Lett. **89**, 147901(2002). Horodecki, M., Sen, A., Sen, U., Horodecki, K.: Local indistinguishability: More nonlocality with less entanglement. Phys. Rev. Lett. **90**, 047902 (2003). De Rinaldis, S.: Distinguishability of complete and unextendible product bases. Phys. Rev. A. **70**, 022309 (2004). Nathanson, M.: Distinguishing bipartitite orthogonal states using LOCC: Best and worst cases. J. Math Phys. **46**, 062103 (2005). Niset, J., Cerf, N. J.: Multipartite nonlocality without entanglement in many dimensions. Phys. Rev. A. **74**, 052103 (2006). Feng, Y., Shi, Y.: Characterizing locally indistinguishable orthogonal product states. IEEE Trans. Inf. Theory **55**, 2799-2806 (2009). Duan, R., Xin, Y., Ying, M.: Locally indistinguishable subspaces spanned by three-qubit unextendible product bases. Phys. Rev. A. **81**, 032329 (2010). Childs, A. M., Leung, D., Mančinska, L.,Ozols, M.: A framework for bounding nonlocality of state discrimination. Commun. Math. Phys. **323**, 1121-1153(2013). Zhang, Z. C., Gao, F., Tian, G. J., Cao, T. Q., Wen, Q. Y.: Nonlocality of orthogonal product basis quantum states. Phys. Rev. A. **90**, 022313 (2014). Wang, Y. L., Li, M. S., Zheng, Z. J., Fei, S. M.: Nonlocality of orthogonal product-basis quantum states. Phys. Rev. A. **92**, 032313 (2015). Zhang, Z. C., Gao, F., Qin, S. J., Yang, Y. H., Wen, Q. Y.: Nonlocality of orthogonal product states. Phys. Rev. A. **92**, 012332 (2015). Zhang, Z. C., Gao, F., Cao, Y., Qin, S. J., Wen, Q. Y.: Local indistinguishability of orthogonal product states. Phys. Rev. A. **93**, 012314 (2016). Xu, G. B., Yang, Y. H., Wen, Q. Y., Qin, S. J., Gao, F.: Constructing locally indistinguishable orthogonal product bases in an $ m\ otimes n $ system(2015). arXiv:1512.06485 Xu, G. B., Wen, Q. Y., Qin, S. J., Yang, Y. H., Gao, F.: Quantum nonlocality of multipartite orthogonal product states. Phys. Rev. A. **93**, 032341 (2016). Wang, Y. L., Li, M. S., Zheng, Z. J., Fei, S. M.: The local indistinguishability of multipartite product states. Quantum Inf. Process. **16**, 5 (2017). Zhang, X., Weng, J., Tan, X., Luo, W.: Indistinguishability of pure orthogonal product states by locc. Quantum Inf. Process. **16**, 168 (2017). Zhang, Z. C., Zhang, K. J., Gao, F., Wen, Q. Y., Oh, C. H.: Construction of nonlocal multipartite quantum states. Phys. Rev. A. **95**, 052344 (2017).
Cohen, S. M.: Understanding entanglement as resource: locally distinguishing unextendible product bases. Phys. Rev. A. **77**, 012304 (2008).
Ekert, A. K.: Quantum cryptography based on Bell¡¯s theorem. Phys. Rev. Lett. **67**, 661 (1991). Gisin, N., Ribordy, G., Tittel, W., Zbinden, H.: Quantum cryptography. Rev. Mod. Phys. **74**, 145 (2002).
Karlsson, A., Bourennane, M.: Quantum teleportation using three-particle entanglement. Phys. Rev. A. **58**, 4394 (1998). Kim, Y. H., Kulik, S. P., Shih, Y.: Quantum teleportation of a polarization state with a complete Bell state measurement. Phys. Rev. Lett. **86**, 1370 (2001).
Wang, C., Deng, F. G., Long, G. L.: Multi-step quantum secure direct communication using multi-particle Green¨CHorne¨CZeilinger state. Opt. Commun. **253**, 15-20 (2005). Tian-Yin, W., Su-Juan, Q., Qiao-Yan, W., Fu-Chen, Z.: Analysis and improvement of multiparty controlled quantum secure direct communication protocol. Acta. Physica Sinica. 57(12), 135¨C137 (2008). Yang, C. W., Tsai, C. W., Hwang, T.: Fault tolerant two-step quantum secure direct communication protocol against collective noises. Sci. China Phys. **54**, 496-501 (2011).
Zhang, Z.C., Gao, F., Cao, T.Q., Qin, S.J., Wen, Q.Y.: Entanglement as a resource to distinguish orthogonal product states. Sci. Rep. 6, 30493 (2016).
Bandyopadhyay, S., Halder, S., Nathanson, M.: Entanglement as a resource for local state discrimination in multipartite systems. Phys. Rev. A. **94**, 022311 (2016). Li, Z.G., Zhao, M., Li, S.M.,Wang, H.,Liu, W.M.: Maximally entangled state can be a mixed state. Quant. Inf. Comput [**12**]{}, 0063-0073 (2012)
|
---
author:
- 'Run-Qiu Yang,'
- 'Chao Niu,'
- 'Cheng-Yong Zhang'
- 'Keun-Young Kim,'
bibliography:
- 'QFTref.bib'
title: Comparison of holographic and field theoretic complexities by time dependent thermofield double states
---
Introduction
============
The applications of the holographic principle on the study of black holes have lead to many new surprising discoveries. In particular, it has been shown that the quantum information may play important roles in the quantum gravity. One important discovery is the connection between entanglement and geometry [@VanRaamsdonk:2010pw; @Ryu:2006bv; @Maldacena:2013xja; @Faulkner2014]. It inspired a viewpoint that a black hole might be highly entangled with a system that was effectively infinitely far away. This viewpoint lead Juan Maldacena and Leonard Susskind to propose a very interesting conjecture named “ER=EPR” [@Maldacena:2013xja] when they considered the wormhole created by an Einstein-Rosen (ER) bridge [@PhysRev.48.73] and a pair of maximally entangled black holes. Here EPR refers to quantum entanglement (Einstein-Podolsky-Rosen paradox). To understand how much difficult sending a signal through the ERB from one side to the other, a new information-theoretic quantity named “complexity” was imported into the holographic duality and quantum gravity [@Susskind:2014rva; @Susskind:2014moa]. Basically, the complexity describes how many fundamental gates or operators are required when we try to obtain a target state from a reference state.
In order to construct holographic models to describe the complexity, let us consider the following thermofield double (TFD) state $$\label{TFD1}
|\text{TFD}\rangle:=Z^{-1/2}\sum_\alpha\exp[-E_\alpha/(2T)]|E_\alpha\rangle_L |E_\alpha\rangle_R \,.$$ The states $|E_\alpha\rangle_L$ and $|E_\alpha\rangle_R$ are defined in the two copy CFTs and $T$ is the temperature of these two CFTs. This state is conjectured to be approximately dual to an eternal AdS black hole with the Hawking temperature which is the same as $T$ of CFTs [@Maldacena:2001kr]. [With the Hamiltonians $H_L$ and $H_R$ at the left and right dual CFTs, respectively, the time evolution of a TFD state $$\label{timesate1}
|\psi(t_L,t_R)\rangle:=e^{-i(t_LH_L+t_RH_R)}|\text{TFD}\rangle\,,$$ can be characterized by the codimension-two surface of $t=t_L$ and $t=t_R$ at the two boundaries of the AdS black hole [@Maldacena:2001kr; @Brown:2015lvg].]{} Two different conjectures were proposed to compute the complexity of $|\psi(t_L,t_R)\rangle$ holographically:[^1] the CV(complexity=volume) conjecture [@Susskind:2014rva; @Stanford:2014jda; @Alishahiha:2015rta] and the CA(complexity= action) conjecture [@Brown:2015bva; @Brown:2015lvg].
The CV conjecture states that the complexity of [$|\psi(t_L,t_R)\rangle$]{} is proportional to the maximal volume of the space-like codimension-one surface which connects the codimension-two time slices denoted by $t_L$ and $t_R$ at two AdS boundaries, i.e. $$\label{CV}
\mathcal{C}_V=\max_{\partial \Sigma=t_L\cup t_R}\left[\frac{V(\Sigma)}{G_N \ell}\right] \,,$$ where $G_N$ is the Newton’s constant. $\Sigma$ is a possible space-like codimension-one surface which connects $t_L$ and $t_R$. $\ell$ is a length scale associated with the bulk geometry such as the horizon radius or AdS radius and so on. However, there is an ambiguity coming from the choice of a length scale $\ell$. This unsatisfactory feature motivated the second conjecture: the CA conjecture [@Brown:2015bva; @Brown:2015lvg], which says the complexity of [$|\psi(t_L,t_R)\rangle$]{} is dual to the action in the Wheeler-DeWitt (WDW) patch associated with $t_L$ and $t_R$, i.e. $$\label{CA}
\mathcal{C}_A=\frac{I_{\text{WDW}}}{\pi\hbar}.$$ The WDW patch associated with $t_L$ and $t_R$ is the collection of all space-like surface connecting $t_L$ and $t_R$ with the null sheets coming from $t_L$ and $t_R$. More precisely it is the domain of dependence of any space-like surface connecting $t_L$ and $t_R$.
Recently, two different methods were proposed by Refs. [@Chapman:2017rqy; @Yang:2017nfn] to define the complexity in quantum field theory.[^2] The method in Ref. [@Chapman:2017rqy], which we will call the FS method in this paper, is based on the Fubini-Study metric and defined the complexity of two states to be the length of the geodesic under the Fubini-Study metric. The method in Ref. [@Yang:2017nfn], which we will call the FG method in this paper, first defined the complexity for operators by the Finsler geometry[^3] and then used this complexity to define the complexity between states. As what we will present later, these two methods are very suitable to compute the complexity between two different TFD states.
Taking the Eqs. , and two quantum field theory proposals in Refs. [@Yang:2017nfn; @Chapman:2017rqy] into account, we have at least four different methods to compute the complexity between two TFD states. The goal of this paper is to compute the complexity in four different methods and see their similarities and differences. It may give us some information to judge which are appropriate methods to compute the complexity among four methods. It may also shed light on a possible connection between two holographic conjectures and two quantum field theory proposals.
The paper is organized as follows. In sec. \[modefCVCA\] we introduce several concepts for complexity in literature and clarify some subtle issues in defining the complexity regarding the reference state and divergence in holographic complexity. In sec. \[CACV\] we use the modified CA and CV conjectures introduce in sec. \[modefCVCA\] to compute complexity between the time dependent TFD state and their corresponding zero temperature vacuum state. In sec. \[fromQTF\], after presenting how to construct a time-dependent TFD state for free field theory explicitly, we use two different quantum field theoretic proposals to compute the complexity between the time-dependent TFD state and the vacuum. Our computations by four different methods are summarized and compared in sec. \[summ\].
#### Note added
While this work was being finished, Ref. [@Carmi:2017jqz] appeared which also studied the complexity growth rate. Our results in sec. \[CACV\] have some overlap with Ref. [@Carmi:2017jqz].
Holographic complexity potential {#modefCVCA}
================================
Before we compute the holographic time-dependent complexity, let us first make some comments on the CV and CA conjectures and present our modified versions of them.
Although these two conjectures satisfy important requirements on the complexity such as the Lloyd’s bound [@Lloyd2000; @Cai:2016xho; @Yang:2016awy; @Cai:2017sjv], it seems that there are a few subtle issues to be clarified. First, the complexity computed by Eq. or Eq. is infinite. Second, there is an ambiguity for the reference state. One may assume that there is an unknown “favorite” reference state which is not found yet[^4], and the divergence of the complexity computed by the CV and CA conjectures show some intrinsic properties of CFT similarly to the case of the entanglement entropy [@Carmi:2016wjl]. Even if we accept this unknown “favorite” reference state exists, it seems that the original CA conjecture has two more issues. First, in principle, the complexity between two states should be non-negative but the value computed by Eq. can be negative. Second, the dynamics will be invariant if we add a constant term into the action so the complexity should also be invariant after adding a constant term into the gravity action. However, the original CA conjecture does not satisfy this property.
To resolve this issue a “modified” complexity was proposed in Ref. [@Yang:2017nfn]. It suggests that the original CV and CA conjectures and give a kind of “complexity potential” rather than the complexity between any two states. When we restrict our considerations to the TFD states, the leading order of complexity between two TFD states $|\text{TFD}_1\rangle$ and $|\text{TFD}_2\rangle$ is given by the following formulas in the CV or CA conjectures[^5] $$\label{newcvca}
\begin{split}
&\mathcal{C}_V(|\text{TFD}_2\rangle,|\text{TFD}_1\rangle)=|\mathcal{C}^{(1)}_{V}-\mathcal{C}^{(2)}_{V}|\,, \\
&\mathcal{C}_A(|\text{TFD}_2\rangle,|\text{TFD}_1\rangle)=|\mathcal{C}^{(1)}_{A}-\mathcal{C}^{(2)}_{A}|\,.
\end{split}$$ Here $\{\mathcal{C}^{(1)}_{V}, \mathcal{C}^{(2)}_{V}\}$ and $\{\mathcal{C}^{(1)}_{A}, \mathcal{C}^{(2)}_{A}\}$ are computed by Eqs. or . This modification does not lose any important physical properties of the original version and seems simpler because it does not need to refer to an unknown reference state[^6]. This modified version has the following basic properties:
- Triangle inequality: $\mathcal{C}(|\text{TFD}_2\rangle,|\text{TFD}_0\rangle)+\mathcal{C}(|\text{TFD}_0\rangle,|\text{TFD}_1\rangle)\geq\mathcal{C}(|\text{TFD}_2\rangle,|\text{TFD}_1\rangle$ for any state $|\text{TFD}_0\rangle$
- Reversibility: $\mathcal{C}(|\text{TFD}_2\rangle,|\text{TFD}_1\rangle)=\mathcal{C}(|\text{TFD}_1\rangle,|\text{TFD}_2\rangle)$
In particular, when Eq. is applied to the TFD states, the results in Eq. agree to the results obtained in a quantum field theory approach in Ref. [@Yang:2017nfn] and also agree to the results computed by the method proposed in Ref. [@Chapman:2017rqy] (which will be presented in subsection \[FSmetric0\]).
The formula can be understood more geometrically. Let us assume a space where all states can be parameterized by $x^a$, which may be the temperature, total charge, mass or any other quantity describing different time slices at two boundaries. Suppose that there is a curve $l: x^a=x^a(\lambda)$ which satisfies $|\text{TFD}(x^a(\lambda_1))\rangle=|\text{TFD}_1\rangle$ and $|\text{TFD}(x^a(\lambda_2))\rangle=|\text{TFD}_2\rangle$ with $\lambda_1\leq\lambda_2$. For any given curve $l$, we can use Eqs. or to compute the $\mathcal{C}_V(\lambda)$ and $\mathcal{C}_A(\lambda)$, which are the functions of $\lambda$ and depend on the choice of the curve $l$. Then the complexity between $|\text{TFD}_1\rangle$ and $|\text{TFD}_2\rangle$ is given by[^7] $$\label{newcvca2v}
\mathcal{C}_V(|\text{TFD}_2\rangle,|\text{TFD}_1\rangle)=\min\left\{\int_{\lambda_1}^{\lambda_2}\left.\left|\frac{{\text{d}}\mathcal{C}_V(\lambda)}{{\text{d}}\lambda} \right|{\text{d}}\lambda~\right|~~\forall~l:x^a=x^a(\lambda)\right\}\,,$$ and $$\label{newcvca2a}
\mathcal{C}_A(|\text{TFD}_2\rangle,|\text{TFD}_1\rangle)=\min\left\{\int_{\lambda_1}^{\lambda_2}\left.\left|\frac{{\text{d}}\mathcal{C}_A(\lambda)}{{\text{d}}\lambda} \right|{\text{d}}\lambda~\right|~~\forall~l:x^a=x^a(\lambda)\right\}\,.$$ These two equations in fact give the Finsler structures such that for any tangent vector $T^a=(\partial/\partial\lambda)^a={\text{d}}x^a/{\text{d}}\lambda$ $$\label{finsler}
F(x^a,T^a):=\left|(\textbf{{\text{d}}}\mathcal{C}_X)_a T^a\right|\,,$$ where $X=V$ or $A$ and $\textbf{{\text{d}}}$ is the exterior differential operator in the parameter space spanned by $x^a$. The integrations in Eqs. and give a holographic version of “complexity geometry”, which is similar to the FG and FS methods in Refs. [@Chapman:2017rqy; @Chapman:2016hwi; @Yang:2017nfn]. Since the absolute value appearing in these two formulas is not convenient, we introduce a positive infinitesimal value $\epsilon$ and arbitrary functions $\omega_{ab}$ so that Eqs. and can be written as the following form $$\label{newcvca2X}
\mathcal{C}_X(|\text{TFD}_2\rangle,|\text{TFD}_1\rangle)=\lim_{\epsilon\rightarrow0^+}\min\left\{\int_{\lambda_1}^{\lambda_2}\sqrt{\left(\frac{\partial\mathcal{C}_X}{\partial x^a}\frac{{\text{d}}x^a}{{\text{d}}\lambda}\right)^2+\epsilon^2\omega_{ab}\frac{{\text{d}}x^a}{{\text{d}}\lambda}\frac{{\text{d}}x^b}{{\text{d}}\lambda}}{\text{d}}\lambda\right\}\,.$$ It is assumed that the limit is well defined and independent of the choices of auxiliary functions $\omega_{ab}$. This means that we have the following “holographic complexity metric” defined in a parameter space spanned by $x^a$, $$\label{holocomg1}
{\text{d}}s_X^2:=\left[\frac{\partial\mathcal{C}_X}{\partial x^a}\frac{\partial\mathcal{C}_X}{\partial x^b}+\epsilon^2\omega_{ab}\right]{\text{d}}x^a{\text{d}}x^b,$$ Since the metric defined in Eq. is a tensor the complexity is diffeomorphism invariant under the reparameterizations on $x^a$. The minimal values of Eq. then is the lengths of geodesics given by metric if $\mathcal{C}_V(x^a)$ and $\mathcal{C}_A(x^a)$ are $C^2$ functions of $x^a$.
To show that Eqs. and are equivalent to Eq. note that $$\label{notemetrics}
\int_{\lambda_1}^{\lambda_2}{\text{d}}\lambda\left|\frac{{\text{d}}\mathcal{C}_{X}}{{\text{d}}\lambda}\right|=\int_l|{\text{d}}\mathcal{C}_X|\geq\left|\int_l{\text{d}}\mathcal{C}_X\right|=|\mathcal{C}_X^{(1)}-\mathcal{C}_X^{(2)}|\,.$$ The equality can be achieved if there is a curve with its tangent $T^a$ satisfying, $$\label{conditionTs}
(\textbf{{\text{d}}}\mathcal{C}_X)_aT^a\leq0, \ \ \forall \lambda\in(\lambda_1,\lambda_2), \quad \text{or} \quad (\textbf{{\text{d}}}\mathcal{C}_X)_aT^a\geq0, \ \ \forall \lambda\in(\lambda_1,\lambda_2)\,.$$ For the case that parameter space has trivial topology and its dimensional is larger than 1, such a curve always can be found, so Eq. shows, $$\label{newcvca3a}
\mathcal{C}_X(|\text{TFD}_1\rangle,|\text{TFD}_2\rangle)=|\mathcal{C}_{X}^{(1)}-\mathcal{C}_{X}^{(2)}|\,,$$ which is Eq. . However, for the cases that parameter space is one dimensional or has non-trivial topology, the condition may not be achieved. The complexity modified definitions and may different from ones in Eq. . In these cases, one have to use definitions and to compute the complexity in holography.
Let us make a summary regarding several concepts for the complexity introduced in literature and this paper. If we can compute the “complexity potential” $\mathcal{C}_V(x^a)$ and $\mathcal{C}_A(x^a)$ by Eqs. and , which is the original proposals, then we can obtain the “modified complexity” which is the complexity between two states through Eq. . The holographic complexity potential has an additional freedom: if we add any term independent of $x^a$, the modified complexity will be invariant. This freedom gives us a possibility to introduce suitable subtraction terms to renormalize the divergent holographic complexity potential defined by Eqs. and . This is the foundation of the “regularized holographic complexity” proposed by Ref. [@Kim:2017lrw], which we will call “renormalized complexity potential” in this paper.
Time dependent complexity of the TFD states: holographic approach {#CACV}
=================================================================
In this section, we will use the modified CV and CA conjectures to compute the complexity between the time-dependent TFD states and their corresponding vacuum states. There are three parameters in the holographic duals: the temperature of the bulk black hole, the time of the left boundary and the time of the right boundary slice i.e. $\{T, t_L, t_R\}$. We first have to compute the complexity potential $\mathcal{C}_V(T,t_L, t_R)$ and $\mathcal{C}_A(T,t_L, t_R)$ which are divergent. To deal with this divergence, Ref. [@Kim:2017lrw] proposed a method to renormalize them by adding some counterterms. In the planner symmetry AdS black holes, the general counterterms have been found and are independent of the values of $T, t_L$ and $t_R$. Therefore, we can use the renormalized complexity potential for the holographic complexity potential and finally find the modified complexity between two TFD states. In the following subsections, we will perform these procedures in both the CA and CV conjectures.
Since we focus on the TFD states which are dual to the planar symmetric Schwarzschild AdS black holes, the problems can be simplified. Thanks to the time translation symmetry, the systems only depend on $t_L+t_R$ so we are left with only two independent parameters $\{T, t_L+t_R\}$.
CA conjecture
-------------
### Total action with the boundary and joint terms {#CAconj}
The central issue in the CA conjecture is the computation of the on-shell action. Because the null boundary and joint terms are involved in this computation, it was a subtle problem at the time when this conjecture was proposed. However, after several careful analysis on the action with null boundaries [@Parattu:2015gga; @Lehner:2016vdi; @Hopfmuller:2016scf; @Jubb:2016qzt], the action in general relativity with suitable boundary and joint terms turned out to be the following form, $$\label{actionull}
I=\frac1{16\pi}\int_{\mathcal{M}}{\text{d}}^{d+1}x\sqrt{-g}\left[R+\frac{d(d-1)}{{\ell_{\text{AdS}}}^2}\right]+\sum_{i}I_{B_i}+\sum_jI_{\mathcal{N}_j}+\sum I_{\mathcal{J}}+\sum I_{\mathcal{J}'}\,.$$ Here $R$ is the scalar curvature in the bulk, ${\ell_{\text{AdS}}}$ is the AdS radius, $I_{B_i}$ is the Gibbons-Hawking-York (GHY) boundary term for the non-null boundary fragments $B_i$, $I_{\mathcal{N}_j}$ is the boundary term for the null boundary fragments $\mathcal{N}_j$, $I_{\mathcal{J}}$ is the joint term defined on the joint of two non-null boundaries and $I_{\mathcal{J}'}$ is the joint term defined on the joint intersected by the null boundaries and others.
The first result for the null boundary term was proposed by Ref. [@Parattu:2015gga] and then represented by Refs. [@Lehner:2016vdi; @Hopfmuller:2016scf; @Jubb:2016qzt]. Refs. [@Parattu:2015gga; @Jubb:2016qzt] showed that the suitable null boundary term should be $$\label{nullbd1}
I_{\mathcal{N}_j}=I_{\mathcal{N}_j}^{(1)}:=-\frac{\text{sign}(\mathcal{N}_j)}{8\pi}\int_{\mathcal{N}_j}{\text{d}}\lambda{\text{d}}^{d-2}x\sqrt{|\sigma|}(\kappa+\Theta)\,,$$ where $\lambda$ is the integral curve of the normal vector $k^\mu$ (future directed) for the null boundary, i.e., $k^\mu=(\partial/\partial\lambda)^\mu$. $\kappa$ is the “surface gravity” of the null surface corresponding to $k^\mu$ and satisfies $k^\mu\nabla_\mu k^\nu=\kappa k^\nu$. $\text{sign}(\mathcal{N}_j)$ is +1 only when it lies on the future boundary of $\mathcal{M}$, otherwise, $\text{sign}(\mathcal{N}_j)$ is $-1$. $\sigma$ is the determinant of the induced metric at the transverse co-dimensional 2 surface orthogonal to $k^\mu$. $\Theta$ is the expansion of the null boundary measured by $\lambda$ and satisfies $\Theta=(\sqrt{|\sigma|})^{-1}k^\mu\partial_\mu\sqrt{|\sigma|}$. Considering the fact that $\Theta$ itself vanishes during the variation, Refs. [@Lehner:2016vdi; @Hopfmuller:2016scf] proposed the “minimal null boundary term” by dropping the expansion term in Eq. , which reads $$\label{nullbd2}
I_{\mathcal{N}_j}=I_{\mathcal{N}_j}^{(2)}:=-\frac{\text{sign}(\mathcal{N}_j)}{8\pi}\int_{\mathcal{N}_j}{\text{d}}\lambda{\text{d}}^{d-2}x\sqrt{|\sigma|}\kappa\,.$$ However, the boundary terms and are both dependent on the choice of $\lambda$ so the re-parameterization of $\lambda$ can lead different values for the null boundary term. To overcome this problem, Refs. [@Lehner:2016vdi; @Reynolds:2016rvl] proposed that we should add an additional term into the boundary term $$\label{nullbd4}
I_{\mathcal{N}_j}=I_{\mathcal{N}_j}^{(3)}:=-\frac{\text{sign}(\mathcal{N}_j)}{8\pi}\int_{\mathcal{N}_j}{\text{d}}\lambda{\text{d}}^{d-2}x\sqrt{|\sigma|}(\kappa+L_0)\,,$$ where $L_0=\Theta\ln(|\Theta|/{\ell_{\text{AdS}}})$ if $\mathcal{N}$ lies to the future boundary of $\mathcal{M}$. Otherwise, $L_0=-\Theta\ln (|\Theta|/{\ell_{\text{AdS}}})$. In this paper, we will use this total boundary term.
The joint term $I_{\mathcal{J}}$ was first found by Ref. [@PhysRevD.47.3275] and then confirmed by Refs. [@Lehner:2016vdi; @Jubb:2016qzt] again recently by different methods. As the CA conjecture will not have such kind of joints, we will not give the detailed form for this kind of joint terms. The joint term for the case that there is at least one null boundary was first found by Ref. [@Lehner:2016vdi], which is in general expressed as $$\label{nulljoint1}
I_{\mathcal{J}'}=\frac{\text{sign}(\mathcal{J'})}{8\pi}\int_{\mathcal{J'}}{\text{d}}^{d-1}x\sqrt{\sigma}a\,.$$ Here $\sigma$ is the determinant of the induced metric on the joint $\mathcal{J'}$. According to the properties of the intersectional surface, the term $a$ can be computed as $$\label{expressa}
a=\left\{
\begin{split}
&\ln(|n^I k_I|)\,,\\
&\ln(|k^I\bar{k}_I|/2)\,,
\end{split}
\right.$$ where $n^I$ is the unit normal vector (outward/future directed) for the non-null intersecting boundary, and $\bar{k}^I$ is the other null normal vector (future directed) for the null intersecting boundary. The value of $\text{sign}(\mathcal{J'})=\pm1$, which can be assigned as follows: “+1” appears only when the WDW patch appears in the future/past of the null boundary component and the joint is at the past/future end of the null boundary fragments.
### Complexity potential
The metric for the general AdS$_{d+1}$($d\geq2$) black hole is $$\label{metricBTZ}
{\text{d}}s^2=-r^2f(r){\text{d}}t^2+\frac{{\text{d}}r^2}{r^2f(r)}+r^2{\text{d}}\Sigma_{d-1}^2 \,,$$ where ${\text{d}}\Sigma_{d-1}^2={\ell_{\text{AdS}}}^{-2}\sum_{i=1}^{d-1}{\text{d}}x_i^2$ is the $(d-1)$ dimensional line element and $\Sigma_{d-1}$ is the volume of the conformal boundary of the AdS black hole. The function $f(r)$ reads $$\label{BTZfr}
f(r)=\frac1{{\ell_{\text{AdS}}}^2}\left(1-\frac{r_h^d}{r^d}\right)\,.$$ The physical total mass (ADM mass) and temperature of this system are $$\label{massAdS}
M=\frac{r_h^d(d-1)\Sigma_{d-1}}{16\pi{\ell_{\text{AdS}}}^2}\,, \qquad T=\frac{r_hd}{4\pi{\ell_{\text{AdS}}}^2}\,.$$
The Penrose diagram and the WDW patch are shown in Fig. \[SAdS1\]. The time direction of the right boundary is the same as the coordinate time $t$ but the time of the left boundary is opposite to the coordinate time $t$. As the space-time has time translation symmetry the on-shell action only depends on the value of $t_L+t_R$.
![Penrose diagrams and WDW patches for AdS$_{d+1}$ black hole ($d\geq3$) when $|t_R|<\Delta t_c$ (left panel) and $|t_R|>\Delta t_c$ (right panel). In the left panel, the past null sheets will meet the singularity at $B_1$ and $B_2$ respectively. In the right panel, the past null sheets will meet each other at $B$ with $r=r_0\in(0,r_h)$. []{data-label="SAdS1"}](SAdS1.pdf "fig:"){width=".49\textwidth"} ![Penrose diagrams and WDW patches for AdS$_{d+1}$ black hole ($d\geq3$) when $|t_R|<\Delta t_c$ (left panel) and $|t_R|>\Delta t_c$ (right panel). In the left panel, the past null sheets will meet the singularity at $B_1$ and $B_2$ respectively. In the right panel, the past null sheets will meet each other at $B$ with $r=r_0\in(0,r_h)$. []{data-label="SAdS1"}](SAdS2.pdf "fig:"){width=".49\textwidth"}
By this property we can fix $t_L=0$ and only study how the complexity depends on the value of $t_R$. In addition, thanks to the time reversal symmetry of the black hole we only consdier the case of $t_R>0$.
In Fig. \[SAdS1\] we see that there is a critical time $\Delta t_c$ distinguishing the left and right panel. Depending on the relationship between $t_R$ and $\Delta t_c$, there are two different types for the WDW patches. One is the case that $|t_R|<\Delta t_c$ shown in the left panel of Fig. \[SAdS1\], where the two future and past null sheets all meet the singularities. The other one is the case that $|t_R|>\Delta t_c$ shown in the right panel of Fig. \[SAdS1\], where the future directed null sheets coming from the boundaries will first meet the singularity $r=0$ but the past directed null sheets coming from the boundaries will meet each other in the inner region of black hole.
Let us introduce the infalling coordinate $v$ and outgoing coordinate $u$ as $$\label{uvcoord1}
v=t+r^*,~~~u=t-r^*\,,$$ where $$\label{AdSrstar}
r^*(r)=\int^{r}\frac{{\text{d}}x}{x^2f(x)}\,.$$ The null dual normal vector field for the null boundaries $AB$ and $CD$ is $k_\mu=-[({\text{d}}t)_\mu+r^{-2}f^{-1}({\text{d}}r)_\mu]$ and the null normal vector field for the null boundaries $BC$ and $AE$ is $\bar{k}_\mu=-[({\text{d}}t)_\mu-r^{-2}f^{-1}({\text{d}}r)_\mu]$. These two null vector are affinely parameterized. The integration in Eq. can be expressed in terms of the hypergeometrical function $$\label{solverstar1}
r^*(r)=\frac{{\ell_{\text{AdS}}}^2}{r}\left[_2F_1\left(1,-\frac{1}{d};1-\frac{1}{d}; \frac{r^d}{r_h^d}\right)-1\right]\,, \qquad r<r_h\,,$$ and $$\label{solverstar2}
r^*(r)=\frac{{\ell_{\text{AdS}}}^2}{r_h}\left[\cot\frac{\pi}{d}-\frac{r_h}{(r^d-r_h^d)^{1/d}} {_2}F_1\left(\frac1d,\frac1d;1+\frac1d;\frac{r_h^d}{r_h^d-r^d}\right)-1\right]\,, \qquad r>r_h\,.$$ The value of $\Delta t_c$ is given by, $$\label{deltatcvalue}
\Delta t_c:=2[r^*(\infty)-r^*(0)]=\frac{2{\ell_{\text{AdS}}}^2}{r_h}\frac{\pi}{d}\cot\frac{\pi}{d}\,.$$ To regularize the WDW patch, we assume the AdS boundaries are located at $r=r_m\gg{\ell_{\text{AdS}}}$. One can see from Fig. \[SAdS1\] that when $t_R\leq\Delta t_c$, the WDW patch is the same as the one of $t_R=0$. This means that the corresponding TFD state is the same one for $t_R\in(0,\Delta t_c)$. So we have $$\label{CAt1}
|\text{TFD}(T,t_R)\rangle= |\text{TFD}(T,0)\rangle,~~\text{if}~|t_R|\leq\Delta t_c\,,$$ and $$\label{CAt10}
\mathcal{C}_{\text{A,ren}}(t_R)=\mathcal{C}_{\text{A,ren},0}\,,$$ where $\mathcal{C}_{\text{A,ren}}(t_R)$ is the renormalized complexity potential for given $t_R$ and $\mathcal{C}_{\text{A,ren},0}$ is its value when $t_R=0$. The value of $\mathcal{C}_{\text{A,ren},0}$ has been given by Ref. [@Chapman:2016hwi; @Kim:2017lrw], $$\label{CAregs0}
\mathcal{C}_{\text{A,ren},0}=\frac{d-2}{d-1}\cot\left(\frac{\pi}{d}\right)\frac{M}{\hbar T}\,.$$
The TFD state begins to evolve after $|t_R+t_L|>\Delta t_c$. In this case, the two past null sheets will meet each other at the joint $B$ with $r=r_0\in(0,r_h)$. We can obtain the equations for all the null boundaries, which are $$\label{lineAB2}
\begin{split}
AB&:~r^*_m=t+r^*(r)\,,\\
CD&:~t_R+r^*_m=t+r^*(r)\,,\\
BC&:~t_R-r^*_m=t-r^*(r)\,,\\
AE&:~-r^*_m=t-r^*(r)\,.
\end{split}$$ By the equations for $AB$ and $BC$, we find that the past null sheets will meet each other at $r=r_0$, where $r_0$ is defined by the following equation $$\label{eqforr0}
r^*(r_0)=r^*_m-\frac{t_R}2\,, \qquad r_0<r_h\,.$$ Eq. has a solution only when $t_R\geq\Delta t_c$. By using Eqs. , and taking $r_m\rightarrow\infty$, we find $r_0$ is given as $$\label{valuer0s1}
\frac{{\ell_{\text{AdS}}}^2}{r_0}\left[_2F_1\left(1,-\frac{1}{d};1-\frac{1}{d}; \frac{r^d_0}{r_h^d}\right)-1\right]=\frac{{\ell_{\text{AdS}}}^2}{r_h}\frac{\pi}{d}\cot\frac{\pi}{d}-\frac{t_R}2\,.$$ For the case $d\geq2$, this equation can be solved only numerically.
Now let us first compute the bulk contribution from the Einstein-Hilbert action. By using the Einstein’s equation, we can write this term as $$\label{intbulk1}
I_{\text{bulk}}=\frac1{16\pi}\int{\text{d}}^{d+1}x\sqrt{-g}\left[R+\frac{d(d-1)}{{\ell_{\text{AdS}}}^2}\right]=-\frac{\Sigma_{d-1}d}{8\pi{\ell_{\text{AdS}}}^2}\iint r^{d-1}{\text{d}}r{\text{d}}t\,.$$ According to the right panel of Fig. \[SAdS1\], the bulk term can be splited into four parts. In the region I, for fixed $r$, the upper and inferior limits for $t$ in the integration are given by the line equations $AB_1$ and $AE$, respectively. Thus we find $$\label{intbulk2a}
I_{\text{bulk,I}}=-\frac{\Sigma_{d-1} d}{8\pi{\ell_{\text{AdS}}}^2}\int_{r_h}^{r_m} r^{d-1}{\text{d}}r\int^{r^*_m-r^*(r)}_{r^*(r)-r^*_m}{\text{d}}t\,.$$ At the region II, the coordinate $r$ can run from $r_0$ to $r_h$. For every given $r$, the upper and inferior limits for $t$ in the integration are given by the line equations $AB$ and $BC$, respectively. Then we have, $$\label{intbulk2b2}
I_{\text{bulk,II}}=-\frac{\Sigma_{d-1} d}{8\pi{\ell_{\text{AdS}}}^2}\int_{r_0}^{r_h} r^{d-1}{\text{d}}r\int^{r^*_m-r^*(r)}_{t_R+r^*(r)-r^*_m}{\text{d}}t\,.$$ For the region III and region IV, we have $$\label{intbulk2c}
I_{\text{bulk,III}}=-\frac{\Sigma_{d-1} d}{8\pi{\ell_{\text{AdS}}}^2}\int_{r_h}^{r_m} r^{d-1}{\text{d}}r\int^{t_R+r^*_m-r^*(r)}_{t_R+r^*(r)-r^*_m}{\text{d}}t=I_{\text{bulk,I}} \,,$$ and $$\label{intbulk2d0}
I_{\text{bulk,IV}}=-\frac{\Sigma_{d-1} d}{8\pi{\ell_{\text{AdS}}}^2}\int_{0}^{r_h} r^{d-1}{\text{d}}r\int^{t_R+r^*_m-r^*(r)}_{r^*(r)-r^*_m}{\text{d}}t\,.$$ Combining Eqs. , , and we find that $$\label{intbulk2d}
\begin{split}
I_{\text{bulk}}(t_R)&=I_{\text{bulk,I}}+I_{\text{bulk,II}}+I_{\text{bulk,III}}+I_{\text{bulk,IV}}\\
&=I_{\text{bulk},0}+\frac{\Sigma_{d-1} d}{8\pi{\ell_{\text{AdS}}}^2}\int_0^{r_0}r^{d-1}{\text{d}}r\int_{t_R+r^*(r)-r^*_m}^{r^*_m-r^*(r)}{\text{d}}t\\
&=I_{\text{bulk},0}+\frac{\Sigma_{d-1}}{4\pi {\ell_{\text{AdS}}}^2}\left[\left.r^d\left(r^*_m-r^*-\frac{t_R}2\right)\right|_0^{r_0}+\int_0^{r_0}r^{d-2}f^{-1}(r){\text{d}}r\right]\\
&=I_{\text{bulk},0}+\frac{\Sigma_{d-1}}{4\pi{\ell_{\text{AdS}}}^2}\int_0^{r_0}r^{d-2}f^{-1}(r){\text{d}}r\,.
\end{split}$$ Here $I_{\text{bulk},0}$ is the bulk on-shell action in the case that $t_R=0$.
Let us turn to the boundary terms. There are four null boundaries and a space-like boundary. As the normal vector fields are affinely parameterized, the only possible boundary terms for the null boundaries are the integration of $L_0$. Based on the results in the Ref. [@Kim:2017lrw], we see that $$\label{IlambdaBZT1}
\begin{split}
I_{\mathcal{N}}(t_R)&= I_{\mathcal{N},0}+\frac{(d-1)\Sigma_{d-1}}{4\pi}\int^0_{r_0}r^{d-2}\left\{1+\ln\left[\frac{(d-1){\ell_{\text{AdS}}}}{r}\right]\right\}{\text{d}}r\\
&=I_{\mathcal{N},0}-\frac{\Sigma_{d-1}}{4\pi}\left\{\ln\left[\frac{(d-1){\ell_{\text{AdS}}}}{r_0}\right]+\frac{1}{(d-1)}\right\}r^{d-1}_0\,. \end{split}$$ Here $I_{\mathcal{N},0}$ is the value of $I_{\mathcal{N}}(t_R)$ at $t_R=0$. The other boundary term comes from the space-like boundary $DE$. This boundary term is given by the GHY boundary term which reads $$\label{BTZGHY}
I_{\text{GHY}}(t_R)=\frac{\Sigma_{d-1}r_h^{d}d}{16\pi{\ell_{\text{AdS}}}^2}[t_R+2r^*_m-2r^*(0)]\,,$$ based on Ref. [@Lehner:2016vdi].
Comparing with the case of $t_R=0$, we only have one new additional null-null joint term at $r_0$ which is $$\label{BTZjoint}
I_{\text{joints}}(t_R)=I_{\text{joints},0}-\frac{r_0^{d-1}\Sigma_{d-1}}{8\pi}\ln[-r_0^2f(r_0)]\,,$$ where $I_{\text{joints},0}$ is the value of $I_{\text{joints}}(t_R)$ at $t_R=0$.
Because of the time translation symmetry, the surface counterterms are the same as the cases of $t_R=0$. Thus the renormalized holographic complexity potential is $$\label{totalCAdS}
\begin{split}
\mathcal{C}_{\text{A,ren}}(T,t_R)&=\mathcal{C}_{\text{A,ren},0}+\frac{\Sigma_{d-1}}{4\pi^2\hbar}\left\{{\ell_{\text{AdS}}}^{-2}\int_0^{r_0}r^{d-2}f^{-1}(r){\text{d}}r+\frac{r_h^{d}(t_R-\Delta t_c)d}{4{\ell_{\text{AdS}}}^2}\right.\\
&\left.-\frac{r_0^{d-1}}2\left[\ln(-r_0^2f(r_0))+2\ln\left(\frac{(d-1){\ell_{\text{AdS}}}}{r_0}\right)+\frac{2}{d-1}\right]\right\}\,,
\end{split}$$ where $r_0$ is the function of $t_R$ and defined by the Eq. with $r_m\rightarrow\infty$. The value of $\mathcal{C}_{\text{A,ren},0}$ is given by Eq. .
### Time dependnent complexity
For convenience we choose $y=r_0/r_h$ and $x=({\ell_{\text{AdS}}}/r_h)^{d-1}$ as two free parameters of complexity potential. In this new coordinate, the state $|\text{TFD}(T,t_R)\rangle$ is $|\text{TFD}(x,y)\rangle$. Let us define a dimensionless renormalized complexity potential $G(x,y)$ such that, $$\label{defineGxs}
\mathcal{C}_{\text{A,ren}}(x,y)=\frac{{\ell_{\text{AdS}}}^{d-1}\Sigma_{d-1}}{4\pi^2\hbar}G(x,y)\,.$$ Comparing with Eq. we find $$\label{totalCAdS2}
G(x,y)=x^{-1}\left[\pi\frac{d-2}{d}\cot\frac{\pi}{d}+h(y)\right]\,,
$$ with $$\label{definhs}
h(y)=\int_0^{y}s^{d-2}f_1^{-1}(s){\text{d}}s+\frac{f_2(y)d}{2}-\frac{y^{d-1}}2\left[\ln(-f_1(y))+2\ln(d-1)+\frac{2}{d-1}\right] \,,$$ and $$\label{definf1f2}
f_1(y)=1-\frac1{y^d},~~f_2(y)=\frac{\pi}{d}\cot\frac{\pi}{d}-\frac{1}{y}\left[_2F_1(1,-1/d;1-1/d;y^d)-1\right]\,.$$ When $d\geq3$, the vacuum state is given by the parameter $x=0,y=0$. Suppose that $x=x_0$ and $y=y_0$ stand for an arbitrary TFD state. According to Eq. , the minimal length connecting $(0,0)$ and $(x_0,y_0)$ is given by $|\mathcal{C}_{\text{A,ren}}(0,0)-\mathcal{C}_{\text{A,ren}}(x_0,y_0)|=|\mathcal{C}_{\text{A,ren}}(x_0,y_0)|$. However, when $d=2$, as Ref. [@Chapman:2016hwi] suggested, the corresponding vacuum state is not the one of $r_h=0$. Instead, the vacuum state is given by $f(r)=1/{\ell_{\text{AdS}}}^2+1/r^2$. The renormalized holographic complexity potential for this vacuum state is $$\label{CAregs0b}
\frac{\pi\hbar\mathcal{C}_{\text{A,BTZ,vac}}}{\Sigma_{1}}=-\frac{\pi{\ell_{\text{AdS}}}}2\,.$$ Finally, we obtain the following complexity between the TFD state and its vacuum state $$\label{AdSCompCA1}
\begin{split}
\mathcal{C}(|\text{TFD}(T,t_R)\rangle,|0\rangle)&=\frac{r_h^{d-1}\Sigma_{d-1}}{4\pi^2\hbar}[G(x_0,y_0)+2\pi^2\delta_{2,d}]\\
&=\frac{d}{\pi^2(d-1)}[G(x_0,y_0)+2\pi^2\delta_{2,d}]\frac{M}{\hbar T}\,,
\end{split}$$ with the relationships $T=x_0^{1/(d-1)}d/(4\pi{\ell_{\text{AdS}}})$ and $t_R=2{\ell_{\text{AdS}}}x^{-1/(d-1)}f_2(y_0)$. The absolute symbol has been dropped because the right side of Eq. is always positive when $d>2$ (we confirmed it numerically from $d=2$ to $d=10$.). The growth rate can be obtained directly from this expression, which reads[^8] $$\label{dCdtAdS1}
\begin{split}
&\frac{{\text{d}}}{{\text{d}}t_R}\mathcal{C}(|\text{TFD}(T,t_R)\rangle,|0\rangle)\\
=&\frac{2M}{\pi\hbar}\left\{1+\frac{y_0^df_1(y_0)}{2\pi\hbar}\left[\ln(-f_1(y_0))+2\ln(d-1)\right)\right\}\,.
\end{split}$$
![The complexity $\mathcal{C}(|\text{TFD}(0,t_R)\rangle,|0\rangle)$ and its growth rate when $d>2$. $\mathcal{C}_0$ is the complexity when $t_R=\Delta t_c$ and $\dot{\mathcal{C}}_m=2M/\pi\hbar$. Higher dimensional cases give the similar results. []{data-label="FigAdS1"}](dcdtAdS1.pdf "fig:"){width=".49\textwidth"} ![The complexity $\mathcal{C}(|\text{TFD}(0,t_R)\rangle,|0\rangle)$ and its growth rate when $d>2$. $\mathcal{C}_0$ is the complexity when $t_R=\Delta t_c$ and $\dot{\mathcal{C}}_m=2M/\pi\hbar$. Higher dimensional cases give the similar results. []{data-label="FigAdS1"}](dcdtAdS2.pdf "fig:"){width=".49\textwidth"}
![The complexity $\mathcal{C}(|\text{TFD}(0,t_R)\rangle,|0\rangle)$ and their growth rates for the BTZ black hole. $\mathcal{C}_0$ is the complexity when $t_R=0$ and $\dot{\mathcal{C}}_m=2M/\pi\hbar$.[]{data-label="FigBTZ1"}](dcdtBTZ1.pdf "fig:"){width=".49\textwidth"} ![The complexity $\mathcal{C}(|\text{TFD}(0,t_R)\rangle,|0\rangle)$ and their growth rates for the BTZ black hole. $\mathcal{C}_0$ is the complexity when $t_R=0$ and $\dot{\mathcal{C}}_m=2M/\pi\hbar$.[]{data-label="FigBTZ1"}](dcdtBTZ2.pdf "fig:"){width=".49\textwidth"}
The time evolution of the complexity $\mathcal{C}(|\text{TFD}(T,t_R)\rangle,|0\rangle)$ and its growth rate are shown in Fig. \[FigAdS1\] and Fig. \[FigBTZ1\]. We find that the relationship between the complexity and $t_R$ is not monotonic. When $t_R$ runs from $\Delta t_c$ to infinite, the value of complexity will first decrease and then increase, so there is a minimal value. For the case that $t_R\rightarrow\Delta t_c$, we have $y_0\rightarrow0^+$. Thus Eq. shows that $$\label{dCdtAdS2}
\frac{{\text{d}}}{{\text{d}}t_R}\mathcal{C}(|\text{TFD}(T,t_R)\rangle,|0\rangle)\rightarrow-\infty,~~~\text{as}~t_R\rightarrow\Delta t_c\,.$$ In the late limit $t_R\rightarrow\infty$, it saturates to the Lloyd’s bound, $$\label{boundAdSa2}
\lim_{t_R\rightarrow\infty}\frac{{\text{d}}}{{\text{d}}t_R}\mathcal{C}(|\text{TFD}(0,t_R)\rangle,|0\rangle)=\lim_{t_R\rightarrow\infty}\dot{\mathcal{C}}_{\text{A,ren}}=\lim_{y_0\rightarrow 1}\dot{\mathcal{C}}_{\text{A,ren}}=\frac{2M}{\pi\hbar}\,.$$ From Fig. \[FigBTZ1\] it is clear that the Lloyd’s bound is violated in the intermediate and large time for the BTZ black hole ($d=2$), but it is not so clear if this is the case also for $d>2$ from Fig. \[FigAdS1\]. To check it we consider the the subleading term from Eq. in the late time limit: $$\label{dCdtAdSs2}
\frac{{\text{d}}}{{\text{d}}t_R}\mathcal{C}(|\text{TFD}(T,t_R)\rangle,|0\rangle)-\frac{2M}{\pi\hbar}=\frac{2M}{\pi\hbar}\frac{y_0^df_1(y_0)}{2\pi\hbar}\ln[-f_1(y_0)]+\mathcal{O}(y_0-1)\,.$$ As $y_0\in(0,1)$, $f_1(y_0)=1-1/y_0^d<0$. In the late time limit, $y_0\rightarrow1^-$ and the first term in the right-side of Eq. dominant, which means that the subleading term is positive. Thus the CA conjecture violates the Lloyd’s bound slightly in the large time.
CV conjecture
-------------
\(I) at (0,0) ;
\(I) +(135:4) coordinate (Iltop) +(-135:4) coordinate (Ilbot) +(0:0) coordinate (Ilmid) +(45:4) coordinate (Irtop) +(-45:4) coordinate (Irbot) +(0:0) coordinate (Irmid) ;
(Iltop) – node\[pos=0.5, below, sloped\] [$\rho=\infty$]{} (Ilbot) – (Ilmid) – (Irtop) – node\[pos=0.5, above, sloped\] [$\rho=\infty$]{} (Irbot) – node\[midway, below, sloped\] (Irmid) – node\[midway, below, sloped\] [$\rho=0$]{} (Iltop) – cycle;
(Iltop) to\[out=-15,in=+195,looseness=1.2\] node\[midway, above, inner sep=2mm\] [$\kappa=\pi/d$]{} (Irtop);
(Ilbot) to\[out=15,in=-195,looseness=1.2\] (Irbot);
(-2.828,2.1) to\[out=0,in=150,looseness=1.3\] node\[pos=-0.15,black\] [$\tilde{t}_B$]{} (-1.828,1.8); (2.828,2.1) to\[out=180,in=30,looseness=1.3\] node\[pos=-0.15,black\] [$\tilde{t}_B$]{} (1.828,1.8); (-1.928,1.86) coordinate to\[out=-30,in=210,looseness=1.36\] node\[black,midway, below\] [$\kappa_0$]{} (1.928,1.86) coordinate; (0,1.1) circle (1.8pt);
($(Iltop)!.0!(Ilbot)$) coordinate to\[out=-40,in=220,looseness=1.4\] node\[black,midway, above\] [$\kappa_m=\pi/2d$]{} ($(Irtop)!.0!(Irbot)$) coordinate; (-2.828,0) – (2.828,0); (0,1.34) circle (1.8pt);
In this subsection, we compute the time-dependent complexity of the AdS$_{d+1}$ Schwarzschild planar black holes in the CV conjecture. Let us rewrite the metric in the following form
$${\text{d}}s^{2}={\ell_{\text{AdS}}}^2(-g^{2}(\rho){\text{d}}\tilde{t}^{2}+{\text{d}}\rho^{2}+h^{2}(\rho)\sum_{i=1}^{d-1}{\text{d}}\tilde{x}_i^2\label{eq:SAdSplanarBB}),$$
where $$h(\rho)=\left(\cosh\frac{d\rho}{2}\right)^{2/d},\ \ g(\rho)=h(\rho)\tanh\frac{d\rho}{2}.$$ Here we introduced dimensionless variables $\tilde{t}=\frac{r_h}{{\ell_{\text{AdS}}}^2}t$, $\tilde{x}_{i}=\frac{r_h}{{\ell_{\text{AdS}}}^2}x_{i}$, $\tilde{r}=\frac{r}{r_h}$ and performed a coordinate transformation ${\text{d}}\rho=\frac{{\text{d}}\tilde{r}}{\tilde{r}\sqrt{1-1/\tilde{r}^{d}}}$. The Penrose diagram is shown in Fig. \[FigAdS\]. Similarly to the CA case, the renormalized complexity potential only depends on $t_L+t_R$.
We can continue (\[eq:SAdSplanarBB\]) into the interior region of Fig. \[FigAdS\] by setting $\rho=i\kappa$ and $\tilde{t}_I=\tilde{t}+i\frac{\pi}{2}$. For the case $\tilde{t}_B\equiv \tilde{t}_R=\tilde{t}_L$, the maximal volume surface is given by the blue line in Fig. \[FigAdS\]. The upper red dotted line is for $\tilde{t}_B=\infty$ and the middle red dotted line is for $\tilde{t}_B=0$. The corresponding volume of this codimension-one surface is described by the following integration $$\begin{aligned}
\label{eq:vol}
V= & \tilde{\Sigma}_{d-1}{\ell_{\text{AdS}}}^d\int h(\rho)^{d-1}\sqrt{-g^{2}(\rho)+(\partial\rho/\partial \tilde{t})^{2}}{\text{d}}\tilde{t},\end{aligned}$$ where $\tilde{\Sigma}_{d-1}\equiv\int d^{d-1} \tilde{x}$ is the volume of the spatial geometry. The volume can be maximized following [@Hartman:2013qma; @MIyaji:2015mia; @Sinamuli:2016rms].
In principle, we should solve the Euler-Lagrangian equation of to find $\rho(\tilde{t})$. Alternatively, following [@Hartman:2013qma] we may find the first integral of the equation of motion of . In other words, because the integrand of is time independent the Hamiltonian is conserved: $$\begin{aligned}
\label{}
\mathcal{H}= & \frac{\partial \mathcal{L}}{\partial \rho'(\tilde{t})}-\mathcal{L}=const.\,,\end{aligned}$$ which yields $$\begin{aligned}
\label{eq:energy}
\frac{g^2 h^{d-1}}{\sqrt{-g^2+(\partial\rho/\partial \tilde{t})^{2}}}= & i g_0 h_0^{d-1}\,,\end{aligned}$$ where $h_0 := h(i\kappa_0)$ and $g_0:= g(i\kappa_0)$ with $\kappa_{0}$ ($0<\kappa_{0}<\frac{\pi}{2d}$) satisfying $\frac{\partial\kappa}{\partial \tilde{t}}| _{\kappa=\kappa_0}=0$. From (\[eq:energy\]), we can write the time $\tilde{t}_B$ in terms of $\kappa_0$ $$\begin{split}
\tilde{t}_B=&\int_{\epsilon}^{\kappa_{0}}\frac{{\text{d}}\kappa}{\left(\cos\frac{d\kappa}{2}\right)^{\frac{2}{d}}\tan\frac{d\kappa}{2}\sqrt{1-\frac{\sin^{2}d\kappa}{\sin^{2}d\kappa_{0}}}}-\int_{\epsilon}^{\infty}\frac{{\text{d}}\rho}{\left(\cosh\frac{d\rho}{2}\right)^{\frac{2}{d}}\tanh\frac{d\rho}{2}\sqrt{1+\frac{\sinh^{2}d\rho}{\sin^{2}d\kappa_{0}}}}\\
=&\int_{0}^{\kappa_{0}}\left(
\frac{\left(\cos\frac{d\kappa}{2}\right)^{-\frac{2}{d}}\cot\frac{d\kappa}{2}}{\sqrt{1-\csc^{2}d\kappa_{0}\sin^{2}d\kappa}}-\frac{\left(\cosh\frac{d\kappa}{2}\right)^{-\frac{2}{d}}\coth\frac{d\kappa}{2}}{\sqrt{1+\csc^{2}d\kappa_{0}\sinh^{2}d\kappa}}\right){\text{d}}\kappa\\
&-\int_{\kappa_{0}}^{\infty}
\frac{\left(\cosh\frac{d\rho}{2}\right)^{-\frac{2}{d}}\coth\frac{d\rho}{2}}{\sqrt{1+\csc^{2}d\kappa_{0}\sinh^{2}d\rho}}{\text{d}}\rho\,.
\end{split}$$ Substituting (\[eq:energy\]) into (\[eq:vol\]), the maximum volume can be expressed in terms of the parameter $\kappa_{0}$, $$V=2\tilde{\Sigma}_{d-1} {\ell_{\text{AdS}}}^d \left(\int_{0}^{\kappa_{0}}\frac{\left(\cos\frac{d\kappa}{2}\right)^{\frac{2(d-1)}{d}}}{\sqrt{\frac{\sin^{2}d\kappa_{0}}{\sin^{2}d\kappa}-1}}{\text{d}}\kappa
+\int_{0}^{\rho_{\infty}}\frac{\left(\cosh\frac{d\rho}{2}\right)^{\frac{2(d-1)}{d}}}{\sqrt{1+\frac{\sin^{2}d\kappa_{0}}{\sinh^{2}d\rho}}}{\text{d}}\rho\right).$$ Here we have introduced the UV cut off $\rho_{\infty}$, IR cut off $\epsilon$ and the factor 2 comes from the symmetry of Fig. \[FigAdS\].
To evaluate the renormalized holographic complexity potantial, we will subtract the surface counterterms which were obtained for $d\ge2$ in Ref. [@Kim:2017lrw]: $$\label{}
\begin{split}
V^{(1)}_{\text{ct}}&=\frac{{\ell_{\text{AdS}}}}{d-1}\int_Bd^{d-1} \tilde{x}\sqrt{\sigma}=\frac{\tilde{\Sigma}_{d-1} {\ell_{\text{AdS}}}^d}{d-1}\left(\cosh\frac{d\rho_{\infty}}{2}\right)^{2(d-1)/d}\,,\\
V^{(n)}_{\text{ct}}&=0,~~~~n>1,
\end{split}$$ where $\sigma$ is the induced metric of the time slice on the boundary. Hence the renormalized holographic complexity potential can be written as $$\label{rcv}
\begin{split}
&\mathcal{C}_{\text{V,ren}}
=\frac{1}{\ell}\lim_{\delta\rightarrow0}(V-2V^{(1)}_{\text{ct}})\\
&= \frac{2\tilde{\Sigma}_{d-1}{\ell_{\text{AdS}}}^d}{\ell}\left( \int_{0}^{\kappa_{0}}\frac{\left(\cos\frac{d\kappa}{2}\right)^{\frac{2(d-1)}{d}}}{\sqrt{\frac{\sin^{2}d\kappa_{0}}{\sin^{2}d\kappa}-1}}{\text{d}}\kappa
+ \int_{0}^{\infty}\left(\frac{\left(\cosh\frac{d\rho}{2}\right)^{\frac{2(d-1)}{d}}}{\sqrt{1+\frac{\sin^{2}d\kappa_{0}}{\sinh^{2}d\rho}}}-\frac{\cosh\frac{d\rho}{2}}{\left(\sinh\frac{d\rho}{2}\right)^{\dfrac{d}{d-2}}}\right){\text{d}}\rho
\right)\,.
\end{split}$$ As in the CA conjecture, the renormalized holographic complexity potential of Schwarzschild AdS black holes at the zero temperature limit are all zeros, the complexity between $|\text{TFD}(T,t_L+t_R)\rangle$ and $|\text{TFD}(0,0)\rangle$ then is[^9] $$\label{CVcomple1}
\mathcal{C}(|\text{TFD}(T,t_L+t_R)\rangle,|0\rangle)=\mathcal{C}_{\text{V,ren}}\,.$$ However, for the BTZ black hole, the vacuum state is not the one of zero horizon. We have to choose the solution $f(r)=1/{\ell_{\text{AdS}}}^2+1/r^2$ for the vacuum state. Then the renormalized holographic complexity potential of this vacuum state is given by $\mathcal{C}_{V,\text{BTZ,vac}}=-4\pi{\ell_{\text{AdS}}}^2/\ell$. Thus we have the following complexity for the BTZ black hole $$\label{CVcomple2}
\mathcal{C}(|\text{TFD}(T,t_L+t_R)\rangle,|0\rangle)=\mathcal{C}_{\text{V,ren}}+4\pi{\ell_{\text{AdS}}}^2/\ell\,.$$ Combining Eqs. and we have an expression $$\label{CVcomple0}
\mathcal{C}(|\text{TFD}(T,t_L+t_R)\rangle,|0\rangle)=\mathcal{C}_{\text{V,ren}}+4\pi\delta_{2,d}{\ell_{\text{AdS}}}^2/\ell\,.$$ The time evolution of the complexity $\mathcal{C}(|\text{TFD}(T,t_L+t_R)\rangle,|0\rangle)$ and its growth rate are shown in Fig. \[Figrcv\] where the relationship between the complexity and $t_L+t_R$ is monotonic contrary to the CA case.
In the early time limit ($\tilde{t}_B\rightarrow0$ or $\kappa_{0}\rightarrow0$ ), we have $$\tilde{t}_B=-\frac{\sin d\kappa_{0}}{2}\int_{0}^{\rho_{\infty}}\frac{{\text{d}}\rho}{\left(\cosh\frac{d\rho}{2}\right)^{\frac{2}{d}}\sinh^{2}\frac{d\rho}{2}},$$ so the complexity $\mathcal{C}(|\text{TFD}(T,t_L+t_R)\rangle,|0\rangle)$ can be written as $$\begin{aligned}
&\mathcal{C}(|\text{TFD}(T,t_L+t_R)\rangle,|0\rangle)=\mathcal{C}_{\text{V,ren}}+4\pi\delta_{2,d}{\ell_{\text{AdS}}}^2\nonumber\\
&\quad=\frac{\tilde{\Sigma}_{d-1}{\ell_{\text{AdS}}}^d}{\ell}\left(
\frac{\sqrt{\pi}(d-2)\Gamma(1+\frac{1}{d})}{(d-1)\Gamma(\frac{1}{2}+\frac{1}{d})}
+\frac{r_h^2d^2\Gamma(\frac{1}{2}+\frac{1}{d})}{8{\ell_{\text{AdS}}}^4\sqrt{\pi}\Gamma(\frac{1}{d})}(t_L+t_R)^2+\dots
\right)+4\pi\delta_{2,d}{\ell_{\text{AdS}}}^2/\ell.\end{aligned}$$ At $t_L+t_R=0$, $$\label{CVComMass}
\mathcal{C}(|\text{TFD}(T,0)\rangle,|0\rangle)= \frac{d\sqrt{\pi}(d-2)\Gamma(1+\frac{1}{d})}{\pi^2(d-1)\Gamma(\frac{1}{2}+\frac{1}{d})}\frac{M}{\hbar T}+\frac{(d-1){\ell_{\text{AdS}}}}{\pi\hbar}\delta_{2,d}\,,$$ where we use Eq. and take the length scale $\ell/{\ell_{\text{AdS}}}=4\pi^2\hbar/(d-1)$.
![The values of $\mathcal{C}(|\text{TFD}(T,t_L+t_R)\rangle,|0\rangle)$ and its growth rate when $d=2,3,4,5$. The higher dimensions give similar results. $\mathcal{C}_0$ is the complexity when $t_L+t_R=0$ and $\dot{\mathcal{C}}_f$ is the Lioyd’s bound of growth rate, which is given by Eq. .[]{data-label="Figrcv"}](cv1 "fig:"){width=".45\textwidth"} ![The values of $\mathcal{C}(|\text{TFD}(T,t_L+t_R)\rangle,|0\rangle)$ and its growth rate when $d=2,3,4,5$. The higher dimensions give similar results. $\mathcal{C}_0$ is the complexity when $t_L+t_R=0$ and $\dot{\mathcal{C}}_f$ is the Lioyd’s bound of growth rate, which is given by Eq. .[]{data-label="Figrcv"}](cv2 "fig:"){width=".45\textwidth"}
In the late time limit ($\tilde{t}_B\rightarrow\infty$ or $\kappa_{0}\rightarrow\kappa_m=\frac{\pi}{2d}$ ), the renormalized complexity becomes $$\begin{split}
&\mathcal{C}(|\text{TFD}(T,t_L+t_R)\rangle,|0\rangle) =\mathcal{C}_{\text{V,ren}} \\
&\quad =\frac{\tilde{\Sigma}_{d-1}{\ell_{\text{AdS}}}^d}{\ell}\left(
\frac{r_h}{2{\ell_{\text{AdS}}}^2}(t_L+t_R) -\int_{\epsilon}^{\frac{\pi}{2d}}\frac{\cos d\kappa~{\text{d}}\kappa} {\left(\cos\frac{d\kappa}{2}\right)^{2/d}\tan\frac{d\kappa}{2}} +\int_{\epsilon}^{\rho_{\infty}}\frac{\coth\frac{d\rho}{2}~{\text{d}}\rho}{\left(\cosh\frac{d\rho}{2}\right)^{2/d}}
-\dfrac{2}{d-1}\right)\\
&\quad =\frac{\tilde{\Sigma}_{d-1}{\ell_{\text{AdS}}}^d}{\ell}\left(\frac{r_h}{2{\ell_{\text{AdS}}}^2}(t_L+t_R)+\text{finite term}\right).
\end{split}$$ Thus the complexity growth rate in the late time limit is given by $$\label{dCdtAdS2CV}
\lim_{t_L+t_R\rightarrow\infty}\frac{{\text{d}}}{{\text{d}}(t_L+t_R)}\mathcal{C}(|\text{TFD}(T,t_L+t_R)\rangle,|0\rangle)=\frac{8\pi{\ell_{\text{AdS}}}M}{\ell(d-1)}\,,$$ where we use Eq. . If we take the length scale $\ell/{\ell_{\text{AdS}}}=4\pi^2\hbar/(d-1)$ we find that the Lloyd’s bound is satisfied in the CV conjecture: $$\label{dCdtAdS2CV}
\frac{{\text{d}}}{{\text{d}}(t_L+t_R)}\mathcal{C}(|\text{TFD}(T,t_L+t_R)\rangle,|0\rangle)<\frac{2M}{\pi\hbar}\,.$$ Numerical results show that this is the maximum value of the growth rate at all time, which is different from the CA conjecture.
Let us make a comparison for the complexity growth rates between the CA and CV conjectures. From Figs. \[FigAdS1\], \[FigBTZ1\] and \[Figrcv\] we see that at early time, two conjectures give different results. In the CA conjecture, we see that the complexity between the TFD state and the vacuum state does not change until $t_R+t_L=\Delta t_c$ for $d>2$. When $t>\Delta t_c$, the CA conjecture predicts the complexity will decrease first and then increase. The growth rate at $t=\Delta t_c$ is negative infinite. In the CV conjecture, we see that the complexity between the TFD state and the vacuum state always increase with the order of $t^2$ when $t$ is small. In the late time limit, two conjectures predict the complexity will increase linearly in $t$ and the slope is proportional to total mass $M$. However, in the large time region, the CA conjecture will approach to $2M$ from a larger value, so it violates the Lloyd’s bound. If we choose the length scale $\ell=4\pi^2\hbar{\ell_{\text{AdS}}}/(d-1)$, the CV conjecture satisfies the Lloyd’s bound at all time and saturates to the Lloyd’s bound in the late time limit.
Time dependent complexity of the TFD states: field theory approach {#fromQTF}
==================================================================
In this section we compute the complexity by the field theoretic methods proposed by Refs. [@Chapman:2017rqy; @Yang:2017nfn]. One is the FS method [@Chapman:2017rqy] based on the Fubini-Study metric and the other is the FG method [@Yang:2017nfn] based on the Finsler geometry. As a common basis of two methods we start with constructing a time-dependent TFD state for free field theory explicitly.
Time evolution of the TFD states {#tTFD}
--------------------------------
Both in the FS and FG methods, a crucial step is to find the transformation from vacuum state to a TFD state. We will follow the method proposed by Ref. [@Yang:2017nfn]. Let us consider a bosonic Hilbert space $\mathcal{H}$ and the occupation number representation. Suppose that $\hat{a}_{\vec{k}_i}$ and $\hat{a}^\dagger_{\vec{k}_i}$ are the annihilation and creation operators, which can annihilate or create a particle of momentum $\vec{k}_i$. The particle number density operator at momentum $\vec{k}_i$ is defined as $$\label{Nakak1}
\hat{N}_{\vec{k}_i}:=\hat{a}^\dagger_{\vec{k}_i}\hat{a}_{\vec{k}_i}\,.$$ As the particle number density operators for different momentum commute each other, their common eigenstates form a complete basis in the Hilbert space $\mathcal{H}$. Let us assume the momentum is discrete and introduce the notation, $$\label{Nbasis}
\prod_{i}|n_i,\vec{k}_i\rangle:=|n_0,\vec{k}_0\rangle|n_1,\vec{k}_1\rangle|n_2,\vec{k}_2\rangle\cdots$$ to stand for one common eigenstate for all the particle number operators. Here the product includes all the possible momentum values. Then any state $|\psi\rangle\in\mathcal{H}$ can be written in the following form $$\label{cn0n1n2}
|\psi\rangle=\sum_{n_0,n_1,\cdots=0}^\infty c_{n_0n_1\cdots}|n_0,\vec{k}_0\rangle|n_1,\vec{k}_1\rangle\cdots\,.$$ To construct a TFD state, one method is using a bogoliubov transformation from the vacuum state defined by any chosen annihilation operators [@Yang:2017nfn]. Let us first decompose the Hilbert space $\mathcal{H}=\mathcal{H}_L\times\mathcal{H}_R$ and define two groups of annihilation and creation operators {$\hat{a}_{\vec{k}_i}^L, \hat{a}_{\vec{k}_i}^R$} and {$\hat{a}_{\vec{k}_i}^{L\dagger}, \hat{a}_{\vec{k}_i}^{R\dagger}$}, which define a vacuum state $|0\rangle$: $$\label{defineA}
\hat{a}_{\vec{k}_i}^L|0\rangle=\hat{a}_{\vec{k}_i}^R|0\rangle=0,~~~\forall~\vec{k}_i\,.$$ However, the decomposition $\mathcal{H}=\mathcal{H}_L\times\mathcal{H}_R$ is not unique. One can choose another decomposition such that $\mathcal{H}=\mathcal{H}_D\times\mathcal{H}_U$ with the annihilation and creation operators {$\hat{b}_{\vec{k}_i}^D, \hat{b}_{\vec{k}_i}^U$} and {$\hat{b}_{\vec{k}_i}^{D\dagger}, \hat{b}_{\vec{k}_i}^{U\dagger}$}. The annihilation operators {$\hat{b}_{\vec{k}_i}^D, \hat{b}_{\vec{k}_i}^U$} also define a vacuum state $|B\rangle$ such that $\hat{b}_{\vec{k}_i}^U|B\rangle=\hat{b}_{\vec{k}_i}^D|B\rangle=0$ for $\forall~\vec{k}_i$. In general, two kinds of decompositions can have no special relationship. However, if we demand that they satisfy the following relationship $$\label{relab}
\left[\begin{matrix}
\hat{b}^D_{\vec{k}}\\
\hat{b}^{U\dagger}_{\vec{k}}
\end{matrix}\right]=c_{\vec{k}}\left[\begin{matrix}
1&-e^{-\pi\omega_{\vec{k}}/a}\\
-e^{-\pi\omega_{\vec{k}}/a}&1\end{matrix}\right]
\left[\begin{matrix}
\hat{a}^L_{\vec{k}}\\
\hat{a}^{R\dagger}_{\vec{k}}
\end{matrix}\right] \,,$$ for a normalization factor $c_{\vec{k}}$ which makes the operators {$\hat{b}_{\vec{k}_i}^D, \hat{b}_{\vec{k}_i}^U$} and {$\hat{b}_{\vec{k}_i}^{D\dagger}, \hat{b}_{\vec{k}_i}^{U\dagger}$} to satisfy the canonical commutation relation, Ref. [@Yang:2017nfn] and Ref. [@Crispino:2007eb] have proven that the vacuum state $|B\rangle$ can be expressed by $$\label{relBnk}
|B\rangle\propto\prod_{\vec{k}_i}\sum_{n=0}^\infty e^{-\pi n\omega_{\vec{k}_i}/a}|n,\vec{k}_i\rangle_L|n,\vec{k}_i\rangle_R\,.$$ There is a non-unitary operator $$\label{relaUaa}
\hat{U}^\dagger_a:=\prod_{\vec{k}_i}\exp\left(e^{-\pi\omega_{\vec{k}_i}/a}\hat{a}^{R\dagger}_{\vec{k}_i}\hat{a}^{L\dagger}_{\vec{k}_i}\right) =\exp\left[\int{\text{d}}^{d-1}ke^{-\pi\omega_{\vec{k}}/a}\hat{a}^{R\dagger}(\vec{k})\hat{a}^{L\dagger}(\vec{k})\right] \,,$$ which can convert the vacuum state $|0\rangle$ into the $|B\rangle$, i.e., $|B\rangle\propto\hat{U}^\dagger_a|0\rangle$. In the second equality, the discrete form has been converted into a continuous form.[^10]
In order to prove the state $|B\rangle$ is a TFD state, the easiest way is to find the reduced density matrix in the projected Hilbert space $\mathcal{H}_L$ or $\mathcal{H}_R$. Ref. [@Crispino:2007eb] has shown that the state in Eq. has the following reduced density matrix $$\label{inducedrho}
\hat{\rho}_L=\hat{\rho}_R=\frac1Z\prod_{\vec{k}_i}\sum_{n=0}^\infty\exp(-2\pi n\omega_{\vec{k}_i}/a)|n,\vec{k}_i\rangle\langle n,\vec{k}_i|\,,$$ where the factor $1/Z$ insures that $\text{Tr}\hat{\rho}_L=\text{Tr}\hat{\rho}_R=1$. We see that this is the density matrix for the system of free bosons with temperature $T=a/2\pi$. Thus, the projected states of $|B\rangle$ in Hilbert space $\mathcal{H}_L$ and $\mathcal{H}_R$ are two thermofield state. This shows that $|B\rangle$ is a TFD state with temperature $T=a/2\pi$.
The time dependent TFD state is given by Eq. so we have $$\label{timeTFD2}
|\text{TFD}(t_L,t_R)\rangle=\exp[-i(\hat{H}_Lt_L+\hat{H}_Rt_R)]|B\rangle\propto\exp[-i(\hat{H}_Lt_L+\hat{H}_Rt_R)]\hat{U}_a^\dagger|0\rangle\,.$$ The Hamiltonian $H_R$ and $H_L$ depend on the dynamic of dual boundary fields. For the free bosons, the Hamiltonian can be expressed by the creation and annihilation operators in the following way[^11] $$\label{Hamilton1}
\hat{H}_Lt_L+\hat{H}_Rt_R=\int{\text{d}}^{d-1}k\omega_{\vec{k}}(\hat{N}_{\vec{k}}^Rt_R+\hat{N}_{\vec{k}}^Lt_L)\,.$$ Although in general we cannot find a function $f(a,\vec{k})$ such that $\exp[-i(\hat{H}_Lt_L+\hat{H}_Rt_R)]\hat{U}_a^\dagger=\exp\left[\int{\text{d}}^{d-1}kf(a,\vec{k})\hat{a}^{R\dagger}(\vec{k})\hat{a}^{L\dagger}(\vec{k})\right]$, we can find a function $f(a,\vec{k})$ satisfying $$\label{timeTFD3}
|\text{TFD}(t_L,t_R)\rangle\propto\exp\left[\int{\text{d}}^{d-1}kf(a,\vec{k})\hat{a}^{R\dagger}(\vec{k})\hat{a}^{L\dagger}(\vec{k})\right]|0\rangle\,.$$ To see this, let us plug Eqs. and into Eq. in the discrete form. Thus we have $$\label{timeTFD4}
\begin{split}
|\text{TFD}(t_L,t_R)\rangle&\propto\prod_{\vec{k}_i}\sum_{n=0}^\infty e^{-\pi n\omega_{\vec{k}_i}/a}\exp[-i(\hat{N}_{\vec{k}}^Rt_R+\hat{N}_{\vec{k}}^Lt_L)]|n,\vec{k}_i\rangle_L|n,\vec{k}_i\rangle_R\\
&\propto\prod_{\vec{k}_i}\sum_{n=0}^\infty e^{-n\omega_{\vec{k}_i}[\pi/a+i(t_R+t_L)]}|n,\vec{k}_i\rangle_L|n,\vec{k}_i\rangle_R \,.
\end{split}$$ Now converting it into the continuous form, we obtain that $$\label{timeTFD5}
|\text{TFD}(t_L,t_R)\rangle\propto\hat{U}^\dagger_a(t_L,t_R)|0\rangle\,.$$ with the time dependent non-unitary operator[^12] $$\label{timeTFD5b}
\hat{U}^\dagger_a(t_L,t_R):=\exp\left[\int{\text{d}}^{d-1}ke^{-\omega_{\vec{k}}[\pi/a+i(t_R+t_L)]}\hat{a}^{R\dagger}(\vec{k})\hat{a}^{L\dagger}(\vec{k})\right]\,.$$ This shows that the function in Eq. is $$\label{funcfak}
f(a,\vec{k})=e^{-\omega_{\vec{k}}[\pi/a+i(t_R+t_L)]}\,.$$ We see that the time dependent TFD state only depends on $t_L+t_R$. For later use we also define $$\label{defrk}
r_{\vec{k}}:= \text{arctanh}[f(a,\vec{k})]=\text{arctanh}e^{-\omega_{\vec{k}}[\pi/a+i(t_R+t_L)]}\,.$$ Eqs. and will play crucial roles when we compute the complexity growth rate in the next subsections.
Fubini-Study (FS) metric {#FSmetric0}
------------------------
Let us first use the method proposed by Ref. [@Chapman:2017rqy] to compute the complexity between $|\text{TFD}(t_L,t_R)\rangle$ and the zero temperature limit vacuum state $|0\rangle$. This method is based on the Fubini-Study metric (see the appendix \[FSmetric\] for some basic introduction and refer to Ref. [@bengtsson2006geometry] for details) and unitary transformations. For a generator set $E=\{M^1,M^2,\cdots\}$, the tangent anti-Hermitian operator $\hat{T}$ can be decomposed as $$\label{decompT}
\hat{T}(s)=\sum_IY_I(s)M^I\,.$$ This tangent operator can generate a unitary operator by a time order exponential map $$\label{mapus1}
\hat{O}(s):=\overleftarrow{P}\exp\left[\int_{0}^s\hat{T}(\tilde{s}){\text{d}}\tilde{s}\right]\,,$$ where $\overleftarrow{\mathcal{P}}$ denotes a time ordering such that the tangent operator at earlier times is applied to the state first. This $s$-dependent operator can induce a curve $c:[0,1]\mapsto\mathcal{H}$ such that $$\label{curvepsi1}
c(s):=\hat{O}(s)|R\rangle\,, \qquad c(0)=|R\rangle \,, \qquad c(1)=|T\rangle\,.$$ This curve is determined by the generator set ($E$) and the coefficients ($Y_I$) of tangent operator which are shown in Eq. . Let us assume the image of the curve is $|\psi(s)\rangle$. We can compute the length of this curve by the Fubini-Study metric $$\label{lenght1}
\mathcal{L}[c]:=\int_0^1[||\partial_s|\psi(s)\rangle|^p-|\langle\psi(s)|\partial_s|\psi(s)\rangle|^p|]^{1/p}{\text{d}}s\,.$$ This paper will focus on the $L^1$ normal, i.e., $p=1$ because it was shown that $p=1$ case leads that the complexity density resembles the divergence structure of holographic complexity [@Chapman:2017rqy].
The complexity between the states $|T\rangle$ and $|R\rangle$ is given by the following optimization problem, $$\label{compFS}
\begin{split}
\mathcal{C}(|T\rangle,|R\rangle):=&\min\left\{\mathcal{L}[c]~|\forall c:[0,1]\mapsto\mathcal{H}, \ c(0)=|R\rangle, \ c(1)=|T\rangle, \right.\\
&\left.\text{and}~\exists\{Y^I\}~\text{such that}~\frac{{\text{d}}}{{\text{d}}s}c(s)=\sum_IY_I(s)M^Ic(s)\right\} \,.
\end{split}$$ By this definition, the choice of generator set $E$ may affect the complexity between two states. So far the generator set $E$ is arbitrary and there may be many possible choices. Finding the complexity in a very general generator set seems to be a too mathematical and technical problem. However, in this subsection, we want to compute the complexity between $|\text{TFD}(t_L,t_R)\rangle$ and $|0\rangle$ which are related by the operators $ \hat{U}_a^\dagger(t_L,t_R)$. Because the TFD states can be generated by some generators which form a su(1,1) Lie algebra, as will be shown in , we choose, as a minimal nontrivial generator set, $$\label{Eset1}
E_L=\bigcup_{\vec{k}}\{\hat{L}^{(\vec{k})}_+,\hat{L}^{(\vec{k})}_-,\hat{L}_0^{(\vec{k})}\} \,,$$ with $$\label{threeLs}
\begin{split}
&\hat{L}_+^{(\vec{k})}:=\hat{a}^{R\dagger}(\vec{k})\hat{a}^{L\dagger}(\vec{k})\,, \\
&\hat{L}_-^{(\vec{k})}:=\hat{a}^{R}(\vec{k}) \hat{a}^{L}(\vec{k})\,, \\
& \hat{L}_0^{(\vec{k})}:=\frac12[\hat{a}^{R}(\vec{k})\hat{a}^{R\dagger}(\vec{k})+\hat{a}^{L}(\vec{k})\hat{a}^{L\dagger}(\vec{k})-1]\,.
\end{split}$$ which satisfies the $su(1,1)$ Lie-algebra $$\label{su11Lie}
[\hat{L}_0^{(\vec{k})},\hat{L}_\pm^{(\vec{k})}]=\pm\hat{L}_\pm^{(\vec{k})}\,, \qquad [\hat{L}_-^{(\vec{k})},\hat{L}_+^{(\vec{k})}]=2\hat{L}_0^{(\vec{k})}\,.$$ In general, the tangent operator $\hat{T}(s)$ has the form $$\label{expressT1}
\hat{T}(s)=\int{\text{d}}^{d-1}k[\alpha_{+}(s)\hat{L}_+^{(\vec{k})}+\alpha_{-}(s)\hat{L}_-^{(\vec{k})}+\alpha_{0}(s)\hat{L}_0^{(\vec{k})}]\,,$$ and we have $$\label{UexpressT1}
\hat{O}(s)=\overleftarrow{P}\exp\left[\int_{0}^s\hat{T}(s){\text{d}}s\right] \,.$$ In order to compute the complexity between $|\text{TFD}(t_L,t_R)\rangle$ and $|0\rangle$, we need an $s$-dependent operator $\hat{O}(s)$ satisfying $$\label{requiU1}
\hat{O}(0)=I, ~~~ \hat{O}(1)|0\rangle=|\text{TFD}(t_L,t_R)\rangle\,.$$ Since, for different $s_1$ and $s_2$, the generators $\hat{T}(s_1)$ and $\hat{T}(s_2)$ do not commute, we cannot drop the time order operator $\overleftarrow{P}$ in . However, as the generator set forms a complete Lie-algebra, there are three functions $b(\vec{k},s), c(\vec{k},s)$ and $d(\vec{k},s)$ so that the operator $ \hat{O}(s)$ can have a “normal decomposition” by using the decomposition formula of the su(1,1) Lie-algebra [@Chapman:2017rqy; @klimov2009a] $$\label{decomp11}
\begin{split}
\hat{O}(s)=&\exp\left[\int{\text{d}}^{d-1}kb(\vec{k},s)\hat{L}_+^{(\vec{k})}\right]\exp\left[\int{\text{d}}^{d-1}kc(\vec{k},s)\hat{L}_0^{(\vec{k})}]\right]\times\\
&\exp\left[\int{\text{d}}^{d-1}kd(\vec{k},s)\hat{L}_-^{(\vec{k})}\right] \,.
\end{split}$$
The requirement $\hat{O}(0)=I$ shows that $b(\vec{k},0)=c(\vec{k},0)=d(\vec{k},0)=0$. One important point of the decomposed form is that $$\label{sampleU1}
|\psi(s)\rangle=\hat{O}(s)|0\rangle=\mathcal{N}(s)\exp\left[\int{\text{d}}^{d-1}k b(\vec{k},s)\hat{L}_+^{(\vec{k})}\right]|0\rangle\,,$$ where $\mathcal{N}(s)$ is a normalization constant factor. The constraint Eq. with Eqs. and yield $$\label{constaint1}
b_+(\vec{k},1)=e^{-\omega_{\vec{k}}[\pi/a+i(t_R+t_L)]}=\tanh r_{\vec{k}}\,,$$ where $r_{\vec{k}}$ is defined by Eq. . Following Ref. [@Chapman:2017rqy], we can find the complexity between $|\text{TFD}(t_L,t_R)\rangle$ and $|0\rangle$ $$\label{FSc1}
\mathcal{C}(|\text{TFD}(t_L,t_R)\rangle,|0\rangle)=\min\left\{\frac{\Sigma_{d-1}}2\int_0^1{\text{d}}s\int{\text{d}}^{d-1}k\frac{|\partial_s b (\vec{k},s)|}{1-|b(\vec{k},s)|^2}\right\}$$ with the constraint Eq. . The solution for this optimization problem has been shown [@Chapman:2017rqy]: $$\label{solutiongamma1}
b(\vec{k},s)=\tanh(r_{\vec{k}}s)=\tanh\{s\cdot\text{arctanh}e^{-\omega_{\vec{k}}[\pi/a+i(t_R+t_L)]}\}$$ and the complexity is given by, $$\label{FSc2}
\Sigma_{d-1}^{-1}\mathcal{C}(|\text{TFD}(t_L,t_R)\rangle,|0\rangle)=\int{\text{d}}^{d-1}k|r_{\vec{k}}|=\int{\text{d}}^{d-1}k \left|\text{arctanh}e^{-\omega_{\vec{k}}[\pi/a+i(t_R+t_L)]}\right| \,.$$
For the full conformal symmetry case, we have $\omega_{\vec{k}}=k$, then Eq. becomes $$\label{FSc3}
\begin{split}
\Sigma_{d-1}^{-1}\mathcal{C}(|\text{TFD}(t_L,t_R)\rangle,|0\rangle)&=S_{d-2}\int_0^\infty{\text{d}}k k^{d-2}\left|\text{arctanh}e^{-k[\pi/a+i(t_R+t_L)]}\right|\\
&=2^{d-1}S_{d-2}T^{d-1}\Xi_d(\tilde{t})\,,
\end{split}$$ where $S_{d-2}$ is the area of $(d-2)$-dimensional sphere and $\tilde{t}:=2(t_L+t_R)T$. $\Xi_d(\tilde{t})$ is a function defined as ($x := k/2T$) $$\label{defineRdt}
\Xi_d(\tilde{t}):=\int_0^\infty x^{d-2}\left|\text{arctanh}e^{-(1+i \tilde{t})x}\right|{\text{d}}x\,,$$ which is finite only when $d\geq2$. It is more convenient to write the result in terms of the total energy of the system. For the free scalar field with conformal symmetry, the total energy $E$ is expressed by $$\label{totalECFT}
\frac{E}{\hbar\Sigma_{d-1}}=\int {\text{d}}k^{d-1}\omega_{\vec{k}}e^{-\omega_{\vec{k}}/(2T)}=S_{d-2}2^dT^d\int_0^\infty x^{d-1}e^{-x}{\text{d}}x=S_{d-2}2^d\Gamma(d+1)T^d\,,$$ so we have $$\label{FSintermE}
\mathcal{C}(|\text{TFD}(t_L,t_R)\rangle,|0\rangle)=\frac{\Xi_d(\tilde{t})}{2\Gamma(d+1)}\frac{E}{\hbar T}\,.$$ The growth rate of the complexity between $|\text{TFD}(t_L,t_R)\rangle$ and $|0\rangle$ can be expressed as $$\label{growthrate1}
\frac{{\text{d}}}{{\text{d}}(t_L+t_R)}\mathcal{C}(|\text{TFD}(t_L,t_R)\rangle,|0\rangle)=\frac{E}{\hbar \Gamma(d+1)}\dot{\Xi}_d(\tilde{t}) \,.$$ For small time $\tilde{t}$ we have the following expansion $$\label{smalltR}
\Xi_d(\tilde{t})=\mathfrak{I}_d^{(0)}-\frac12\mathfrak{I}_d^{(1)}\tilde{t}^2+\mathcal{O}(\tilde{t}^4)\,,$$ where $\mathfrak{I}_d^{(0)}=\Gamma(d)(2^d-1)\zeta(d)/[2^d(d-1)]$ and $\mathfrak{I}_d^{(1)}>0$. It is not easy to write down the analytic formula for $\mathfrak{I}_d{(1)}$ so the numerical computations shows that $$\mathfrak{I}_3^{(1)}\approx0.07565\,, \qquad \mathfrak{I}_4^{(1)}\approx0.1639.$$ For large $\tilde{t}$ limit, i.e., in the late time limit, we can see that the phase factor $ik\tilde{t}$ makes a rapidly oscillation so the complexity becomes constant $$\label{largetR}
\lim_{t\rightarrow\infty}\Xi_d(\tilde{t})=\mathfrak{I}_d^{(0)}\vartheta \,,$$ with a positive constant $\vartheta\approx0.986$. Thus we conclude $$\label{limitvalues1}
\frac{{\text{d}}}{{\text{d}}(t_L+t_R)}\mathcal{C}(|\text{TFD}(t_L,t_R)\rangle,|0\rangle)=\left\{
\begin{split}
&-\frac{E}{\hbar\Gamma(d+1)}\mathfrak{I}_d^{(1)}\tilde{t} ~~ \quad \text{for}~|\tilde{t}|\ll1,\\
&\qquad \qquad 0 \qquad \qquad \ \, \quad \text{for}~|\tilde{t}|\rightarrow\infty.
\end{split}\right.$$ In Fig. \[Xit1\], the values of $\mathcal{C}(|\text{TFD}(t_L,t_R)$ and its growth rates for different $\tilde{t}$ are shown.
![The numerical values of $\mathcal{C}$ and $\dot{\mathcal{C}}$ at $d=2,3,4,5$. Here $\mathcal{C}_0$ is the complexity when $\tilde{t}=0$ and $\dot{\mathcal{C}}_f=-\dot{\mathcal{C}}_{\min}$. They show that $\dot{\mathcal{C}}$ will first decease linearly with respective to time $\tilde{t}$ and then increase later. Finally, the $\dot{\mathcal{C}}$ goes to zero. []{data-label="Xit1"}](Ct1.pdf "fig:"){width=".49\textwidth"} ![The numerical values of $\mathcal{C}$ and $\dot{\mathcal{C}}$ at $d=2,3,4,5$. Here $\mathcal{C}_0$ is the complexity when $\tilde{t}=0$ and $\dot{\mathcal{C}}_f=-\dot{\mathcal{C}}_{\min}$. They show that $\dot{\mathcal{C}}$ will first decease linearly with respective to time $\tilde{t}$ and then increase later. Finally, the $\dot{\mathcal{C}}$ goes to zero. []{data-label="Xit1"}](dXidt1.pdf "fig:"){width=".49\textwidth"}
We see that in the Fubini-Study metric method, the complexity growth rate between a TFD state and its corresponding vacuum state is negative for small time and increases later, finally goes to zero in the large time limit. This is very different from the CV and CA conjectures and also the FS method, which will be considered in the following subsection.
Finsler geometry (FG)
---------------------
In this subsection, we will use another field theoretic method proposed by Ref. [@Yang:2017nfn] to compute the time dependence of the complexity in the TFD state. Ref. [@Yang:2017nfn] first try to define the complexity for an operator and then define the complexity between two states. Let us first make a brief review on this method.
For a given generator set $E=\{M^1,M^2,\cdots\}$, all the operators generated by (Eqs. and ) $$\label{mapus122}
\hat{O}(s):=\overleftarrow{P}\exp\left[\int_{0}^s\hat{T}(\tilde{s}){\text{d}}\tilde{s}\right]\,, \qquad \hat{T}(s)=\sum_IY_I(s)M^I\,,$$ form an operator set $\mathcal{O}$ where the identity operator $I$ is also included. Eq. defines a curve, $c(s)$, in $\mathcal{O}$, $c: [0,1]\mapsto\mathcal{U}$. The length of the curve may be defined as $$\label{lengthLc}
L[c]:=\int_0^1{\text{d}}s F[c(s),\hat{T}(s)]\,,$$ where the Finsler structure $F[c(s),\hat{T}(s)]$ is always positive and some functional of $Y_I(s)$, which depends on the choice of the generators $\hat{T}(s)$ of the curve $c(s)$. The explicit form of the Finsler structure will be explained later on.
Once the length is defined, the complexity of any operator $\hat{O}$ belonging to $\mathcal{O}$ is given by the minimal length from the identity: $$\label{complU1}
\mathcal{C}(\hat{O}):=\min\{L[c]~|~\forall c: [0,1]\mapsto\mathcal{O}, \ s.t., \ c(0)=I~\text{and}~\exists\lambda\neq0,~c(1)=\lambda\hat{O}\}\,.$$ After defining the complexity of an operator, we may define the complexity from one state to another state as $$\label{FCompstates}
\mathcal{C}(|\psi_2\rangle,|\psi_1\rangle):=\min\{\mathcal{C}(\hat{O})~|~\forall\hat{O}\in\mathcal{O}, s.t., \hat{O}|\psi_1\rangle\sim|\psi_2\rangle\}\,,$$ where the notation $\sim$ means that two state can be different by a nonzero complex constant. Thus, there are three steps to find the complexity between two states. Firstly, we have to find the complexity of all operators in $\mathcal{O}$. Then, we have to find all the operators which can change the reference state to the target state. Finally, we need to find the minimal complexity of these operators. For some cases where we only care about the complexity between states it is not necessary to compute the complexity of all the operators. Instead, we can directly solve the following optimization problem $$\label{FCompstates2}
\mathcal{C}(|\psi_2\rangle,|\psi_1\rangle):=\min\{L[c]~|~\forall c:[0,1] \rightarrow \mathcal{O}, s.t., \hat{O}(s)=c(s),~\hat{O}(0)=I~\text{and}~\hat{O}(1)|\psi_1\rangle\sim|\psi_2\rangle\}\,.$$ In some cases, this optimization problem is easier to handle than finding complexity of operators.
In Ref. [@Yang:2017nfn], a very general generator set formed by creation and annihilation operators is considered. Although this makes the generator set big enough, it makes the optimization problems and hard to solve exactly. In order to make a comparison with the results in the Fubini-Study metric, we will use a smaller generator set, which is defined by . In this case, the operator set $\mathcal{O}$ is just the infinite direct product of $SU(1,1)$ group. Any operator in $\mathcal{O}$ can be parameterized uniquely by three complex-valued functions $\gamma_+(\vec{k}),\gamma_-(\vec{k})$ and $\gamma_0(\vec{k})$. $$\label{decomp1}
\begin{split}
\hat{U}[\gamma_+(\vec{k}),\gamma_-(\vec{k}),\gamma_0(\vec{k})]=&\exp\left[\int{\text{d}}^{d-1}k\gamma_+(\vec{k})\hat{L}_+^{(\vec{k})}\right]\exp\left[\int{\text{d}}^{d-1}k\ln\gamma_0(\vec{k})\hat{L}_0^{(\vec{k})}]\right]\times\\
&\exp\left[\int{\text{d}}^{d-1}k\gamma_-(\vec{k})\hat{L}_-^{(\vec{k})}\right]\,.
\end{split}$$ We find from that $|\text{TFD}(t_L,t_R)\rangle\sim \hat{U}[\gamma_+(\vec{k}),\gamma_-(\vec{k}),\gamma_0(\vec{k})]|0\rangle$ if and only if $\gamma_+=e^{-\omega_{\vec{k}}[\pi/a+i(t_R+t_L)]}$. Let us take $$\label{Ugamma}
\hat{U}_a(t_L,t_R):=\exp\left[\int{\text{d}}^{d-1}ke^{-\omega_{\vec{k}}[\pi/a+i(t_R+t_L)]}\hat{L}_+^{(\vec{k})}\right]\,.$$ Thus the set of all the operators which can change from $|0\rangle$ to $|\text{TFD}(t_L,t_R)\rangle$ is $$\label{goodset}
\mathcal{D}:=\left\{\left.\hat{U}[\gamma_+(\vec{k}),\gamma_-(\vec{k}),\gamma_0(\vec{k})]~\right|~\forall~\gamma_-,\gamma_0\in\mathbb{C}, \gamma_+=e^{-\omega_{\vec{k}}[\pi/a+i(t_R+t_L)]}\right\} \,.$$ The complexity between $|0\rangle$ to $|\text{TFD}(t_L,t_R)\rangle$ is given by $$\label{FGTFDA2}
\begin{split}
\mathcal{C}(|\text{TFD}(t_L,t_R)\rangle,&|0\rangle)=\min\left\{\mathcal{C}(\hat{U})|~\forall \hat{U}\in\mathcal{D}\right\}
\end{split}$$ In order to proceed, we need to obtain the explicit form of the Finsler structure in the generator set $E_L$. Relegating technical details to appendix \[FinSEL\] we here present a final result. For any generator $\hat{T}(s)$ expanded in the basis $E_L$ $$\label{FGgenerator0tt}
\hat{T}(s)=\int{\text{d}}^{d-1}k[\alpha_+(s,\vec{k})\hat{L}_+^{(\vec{k})}+\alpha_0(s,\vec{k})\hat{L}_0^{(\vec{k})} +\alpha_-(s,\vec{k})\hat{L}_-^{(\vec{k})}]\,.$$ the Finsler structure is given by $$\label{defFp20tt}
F|_{E_L}=\ell_0\Sigma_{d-1}\int{\text{d}}^{d-1}k[\parallel\alpha_+(s,\vec{k})\parallel+\parallel\alpha_-(s,\vec{k})\parallel +\parallel\alpha_0(s,\vec{k})\parallel]\,,$$ where $\ell_0$ is a free parameter to be chosen later. Based on the detailed computation in appendix \[relga\], it turns out that $$\label{FGTFDA4tt}
\mathcal{C}(|\text{TFD}(t_L,t_R)\rangle,|0\rangle)=\ell_0 \Sigma_{d-1} \int{\text{d}}^{d-1}k\parallel\gamma_+(\vec{k}) \parallel \,.$$ Therefore, we have $$\label{valueFGC2}
\mathcal{C}(|\text{TFD}(t_L,t_R)\rangle,|0\rangle) = \ell_0\Sigma_{d-1}\int_0^\infty k^{d-2}\parallel e^{-\omega_{\vec{k}}[\pi/a+i(t_R+t_L)]}\parallel{\text{d}}k\,.$$ Using the definition $\parallel\cdot\parallel$ in Eq. , we finally obtain the complexity between $|0\rangle$ to $|\text{TFD}(t_L,t_R)\rangle$: $$\label{valueFGC3}
\mathcal{C}(|\text{TFD}(t_L,t_R)\rangle,|0\rangle)=\ell_0\Sigma_{d-1}S_{d-2}2^{d-1}T^{d-1}\Omega_d(\tilde{t})=\ell_0\frac{\Omega_d(\tilde{t})}{2\Gamma(d+1)}\frac{E}{\hbar T}\,,$$ where $S_{d-2}$ is the area of $(d-2)$-dimensional sphere, $\tilde{t}:=2(t_L+t_R)T$, $$\label{defOmegat}
\Omega_d(\tilde{t}):=\int_0^\infty x^{d-2}e^{-x}(|\cos x \tilde{t}|+|x \tilde{t}|\cdot|\sin x \tilde{t}|){\text{d}}x\,,$$ and the total energy $E$ is given by Eq. .
For small $\tilde{t}$ $$\label{smalltOmega}
\Omega_d(\tilde{t})=\Gamma(d-1)+\frac{\Gamma(d+1)}2\tilde{t}^2+\mathcal{O}(\tilde{t}^4)\,,$$ and for large $\tilde{t}$ $$\label{largetOmega}
\Omega_d(\tilde{t})=\frac2{\pi}\left[\Gamma(d-1)+\Gamma(d)\tilde{t}\right][1+\mathcal{O}(1/\tilde{t})]\,.$$ Thus we see that $$\label{limitvalues3}
\frac{{\text{d}}}{{\text{d}}(t_L+t_R)}\mathcal{C}(|\text{TFD}(t_L,t_R)\rangle,|0\rangle)=\left\{
\begin{split}
&\ell_0E\tilde{t}/\hbar~~~~\text{for}~|\tilde{t}|\ll1,\\
&\ell_0\frac{E}{\hbar d}~~~~~~~\text{for}~|\tilde{t}|\gg1.
\end{split}\right.$$ For the intermediate time, we can compute $\Omega_d(\tilde{t})$ analytically but it is not so illuminating. Therefore, we show a numerical plot for $\Omega_4(\tilde{t})$ and $\dot{\Omega}_4(\tilde{t})$ in the Fig. \[valuesW\]. For lager $d>4$, the behavior is similar.
Note that the linear $\tilde{t}$ dependence of the complexity in the late time limit comes from $|x \tilde{t}|$ in Eq. , which is due to our definition of $\parallel\cdot\parallel$ in Eq. . If we use the definition $\parallel\cdot\parallel$ in Eq. then the complexity will be constant independent of time, which is the same as the FS case. Therefore, our result here is not so robust. It should be understood as one example to define the field theory complexity showing the linear-time complexity.
![The numerical values of complexity and its growth rate at $d=2,3,4,5$. $\mathcal{C}_0$ is the complexity at $\tilde{t}=0$ and $\dot{\mathcal{C}}_f$ is the growth rate at the late time limit. From the left panel, we see that $\mathcal{C}$ will monotonously increase with respective to time $\tilde{t}$. For small $\tilde{t}$, $\dot{\mathcal{C}}$ will linearly depends on $\tilde{t}$. For large $\tilde{t}$, $\dot{\mathcal{C}}$ will tend to increase linearly with respective to $\tilde{t}$ and $\dot{\mathcal{C}}$ tends to a constant. For large $d$, we obtain similar behaviours. []{data-label="valuesW"}](Ct2.pdf "fig:"){width=".49\textwidth"} ![The numerical values of complexity and its growth rate at $d=2,3,4,5$. $\mathcal{C}_0$ is the complexity at $\tilde{t}=0$ and $\dot{\mathcal{C}}_f$ is the growth rate at the late time limit. From the left panel, we see that $\mathcal{C}$ will monotonously increase with respective to time $\tilde{t}$. For small $\tilde{t}$, $\dot{\mathcal{C}}$ will linearly depends on $\tilde{t}$. For large $\tilde{t}$, $\dot{\mathcal{C}}$ will tend to increase linearly with respective to $\tilde{t}$ and $\dot{\mathcal{C}}$ tends to a constant. For large $d$, we obtain similar behaviours. []{data-label="valuesW"}](dXidt2.pdf "fig:"){width=".49\textwidth"}
The complexity growth rate is positive and linearly dependent of $t$ at the early time ($(t_L+t_R)T\ll1$), which is the same as the prediction of the CV conjecture. In the late time limit ($(t_L+t_R)T\gg1$), Eq. is constant and proportional to $T^{d}$. In the planar symmetry asymptotic AdS black hole, the total ADM mass $M$ is also proportional to $T^{d}$. Thus we see that the complexity growth rate is similar to the predictions of both the CV and CA conjectures in the late time limit. The free parameter $\ell_0$ in Eq. can be determined if we require that the complexity growth rate saturate to the Lloyd’s bound at $t_L+t_R\rightarrow\infty$. We see that if we take $$\ell_0=\frac{2d}\pi\,,$$ the complexity growth rate at the late time limit saturates to the Lloyd’s bound. However, it turned out that the subleading term in Eq. is positive so the complexity growth rate will approach to the limiting value from the larger value. Thus, like the CA conjecture, at large time region, the complexity growth rate violate the Lloyd’s bound with this choice of $\ell_0$.
Summary {#summ}
=======
In this paper, we have computed the complexity of the time dependent TFD states and their growth rates by four different methods, two holographic and two field theory methods. Two holographic methods are based on the “complexity-action” (CA) conjecture or “complexity-volume” (CV) conjecture. Two quantum field theoretic methods are based on the Fubini-Study metric (FS) or the Finsler geometry (FG). In particular, for holographic computation, we have proposed a modified CA and CV conjectures between two TFD states, $|\text{TFD}_2\rangle$ and $|\text{TFD}_1\rangle$ $$\begin{split}
&\mathcal{C}_V(|\text{TFD}_2\rangle,|\text{TFD}_1\rangle)\equiv |\mathcal{C}^{(1)}_{V}-\mathcal{C}^{(2)}_{V}|\,, \\
&\mathcal{C}_A(|\text{TFD}_2\rangle,|\text{TFD}_1\rangle)\equiv |\mathcal{C}^{(1)}_{A}-\mathcal{C}^{(2)}_{A}|\,,
\end{split}$$ where ${\mathcal{C}}_{V}^{(i)}$ and ${\mathcal{C}}_{A}^{(i)}$ are the original CV and CA conjectures for the $|\text{TFD}_i\rangle$ state. It is similar to the ‘complexity of formation’ proposed in [@Chapman:2016hwi] but there is a subtle difference in that here we do not assume any reference state [@Yang:2017nfn]. These modified versions yield finite values agreeing to the field theory computation for a static case [@Yang:2017nfn]. For a concrete example in this paper we consider the complexity between the time-dependent TFD state and its corresponding vacuum state. We call it ‘complexity’ for simplicity.
Our main results for the time dependent complexity for the TFD states are summarized in Table \[tab1\]. As a companion to Table \[tab1\], for readers’ convenience, we show a schematic plot, Fig. \[summaryfig\] of which precise information can be found in Figures \[FigAdS1\], \[FigBTZ1\], \[Figrcv\], \[Xit1\], and \[valuesW\]. We define a common time $\bar{t} = t_L+t_R$ for all cases.
CA CV FS FG
--------------------------- ----------------------------------------- ----------------------------------------- --------------------------------------- -----------------------------------------
$t_L+t_R=0$ $\mathcal{C}\propto\frac{E}{\hbar T}$ $\mathcal{C}\propto\frac{E}{\hbar T}$ $\mathcal{C}\propto\frac{E}{\hbar T}$ $\mathcal{C}\propto\frac{E}{\hbar T}$
early time $\dot{\mathcal{C}}=0$ if $d>2$ $\dot{\mathcal{C}}\propto \bar{t}$ $\dot{\mathcal{C}}\propto-\bar{t}$ $\dot{\mathcal{C}}\propto \bar{t}$
$\dot{\mathcal{C}}=-\infty$ if $d=2$
late time $\dot{\mathcal{C}}=\frac{2E}{\pi\hbar}$ $\dot{\mathcal{C}}=\frac{2E}{\pi\hbar}$ $\dot{\mathcal{C}}=0$ $\dot{\mathcal{C}}=\frac{2E}{\pi\hbar}$
sign$(\dot{\mathcal{C}})$ indefinite + indefinite +
Lloyd’s bound broken satisfied saisfied broken
: The summary of the complexity between the time-dependent TFD state and its corresponding vacuum state in four different methods. $\bar{t}=t_L+t_R$ and $E$ is the total energy of the system. We have set $\ell/{\ell_{\text{AdS}}}=4\pi^2\hbar/(d-1)$ for the CV conjecture, $\ell_0=2d/\pi$ for the FG method, and the speed of light $c=1$.[]{data-label="tab1"}
![Schematic plots for the complexity by four methods: the holographic CV and CA conjecture and the field theoretic FS and FG methods. It is obtained from Figures \[FigAdS1\], \[FigBTZ1\], \[Figrcv\], \[Xit1\], and \[valuesW\]. $\bar{t}= t_L + t_R$. Left: ${\mathcal{C}}/{\mathcal{C}}_0$, where ${\mathcal{C}}_0$ is the complexity at $\bar{t}=0$. Right: $\dot{\mathcal{C}}/\dot{\mathcal{C}}_f$, where $\dot{\mathcal{C}}_f$ is the growth rate at $\bar{t} \rightarrow\infty$ in CA,CV and FG methods, and $\dot{\mathcal{C}}_f=-\dot{\mathcal{C}}_{\min}$ in the FS method.[]{data-label="summaryfig"}](summaryfig.pdf "fig:"){width=".49\textwidth"} ![Schematic plots for the complexity by four methods: the holographic CV and CA conjecture and the field theoretic FS and FG methods. It is obtained from Figures \[FigAdS1\], \[FigBTZ1\], \[Figrcv\], \[Xit1\], and \[valuesW\]. $\bar{t}= t_L + t_R$. Left: ${\mathcal{C}}/{\mathcal{C}}_0$, where ${\mathcal{C}}_0$ is the complexity at $\bar{t}=0$. Right: $\dot{\mathcal{C}}/\dot{\mathcal{C}}_f$, where $\dot{\mathcal{C}}_f$ is the growth rate at $\bar{t} \rightarrow\infty$ in CA,CV and FG methods, and $\dot{\mathcal{C}}_f=-\dot{\mathcal{C}}_{\min}$ in the FS method.[]{data-label="summaryfig"}](summaryfig2.pdf "fig:"){width=".49\textwidth"}
If $\bar{t} =0$, four different methods give similar results but give different predictions on the time evolution of the complexity. At early time, both the CV conjecture and FG method predict that the complexity will increase as $\bar{t}^2$ while the FS method predicts that the complexity will decrease as $-\bar{t}^2$. The CA conjecture says that for $d>2$ the complexity does not change until a critical time and after then it will decrease. For $d=2$, it decreases first and increases as time goes on.
In the late time limit, the CA conjecture, CV conjecture and FG method predict that the complexity will increase linearly in $\bar{t}$ and the growth rate will be proportional to the total energy of the system. On the contrary, the FS method shows that the complexity in the late time limit will keep constant rather than increasing. The CV conjecture and FG method show that the complexity will monotonically increase for $\bar{t}>0$ while the CA conjecture and FS method show that the complexity first decreases and then increases. The Lloyd’s bound is satisfied only for the CV conjecture and FS method. The Lloyd’s bound is also satisfied for the CA conjecture and FG method in the late time limit, but it is weakly violated in the intermediate time. We have set $\ell/{\ell_{\text{AdS}}}=4\pi^2\hbar/(d-1)$ for the CV conjecture, $\ell_0=2d/\pi$ for the FG method. With other choices, the growth rate saturate to some value which is not the Lloyd’s bound.
The results summarized in Table \[tab1\] seem to give us some pieces of information to judge which are appropriate methods to compute the complexity among two holographic conjectures and two quantum field theory proposals. For examples, if we expect that the complexity should increase with time then it seems that the CV conjecture and FG method are favored. The similarity between the CV conjecture and the FG method in the early and late time limit seems to show these two proposal are more correlated while they are different in the intermediate time regime. Note that the FS method is quite different from all the other methods. In particular, the FS method shows the complexity will keep constant in the late time limit. Because the linear growth of the complexity in the late time limit has much evidence both in quantum information theory and holography [@Susskind:2014moa; @Susskind:2014rva; @Stanford:2014jda; @Brown:2015lvg; @Brown:2017jil; @Hashimoto:2017fga; @Qaemmaqami:2017lzs] it seems to be a challenge to the FS method.
However, there is also a caveat in the FG method. The results of the FG method depend on the definition of the Finsler structure. Our result here should be understood as just one example to define the Finsler structure displaying the linear-time complexity in the late time limit and showing similar behaviors to holographic complexity. For both the FS and FG method, the complexity also depend on the generator set. We have chosen a small generator set to make the TFD states so that we can compute the complexity analytically. If we choose another generator set the complexity may or may not change. Therefore, it will be interesting to investigate how much our results are robust under different choices of generator sets and/or different choices of the Finsler structures.
The work of K.-Y. Kim and C. Niu was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT $\&$ Future Planning(NRF- 2017R1A2B4004810) and GIST Research Institute(GRI) grant funded by the GIST in 2017. C.Y. Zhang is supported by National Postdoctoral Program for Innovative Talents BX201600005. We also would like to thank the APCTP(Asia-Pacific Center for Theoretical Physics) focus program, “Geometry and Holography for Quantum Criticality” in Pohang, Korea for the hospitality during our visit, where part of this work was done.
Fubini-Study metric {#FSmetric}
===================
Let us consider an $n$-dimensional Hilbert space $\mathcal{H}$. Any two vector $|\psi_1\rangle$ and $|\psi_2\rangle$ describe the same state if there is a nonzero complex number $c$ such that $|\psi_1\rangle=c|\psi_2\rangle$. This means that the different states of the Hilbert space $\mathcal{H}$ form a complex projective space $\mathbb{C}$P$^n$. As $\mathbb{C}$P$^n=S^{2n+1}/S^1$, we can use the length of geodesic curve in $S^{2n+1}/S^1$ to build the distance of two states. It turns out that this distance is the Fubini-Study distance, which is $$\label{FSdist1}
D_{FS}(|\psi\rangle,|\phi\rangle):=\arccos(|\langle\psi|\phi\rangle|)\in[0,\pi/2]\,.$$ In this equation, the state vectors should satisfy $\langle\psi|\psi\rangle=\langle\phi|\phi\rangle=1$. This expression can be generalized to the infinite dimensional cases. To obtain the line element in a local form, let us assume $$\label{dpsi1}
|\phi\rangle= \mathcal{N}\left( |\psi\rangle+{\text{d}}|\psi\rangle \right)$$ and expand to second order in the vector ${\text{d}}|\psi\rangle$. $\mathcal{N}$ is the normalization factor for $\langle\phi|\phi\rangle=1$. The result is the Fubini-Study metric $$\label{FSmetric1}
{\text{d}}s_{FS}^2(|\psi\rangle)=\langle{\text{d}}\psi|{\text{d}}\psi\rangle^2-|\langle{\text{d}}\psi|\psi\rangle|^2\,.$$ For any curve $l:[s_i,s_f]\mapsto\mathcal{H}$ such that $l(t)=|\psi(t)\rangle$, its length is defined by $$\label{lengthl}
L^2[l]:=\int_{s_i}^{s_f}{\text{d}}s_{FS}(|\psi(t)\rangle)=\int_{s_i}^{s_f}{\text{d}}t\sqrt{|\partial_t|\psi(t)\rangle|^2-|\langle\psi(t)|\partial_t|\psi(t)\rangle|^2}$$ If there is no restriction for the curve, the length of the geodesic connecting any two states is given by Eq. . We can define the complexity for two states $$\label{defC1}
\mathcal{C}(|\psi\rangle,|\phi\rangle):=D_{FS}(|\psi\rangle,|\phi\rangle)=\arccos(|\langle\psi|\phi\rangle|)\in[0,\pi/2]\,.$$ However, in the case that the curve can only be generated by some appointed generator set $E$, the minimal length of the curves may be different from the Eq. and we have to solve the following optimization problem $$\label{defC2}
\mathcal{C}(|\psi\rangle,|\phi\rangle,E):=\min_{E}\{L^2[l]~|~l(s_i)=|\psi\rangle,l(s_f)=|\phi\rangle\}\,.$$ As noted by Ref. [@Chapman:2017rqy], it is not necessary to restrict the line element in $L^2$ normal. Then for more general case, we can define the general Fubini-Study metric by $L^p$ normal, which reads $$\label{lengthl}
{\text{d}}s:=[||\partial_s|\psi(s)\rangle|^p-|\langle\psi(s)|\partial_s|\psi(s)\rangle|^p|]^{1/p} \,.$$
Finsler structure in the generator set $E_L$ {#FinSEL}
============================================
In this appendix, we explain how to obtain the explicit functional form of the Finsler structure. We start with the proposal for a more general case in Ref. [@Yang:2017nfn] and we restrict ourselves to the generator set $E_L$.
In Ref. [@Yang:2017nfn], the generator set is chosen as the general enveloping algebra of Heisenberg-Weyl Lie algebra. Let us first define the fundamental generator set $E^{0}$ to be the collection of all the creation and annihilation operators $$\label{defE0}
E^{0}:=\bigcup_{i}\{\hat{a}_i^\dagger,\hat{a}_i, \hat{\mathbb{I}}\}\,.$$ Here index $i$ stands for different creation and annihilation operators (in our context, $i$ may stand for $\vec{k}$ and the superscript $R$ and $L$.). $\hat{\mathbb{I}}$ satisfies $[\hat{a}_i,\hat{a}_j^\dagger]=\hat{\mathbb{I}}\delta_{ij}$ and $\hat{\mathbb{I}}\hat{e}=\hat{e}$ for $\forall\hat{e}\in E^{0}$. This fundamental operator set forms a Heisenberg-Weyl Lie algebra. Because this generator set is not big enough Ref. [@Yang:2017nfn] extends it to a larger set $E$ by $$\label{defElarge}
E:=\bigcup_{n}(E^{0})^n,~~~\text{with}~(E^0)^n:=\{\hat{M}^{i_1i_2\cdots i_n}=:\hat{e}_{i_1}\hat{e}_{i_2}\cdots \hat{e}_{i_n}:|\forall \hat{e}_{i_1},\hat{e}_{i_2},\cdots,\hat{e}_{i_n}\in E^0\}\,.$$ Here the “: :” stands for the normal ordering, e.g., $:\hat{a}_{i}\hat{a}^\dagger_{j}:=\hat{a}^\dagger_{j}\hat{a}_{i}$ for $\forall i,j$. In the definition , $\hat{e}_{i_1}, \hat{e}_{i_2}, \cdots, \hat{e}_{i_n}$ do not need to be different from each others. This extended operator set forms the general enveloping algebra of Heisenberg-Weyl Lie algebra. Any generator $\hat{T}(s)$ can be expand it by basis $E$ as follows: $$\label{decompTs1}
\hat{T}(s)=T_0(s)\hat{\mathbb{I}}+\sum_i Y_i(s)\hat{M}^i+\sum_{ij}Y_{ij}(s)\hat{M}^{ij}+\cdots+\sum_{i_1i_2\cdots i_n}Y_{i_1i_2\cdots i_n}(s)\hat{M}^{i_1i_2\cdots i_n}+\cdots\,.$$ Here $T_0(s), Y_i(s), Y_{ij}(s),\cdots$ are complex numbers, $\hat{M}^i, \hat{M}^{ij},\cdots$ are the elements in $E$ except for $\hat{\mathbb{I}}$. Then the Finsler structure expressed in the basis is given by [@Yang:2017nfn] $$\label{defFp10}
F|_E=\ell_s\left[\sum_i \parallel Y_i(t)\parallel +2\sum_{ij}\parallel Y_{ij}(t)\parallel +\cdots+n\sum_{i_1i_2\cdots i_n}\parallel Y_{i_1i_2\cdots i_n}(t)\parallel +\cdots\right]\,.$$ Here $\ell_s$ is a dimensionless positive constant. The meaning of $\parallel\cdot\parallel$ will clarified later on. For the continuous index case, the summation in the right-hand of Eq. should be replaced by integration. Here the index $E$ is added into $F$ to explicitly show that the right-hand of Eq. is valid only when we use the set $E$ to expand $\hat{T}(s)$.
In this paper, the generator set $E_L$ is neither $E$ nor its subsect, we cannot directly use formula to obtain the functional form of the Finsler structure in the basis $E_L$ as the functional form of a Finsler structure depends on the basis. However, the generator set $E_L$ is just the linear combinations of some elements in $E$, $$\label{transforEEL}
\left[\begin{matrix}
\hat{L}_+^{(\vec{k})}\\
\hat{L}_-^{(\vec{k})}\\
\hat{L}_0^{(\vec{k})}
\end{matrix}\right]=\left[\begin{matrix}1,0,0,0,0\\
0,1,0,0,0\\
0,0,\frac12,\frac12,\frac12
\end{matrix}\right]
\left[\begin{matrix}
\hat{a}^{R\dagger}(\vec{k})\hat{a}^{L\dagger}(\vec{k})\\
\hat{a}^{R}(\vec{k})\hat{a}^{L}(\vec{k})\\
\hat{a}^{R\dagger}(\vec{k})\hat{a}^{R}(\vec{k})\\
\hat{a}^{L\dagger}(\vec{k})\hat{a}^{L}(\vec{k})\\
\hat{\mathbb{I}}
\end{matrix}\right]$$ Let us consider a generator $\hat{T}(s)$ expanded it in the basis $E_L$ $$\label{FGgenerator0}
\hat{T}(s)=\int{\text{d}}^{d-1}k[\alpha_+(s,\vec{k})\hat{L}_+^{(\vec{k})}+\alpha_0(s,\vec{k})\hat{L}_0^{(\vec{k})} +\alpha_-(s,\vec{k})\hat{L}_-^{(\vec{k})}]\,.$$ Using the basis transformation formula in Eq. (A.5) of Ref. [@Yang:2017nfn], we find that the functional form of the Finsler structure is $$\label{defFp20}
\begin{split}
F|_{E_L}&=\frac{2\ell_s\Sigma_{d-1}}{(2\pi)^{d-1}}\int{\text{d}}^{d-1}k[\parallel\alpha_+(s,\vec{k})\parallel+\parallel\alpha_-(s,\vec{k})\parallel +\frac12\parallel\alpha_0(s,\vec{k})\parallel+\frac12\parallel\alpha_0(s,\vec{k})\parallel]\\
&=\ell_0\Sigma_{d-1}\int{\text{d}}^{d-1}k[\parallel\alpha_+(s,\vec{k})\parallel+\parallel\alpha_-(s,\vec{k})\parallel +\parallel\alpha_0(s,\vec{k})\parallel]\,,
\end{split}$$ where we define that $\ell_0=2\ell_s/(2\pi)^{d-1}$.
The notation $\parallel\cdot\parallel$ was introduced in Ref. [@Yang:2017nfn]. Let’s explain again why it is used in the Finsler structure . For a complex number $Y^I=\rho^I e^{i\theta^I}$ one may want to use $\parallel Y^I\parallel =\rho^I$ but this will lead to an inconsistent: the “rotation” caused by $\theta^I$ will change the operator $ \hat{O}(s)$ but it does not change the complexity. One simple modification will be $$\label{complexY0}
\parallel Y^I(s)\parallel = \rho^I(|\cos\theta^I|+|\sin\theta^I|) = |\text{Re}Y^I(s)|+|\text{Im}Y^I(s)| \,.$$ Another possibility is [@Yang:2017nfn] $$\label{complexY}
\parallel Y^I(s)\parallel :=|\text{Re}Y^I(s)|+|\theta^I(s)|\cdot|\text{Im}Y^I(s)|\,.$$ where $\theta(0)\in[-\pi,\pi)$ and $\theta(s)$ is continuous for $s\in[0,1]$. In this paper, we choose the as we can see that it can give the linear growth rate of the complexity at the late time limit.
Complexity of operator generated by $E_L$ {#relga}
=========================================
In this appendix, we will give the method to compute the complexity for any element in operators set $\mathcal{U}$ which is generated by generator set . Any operator in $\mathcal{U}$ can be parameterized uniquely by three complex-valued functions $\gamma_+(\vec{k}),\gamma_-(\vec{k})$ and $\gamma_0(\vec{k})$ by Eq. . In order to find its complexity, we have to compute the lengths of all curves connecting $\hat{U}$ and identity in $\mathcal{U}$, and then find the minimal value of them. In $\mathcal{U}$, any curve starting from identity can be given by an $s$-dependent operator $\hat{O}(s)$ as $$\label{TandU1}
\hat{O}(s)=\overleftarrow{P}\exp\left[\int_0^s\hat{T}(\tilde{s}){\text{d}}\tilde{s} \right]\,,$$ where $$\label{FGgenerator}
\hat{T}(\tilde{s})=\int{\text{d}}^{d-1}k[\alpha_+(\tilde{s},\vec{k})\hat{L}_+^{(\vec{k})}+\alpha_0(\tilde{s},\vec{k})\hat{L}_0^{(\vec{k})} +\alpha_-(\tilde{s},\vec{k})\hat{L}_-^{(\vec{k})}]\,.$$ As $[\hat{T}(\tilde{s}_1),\hat{T}(\tilde{s}_2)]\neq0$ in general when $s_1\neq s_2$, the time-order operator cannot be neglected. Different choices of functions $\{\alpha_\pm(\tilde{s},\vec{k}), \alpha_0(\tilde{s},\vec{k})\}$ give different curves. We need this curve to end at $\hat{U}$ when $s=1$, i.e., $\hat{O}(1)=\hat{U}$. Let us find the relationship between $\{\gamma_\pm(\vec{k}), \gamma_0(\vec{k})\}$ defined in Eq. and $\{\alpha_\pm(s,\vec{k}),\alpha_0(s,\vec{k})\}$ when we require that $\hat{O}(1)=\hat{U}$.
It is more convenient to consider the problem in the discrete momentum system. As the elements with different momentum in are commutative to each others, we can see that $$\label{TandU1b}
\hat{O}(s):=\prod_{\vec{k}_i}\hat{O}_{\vec{k}_i}(s)\,,$$ where $$\label{defUk}
\hat{O}_{\vec{k}_i}(s):=\overleftarrow{P}\exp\left[\int_0^s\hat{T}_{\vec{k}_i}(\tilde{s}){\text{d}}\tilde{s} \right]\,,$$ $$\label{FGgenerator}
\hat{T}_{_{\vec{k}_i}}(s)=\alpha_+(\tilde{s},\vec{k}_i)\hat{L}_+^{(\vec{k}_i)}+\alpha_0(\tilde{s},\vec{k}_i)\hat{L}_0^{(\vec{k}_i)} +\alpha_-(\tilde{s},\vec{k}_i)\hat{L}_-^{(\vec{k}_i)}\,.$$ The operator $\hat{O}_{\vec{k}_i}(s)$ has also a normal decomposition by three functions $b_{\vec{k}_i}(s), c_{\vec{k}_i}(s)$ and $d_{\vec{k}_i}(s)$ $$\label{discdecomp111}
\hat{O}_{\vec{k}_i}(s)=\exp[b_{\vec{k}_i}(s)\hat{L}_+^{(\vec{k}_i)}]\exp[c_{\vec{k}_i}(s)\hat{L}_0^{(\vec{k}_i)}]\exp[d_{\vec{k}_i}(s)\hat{L}_-^{(\vec{k}_i)}] \,.$$
Differentiating both and with respective to $s$, we have $$\label{deriva111}
\begin{split}
\hat{T}_{{\vec{k}_i}}\hat{O}_{\vec{k}_i}&=b'_{\vec{k}_i}\hat{L}_+^{(\vec{k}_i)}\exp[b_{\vec{k}_i}\hat{L}_+^{(\vec{k}_i)}]\exp[c_{\vec{k}_i}\hat{L}_0^{(\vec{k}_i)}]\exp[d_{\vec{k}_i}\hat{L}_-^{(\vec{k}_i)}]\\
&+c_{\vec{k}_i}'\exp[b_{\vec{k}_i}\hat{L}_+^{(\vec{k}_i)}]\hat{L}_0^{(\vec{k}_i)}\exp[c_{\vec{k}_i}\hat{L}_0^{(\vec{k}_i)}]\exp[d_{\vec{k}_i}\hat{L}_-^{(\vec{k}_i)}]\\
&+d_{\vec{k}_i}'\exp[b_{\vec{k}_i}\hat{L}_+^{(\vec{k}_i)}]\exp[c_{\vec{k}_i}\hat{L}_0^{(\vec{k}_i)}]\hat{L}_-^{(\vec{k}_i)}\exp[d_{\vec{k}_i}\hat{L}_-^{(\vec{k}_i)}]\,.
\end{split}$$ There is a very useful formula when we compute the right hand of . For any two operators $\hat{A},\hat{B}$, let us define $[^{(0)}A,B]:=B$ and $[^{(n+1)}A,B]:=[A,[^{(n)}A,B]]$. Then we can find $$\label{adjointf}
e^{\hat{A}}\hat{B}e^{-\hat{A}}=\sum_{n=0}^{\infty}\frac1{n!}[^{(n)}\hat{A},\hat{B}].$$ With this formula and the commutation relation , Eq. becomes $$\begin{aligned}
\hat{T}_{{\vec{k}_i}}(s)\hat{O}_{\vec{k}_i}&&=\left\{b'_{\vec{k}_i}\hat{L}_+^{(\vec{k}_i)}+c'_{\vec{k}_i}[\hat{L}_0^{(\vec{k}_i)}-b_{\vec{k}_i}\hat{L}_+^{(\vec{k}_i)}] +d_{\vec{k}_i}'e^{c_{\vec{k}_i}}[\hat{L}_-^{(\vec{k}_i)}-2b_{\vec{k}_i}\hat{L}_0^{(\vec{k}_i)}+b_{\vec{k}_i}^2\hat{L}_+^{(\vec{k}_i)}]\right\}\hat{O}_{\vec{k}_i} \nonumber \\
&&=\left\{[b'_{\vec{k}_i}-c'_{\vec{k}_i}b_{\vec{k}_i}+b_{\vec{k}_i}^2d_{\vec{k}_i}'e^{c_{\vec{k}_i}}]\hat{L}_+^{(\vec{k}_i)} +[c'_{\vec{k}_i}-2d_{\vec{k}_i}'e^{c_{\vec{k}_i}}b_{\vec{k}_i}]\hat{L}_0^{(\vec{k}_i)}+d_{\vec{k}_i}'e^{c_{\vec{k}_i}}\hat{L}_-^{(\vec{k}_i)}\right\}\hat{O}_{\vec{k}_i} \nonumber \\
&&=[\alpha_+(s,\vec{k}_i)\hat{L}_+^{(\vec{k}_i)}+\alpha_0(s,\vec{k}_i)\hat{L}_0^{(\vec{k}_i)} +\alpha_-(s,\vec{k}_i)\hat{L}_-^{(\vec{k}_i)}]\hat{O}_{\vec{k}_i}\,. \nonumber
$$ Thus we obtain the following differential equations $$\label{odebcd1}
\begin{split}
&b'_{\vec{k}_i}(s)=\alpha_+(s,\vec{k}_i)+\alpha_0(s,\vec{k}_i) b_{\vec{k}_i}(s) + b^2_{\vec{k}_i}(s)\alpha_-(s,\vec{k}_i),\\
&c'_{\vec{k}_i}(s)=\alpha_0(s,\vec{k}_i)+ 2b_{\vec{k}_i}(s)\alpha_-(s,\vec{k}_i),\\
&d'_{\vec{k}_i}(s)=\alpha_-(s,\vec{k}_i)e^{-c_{\vec{k}_i}}\,.
\end{split}$$ They should satisfy the following boundary conditions: $$\label{reqonbcd}
\begin{split}
&b_{\vec{k}_i}(0)=c_{\vec{k}_i}(0)=d_{\vec{k}_i}(0)=0,\\
&b_{\vec{k}_i}(1)=\gamma_+(\vec{k}_i)\,,\quad c_{\vec{k}_i}(1)=\ln(\gamma_0(\vec{k}_i))\,, \quad d_{\vec{k}_i}(1)=\gamma_-(\vec{k}_i)\,.
\end{split}$$ The first line comes from the requirement $\hat{O}_{\vec{k}_i}(0)=I$ and the second line comes from $\hat{O}(1)=\hat{U}$ or $\hat{O}_{\vec{k}_i}(1)=\hat{U}_{\vec{k}_i}$, where $\hat{U}_{\vec{k}_i}$ is the discrete form of Eq. : $$\hat{U}=\prod_{i}\hat{U}_{\vec{k}_i} \,,$$ with $$\label{discdecomp1}
\hat{U}_{\vec{k}_i}:=\exp[\gamma_+(\vec{k}_i)\hat{L}_+^{(\vec{k}_i)}]\exp[\ln\gamma_0(\vec{k}_i)\hat{L}_0^{(\vec{k}_i)}]\exp[\gamma_-(\vec{k}_i)\hat{L}_-^{(\vec{k}_i)}] \,.$$
Based on the function form in Eq. , the complexity for a particular operator $\hat{U}$ defined in Eq. can be obtained by following optimization problem $$\label{complCU2}
\begin{split}
&\Sigma_{d-1}^{-1}\mathcal{C}(\hat{U}[\gamma_+(\vec{k}),\gamma_-(\vec{k}),\gamma_0(\vec{k})])\\
=&\min\left\{\ell_0\int{\text{d}}^{d-1}k\int_0^1{\text{d}}s[\parallel\alpha_+(s,\vec{k})\parallel +\parallel\alpha_0(s,\vec{k})\parallel+\parallel\alpha_-(s,\vec{k})\parallel]\right\}
\end{split}$$ with the restrictions given by Eq. and Eq. . As these restrictions are independent for different $\vec{k}$, we can further write Eq. as $$\label{complCU3}
\begin{split}
&\Sigma_{d-1}^{-1}\mathcal{C}(\hat{U}[\gamma_+(\vec{k}),\gamma_-(\vec{k}),\gamma_0(\vec{k})])\\
=&\ell_0\int{\text{d}}^{d-1}k\left[\min\int_0^1{\text{d}}s\{\parallel\alpha_+(s,\vec{k})\parallel +\parallel\alpha_0(s,\vec{k})\parallel+\parallel\alpha_-(s,\vec{k})\parallel\}\right]
\end{split}$$ As the differential equations is highly nonlinear, for general values of $\gamma_+(\vec{k}),\gamma_-(\vec{k})$ and $\gamma_0(\vec{k})$, the optimization problem is not easy to solve. However, it is possible to find the complexity presented in Eqs. and . As the Eq. finds the minimal length in all the possible values of $\gamma_-(\vec{k})$ and $\gamma_0(\vec{k})$, the two of three functions $\alpha_\pm(s,\vec{k})$ and $\alpha_0(s,\vec{k})$ will be free. We can choose that $\alpha_-(s,\vec{k})$ and $\alpha_0(s,\vec{k})$ are free. Then Eqs. becomes, $$\label{FGTFDA3}
\begin{split}
\Sigma_{d-1}^{-1}\mathcal{C}(|\text{TFD}(t_L,t_R)\rangle,|0\rangle)&=\ell_0\int{\text{d}}^{d-1}k\left[\min\int_0^1{\text{d}}s\{\parallel\alpha_0(s,\vec{k})\parallel+\parallel\alpha_-(s,\vec{k})\parallel\right.\\
&+\left.\parallel b'_{\vec{k}}-\alpha_0(s,\vec{k}) b_{\vec{k}}+b^2_{\vec{k}}\alpha_-(s,\vec{k})\parallel\}\right] \,,
\end{split}$$ for arbitrary functions $\alpha^{(\vec{k})}_-, \alpha^{(\vec{k})}_0$ and $b_{\vec{k}_i}$ with $b_{\vec{k}_i}(0)=0$ and $b_{\vec{k}_i}(1)=\gamma_+(\vec{k})$. When $t=0$ in Eq. , the solution of can be obtain by the following method. As $\gamma_+(\vec{k})\in\mathbb{R}$ and $0<\gamma_+(\vec{k})\leq1$, we can naturally expect that the solution of Eq. is given in the case $ b_{\vec{k}}(s)\in\mathbb{R}$ and $|b_{\vec{k}}(s)|\leq1$. Then one can see that $$\label{ineqs1}
\begin{split}
&\int_0^1{\text{d}}s\{\parallel\alpha_0^{(\vec{k})}(s)\parallel+\parallel\alpha_-^{(\vec{k})}(s)\parallel+\parallel b'_{\vec{k}}(s)-\alpha_0(s,\vec{k}) b_{\vec{k}}(s)-b^2_{\vec{k}}(s)\alpha_-(s,\vec{k})\parallel\}\\
\geq&\int_0^1{\text{d}}s\{\parallel b_{\vec{k}}(s)\alpha_0^{(\vec{k})}(s)\parallel+\parallel b^2_{\vec{k}}(s)\alpha_-^{(\vec{k})}(s)\parallel+\parallel b'_{\vec{k}}(s)-\alpha_0(s,\vec{k}) b_{\vec{k}}(s)-b^2_{\vec{k}}(s)\alpha_-(s,\vec{k})\parallel\}\\
\geq&\int_0^1{\text{d}}s\{\parallel b_{\vec{k}}(s)\alpha_0^{(\vec{k})}(s)+b^2_{\vec{k}}(s)\alpha_-^{(\vec{k})}(s)+ b'_{\vec{k}}(s)-\alpha_0(s,\vec{k}) b_{\vec{k}}(s)-b^2_{\vec{k}}(s)\alpha_-(s,\vec{k})\parallel\}\\
=&\int_0^1{\text{d}}s|b_{\vec{k}}'(s)|\geq|\int_0^1b_{\vec{k}}'(s){\text{d}}s |=|\gamma_+(\vec{k})|\,.
\end{split} \nonumber$$ The final equality can be satisfied only when $\alpha_0(s,\vec{k})=\alpha_-(s,\vec{k})=0$ and $b_{\vec{k}}'(s)<0$ or $b_{\vec{k}}'(s)>0$ for $\forall s\in(0,1)$. When $t_L+t_R\neq0$, we see Im$\gamma_+(\vec{k})\neq0$. In this case, we have to separate every variable into the real part and the imaginary part firstly. Then we use the definition in Eq. to convert $\parallel\cdot\parallel$ into the usual absolute symbol. After that, we can use the Euler-Lagrange equation to find the minimal value in Eq. . The result still shows that the minimal value can be reached if $\alpha_0(s,\vec{k})=\alpha_-(s,\vec{k})=0$. Hence, we find that, $$\label{FGTFDA4}
\Sigma_{d-1}^{-1}\mathcal{C}(|\text{TFD}(t_L,t_R)\rangle,|0\rangle)=\ell_0\int{\text{d}}^{d-1}k\parallel\gamma_+(\vec{k})\parallel$$
[^1]: There are also other holographic proposals for complexity, see Refs. [@Alishahiha:2015rta; @Ben-Ami:2016qex; @Couch:2016exn] for examples.
[^2]: Refs. [@Caputa:2017urj; @Caputa:2017yrh] also give a definition for the complexity in conformal field theory by the Liouville action. It is also interesting to see what the results are if this method is applied to the time-dependent TFD states.
[^3]: The Finsler geometry was first introduced to investigate the computational complexity by Refs. [@Nielsen1133; @Nielsen:2006:GAQ:2011686.2011688; @Dowling:2008:GQC:2016985.2016986] and recently drew attention again in Ref. [@Jefferson:2017sdb].
[^4]: Because the complexity from any state to itself is zero, if this reference state is any state dual to an asymptotic AdS black hole then we can find Eq. or should be zero at this black hole. However, Ref. [@Carmi:2016wjl] has proven that Eqs. or are divergent for all asymptotically AdS black holes, so this reference state is not dual to any AdS black hole.
[^5]: Formally, Eq. is similar to the “complexity of formation” proposed by Ref. [@Chapman:2016hwi]. However, they are not always the same. A detailed discussion about it can be found in Ref. [@Yang:2017nfn].
[^6]: In principle, a well-defined complexity should satisfy that $\mathcal{C}(|\text{TFD}_2\rangle,|\text{TFD}_1\rangle)=0$ if and only if $|\text{TFD}_2\rangle=|\text{TFD}_1\rangle$. However, it may be possible that the modified version in Eq. vanish even with two different states. This can appear if the system has multiple different solutions for given physical conditions, for example, in the cases which contain phase transitions. Another possibility stems from the the fact that the TFD states is only approximately dual to the eternal AdS black holes so we can expect the original CA and CV conjectures in Eq. and may lose some subleading contributions.
[^7]: We assume that the curve is regular, which means that $0\leq|\partial x^a/\partial\lambda|<\infty$.
[^8]: When this paper was finished, Ref. [@Carmi:2017jqz] appeared which also studied the complexity growth rate. Our result Eq. is the same as Eq. (E.9) in Ref. [@Carmi:2017jqz].
[^9]: We have substituted $t_L + t_R$ for $2 t_B$ for comparison with the results obtained by other methods.
[^10]: Ref. [@Yang:2017nfn] proves that $\hat{U}^\dagger_a$ has a unitary partner, $$\label{defineGa}
\hat{G}_a:=\exp\left\{\int\text{arctanh}e^{-\pi\omega_{\vec{k}}/a}[\hat{a}^{R\dagger}(\vec{k})\hat{a}^{L\dagger}(\vec{k})-\hat{a}^{R}(\vec{k})\hat{a}^{L}(\vec{k})]{\text{d}}^{d-1}k\right\}\,,$$ which can also realize $|B \rangle\propto\hat{G}_a|0\rangle$.
[^11]: The zero point energy has been neglected, as it only contribute a constant factor on the state.
[^12]: Note that $\exp[-i(\hat{H}_Lt_L+\hat{H}_Rt_R)]\hat{U}_a^\dagger\neq\hat{U}^\dagger_a(t_L,t_R)$. The non-unitary operator $\hat{U}^\dagger_a(t_L,t_R)$ has a unitary partner: $$\label{defineGa2}
\hat{G}_a(t_L,t_R):=\exp\left\{\int[r_{\vec{k}}\hat{a}^{R\dagger}(\vec{k})\hat{a}^{L\dagger}(\vec{k})-r_{\vec{k}}^*\hat{a}^{R}(\vec{k})\hat{a}^{L}(\vec{k})]{\text{d}}^{d-1}k\right\}\,,$$ Thus the time evolution of the TFD state can be generated by two ways: $ |\text{TFD}(t_L,t_R)\rangle\propto\hat{U}^\dagger_a(t_L,t_R)|0\rangle\propto \hat{G}_a(t_L,t_R)|0\rangle$.
|
---
abstract: 'Hard and soft QCD dynamics are both important in charmonium hadroproduction, as presented here through a next-to-leading order QCD matrix element calculation combined with the colour evaporation model. Observed $x_F$ and $p_\perp$ distributions of $J/\psi$ in hadroproduction are reproduced. Quite similar results can also be obtained with a Monte Carlo event generator where [$c\bar{c}$]{} pairs are instead produced through leading order matrix elements and the parton shower approximation of higher order processes. The soft dynamics may alternatively be described by the soft colour interaction model. We also discuss the relative rates of different charmonium states and introduce an improved model for mapping the continuous [$c\bar{c}$]{} mass spectrum on the physical charmonium resonances.'
author:
- '[[C. Brenner Mariotto]{}]{}'
- 'M.B. Gay Ducati'
- 'G. Ingelman'
title: 'Soft and hard QCD in charmonium production [^1] '
---
[ address=[Inst. of Physics, Univ. Fed. do Rio Grande do Sul, Box 15051, CEP 91501-960 Porto Alegre, Brazil]{} ]{}
[ address=[Inst. of Physics, Univ. Fed. do Rio Grande do Sul, Box 15051, CEP 91501-960 Porto Alegre, Brazil]{} ]{}
[ address=[High Energy Physics, Uppsala University, Uppsala, Sweden and DESY, Hamburg, Germany]{} ]{}
The theoretical description of charmonium production separates the hard and soft parts of the process based on the factorisation theorem in QCD. Thus, we first consider the perturbative production of a [$c\bar{c}$]{} pair at the parton level and then the non-perturbative formation of a bound charmonium state [@epjc].
Perturbative QCD (pQCD) should be applicable for [$c\bar{c}$]{} production, since the charm quark mass $m_c$ is large enough to make $\alpha_s(m_c^2)$ a small expansion parameter. The leading order (LO) processes are $gg \rightarrow c\bar{c}$ and $q\bar{q} \rightarrow c\bar{c}$. The next-to-leading order (NLO) processes, [[*i.e.*]{}]{} ${\cal O}(\alpha_s^3)$, include the emission of a third parton and virtual corrections (where divergences are properly cancelled). The full NLO matrix elements, with explicit charm quark mass, are available in a computer program [@NLO] giving total and differential cross sections.
An alternative description of the pQCD production of [$c\bar{c}$]{} pairs is given by the [[Pythia]{}]{} [@Pythia] Monte Carlo, where all LO QCD $2\to 2$ processes are included with their corresponding matrix elements and the incoming and outgoing partons may branch as described by the DGLAP equations. A [$c\bar{c}$]{} pair can then be produced as described by the LO matrix elements for $q\bar{q} \rightarrow c\bar{c}$ and $gg \rightarrow c\bar{c}$ (with explicit $m_c$ dependence) or in a gluon splitting $g \rightarrow c\bar{c}$ in the parton shower.
The main free parameter is the charm quark mass $m_c$, taken as $m_c=1.5$ GeV in the NLO program and $m_c=1.35$ GeV in [[Pythia]{}]{}. In both approaches, the factorization and renormalization scales are taken as the average transverse mass of the $c$ and $\bar{c}$.
The formation of bound hadron states occurs through processes with small momentum transfers such that $\alpha_s$ is large and prevents the use of perturbation theory. The lack of an appropriate method to calculate non-perturbative processes, forces us to use phenomenological models to describe the formation of charmonium states from perturbatively produced [$c\bar{c}$]{} pairs. The Color Evaporation Model (CEM) [@HALZENquantit] and the Soft Colour Interaction (SCI) model [@sci; @sci-onium] are based on a similar phenomenological approach, where soft colour interactions can change the colour state of a [$c\bar{c}$]{} pair from an octet to a singlet. They employ the same hard pQCD processes to produce a [$c\bar{c}$]{} pair regardless of its spin state. A colour singlet [$c\bar{c}$]{} pair with an invariant mass below the threshold for open charm ($m_{c\bar{c}}<2m_D$) will then form a charmonium state.
In CEM [@HALZENquantit; @satz; @schuler; @CEMnosso] the exchange of soft gluons is assumed to give a randomisation of the colour state. This implies a probability $1/9$ that a $c\bar{c}$ pair is in a colour singlet state and produces charmonium if its mass is below the threshold for open charm production, $m_{c\bar{c}}<2m_D$. The fraction of a specific charmonium state $i$, relative to all charmonia, is given by a non-perturbative parameter $\rho_{i}$ ($\rho_{J/\psi}=0.4-0.5$) [@HALZENquantit].
In SCI [@sci; @sci-onium; @gaps] it is assumed that colour-anticolour, corresponding to non-perturbative gluons, can be exchanged between partons emerging from a hard scattering and hadron remnants. The unknown probability to exchange a soft gluon between parton pairs is given by a phenomenological parameter $R$. These colour exchanges lead to different topologies of the confining colour string-fields and thereby to different hadronic final states after hadronisation. The mapping of $c\bar{c}$ pairs, with mass below the threshold for open charm production, is here made based on spin statistics resulting in a fraction of a specific quarkonium state $i$ with total angular momentum $J_i$ given by $f_i = \frac{\Gamma_i}{\sum_k \Gamma_k} $, where $\Gamma = (2J_i+1)/n_i$ including a suppression of radially excited states through the main quantum number $n_i$. This model was found to give a correct description of the different heavy quarkonium states observed at the Tevatron [@sci-onium].
The complete models are formed by adding the CEM or SCI models for the soft processes to any of the descriptions for the hard pQCD processes. The first model we label [**CEM-NLO**]{} and is the combination of the CEM model with the NLO program. The second model is [**CEM-PYTHIA**]{}, where CEM has been implemented in [[Pythia]{}]{} version 5.7 [@Pythia]. The third model, [**SCI-PYTHIA**]{}, is to use the SCI model as implemented in [[Pythia]{}]{} 5.7. Further ingredients are the intrinsic $k_{\perp}$, due to the Fermi motion of partons inside the initial state hadrons, and soft $p_T$ in soft gluon exchange that neutralize color. Both effects are modelled by a gaussian distribution of width $0.6 - 0.8$ GeV used in [[Pythia]{}]{} and in the NLO program.
Comparing these three models we can separate different effects. With CEM implemented in the NLO program and in [[Pythia]{}]{}, we can compare the pQCD contributions, namely NLO versus LO plus the parton shower approximation of higher orders. Having SCI and CEM implemented in [[Pythia]{}]{}, we can explicitly compare these two non-perturbative models and see to what extent they can account for observed soft effects.
Detailed comparisons between the models have been done as well as extensive comparison with data, both from fixed target experiments and the Tevatron collider [@epjc]. Here we limit ourselves to proton beams. The targets are different nuclei, but the experimental results are rescaled to the cross section per nucleon. Thus we compare directly with our models which do not include any nuclear effects but treat hadron-nucleon interactions.
Fig. \[pp\] shows $x_F$ and $p_\perp$ distributions of the produced $J/\psi$ for proton beams of different energies. As can be seen, the data are approximately reproduced, both in shape and normalization, by all three models. Looking into the details of the [$x_{F} \,$]{} distributions, one can observe that the model curves fall less steeply than the data and therefore overshoot somewhat at large [$x_{F} \,$]{}. The observed [$p_{\perp} \,$]{}distribution is better reproduced, with only small differences between the models.
![Distributions in $x_F$ and $p_\perp^2$ of $J/\psi$ produced with proton beams of energies $800$, $530$ and $300\, GeV$ on fixed target. Data [@pp800; @300; @pp800k; @pp530] compared to CEM based on NLO pQCD matrix elements, and CEM and SCI based on LO matrix elements plus parton showers in the [[Pythia]{}]{} Monte Carlo.\[pp\]](fig.3.pplcemsci.eps){height=".625\textheight"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![Distributions in $x_F$ and $p_\perp^2$ of $J/\psi$ (in 800 GeV proton on proton as in Fig. \[pp\]) from variations of the pQCD treatment.\[fig:xfteori\] CEM based on the NLO program with $m_c=1.5$ GeV: NLO and LO matrix elements, NLO with no intrinsic $k_\perp$. CEM based on [[Pythia]{}]{} with $m_c=1.35$ GeV: LO matrix elements plus parton showers (PS) and PS contribution shown separately.](fig.4a.xfpp800teoril.eps "fig:"){height=".263\textheight"} ![Distributions in $x_F$ and $p_\perp^2$ of $J/\psi$ (in 800 GeV proton on proton as in Fig. \[pp\]) from variations of the pQCD treatment.\[fig:xfteori\] CEM based on the NLO program with $m_c=1.5$ GeV: NLO and LO matrix elements, NLO with no intrinsic $k_\perp$. CEM based on [[Pythia]{}]{} with $m_c=1.35$ GeV: LO matrix elements plus parton showers (PS) and PS contribution shown separately.](fig.4b.ptpp800teoril.eps "fig:"){height=".263\textheight"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
Having CEM combined with different treatments of the pQCD production of [$c\bar{c}$]{}, we can now investigate pQCD effects in more detail. Fig. \[fig:xfteori\] illustrates this for the case of 800 GeV proton energy, similar conclusions can also be drawn for other energies and beam particles. For the [$x_{F} \,$]{} distribution in Fig. \[fig:xfteori\]a, the full NLO result and that based on LO+PS agree reasonably well. The NLO corrections are very important, as we see by comparing the LO and the full NLO results. In the LO+PS result, however, the PS contribution is unimportant for the overall cross section which is dominated by the LO [$c\bar{c}$]{} production. The agreement with the NLO result is here obtained by using a lower charm mass, $m_c=1.35$ GeV. We have cross-checked this within the NLO program, where the full result is essentially reproduced by the LO part if this lower mass value is used. This demonstrates that the NLO correction is essentially an overall $K$-factor from soft and virtual corrections. For the [$p_{\perp} \,$]{}distributions in Fig. \[fig:xfteori\]b, the NLO program gives a [$p_{\perp} \,$]{}distribution with a much larger tail at large [$p_{\perp} \,$]{}, but it is still substantially affected by the inclusion of the intrinsic $k_\perp$ at the limited values of [$p_{\perp} \,$]{}accessible at fixed target energies. The [$p_{\perp} \,$]{}distribution resulting from the LO+PS in the [[Pythia]{}]{} approach, is at high-[$p_{\perp} \,$]{}dominated by [$c\bar{c}$]{} from gluon splittings in the partons showers, whereas the bulk of the cross section comes from the low-[$p_{\perp} \,$]{}region where the LO diagrams dominate. The total LO+PS result, which also includes a gaussian intrinsic $k_\perp$ with the same width 0.6 GeV, agrees quite well with the NLO result.
Data on [${\psi}^{\prime}$]{} production provide an additional testing ground for the models, which produce all charmonium states with the same dynamics. A comparison made in [@epjc] shows that all models account quite well for the shape of the distributions. The proper normalization of CEM is obtained by chosing $\rho_{\psi^{\prime}}=0.066$. The spin statistics used in SCI predicts only a factor two suppression of [${\psi}^{\prime}$]{}, and must be lowered by an additional factor four in order to reproduce the data. This has prompted us to develop a more elaborate model for turning [$c\bar{c}$]{} pairs into different charmonium resonances [@epjc], which is briefly described here.
The [$c\bar{c}$]{} pair is produced in a pQCD process with a continuous distribution of its invariant mass $m_{c\bar{c}}$ and must be mapped onto the discrete spectrum of charmonium states. The soft interactions that turn the pair into a colour singlet and form the state, may very well change its mass by a few hundred MeV, which is the typical scale of the soft interactions. We model this by a gaussian smearing of a few hundred MeV. The probability to end up in a specific resonance, shown in Fig. \[dQ2PsDserr\], is then proportional to the superposition of this gaussian with the resonance peak, times the corresponding spin-statistics factor. The smearing of $m_{c\bar{c}}$ across the threshold $2m_D$ for open charm, implies non-zero contributions for charmonium also above the $D\overline{D}$ threshold as well as some open charm production for $m_{c\bar{c}}$ originally below this threshold.
![Probability distributions for the different charmonium states as obtained in the model with gaussian smearing ($\sigma_{sme}=400\, MeV$). The resulting total probability for charmonium production and the remainder as open charm production are also shown.[]{data-label="dQ2PsDserr"}](fig.9.probress.eps){height=".3\textheight"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![The ratio of [${\psi}^{\prime}$]{} to [$J/\psi$]{} production (times their branching ratios for decay into $\mu^+\mu^-$) (left) and fractions of [$J/\psi$]{} produced directly, and coming from the decay of $\chi_c$ and $\psi^{\prime}$ states (right) in hadron-hadron interactions of cms energy $\sqrt{s}$. Data [@Na38p450; @high-pt-data; @ppi300E705; @pi515koreshev] compared to simple spin statistics and to our model with different gaussian smearing widths applied to CEM.[]{data-label="dQ2gaussfract"}](fig.10.primepsi.eps "fig:"){height=".30201\textheight"} ![The ratio of [${\psi}^{\prime}$]{} to [$J/\psi$]{} production (times their branching ratios for decay into $\mu^+\mu^-$) (left) and fractions of [$J/\psi$]{} produced directly, and coming from the decay of $\chi_c$ and $\psi^{\prime}$ states (right) in hadron-hadron interactions of cms energy $\sqrt{s}$. Data [@Na38p450; @high-pt-data; @ppi300E705; @pi515koreshev] compared to simple spin statistics and to our model with different gaussian smearing widths applied to CEM.[]{data-label="dQ2gaussfract"}](fig.11.rates.eps "fig:"){height=".39401\textheight"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
By folding these charmonium probability functions with the distribution in $m_{c\bar{c}}$ obtained from pQCD, one gets the cross section for a given charmonium state. Applying this mapping procedure to the CEM model we obtain the results in Fig. \[dQ2gaussfract\]. As opposed to the simple spin statistics factor, this model gives a reasonable description of the observed ratio of [${\psi}^{\prime}$]{} to [$J/\psi$]{} production and fractions of [$J/\psi$]{} produced directly, coming from decays of $\chi_c$ states and from $\psi^{\prime}$. In particular, the model gives a characteristic energy dependence of the kind indicated by the data.
In summary, both hard and soft QCD dynamics play important roles in the production of charmonium states in hadronic interactions. The [$c\bar{c}$]{} pair production in pQCD have substantial higher order contributions, with a factor two increase of the total cross section from NLO corrections. These come mainly from soft and collinear gluon emissions combined with virtual corrections and can be effectively accounted for by an overall $K$-factor. This supports to the use of the [[Pythia]{}]{} Monte Carlo with LO matrix elements and a reduced charm quark mass to increase the cross section correspondingly. The high [$p_{\perp} \,$]{}tail of the cross section is, however, dominated by higher order tree diagrams in the NLO matrix elements and in the parton showers of the Monte Carlo approach.
The non-perturbative formation of [$J/\psi$]{}, can be described by the Colour Evaporation Model and the Soft Colour Interaction model, where [$c\bar{c}$]{} pairs in a colour octet state can turn into a colour singlet state by soft gluon exchange. A simple spin statistics factor is not sufficient for a proper description of other charmonium states, but our more elaborated model to map [$c\bar{c}$]{} pairs onto the physical charmonium states improves this situation.
To conclude, the main features of hadroproduction of charmonium can be described in these models combining pQCD and effects of soft colour exchanges. This shows, in particular, that these models for the soft QCD dynamics contain the essential effects and therefore improve our understanding of non-perturbative QCD.
This work was partially financed by Fundação Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Brazil, and by the Swedish Research Council.
C. Brenner Mariotto, M.B. Gay Ducati, G. Ingelman, Eur. Phys.J. C [**23**]{}, (2002) 527
M.L. Mangano, P. Nason, G. Ridolfi, Nucl. Phys. B [**373**]{}, (1992) 295
T. Sjöstrand, Computer Phys. Commun. [**82**]{}, (1994) 74
J.F. Amundson [[*et al.*]{}]{}, Phys. Lett. B [**390**]{}, (1997) 323; O.J.P. Éboli, E.M. Gregores, F. Halzen, Phys. Rev. D [**60**]{}, (1999) 117501
A. Edin, G. Ingelman, J. Rathsman, Phys. Lett. B [**366**]{}, 371 (1996); Z. Phys. C [**75**]{} (1997) 57
A. Edin, G. Ingelman, J. Rathsman, Phys. Rev. D [**56**]{}, (1997) 7317
R. Gavai [*et al.*]{}, Int. J. Mod. Phys. A [**10**]{}, (1995) 3043
G.A. Schuler, R. Vogt, Phys. Lett. B [**387**]{}, (1996) 181
M.B. Gay Ducati, C.B. Mariotto, Phys. Lett. B [**464**]{}, (1999) 286\
M.B. Gay Ducati, V.P. Goncalves, C.B. Mariotto, Phys. Rev. D [**65**]{} (2002) 037503
R. Enberg, G. Ingelman, N. Timneanu, J. Phys. G: Nucl. Part. Phys. [**26**]{}, (2000) 712; Phys. Rev. D [**64**]{}, (2001) 114015
M.H. Schub [*et al*]{}, Phys. Rev. D [**52**]{}, (1995) 1307; T. Alexopoulos [*et al*]{}, Phys. Rev. D [**55**]{}, (1997) 3927
L. Antoniazzi [*et al*]{}, Phys. Rev. D [**46**]{}, (1992) 4828
M.S. Kowitt [*et al*]{}, Phys. Rev. Lett. [**72**]{}, (1994) 1318
A. Gribushin [*et al*]{}, Phys. Rev. D [**62**]{}, (2000) 012001
M.C. Abreu [*et al*]{}, Phys. Lett. B [**444**]{}, (1998) 516, and references therein
F. Abe [*et al*]{}, Phys. Rev. Lett. [**75**]{}, (1995) 4358; [**79**]{}, (1997) 572; [**79**]{},(1997) 578
L. Antoniazzi [*et al*]{}, Phys. Rev. Lett. [**70**]{}, (1993) 383
V. Koreshev [*et al*]{}, Phys. Rev. Lett. [**77**]{}, (1996) 4294
[^1]: Based on the talk presented by C. Brenner Mariotto at the Pan American Advanced Studies Institute (PASI 2002), Campos do Jordão, Brazil, January 7-18, 2002.
|
---
abstract: 'When dealing with time series with complex non-stationarities, low retrospective regret on individual realizations is a more appropriate goal than low prospective risk in expectation. Online learning algorithms provide powerful guarantees of this form, and have often been proposed for use with non-stationary processes because of their ability to switch between different forecasters or “experts”. However, existing methods assume that the set of experts whose forecasts are to be combined are all given at the start, which is not plausible when dealing with a genuinely historical or evolutionary system. We show how to modify the “fixed shares” algorithm for tracking the best expert to cope with a steadily growing set of experts, obtained by fitting new models to new data as it becomes available, and obtain regret bounds for the growing ensemble.'
author:
- |
Cosma Rohilla Shalizi\
Carnegie Mellon University\
Pittsburgh, PA\
`cshalizi@cmu.edu`\
Abigail Z. Jacobs\
Northwestern University\
Evanston, IL\
`azjacobs@northwestern.edu`\
Kristina Lisa Klinkner\
Amazon.com\
Seattle, WA\
`klinkner@amazon.com`\
Aaron Clauset\
University of Colorado\
Boulder, CO\
`aaron.clauset@colorado.edu`\
bibliography:
- 'locusts.bib'
title: 'Adapting to Non-stationarity with Growing Expert Ensembles'
---
Introduction
============
Non-stationarity is ubiquitous in the study of real time series; macroeconomic statistics, climate records and gene expression levels are all prominent examples, as are important engineering problems of signal processing and anomaly detection. Sometimes the nonstationarity is harmless, as when the data come from a homogeneous Markov process, or more generally from a conditionally stationary [@Caires-Ferreira-prediction-of-cond-stat-seq] source, since then the best prediction for each historical context is invariant, though various contexts become more or less common. More generally, however, non-stationary processes have trends, so the predictive implications of any given historical context changes over time.
Time series textbooks (such as [@Shumway-Stoffer]) advise turning non-stationary processes into stationary ones, by, e.g., subtracting off trends and then analyzing the residuals as a stationary process. If there are multiple independent replicas of the process, all with the same trend, the latter could be estimated non-parametrically. If there is only one realization, systematically estimating the trend needs a well-specified parametric model embracing both trend and fluctuations. Time series from complex systems, however, typically lack parametric models of trends deserving much credence. In macroeconomics, for instance, the state-of-the-art models are all for stationary fluctuations, and trends are identified by [*ad hoc*]{} procedures, most often spline smoothing[^1] [@DeJong-Dave-structural-macro].
The fundamental problem is that in many complex, evolving systems, the low-dimensional variables we happen to measure may develop in basically unpredictable ways. Old patterns may become completely irrelevant, even actively misleading. We could try to identify change-points and start modeling afresh at each break, but there is often little [*a priori*]{} reason to think that non-stationarities [*will*]{} take the form of abrupt breaks, as opposed to more gradual transitions, to say nothing of all of the difficulties which plague change-point detection.
While a non-stationary process [*could*]{} evolve in a totally capricious fashion, more often there are at least local periods where the predictive relationship between history and future does not change too rapidly. For stationary processes, this relationship is fixed and can be learned nonparametrically [@Ornstein-Weiss-how-sampling-reveals-a-process; @Algoet-universal-schemes], leading to forecasts with low risk, i.e., low expected loss on new data. In contrast, we follow the individual-sequence forecasting literature [@prediction-learning-and-games] in wanting to have low **regret** relative to a given collection of models — no matter what sample path the process realizes, we want to have done nearly as well the model, or sequence of models, which in hindsight proves to have forecast best. It is hard to see how we could go beyond bounds on retrospective regret to bounds on prospective risk in the face of arbitrary, unknown non-stationarities; to do so would be tantamount to solving the problem of induction.
We start from algorithms for “prediction with expert advice,” which adaptively combine the forecasts of an ensemble of models or “experts” so as to guarantee low regret. We focus on versions of the “exponentially-weighted average forecaster” [@Littlestone-Warmuth-WM; @Vovk-aggregating] (or “multiplicative weight training” [@Arora-Hazan-Kale-multiplicative-weights]), which forecasts a weighted combination of the predictions of the experts in its ensemble, with weights being multiplied up or down as experts do better or worse than the ensemble average. Using ${q}$ experts over $n$ rounds, this guarantees a regret, with respect to the retrospectively-best single expert, of no more than $O(\sqrt{n \ln{{q}}})$.
If instead of combining individual experts, we combine [*sequences*]{} drawn from an ensemble of base experts, we can “track the best expert”. Specifically, the regret compared to a sequence where the base expert is switched at most $m$ times follows the same form, but with the number of such sequences in place of the number of base experts ${q}$; some combinatorics [@prediction-learning-and-games] gives a bound of $O(\sqrt{n\left((m+1)\ln{{q}} + (n-1)H(\frac{m}{n-1})\right)})$, with $H(p) = -p\ln{p}-(1-p)\ln{(1-p)}$ being the binary entropy function, appearing here via Stirling’s formula. The “fixed shares” algorithm introduced by [@Herbster-Warmuth-tracking-the-best] implements this with only ${q}$ weights, not a combinatorially-large number.
These algorithms are not quite suitable for the problems we have in mind, however, because they presume that all experts in the ensemble are present at the start. Low regret relative to such an ensemble is not very comforting: none of the experts might be much good, because one is faced with conditions very different from any anticipated when the experts were set up. One could allow each expert to adapt — rather than being a fixed forecasting rule, regard each expert as estimating some statistical model from (some part) of the sequence, and then forecasting on that basis. This actually requires [*no*]{} change to results for, for example, the fixed-share forecaster (see below), because the conditions of the theorems put no limit on how the experts’ forecasts depend on the past, just that they do (measurably).
Our proposal therefore is to [*grow*]{} the ensemble, adding a new expert every $\tau$ time-steps. To cope with non-stationarity, which would mean that old data becomes irrelevant, the expert added at time $k\tau$ is fitted to the data from $(k-1)\tau+1$ onwards[^2], and thereafter is free to keep on updating its parameter estimates and, of course, its predictions, using new data. As the ensemble grows, the oldest model is always fitted to the complete time series, followed by successively younger models which omit more and more of the oldest data, until the very youngest model only fits to the last $\tau$ steps or less. There is thus always an expert which is fitted to the whole data stream, and at the other extreme an expert fitted only to the most recent data. If we can prove a regret bound for this growing ensemble, we will have something which performs (nearly) as well as a rule which uses the whole of the data, presumably optimal in the stationary case, as well as performing (nearly) as well as an expert using only the last $\tau$ observations, as would be suitable in case of a profound change-point or structural break.
We show how to modify the fixed shares algorithm to efficiently work with such an ensemble, while still providing an $o(n)$ bound on tracking regret. §\[sec:setting\] fixes the setting and notation. §\[sec:growing-ensemble\] introduces the exponentially-weighted forecaster over expert sequences drawn from a growing ensemble, and our modification of the fixed-shares forecaster. The major results, the equivalence of the two forecasters and the regret bound, follow in §§\[sec:equivalence\]–\[sec:regret-bounds\]. §\[sec:example\] presents an empirical example from macroeconomics. §\[sec:discussion\] contrasts our approach with previous work, and discusses its methodological significance.
Setting and Notation {#sec:setting}
====================
We follow the usual setting of individual-sequence forecasting [@prediction-learning-and-games]. At each discrete time $t \in 1, 2,
\ldots n$, Nature produces an observation $y_t \in \mathcal{Y}$. Nature may be deterministic, stochastic, or even a clever and deceitful Adversary. Our forecaster has access to a set of experts (for us, a set depending on $t$), with the $i^{\mathrm{th}}$ expert predicting $f_{i,t} \in \mathcal{D}$. (The “prediction” could be an action, but for concreteness we will only talk about predictions.) The forecaster also has available the data $y_1, y_2, \ldots
y_{t-1}$, and combines this, along with the advice of the experts, to give a prediction $\hat{p}_t \in \mathcal{D}$. After the forecaster makes its prediction, it learns $y_t$, leading to losses $\ell(f_{i,t},y_t)$ for the experts and $\ell(\hat{p}_t,y_t)$ for the forecaster.
The aim of the forecaster is to have predicted almost as well as the best expert, or even the best sequence of experts, no matter what Nature does. The [**tracking regret**]{} of the forecaster with respect to a sequence of experts $i_1, i_2, \ldots i_n$ is the difference in their cumulative losses: $$R(i_1, i_2, \ldots i_n) = \sum_{t=1}^{n}{\ell(\hat{p}_{t}, y_{t})} - \sum_{t=1}^{n}{\ell(f_{i_t,t},y_{t})}$$ Good forecasting strategies have regrets which can be bounded uniformly over both expert sequences and observation sequences $y_1, \ldots y_n$. Ideally the bound would be $o(n)$, so that the regret per unit time goes to zero; in that case the forecaster is “Hannan-consistent.”
Some convenient abbreviations: $y_s^t$ is the sub-sequence of observations $y_s, y_{s+1}, \ldots y_{t-1}, y_t$, and likewise for other sequence variables. Further abbreviate $\ell(f_{i,t},y_t)$ by $\ell(i_{t},y_t)$, and $\sum_{r=s}^{t}{\ell(i_r, y_r)}$ by $\ell(i_s^t,Y_s^t)$. Regret is then $$R(i_1^n) = \ell(\hat{p}_1^n,y_1^n) - \ell(i_1^n,y_1^n)$$
The basic forecasters
---------------------
#### The exponentially weighted average forecaster
[@Littlestone-Warmuth-WM; @Vovk-aggregating] Given an ensemble of ${q}$ experts, initial (positive) weights $w_{i,0}$, and a learning rate $\eta > 0$, this forecaster predicts by a weighted average[^3], $$\hat{p}_t = \frac{\sum_{i=1}^{{q}}{w_{i,t-1} f_i,t}}{\sum_{j=1}^{{q}}{w_{j,t-1}}}$$ and updates the weights by $$w_{i,t} = w_{i,t-1}e^{-\eta\ell(f_{i,t},y_t)}$$ This can be seen as a version of reinforcement learning, or as Bayes’s rule (if $\ell$ is negative log-likelihood), or as the evolutionary replicator dynamic, with time-dependent fitness function $e^{-\eta\ell(f_{i,t},y_t)}$ \[removed for anonymous submission\].
As mentioned above, the regret of the EWAF is $O(\sqrt{n \ln{{q}}})$ [@prediction-learning-and-games]. If each member of the EWAF’s ensemble is actually a sequence over some class of base experts, we get a forecaster which can keep low regret even if the best expert to use changes; the cost, however, is keeping around a combinatorially-large number of weights. The fixed shares forecaster, described next, achieves the same results with only one weight for each base expert, by modifying the manner in which weights are updated.
#### The fixed shares forecaster
[@Herbster-Warmuth-tracking-the-best] We have ${q}$ experts, each with a time-varying weight, and the forecast is, as before, a convex combination: $$\hat{p}_{t} = \frac{\sum_{i=1}^{{q}}{w_{i,t-1} f_{i,t}}}{\sum_{j=1}^{{q}}{w_{j,t-1}}}$$ Initially, all weights are equal, $w_{i,0} = 1/{q}$. The update equations are $$w_{i,t} = (1-\alpha)v_{i,t} + \alpha \frac{\sum_{i=1}^{{q}}v_{i,t}}{{q}}$$ where $$v_{i,t} = w_{i,t-1}e^{-\eta\ell(i,y_{t})}$$ and $\alpha \in [0,1]$ is another control setting. In words, weights update almost exactly as in the EWAF, except that weight is shared so that no expert ever falls below a fraction $\alpha$ of the total weight. As shown in [@Herbster-Warmuth-tracking-the-best], this matches the behavior of exponential weighting over expert sequences, provided the initial weights of sequences are [*not*]{} all equal; the weights are in fact chosen to depend on the number of times the sequence changes expert, peaking when the number of switches is about $\alpha n$.
Growing Ensemble Forecasters {#sec:growing-ensemble}
============================
The Growing Ensemble
--------------------
We start with a single expert. We divide the time series into “epochs,” each of length $\tau$, and add a new expert at the beginning of each epoch.[^4] When added, the new expert is trained only on the data in the previous epoch. The number of experts at time $t$ is ${q}_t = 1 + \lfloor t/\tau
\rfloor$. By time $n$, when the ensemble has ${q}_n = 1+ \lfloor
\frac{n}{\tau}\rfloor$ experts, one is trained over all data from time 1 to $n$, one on data from $1+\tau$ to $n$, one on $1+2\tau$ to $n$, and so on, down to one trained on the last $n \mod \tau$ observations. The hope is that this will let us cope with abrupt structural breaks (within at most $\tau$ time-steps), gradual drift, and, of course, actual stationarity.
Exponentially-Weighted Averaging over the Growing Ensemble
----------------------------------------------------------
To obtain a low tracking regret, we wish to run EWAF over sequences of experts from the growing ensemble, limiting it, of course, to only using experts which are currently available. During the first $\tau$ time steps, there is only one expert, but either of two experts can be used at any time from $t=1+\tau$ to $t=2\tau$, any of three experts from $t=1+2\tau$ to $3\tau$, etc. Even limiting ourselves to sequences which switch experts no more than $m$ times still leaves a combinatorially-large number of base-expert sequences, though smaller than what would be the case if all ${q}_n$ final experts were available from the beginning.
We will write the weight of the expert sequence $i_1^n$ at time $t$ as ${\phi}_t(i_1^n)$. It is of course $${\phi}_t(i_{1}^n) = {\phi}_{t-1}(i_1^n)e^{-\eta\ell\left(i_t,y_t\right)} = {\phi}_0(i_{1}^n)e^{-\eta\ell\left(i_{1}^t, y_{1}^t\right)}
\label{eqn:ewa-updating}$$ We may regard ${\phi}_t$ as a measure on the space of expert sequences of length $n$, which defines measures on sub-sequences by summation; by a slight abuse of notation we will also write them as ${\phi}_t$, so ${\phi}_t(i_s^t) = \sum_{i_1^{s-1},i_{t+1}^n}{{\phi}_t(i_1^{s-1},
i_s^t, _{t+1}^n)}$.
We propose the following scheme of initial weights ${\phi}_0(i_1^n)$. Its main virtue is that it can be emulated by a direct modification of the fixed-shares forecaster, described immediately below. We prove the emulation result as Theorem \[thm:fixed-shares-emulates-compound-experts\]. $$\frac{{\phi}_0(i_{1}^{t + 1})}{{\phi}_0(i_{1}^{t})}
= \left\{
\begin{array}{ll}
& 0 ~\mathrm{if} ~ i_{t+1} > {q}_{t+1}\\
& \beta_t ~\mathrm{if} ~ t\bmod\tau = 0 ~\mathrm{and}~ i_{t+1} = {q}_{t+1}\\
& \frac{\alpha}{{q}_{t+1}} + (1 - \alpha)\mathbf{1}_{\{ i_{t + 1} = i_{t}\}} ~\mathrm{otherwise}
\end{array}
\right.$$ That is, $\beta_t$ controls the weight assigned to an expert when it enters the ensemble (and has no track record of losses). The choice of $\beta$ is an important issue, to which we return in the conclusion. For the rest of this paper, however, we set $\beta = \frac{\alpha}{{q}_{t+1}}$, so that $$\frac{{\phi}_0(i_{1}^{t + 1})}{{\phi}_0(i_{1}^{t})}
= \frac{\alpha}{{q}_{t+1}} + (1 - \alpha)\mathbf{1}_{\{ i_{t + 1} = i_{t}\}}.$$ We abbreviate $\mathbf{1}_{\{ i_{t + 1} = i_{t}\}}$ by $\chi_t$, suppressing explicit dependence on the sequence of actions. Setting the base condition ${\phi}_0(1) = 1$ (because every sequence must begin with the single expert available at the start), this recursively defines the initial weights for all sequences of experts: $$\begin{aligned}
\label{eqn:initial-initial-weight}
{\phi}_0(1) & = & 1\\
\label{eqn:initial-weight-recursion}
{\phi}_0(i_1^{t+1}) & = & {\phi}_0(i_1^t)\left(\frac{\alpha}{{q}_{t+1}} + (1-\alpha)\chi_t\right)\end{aligned}$$ with the restriction $i_t \leq {q}_t$ understood.
Growing-ensemble fixed shares forecaster
----------------------------------------
The number of sequences of length $n$ from the growing ensemble is too large to keep track of weights for each one, so, following the lead of [@Herbster-Warmuth-tracking-the-best], we introduce a fixed-shares procedure which will turn out to match the weights induced by Eqs.\[eqn:ewa-updating\] and \[eqn:initial-weight-recursion\].
At time $t$, each of the ${q}_t$ experts has a weight ${w}_{i,t}$. Initially, ${w}_{1,0} = 1$. Thereafter, weights update following the static ensemble procedure almost exactly. For $ 1 \leq i \leq {q}_t$, $$\begin{aligned}
\label{eqn:growing-ensemble-fixed-shares-update-1}
v_{i,t} & = & w_{i,t-1}e^{-\eta\ell(i,y_{t})}\\
\label{eqn:growing-ensemble-fixed-shares-update-2}
{w}_{i,t} & = & (1-\alpha)v_{i,t} + \frac{\alpha}{{q}_t} \sum_{i=1}^{{q}_t}{v_{i,t}}\end{aligned}$$ and $w_{i,t} = 0$ for $i > {q}_t$. Prediction, as always, is a convex combination, $\hat{p}_t = \sum_{i=1}^{{q}_t}{{w}_{i,t-1} f_{i,t}}/\sum_{j=1}^{{q}_t}{{w}_{j,t-1}}$.
Equivalence of Fixed Shares and Exponentially-Weighted Sequences {#sec:equivalence}
----------------------------------------------------------------
Following [@Herbster-Warmuth-tracking-the-best], we show that the fixed shares algorithm assigns the same weight to any given base expert, at any given time, as it gets from the exponentially-weighted averaging forecaster applied to base-expert sequences. This implies that they have the same behavior, and in particular the same regret bounds. Our proof is based on that from [@prediction-learning-and-games Theorem 5.1, p. 103].
Let ${\phi}_{j,t} = \sum_{i_1^n: i_{t+1} =j}{{\phi}_t(i_1^n)}$. If ${\phi}_0$ is set by Eq. \[eqn:initial-weight-recursion\], and ${w}$ updates by Eqs. \[eqn:growing-ensemble-fixed-shares-update-1\]–\[eqn:growing-ensemble-fixed-shares-update-2\], then for all $j$ and $t$ and $y_1^n$, ${\phi}_{j,t} = {w}_{j,t}$. \[thm:fixed-shares-emulates-compound-experts\]
[*Comment:*]{} Recall that the EWAF will use ${\phi}_{t-1}$, not ${\phi}_t$, to make its forecast at time $t$.
#### Proof
By induction on $t$. When $t=0$, by construction, ${w}_{1,0} = 1$, and ${w}_{j,0} = 0$ for all $j > 1$. But this is true for ${\phi}_{j,0}$ as well, by Eq. \[eqn:initial-initial-weight\]. For the inductive step from $t-1$ to $t$, assume ${w}_{j,s} =
{\phi}_{j,s}$ for all $j$ and for all $s < t$. Write ${\phi}_{i,t}$ as a “sum over histories”, using Eq. \[eqn:ewa-updating\]. $$\begin{aligned}
{\phi}_{i,t}
& = & \sum_{i_{1}^{t}, i_{t + 2}^{n}}{ {\phi}_t(i_{1}^{t}, i, i_{t+2}^{n})}
= \sum_{i_{1}^{t}}{e^{- \eta \ell\left(i_{1}^t,Y_{1}^t\right)} {\phi}_0(i_{1}^{t}, i)}
= \sum_{i_{1}^{t}}{e^{- \eta \ell\left(i_{1}^t,Y_{1}^t \right)} {\phi}_0(i_{1}^{t}) \frac{{\phi}_0(i_{1}^{t}, i)}{{\phi}_0(i_{1}^{t})}}\\
& = & \sum_{i_{1}^{t}} {e^{- \eta \ell\left(i_{1}^t,Y_{1}^t \right)} {\phi}_0(i_{1}^{t})\left(\frac{\alpha}{{q}_{t}} + (1 - \alpha)\chi_t \right)}\end{aligned}$$ by Eq. \[eqn:initial-weight-recursion\]. Moving the losses through time $t-1$ into the weights, $$\begin{aligned}
& = & \sum_{i_{1}^{t}}{{\phi}_{t - 1}(i_{1}^{t})e^{- \eta\ell\left(i_{t}, y_{t}\right)}\left(\frac{\alpha}{{q}_{t}} + (1 - \alpha)\chi_t \right)}
= \sum_{i_{t}}{{\phi}_{i_{t}, t-1} e^{-\eta\ell\left(i_{t}, y_{t}\right)}\left(\frac{\alpha}{{q}_{t}} + (1 - \alpha)\chi_t \right)}\\
&= & \sum_{i_{t}}{{w}_{i_{t}, t-1} e^{- \eta\ell\left(i_{t}, y_{t}\right)}\left(\frac{\alpha}{{q}_{t}} + (1 - \alpha)\chi_t \right)}\end{aligned}$$ where we replace ${\phi}_{i_{t}, t-1}$ with ${w}_{i_{t}, t-1}$ by the inductive hypothesis. $$= \sum_{i_{t}}{v_{i,t}\left(\frac{\alpha}{{q}_{t}} + (1 - \alpha)\chi_t \right)} =
{w}_{i,t}$$ by Eqs. \[eqn:growing-ensemble-fixed-shares-update-1\]–\[eqn:growing-ensemble-fixed-shares-update-2\]. $\Box$
Regret bounds for the modified forecasters {#sec:regret-bounds}
------------------------------------------
The [**size**]{} $\sigma(i_1^n)$ of a sequence of experts is the number of times the expert used changes, $\equiv \sum_{t=1}^{n-1}{\chi_t}$. We require this to be $\leq m$. Let $m_k$ be the number of switches within the $k^{\mathrm{th}}$ epoch, i.e., $\sigma(i_1^{k\tau}) - \sigma(i_1^{(k-1)\tau})$, so $\sum_{k=1}^{{q}_n}{m_{k}} = m$.
For all $n \geq 1$ and $y_1^n$, the tracking regret of the growing ensemble fixed shares forecaster is at most $$R(i_{1}^n)
\leq
\frac{m}{\eta}\ln{{q}_n} - \frac{1}{\eta}\ln{\alpha^{m}(1-\alpha)^{n-m}} + \frac{\eta}{8}n$$ for all expert sequences $i_{1}^{n}$ where $m = \sigma(i_{1}^{n})$. \[thm:main-regret-bound\]
<span style="font-variant:small-caps;">Proof</span> (after [@prediction-learning-and-games], Theorem 5.2): The key observation, proved as e.g. Lemma 5.1 in [@prediction-learning-and-games], is a general bound for exponentially-weighted forecasters with unequal initial weights, which relates their loss to the sum of the weights: $$\ell(\hat{p}_{1}^{t}, y_{1}^{t})
\leq -\frac{1}{\eta}\ln{\sum_{i_1^n}{{\phi}_{n}(i_1^n)}} + \frac{\eta}{8}n$$ Since weights are non-negative and $\ln$ is an increasing function, this implies $$\ell(\hat{p}_{1}^{t}, y_{1}^{t}) \leq -\frac{1}{\eta}\ln{{\phi}_n(i_{1}^{n})} + \frac{\eta}{8}n
\label{eqn:mixture-loss-in-terms-of-final-weights}$$ for any action sequence $i_1^n$. By the construction of the exponentially weighted forecaster, $$\ln{{\phi}_n(i_{1}^{n})} = \ln{{\phi}_0(i_{1}^{n})} - \eta\ell(i_{1}^n, y_{1}^n)
\label{eqn:final-weights}$$ Assuming $\sigma(i_1^n) \leq m$, the initial log weight is bounded by construction: $${\phi}_0(i_{1}^{n})
= \prod_{k = 1}^{{q}_n}{{\left(\frac{\alpha}{k}\right)}^{m_k}{\left(\frac{\alpha}{k} + 1 - \alpha\right)}^{\tau - m_{k}}}
\geq \prod_{k = 1}^{{q}_n}{{\left(\frac{\alpha}{k}\right)}^{m_{k}}{(1 - \alpha)}^{\tau - m_{k}}}
\geq {\left(\frac{\alpha}{{q}_n}\right)}^{m}(1 - \alpha)^{n-m}
\label{eqn:initial-weights-bound}$$ Substituting Eq. \[eqn:initial-weights-bound\] into Eq.\[eqn:final-weights\], and the latter into Eq.\[eqn:mixture-loss-in-terms-of-final-weights\], we get that $$\ell(\hat{p}_{1}^{t}, y_{1}^{t}) \leq \ell(i_{1}^n, y_{1}^n) + \frac{m}{\eta}\ln{{q}_n} - \frac{1}{\eta}\ln{\alpha^{m}(1-\alpha)^{n-m}} + \frac{\eta}{8}n$$ and the theorem follows. $\Box$
The familiar regret bound for the exponentially-weighted forecaster with equal initial weights is that $R$ is at most $O(\sqrt{n\ln{N}})$, with $N$ being the size of the ensemble. Since the number of allowable expert sequences of length $n$ with $m$ switches is at most $({q}_n)^m {n-1 \choose m}$, we would be doing well to achieve a regret bound of $O(\sqrt{n ( \log{{n-1 \choose
m}} + m \ln{{q}_n})})$. This can in fact be done by tuning $\alpha$ and $\eta$.
Fix $n$ and $m$, and run the modified fixed share forecaster with $\widehat{\alpha} = \frac{m}{n-1}$, and $$\textstyle
\widehat{\eta} = \sqrt{\frac{8}{n}\left((n-1)H(\widehat{\alpha}) - \ln(1-\widehat{\alpha}) + m\ln {q}_n \right)},$$ then $$R(i_{1}^{n}) \le \sqrt{\frac{n}{2}\left( (n-1)H\left(\widehat{\alpha}\right) - \ln{\left(1-\widehat{\alpha}\right)} + m\ln{{q}_n} \right)}$$ for any action sequence $i_{1}^{n}$ making at most $m$ switches. \[cor:optimal-control-settings\]
#### Proof
(After [@prediction-learning-and-games Cor. 5.1, p.105]): Let $\alpha = \frac{m}{n-1}$. Then: $$\begin{aligned}
\ln{\left(\frac{1}{\widehat{\alpha}^{m}(1 - \widehat{\alpha})^{n-m}} \right)}
& = & - m \ln{\widehat{\alpha}} - (n - m)\ln{(1 - \widehat{\alpha})}\\
& = & -m\ln{\widehat{\alpha}} - (n-m-1)\ln{(1-\widehat{\alpha})} -\ln{(1-\widehat{\alpha})}\\
&= & (n - 1)H(\widehat{\alpha}) - \ln{\left(1-\widehat{\alpha}\right)}.\end{aligned}$$
Substituting $\eta$ into the regret bound, and using this equality, we are done. $\Box$
[*Remark 1:*]{} Notice that for fixed $m$, $\widehat{\alpha} \rightarrow 0$ as $n\rightarrow\infty$, so that $H(\widehat{\alpha}) \rightarrow 0$ and the over-all bound is $o(n)$.
[*Remark 2:*]{} The learning rate and minimum share could probably be tuned better by more careful counting of the number of size-$m$ sequences from the growing ensemble, but since this will only improve the comparatively-small $m\ln{{q}_n}$ term, we omit the combinatorics here.
Example: GDP Forecasting {#sec:example}
========================
We illustrate our approach by predicting a non-stationary time series of great practical importance, the gross domestic product (GDP) of the United States, recorded quarterly from the second quarter of 1947 to the first quarter of 2010 (from the FRED data service of the Federal Reserve Bank of St. Louis). After following the common practice of converting this to quarterly growth rates, this gives $n=252$ observations. Somewhat arbitrarily, we made all of our models linear autoregressions of order 12 (i.e., AR(12) models), set the epoch length $\tau$ to 16 quarters or 4 years, and allowed $m=15$ switches of expert, with $\alpha$ and $\eta$ then following by Corollary \[cor:optimal-control-settings\].
Figure \[fig:gdp\]$a$ shows the evolution of GDP growth (clearly non-stationary), as well as the evolution of the ensemble and its weighted-average prediction, which does quite well despite the fact that AR models are both the simplest possible predictors here, and are all assured mis-specified[^5]. State-of-the-art economic forecasts rely on complicated multivariate state-space models called DSGEs [@DeJong-Dave-structural-macro], after de-trending with a smoothing spline. For GDP, however, the predictions of DSGEs are close to those of a simple autoregressive moving average (ARMA) model, so we fit one to spline residuals; AIC order selection [@Shumway-Stoffer] gave us an ARMA(8,7). Figure \[fig:gdp\]$b$ shows the accumulated loss of the ensemble compared to this model; it is both small and growing sub-linearly, despite the fact that the ARMA model has much more memory than the ARs (because of the moving-average component), and it takes advantage of the flexibility of non-parametric (and indeed non-causal) smoothing in the spline. Calculating regret against the best sequence of models from the ensemble, allowing $m=n$, produced a similar profile over time (not shown), but even smaller comparative losses.
$a$ ![$a$: Quarterly growth rate of US GDP, 1948–2010, with predictions of the growing ensemble of AR(12) models and the weighted ensemble forecast. $b$ Accumulated regret of the ensemble, compared to combining global spline smoothing with an ARMA(8,7) model.[]{data-label="fig:gdp"}](GDP-data-vs-predictions.pdf "fig:"){width="45.00000%"} $b$ ![$a$: Quarterly growth rate of US GDP, 1948–2010, with predictions of the growing ensemble of AR(12) models and the weighted ensemble forecast. $b$ Accumulated regret of the ensemble, compared to combining global spline smoothing with an ARMA(8,7) model.[]{data-label="fig:gdp"}](GDP-regret-vs-conventional-model.pdf "fig:"){width="45.00000%"}
Discussion {#sec:discussion}
==========
Related Work
------------
The closest approach to our method is that of Hazan and Seshadhri [@Hazan-Seshadhri], who also work within the family of variants on multiplicative weight training. They introduce a new expert at each time step, whose initial weight is a fixed function of time, and do not otherwise implement a “fixed share” of weights, i.e., a minimum weight for each expert. Maintaining such a fixed share is extremely useful when a pre-existing model [*becomes*]{} one of the best, drastically cutting the time needed for it to dominate the ensemble. [@Hazan-Seshadhri] also does not use tracking regret, but rather the maximum regret against any single expert attained over any contiguous time interval. This time-uniform regret is attractive, and they prove bounds on it, but only by assuming that each individual expert itself has a low, time-uniform regret (in the ordinary sense); some of their results even require low losses, not just low regrets. Our approach, by contrast, is able to accommodate the much more realistic situation where each individual expert may indeed have high loss, or even high regret, because the process is hard to predict and no one model is uniformly applicable.
Turning to more conventional approaches, econometrics has a large literature on detecting non-stationarity (of the basically-harmless “integrated” type characteristic of random walks), and finding “structural breaks” (change points), after which models must be re-estimated or re-specified [@Clements-Hendry-forecasting-nonstationary]. Economists do not seem to have considered an ensemble method like ours, perhaps due to their laudable (if unfulfilled) ambition to capture the exact data-generating process in a single parsimonious model. Similarly, most work on data-set shift and concept drift in machine learning [@dataset-shift-in-ML] deals with how a single model should be learned (or modified) so as to be robust to various changes in the joint distribution of inputs and outputs. Unlike all these approaches, we do not have to assume that [*any*]{} of our models are well-specified, nor assume anything about the nature of the data-generating process or how it changes over time.
There are some ensemble methods which are reminiscent of aspects of our proposal, such as Kolter and Maloof’s “additive expert ensemble” algorithm `AddExp` [@Kolter-Maloof-AddExp], the incremental-learning SEA algorithm [@Street-Yim-SEA], and adaptive time windows algorithms (e.g. [@Scholz-Klinkenberg-drifting-concepts]). None of these allow the full combination of a growing ensemble with temporally-specialized experts and adaptive weights. Consequently, while some of them can handle mild non-stationarities if the base models are close to well-specified, none of them are able to make strong individual-sequence prediction guarantees like those of Theorem \[thm:main-regret-bound\].
Conclusion
----------
We have introduced the growing-ensemble method, and shown that it leads to a modification of the conventional fixed-shares forecaster which is still Hannan-consistent, with $o(n)$ regret over $n$ time-steps compared to the retrospectively-best sequence of experts. This bound takes into account the fact that the ensemble grows continually and that individual experts can be arbitrarily bad, while the time series can have arbitrary non-stationarities. There are several interesting technical directions in which to take this (Can the counting of experts be replaced with variation of losses across the ensemble, as in [@Hazan-Kale-variation-in-costs]? Would it help to vary the weight with which new experts get introduced? Is there an optimal epoch length $\tau$?), the real importance of this work is methodological.
Complex systems tend to produce time series which are not just non-stationary but genuinely evolutionary — even if there is, in some sense, a fixed high-dimensional generative model, the dynamics of the low-dimensional variables we deal with changes in character over time. Tractable prediction models for such time series are at best local and transient approximations, no single one of which will work well for long. It is implausible even to come up with a fixed collection of models before we see how the system actually develops. Our growing ensemble method accommodates arbitrary dynamics, without assuming well-specified models, trends that can be extrapolated, stationary behavior punctuated by well-defined structural breaks, or other such props supporting previous work. Giving up the desire for the One True Model, of minimal risk, in favor of a growing ensemble of imperfect models, means we adapt automatically to arbitrary, historically evolving non-stationarities — including stationarity.
[^1]: Under the alias of the “Hodrick-Prescott filter” [@Paige-Trindade-on-HP-filter].
[^2]: The very first expert is initialized with some default parameter setting.
[^3]: If convex combinations over $\mathcal{D}$ do not make sense, we use a randomized forecaster, complicating the notation a little and requiring us to bound expected regret rather than actual regret. (By Markov’s inequality, low expected regret implies low realized regret with high probability.)
[^4]: Only trivial changes are required to begin with ${q}_0$ experts and add $c$ new experts every epoch.
[^5]: They are confidentaly rejected by Box-Ljung tests [@Shumway-Stoffer].
|
---
abstract: 'This article introduces spotlight tiling, a type of covering which is similar to tiling. The distinguishing aspects of spotlight tiling are that the “tiles” have elastic size, and that the order of placement is significant. Spotlight tilings are decompositions, or coverings, and can be considered dynamic as compared to typical static tiling methods. A thorough examination of spotlight tilings of rectangles is presented, including the distribution of such tilings according to size, and how the directions of the spotlights themselves are distributed. The spotlight tilings of several other regions are studied, and suggest that further analysis of spotlight tilings will continue to yield elegant results and enumerations.'
address: 'Department of Mathematical Sciences, DePaul University, 2320 North Kenmore Avenue, Chicago, IL 60614, USA'
author:
- Bridget Eileen Tenner
date: 'June 25, 2008'
title: Spotlight Tiling
---
Introduction
============
Domino tilings, and relatedly perfect matchings, are well studied objects in combinatorics and statistical mechanics. In the typical setup, there is a finite set $S$ of distinct tiles which may be used repeatedly to tile a particular region or family of regions. It is then natural to count the number of ways a particular region can be tiled by elements of $S$, or, more fundamentally, to determine if any such tiling is even possible. The number of domino tilings of a rectangle, the most elementary region, was computed by Kasteleyn in [@kasteleyn].
The number of tilings of an $m \times n$ rectangle can become much simpler if certain restrictions are imposed. For example, suppose that the region $R$ is colored as a checkerboard having a black upper-left square, with alternating black and white squares in each column or row. Restrict the set $S$ to contain vertical dominos of both colorings (one with a white top square and one with a black top square), and only the horizontal domino with a black left square. Then, it is straightforward to show that the number of such tilings of an $m \times n$ region $R$ by elements of $S$ is
$$\left\{\begin{array}{c@{\quad:\quad}l}
0\phantom{^{n/2}} & m \text{ and } n \text{ are both odd};\\
1\phantom{^{n/2}} & m \text{ is even};\\
\left(\frac{m+1}{2}\right)^{n/2} & m \text{ is odd and } n \text{ is even}.
\end{array}\right.$$
These numbers are sequence A133300 of [@oeis]. There is a rich literature concerning domino tilings, as well as tilings by shapes which are generalizations of dominoes in some aspect. For example, see [@golomb; @kasteleyn; @kenyon; @propp].
Typical tiling results do not depend on the order in which the tiles are placed. Because the set $S$ of allowable tiles does not change as each tile is placed in the region, tiles may be considered to be placed simultaneously.
This article introduces a method of covering regions, somewhat related to tilings, and provides a sample of results answering the most basic questions about this method. There are two significant differences between this and previous tiling methods: the shape of the “tiles” here is elastic, and the order in which they are positioned is important. One interpretation of these differences is that the method studied here is a *dynamic* covering model, while other methods, such as domino tiling, would be static.
Henceforth, the “tiles” in this paper will be called *spotlights* to emphasize their elastic nature and to avoid confusion with more customary notions of tiling.
In this initial foray into the dynamic spotlight tiling model, the rules for placing the spotlights will be somewhat strict, requiring that each spotlight originate in the same type of corner. Relaxing this restriction leads to other interesting questions, discussed in the last section of the paper.
As mentioned above, spotlights are placed in the region sequentially, and after each placement the set of allowable spotlights may change. To be specific, first a particular corner direction is specified (*northwest* for the duration of this article). At each stage a spotlight is placed with one end point in a “corner,” as defined by the chosen direction, and the spotlight must extend as far as possible from this corner either horizontally or vertically. This type of covering is called a *spotlight tiling*, in reference to the fact that it is like placing a spotlight in one of the specified corners and turning it to point horizontally or vertically so that it shines as far as possible until it reaches an obstruction.
Spotlight tilings of rectangles are examined thoroughly below, including a description of various statistics, such as the number of spotlights needed and the average number of spotlights used in a spotlight tiling of the rectangle. Additionally, spotlight tilings of regions which are similar to rectangles are studied. The nature of spotlight tiling means that many of the proofs used to obtain the results below are recursive in nature.
The most basic region is an $m \times n$ rectangle. Therefore, in this introductory analysis of spotlight tiling, attention is primarily focused on rectangles, in terms of their enumeration and their properties. This will be the substance of Section \[sec:tiling rectangles\]. For example, in addition to determining the number of spotlight tilings of an $m \times n$ rectangle, more detailed statistics will be studied. Unlike other sorts of tilings, where the number of tiles required to cover a region is fixed, the number of spotlights used depends on the particular spotlight tiling itself. The distribution of the number of these tiles will be part of the discussion in Section \[sec:tiling rectangles\]. Following this discussion, in Section \[sec:other regions\], attention will be turned to spotlight tilings of regions which are formed from rectangles by removing squares at the corners. The recursive nature of these spotlight tilings leads naturally to recursive enumeration formulae. In some cases, these equations will be left in a recursive format, as it is simpler to read them in this manner. In other situations, when a closed form itself is quite elegant, both the recursive and the closed formulae will be given. Finally, in Section \[sec:frames\], the spotlight tilings of a certain family of frame-like regions is explored. The paper concludes with a brief discussion of how spotlight tilings may be studied further.
Definitions
===========
The basic definitions and notation of this article are outlined below.
A *region* is the dual of a finite connected induced subgraph of $\mathbb{Z}^2$.
Spotlight tilings rely on the choice of a particular direction and type of corner, in this case a northwest corner.
A *northwest corner* in a region is a square belonging to the region that is bound above and on the left by the boundary edge of the region.
For example, the four northwest corners of the region in Figure \[fig:nwcorners\] have been shaded.
As discussed in the introduction, spotlight tilings differ in nature from static tilings. Instead of choosing from a finite set of tiles, the possible spotlights themselves are defined by the region and any spotlights that have been positioned previously.
\[defn:spotlight tile\] A *spotlight* with an endpoint in a northwest corner $s$ extends as far east horizontally or south vertically from $s$ as possible, terminating at the boundary of the region, or when it encounters a spotlight that has already been placed.
\[defn:spotlight tiling\] Given a region $R$, a *spotlight tiling* of $R$ is defined recursively as follows. Choose any northwest corner $s \in R$. Place a spotlight tile with an endpoint in $s$, extending either horizontally (east) or vertically (south) as far as possible. Let $R'$ be the collection of disjoint regions remaining after placing this spotlight in $R$. The spotlight tiling of $R$ is completed by finding spotlight tilings of each connected component of $R'$.
A spotlight tiling of a $3 \times 4$ rectangle is depicted in Figure \[fig:3x4\]. The complete tiling is the last image in the figure, having been built successfully from the previous images.
$\Rightarrow$ $\Rightarrow$ $\Rightarrow$ $\Rightarrow$
Although spotlight tiles are placed sequentially in a region, two spotlight tilings are considered distinct only if they look different once all the spotlights are in place. In other words, if there is more than one order in which the spotlights can be placed in the region, this alone does not distinguish one tiling from another. Moreover, the direction (horizontal or vertical) of a spotlight is obvious except in certain cases of tiles of length one, where the direction of such a spotlight will not be specified as uniquely horizontal or vertical. Ignorance of the orientation of this spotlight maintains consistency with the fact that two spotlight tilings differ only if they look different. However, the enumerations of this paper could be reformulated without this stipulation, and similarly nice results would ensue.
The order in which spotlights are placed in a spotlight tiling of a region $R$ can be recovered in some cases. More precisely, a complete recovery is possible if the region $R$ has only one northwest corner and does not have any holes. If $R$ did have holes, then it could be possible to place some number of spotlights in $R$ and yield an untiled subregion having more than one northwest corner.
If the last spotlight placed in a spotlight tiling has length $1$, it is a *HV-spotlight*, referring to the fact that the spotlight’s direction could be considered to be either horizontal or vertical.
The seven different spotlight tilings of a $2 \times 3$ rectangle are depicted in Figure \[fig:2x3\].
\
Let $R_{m,n}$ denote an $m \times n$ rectangle. The set of spotlight tilings of $R_{m,n}$ is denoted $\mathcal{T}_{m,n}$, and $T_{m,n} = |\mathcal{T}_{m,n}|$. For all $m,n > 0$, set $T_{m,0} = T_{0,n} = 1$.
As depicted in Figure \[fig:2x3\], $T_{2,3} = 7$.
The recursive definition of spotlight tiling means that $$\label{eqn:set recursion}
\begin{split}
\mathcal{T}_{m,n} =& \left\{\text{one } (1 \times n)\text{-spotlight together with } t \mid t \in \mathcal{T}_{m-1,n}\right\}\\
&\cup \left\{\text{one } (m \times 1)\text{-spotlight together with } t \mid t \in \mathcal{T}_{m,n-1}\right\}.
\end{split}$$
Spotlight tilings of rectangles {#sec:tiling rectangles}
===============================
The first goal of this examination of spotlight tilings is a thorough understanding of spotlight tilings of rectangles. Since the definition of a spotlight tiling gives no preference to horizontal or vertical spotlights, all results in this section should be symmetric with respect to $m$ and $n$. In particular, it should be the case that $T_{m,n} = T_{n,m}$.
A precise formula for $T_{m,n}$ is straightforward to compute, based on the recursive nature of Definition \[defn:spotlight tiling\].
\[thm:total rect\] For all $m,n \ge 1$, $$\label{eqn:rectangle value}
T_{m,n} = \binom{m+n}{m} - \binom{m+n-2}{m-1}.$$
Definition \[defn:spotlight tiling\] gives the recursive formula $$\label{eqn:rectangle recursion}
T_{m,n} = T_{m-1,n} + T_{m,n-1}$$ for all positive $m$ and $n$ such that $mn >1$. Since $T_{1,1} = 1$, equation is satisfied for $m=n=1$. Supposing inductively that the result holds whenever the dimensions of the rectangle sum to less than $k$, consider an $m \times n$ rectangle where $m + n = k$. Then, using equation , $$\begin{aligned}
T_{m,n} &=& T_{m-1,n} + T_{m,n-1}\\
&=& \binom{m + n - 1}{m-1} - \binom{m + n - 3}{m - 2} + \binom{m + n - 1}{m} - \binom{m + n - 3}{m - 1}\\
&=& \binom{m + n}{m} - \binom{m + n - 2}{m - 1},\end{aligned}$$ Thus the result holds for all $m, n \ge 1$.
Notice that equation is symmetric in $m$ and $n$, as required. The values of $T_{m,n}$ for small $m$ and $n$ are displayed in Table \[table:T\_[m,n]{}\]. Additionally, these are sequence A051597 of [@oeis].
[c|ccccccc]{}
------------------------------------------------------------------------
$T_{m,n}$ & $n=1$ & 2 & 3 & 4 & 5 & 6 & 7\
------------------------------------------------------------------------
$m=1$ & 1 & 2 & 3 & 4 & 5 & 6 & 7\
2 & 2 & 4 & 7 & 11 & 16 & 22 & 29\
3 & 3 & 7 & 14 & 25 & 41 & 63 & 92\
4 & 4 & 11 & 25 & 50 & 91 & 154 & 246\
5 & 5 & 16 & 41 & 91 & 182 & 336 & 582\
6 & 6 & 22 & 63 & 154 & 336 & 672 & 1254\
7 & 7 & 29 & 92 & 246 & 582 & 1254 & 2508\
As demonstrated in Figure \[fig:2x3\], the number of spotlights in a particular spotlight tiling of $R_{m,n}$ is not fixed. For example, a spotlight tiling of $R_{2,3}$ can consist of $2$, $3$, or $4$ spotlights. Therefore, to better understand spotlight tilings of rectangles, it is important to understand how many spotlights may (likewise, “must” and “can”) be used in a spotlight tiling of $R_{m,n}$, and how many spotlight tilings of the rectangle use exactly $r$ spotlights. There are additional aspects of spotlight tilings using the minimal or maximal number of spotlights that are of interest as well.
For a spotlight tiling $t$ of a region $R$, let $|t|$ be the number of spotlights used in $t$, known as the *size* of $t$.
Let $t^-_{m,n}$ denote the minimum number of spotlights needed in a spotlight tiling of $R_{m,n}$, and let $t^+_{m,n}$ denote the maximum number of spotlights that can be used in a spotlight tiling of $R_{m,n}$. That is, $$\begin{aligned}
t^-_{m,n} = \min_{t \in \mathcal{T}_{m,n}} |t|\\
t^+_{m,n} = \max_{t \in \mathcal{T}_{m,n}} |t|\\\end{aligned}$$ An element of $\mathcal{T}_{m,n}$ using $t^-_{m,n}$ spotlights is a *minimal* spotlight tiling, while one that uses $t^+_{m,n}$ spotlights is a *maximal* spotlight tiling.
\[prop:bounds\] For all $m, n \ge 1$, $$\begin{aligned}
t^-_{m,n} &=& \min\{m,n\};\label{eqn:t^-}\\
t^+_{m,n} &=& m + n - 1.\label{eqn:t^+}\end{aligned}$$
By the definition of spotlight tilings, it is clear that the minimum number of spotlights needed depends on the minimum dimension of $R_{m,n}$. Suppose, without loss of generality, that $m \le n$. If fewer than $m$ spotlights are placed in $R_{m,n}$, then at least one row and at least one column are not completely covered. Thus, $t^-_{m,n}$ can be no less than $m$. Additionally, one spotlight tiling of the rectangle consists of $m$ horizontal spotlights, so $t^-_{m,n} = m$. This proves equation .
Equation implies that $t^+_{m,n} = \max\{1 + t^+_{m-1,n}, 1 + t^+_{m,n-1}\}$. Then, since $t^+_{1,1} = 1$ and $t^+_{m,1} = m$, the rest of the proof of equation follows inductively.
Note that $t^-_{m,n} = t^+_{m,n}$ if and only if $m = n = 1$. Therefore, in anything larger than a $1\times 1$ square, there will be variation in the number of spotlights used.
The number of minimal spotlight tilings of an $m \times n$ rectangle is necessarily $1$ or $2$, depending on whether $m \neq n$ or $m = n$. This will be included in a more general argument in Theorem \[thm:counting rect\].
On the other hand, the number of maximal spotlight tilings is somewhat specialized and will first be treated independently.
\[thm:max rect\] The number of maximal spotlight tilings of $R_{m,n}$ is $$\binom{m + n - 2}{m - 1}.$$
Equations and imply that once the first spotlight has been placed in the rectangle, this can (and, in fact, must) be completed to a maximal tiling of the rectangle by finding a maximal spotlight tiling of the resulting sub-rectangle ($R_{m-1,n}$ or $R_{m,n-1}$, depending on whether the first spotlight was horizontal or vertical).
There is a single element in the set $\mathcal{T}_{1,1}$, and it consists of a single HV-spotlight. Therefore, using equation , the last spotlight placed in a maximal spotlight tiling must be an HV-spotlight. In fact, if $m$ and $n$ are not both equal to $1$, then the penultimate spotlight placed in a maximal spotlight tiling of $R_{m,n}$ must also have length $1$, although this will not be an HV-spotlight since its direction must be specified.
The result follows immediately by induction.
Alternatively, Theorem \[thm:max rect\] can also be proved bijectively in the following manner. By nature of spotlight tiling, there cannot be more than $m$ horizontal spotlights or $n$ vertical spotlights in an element of $\mathcal{T}_{m,n}$. If the last spotlight is an HV-spotlight, than of the previous $m + n - 2$ spotlights in a maximal spotlight tiling, at most $m - 1$ can be horizontal and at most $n - 1$ can be vertical. Consequently, of these $m + n - 2$ spotlights, exactly $m - 1$ are horizontal and exactly $n - 1$ are vertical. Consider an initial set of spotlights in $R_{m,n}$, consisting of at most $m-1$ horizontal spotlights and at most $n-1$ vertical spotlights. Any such initial spotlight tiling can be completed to a maximal spotlight tiling. Therefore the number of maximal spotlight tilings depends only on which $m-1$ of the first $m+n-2$ spotlights are horizontal, and thus is $$\binom{m + n - 2}{m-1}.$$
Let $t^r_{m,n}$ be the number of spotlight tilings of $R_{m,n}$ that use $r$ spotlights. That is, $t^r_{m,n} = |\{t \in \mathcal{T}_{m,n} \mid |t| = r\}|$. Set $t^r_{m,0} = t^r_{0,n} = \delta_{0r}$, where $\delta_{0r}$ is the Kronecker delta.
\[thm:counting rect\] For all integers $r < m + n - 1$, $$t^r_{m,n} = \binom{r-1}{m-1} + \binom{r-1}{n-1}.$$
Note that if $r < \max\{m,n\}$, then at least one of the binomial coefficients in the statement of the theorem is $0$, by the convention that $\binom{j}{i} = 0$ if $i > j$.
As in the proof of Theorem \[thm:total rect\], the values $t^r_{m,n}$ satisfy a recurrence relation. That is, for all $m,n,r > 0$ such that $mn > 1$, $$t^r_{m,n} = t^{r-1}_{m-1,n} + t^{r-1}_{m,n-1}.$$
The base case $t^1_{1,1} = 1$ is easy to calculate, and the result follows by induction.
Therefore, Theorems \[thm:max rect\] and \[thm:counting rect\] and Proposition \[prop:bounds\] can be combined in the following equation: $$t^r_{m,n} =
\begin{cases}
\rule[-3mm]{0mm}{7mm}\binom{r-1}{m-1} + \binom{r-1}{n-1} & r < m+n-1;\\
\binom{m+n-2}{m-1} & r = m+n-1.
\end{cases}$$
Observe that $t^{m+n-1}_{m,n}$ is exactly half of $\binom{m+n-1-1}{m-1} + \binom{m+n-1-1}{n-1}$, which would have been the value if Theorem \[thm:counting rect\] had applied. This differences arises from the HV-spotlight present in any maximal spotlight tiling. If the orientation of such a spotlight could be distinguished, then there would be twice as many maximal spotlight tilings of the rectangle. As suggested earlier, the convention in this paper that an HV-spotlight lose its orientation supports the idea that these dynamic spotlight tilings should be considered as coverings of a region, and so are only distinguished if they actually look different. However, analogously concise enumeration results will arise if this convention is dropped.
In fact, if $(m,n) \neq (1,1)$, then $t^{m+n-2}_{m,n} = t^{m+n-1}_{m,n}$, and the values $t^r_{m,n}$ are strictly increasing on the interval $r \in [\min\{m,n\}, m+n-2]$. More specifically, for $r \in [\min\{m,n\}+1, m+n-2]$, $$\begin{aligned}
t^r_{m,n} - t^{r-1}_{m,n} &=& \binom{r-1}{m-1} + \binom{r-1}{n-1} - \binom{r-2}{m-1} - \binom{r-2}{n-1}\\
&=& \binom{r-2}{m-2} + \binom{r-2}{n-2} = t^{r-1}_{m-1,n-1}.\end{aligned}$$
Moreover, it is straightforward to check that $$\sum_{r \ge 1} t^r_{m,n} = \binom{m+n}{m} - \binom{m+n-2}{m-1},$$ confirming Theorem \[thm:total rect\].
Given Theorems \[thm:max rect\] and \[thm:counting rect\], it is straightforward now to compute the average number of spotlights used in a spotlight tiling of an $m \times n$ rectangle.
The average number of spotlights used in a spotlight tiling of $R_{m,n}$, that is, the average size of an element of $\mathcal{T}_{m,n}$, is $$\label{eqn:average}
\frac{mn(m+n-1)}{(m+n)(m+n-1)-mn}\left(1 + \frac{n-1}{m+1} + \frac{m-1}{n+1}\right).$$
This average is computed by evaluating $$\begin{aligned}
\frac{\sum\limits_{r=1}^{m+n-1}r\cdot t^r_{m,n}}{\binom{m+n}{m} - \binom{m+n-2}{m-1}} &=& \frac{(m+n-1)\binom{m+n-2}{m-1} + \sum\limits_{r=1}^{m+n-2}\left[r\binom{r-1}{m-1} + r\binom{r-1}{n-1}\right]}{\binom{m+n}{m} - \binom{m+n-2}{m-1}}\\
&=& \frac{(m+n-1)\binom{m+n-2}{m-1} + m\binom{m+n-1}{m+1} + n\binom{m+n-1}{n+1}}{\binom{m+n}{m} - \binom{m+n-2}{m-1}}\\
&=& \frac{mn(m+n-1)}{(m+n)(m+n-1)-mn}\left(1 + \frac{n-1}{m+1} + \frac{m-1}{n+1}\right).\end{aligned}$$
The growth of the expression in can be seen in Table \[table:averages\], which displays the expected number of spotlights in a random spotlight tiling of $R_{m,n}$ for small values of $m$ and $n$. Additionally, the average number of spotlights used in a spotlight tiling of the square $R_{n,n}$ approaches $2n-7/3$ as $n$ increases, as reflected in the table.
[c|ccccccc]{}
------------------------------------------------------------------------
& $n=1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$\
------------------------------------------------------------------------
$m=1$ &$1$ & $1.5$ & $2$ & $2.5$ & $3$ & $3.5$ & $4$\
$2$ & $1.5$ & $2.5$ & $3.286$ & $4$ & $4.688$ & $5.364$ & $6.034$\
$3$ & $2$ & $3.286$ & $4.286$ & $5.16$ & $5.976$ & $6.762$ & $7.533$\
$4$ & $2.5$ & $4$ & $5.16$ & $6.16$ & $7.077$ & $7.948$ & $8.793$\
$5$ & $3$ & $4.688$ & $5.976$ & $7.077$ & $8.077$ & $9.018$ & $9.923$\
$6$ & $3.5$ & $5.364$ & $6.762$ & $7.948$ & $9.018$ & $10.018$ & $10.974$\
$7$ & $4$ & $6.034$ & $7.533$ & $8.793$ & $9.923$ & $10.974$ & $11.934$\
In a maximal spotlight tiling of $R_{m,n}$, there are $m-1$ horizontal spotlights, $n-1$ vertical spotlights, and $1$ HV-spotlight. Moreover, a spotlight tiling $t \in \mathcal{T}_{m,n}$ contains an HV-spotlight if and only if $t$ is maximal. The breakdown of spotlight directions is immediate for maximal spotlight tilings, but the question is more subtle for non-maximal spotlight tilings.
For a spotlight tiling $t$ with no HV-spotlights, let $h(t)$ be the number of horizontal spotlights in $t$, and let $v(t)$ be the number of vertical spotlights in $t$.
Define the generating function $$G_{m,n}(H,V) = \sum_{\genfrac{}{}{0pt}{}{\text{non-maximal}}{t \in \mathcal{T}_{m,n}}} H^{h(t)} V^{v(t)}.$$
Notice that $G_{1,1}(H,V) = 0$, because the only spotlight tiling of a $1 \times 1$ rectangle is maximal, yielding an empty sum.
\[thm:rect HV count\] For all $m, n \ge 1$, where $(m,n) \neq (1,1)$, $$G_{m,n}(H,V) = H^m \sum_{r=0}^{n-2} \binom{r+m-1}{m-1} V^r + V^n \sum_{r=0}^{m-2} \binom{r+n-1}{n-1} H^r.$$
Consider a non-maximal spotlight tiling of $R_{m,n}$ using $r$ spotlights. In the successive iterations of the spotlight tiling procedure, the last untiled sub-rectangle will be covered either by a horizontal or by a vertical spotlight. Thus, after placing the first $r-1$ spotlights, what remains must be a rectangle of dimensions $1 \times (m + n - r)$ or $(m + n - r) \times 1$. In the former case, the final spotlight is horizontal, and in the latter case the final spotlight is vertical.
In the case of a final horizontal spotlight, there are $m-1$ of the first $r-1$ spotlights which are horizontal, and the remaining $r - m$ are vertical. The recursive nature of spotlight tiling means that these horizontal and vertical spotlights can occur in any order. Thus there are $\binom{r-1}{m-1}$ ways for the last spotlight to be horizontal in a non-maximal element of $\mathcal{T}_{m,n}$ with $r$ spotlights. Similarly, there are $\binom{r-1}{n-1}$ ways for the last spotlight to be vertical in a non-maximal element of $\mathcal{T}_{m,n}$ with $r$ spotlights.
Therefore, $$\begin{aligned}
G_{m,n}(H,V) &=& \sum_{\genfrac{}{}{0pt}{}{\text{non-maximal}}{t \in \mathcal{T}_{m,n}}} H^{h(t)} V^{v(t)}\\
&=& \sum_{r = \min\{m,n\}}^{m+n-2} \binom{r-1}{m-1}H^{m-1}V^{r-m}\cdot H\\
& & \hspace{.4in} + \sum_{r = \min\{m,n\}}^{m+n-2} \binom{r-1}{n-1}V^{n-1}H^{r-n}\cdot V\\
&=& \sum_{r = m}^{m+n-2} \binom{r-1}{m-1}H^mV^{r-m} + \sum_{r = n}^{m+n-2} \binom{r-1}{n-1}V^nH^{r-n}\\
&=& H^m \sum_{r=0}^{n-2} \binom{r+m-1}{m-1} V^r + V^n \sum_{r=0}^{m-2} \binom{r+n-1}{n-1} H^r.\end{aligned}$$
One consequence of Theorem \[thm:rect HV count\] is that in any non-maximal spotlight tiling of $R_{m,n}$, there are either exactly $m$ horizontal spotlights or exactly $n$ vertical spotlights. In the former case, there can be between $0$ and $n-2$ vertical spotlights, and in the latter case there can be between $0$ and $m-2$ horizontal spotlights.
Substituting $x$ for both $H$ and $V$ in $G_{m,n}(H,V)$ gives the generating function for the numbers $t^r_{m,n}$ when $r < m + n - 1$, and in fact the coefficient $[x^r]G_{m,n}(x,x)$ is equal to $\binom{r-1}{m-1} + \binom{r-1}{n-1}$, confirming Theorem \[thm:counting rect\].
Spotlight tilings of rectangles with missing corners {#sec:other regions}
====================================================
The recursive nature of spotlight tilings means that enumerating the spotlight tilings of certain families of regions can be done without difficulty. For the most part, the regions considered in this section are variations on rectangles, in particular rectangles missing squares at the corners. Because the northwest corner is specified in spotlight tilings, the enumeration of the spotlight tilings of these regions depends on which corner was removed.
It should be noted that it is possible to obtain formulae for the number of spotlight tilings of other regions as well, due to the iterative definition of this method. For example, the number of spotlight tilings of a rectangle with a single square removed from somewhere in the interior is not difficult to obtain, particularly if this square is parameterized by its position relative to the southeast corner of the rectangle, which does not change when spotlights are placed.
Fix integers $m, n \ge 2$. Let $R_{m,n}^{\sf{NW}}$ (respectively, $R_{m,n}^{\sf{NE}}$, $R_{m,n}^{\sf{SW}}$, and $R_{m,n}^{\sf{SE}}$) be an $m \times n$ rectangle whose northwest (respectively, northeast, southwest, and southeast) corner has been removed. The set $\mathcal{T}_{m,n}^*$ consists of all spotlight tilings of the region $R^*_{m,n}$, and $T_{m,n}^* = |\mathcal{T}_{m,n}^*|$.
The most difficult of these spotlight tilings to enumerate, and the one with the least elegant answer, is for the region $R_{m,n}^{\sf{NW}}$. That this case differs from the others is no surprise, since there are two northwest corners in the new region, and thus spotlights can start from two different squares.
\[prop:northwest corner\] For all $m, n \ge 2$, $$\begin{aligned}
T_{m,n}^{\sf{NW}} &=& T_{m-1,n-1} + T_{1,n-1}T_{m-2,n} + T_{m-1,1}T_{m,n-2}\\
&=& T_{m-1,n-1} + (n-1)T_{m-2,n} + (m-1)T_{m,n-2}\\
&=& \binom{m+n-2}{m-1} \left[1 + (m-1)(n-1)\left(\frac{1}{m} + \frac{1}{n} - \frac{1}{m+n-2}\right)\right]\end{aligned}$$
Just as Proposition \[prop:northwest corner\] computes $T_{m,n}^{\sf{NW}}$, the spotlight tilings of $R_{m,n}^{\sf{NE}}$, $R_{m,n}^{\sf{SW}}$, and $R_{m,n}^{\sf{SE}}$ can also be enumerated. In fact, these enumerations are significantly more elegant, due to the fact that the missing corner does not affect where spotlights may begin. The proofs of these results are inductive, and use the recursion inherent to spotlight tilings.
\[prop:northeast/southwest corner\] For all $m, n \ge 2$, the number of spotlight tilings of an $m \times n$ rectangle missing either its northeast or its southwest corner is $$\begin{aligned}
\label{eqn:T_{m,n}^{NE}}
T_{m,n}^{\sf{NE}} = T_{m,n}^{\sf{SW}} &=& T_{m,n} - 1\\
&=& \binom{m+n}{m} - \binom{m+n-2}{m-1} - 1.\end{aligned}$$
\[prop:southeast corner\] For all $m, n \ge 2$, the number of spotlight tilings of an $m \times n$ rectangle missing its southeast corner is $$\begin{aligned}
T_{m,n}^{\sf{SE}} &=& T_{m,n} - \binom{m+n-2}{m-1}\\
&=& \binom{m+n}{m} - 2 \binom{m+n-2}{m-1}.\end{aligned}$$
The number of spotlight tilings of $R_{m,n}^{\sf{SE}}$ is the number of spotlight tilings of $R_{m,n}$, minus the number of maximal spotlight tilings of $R_{m,n}$.
The numbers described in Proposition \[prop:southeast corner\] are sequence A051601 in [@oeis].
While the symmetry $T_{m,n}^{\sf{NE}} = T_{n,m}^{\sf{SW}}$ in Proposition \[prop:northeast/southwest corner\] is not surprising, the fact that $T_{m,n}^{\sf{NE}}$ (and $T_{m,n}^{\sf{SW}}$) is symmetric with respect to $m$ and $n$ is intriguing. Similarly, the fact that the results of Propositions \[prop:northeast/southwest corner\] and \[prop:southeast corner\] are so similar to $T_{m,n}$ indicates that removing one of these corners does not drastically alter the spotlight tilings of the original rectangle.
In fact, Proposition \[prop:northeast/southwest corner\] could also be proved in another fashion, which highlights a more general trend in spotlight tilings.
\[defn:R\[r\]\] Suppose that $R$ is a region as in the following figure, where the only requirement of $R$ in the dashed portion is that it have no northwest corners there. $$\PandocStartInclude{NEwithR.pstex_t}\PandocEndInclude{input}{436}{24}$$ Let $R[r]$ be the region obtained from $R$ be removing the top $r$ squares in the rightmost column specified in $R$. That is, $R[r]$ is the region displayed below. $$\PandocStartInclude{NEwithoutR.pstex_t}\PandocEndInclude{input}{440}{27}$$ The column of $r$ squares which gets removed from $R$ to form $R[r]$ is the *difference column*.
By this definition, $R_{m,n}^{\sf{NE}} = R_{m,n}[1]$.
\[prop:corner column\] Let $R$ and $R[r]$ be regions defined as in Definition \[defn:R\[r\]\], keeping the meaning of $r$ and $n$. Then $$\#\{\text{spotlight tilings of } R[r]\} = \#\{\text{spotlight tilings of } R\} - \sum_{k=0}^{r-1} \binom{n-1}{k}.$$
Consider the ways that the difference column might be tiled by spotlights in $R$. It can consist of the ends of $r$ horizontal spotlights, or the ends of $k$ horizontal spotlights atop a vertical spotlight, where $0 \le k \le r-1$. If a vertical spotlight is involved, then this spotlight would continue down below the difference column into $R[r] \subset R$. Additionally, if a vertical spotlight is used to cover the difference column, then there must be $n-1$ other vertical spotlight tiles positioned to the left of the difference column in $R$. The placement of these $n-1$ vertical spotlight tiles and the $k$ horizontal spotlight tiles can be done in any order.
A given spotlight tiling of $R[r]$ can be extended to a spotlight tiling of $R$ by filling the difference column with horizontal spotlights (if the spotlight tiling of $R[r]$ included a horizontal terminating at the difference column in some row, then glue an extra square to the end of this spotlight tile). This will yield all spotlight tilings of $R$ except those which cover some portion of the difference column with a vertical spotlight tile. This concludes the proof.
Notice that Proposition \[prop:corner column\] agrees with Proposition \[prop:northeast/southwest corner\] by setting $r = 1$.
Also notice that the symmetry of spotlight tilings indicates that Proposition \[prop:corner column\] would also be true if the figures in Definition \[defn:R\[r\]\] were reflected across the northwest-southeast diagonal.
One specific corollary to Proposition \[prop:corner column\] is presented below, although this could also have been shown in a straightforward proof using the recursion inherent to spotlight tilings.
Fix integers $m, n \ge 3$. Let $R_{m,n}^{\sf{NE,SE}}$ be the region obtained from $R_{m,n}$ by removing the northeast and southeast corners. Likewise, $R_{m,n}^{\sf{NE,SW,SE}}$ is an $m \times n$ rectangle whose northeast, southwest, and southeast corners have been removed. Other regions are defined analogously, and $\mathcal{T}_{m,n}^*$ and $T_{m,n}^*$ have their customary definitions.
For all $m, n \ge 3$ $$\begin{aligned}
T_{m,n}^{\sf{NE,SW}} &=& T_{m,n} - 2\\
&=& \binom{m+n}{m} - \binom{m+n-2}{m-1} - 2;\\
\\
T_{m,n}^{\sf{NE,SE}} = T_{m,n}^{\sf{SW,SE}} &=& T_{m,n}^{\sf{SE}} - 1\\
&=& \binom{m+n}{m} - 2\binom{m+n-2}{m-1} - 1;\\
\\
T_{m,n}^{\sf{NE,SW,SE}} &=& T_{m,n}^{\sf{SE}} - 2\\
&=& \binom{m+n}{m} - 2\binom{m+n-2}{m-1} - 2.\end{aligned}$$
There are several regions $R_{m,n}^*$ whose spotlight tilings have not yet been enumerated. In these, the northwest corner has been removed, along with at at least one other corner. Six of these seven cases are treated in Corollary \[cor:missing corners\], and the remaining case (when all four corners have been removed) appears independently below. The results of Corollary \[cor:missing corners\] are not written in closed form, although it would not be hard to do so.
\[cor:missing corners\] For $m, n \ge 3$, $$\begin{aligned}
T_{m,n}^{\sf{NW,SE}} &=& T_{m-1,n-1}^{\sf{SE}} + (n-1)T_{m-2,n}^{\sf{SE}} + (m-1)T_{m,n-2}^{\sf{SE}};\\
\\
T_{m,n}^{\sf{NW,NE}} &= &T_{n,m}^{\sf{NW,SW}}\\
&=& T_{m-1,n-1} + (n-2)T_{m-2,n} +(m-1)T_{m,n-2} - m + 1;\\
\\
T_{m,n}^{\sf{NW,NE,SE}} &=& T_{n,m}^{\sf{NW,SW,SE}}\\
&=& T_{m-1,n-1}^{\sf{SE}} + (n-2)T_{m-2,n}^{\sf{SE}} + (m-1)T_{m,n-2}^{\sf{SE}} - m + 1;\\
\\
T_{m,n}^{\sf{NW,NE,SW}} &=& T_{m-1,n-1} + (n-2)T_{m-2,n} + (m-2)T_{m,n-2} - m - n + 4.\end{aligned}$$
For $m, n \ge 3$, let $R_{m,n}^{\circ}$ be the region obtained from $R_{m,n}$ by removing the northwest, northeast, southwest, and southeast corner squares. Let $\mathcal{T}_{m,n}^{\circ}$ be the set of spotlight tilings of $R_{m,n}^{\circ}$, and $T_{m,n}^{\circ} = |\mathcal{T}_{m,n}^{\circ}|$.
The following formula for $T_{m,n}^{\circ}$ is not difficult to compute, using the inductive definition of spotlight tilings.
\[cor:missing all corners\] For all $m,n \ge 3$, $$T_{m,n}^{\circ} = T_{m-1,n-1}^{\sf SE} + (n-2)T_{m-2,n}^{\sf SE} + (m-2)T_{m,n-2}^{\sf SE} - m - n + 4.$$
The similarities between the results in Corollaries \[cor:missing corners\] and \[cor:missing all corners\] are striking, and suggest that the iterative nature of spotlight tiling respects certain substructures of a region.
Spotlight tilings of frame-like regions {#sec:frames}
=======================================
This section explores the spotlight tilings of a family of regions that are formed by making a large hole in the center of a rectangle. To give a flavor for these results, this discussion studies only those cases where the remaining region has width $1$, although it would not be difficult to generalize to wider frames.
Fix $m, n \ge 3$. Let $F_{m,n}$ be the region formed by removing a centered $(m-2) \times (n-2)$ rectangle from the rectangle $R_{m,n}$. Let $f_{m,n}$ be the number of spotlight tilings of $F_{m,n}$.
In other words, the region $F_{m,n}$ looks like an $m \times n$ picture frame of width $1$. To understand $f_{m,n}$, it is helpful first to enumerate the spotlight tilings of some related regions.
Fix $m, n \ge 1$. Let $C_{m,n}^{\sf{NW}}$ be the region of $m + n -1$ squares formed by overlapping the north-most square of a column of length $m$ and the west-most square of a row of length $n$. Let $c_{m,n}^{\sf{NW}}$ be the number of spotlight tilings of $C_{m,n}^{\sf{NW}}$. The regions $C_{m,n}^{\sf{NE}}$, $C_{m,n}^{\sf{SW}}$, and $C_{m,n}^{\sf{SE}}$ and their enumerations are defined analogously.
\[prop:corners\] For $m, n \ge 1$, $$\begin{aligned}
c_{m,n}^{\sf{NW}} &=& m + n -2\\
c_{m,n}^{\sf{NE}} = c_{n,m}^{\sf{SW}} &=& n(m-1) + 1\\
c_{m,n}^{\sf{SE}} &=& 2(m-1)(n-1) + 1\end{aligned}$$
Each of these quantities can be computed by careful counting, together with the fact that $T_{1,p} = T_{p,1} = p$.
\[thm:frames\] For $m, n \ge 3$, $$f_{m,n} = 2(m-2)(n-2)(m+n-2) + (m-2)(m+1) + (n-2)(n+1).$$
Initially, there is only one northwest corner in the region $F_{m,n}$. This can be covered with a horizontal spotlight of length $n$ or a vertical spotlight of length $m$. Either way, the remaining region has two northwest corners, and careful applications of Proposition \[prop:corners\] and the inclusion-exclusion property give the answer.
The values of $f_{m,n}$ for small $m$ and $n$ are displayed in Table \[table:f\_[m,n]{}\]. These values are sequence A132370 of [@oeis].
[c|ccccc]{}
------------------------------------------------------------------------
$f_{m,n}$ & $n=3$ & 4 & 5 & 6 & 7\
------------------------------------------------------------------------
$m=3$ & 16 & 34 & 58 & 88 & 124\
4 & 34 & 68 & 112 & 166 & 230\
5 & 58 & 112 & 180 & 262 & 358\
6 & 88 & 166 & 262 & 376 & 508\
7 & 124 & 230 & 358 & 508 & 680
Further directions {#sec:further}
==================
The preceding sections have examined the spotlight tilings of several families of regions. In each case, the enumeration of these spotlight tilings had a concise and often illuminating form. For the rectangle, more refined analysis was also performed, and yielded results whose simplicity and elegance may not have been anticipated.
The obvious analogue of spotlight tiling in higher dimensions may also yield fruitful results. Additionally, the questions particular to spotlight tiling (such as the distribution of the number of spotlights in a given spotlight tiling) may give rise to new aspects of this and other tilings methods which warrant further study.
This work can be extended by studying variations on the spotlight tilings described here. For example, in this article, every spotlight has started in a northwest corner. If this restriction were removed, and spotlights were allowed to start in any square and continue in any direction until reaching a barrier, then the resulting model would be an expansion of this type of dynamic tiling.
Another generalization would be to allow tiles to expand as much as possible in two directions, instead of only horizontally or only vertically. Such a tile would create an $a \times b$ rectangle, instead of $a \times 1$ or $1 \times b$. Continuing the imagery of this article, these new tiles could be called *floodlights*, and dynamic floodlight tiling might have interesting enumerative results as well. It should be noted that the region $R_{m,n}$ has exactly $1$ floodlight tiling, and, consequently, more complicated regions need to be studied in order to gain an understanding of the model.
[99]{}
S. W. Golomb, Tiling with sets of polyominoes, *J. Combin. Theory* 9 (1970) 60–71.
P. W. Kasteleyn, The statistics of dimers on a lattice, I. The number of dimer arrangements on a quadratic lattice, *Physica* 27 (1961) 1209–1225.
R. W. Kenyon, J. G. Propp, and D. B. Wilson, Trees and matchings, *Electron. J. Combin.* 7(1) (2000) R25.
J. Propp, Enumeration of Matchings: Problems and Progress, in *New Perspectives in Geometric Combinatorics*, L. Billera, A. Björner, C. Greene, R. Simion, and R. P. Stanley, eds., MSRI Publications, vol. 38, Cambridge University Press, Cambridge, 1999, pp. 255–291.
N. J. A. Sloane, The on-line encyclopedia of integer sequences, published electronically at .
|
---
abstract: 'We analyze a new gravitational lens [OAC–GL J1223-1239]{}, serendipitously found in a deep I$_{814}-$band image of the Hubble Space Telescope (HST) Advanced Camera for Surveys (ACS). The lens is a $L^*$, edge-on S0 galaxy at $z_{\rm l}=0.4656$. The gravitational arc has a radius of $0''''.42 \simeq 1.74 \, h^{-1} \, {\rm kpc}$. We have determined the total mass and the dark matter (DM) fraction within the Einstein radius as a function of the lensed source redshift, which is presently unknown. For $z_{\rm s}\sim 1.3$, which is in the middle of the redshift range plausible for the source according to some external constraints, we find the central velocity dispersion to be $\sim 180 \, {\rm km \, s}^{-1}$. With this value, close to that obtained by means of the Faber-Jackson relation at the lens redshift, we compute a 30% DM fraction within $\RE\ $ (given the uncertainty in the source redshift, the allowed range for the DM fraction is 25-35 % in our lensing model). When compared with the galaxies in the local Universe, the lensing galaxy [OAC–GL J1223-1239]{} seems to fall in the transition regime between massive DM dominated galaxies and lower-mass, DM deficient systems.'
author:
- 'G. Covone,$^{1,2,3}$ M. Paolillo,$^{1,3}$ N.R. Napolitano,$^{2}$ M. Capaccioli,$^{1,4}$ G. Longo,$^{1}$ J.-P. Kneib,$^{5}$ E. Jullo, $^{5,6}$ J. Richard, $^{7}$ O. Khovanskaya,$^{8}$ M. Sazhin$^{8}$ N.A. Grogin,$^{9}$ and E. Schreier,$^{10}$'
title: 'Gauging the dark matter fraction in a $L_*$ S0 galaxy at $z=0.47$ through gravitational lensing from deep HST/ACS imaging [^1] '
---
Introduction
============
Strong gravitational lensing (GL) is a valuable astrophysical tool to investigate the structure and evolution of early-type galaxies up to a redshift of $\sim 1$ (e.g., Keeton et al. 1998; Rusin et al. 2003, Treu et al. 2005) and to gauge the DM content at various galaxian scales (e.g., Treu et al. 2006, Jiang & Kochanek 2007). This tool offers the advantage of constraining the total mass within the Einstein radius independently of the dynamical status of the lensing galaxy. The technique is challenging to apply to low-luminosity lenses (LLLs), however, as the GL cross section is proportional to the fourth power of the central velocity dispersion (e.g., Covone et al. 2005). This is why LLLs appear far more rare than more massive systems. Their systematically smaller Einstein radii make strong lensing arcs around them hard to find with optical surveys. For instance, a lensing galaxy with velocity dispersion $\sL\sim 200
\, {\rm km \, s^{-1}}$ at $\zl \sim 0.5$, coupled to a source at $\zs \sim 1.0$, produces an Einstein ring with radius $\theta_{\rm E} \sim 0''.5$. As a comparison, in a visual search for gravitational lenses in the COSMOS survey (Faure et al. 2008)[^2], no lensing galaxy was found with central velocity dispersion smaller than $200~$km s$^{-1}$, out of a sample of 20 secure systems. Although difficult to discover, LLLs are of great interest regarding recent claims of dark-matter deficient $L_*$ galaxies in the local Universe (see, e.g., Capaccioli et al. 2003, Romanowsky et al. 2003, Napolitano et al. 2005, N+05 hereafter). Furthermore, these systems occupy as region of the luminosity/mass distribution in between boxy/slow-rotator systems and disky/fast-rotator systems (Nieto & Bender 1989, Capaccioli et al. 1992). The dichotomy of these systems appears to involve also their DM properties, either at effective radii scales (Cappellari et al. 2006) or beyond (Capaccioli et al. 2003, N+05).
This paper analyses a $L_*$ edge-on S0 lensing galaxy at $z_{\rm
l}=0.4656\pm 0.0004$, hereafter named [OAC–GL J1223-1239]{}, serendipitously discovered in the HST/ACS follow-up imaging of a former candidate lensing cosmic string (Sazhin et al. 2007). The mean radius of the gravitational arc is $ \theta_{\rm E} = 0''.42$, corresponding to a linear radius of $1.74 \, h^{-1} \, {\rm kpc}$, slightly larger than the estimated value of the effective (half-light) radius $\reff$ of the lens bulge (i.e. $\theta_{\rm E} \sim 1.15 \, \reff
$). This compact arc offers the rare opportunity to gauge the mass distribution within the effective radius of a $L_*$ galaxy at $z
\sim 0.5$. Note that, among the 20 strong GLs found in the 1.64 square degrees of the COSMOS survey by Faure et al. (2008), selected in the range $ 0.2 < z < 1.0$ and median redshift 0.71, only one lens exhibits an arc with an angular radius smaller than $0''.40$.
As the redshift of the lensed source is yet unknown, despite two spectroscopic runs (see Sect. 2), the DM content of the lens cannot be unambiguously determined. However, by requiring that the lensing galaxy follows the Faber-Jackson relation (Faber & Jackson 1976), and exploiting the decoupled geometry of the luminous and the total mass, we can then constrain the DM fraction within the Einstein radius. Toward this end, in Sect. 3 we present a lensing model, and in Sect. 4 we infer the luminous mass from the analysis of the spectro-photometric data, in order to disentangle the lensing contribution of the DM from that of the total mass. In Sect. \[solution\], we discuss the best mix of dark and luminous mass which produces the measured total mass as a function of the (unknown) source redshift, and draw conclusions in Sect. \[concl\].
Throughout the paper we will assume a cosmological model with $\Omega_{\rm m}=0.27$, $\Omega_{\Lambda}=0.73$ and $h \equiv H_{0}
/ (100 \, {\rm km} \, {\rm s}^{-1}$Mpc$^{-1}) = 0.7$. At the lens redshift, $1''$ corresponds to $4.15 \, h^{-1}$ kpc (Komatsu et al. 2008). Magnitudes are in the AB system.
The data {#data}
========
The gravitational lens [OAC–GL J1223-1239]{} (RA$_{\rm J2000} = $ 12h 23m 32.65s; Dec$_{\rm J2000} = -12^{\rm o}~39'~40''.7$) falls in the Osservatorio Astronomico di Capodimonte Deep Field (OACDF, Alcalá et al. 2004), a survey performed with the 2.2m Wide Field Imager (WFI) in three broad bands (B, V, R) and six intermediate filters over the wavelength range 7730 - 9130 Å. OACDF broadband photometry and coordinates of the lens are given in Table \[table:oacdf\]. The spatial resolution ($FWHM\sim 1''$) of the best stacked OACDF image (R band, limiting magnitude $R = 25.1$) was not sufficient to resolve the gravitational arc, which was instead discovered serendipitously by HST observations.
The latter are deep HST/ACS WFC follow-up images[^3] collected through the F814W filter ($I_{814}$ in the following) for a total exposure time of 15 ks. A 1/3 pixel dither pattern (005 per pixel) was adopted for sub-pixel sampling of the HST PSF and for cosmic ray rejection. The stacked image, obtained by combining the 12 exposures through the software Multidrizzle (Koekemoer et al. 2002), has $0\farcs025$ per pixel sampling with spatial resolution of $0''.10$ and 5$\sigma$ detection limit of $\sim 27.3$ mag arcsec$^{-2}$ (see details in Sazhin et al. 2007). The HST/ACS image of the lensing system is shown in the left panel of Fig. \[fig:one\]. In the right panel of Fig. \[fig:one\] we show the residuals after subtracting the lens photometric model in order to enhance the details of the lensed images. The model is based on the surface photometry performed with the tool [galfit]{} (Peng et al. 2002). The best fit model requires two photometric components: a $r^{1/4}$ (de Vaucouleurs 1948) bulge combined with an exponential disk. The photometric parameters are given in Table \[table:phot\]. Both the photometric model and the spectroscopic data (discussed below) confirm that the lensing galaxy is an edge-on S0.
The gravitational arc, subtending $\sim 150 \, {\rm degree}$, is located at $0''.42$ from the lens centre, perpendicular to the minor axis of the galaxy. A candidate counter-image appears on the opposite side at $0''.65 $. The nature of the other sources around the galaxy is unclear: they could be part of the S0 galaxy and/or could be background objects, possibly linked with the strongly lensed source.
\[table:oacdf\]
band total mag
---------- --------------------- --
WFI B $ 22.74 \pm 0.05 $
WFI V $ 21.86 \pm 0.04 $
WFI R $ 20.85 \pm 0.03 $
ACS F775 $ 20.01 \pm 0.06 $
ACS F814 $ 20.04 \pm 0.05 $
: Broad-band photometry of the lens OAC–GL J1223-1239
The spectroscopic information of the [OAC–GL J1223-1239]{} rests on data collected in two distinct observing runs. A low-resolution spectrum was obtained in April 2000 with the ESO Multi-Mode Instrument at the New Technology Telescope (NTT) in the multiobject spectroscopy mode, within a survey of color-selected early-type galaxies at intermediate redshift. Recently (while the paper was under revision), a long-slit medium resolution spectrum has been secured with the Low Resolution Imaging Spectrometer (LRIS) at the Keck I Telescope.
The total exposure time of the NTT spectrum was $3 \times 2400$ sec, with a slit of 1 arcsec and the grism No. 3, yielding a dispersion of 2.3 Å/pixel. The spectral resolution is $\simeq 10$ Å; the signal-to-noise about 8. Details of the data reduction are given in Alcalá et al. (2004). The NTT spectrum is shown in Fig. \[fig:spec\], with the OACDF broad- and narrow-band imaging photometry overplotted. Some features typical of an early-type galaxy are apparent (the 4000 Å break, the H and K Ca II absorption bands), thus confirming the S0 classification and providing the redshift estimate $\zl=0.466\pm0.005$. No clear feature from the lensed source was observed because of the low S/N: the average surface brightness of the arc, $\mu_{\rm 814} = 22.3 \, {\rm mag \,
arcsec}^{-2}$, is too faint to provide identifiable spectral features other than strong emission lines.
In order to measure the redshift of the gravitational arc, on May 9th 2008 we obtained a spectrum of [OAC–GL J1223-1239]{} with an exposure time of $3\times1200$ sec using the 300 lines/mm grating of Keck/LRIS and a slit $1''$ wide centered on the arc and aligned along the disk axis.
The blue side of the instrument used a 300 lines/mm grating blazed at 5000 Å; the red side used a 600 lines/mm grating blazed at 7500 Å. A dichroic at 6800Å allows a single exposure to sample the wavelength range 3500-9400 Å. The spectral resolution in the blue and red parts of the spectrum was 8.5 and 3.8 Å, respectively. Despite the good seeing conditions ($\sim\ 0''.7$), the spectrum does not allow us to disentangle the arc from the lens. Also, no clear feature from the arc is apparent, owing possibly to the abundance of atmospheric emission toward the red end of the spectrum. The superior spectral resolution of Keck/LRIS gives a more precise redshift for the lensing galaxy of $\zl = 0.4656 \pm 0.0004$.
{width="50.00000%"} {width="50.00000%"}
![Spectral energy distribution of the lensing galaxy: the NTT spectrum (black) and the total magnitudes (red points). Dashed vertical lines show the positions of the strongest spectral features (Ca H and K bands and G band). The horizontal green bar gives the approximate comon wavelength range covered by the F775 and F814 filters.[]{data-label="fig:spec"}](f3.eps){width="50.00000%"}
The rest-frame absolute magnitude of the lens galaxy, derived from the broad-band photometry (see Table \[table:oacdf\]) assuming a negligible extinction[^4] is M$_B = -20.7
\pm 0.1$. Thus [OAC–GL J1223-1239]{} is an intermediate luminosity galaxy (in the sense of the Schechter luminosity function’s parameter $L_*$) since, at $z \sim 0.45$, the absolute magnitude of $L_*$ galaxies is $M_B^* = -20.78 \pm 0.17 $ (Bell et al. 2004).
The color of [OAC–GL J1223-1239]{}, ($V-I) = 1.81$, is close to that of early-type galaxies in clusters at the same redshift. For instance, in the sample of objects at $z \sim 0.47$ within the ESO Distant Cluster Survey (De Lucia et al. 2007), galaxies at $I \sim 20$ on the red sequence have $(V-I) \simeq 1.65$.
\[table:phot\]
[l l l l l l ]{} & mag & $q$ & PA & $\reff$ & $\reff$ \
& & & \[deg\] & & \[kpc\]\
\
bulge & 20.45 & 0.35 & -13.6 & $0''.37$ & 1.54 \
disk & 21.59 & 0.25 & -8.4 & $0''.77$ & 3.16 \
The lensing model and the total mass within {#lensing}
============================================
The mass distribution of the lens component of [OAC–GL J1223-1239]{}, which must reproduce the morphology of the arc and of the counter-image, was modeled by a singular isothermal ellipse (SIE) plus a contribution from an external shear $\gamma$ to take into account the effects of the galaxies close to the line-of-sight. The $\sim
r^{-2}$ total mass density profile of the our model is motivated by several statistical studies of lens samples. For instance, Koopmans et al. (2006) find that the slope of the total mass profile is isothermal at any redshift in the sample of SLACS lensing galaxies. The parameters of the model were fit by means of the [lenstool]{} software[^5] (Kneib 1993), based upon a Bayesian Monte Carlo Markov Chain optimization method (see Jullo et al. 2007 for details). The critical lines of the best SIE model are drawn in Fig. \[fig:one\], and the parameters listed in Table \[table:model\].
\[table:model\]
------------------------------ ------------------- --
$\chi^2 / {\rm n.d.f.}$ 1.28
axis ratio 0.55 $\pm$ 0.06
P.A. \[deg\] $-26 \pm 5$
external shear $\gamma$ $0.080 \pm 0.026$
$\theta_{\gamma}$ (deg) $ -72 \pm 20$
magnification $\sim 12$
($z_s$=0.9) \[km s$^{-1}$\] 204 $\pm 3 $
($z_s$=1.6) \[km s$^{-1}$\] 170 $\pm 3 $
------------------------------ ------------------- --
: Parameters of the best gravitational lensing model
![Color-magnitude diagram of the galaxies within 1 Mpc of the gravitational lens (black boxes), compared with the sources from the whole OACDF survey (red cross). Galaxies at redshift $z = 0.46 \pm 0.01$ in the same spatial region are shown as green boxes. The overplotted line is the fitted red sequence in galaxy clusters at $z\sim0.45$ observed by De Lucia et al. (2007).[]{data-label="fig:cm"}](f4.eps){width="50.00000%"}
The best lensing model includes some external shear, possibly due to a local galaxy overdensity. The presence of a group of galaxies at $\zl$ is supported by both photometric and spectroscopic data. A color-magnitude diagram of the galaxies within 4 arcmin (i.e., a projected radius of $\sim
1$ Mpc) from the lensing galaxy, compared with the distribution of galaxies from the whole OACDF survey (see Fig. \[fig:cm\]), shows a dozen $L>L_* $ galaxies located along a red sequence close to the one observed at $z \sim 0.45$ by De Lucia et al. (2007). Six galaxies have spectroscopic redshifts in the range $z = 0.46 \pm 0.01$, including the [OAC–GL J1223-1239]{} and a system of two bright ellipticals (CSL1, $z = 0.463$, Paolillo et al. 2008) located at $53''$ (220 kpc $\, h^{-1}$) from the lensing galaxy. Note that the spectroscopic survey of early-type galaxies in the OACDF was not spatially complete and did not cover the whole survey field. In particular, only $\sim 1/3$ of the region within 4.0 arcmin from the lens was covered by the masks.
The lensing model exhibits a small difference, $ \Delta \theta =
-13^{\rm o} \pm 5^{\rm o}$, in the orientation of the total mass model (i.e., dark and luminous matter) with respect to the light distribution (see Table \[table:phot\] and \[table:model\]), which does not disappear by forcing a stronger external shear. This result is marginally consistent with Koopmans et al. (2006) who find, for lenses with velocity dispersion $ > 200 ~{\rm km \,
s}^{-1}$, that the total mass is aligned with the light to within $10^{o}$ (see also Kochanek 2002). Furthermore, the total mass distribution is more circularized ($q_{\rm SIE} = 0.55$) than the light ($q_{\rm star}=0.35$).
This modeling points towards a geometry of the total mass different from that of the stellar mass, which in turn suggests the presence, within , of a dark matter component with a different spatial distribution than the light. By an iterative double-component approach with a spherical DM halo coupled to a flattened ($q_{\rm
star}=0.35$) stellar bulge, we find that, within , a dark matter mass of the order of 25% of the total mass is sufficient to account for the ratio $q_{\rm SIE}/q_{\rm star}$ and to reproduce the observed image configuration. A larger DM fraction ($\sim 40$%) is found for a non-spherical ($q
\sim 0.8$) dark halo, in agreement with N-body simulations at this mass scale (Bullock et al. 2001) as well as weak lensing measurements (Hoekstra et al. 2004). It reduces to $\sim 35$% by forcing the halo to match the observed tilt.\
In conclusion, this analysis constrains the dark mass fraction within the Einstein radius to 25-35% of .
Our model additionally provides the central velocity dispersion of the lensing galaxy as a function of the redshifts of the source and of the lens: $$\sL = 144 \, \times \, \sqrt{\frac{D_{\rm s} \, (z_{\rm
s})}{D_{\rm ls} \, (z_{\rm l}, z_{\rm s})} \, } \, {\rm km} \, {\rm s}^{-1} \, ,$$ where $D_{\rm s}$ and $\, D_{\rm ls}$ are the angular diameter distances to the source and from the lens to the source respectively (see also Fig. \[fig:sigma1\]). The corresponding mass is: $$\TM = 4.7 \, \times 10^{10} \, \frac{D_{\rm s} \, (z_{\rm
s})}{D_{\rm ls} \, (z_{\rm l}, z_{\rm s})} \, {\rm M}_{\odot}.$$ For instance, with a source redshift ($z_{\rm s}=1.6$) corresponding to the maximum strong lensing cross section of a lens at $\zl=0.466$, one obtains $ \sL = 170 \, {\rm km \, s}^{-1}$ (see Table \[table:model\]) and $\TM= 7.0\times 10^{10} M_\odot$.
We can compare the trend of with the value of the velocity dispersion derived from the Faber & Jackson relation (1976; FJ) applied to [OAC–GL J1223-1239]{}, $$\frac{ L}{L_*} = \, \left(\frac{\sFJ}{\sigma_*}\right)^{\gamma} \,
\times 10^{\gamma_{\rm ev} z}$$ corrected for galaxy passive evolution. Following Rusin et al. (2003), we adopt a B-band FJ slope $\gamma = 3.29 $, $\sigma_* = 225 \, {\rm km/s}$, and a passive evolution term $\gamma_{\rm ev} = -0.41$. The result $\sFJ = 204 \pm 28 \, {\rm
km \, s}^{-1}$ is larger than the value obtained from the lens model with $z_{\rm s}=1.6$, and requires a source redshift as low as $z_{\rm s}=0.9$, implying $\TM= 10.1\times 10^{10}M_\odot$.
We note, however, that a large uncertainty still remains on the value of the critical parameter $\sigma_*$. For instance, Davis et al. (2003) argue in favor of a significantly lower value, $\sigma_* \sim 185 \pm 15 \, {\rm km \, s}^{-1}$, by which we obtain $\sFJ = 167 \pm 23 \, {\rm km \, s}^{-1}$, identical to the value of with $z_{\rm s}=1.6$.
In summary, the FJ relation constrains $z_{\rm s}$ in the wide range 0.9 - 1.6, equivalent to a $\sim$30% variation of the total mass within the . This is three times larger than typical mass uncertainties at these radii (Kochanek et al. 2000) and does not independently allow to draw any firm conclusion on the DM content of the system.
However, besides the geometrical constraint discussed above ($\DMM\simeq 25-35\% \, \TM$) and that given by the Faber-Jackson relation ($144 < \sL < 232 \, {\rm km \, s}^{-1}$, including errors), we have still another card to play: we may estimate the stellar mass within from the spectro-photometric data.
![Velocity dispersion of the best SIE lens model as a function of the source redshift. Dashed lines correspond to the two FJ estimates (see text for details).[]{data-label="fig:sigma1"}](f5.eps){width="50.00000%"}
The Keck/LRIS spectroscopic data allow us to measure the average velocity dispersion $\overline{\sigma}$ of the lensing galaxy over the region covered by the slit, which is $\sim 3 \reff$ wide and slightly off-center with respect to the galaxy nucleus (see Fig. 1). The fit of the $\lambda5892$ absorption line profile gives $\overline{\sigma}=150 \pm 11 \, {\rm km s}^{-1}$. By modeling the galaxy as in Napolitano et al. (2008, in preparation), this figure returns an estimate of the central velocity dispersion in agreement with the lower value predicted by the FJ relation, i.e. $\sFJ = 170 \pm 15 \, {\rm km s}^{-1}$. Mindful of the uncertainty intrinsic to this extrapolation, in the following analysis we will consider the whole range of values allowed by the FJ relation.
The stellar mass content within $R_{\rm E}$
===========================================
The stellar mass-to-light ratio, , has been derived by fitting the low-resolution spectrum (Fig. 2) with the library of synthetic spectra by Bruzual & Charlot (2003). Since the choice of the initial mass function (IMF) is a critical point, we shall explore both Salpeter (1955) and Chabrier (2003) IMFs. A Salpeter IMF gives a global stellar mass-to-light ratio $M_* / L = 4.0 \, M_{\odot} /
L_{\odot}$, in the rest-frame B-band, with an age of 8 Gyr and a sub-solar metallicity ($Z = 0.004 \, Z_{\odot}$). If is uniform, from the total luminosity $L_{\rm B} = 2.7 \times 10^{10}
\, L_{\odot}$ one obtains a total stellar mass $ M_* = 1.1 \times
10^{11} $ M$_{\odot}$.
In order to compute $\SM$, the stellar mass within , it is prudent to disentangle the contributions of the two components, disk and bulge, which have different light profiles, isophotal geometry (see Table 2), and possibly mass-to-light ratios. Evolutionary synthesis models of template S0 galaxies of age $\sim
8$ Gyr predict a bulge mass-to-light ratio, $(M_*/L)_{\rm b}$, in excess of up to 1.5 times that of the disk, $(M_*/L)_{\rm d}$ (Buzzoni 2005). We use this upper limit to separate the contribution of the two components.
By definition, $$M_*= \mlBulge \times L_{\rm bulge}+ \mlDisk \times L_{\rm disk}
\label{eq:ml}$$ where $ L_{\rm bulge} = 1.9 \times 10^{10} L_{\odot} $ and $L_{\rm
disk} = 0.8 \times 10^{10} L_{\odot}$ from the measured bulge-to-disk ratio $B/D \sim 2.2$ (Table 2). Solving Eq. (\[eq:ml\]) for a Salpter IMF, we obtain $\mlDisk = 2.9 \,
M_{\odot} / L_{\odot}$ and $\mlBulge = 1.5 \times M/L_{\rm d} = 4.4
\, M_{\odot} / L_{\odot}$. Thus, the bulge and disk masses within $R_{\rm E}$ are $M^E_{\rm b}=5.4 \times 10^{10} M_{\odot}$ and $M^E_{\rm d}=0.3 \times 10^{10} M_{\odot}$ respectively, with a total stellar mass $\SM=5.7 \times 10^{10} \, M_{\odot}$.
A simpler model with a single value of $M_*/L$ gives a total mass (bulge+disk) within $R_{\rm E}$ of $5.2 \times 10^{10} \, M_{\odot}$ (Table 4).
Repeating the calculations with a Chabrier IMF, the stellar ratio is $\sim 1.8$ smaller than with Salpeter ($\SMLR = 2.2 \,
M_{\odot} / L_{\odot}$ for the single population model, and $M/L_{\rm b}= 2.6$, $M/L_{\rm d}=1.6\, M_{\odot}/L_{\odot}$ for the two population model), with the same age and metallicity assumed above. The stellar masses of bulge and disk are 1.8 times smaller, accordingly (Table 4).
[llllll]{}\
model & $M/L_{\rm b}$ & $M_{\rm b}$& $M/L_{\rm b}$ & $M_{\rm d}$ & $M_* (R_{\rm E})$ \
$M/L_{\rm b} = 1.5 \times M/L_{\rm d}$ & 4.4 & 5.4 & 2.9 & 0.3 & 5.7\
$M/L_{\rm b}=M/L_{\rm d}$ & 4.0 & 4.8 & 4.0 & 0.4 & 5.2 \
\
\
model & $M/L_{\rm b}$ & $M_{\rm b}$ & $M/L_{\rm b}$ & $M_{\rm d}$ & $M_* (R_{\rm E})$\
$M/L_{\rm b} = 1.5 \times M/L_{\rm d}$ & 2.6 & 3.0 & 1.6 & 0.2 & 3.2\
$M/L_{\rm b}=M/L_{\rm d}$ & 2.2 & 2.7 & 2.2 & 0.2 & 2.9 \
Comparing lensing and stellar masses {#solution}
====================================
Since the quantity $\TM$ remains parametrised with the source redshift (see Sect. 3), we now use the $\SM$ values derived in the previous Section to infer the range for $z_{\rm s}$ compatible with the other constraints. Figure \[fig:tre\] shows the trend for the projected DM fraction, $\fracDM (R_{\rm E})=\DMM/\TM$, as a function of the $z_{\rm s}$, for both Salpeter and Chabrier IMFs, together with the constraints by the FJ relation and by the geometry of the system (green region). The redshift range allowed by all the constraints is $z_{\rm s}= 1.2-1.5$ in the case of a single population model and $z_s=1.1-1.3$, in the case of a composite disk+bulge population. The Chabrier IMF is highly inconsistent with the given limits.
In conclusion, considering a DM fraction (within ) in the range 25-35%, the most likely solutions correspond to the redshift range $z_{\rm s} = 1.3 \pm 0.2$ and favor the Salpeter IMF.
Interpreting the favored lens model in the $\Lambda$CDM framework
-----------------------------------------------------------------
The total $M/L$ within varies from $4.7 \, (z_s=1.5)$ to $6.1 \, (z_s=1.1)$, with an average value of $M/L(R_{\rm E})=5.4$–5.5 for $z_{\rm s}=1.3$. As shown in Sect. 4, considering just the baryons contribution, $M/L_*=4.0$–4.4, allowing for the two-population model (Table 2); that is, the total $M/L$ is a factor $\sim 1.2-1.5$ larger than the stellar $M/L$ over the whole redshift range. These variations can be interpreted with the toy model from N+05 to infer the global properties of the virial DM content of the lensing system in the $\Lambda$CDM cosmology (see also Ferreras et al. 2005 for an application to a sample of lensing galaxies).
We consider a multi-component galaxy model with bulge stars distributed following a Hernquist (1990) profile with effective radius and total luminosity as from the first row of Table 2, disk stars following an exponential profile, with parameters as in the second row of Table 2, and a spherical DM halo with a Navarro, Frenk & White (NFW; 1998) density profile. As in N+05 and Bullock et al. (2001), we adopt the following concentration-mass relation for the DM halo: $$c_{\rm dm}(M_{\rm dm})\simeq \frac{1}{1+z} 17.1
\left( \frac{M_{\rm dm}}{10^{11} M_{\odot} \, h^{-1} }\right )^{-0.125} \, ,
\label{cMvir}$$ where the concentration parameter is $c_{\rm dm} \equiv r_{\rm
vir}/r_s$, with $r_{\rm s}$ the characteristic scale of the NFW profile, and $M_{\rm dm}$ is the total dark halo mass at the virial radius $r_{\rm vir}$. The cumulative mass profile for a NFW dark halo is $$M_{\rm dm}(r) = M_{\rm dm} \frac{A(r/r_s)}{A(c_{\rm dm})} \, ,
\label{MNFW}$$ where $$A(x)=\ln (1+x)-\frac{x}{1+x} \, .$$
We can now estimate the ratio $f_{\rm vir} \, \equiv \massDM
/M_{*}$, computing the NFW halo mass from Eq. (\[cMvir\]) and Eq. (\[MNFW\]), and the stellar mass associated to the Hernquist model by using the photometric parameters of bulge and disk (see Table \[table:phot\]). Imposing that the modeled $M/L$ is a factor $\sim 1.2-1.5$ larger than the stellar $M/L$ within $R_{\rm E} (\sim 1.15 \,
\reff$)[^6], we find $f_{\rm vir}
\, \sim 6 \pm 4$, for the whole galaxy. Here the main uncertainty comes from the choice of the stellar $M/L$, rather than from the halo concentration from the adopted c$_{\rm dm}-$M$_{\rm dm}$ (see, e.g., Bullock et al. 2001) and the total $M/L(R_{\rm E})$ estimates. In Fig. \[fig:tre\] the projected DM fraction is shown together with the corresponding values of $f_{\rm vir}$ as a function of the source redshift. If we consider the whole allowed range of $z_{\rm
s}$ and the effect of the choice of the stellar $M/L$ on the DM fractions, we see that the largest $f_{\rm vir}$ value is $\sim$20 for $z_{\rm s}$=1.1 (i.e., the lowest redshift allowed), while $f_{\rm vir}<20$ for $z>1.1$, and $f_{\rm vir}<10$ for $z_{\rm
s}>1.2$.
These values are consistent with the low global DM fraction regime at $z=0$ discussed in N+05. These authors found a transition mass around $M_{\rm *}=1.6\times 10^{11} M_\odot$ between more massive DM dominated systems and low-mass, DM poorer systems. Incidentally, [OAC–GL J1223-1239]{} is exactly located in this transition regime.
Discussion and conclusions {#concl}
==========================
![DM fraction at $R_{\rm E}$ versus the (unknown) redshift of the lensed source. On the right the corresponding (mean) $f_{\rm vir}$ values; see text for details. Empty boxes are the DM fractions for the single bulge+disk population, full boxes for the two population models, assuming Salpeter IMF for the stellar $M/L$. Diamonds represent the same models for the Chabrier IMF. Grey denotes the regions excluded by the constraints on the $\fracDM$ from the lensing model geometry, cyan are the regions excluded by the Faber-Jackson relation. In green, the range of the parameter space allowed by all the contraints, accounting also for the uncertainty associated to the population models $M/L$.[]{data-label="fig:tre"}](f6.eps){width="50.00000%"}
[OAC–GL J1223-1239]{}, a $\sim L_*$ S0 galaxy at redshift $z=0.466$, is among the least massive lenses at high redshift known to date. Although the source redshift is still not known, the observed phenomenology unambiguously supports the strong-lensing interpretation. The lensed source produces a bright arc (see Fig. 1) with radius $0''.42$ ($\sim 1.15 \, \reff$ of the lens bulge). A candidate counter-image is found at $0''.65$ from the lens center.
The total mass distribution, modeled by a singular isothermal ellipse, is found to be rounder than the light at the Einstein radius, with a ratio $ q_{\rm SIE} / q_{*} = 1.84$, similar to the value found for galaxies with velocity dispersions lower than $\sim
200 \, {\rm km \, s}^{-1}$. We constrain the redshift of the source by coupling the value of the ratio $(\massDM /M_*) (\RE)$, obtained via the lens model, with the lens total mass within $R_{\rm E}$, once the corresponding stellar mass is evaluated by an assumption on the baryonic component and corresponding stellar $M/L$. We obtain $z_{\rm s} \sim 1.3 \pm 0.2$ with a Salpeter IMF, the Chabrier IMF being ruled out. In this case the velocity dispersion from the lens model (a singular isothermal ellipse with an external shear) is $\sL\ = 177^{+10}_{-6}$ km s$^{-1}$ (with the given uncertainty derived from the allowed $z_{\rm s}$ range). This value is consistent with the limits imposed by the FJ relation.
[OAC–GL J1223-1239]{} offers the possibility to investigate the mass distribution of the very inner region in an $L_*$ galaxy at $z \sim 0.5$. Indeed, while there is growing evidence for a picture in which massive galaxies (i.e., above $\sim 10^{11.5} M_{\odot}$) are well described by an isothermal mass profile and show no structural evolution since $z\sim1$ (see, for instance, Treu 2007), we still lack a large sample of $L_*$ galaxies at $z\gtrsim0.5$ with a robust determination of the inner DM fraction in order to probe mass assembly and galaxy evolution at lower mass scales.
We find that [OAC–GL J1223-1239]{}(assuming $z_{\rm s} = 1.3$) has a projected DM fraction $\sim 30$% within the Einstein radius. Lower values of the source redshift require a larger DM fraction in a non-spherical halo or a different IMF (e.g., Fig. \[fig:tre\]). By matching the measured mass-to-light ratio at the with the one expected from a double-component model formed by a NFW spherical halo and a Hernquist spheroidal light distribution, we derive the total dark-to-luminous mass ratio $f_{\rm vir} \equiv \massDM /M_{*} = 6 \pm 4$. This value is close to the one obtained for galaxies in the local Universe (N+05) located in the transition regime between massive DM dominated galaxies and lower-mass, DM deficient systems.
A firmer estimate of the inner DM and stellar content in this $L_*$ galaxy requires a further observational effort to determine the redshift of the lensed source. A possible strategy would involve, for instance, near-infrared spectroscopy aimed at detecting the H$_{\alpha}$ emission line.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank J.M. Alcalá and M. Pannella for allowing them to use the NTT spectrum of the lensing galaxy, C. Tortora for discussions on the derivation of the stellar mass from spectroscopic data, the Director of the Space Telescope Science Institute for granting Director’s Discretionary Time, and the referee for constructive comments. GC and NRN acknowledge funding from EC through the FP6-European Reintegration Grants MERG-CT-2005-029159 and MERG-CT-2005-014774, respectively. JPK acknowledge support from CNRS. OK acknowledges INTAS grant Ref.Nr. 05-109-4793. NAG acknowledges support from grant HST-GO-10715.11-A.
Alcalá, J.M., Pannella, M., Puddu, E. et al. 2004, A&A 428, 339
Bell, E. et al. 2004, ApJ 608, 752
Bruzual & Charlot 2003, MNRAS 344, 1000
Bullock, J. S., Kolatt, T. S., Sigad, Y., Somerville, R. S., Kravtsov, A. V., Klypin, A. A., Primack, J. R., & Dekel, A. 2001, , 321, 559
Burstein, D. & Heiles, C. 1982, AJ 87, 1165
Buzzoni, A. 2005, MNRAS 361, 725
Capaccioli M., Caon N., D’Onofrio M., 1992, ESO ESP/EIPCWorkshop on Structure, Dynamics and Chem- ical Evolution of Early-type Galaxies (Elba), eds. J. Danziger, W. W. Zeilinger, and K. Kjar, ESO: Garching, 43
Capaccioli, M., Napolitano, N. R., Arnaboldi, M. 2003, in Proceedings for the Sakharov Conference of Physics (Moskow), astro-ph:0211323
Cappellari, M. et al. 2006, MNRAS 366, 1126
Chabrier, G. 2003, PASP 115, 763
Covone, G, Sereno, M., de Ritis, R. 2005, MNRAS 357, 773
De Lucia, G., Poggianti, B.M., Aragón-Salamanca, A. et al. 2007, MNRAS 374, 809
de Vaucouleurs, G. 1948, Annales d’Astrophysique 11, 247
Faber, S.M. & Jackson, R.E. 1976, ApJ 204, 668
Faure, C., Kneib, J.-P., Covone, G. e al. 2008, ApJ, in press, arXiv:0802.2174
Ferreras, I., Saha, P., Williams, L. L. R. 2005, ApJ 623, L5
Hernquist, L. 1990, ApJ, 356 359
Jiang, G. & Kochanek, C. 2007, ApJ 671, 1568
Hoekstra, H. et al. 2004, ApJ 606, 67
Keeton, C.R., Kochanek, C.S., Falco, E.E. 1998, ApJ 509, 561
Kneib, J.-P. 1993, PhD thesis, Toulouse
Kochanek, C.S. 2002, in “The Shapes of Galaxies and Their Dark Matter Halos”, New Haven, Connecticut, USA, 28-30 May 2001. Edited by P. Natarajan, World Scientific
Kochanek, C.S., Keeton, C.R. & McLeod, B.A. 2001, ApJ 547, 50
Kochanek, C. S., et al. 2000, , 543, 131
Koekemoer, A.M., Fruchter, A.S., Hook, R., Hack, W. 2002 HST Calibration Workshop, 337.
Komatsu, E. et al. 2008, submitted, arXiv:0803.0547
Koopmans, L.V.E., Treu, T., Bolton, A.S. et al. 2006, ApJ 649, 599
Jullo, E., Kneib, J.-P., Limousin, M. et al. 2007, “Gravitational Lensing” Focus Issue of the New Journal of Physics, in press, arXiv:0706.0048v1
Napolitano, N.R., Capaccioli, M., Romanowsky, A. J. et al. 2005, MNRAS 357, 691 (N+05)
Nieto, J.L., Bender, R., 1989, A&A, 215, 266
Peng, C.Y. et al. 2002, AJ 124, 266
Romanowsky, A.J., Douglas, N.G., Arnaboldi, M. et al. 2003, Science 5640, 1696
Rusin, D., Kochanek, C.S., Falco, E.E. et al. 2003, ApJ 587, 143
Salpeter, E. E. 1955, ApJ 121, 161
Sazhin, M. V., Khovanskaya, O. S., Capaccioli, M., Longo, G., Paolillo, M., Covone, G., Grogin, N. A., & Schreier, E. J. 2007, , 376, 1731
Treu, T. et al. 2005, ApJ 633, 174
Treu, T. et al. 2006, ApJ 640, 662
Treu, T. 2007, in “Galaxy Evolution Across the Hubble Time”, Edited by F. Combes and J. Palous, Cambridge University Press, 2007, p.12
[^1]: Based on observations made with ESO Telescopes at the La Silla Observatories and the NASA/ESA Hubble Space Telescope.
[^2]: [http://cosmosstronglensing.uni-hd.de/]{}
[^3]: Proposal ID 10715, PI: M. Capaccioli. The same field was also observed with HST/ACS by the program 10486, using filters F775W and F606W and with shorter exposures (2409 and 5062 s, respectively). However, the filter F775W covers very similar wavelength range to the I$_{814}$ image, offering no additional information, and in the F606W-band data our target is a few arcsec out of the field-of-view.
[^4]: ${\rm E(B-V)}$ is below 0.03 all over the OACDF; see Burstein & Heiles (1982).
[^5]: Publicly available at [www.oamp.fr/cosmology/lenstool/]{}.
[^6]: This corresponds to a logarithmic $M/L$ gradient, $\nabla \Upsilon\equiv \frac{\reff \Delta \Upsilon}{\Upsilon_{\rm
in} \Delta R}\sim0.2-0.3$ (see N+05) which is typical of local low-$\nabla \Upsilon$, i.e. low-DM systems.
|
[9999 .5@ .5@]{}
Behavior of local cohomology modules under polarization
Mitsuhiro Miyazaki[^1]
Let $S=k[x_1,\ldots, x_n]$ be a polynomial ring over a field $k$ with $n$ variables $x_1$, …, $x_n$, ${\mathfrak{m}}$ the irrelevant maximal ideal of $S$, $I$ a monomial ideal in $S$ and $I'$ the polarization of $I$ in the polynomial ring $S'$ with $\rho$ variables. We show that each graded piece $H_{\mathfrak{m}}^i(S/I)_\aaa$, $\aaa\in\ZZZ^n$, of the local cohomology module $H_{\mathfrak{m}}^i(S/I)$ is isomorphic to a specific graded piece $H_{{\mathfrak{m}}'}^{i+\rho-n}(S'/I')_\alphaaa$, $\alphaaa\in\ZZZ^\rho$, of the local cohomology module $H_{{\mathfrak{m}}'}^{i+\rho-n}(S'/I')$, where ${\mathfrak{m}}'$ is the irrelevant maximal ideal of $S'$.
local cohomology, monomial ideal, polarization, Hochster’s formula
Introduction
============
Let $S=k[x_1,\ldots, x_n]$ be a polynomial ring over a field $k$ with $n$ variables $x_1$, …, $x_n$ and ${\mathfrak{m}}$ the irrelevant maximal ideal of $S$. For a monomial ideal $I$ in $S$, the local cohomology modules $H_{\mathfrak{m}}^i(S/I)$ have $\ZZZ^n$-graded structure.
Hochster described each graded piece of $H_{\mathfrak{m}}^i(S/I)$ by using the reduced cohomology group of a simplicial complex related to $I$ when $I$ is square-free (see [@sta II 4.1 Thoerem]). Takayama [@tak Theorem 1] generalized this result to the case where $I$ is not necessarily square-free.
On the other hand, there is a technique which associates a not necessarily square-free monomial ideal $I$ with a square-free monomial ideal, called the polarization of $I$, sharing many ring theoretical properties with $I$.
In this note, we show that the each graded piece of the local cohomology modules $H_{\mathfrak{m}}^i(S/I)$ is isomorphic to a specific graded piece of the local cohomology module of the polarization of $I$.
Preliminaries
=============
Let $S=k[x_1,\ldots, x_n]$ be a polynomial ring over a field $k$ with $n$ variables $x_1$, …, $x_n$ and ${\mathfrak{m}}$ the irrelevant maximal ideal of $S$ and $I$ a monomial ideal of $S$.
For a monomial $m=x_1^{b_1}\cdots x_n^{b_n}$ in $S$, we set $\nu_i(m)=b_i$. We denote by $G(I)$ the minimal set of monomial generators of $I$. Set $\rho_i=\max\{\nu_i(m)\mid m\in G(I)\}$ for $i=1$, $2$, …, $n$ and $\rho=\rho_1+\cdots+\rho_n$. Then the polarization of $I$ is defined as follows.
Let $S'=k[x_{ij}\mid 1\leq i\leq n$, $1\leq j\leq \rho_i]$ be the polynomial ring with $\rho$ variables $\{x_{ij}\}$. For a monomial $m$ in $S$, we set $m'=\prod_{i=1}^n\prod_{j=1}^{\nu_i(m)}x_{ij}$. Then the polarization $I'$ of $I$ is defined by $I'=(m'\mid m\in G(I))S'$. It is clear from the definition that $I'$ is a square-free monomial ideal. Furthermore, it is known that $
\{x_{ij}-x_{i1}\mid 1\leq i\leq n, 2\leq j \leq \rho_i\}
$ is an $S'/I'$-regular sequence in any order and $$S'/(I'+(x_{ij}-x_{i1}\mid 1\leq i\leq n, 2\leq j \leq \rho_i))\simeq S/I.$$
For vectors $\aaa=(a_1,\ldots, a_n)$ and $\bbb=(b_1,\ldots, b_n)$, we denote $\aaa\leq\bbb$ to express that $a_i\leq b_i$ for $i=1$, …, $n$. And we define ${\mathrm{supp}}_-\aaa=\{i\mid a_i<0\}$ and call the negative support of $\aaa$. We set $\zerovec=(0,0,\ldots,0)$, $\onevec=(1,1,\ldots,1)$ and $\rhooo=(\rho_1,\rho_2,\ldots, \rho_n)$.
We denote the cardinality of a finite set $X$ by $|X|$ and the set $\{1,2, \ldots, n\}$ by $[n]$.
Here we recall the result of Takayama.
\[tak thm\] Let $S$ and $I$ be as above and $\aaa\in\ZZZ^n$. Set $\Delta_\aaa=\{F\setminus{\mathrm{supp}}_-\aaa\mid
[n]\supset F\supset{\mathrm{supp}}_-\aaa,
\forall m\in G(I)\exists i\in[n]\setminus F;a_i<\nu_i(m)\}$. Then $$H_{\mathfrak{m}}^i(S/I)_\aaa\simeq\tilde H^{i-|{\mathrm{supp}}_-\aaa|-1}(\Delta_\aaa;k).$$
Note that $\Delta_\aaa$ is a simplicial complex with vertex set $[n]$. We call $\Delta_\aaa$ the Takayama complex. Note also that if $a_i\geq \rho_i$, then $\Delta_\aaa$ is a cone over $i$ and all the reduced cohomology modules vanish. Therefore $H_{\mathfrak{m}}^i(S/I)_\aaa=0$ if $\aaa\not\leq\rhooo-\onevec$.
For a $\ZZZ^n$-graded module $M$, we define the degree shifted module $M(\aaa)$ by $M(\aaa)_\bbb=M_{\aaa+\bbb}$.
Main theorem
============
Now we state the main result of this paper.
\[main thm\] With the notation in previous section, assume that $\aaa\leq\rhooo-\onevec$. Set $$\alpha_i=
\left\{
\begin{array}{ll}
(\underbrace{0,\ldots,0}_{a_i+1},\underbrace{-1,\ldots,-1}_{\rho_i-a_i-1})&
\quad\mbox{if $a_i\geq 0$,}\\
(\underbrace{-1,\ldots,-1}_{\rho_i})&
\quad\mbox{if $a_i< 0$}
\end{array}
\right.$$ and $\alphaaa=(\alpha_1,\alpha_2,\ldots,\alpha_n)\in\ZZZ^\rho$. Then $$H_{\mathfrak{m}}^i(S/I)_\aaa\simeq
H_{{\mathfrak{m}}'}^{i+\rho-n}(S'/I')_\alphaaa,$$ where ${\mathfrak{m}}'$ is the irrelevant maximal ideal of $S'$.
[**proof.**]{} [**Step 1.**]{} We first consider the case where $\aaa=\rhooo-\onevec$.
First note that local cohomology modules $H_{{\mathfrak{m}}'}^i(S'/I')$ have not only the $\ZZZ^\rho$-grading but also the $\ZZZ^n$-grading by setting $\deg x_{ij}=\eee_i$, where $\eee_i$ is the $i$-th fundamental vector in $\ZZZ^n$.
We denote by $C^\bullet$ the Čech complex with respect to $\{x_{ij}\mid 1\leq i\leq n$, $1\leq j\leq \rho_i\}$ and $K_\bullet$ the Koszul complex with respect to $\{x_{ij}-x_{i1}\mid 1\leq i\leq n, 2\leq j \leq \rho_i\}$. Set $M^{p,q}=C^p\otimes_{S'}K_{\rho-n-q}\otimes_{S'}S'/I'$. Then $M^{\bullet,\bullet}$ is a third quadrant double complex which has a $\ZZZ^n$-graded structure. Set $\{'E_r\}$ and $\{''E_r\}$ be the spectral sequences arising from $M^{\bullet,\bullet}$.
Since $\{x_{ij}-x_{i1}\mid 1\leq i\leq n, 2\leq j \leq \rho_i\}$ is an $S'/I'$-regular sequence, we see that $$'E_1^{p,q}
\simeq
\left\{
\begin{array}{ll}
C^p\otimes S/I&\quad\mbox{$q=\rho-n$,}\\
0&\quad\mbox{otherwise.}
\end{array}
\right.$$ And the horizontal complex $'E_1^{\bullet,\rho-n}$ is isomorphic to the Čech complex with respect to $\underbrace{x_1, x_1, \ldots, x_1}_{\rho_1}$, $\underbrace{x_2, x_2, \ldots, x_2}_{\rho_2}$, …, $\underbrace{x_n, x_n, \ldots, x_n}_{\rho_n}$. Therefore $$'E_2^{p,q}
\simeq
\left\{
\begin{array}{ll}
H_{\mathfrak{m}}^p(S/I)\quad&\mbox{$q=\rho-n$,}\\
0&\mbox{otherwise.}
\end{array}
\right.$$ So the spectral sequence $\{'E_r\}$ collapses and we see that $$H^i({\mathrm{Tot}}(M^{\bullet,\bullet}))\simeq
{}'E_2^{i-\rho+n,\rho-n}\simeq
H_{\mathfrak{m}}^{i-\rho+n}(S/I)$$ for any $i\in\ZZZ$, where ${\mathrm{Tot}}(M^{\bullet,\bullet})$ is the total complex of $M^{\bullet,\bullet}$.
Next we consider $\{''E_r\}$. It is clear that $''E_1^{p,q}\simeq K_{\rho-n-q}\otimes_{S'}H_{{\mathfrak{m}}'}^p(S'/I')$. Since $I'$ is square-free, $H_{{\mathfrak{m}}'}^p(S'/I')_{\alphaaa}=0$ if $\alphaaa\not\leq\zerovec$ by the remark after Theorem \[tak thm\]. Therefore $H_{{\mathfrak{m}}'}^p(S'/I')_l=0$ if $l>0$, $l\in\ZZZ$, where $H_{{\mathfrak{m}}'}^p(S'/I')_l$ denotes the total degree $l$ piece of $H_{{\mathfrak{m}}'}^p(S'/I')$.
Since $K_{\rho-n-q}$ is a free $S'$-module with free basis consisting of total degree $\rho-n-q$ elements, we see that $$(''E_1^{p,q})_{\rho-n}=0\quad\mbox{if $q\neq0$.}$$ Therefore total degree $\rho-n$ piece of $\{''E_r\}$ collapses and we see that $$H^i({\mathrm{Tot}}(M^{\bullet,\bullet}))_{\rho-n}\simeq
(''E_1^{i,0})_{\rho-n}\simeq
H_{{\mathfrak{m}}'}^i(S'/I')_0$$ for any $i\in\ZZZ$.
So $$H_{{\mathfrak{m}}}^{i-\rho+n}(S/I)_{\rho-n}\simeq
H_{{\mathfrak{m}}'}^i(S'/I')_0$$ for any $i\in\ZZZ$. By the remark after Theorem \[tak thm\], we see that $H_{\mathfrak{m}}^j(S/I)_{\rho-n}=H_{\mathfrak{m}}^j(S/I)_{\rhooo-\onevec}$ and $H_{{\mathfrak{m}}'}^j(S'/I')_0=H_{{\mathfrak{m}}'}^j(S'/I')_\zerovec$ for any $j\in\ZZZ$. This means $$H_{\mathfrak{m}}^i(S/I)_{\rhooo-\onevec}\simeq
H_{{\mathfrak{m}}'}^{i+\rho-n}(S'/I')_\zerovec$$ and this is what we wanted to prove.
[**Step 2.**]{} Next we consider the case where $a_i=\rho_i-1$ or $a_i<0$ for any $i=1$, …, $n$.
By changing the subscripts, we may assume that $a_i=\rho_i-1$ for $i=1$, …, $m$ and $a_i<0$ for $i=m+1$, …, $n$. Set $S_+=k[x_1,\ldots, x_m]$, $I_+=IS[x_{m+1}^{-1},\ldots, x_n^{-1}]\cap S_+$ and ${\mathfrak{m}}_+$ the irrelevant maximal ideal of $S_+$. Note that $I_+$ is the monomial ideal of $S_+$ generated by the monomials obtained by substituting 1 to $x_{m+1}$, …, $x_n$ of the monomials in $I$. Note also that if we denote the Takayama complex with respect to $I_+$ and $(a_1,\ldots, a_m)$ by $\Delta_+$, then $\Delta_+=\Delta_\aaa$. Therefore $$\begin{array}{rcl}
H_{\mathfrak{m}}^i(S/I)_\aaa
&\simeq&\tilde H^{i-|{\mathrm{supp}}_-\aaa|-1}(\Delta_\aaa;k)\\
&\simeq&\tilde H^{i-(n-m)-1}(\Delta_+;k)\\
&\simeq&H_{{\mathfrak{m}}_+}^{i-n+m}(S_+/I_+)_{(a_1,\ldots,a_m)}
\end{array}$$ since $|{\mathrm{supp}}_-\aaa|=n-m$ and $|{\mathrm{supp}}_-(a_1,\ldots,a_m)|=0$.
We also set $S'_+=k[x_{ij}\mid 1\leq i\leq m, 1\leq j\leq \rho_i]$, $I'_+=I'S'[x_{ij}^{-1}\mid m+1\leq i\leq n, 1\leq j\leq \rho_i]\cap S'_+$ and ${\mathfrak{m}}'_+$ the irrelevant maximal ideal of $S'_+$. Then it is easily verified that $I'_+$ is the polarization of $I_+$. And we see that $H_{{\mathfrak{m}}'}^{i+\rho-n}(S'/I')_\alphaaa
\simeq H_{{\mathfrak{m}}'_+}^{i+\rho_1+\cdots+\rho_m-n}(S'_+/I'_+)_{(\alpha_1,\ldots,\alpha_m)}$ by the same argument above. Therefore, we can reduce this case to the case of step 1.
[**Step 3.**]{} Finally, we consider the general case.
Assume that $0\leq a_n<\rho_n-1$. Consider the “partial polarization with respect to $x_n$”. That is, set $S''=S[x_{nj}\mid a_n+2\leq j\leq \rho_n]$ and for a monomial $m$ in $S$, set $$m''=
\left\{
\begin{array}{ll}
m&\mbox{if $\nu_n(m)\leq a_n+1$,}\\
\prod_{i=1}^{n-1}x_i^{\nu_i(m)}x_n^{a_n+1}\prod_{j=a_n+2}^{\nu_n(m)}x_{nj}
\quad&\mbox{if $\nu_n(m)\geq a_n+2$.}
\end{array}
\right.$$ Set also $I''=(m''\mid m\in G(I))$. Then the polarization of $I''$ is the same as that of $I$.
And if we denote the Takayama complex with respect to $I''$ and $(a_1,\ldots, a_{n-1}, a_n, \underbrace{-1,\ldots,-1}_{\rho_n-1-a_n})$ by $\Delta''$, then we see that $\Delta''=\Delta_\aaa$. Therefore $$\begin{array}{rcl}
H_{\mathfrak{m}}^i(S/I)_\aaa
&\simeq&\tilde H^{i-|{\mathrm{supp}}_-\aaa|-1}(\Delta_a;k)\\
&\simeq&\tilde H^{i-|{\mathrm{supp}}_-(a_1,\ldots, a_{n-1}, a_n, -1,\ldots,-1)|-1+(\rho_n-1-a_n)}(\Delta'';k)\\
&\simeq&H_{{\mathfrak{m}}''}^{i+\rho_n-(a_n+1)}(S''/I'')_
{(a_1,\ldots, a_{n-1}, a_n, -1,\ldots,-1)}.
\end{array}$$ So we can reduce the proof to the case where $a_n=\rho_n-1$.
Using this argument repeatedly, we may assume that $a_i\geq 0\Rightarrow a_i=\rho_i-1$, i.e., we can reduce the proof of the theorem to the case of step 2.
------------------------------------------------------------------------
Step 1 of the proof of Theorem \[main thm\] can also be proved by using [@sba Corollary 5.2], insted of the spectral sequence argument.
[Tak]{}
Sbarra, E.: [*Upper bounds for local cohomology for rings with given Hilbert function.*]{} Communications in Algebra [**29**]{} (2001), 5383–5409.
Stanley, R. P.: “Combinatorics and Commutative Algebra", Progress in Math., Vol.41, Birkh$\ddot{\mbox{a}}$user, Boston/ Basel/ Stuttgart, 1983
Takayama, Y.: [*Combinatorial characterizations of generalized Cohen-Macaulay monomial ideals.*]{} preprint, available from [http://front.math.ucdavis.edu/math.AC/0503111]{}
[^1]: Dept. Math.Kyoto University of Education, Fushimi-ku, Kyoto, 612-8522 Japan E-mail: g53448@kyokyo-u.ac.jp
|
---
author:
- Petar Simidzija
- 'Eduardo Martín-Martínez'
bibliography:
- 'references.bib'
title: The information carrying capacity of a cosmological constant
---
Introduction\[sec:intro\]
=========================
In recent years, there have been a number of results highlighting the relationship between the physics of information and fundamental topics in quantum field theory and gravitation. For example, quantum entanglement in vacuum fluctuations has been linked to phenomena like Hawking radiation and the Unruh effect [@Hotta2015]. Entanglement in the vacuum state of a quantum field can also be viewed as a resource in protocols of *quantum energy teleportation* [@Hotta2009; @Hotta2011], and can be *harvested* [@Valentini1991; @Reznik2003; @Reznik2005; @Franson2008; @Lin2009; @Pozas2015; @Pozas2016], or *farmed* [@Martin-Martinez2013a], by particle detectors which locally couple to the field. These detectors can become entangled with one another, even if they are spacelike separated. Interestingly, entanglement harvesting has been proven to be sensitive to spacetime curvature [@Steeg2009; @Nambu2011; @Hu2013; @Martin-Martinez2012; @Martin-Martinez2014; @Tian2014; @Salton2015] and even spacetime topology [@Martin-Martinez2016]. However, although two spacelike separated detectors may become correlated just out of their interaction with the vacuum, superluminal broadcasting of information between them is, of course, not possible.
In this context, it is relevant to ask what is the information carrying capacity of a quantum field. From a fundamental point of view, when we want to transmit information through a quantum field—whether in telecommunication or in an attempt to gather information about the early Universe—a necessary (but not sufficient) condition for communication is that the field commutator between the spacetime events of sending and receiving the message, does not vanish [@Cliche2010; @Benincasa2014; @Martin-Martinez2015].
The (expectation value of the) commutator of a quantum field is given by the classical radiation Green’s function [@Blasco2016] (difference between the retarded and the advanced Green’s functions). In this regard, the strong Huygens principle [@McLenaghan1974] states that the support of the radiation Green’s function of a massless field is restricted to lightlike separated events, implying that only lightlike signaling is possible. This is consistent with our everyday intuition: if we beam an empty chair with a laser and no one is there to receive the message, the information is gone, and it is not recoverable by a late receiver that sits on the chair the next day. However, for general spacetimes, the strong Huygens principle can be violated [@McLenaghan1974; @Blanchet1988; @Blanchet1992; @Faraoni1992; @Bombelli1994; @Gundlach1994; @Faraoni1999]. In these cases, massless field commutators can have support for timelike separated events. In fact, even for a simple massless scalar field, these violations are extremely common: the strong Huygens principle is violated in almost any curved spacetime, and in flat spacetimes of (1+1) and (2$n$+1)-dimensions [@McLenaghan1974; @Jonsson2015; @Blasco2016].
Note that violations of the strong Huygens principle are not enough to guarantee timelike separated observers the ability to communicate. It was shown in [@Jonsson2015] however, that if, additionally, the observers operate quantum antennas initialized to coherent superpositions of ground and excited eigenstates, a timelike signaling protocol can be established. Furthermore, this protocol allows for the possibility of broadcasting a message to an arbitrary number of timelike receivers, with the energy cost of transmitting the message being paid for by the receivers themselves. Because of this, this protocol received the name *quantum collect calling*.
This method of information broadcasting has been studied in great detail in [@Blasco2015; @Blasco2016] for a polynomially expanding, Fridmann-Robertson-Walker (FRW) cosmology generated by matter, in which reside two comoving observers: an early time signal emitter, Alice, and a later time receiver, Bob. It was shown that in this universe the timelike communication channel capacity is independent of the observers’ spatial separation, but decays as the instant that Bob attempts to retrieve Alice’s message (by coupling his antenna to the field) goes further into the future.
In the present work, we analyze the ability of timelike separated observers to communicate in an FRW universe dominated by a cosmological constant, which expands exponentially in comoving time, and we compare this to the matter-dominated case. We consider and minimal coupling of the massless scalar quantum field to the geometry. We supply Alice with a particle detector with which she can couple to the field, thereby leaving behind information which Bob can recover at a later time by coupling his own detector to the field.
We will show that timelike communication in an exponentially expanding universe displays unexpected and remarkable features that fundamentally impact the channel capacity. Namely, while we may expect that in an exponential expansion less information will reach Bob than in a polynomial one—due to the information being dispersed more in the faster expanding case—we show that, in fact, the opposite occurs. In the exponentially expanding universe, Bob’s ability to recover Alice’s message remains the same regardless of how long he waits before reading it out, in stark contrast with the decay present in the polynomially expanding cosmology. What is more, we find that Alice can broadcast more information to Bob the faster the exponential expansion of space is.
Not only does this imply that in principle more information is available to Bob through the timelike channel than through a light signal (which was proven in [@Blasco2015; @Blasco2016] to decay with the distance from the source), but it also means that Bob’s ability to access Alice’s information remains the same no matter how long Bob waits to switch on his antenna.
The outline of this article is as follows: Sec. \[sec:geometry\_field\] introduces the field-detector setup, along with the background spacetime geometry. In Sec. \[sec:communication\], the communication protocol is defined, and the ability of Alice to signal Bob is quantified through their channel capacity. Sec. \[sec:timelike\_communication\] particularizes the channel capacity in each cosmology to the case of timelike separated observers, and compares the two models within this causal regime. In this section we also look at the dependence of the channel capacity on whether we keep constant the proper or the comoving distance separating Alice and Bob. In Sec. \[sec:conclusions\], we present our conclusions. Natural units $\hbar=c=1$ are used throughout.
Background setup\[sec:geometry\_field\]
=======================================
We will consider a spatially flat Friedmann-Robertson-Walker cosmology given by the metric $$\begin{aligned}
\dif s^2 &= -\dif t^2 + a(t)^2(\dif r^2 + r^2 \dif \Omega^2) \notag \\
&= a(\eta)^2(-\dif \eta^2 + \dif r^2 + r^2 \dif \Omega^2), \label{eq:metric}
\end{aligned}$$ where we define the conformal time $\eta$ in terms of the comoving time $t$ (proper to observers comoving with the Hubble flow, also called *cosmological time*) as . The scale factor $a(t)$ quantifies the spatial expansion of the Universe, and its precise form depends on the stress-energy density which generates the spacetime. We will consider a universe generated by a perfect fluid with density $\rho$ and pressure $p=w\rho$. Specifically, we will focus on the two cases $w=0$ and $w=-1$, which correspond to (dust) matter and cosmological constant-dominated universes respectively. From the Friedmann equations, we obtain for the matter-dominated case that $$\label{eq:w=0}
a(t)=(9\kappa_1 t^2)^{1/3},\qquad \eta(t)=\left( \frac{3t}{\kappa_1} \right)^{1/3},$$ where $t,\eta\in[0,\infty)$. Doing the same for the cosmological constant-dominated case, we get $$\label{eq:w=-1}
a(t)=\kappa_2 e^{{\ensuremath{\sqrt{|\Lambda|}}}t},\qquad \eta(t)=-\frac{1}{{\ensuremath{\sqrt{|\Lambda|}}}\kappa_2}e^{-{\ensuremath{\sqrt{|\Lambda|}}}t},$$ where $t\in(-\infty,\infty)$ and $\eta\in(-\infty,0)$. $\kappa_1$ and $\kappa_2$ are constants of integration. We see that the matter-dominated universe is born out of a Big-Bang singularity and it experiences a polynomial spatial expansion. On the other hand, the cosmological constant-dominated universe does not originate with a Big-Bang (understood as a cancellation of $a(t)$ for a finite value of $t$) and expands exponentially in comoving time $t$.
Let us introduce a test massless scalar quantum field $\phi$. The equation of motion for the field is $$\label{eq:wave_eqn}
(\Box-\xi\mathcal{R})\phi=0,$$ where $\xi$ is the coupling to the Ricci scalar $$\mathcal{R}=\frac{6}{a^3}\frac{{ \mathrm{d} \ifx\relax2\relax\else
\rule{-0.02em}{1.5ex}^{2}\rule{0.08em}{0ex}\!
\fi
a\,
}}{\dif\eta^2},$$ and where the d’Alambertian operator in the FRW spacetime is given by $$\Box=-\frac{1}{a^4}\frac{\dif}{\dif\eta}\left(a^2\frac{\dif}{\dif\eta}\right)+
\frac{1}{a^2}{\ensuremath{\nabla^2}}.$$ Same as in [@Blasco2016], for computational purposes, we will choose the quantization scheme that corresponds to the adiabatic vacuum (see e.g., [@Junker2002; @birrell_quantum_1982]). This quantization scheme is particularly useful since the adiabatic vacuum corresponds to the field state for which the creation of particles due to the expansion of spacetime is finite and the smallest possible [@birrell_quantum_1982]. Furthermore, as shown in [@Cortez2011; @Fernandez-Mendez2012; @Cortez2012]. As rigorously discussed in these papers, for conformally flat compact spacetimes there exist natural criteria that select a unique equivalence class of vacua, which includes the adiabatic vacuum. Notice however, that we do not assume that the field is initially prepared in the adiabatic vacuum, and instead allow the field to be in any (non ill-defined) state. Indeed, for the same reasons as in [@Jonsson2015; @Blasco2015; @Blasco2016], the results in this paper will be independent of the initial state of the field, as long as it is well defined.
It was shown in [@Blasco2016] that if the field is minimally coupled to the curvature ($\xi=0$), two timelike separated observers gain the ability to communicate through the massless field without exchanging field quanta, taking advantage of the violations of the Strong Huygens Principle [@Jonsson2015]. In fact this is true not only for minimal coupling, but rather for any coupling to curvature that breaks conformal invariance. For concreteness, in this paper we will focus on the minimally coupled case.
Communication through detectors coupled to the field\[sec:communication\]
=========================================================================
Let us consider the following communication scenario: An observer in the early universe, Alice, wants to communicate with an observer living at a later cosmological epoch, Bob. We suppose that both Alice and Bob are fundamental (comoving) observers, meaning that they move with the Hubble flow and experience the Universe to be isotropic through its evolution. This assumption seems reasonable considering that all distant galaxies have small peculiar velocities with respect to local fundamental observers. Significantly, Earth bound observers are nearly fundamental as is evident by the observed dominant isotropy of the cosmic microwave background and of galactic densities on large scales. Hence in the above described picture we can think of ourselves as Bob, trying to detect a signal from an early Universe emitter Alice.
Let us assume that Alice operates a *radio emitter* that locally couples to the field, and Bob a *radio receiver* with which he tries to recover the information encoded in the field by Alice. We will model Alice and Bob’s antennas as two-level quantum systems (particle detectors) that couple locally to the quantum field through the Unruh-DeWitt interaction Hamiltonian [@DeWitt1980] $$\label{eq:HI}
H_{\textsc{i},v}=\lambda_\nu\chi_\nu(t)\mu_\nu(t)\int{ \mathrm{d} \ifx\relax3\relax\else
\rule{-0.02em}{1.5ex}^{3}\rule{0.08em}{0ex}\!
\fi
\bm{x}\,
} a(t)^3 F_\nu[\bm{x}-\bm{x}_\nu(t),t]\phi[\bm{x},\eta(t)],$$ (where $\text{d}^3\bm x=r^2\text{d}r\,\text{d}\Omega$ and $\bm x$ is a spatial 3-vector), which has been shown to capture the fundamental features of the light-matter interaction when there is no exchange of orbital angular momentum [@Pozas2016; @Alhambra2014]. Here $\nu\in\{\text{A,B}\}$ labels Alice and Bob’s detectors, and $\mu_\nu(t)$ is the monopole moment of detector $\nu$, $$\mu_\nu(t)=\sigma_\nu^+e^{i\Omega_\nu t}+\sigma_\nu^-e^{-i\Omega_\nu t}.$$ $\sigma_\nu^+=|e_\nu \rangle \langle g_\nu|$ and $\sigma_\nu^-=|g_\nu \rangle \langle e_\nu|$ are the $\text{SU}(2)$ raising and lowering operators, with $|g_\nu \rangle$ and $|e_\nu \rangle$ the ground and excited states, separated by an energy gap $\Omega_\nu$. The detector-field coupling (for detector $\nu$) is characterized by the coupling strength $\lambda_\nu$ and the switching function $\chi_\nu$, which for simplicity we consider to be the characteristic function $$\label{eq:switching}
\chi_\nu(t)=
\begin{cases}
1 & t\in[T_{i\nu},T_{f\nu}] \\
0 & t\not\in[T_{i\nu},T_{f\nu}]
\end{cases}.$$ $F_\nu[\bm{x}-\bm{x}_\nu(t),t]$ is a smearing function characterizing the geometry of detector $\nu$, centered along its trajectory $\bm{x}_\nu$. We consider comoving detectors, $\bm{x}_\nu=\text{const}$, and for now keep the detector smearing general.
Let each detector start out in the pure state , where ${| {\psi_{0,\nu}} \rangle}=\alpha_\nu{| {e_\nu} \rangle}+\beta_\nu{| {g_\nu} \rangle}$, and let the field start out in the arbitrary state $\rho_{0,\phi}$. Hence, the initial state of the system is $$\rho_0=\rho_{0,\textsc{a}} \otimes \rho_{0,\textsc{b}} \otimes \rho_{0,\phi}.$$ Allowing the system to evolve under the full interaction Hamiltonian $H_\textsc{i}(t)=H_{\textsc{i},\textsc{a}}(t)+H_{\textsc{i},\textsc{b}}(t)$ for a time $T$ results in the state $\rho_{_T}=U\rho_0 U^\dagger$, where U is the time evolution operator $$U=\mathcal{T}\exp\left[-{\mathrm{i}}\int_{-\infty}^\infty \!\!\dif t H_\textsc{i}(t)\right],$$ and $\mathcal{T}$ denotes time-ordering of the exponential. The final state of Bob’s detector is obtained by tracing out the degrees of freedom corresponding to the field and the state of Alice: $$\label{eq:rho_t_b}
\rho_{_{T,\text{B}}}={\text{Tr}}_{\phi,\textsc{a}}(\rho_{_T}).$$ The excitation probability of Bob’s detector at time $T$ is given by [@Martin-Martinez2015; @Blasco2015; @Jonsson2015; @Jonsson2014] $$\label{eq:Pe}
P_e={\langle {e_\textsc{b}} |}\rho_{_{T,\text{B}}}{| {e_\textsc{b}} \rangle}
=|\alpha_\textsc{b}|^2+R+S,$$ where $R$ is the local correction to the excitation probability of Bob (independent of $\lambda_\textsc{a}$), and $S$ is the signaling term (dependent on $\lambda_\textsc{a}$) that captures the influence of Alice’s detector on the excitation probability of Bob [@Martin-Martinez2015; @Jonsson2015]. We call $S$ the *signaling contribution* to Bob’s excitation probability. A power series expansion in the coupling strengths gives $$S=\lambda_\textsc{a}\lambda_\textsc{b}S_2+\mathcal{O}(\lambda_\nu^4),$$ where the lowest order term, $S_2$, takes the form[^1] [@Blasco2015; @Blasco2016; @Jonsson2015] $$\begin{aligned}
&S_2=4\!\int\!\dif v\!\int\!\dif v' \chi_\textsc{a}(t)\chi_\textsc{b}(t')
F_\textsc{a}(\bm{x}-\bm{x}_\textsc{a},t)
F_\textsc{b}(\bm{x'}-\bm{x}_\textsc{b},t') \notag \\
&\!\times \! {\operatorname{Re}{(\alpha_\textsc{a}^*\beta_\textsc{a}e^{{\mathrm{i}}\Omega_\textsc{a}t})}}
{\operatorname{Re}{\Big(\alpha_\textsc{b}^*\beta_\textsc{b}e^{{\mathrm{i}}\Omega_\textsc{b}t'}\Big\langle [\phi(\bm{x}_\textsc{a},t),\phi(\bm{x}_\textsc{b},t')]\Big\rangle_{\!\!\rho_{_{0,\phi}}}\!\!\Big)}}, \label{eq:S2}
\end{aligned}$$ with $\dif v=a(t)^3{ \mathrm{d} \ifx\relax3\relax\else
\rule{-0.02em}{1.5ex}^{3}\rule{0.08em}{0ex}\!
\fi
\bm{x}\,
}
\rule{-0.2em}{0ex}\dif t$ being the FRW volume element. Notice that, since the field commutator is a c-number (multiple of the identity), its expectation value is independent of $\rho_{0,\phi}$.
Let us now, for simplicity, particularize the discussion to the limit of point-like detectors, characterized by the smearing function $$F_\nu(\bm{x},t)=\delta(\bm{x}).$$ Although the use of detectors in this limit along with sudden switching functions is known to cause UV divergences in the excitation probability [@Louko2008], $S_2$ was proven to be UV-safe [@Blasco2016; @Martin-Martinez2015]. In this limit, becomes $$\begin{aligned}
S_2&=4\int\dif t\int\dif t' \chi_\textsc{a}(t)\chi_\textsc{b}(t')
{\operatorname{Re}{(\alpha_\textsc{a}^*\beta_\textsc{a}e^{{\mathrm{i}}\Omega_\textsc{a}t})}} \notag \\
&\times
{\operatorname{Re}{\Big(\alpha_\textsc{b}^*\beta_\textsc{b}e^{{\mathrm{i}}\Omega_\textsc{b}t'}\Big\langle [\phi(\bm{x}_\textsc{a},t),\phi(\bm{x}_\textsc{b},t')]\Big\rangle_{\!\!\rho_{_{0,\phi}}}\Big)}}, \label{eq:S2_2}
\end{aligned}$$
For matter and cosmological constant-dominated universes in the case of minimal coupling of the field to the geometry, the field commutator between two events, $x=(\bm{x}_\textsc{a},t)$ and $x'=(\bm{x}_\textsc{b},t')$, is (see details in Appendix \[appendix:commutators\]) $$\begin{gathered}
\label{eq:commutator}
\langle[\phi(x),\phi(x')]\rangle_{\rho_\phi}=\frac{{\mathrm{i}}}{4\pi}
\Bigg[
\frac{\delta(\Delta\eta+R)-\delta(\Delta\eta-R)}{a(t)a(t')R}\\
+\frac{\theta(-\Delta\eta-R)-\theta(\Delta\eta-R)}{a(t)a(t')|\eta(t)\eta(t')|}
\Bigg],
\end{gathered}$$ where $\Delta\eta=\eta-\eta'=\eta(t)-\eta(t')$ and $R=\|\bm{x}_\textsc{a}-\bm{x}_\textsc{b}\|$. Immediately, due to the presence of the Heaviside $\theta$-function, we see that the support of the field commutator is not limited solely to boundaries of the light cone $\Delta\eta=\pm R$, and so we expect timelike signaling to be possible.
Let us set the initial states of the detectors to be $$\begin{aligned}
&{| {\psi_{0,\textsc{a}}} \rangle}=\frac{1}{\sqrt{2}}({| {e_\textsc{a}} \rangle}-{| {g_\textsc{a}} \rangle}),\notag\\
&{| {\psi_{0,\textsc{b}}} \rangle}=\frac{1}{\sqrt{2}}({| {e_\textsc{b}} \rangle}+{\mathrm{i}}{| {g_\textsc{b}} \rangle}).\label{eq:initstates}
\end{aligned}$$ We make this choice since it maximizes the signaling estimator in the case of zero gap detectors. Nevertheless, this choice is arbitrary, and any other initialization of detectors would lead to the same qualitative results.
Using the initial detector states and the field commutator , the expression for $S_2$ becomes $$\label{eq:S2_final}
S_2=\frac{1}{4\pi}(I_\delta+I_\theta),$$ where $$\begin{aligned}
I_\delta&=\frac{1}{R}\int_{\eta_{i\textsc{b}}}^{\eta_{f\textsc{b}}}\!\dif\eta\,\chi_A(\eta-R)
\cos[\Omega_\textsc{b}t(\eta)] \cos[\Omega_\textsc{a}t(\eta-R)], \label{eq:Idelta}\\
I_\theta&=\int_{\eta_{i\textsc{b}}}^{\eta_{f\textsc{b}}}\frac{\dif\eta_2}{|\eta_2|}
\theta[\min(\eta_{f\textsc{a}},\eta_2-R)-\eta_{i\textsc{A}}]\cos[\Omega_\textsc{b}t(\eta_2)] \notag\\
&\times\int_{\eta_{i\textsc{A}}}^{\min(\eta_{f\textsc{a}},\eta_2-R)}\frac{\dif\eta_1}{|\eta_1|}
\cos[\Omega_\textsc{a}t(\eta_1)]. \label{eq:Itheta}
\end{aligned}$$ Here $\eta_{i\nu}=\eta(T_{i\nu})$ and $\eta_{f\nu}=\eta(T_{f\nu})$.
Let us analyze the simple communication protocol laid out in [@Blasco2015; @Jonsson2015]: Alice encodes the bit “1" by coupling her detector to the field at time $T_{i\textsc{a}}$ and decoupling at time $T_{f\textsc{a}}=T_{i\textsc{a}}+\Delta$, and the bit “0" by remaining uncoupled. To later decode the message, Bob couples to the field at time $T_{i\textsc{b}}$, decouples at time $T_{f\textsc{b}}=T_{i\textsc{b}}+\Delta$, and measures his energy eigenstate. If he is excited, he interprets it as “1", and “0" otherwise. Notice that for simplicity we are keeping Alice and Bob’s detectors switched on for an equal proper time interval $\Delta$, and recall that we are considering sudden switching of detectors, as given in . The number of bits per use of this binary communication channel that Alice can transmit to Bob is given by the Shannon capacity [@Silverman1955], which was shown in [@Jonsson2015] to be $$\label{eq:cc_general}
C=\frac{\lambda_\textsc{a}^2\lambda_\textsc{b}^2}{8\ln 2}
\left(\frac{S_2}{|\alpha_\textsc{b}||\beta_\textsc{b}|}\right)^2+\mathcal{O}(\lambda_\nu^6).$$ For a matter or cosmological constant-dominated universe, with minimal coupling of the field to the curvature, and with initial detector states , $S_2$ is given by . Hence the channel capacity becomes $$\label{eq:cc}
C=\frac{\lambda_\textsc{a}^2\lambda_\textsc{b}^2}{32\pi^2\ln2}
(I_\delta+I_\theta)^2+\mathcal{O}(\lambda_\nu^6).$$ We will study the form of the channel capacity when Alice and Bob are strictly timelike separated. The matter-dominated case was thoroughly analyzed in [@Blasco2015; @Blasco2016]. However, the cosmological constant-dominated scenario remains unexplored. Despite the mathematical similarities between them, we will show that there are fundamental and unintuitive physical differences in the abilities of timelike separated observers to communicate within the two cosmologies. Namely, timelike signals can carry considerably more information about the early universe when the spatial expansion is exponential as opposed to polynomial.
Timelike communication in polynomially and exponentially expanding cosmologies\[sec:timelike\_communication\]
=============================================================================================================
![\[fig:causal\]The five possible causal relationships between the switching periods of Alice and Bob’s detectors. In conformal time $\eta$ and comoving distance $R$, the boundaries of the light cones (diagonal lines) have slopes of $c=1$.](causal.pdf){width="50.00000%"}
The form of Alice and Bob’s communication channel capacity depends on the causal relationship between the supports of their switching functions. Fig. \[fig:causal\] shows the five possible causal relationships. When Bob is strictly spacelike separated from Alice, as in B1, both $I_\delta$ and $I_\theta$ in the channel capacity vanish, hence superluminal communication between the observers is indeed impossible. In the cases B2, B3 and B4, there is partial lightlike contact with Alice, so $I_\delta$ does not vanish entirely. As expected, lightlike communication is possible through a massless scalar field. Note that in the case of B4, the $I_\theta$ term also contributes to the channel capacity, meaning that communication is due to both lightlike and timelike signals. Timelike signaling is most evident when we consider detector B5, which is strictly within Alice’s future light cone. Here, while $I_\delta$ vanishes, $I_\theta$ does not: in matter and cosmological constant-dominated universes with minimal coupling of the field to the curvature, slower than light communication is possible.
\[sec:signal\_timing\]Signal timing
-----------------------------------
Before we study the channel capacity of timelike separated observers, let us discuss when Alice and Bob are timelike separated. We hold constant the switching times of Alice’s detector, and ask the following question: where and when can Bob switch his detector on such that during his interaction with the field he is strictly within Alice’s future light cone? That is, we fix $T_{i\textsc{a}}$ and $T_{f\textsc{a}}$ and look for the values of $T_{i\textsc{b}}$ and $R$ for which the two detectors, while switched on, are strictly timelike separated. From Fig. \[fig:causal\] it is evident that this occurs when $$\eta_{i\textsc{b}}> \eta_{f\textsc{a}}+R.$$ If we keep the comoving separation between the detectors ($R$) constant, then $$\label{eq:tib0}
T_\text{min}^R=t(\eta(T_{i\textsc{a}}+\Delta)+R)$$ is the smallest value of $T_{i\textsc{b}}$ at which there is strict timelike contact between the detectors. If instead we keep constant the time $T_{i\textsc{b}}$ at which Bob switches on his detector, then $$\label{eq:R0}
R_\text{max}=\eta(T_{i\textsc{b}})-\eta(T_{i\textsc{a}}+\Delta)$$ is the largest comoving separation between Alice and Bob for which the two are fully timelike separated. One can trivially particularize $T_\text{min}^R$ and $R_\text{max}$ for the cosmologies generated by matter and a cosmological constant by using the appropriate forms of $\eta(t)$ from and —and their inverses $t(\eta)$—in equations and .
The comoving distance, $R(t)$, is not usually the measure considered when discussing the spatial separation between us and distant cosmic objects. In astronomical terms, such separations are typically given in terms of the proper distance (i.e. the physical length of a measuring tape extended between us and the distant object as measured by us at time $t$). As a function of the comoving distance, the proper distance is given by $$\label{eq:proper}
P(t)=a(t)R(t).$$ While the comoving distance between observers moving with the Hubble flow is independent of time, the proper distance between these observers increases as the universe expands.
Alternatively to what was done in [@Blasco2015; @Blasco2016], instead of keeping the comoving distance between Alice and Bob constant, we can keep constant the proper distance. This requires at least one of Alice or Bob to be non-comoving. However, it is convenient to assume that both observers are comoving during their interaction time with the field, in order to obtain an analytic expression for the field commutator . For this reason we approximate the channel capacity at a constant proper separation, $P$, by the capacity at a constant *comoving* separation, $$\label{eq:R0proper}
R(T_{i\textsc{b}})=\frac{P}{a(T_{i\textsc{b}})}.$$ This is a valid approximation as long as Alice and Bob’s detector-field interaction times are much shorter than their temporal separation ($\Delta\ll T_{i\textsc{b}}-T_{f\textsc{a}}$). That is, we consider the expansion of the universe during the interaction time of the detectors with the field to be negligible, but we consider the full dynamics of the background spacetime between the emission and reception events. This is reasonable to expect if Bob is us and Alice is an early universe observer.
The earliest time $T_\text{min}^P$ that Bob can switch on his detector while remaining strictly in Alice’s timelike future and maintaining a constant proper separation $P$, is found by solving $$\label{eq:tib0proper}
\eta(T_\text{min}^P)=\eta(T_{i\textsc{a}}+\Delta)+\frac{P}{a(T_\text{min}^P)}.$$ In a matter-dominated universe , the solution is given by the single real root of the cubic equation $$\left(T_\text{min}^P-\frac{R_0}{3}\right)^3=T_{f\textsc{a}}(T_\text{min}^P)^2,$$ while in a cosmological constant-dominated universe , the solution to is $$T_\text{min}^R=\frac{1}{{\ensuremath{\sqrt{|\Lambda|}}}}[\ln(1+P{\ensuremath{\sqrt{|\Lambda|}}})+T_{f\textsc{a}}].$$
Finally, we can keep $T_{i\textsc{b}}$ constant and vary the proper distance between Alice and Bob. The largest value of $P$ for which the observers are strictly timelike separated is given by multiplying the comoving distance $R_\text{max}$ by the appropriate scale factor or .
\[sec:cc\]Channel capacity
--------------------------
The capacity of Alice and Bob’s communication channel is given in expression . In the region of strict timelike contact of the detectors, the $I_\delta$ integral vanishes identically, while $I_\theta$ becomes $$\label{eq:Itheta2}
I_{\theta}=
\int_{\eta_{i\textsc{A}}}^{\eta_{f\textsc{a}}}\mspace{-15mu}\dif\eta_1\frac{\cos[\Omega_\textsc{a}t(\eta_1)]}{|\eta_1|}
\int_{\eta_{i\textsc{b}}}^{\eta_{f\textsc{b}}}\mspace{-15mu}\dif\eta_2\frac{\cos[\Omega_\textsc{b}t(\eta_2)]}{|\eta_2|}.$$ Changing the integration variable to comoving time one obtains $$\label{eq:Itheta3}
I_{\theta}=
\int_{T_{i\textsc{a}}}^{T_{i\textsc{a}}+\Delta}\mspace{-20mu}\dif t_1 \frac{\cos(\Omega_\textsc{a}t_1)}{a(t_1)|\eta(t_1)|}
\int_{T_{i\textsc{b}}}^{T_{i\textsc{b}}+\Delta}\mspace{-20mu}\dif t_2 \frac{\cos(\Omega_\textsc{b}t_2)}{a(t_2)|\eta(t_2)|}.$$ We will particularize this expression to the two cosmologies that we are considering.
### \[sec:cc\_matter\]Matter-dominated cosmology
In the matter-dominated universe ($w=0$), using we obtain $$\label{eq:a_eta_w=0}
a(t)|\eta(t)|=3t,$$ which is the proper *particle horizon* of the observer at time $t$, i.e. the maximal proper distance that light could have traveled to the observer in the age of the Universe. Notice that in this case also corresponds to twice the Hubble radius at time $t$. For the case of non-zero gap detectors, $\Omega_\nu>0$, equation becomes $$\begin{aligned}
\label{eq:Itheta_w=0}
I_{\theta}^{w=0}=\frac{1}{9}
&({\text{Ci}}[\Omega_\textsc{a}(T_{i\textsc{a}}+\Delta)]-{\text{Ci}}[\Omega_\textsc{a}T_{i\textsc{a}}]) \notag \\
\times&({\text{Ci}}[\Omega_\textsc{b}(T_{i\textsc{b}}+\Delta)]-{\text{Ci}}[\Omega_\textsc{b}T_{i\textsc{b}}]),
\end{aligned}$$ where ${\text{Ci}}$ is the cosine integral function, $${\text{Ci}}(z)=\int_z^\infty \dif t \frac{\cos t}{t}.$$ If we assume that $\Delta\ll T_{i\textsc{a}}<T_{i\textsc{b}}$, simplifies to $$\label{eq:Itheta_w=0_2}
I_{\theta}^{w=0}\simeq \frac{\Delta^2}{9}\frac{\cos(\Omega_\textsc{a}T_{i\textsc{a}})}{T_{i\textsc{a}}}
\frac{\cos(\Omega_\textsc{b}T_{i\textsc{b}})}{T_{i\textsc{b}}}.$$ This assumption is reasonable since, as mentioned above, we expect the time scale of the detectors being switched on to be much smaller than the cosmological time scale on which the universe evolves.
Note that taking the limit $\Omega_\nu\to 0$ in one obtains $$\label{eq:Itheta_gapless}
\lim_{\Omega_\nu\to 0}
\left(I_{\theta}^{w=0}\right)=
\frac{1}{9}
\ln\left(\frac{T_{i\textsc{a}}+\Delta}{T_{i\textsc{a}}}\right)
\ln\left(\frac{T_{i\textsc{b}}+\Delta}{T_{i\textsc{b}}}\right),$$ which is the result derived in [@Blasco2016] for gapless detectors.
Therefore, for strictly timelike separated observers in the matter-dominated universe, the channel capacity becomes $$\begin{aligned}
\label{eq:cc_w=0}
C^{w=0}_{\Omega_\nu>0}&=\frac{\lambda_\textsc{a}^2\lambda_\textsc{b}^2}{2592\pi^2\ln2}
\left({\text{Ci}}[\Omega_\textsc{a}(T_{i\textsc{a}}+\Delta)]-{\text{Ci}}[\Omega_\textsc{a}T_{i\textsc{a}}]\right)^2 \notag \\
&\mspace{96mu}
\times \left({\text{Ci}}[\Omega_\textsc{b}(T_{i\textsc{b}}+\Delta)]-{\text{Ci}}[\Omega_\textsc{b}T_{i\textsc{b}}]\right)^2 \notag \\
&\simeq\frac{\lambda_\textsc{a}^2\lambda_\textsc{b}^2\Delta^4}{2592\pi^2\ln2}
\left(\frac{\cos(\Omega_\textsc{a}T_{i\textsc{a}})}{T_{i\textsc{a}}}
\frac{\cos(\Omega_\textsc{b}T_{i\textsc{b}})}{T_{i\textsc{b}}}\right)^2, \\
C^{w=0}_{\Omega_\nu=0}&=\frac{\lambda_\textsc{a}^2\lambda_\textsc{b}^2}{2592\pi^2\ln2}
\left(
\ln\left(\frac{T_{i\textsc{a}}+\Delta}{T_{i\textsc{a}}}\right)
\ln\left(\frac{T_{i\textsc{b}}+\Delta}{T_{i\textsc{b}}}\right)
\right)^2\!\!\!, \notag
\end{aligned}$$ where we used equations , and for $I_\theta^{w=0}$.
### \[sec:cc\_lambda\]-dominated cosmology
In the cosmological constant-dominated universe (), we see from that the denominators of the integrands of become $$\label{eq:a_eta_w=-1}
a(t)|\eta(t)|=\frac{1}{{\ensuremath{\sqrt{|\Lambda|}}}},$$ which in this case coincides with both the Hubble radius and the proper *event horizon* of the observer at time $t$, i.e. the proper distance that light emitted at time $t$ would travel in the lifetime of the Universe. Critically, as opposed to the particle horizon in the matter-dominated universe , equation does not depend on time. In the $\Lambda$-dominated cosmology, becomes $$\begin{aligned}
I_{\theta}^{w=-1}&=\frac{4|\Lambda|}{\Omega_\textsc{a}\Omega_\textsc{b}}
\sin\left(\frac{\Omega_\textsc{a}\Delta}{2}\right)
\sin\left(\frac{\Omega_\textsc{b}\Delta}{2}\right) \notag \\
&\times\cos\left[\Omega_\textsc{a}\left(T_{i\textsc{a}}+\frac{\Delta}{2}\right)\right]
\cos\left[\Omega_\textsc{b}\left(T_{i\textsc{b}}+\frac{\Delta}{2}\right)\right],
\label{eq:Itheta_w=-1}
\end{aligned}$$ and the channel capacity acquires the form, $$\begin{aligned}
C^{w=-1}&=
\frac{\lambda_\textsc{a}^2\lambda_\textsc{b}^2\Lambda^2}{162\pi^2\ln2\,\Omega_\textsc{a}^2\Omega_\textsc{b}^2}
\sin^2\left(\frac{\Omega_\textsc{a}\Delta}{2}\right)
\sin^2\left(\frac{\Omega_\textsc{b}\Delta}{2}\right) \notag \\
&\times \cos^2\left[\Omega_\textsc{a}\left(T_{i\textsc{a}}+\frac{\Delta}{2}\right)\right]
\cos^2\left[\Omega_\textsc{b}\left(T_{i\textsc{b}}+\frac{\Delta}{2}\right)\right]
\label{eq:cc_w=-1}.
\end{aligned}$$ We are now ready to compare the abilities of timelike separated observers to communicate within the two cosmologies.
\[sec:results\]Results
----------------------
Let us now compare the communication channel capacities between an early-universe signal emitter, Alice, and a late-time receiver, Bob, in cosmologies generated by matter and a cosmological constant.
We will focus on the channel capacities when Alice and Bob’s detectors are strictly timelike separated. We recall that, since the real quanta of the massless scalar field travel at the speed of light, one may intuitively have expected that the channel capacities in this causal regime are zero. However, as explained above, this is the relevant case of *quantum collect calling*, where slower than light communication through the massless scalar field is possible if Alice and Bob’s detectors are prepared in coherent superpositions of their excited and ground states in scenarios where the strong Huygens principle is violated [@Jonsson2015; @Blasco2016], as is the case for minimally coupled fields in FRW backgrounds [@Blasco2015; @Blasco2016].
The initial states of the qubit detectors with which Alice and Bob couple to the field are defined in . To facilitate a comparison with the results in [@Blasco2016; @Blasco2015], the initialization is chosen to maximize the channel capacity in the case of zero-gap detectors. We will not particularize to the zero-gap case, but we will for simplicity consider the energy gaps of the two detectors to be equal: $\Omega_\textsc{a}=\Omega_\textsc{b}=\Omega$. Recall that the detectors are switched on and off suddenly, according to .
To elucidate the effects of cosmological expansion on the ability of observers to communicate, we use our freedom to choose a reference scale for the constant factors to set the two spacetimes and their rates of expansion to be equal at a given initial time, which in our case will be the time at which Alice’s detector is switched on, $T_{i\textsc{a}}$. To that effect, we set: $$\begin{aligned}
&a_{w=0}(T_{i\textsc{a}})=a_{w=-1}(T_{i\textsc{a}})=1, \notag \\
&\dot{a}_{w=0}(T_{i\textsc{a}})=\dot{a}_{w=-1}(T_{i\textsc{a}})=1. \label{eq:init_cond_a}
\end{aligned}$$ This is done by setting $\kappa_1=1/4$, $\kappa_2=\exp(-2/3)$, ${\ensuremath{\sqrt{|\Lambda|}}}=1$ and $T_{i\textsc{a}}=2/3$. The effects of this choice in both dynamics can be seen in Fig. \[fig:bounds\].
![\[fig:bounds\]Scale factors governing the expansion of the matter ($w=0$) and cosmological constant ($w=-1$) dominated universes, plotted as functions of time. The four rightmost vertical lines show the earliest times $T_{i\textsc{b}}$ for which Alice and Bob are strictly timelike separated, while keeping constant their spatial separation. When comoving separation is held constant, $R=1/2$, and when proper separation is approximated as constant, $P(T_{i\textsc{b}})=1/2$. Here $T_{i\textsc{a}}=2/3$, $\Delta=1/100$, $\kappa_1=1/4$, $\kappa_2=\exp(-2/3)$ and ${\ensuremath{\sqrt{|\Lambda|}}}=1$.](bounds.pdf){width="45.00000%"}
The comoving time for which each detector is switched on is set to $\Delta=1/100$. For the values of $T_{i\textsc{b}}$ that we will work with, this ensures that , which we require in order to approximate a constant proper separation between the detectors, as discussed in Sec. \[sec:signal\_timing\].
We see from equations and that the timelike channel capacities in the two cosmologies are both independent of the distance (proper or comoving) separating Alice and Bob. It was pointed out in [@Blasco2015; @Blasco2016] (for the matter-dominated case) that this fact allows timelike channels to potentially convey more information from spatially distant events than light signals due to the fact that the timelike channel capacity does not decay with the distance to the source. We see that this is also true in the cosmological constant-dominated universe.
Remarkably, we find critical differences in the ability of Alice to communicate with Bob through timelike channels in the two cosmologies. Namely, the timelike *afterglow* of Alice’s interaction with the field remains constant (up to oscillations) before reaching Bob in the $\Lambda$-dominated universe, in stark contrast with the time decay present in the matter-dominated case.
This can be seen in Fig. \[fig:cc\], where we plot the channel capacities as functions of the instant that Bob’s detector is switched on, $T_{i\textsc{b}}$. We consider two different situations: 1) constant comoving separation between Alice and Bob, $R=1/2$, as in [@Blasco2015; @Blasco2016], and 2) constant proper separation, $P(T_{i\textsc{b}})=1/2$. We see that there are no relevant qualitative differences between the situations 1) and 2). The only difference in communication that the choice of distance measure effects, is the values of $T_{i\textsc{b}}$ for which the detectors are strictly timelike separated. Namely, keeping constant the proper separation results in strictly timelike separated detectors at lower $T_{i\textsc{b}}$ than when maintaining the same comoving distance constant. This is due to our choice of reference scale when normalizing the scale factors : since $a(T_{i\textsc{b}})>1$, at time $T_{i\textsc{b}}$ a given comoving separation is physically larger (and hence takes light longer to traverse) than the same proper separation. The distance measure that we choose to keep constant therefore affects the relative spacetime positioning of Alice and Bob, displacing the positions of the timelike connected regions in Fig. \[fig:cc\].
Notice that the magnitudes of the channel capacities in Fig. \[fig:cc\]a are much smaller than those reported in [@Blasco2016]. This is mainly due to us considering a detector-field interaction time, $\Delta$, that is several orders of magnitude less than that in [@Blasco2016] (note from that $C^{w=0}\propto\Delta^4$). Indeed, as expected, the longer Alice interacts with the field, the more information she encodes for Bob to later recover.
If we look at Fig. \[fig:cc\]a, we see that in the matter-dominated universe the channel capacity has a polynomial decay in time: Bob’s ability to retrieve Alice’s signal is suppressed the longer he waits to do so.
Remarkably, Fig. \[fig:cc\]b shows that the channel capacity in the exponentially expanding cosmology does not decay as the time that Bob waits to read out the signal increases: even if Bob waits the age of the universe to recover the signal, the channel capacity between him and Alice will remain the same (up to oscillations).
The behaviour shown in Fig. \[fig:cc\], stems from the time dependence of the equations for the channel capacities in the matter and $\Lambda$-dominated cosmologies, and , respectively. If, for illustration, we look at the approximated form of (which applies to the results in the figures since ), we see that $C^{w=0}\propto T_{i\textsc{b}}^{-2}$. The capacity in the $\Lambda$-dominated case exhibits no such decay with $T_{i\textsc{b}}$.
This result seems contrary to the physical intuition that, since an exponential expansion is faster than a polynomial one, the information encoded in the field by Alice in the former case should get dispersed more, resulting in a faster decaying channel capacity, as is the case with lightlike signals. What is more, not only does the channel capacity in the $\Lambda$-dominated cosmology not decay, but it actually grows as $\Lambda^2$, meaning that more information can in principle be broadcast from Alice to Bob the faster the exponential expansion of the Universe is.
Along with the decay (or lack thereof) discussed above, both channel capacities also exhibit oscillations with $T_{i\textsc{a}}$ and $T_{i\textsc{b}}$ at frequencies equal to the energy gap of the detectors, $\Omega$.
Conclusions\[sec:conclusions\]
==============================
By using the protocols of *quantum collect calling* [@Blasco2016; @Jonsson2015] it is possible to detect signals broadcast by early universe observers in our timelike past (when there is no light contact), greatly increasing both the volume of observable spacetime and the amount of recoverable information from that available through classical observational methods. This ability of timelike separated observers to communicate is fundamentally dependent on 1) the coupling of the field to the underlying geometry, 2) the dimensionality of spacetime, and 3) the geometry of spacetime. We focused here on the case of minimal coupling in (3+1)-dimensions, which was shown in [@Blasco2016; @Blasco2015] to be a viable setup for timelike signaling in the case of a polynomially-expanding, matter-dominated cosmology. In this paper, we have analyzed the exponentially-expanding, cosmological constant-dominated universe, and we found unexpected fundamental differences between the two cases.
We quantified the ability of timelike separated observers, Alice and Bob, to exchange information in the two cosmologies. To do so, we computed a lower bound to the Shannon capacity of the channel established when they communicate using antennas coupled to the quantum field. We showed that, as in the matter-dominated cosmology, the channel capacity in the $\Lambda$-dominated case is independent of the spatial and temporal separations between Alice and Bob.
Most interestingly, we also found that in the exponentially expanding universe, there is no decay of the channel capacity with Alice and Bob’s individual coupling times. This means that Bob can wait as long as he wants and the amount of information that he can recover from Alice will not change. What is more, we find that the channel capacity is proportional to $\Lambda^2$. This implies that the faster the expansion of the Universe is, the greater the ability of Bob to recover the information sent by Alice through timelike communication.
This is contrary to the polynomial decay present in the matter-dominated universe and studied in previous literature [@Blasco2015; @Blasco2016], and it challenges the—perhaps intuitive—physical expectation that a faster spatial expansion results in less information reaching an observer, since it would be dispersed as the Universe expands.
The unintuitive lack of decay in a $\Lambda$-dominated cosmology is made even more interesting when we note that our own Universe seems to have been exponentially expanding at very early times in its history, and appears to be currently dominated by a cosmological constant as well. This opens up fascinating possibilities of applying the theory presented here, at least in principle, to observe our distant timelike past, or to send signals to observers in our timelike future.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors are very thankful to Ana Blasco, Luis J. Garay and Mercedes Martín-Benito for helpful discussions and invaluable feedback. The authors gratefully acknowledge the financial support provided by the NSERC Discovery and USRA Programs.
Field commutator for minimal coupling of the field to the geometry {#appendix:commutators}
==================================================================
In this appendix, we review the calculations originally outlined in [@Blasco2015; @Blasco2016; @Poisson2011]. We start with the expression for the expectation value of the field commutator between two events, $x=(\bm{x}_\textsc{a},t)$ and $x'=(\bm{x}_\textsc{b},t')$, in terms of the advanced and retarded Green functions, $G_-$ and $G_+$, respectively: $$\label{eq:commutator_green}
\langle[\phi(x),\phi(x')]\rangle={\mathrm{i}}\frac{G_-(x,x')-G_+(x,x')}{4\pi}.$$ The $G_\pm$ are solutions to the wave equation with a point-like source $$\label{eq:wave_eqn2}
(\Box-\xi\mathcal{R})G_\pm(x,x')=-\frac{4\pi}{a(\eta)^4}\delta(\eta-\eta')\delta^3(\bm{x}-\bm{x}').$$ Rescaling by $a(\eta)a(\eta')$ and introducing the Fourier transform $\hat{g}$, we can rewrite $G_\pm$ as $$G_\pm(x,x')=\frac{\pm\theta(\pm\eta\mp\eta')}{(2\pi)^3 a(\eta)a(\eta')}\int{ \mathrm{d} \ifx\relax3\relax\else
\rule{-0.02em}{1.5ex}^{3}\rule{0.08em}{0ex}\!
\fi
\bm{k}\,
}e^{{\mathrm{i}}\bm{k}\cdot(\bm{x}-\bm{x}')}\hat{g}(\eta,\eta',k),$$ which upon substitution into gives the auxiliary differential equation $$\label{eq:wave_eqn3}
\left(\frac{{ \mathrm{d} \ifx\relax2\relax\else
\rule{-0.02em}{1.5ex}^{2}\rule{0.08em}{0ex}\!
\fi
}\,
}{\dif\eta^2}+k^2-(1-6\xi)\frac{\alpha^2-1/4}{\eta^2}\right)\hat{g}(\eta,\eta',k)=0,$$ with boundary conditions $$\hat{g}(\eta=\eta',k)=0,\quad \frac{\dif\hat{g}}{\dif\eta}(\eta=\eta',k)=4\pi.$$ Here, we have defined $\alpha=\left|(3-3w)/(6w+2)\right|$, where we recall that $w=p/\rho$ is the pressure to density ratio of the perfect fluid generating our spacetime.
In the case of minimal coupling, $\xi=0$. Then, the solution $\hat{g}_\alpha(\eta,\eta',k)$ (where we have explicitly denoted the $\alpha$ dependence) to is given by equation (55) in [@Blasco2016]. The commutator then becomes $$\begin{aligned}
\label{eq:commutator_int}
\langle[\phi(x),\phi(x')]\rangle&={\mathrm{i}}\frac{\theta(-\Delta(\eta))-\theta(\Delta(\eta))}{\pi^2 a(t)a(t')R} \notag \\
&\times \int_0^\infty\dif k\sin(kR)\hat{g}_\alpha(\eta(t),\eta(t'),k),
\end{aligned}$$ where $\Delta(\eta)=\eta(t)-\eta(t')$ and $R=\|\bm{x}_\textsc{a}-\bm{x}_\textsc{b}\|$. In both matter ($w=0$) and $\Lambda$ ($w=-1$) dominated cosmologies, $\alpha=3/2$, and the integral in can be solved analytically, yielding expression .
[^1]: For a step-by-step derivation of , see [@Martin-Martinez2015], Eq. (5) to (25).
|
---
author:
- 'Y.Q. Chen'
- 'P.E. Nissen'
- 'G. Zhao'
date: 'Received ..... / Accepted ......'
title: 'The \[Zn/Fe\] – \[Fe/H\] trend for disk and halo stars [^1]'
---
Introduction {#intro}
============
The determination of zinc abundances in Galactic stars is of high interest in astrophysics for at least two reasons. Firstly, the nucleosynthesis of zinc is not well understood. According to Woosley & Weaver ([@Woosley95]) Zn is produced by supernovae (SNe) of Type II from two processes: i) neutron capture on iron group nuclei during He and C-burning (the weak $s$-process; Langer et al. [@Langer89]), and ii) alpha-rich freeze-out following nuclear statistical equilibrium in layers heated to more than 5 $\times 10^9$K. However, as shown by Timmes et al. ([@Timmes95]), and Goswami & Prantzos ([@Goswami00]), the corresponding yields underpredicts Zn/Fe by about a factor of two compared to the observed ratio in Galactic halo stars. Furthermore, the yield corresponding to the first process is metallicity dependent leading to a predicted rise of as a function of \[Fe/H\] for disk stars in disagreement with the rather flat trend of vs. observed by Sneden et al. ([@Sneden91]). In order to get a better agreement with the observed trend, Matteucci et al. ([@Matteucci93]) argued that Type Ia SNe give a very significant contribution to the production of Zn, but according to Iwamoto et al. ([@Iwamoto99]) standard models of Type Ia SNe produce little Zn. As an alternative source of Zn, Hoffman et al. ([@Hoffman96]) suggested that a significant amount of zinc (together with light $p$-process nuclei) could be produced in the neutrino-powered wind of 10 to 20 solar mass SNe, essentially the same site as proposed for the $r$-process (Woosley et al. [@Woosley94]) but at earlier times after the explosion. Furthermore, Umeda & Nomoto ([@Umeda02]) have discussed how the produced Zn/Fe ratio in massive metal-poor SNe depends on the mass cut, neutron excess and explosion energy, in an attempt to explain the high values observed for the most metal-poor stars (Primas et al. [@Primas00], Cayrel et al. [@Cayrel04]).
Secondly, zinc is a key element in studies of abundances in damped Ly$\alpha$ (DLA) systems, because it is the only element in the iron-peak group, which is undepleted onto dust in the interstellar medium. With reference to the data of Sneden et al. ([@Sneden91]), Zn is often taken as a proxy for Fe in DLA studies to derive dust depletion factors (e.g. Vladilo [@Vladilo02]) and to date the the star formation process at high $z$ from \[$\alpha$/Fe\], i.e. the logarithmic ratio between the abundance of alpha-elements (O, Ne, Mg, Si, S, Ca, Ti) and the abundance of iron (e.g. Pettini et al. [@Pettini99], Centurión et al. [@Centurion00]). Given the uncertainty about the nucleosynthetic origin of zinc doubts have, however, been raised about the reliability of this method (Prochaska [@Prochaska03], Fenner et al. [@Fenner04]).
The often cited conclusion by Sneden et al. ([@Sneden91]) that zinc abundances closely track the overall metallicity with no discernible change in \[Zn/Fe\] in the range $-2.9 < {\mbox{\rm [Fe/H]}}< -0.1$ has been challenged in a number of recent studies. Prochaska et al. ([@Prochaska00]) found a mild enhancement of Zn relative to Fe (${\mbox{\rm [Zn/Fe]}}\sim +0.1$) in ten thick disk stars with metallicities between $-1.2$ and $-0.4$. Mishenina et al. (2002) have published a survey of Zn abundances in 90 disk and halo stars, and although they conclude that the data “confirms the well-known fact that is almost solar at all metallicities”, Nissen ([@Nissen04b]) notes that there is a tendency for thick disk stars in the metallicity range $-1.0 < {\mbox{\rm [Fe/H]}}< -0.5$ to be overabundant in Zn. In addition, Reddy et al. ([@Reddy03]) have found a slightly increasing with decreasing metallicity for 181 thin disk stars in the metallicity range $-0.8 < {\mbox{\rm [Fe/H]}}< +0.1$, and Bensby et al. ([@Bensby03]) found evidence of a separation of between thin and thick disk stars as well as a tendency of an uprising at ${\mbox{\rm [Fe/H]}}> 0$. Among halo stars, Nissen et al. ([@Nissen04b]) derived ${\mbox{\rm [Zn/Fe]}}\sim +0.1$ for stars in the metallicity range $-2.5 < {\mbox{\rm [Fe/H]}}< -2.0$, and as noted above Primas et al. ([@Primas00]) and Cayrel et al. ([@Cayrel04]) have found clear evidence of increasing values below ${\mbox{\rm [Fe/H]}}\simeq -2.5$ with reaching a value of +0.5 dex at ${\mbox{\rm [Fe/H]}}\simeq -4.0$.
In most of these works, the zinc abundances are primarily based on the $\lambda \lambda$4722.16, 4810.54Å lines, although the weak line at 6362.35Å is included in some of the disk star studies. A problem with the $\lambda \lambda$4722.16, 4810.54Å lines is that they are rather strong at solar metallicities ($EW \sim 70 - 80$mÅ in the solar flux spectrum) and blended by several weak lines in the wings. This makes it difficult to set a reliable continuum and to measure their equivalent widths accurately. The derived Zn abundances depend furthermore critically on the assumed value of the Van der Waals damping constant. This means that the trend of versus for disk stars becomes rather uncertain if the $\lambda \lambda$4722.16, 4810.54Å lines are included in the analysis.
In the present paper we derive zinc abundances exclusively from the weak 6362.35Å line, for which the equivalent width ranges from a few mÅ in the most metal-poor disk stars to about 30mÅin the metal-rich stars. Hence, the line is rather ideal for accurate abundance determinations being insensitive to microturbulence and Van der Waals damping parameters. In the following Sect. \[obs\], high resolution observations of this line is presented. Sect. \[analysis\] describes the model atmosphere analysis of the data and possible errors are discussed in Sect. \[errors\]. The results are discussed in Sect. \[results\].
Observations and data reduction {#obs}
===============================
Two sets of stars are included in this paper. The first set consists mainly of thin disk stars from Chen et al. ([@Chen00], [@Chen02]), who determined abundances of solar-type dwarfs selected from the $uvby - \beta$ photometric catalogues of Olsen ([@Olsen83], [@Olsen93], [@Olsen94]). The stars have $5600 \leq {\mbox{$T_{\rm eff}$}}\leq 6400$ K, ${\mbox{{\rm log}\,$g$}}\geq 3.8$ and $-1.0 \leq {\mbox{\rm [Fe/H]}}\leq +0.3$. In Chen et al. ([@Chen00]) chemical abundances of O, Na, Mg, Al, Si, K, Ca, Ti, V, Cr, Fe, Ni and Ba were derived from high-resolution spectra of 90 disk stars. Thirty of these spectra are of sufficiently high quality to allow measurements of the equivalent width of the 6362.35Å line with good accuracy. In Chen et al. ([@Chen02]) S, Si and Fe abundances were derived for 26 disk stars; 14 of these are included in the present paper. As described in the above papers, the spectra were obtained with the Coudé Echelle Spectrograph attached to the 2.16m telescope at the National Astronomical Observatories (Xinglong, China). The resolution is 37000 and the signal-to-noise ratio is generally over 200 in the region of the 6362.35Å line.
The second set of stars consists of 19 disk and halo stars with overlapping metallicities in the range $-1.0 < {\mbox{\rm [Fe/H]}}< -0.4$ taken from the work of Nissen & Schuster ([@Nissen97]), who determined various abundance ratios from high resolution ($R$ = 60000, $S/N \sim$ 150 - 200) spectra obtained with the EMMI spectrograph on the ESO 3.5m NTT telescope. The stars were originally selected from the catalogues of $uvby - \beta$ photometry by Schuster & Nissen ([@Schuster88]) and Schuster et al. ([@Schuster93]), and have atmospheric parameters in the range $5400 \leq {\mbox{$T_{\rm eff}$}}\leq 6300$ K and $4.0 < {\mbox{{\rm log}\,$g$}}< 4.6$. According to the value of the Galactic rotational velocity component, $V_{rot}$, the stars were classified as belonging either to the Galactic halo ($V_{rot} < 50$) or the thick disk ($V_{rot} > 150$).
For details about the reduction of the spectra we refer to the papers by Chen et al. ([@Chen00], [@Chen02]) and Nissen & Schuster ([@Nissen97]). A special problem with the $\lambda$6362.35Å line is that it lays in the midst of a very broad and shallow absorption feature identified by Michell & Mohler ([@Mitchell65]) as due to a auto-ionization line with a central wavelength of 6361.8Å. In the solar flux spectrum (Kurucz et al. [@Kurucz84]) the central depth of the line is about 5% and the total width is more than 15Å. By fitting a polynomial function to this broad line we have rectified the spectra around the $\lambda$6362.35Å line. Fig. \[fig:sp.xin\] shows the resulting spectra for three stars of different metallicities observed with the 2.16m telescope at Xinglong, and Fig. \[fig:sp.eso\] shows two representative spectra observed with the ESO NTT.
The equivalent width of the $\lambda$6362.35Å line was measured in the rectified spectra by Gaussian fitting so that the contribution from the nearby FeI/CrI line at 6362.88Åcan be nearly avoided. The equivalent width for the Sun ($EW = 22.4$mÅ), was obtained from a Moon spectrum observed at Xinglong in the same way as the programme stars. This value is consistent with the equivalent width of the line obtained from the the solar flux spectrum of Kurucz et al. ([@Kurucz84]).
The measured equivalent widths are given in Tables \[tb:ZnXin\] and \[tb:ZnEMMI\]. Some of the stars are included in the ELODIE public library of high resolution spectra (Prugniel & Soubiran [@Prugniel01]). With a resolution of $R$ = 42000 and a $S/N \sim$ 150 - 200 these spectra are of similar quality as our spectra and as a check, equivalent widths of the $\lambda$6362.35Å line have been measured in the ELODIE spectra also and are listed in Tables \[tb:ZnXin\] and \[tb:ZnEMMI\]. Figure \[fig:w-w\] shows a comparison between the two sets of equivalent widths. As seen the agreement is quite satisfactory. The rms scatter around the 1:1 line is 1.8mÅ suggesting that the error of our equivalent measurements is around 1.3mÅ.
Analysis
========
Stellar parameters, and , are taken from the above-mentioned papers from which the stars were selected. In Chen et al. ([@Chen00], [@Chen02]), was determined from the $b-y$ colour index using the IRFM calibration of Alonso et al. ([@Alonso96]) and was calculated from the Hipparcos parallax. In Nissen & Schuster ([@Nissen97]), was determined from the excitation balance of lines but it was checked that the values agree quite well with effective temperatures derived from $b-y$. The average difference, $T_{\rm exc.}-T_{b-y}$, is 65K and the rms scatter of the difference is $\pm55$K. As discussed later (see Table \[tb:abuerr\]) this systematic difference in corresponds to a change in of about 0.02dex - quite negligible compared to other error sources. Concerning the gravity parameter, we note that some stars in Nissen & Schuster are too distant to have accurate parallaxes. Hence, was determined from the ionization balance of and lines. Fourteen of the 19 stars in Table \[tb:ZnEMMI\] have, however, Hipparcos parallaxes with a relative error less than 25%, corresponding to a error less than about 0.20dex. The mean difference between the spectroscopic and parallax-derived gravity is 0.03dex with a rms scatter of $\pm0.15$ dex. The corresponding difference in is negligible. Hence, we conlcude that the different ways of deriving stellar parameters will not introduce significant effects in the abundance determination. Furthermore, we note that one star that is in common between the two samples, (=), has exactly the same abundance ratios in Tables \[tb:ZnXin\] and \[tb:ZnEMMI\], but this is of course a fortuitous agreement.
------ ----- ------ ------ --------- ------- ------- --------- ---------
Star Pop $W_a$ $W_b$
K mÅ mÅ
Sun D 5780 4.44 0.00 22.4 0.00 0.00
D 6173 4.11 $-$0.30 14.5 13.0 $-$0.03 $-$0.06
D 6119 4.12 0.12 33.7 0.05 $-$0.06
D 5776 4.13 $-$0.12 19.7 $-$0.06 0.09
D 6368 3.96 $-$0.46 13.0 0.08 $-$0.03
D 6301 4.12 $-$0.43 13.9 0.10 $-$0.06
D 6228 4.27 0.01 21.0 $-$0.09 $-$0.05
D 5915 4.03 $-$0.16 24.1 23.5 0.06 0.07
D 5962 4.30 $-$0.21 17.7 $-$0.01 $-$0.04
D 5773 4.02 $-$0.09 29.3 28.6 0.15 0.12
D 6202 3.83 $-$0.38 14.5 0.02 $-$0.07
D 5805 4.29 $-$0.18 16.9 18.9 $-$0.08 $-$0.04
D 5767 4.06 $-$0.10 27.1 25.2 0.11 0.12
D 5859 4.23 $-$0.30 16.7 0.02 $-$0.01
D 5802 4.36 $-$0.32 15.2 0.01 $-$0.03
D 6149 4.22 $-$0.56 14.4 0.21 0.05
D 6280 4.27 $-$0.16 21.2 0.06 $-$0.01
TD 5867 4.24 $-$0.80 7.9 0.11 0.01
D 6227 4.16 $-$0.15 21.6 0.04 0.06
D 6243 4.28 $-$0.32 16.4 0.07 0.03
D 6130 4.32 0.00 27.6 0.03 $-$0.09
D 6004 4.21 $-$0.44 17.8 0.18 $-$0.01
D 6329 4.15 $-$0.46 13.4 0.10 $-$0.03
D 5874 4.06 $-$0.09 27.6 0.09 0.08
D 6308 3.84 $-$0.13 18.4 $-$0.06 $-$0.12
D 5813 4.12 0.13 28.6 $-$0.04 $-$0.06
D 6051 4.36 $-$0.17 15.2 $-$0.14 $-$0.04
TD 6020 4.15 $-$0.25 18.9 0.05 0.03
D 6063 4.10 $-$0.06 30.6 0.14 0.13
D 5731 4.16 $-$0.07 24.2 23.5 0.01 $-$0.02
D 5866 4.12 $-$0.19 24.0 0.11 0.05
D 6102 4.09 $-$0.51 11.2 0.02 0.03
D 5905 4.10 $-$0.59 12.3 0.12 $-$0.07
D 5877 4.24 $-$0.05 22.1 26.9 $-$0.05 0.00
D 5866 4.03 $-$0.02 24.4 26.3 $-$0.07 0.01
D 6061 3.93 $-$0.63 9.7 9.2 0.03 $-$0.02
D 5925 4.30 $-$0.05 27.3 0.08 $-$0.04
D 5935 4.32 $-$0.46 13.8 0.08 0.06
D 6059 4.12 $-$0.26 19.1 0.04 0.07
D 6298 4.15 $-$0.02 19.3 $-$0.12 $-$0.10
D 5888 4.26 $-$0.30 18.1 17.3 0.06 0.03
D 6141 4.18 $-$0.44 12.7 0.02 $-$0.05
D 6158 3.96 $-$0.24 17.5 15.8 $-$0.01 $-$0.02
D 5923 3.74 $-$0.15 19.5 $-$0.09 $-$0.13
D 6178 4.08 $-$0.13 19.0 19.0 $-$0.05 $-$0.03
------ ----- ------ ------ --------- ------- ------- --------- ---------
: Stellar parameters, equivalent widths of the 6362.35Å line and derived abundance ratios and for disk stars observed with the Xinglong 2.16m telescope. $W_a$ is the equivalent width measured from the Xinglong spectra and $W_b$ the value obtained from ELODIE spectra. The stars are classified as belonging to the thin disk (D) or the thick disk population (TD) according to their kinematics.[]{data-label="tb:ZnXin"}
------ ----- ------ ------ --------- ------- ------- --------- ---------
Star Pop $W_a$ $W_b$
K mÅ mÅ
TD 5714 4.44 $-$0.88 7.5 0.25 0.00
TD 5807 4.24 $-$0.67 11.4 0.20 0.03
H 5863 4.24 $-$0.83 9.4 0.26 0.02
TD 5734 4.38 $-$0.76 9.6 0.24 $-$0.01
TD 5851 4.36 $-$0.82 7.3 7.7 0.16 $-$0.01
TD 5900 4.37 $-$0.63 10.3 0.14 0.02
TD 5886 4.33 $-$0.85 6.6 0.12 $-$0.02
H 5933 4.26 $-$0.90 4.1 $-$0.04 $-$0.18
H 5672 4.57 $-$0.83 4.1 0.00 $-$0.17
TD 5967 4.26 $-$0.80 7.1 0.11 0.01
TD 5914 4.23 $-$0.85 6.7 7.0 0.12 0.01
H 6021 4.44 $-$0.75 6.2 0.04 $-$0.12
H 6029 4.32 $-$0.79 5.2 $-$0.02 $-$0.11
H 5831 4.36 $-$0.80 6.2 0.06 $-$0.06
TD 6269 4.51 $-$0.61 7.6 5.8 0.05 0.00
D 6272 4.03 $-$0.42 11.1 0.01 0.00
H 5720 4.14 $-$0.65 10.1 0.18 0.03
D 5396 4.38 $-$0.93 5.0 0.12 $-$0.01
H 5686 4.40 $-$0.70 9.4 0.17 0.00
------ ----- ------ ------ --------- ------- ------- --------- ---------
: Same as Table \[tb:ZnXin\] but for stars from Nissen & Schuster ([@Nissen97]) and $W_a$ is measured from the ESO NTT spectra.[]{data-label="tb:ZnEMMI"}
Our determination of abundances is based on 1D model atmospheres computed with the MARCS code using updated continuous opacities (Asplund et al. [@Asplund97]) and including UV line blanketing by millions of absorption lines. LTE is assumed both in constructing the models and in deriving abundances. For all stars the microturbulence parameter $\xi_{t}$ was determined by requesting that the iron abundance derived from lines should be independent of the equivalent width.
For stars selected from Chen et al. ([@Chen02]) we have adopted the values given in that paper. These iron abundances are based on equivalent widths of 18 weak lines analyzed in a differential way with respect to the Sun. For stars in Chen et al. ([@Chen00]), we redetermined the iron abundances using equivalent widths of a subset of 8 of the 18 ${\ion{Fe}{ii}}$ lines. The abundances are nearly the same as those presented in Chen et al. ([@Chen00]) based on lines. For the Nissen & Schuster ([@Nissen97]) stars we use their values as determined from 104 and 12 lines (forced to give the same iron abundance due to the way the gravity was determined).
In deriving zinc abundances from the $\lambda$6362.35Å ${\ion{Zn}{i}}$ line ($\chi_{\rm exc} = 5.79$eV), we have adopted log$gf = 0.14$ as determined by Biémont & Godefroid ([@Biemont80]) from multi-configurational Hartree-Fock calculations. Using the MARCS model atmosphere for the Sun and an equivalent width of 22.4mÅ of the 6362.35Å line as measured from the Moon spectrum observed in the same way as the program stars, we derive a solar zinc abundance of log$\epsilon$(Zn) = 4.52. This is considerably below the meteoritic Zn abundance of 4.67 $\pm 0.04$ (Grevesse & Sauval [@Grevesse98]). The same problem was encountered by Biémont & Godefroid ([@Biemont80]), who derived a solar photospheric zinc abundance of log$\epsilon$(Zn) = 4.54 from the $\lambda$6362.35Å line using the Holweger & Müller ([@Holweger74]) model of the Sun. This difference between photospheric and meteoritic Zn abundances may well be due to an error in the $\log gf$ value of the line. However, when determining differential Zn abundances with respect to the Sun, i.e. , using our solar photospheric abundance of $\log \epsilon$(Zn) = 4.52 as a reference, the possible error in $\log gf$ cancels. The resulting values of are given in Tables \[tb:ZnXin\] and \[tb:ZnEMMI\]. The two tables also include values of as derived from more than 20 lines (see Chen et al. [@Chen00] and Nissen & Schuster [@Nissen97])
Errors of the derived abundance ratios {#errors}
======================================
The uncertainties of the model parameters are estimated to be $\pm 100$K in temperature, $\pm 0.15$dex in gravity, $\pm 0.1$dex in metallicity and $\pm 0.3$ in microturbulence. The dependence of the derived and abundance ratios on the stellar parameters is calculated by altering temperature, gravity, metallicity and microturbulence for , and the Sun, representing most of our metallicity range. As seen from Table \[tb:abuerr\], is most sensitive to gravity, but the combined error is small - less than 0.06dex if the individual errors are added quadratically. We also checked that the derived ratio is only marginally affected ($<\!0.02$dex), when alpha-element enhanced models (\[$\alpha$/Fe\] = +0.4, $\alpha$ = O, Ne, Mg, Si, S, Ca, and Ti) are used instead of models with \[$\alpha$/Fe\] = 0.0. Taking into account the uncertainty of the measured equivalent width of the $\lambda$6362.35Å line ($\pm$ 1.3mÅ) we then estimate a total error of about 0.07dex for and . A similar error is estimated for and .
[lrrrrrrrr]{} & & &\
& & &\
& $\Delta {\mbox{${\rm [\frac{Fe}{H}]}$}}$ & $\Delta {\mbox{${\rm [\frac{Zn}{Fe}]}$}}$ & $\Delta {\mbox{${\rm [\frac{Fe}{H}]}$}}$ & $\Delta {\mbox{${\rm [\frac{Zn}{Fe}]}$}}$ & $\Delta {\mbox{${\rm [\frac{Fe}{H}]}$}}$ & $\Delta {\mbox{${\rm [\frac{Zn}{Fe}]}$}}$\
$\Delta {\mbox{$T_{\rm eff}$}}= 100$ K & $-$0.01 & 0.02 & $-$0.01 &0.03 & $-$0.04 & 0.02\
$\Delta {\mbox{{\rm log}\,$g$}}= 0.15$ & 0.06 & $-$0.03 & 0.06 & $-$0.03 & 0.06 & $-$0.03\
$\Delta {\mbox{\rm [Fe/H]}}$= 0.1 & 0.01 & $-$0.01 & 0.01 & $-$0.01 & 0.03 & $-$0.01\
$\Delta \xi_t= 0.3 $ km/s & $-$0.02 & 0.02 & $-$0.04 & 0.03 & $-$0.03 & 0.01\
$E_{\gamma} = 1.0 \rightarrow 2.0$ & & 0.00 & & 0.00 & & $-$0.02\
\
\
\
In calculating zinc abundances from the 6362.35Å line we adopted the Unsöld ([@Unsold55]) approximation to the Van der Waals interaction constant with an enhancement factor $E_{\gamma} = 1.5$. As seen from Table \[tb:abuerr\], the derived zinc abundance depends only slightly on the value of $E_{\gamma}$; the effect of increasing the enhancement factor from 1.0 to 2.0 is less than 0.03dex even in the most metal rich stars. This is in stark contrast to the effect of $E_{\gamma}$ on the stronger lines at 4722.2 and 4810.5Å, for which an increase of $E_{\gamma}$ from 1.0 to 2.0 leads to a decrease of the derived Zn abundance for the Sun with about 0.13dex, whereas the effect for the metal-poor disk stars is a decrease of 0.02dex only. Hence, in metal-poor disk stars becomes very sensitive to the adopted value of $E_{\gamma}$ if the stronger lines are used. This is the main reason that we have avoided these lines in the present paper.
As mentioned in Sect. \[obs\] the 6362.35Å line lays in the midst of a $\sim \! 15$Å broad and shallow absorption feature with a central depth of a few percent identified by Michell & and Mohler ([@Mitchell65]) as due to a auto-ionization line. The equivalent width of the line was measured relative to the local apparent continuum but the small contribution of the auto-ionization line to the absorption coefficient has been neglected in our analysis. We have investigated the effect of this auto-ionization line on the derived Zn abundance by synthesizing the solar flux spectrum in the spectral region 6356 - 6368Å. In order to fit the large width, we adopt a very high radiation damping constant (1.4E+12) of the auto-ionization line and vary its (unknown) log$gf$ value so that the right equivalent width is obtained. In the case of the Sun, the effect of including the auto-ionization line is to increase the Zn abundance by +0.03 dex, i.e. from log$\epsilon$(Zn) = 4.52 to 4.55. For a typical metal-poor thick disk star, , with ${\mbox{\rm [Fe/H]}}= -0.80$ and ${\mbox{\rm [Ca/Fe]}}\sim +0.2$, the correction of the Zn abundance is of the order of +0.01 dex. Differentially with respect to the Sun, the largest effect of including the opacity contribution from the auto-ionization line would then be a decrease of for the most metal-poor stars by about 0.02 dex. This is quite negligible compared with the other error sources.
As described in Sect. \[analysis\] we have derived the abundances on the basis of plane parallel, homogeneous (1D) model atmospheres and the assumption of LTE. Asplund et al. ([@Asplund99]) have shown that there may be significant effects from using three-dimensional (3D) hydrodynamical models instead, especially for metal-poor stars and for lines formed in the upper part of the atmosphere such as molecular lines (Asplund & García P[é]{}rez [@Asplund01]). For lines formed deep in the atmosphere, such as the and lines which we have used, the 3D effects are, however, small (Nissen et al. [@Nissen04b]) and go in the same direction. Hence, we don’t expect that the derived is subject to any significant 3D corrections.
Regarding non-LTE effects on we first note that for the Chen et al. sample the iron abundances are based on lines for which departures from LTE are small according to the computations of Thévenin & Idiart ([@Thevenin99]). For the Nissen & Schuster([@Nissen97]) stars we adopted iron abundances based on lines and determined the gravities by requiring that and lines should provide the same Fe abundance. Hence, one might expect that the derived iron abundances are too low due to an overionization of as predicted by Thévenin & Idiart ([@Thevenin99]) for metal-poor stars. However, as discussed in Sect. \[analysis\], gravities derived from Hipparcos parallaxes for a subsample of the Nissen & Schuster stars agree very well with the spectroscopic gravities. Hence, we conclude that the derived iron abundances are not significantly different from the true Fe abundances. This should not be taken as evidence against the predicted over-ionization of Fe for metal-poor stars. Rather, it reflects the way the $gf$-values were determined by Nissen & Schuster, namely by an “inverted" abundance analysis of the two bright stars, and , with their parameters and abundances taken from Edvardsson et al. ([@Edvardsson93]).
It is more unclear if there are non-LTE effects on the derived Zn abundances. With an ionization potential, $\chi_{\rm ion} ({\ion{Zn}{i}})$ = 9.39eV, there are approximately equal numbers of neutral and ionized zinc atoms at the temperatures and electron pressures of the line forming regions in the atmospheres of our stars. Hence, an overionization of relative to LTE, as is the case for (Thévenin & Idiart [@Thevenin99]), would lead to an underestimate of the abundance of Zn in the LTE analysis of lines. However, such an over-ionization effect on is likely to be smaller than the equivalent effect for , since $N_{\ion{Zn}{i}}\sim N_{\ion{Zn}{ii}}$ whereas $N_{\ion{Fe}{i}}<< N_{\ion{Fe}{ii}}$, where N denotes the number density of the atoms and ions. Furthermore, we note that the 6362.35Å line, on which our Zn abundances is based, is a weak, high excitation potential ($\chi_{\rm exc} = 5.79$eV) line formed deep in the atmosphere, where departures from LTE tend to be small due to the high density. Nevertheless, it would be very desirable to perform a thorough study of non-LTE effects on the determination of Zn abundances.
Results and discussion {#results}
======================
Following Fuhrmann ([@Fuhrmann98], [@Fuhrmann00]) we have used the total space velocity, ${\mbox{$V_{\rm tot}$}}= (U^2 + V^2 +W^2)^{\frac{1}{2}}$, with respect to the Local Standard of Rest (LSR) to classify the stars in three main populations: i) thin disk stars with ${\mbox{$V_{\rm tot}$}}< 85$, ii) thick disk stars with $85 < {\mbox{$V_{\rm tot}$}}< 180$, and iii) halo stars with ${\mbox{$V_{\rm tot}$}}> 180$. For stars in Table \[tb:ZnXin\], the $U,V,W$ velocity components with respect to the LSR are taken from Chen et al. ([@Chen00], [@Chen02]). For stars in Table \[tb:ZnEMMI\] we have updated the velocity components calculated by Nissen & Schuster ([@Nissen97]) using Hipparcos (ESA [@ESA97]) or Tycho-2 (H[ø]{}g et al. [@Hog00]) proper motions and adopting the most recent values for the solar motion with respect to the LSR (($U_{\sun}, V_{\sun}, W_{\sun}$) = ($-10.0$, +5.2, +7.2)[^2] (Dehnen & Binney [@Dehnen98]). The changes with respect to Nissen & Schuster are relatively small except for one star, () that was classified as a halo star with an unusually high metallicity (${\mbox{\rm [Fe/H]}}= -0.42$) by Nissen & Schuster. With the new proper motion values from the Tycho catalogue, it turns out to be an ordinary thin disk star with velocity components ($U, V, W$) = (35, 30, 7).
The large majority of stars in Table \[tb:ZnXin\] have thin disk kinematics, whereas those in Table \[tb:ZnEMMI\] are mainly thick disk or halo stars. As seen from the Toomre diagram in Fig. \[fig:toomre\], the thick disk and the halo stars are very well separated in $V$ in accordance with the way the two groups were selected by Nissen & Schuster ([@Nissen97]). The thin and thick disk stars have a slight overlap in $V$, but all thick disk stars except one have a rotational lag in the range $-100 < V < -50$, whereas all thin disk stars except one have $V > -50$.
Figure \[fig:ZnFe\] shows versus with different symbols for the various populations. Looking first at the thin disk stars, we see a slight increasing trend of with decreasing metallicity; a straight line least squares fit to the disk star data gives: $$\begin{aligned}
{\mbox{\rm [Zn/Fe]}}= -0.01 \,(\pm 0.02) - 0.16 \,(\pm 0.05) \cdot {\mbox{\rm [Fe/H]}}\nonumber\end{aligned}$$ with a reduced chi-square, $\chi^2_{red}$ = 1.11, when the estimated error of 0.07dex on and is adopted. This relation corresponds to ${\mbox{\rm [Zn/Fe]}}\simeq +0.1$ for a thin disk metallicity of ${\mbox{\rm [Fe/H]}}= -0.6$, and the relation agrees quite well with the trend of versus found by Reddy et al. ([@Reddy03]) and Bensby et al. ([@Bensby03]) for thin disk stars.
Bensby et al. ([@Bensby03]) have found evidence of a difference of about 0.1dex in between thick and thin disk stars in the metallicity range $-0.6 < {\mbox{\rm [Fe/H]}}< -0.3$. As seen from Fig. \[fig:ZnFe\], we have no thick disk stars in this metallicity range. Our ten thick disk stars in the metallicity range $-0.9 < {\mbox{\rm [Fe/H]}}< -0.6$ have a mean of 0.15dex, which is only slightly higher than the value predicted from the thin disk relation given above. The explanation may be that our thick disk stars lie in a metallicity range, ${\mbox{\rm [Fe/H]}}< -0.6$, where the thick and the thin disk are not chemically well separated. The same is seen for e.g. the ratio between alpha-capture elements and Fe, i.e. \[$\alpha$/Fe\]. In the metallicity range $-0.6 < {\mbox{\rm [Fe/H]}}< -0.3$ \[$\alpha$/Fe\] is different for thick and thin disk stars (Bensby et al. [@Bensby03], [@Bensby04]), whereas the two populations start to merge together in \[$\alpha$/Fe\] for ${\mbox{\rm [Fe/H]}}\simeq -0.7$.
Among the eight halo stars shown in Fig. \[fig:ZnFe\], three have an enhanced Zn/Fe ratio at the same level as the thick disk stars, but the other five have a solar Zn/Fe ratio. As shown in Figs. \[fig:ZnFeTe\] and \[fig:ZnFeLg\], the eight halo stars and the ten thick disk stars have similar temperatures and gravities, and there is no significant trend of with the stellar parameters. Thus, the scatter of among the halo stars seems to be real. Furthermore, the five halo stars with low are among the group of so-called alpha-poor halo stars that were found by Nissen & Schuster ([@Nissen97]) to have a solar-like ratio between the alpha-capture elements (O, Mg, Si, Ca and Ti) and Fe. Hence, Zn tends to follow the alpha-capture elements in two respects: by being overabundant in metal-poor disk stars and by having a solar-like for the alpha-poor halo stars. The amplitude of the variations is, however, smaller than in the case of and , i.e. $\sim \! 0.15$dex instead of $\sim \! 0.3$dex.
Nissen & Schuster ([@Nissen97]) also found the alpha-poor halo stars to have unusual low abundances of Na and Ni with respect to Fe. varies from $-0.4$ to 0.0 and varies from $-0.2$ to 0.0 with a tight correlation between the two ratios. As discussed by Nissen ([@Nissen04a]) recent abundance studies of giant stars in dwarf spheroidal galaxies (Shetrone et al. [@Shetrone03]) show that these stars also have underabundant values of and and fit the - relation of alpha-poor halo stars very well. This supports the suggestion of Nissen & Schuster ([@Nissen97]) that the alpha-poor halo stars have been accreted from dwarf galaxies with a different star formation history and/or initial mass function than the Milky Way. The deficiency of Na and Ni may be connected to the fact that the yields of both Na and the dominant Ni isotope ($^{58}$Ni) depend upon the neutron excess (Thielemann et al. [@Thielemann90]). The deficiency of Zn in the alpha-poor halo stars may then be explained by the fact that the production of Zn in Type II SNe also depends on the neutron excess (Timmes et al. [@Timmes95]). Indeed, Fig. \[fig:ZnNi\] shows that the alpha-poor halo stars do not stand out in a plot of versus .
{width="17cm"}
The $\lambda$6362.25Å line is too weak to be used for deriving Zn abundances in stars with ${\mbox{\rm [Fe/H]}}< -1.0$. For such stars the $\lambda \lambda$4722.16, 4810.54Å lines are, however, quite ideal having equivalent widths ranging from a few mÅ to about 50mÅ when increases from $-2.5$ to $-0.8$. Figure \[fig:ZnFe.all\] includes such data for 29 stars from Nissen et al. ([@Nissen04b]). Three of the stars, , and are in common with the present paper; their symbols are connected with straight lines in Fig.\[fig:ZnFe.all\]. As seen, there is a reasonable good agreement between the two sets of data. The mean difference in (Nissen et al. $-$ present paper) is $-0.06$dex suggesting perhaps a slight offset of the data based on the $\lambda \lambda$4722.16, 4810.54Å lines, which may occur because the lines are quite strong in the Sun causing to be sensitive to the adopted value of the damping constant as discussed in Sect.\[errors\].
From Fig.\[fig:ZnFe.all\] it is seen that although the overall trend is quite flat over the metallicity range $-2.5 < {\mbox{\rm [Fe/H]}}< +0.2$, subtle deviations of from zero seem to occur. The most metal-poor halo stars have ${\mbox{\rm [Zn/Fe]}}\sim +0.1$ and metal-poor thin disk and thick disk stars have ${\mbox{\rm [Zn/Fe]}}\sim$ +0.10 to +0.15. One might argue that such small deviations could be due to errors in the analysis, because non-LTE effects were not taken into account. On the other hand, it seems very difficult to explain the systematic difference of about 0.15dex in between thick disk stars and alpha-poor halo stars at the [*same metallicity*]{} as a non-LTE effect. We are here comparing abundances of stars with nearly the same atmospheric parameters, , and . Hence, any non-LTE effects are expected to cancel when determining the difference in .
We therefore think that the small deviations of from zero are real and contain important information about the complicated nucleosynthesis of Zn. For the most metal-poor halo stars Cayrel et al. ([@Cayrel04]) also find a slight overabundance of at ${\mbox{\rm [Fe/H]}}= -2.5$ increasing to ${\mbox{\rm [Zn/Fe]}}\sim +0.5$ at ${\mbox{\rm [Fe/H]}}= -4.0$. As discussed by Umeda & Nomoto ([@Umeda02]) this trend and other non-solar abundance ratios for the iron-peak elements may be explained with high-energy (hypernovae) models for core collapse explosions of massive Population III stars. In the case of the thick disk and metal-poor thin disk stars, the overabundance of Zn/Fe indicates that there is a source of zinc production in addition to the weak $s$-process and alpha-rich freezeout in Type II SNe. This additional contribution may be due to zinc production in the neutrino-powered wind of 10 to 20 solar mass SNe as suggested by Hoffman et al. ([@Hoffman96]), essentially at the same site as proposed for the $r$-process (Woosley et al. [@Woosley94]). In this connection, we note that the $r$-process elements are also known to be overabundant with respect to Fe in thick disk and metal-poor thin disk stars (Mashonkina et al. [@Mashonkina03]). Furthermore, it is interesting that it is the $^{64}$Zn isotope, which is produced in the neutrino-powered wind according to Hoffman et al. ([@Hoffman96]). $^{64}$Zn is the dominant isotope in the solar system (Anders & Grevesse [@Anders89]) but is underproduced by a factor of three in traditional calculations of nucleosynthesis of Zn in type II SNe (Timmes et al. [@Timmes95], Fig. 3).
As seen from Fig.\[fig:ZnFe.all\], the halo stars in the metallicity range $-1.8 < {\mbox{\rm [Fe/H]}}< -1.0$ have ${\mbox{\rm [Zn/Fe]}}\sim 0.0$. They form a sequence in the figure that fit well to the alpha-poor halo stars at ${\mbox{\rm [Fe/H]}}\sim -0.8$. The thick disk and the halo stars at ${\mbox{\rm [Fe/H]}}\sim -0.8$ with ${\mbox{\rm [Zn/Fe]}}\sim +0.15$ are apparently subject to a different evolution in . The explanation may be that the Galactic halo consists of two components as also suggested by Gratton et al. ([@Gratton03]): A dissipative component with high and \[$\alpha$/Fe\] connected to the thick disk and an accreted component with low and \[$\alpha$/Fe\].
Although these possible variations of as a function of are of importance when interpreting abundance data for DLAs, we note that the amplitude of is smaller than in the case of \[$\alpha$/Fe\]. Sulphur is of particular interest in this connection, because S like Zn is practically undepleted onto dust so that the observed S/Zn interstellar gas ratio in DLAs equals the intrinsic abundance ratio between the two elements. As shown by Nissen et al. ([@Nissen04b]), the overabundance of S in halo stars corresponds to ${\mbox{\rm [S/Fe]}}\sim +0.3$ to +0.4dex. Hence, when plotting versus we still get a trend that looks like versus (see Nissen et al. [@Nissen04b], Figs. 6 and 8). This suggests that may be used as a “chemical clock” to date the star formation process at high $z$ albeit with a lower sensitivity than . As discussed by Fenner et al. ([@Fenner04]), a better understanding of the nucleosynthesis of Zn is, however, needed in order to derive the past history of star formation in DLAs from the observed S/Zn ratio.
Conclusions
===========
We have determined Zn abundances for 62 dwarf stars with metallicities in the range $-1.0 < {\mbox{\rm [Fe/H]}}< +0.2$. The abundances are based on equivalent widths of the weak $\lambda 6362.35$Å line and are relatively insensitive to possible errors in the atmospheric parameters and the damping constant of the line; thus they are more reliable than Zn abundances based on the stronger $\lambda \lambda$4722.16, 4810.54Å lines, which have been applied in other studies of Zn abundances.
The stars were grouped into thin disk, thick disk and halo populations according to their kinematics. It is found that in thin disk stars shows a slight increasing trend with decreasing metallicity reaching a value of ${\mbox{\rm [Zn/Fe]}}\sim \! +0.1$ at ${\mbox{\rm [Fe/H]}}= -0.6$ in agreement with Reddy et al. ([@Reddy03]) and Bensby et al. ([@Bensby03]). Ten thick disk stars in the metallicity range $-0.9 < {\mbox{\rm [Fe/H]}}< -0.6$ have an average ${\mbox{\rm [Zn/Fe]}}\ = +0.15$dex, whereas five alpha-poor halo stars in the same metallicity range have ${\mbox{\rm [Zn/Fe]}}\ \simeq 0.0$dex. Interestingly, the same five stars are also underabundant in Ni having an average ${\mbox{\rm [Ni/Fe]}}\simeq -0.13$.
These results suggest that Zn is not an exact tracer of Fe as often assumed in abundance studies of DLA systems. The overabundance of Zn/Fe in metal-poor thin and thick disk stars may be explained by Zn production in the neutrino-powered wind of 10 to 20 solar mass SNe as suggested by Hoffman et al. ([@Hoffman96]). The deviations of from zero are, however, considerably smaller than in the case of , and ; versus for DLA systems may therefore still be used to obtain information on the star formation history in these objects if the nucleosynthesis of zinc can be better understood.
In order to improve on the study of the - trend for Galactic stars even more accurate data for the equivalent width of Zn lines should be obtained in larger samples of disk and halo stars. At the same time a thorough study of non-LTE effects on the determination of Zn abundances is needed, as well as further studies of the nucleosynthesis of zinc in various types of objects.
This work is partly supported by the National Natural Science Foundation of China and grant NKBRSF G1999075406, and by the Danish Natural Science Research Council, grant 21-01-0523.
Alonso, A., Arribas, S., & Martínez-Roger, C. 1996, A&A, 313, 873 Anders, E., & Grevesse, N. 1989, Geochim. Cosmochim. Acta, 53, 197 Asplund, M., & García Pérez, A.E. 2001, A&A, 372, 601 Asplund, M., Gustafsson, B., Kiselman, D., & Eriksson, K. 1997, A&A 318, 521 Asplund, M., Nordlund, Å., Trampedach, R., & Stein, R.F. 1999, A&A, 346, L17 Bensby, T., Feltzing, S., & Lundström, I., 2003, A&A, 410, 527 Bensby, T., Feltzing, S., & Lundström, I., 2004, A&A, 415, 155 Biémont, E., & Godefroid, M. 1980, A&A 84, 361 Cayrel, R., Depagne, E., Spite, M., et al. 2004, A&A, 416, 1117 Centurión, M., Bonifacio, P., Molaro, P., & Vladilo, G. 2000, ApJ, 536, 540 Chen, Y.Q., Nissen, P.E., Zhao, G., Zhang, H.W., & Benoni, T. 2000, A&AS, 141, 491 Chen, Y.Q., Nissen, P.E., Zhao, G., & Asplund, M. 2002, A&A, 390, 225 Dehnen, W., & Binney, J.J. 1998, MNRAS, 298, 387 Edvardsson, B., Andersen, J., Gustafsson, B., Lambert, D.L., Nissen, P.E., & Tomkin, J. 1993, A&A, 275, 101 ESA 1997, The Hipparcos and Tycho Catalogues, ESA SP-1200 Fenner, Y., Prochaska, J.X., & Gibson, B.K. 2004, ApJ, 606, 116 Fuhrmann, K. 1998, A&A, 338, 161 Fuhrmann, K. 2000, http://www.xray.mpe.mpg.de/$\sim$fuhrmann Goswami, A., & Prantzos, N. 2000, A&A 359, 191 Gratton, R.G., Carretta, E., Desidera, S. et al. 2003, A&A, 406, 131 Grevesse, N., & Sauval, A.J. 1998, Space Science Reviews 85, 161 Hoffman, R.D., Woosley, S.E., Fuller, G.M., & Meyer, B.S. 1996, ApJ, 460, 478 Holweger, H., & Müller, E.A. 1974, Solar Phys., 39, 19 H[ø]{}g, E., Fabricius, C., Makarov, V.V., et al. 2000, A&A, 355, L27 Iwamoto, K., Brachwitz, F., Nomoto, K., et al. 1999, ApJS, 125, 439 Kurucz, R.L., Furenlid, I., Brault, J., & Testerman, L. 1984, Solar Flux Atlas from 296 to 1300 nm, National Solar Observatory, Sunspot, New Mexico Langer, N., Arcoragi, J.-P., & Arnould, M. 1989, A&A, 210, 187 Mashonkina, L., Gehren, T., Travaglio, C., & Borkova, T. 2003, A&A, 397, 275 Matteucci, F., Raiteri, C.M., Busso, M., Gallino, R., Gratton, R. 1993, A&A, 272, 421 Mitchell, W.E., & Mohler, O.C. 1965, ApJ, 141, 1126 Mishenina, T.V., Kovtyukh, V.V., Soubiran, C., Travaglio, C., & Busso, M. 2002, A&A, 396, 189 Nissen, P.E. 2004, in Carnegie Observatories Astrophysics Series, Vol. 4: Origin and Evolution of the Elements, ed. A. McWilliam & M. Rauch, (Cambridge: Cambridge Univ. Press), in press Nissen, P.E., Chen Y.Q., Asplund M., & Pettini M. 2004, A&A, 415, 993 Nissen, P.E., & Schuster, W.J. 1997, A&A, 326, 751 Olsen, E.H. 1983, A&AS, 54, 55 Olsen, E.H. 1993, A&AS, 102, 89 Olsen, E.H. 1994, A&AS, 106, 257 Pettini, M., Ellison, S.L., Steidel, C.C., & Bowen, D.V. 1999, ApJ, 510, 576 Primas, F., Reimers D., Wisotzki et al. 2000, in The First Stars, eds. A. Weiss, T.Abel, & V. Hill (Berlin: Springer), p. 51 Prochaska, J.X., Naumov, S.O., Carney, B.W., McWilliam, A., & Wolfe, A.M. 2000, AJ, 120, 2513 Prochaska, J.X. 2003, in Carnegie Observatories Astrophysics Series, Vol. 4: Origin and Evolution of the Elements, ed. A. McWilliam & M. Rauch (Cambridge: Cambridge Univ. Press), in press Prugniel, P., & Soubiran, C. 2001, A&A, 369, 1048 Reddy, B.E., Tomkin, J., Lambert, D.L., & Allende Prieto, C., 2003, MNRAS, 340, 304 Shetrone, M., Venn, K. A., Tolstoy, E., et al. 2003, AJ, 125, 684 Schuster, W.J., & Nissen, P.E. 1988, A&AS 73, 225 Schuster, W.J., Parrao, L., & Contreras Martínez, M.E. 1993, A&AS, 97, 951 Sneden, C., Gratton, R.G., & Crocker D.A. 1991, A&A, 246, 354 Thévenin, F., & Idiart, T. 1999, ApJ, 521, 753 Thielemann, F.-K., Hashimoto, M., & Nomoto, K. 1990, ApJ, 349, 222 Timmes, F.X., Woosley, S.E., & Weaver, T.A. 1995, ApJS, 98, 617 Umeda, H., & Nomoto, K. 2002, ApJ, 565, 385 Unsöld, A. 1955, Physik der Sternatmosphären, Springer Verlag, Berlin Vladilo, G. 2002, A&A, 391, 407 Woosley, S.E., & Weaver, T.A. 1995, ApJS, 101, 181 Woosley, S.E., Wilson, J.R., Mathews, G.J., et al. 1994, ApJ, 433, 229
[^1]: Based on observations collected at the National Astronomical Observatories, Xinglong, China and the European Southern Observatory, La Silla, Chile (ESO No. 67.D-0106).
[^2]: In the present work, $U$ is defined to be positive in the anticentre direction
|
---
abstract: 'We consider infinite harmonic chain with $l_{2}$-initial conditions and deterministic dynamics (no probability at all). Main results concern the question when the solution will be uniformly bounded in time and space in the $l_{\infty}$-norm.'
author:
- 'A.A. Lykov[^1]'
- 'M.V. Melikian'
title: 'Long Time Behavior of Infinite Harmonic Chain with $l_{2}$ Initial Conditions'
---
Introduction
============
Consider a countable system of point particles with unit masses on $\mathbb{R}$ with coordinates $\{x_{k}\}_{k\in\mathbb{Z}}$ and velocities $\{v_{k}\}_{k\in\mathbb{Z}}$. We define formal energy (hamiltonian) by the following formula: $$H=\sum_{k\in\mathbb{Z}}\frac{v_{k}^{2}}{2}+\frac{\omega_{0}^{2}}{2}\sum_{k\in\mathbb{Z}}(x_{k}(t)-ka)^{2}+\frac{\omega_{1}^{2}}{2}\sum_{k\in\mathbb{Z}}(x_{k}(t)-x_{k-1}(t)-a)^{2},$$ with parameters $a>0,\ \omega_{1}>0,\ \omega_{0}\geqslant0$. Particle dynamics is defined by the infinite system of ODE: $$\begin{aligned}
\ddot{x}_{k}(t)&=-\frac{\partial{H}}{\partial{x_{k}}}=-\omega_{0}^{2}(x_{k}(t)-ka)+\omega_{1}^{2}(x_{k+1}(t)-x_{k}(t)-a) \notag \\
&\quad {} -\omega_{1}^{2}(x_{k}(t)-x_{k-1}(t)-a),\quad k\in\mathbb{Z}\label{mainEqX}\end{aligned}$$ with initial conditions $x_{k}(0),v_{k}(0)$. The equilibrium state (minimum of the energy) is $$x_{k}=ka,\quad v_{k}=0,\quad k\in\mathbb{Z}.$$ This means that if the initial conditions are in the equilibrium state then the system will not evolve, i.e. $x_{k}(t)=ka,\ v_{k}(t)=0$ for all $t\geqslant0$. Let us introduce deviation variables: $$q_{k}(t)=x_{k}-ka,\quad p_{k}(t)=\dot{q}_{k}(t)=v_{k}(t).$$ Our main assumption is $q(0)=\{q_{k}(0)\}_{k\in\mathbb{Z}}\in l_{2}(\mathbb{Z}),\ p(0)=\{p_{k}(0)\}_{k\in\mathbb{Z}}\in l_{2}(\mathbb{Z})$. In the present article we study the long time behavior of $q_{k}(t)$ depending on initial conditions and parameters $a,\omega_{0},\omega_{1}$. Namely, we are interested in the uniform boundedness (in $k$ and $t$), the order of growth (in $t$) and exact asymptotic behavior of $q_{k}(t)$.
It is easy to see that $q_{k}(t)$ satisfies the following system of ODE: $$\ddot{q}_{k}=-\omega_{0}^{2}q_{k}+\omega_{1}^{2}(q_{k+1}-q_{k})-\omega_{1}^{2}(q_{k}-q_{k-1}),\quad k\in\mathbb{Z}.\label{mainEqForQ}$$ The system of coupled harmonic oscillators (\[mainEqForQ\]) and its generalizations is a classical object in mathematical physics. The existence of solution and its ergodic properties were studied in [@LanfordLebowitz]. There has been an extensive research of convergence to equilibrium for infinite harmonic chain coupled with a heat bath [@Bogolyubov; @DudKomechSpohn; @SpohnLebowitz; @BPT]. The property of uniform boundedness (by time $t$ and index $k)$ is crucial in some applications. For instance, uniform boundedness in finite harmonic chain allows to derive Euler equation and Chaplygin gas without any stochastics (see [@LM1]). Uniform boundedness of a one-side non-symmetrical harmonic chain play important role in some traffic flow models [@LMM]. We should note some physical papers [@Hemmen; @Fox; @FlorencioLee]. The most closely related works to ours are [@Dud1; @Dud2], where the author studied weighted $l_{2}$ norms of infinite harmonic chains, whereas our main interest is a max-norm.
The paper is organized as follows. Section 2 contains definitions and formulation of the main results, remainder sections contain detailed proofs.
Model and results
=================
If $q(0),p(0)\in l_{2}(\mathbb{Z})$ then there exists unique solution $q(t),p(t)$ of (\[mainEqForQ\]) which belongs to $l_{2}(\mathbb{Z})$, i.e. $q(t),p(t)\in l_{2}(\mathbb{Z})$ for all $t\geqslant0$.
This assertion is well known (see [@LanfordLebowitz; @DalKrein; @Deimling]), and easily follows from the boundedness of the operator $W$ on $l_{2}(\mathbb{Z})$: $$(Wq)_{k}=-\omega_{0}^{2}q_{k}+\omega_{1}^{2}(q_{k+1}-q_{k})-\omega_{1}^{2}(q_{k}-q_{k-1}).$$
Uniform boundedness
-------------------
The first question of our interest is uniform boundedness (in $k$ and time $t\geqslant0$) of $|q_{k}(t)|$. Define the max-norm of $q_{k}(t)$: $$M(t)=\sup_{k}|q_{k}(t)|.$$
We shall say that the system has the property of uniform boundedness if: $$\sup_{t\geqslant0}M(t)<\infty.$$
\[uniBoundTh\] The following assertions hold:
1. If $\omega_{0}>0$, then: $$\sup_{t\geqslant0}M(t)<\infty.$$
2. If $\omega_{0}=0$ then we have the following results.
1. For all $t\geqslant0$ the following inequality holds: $$M(t)\leqslant\frac{2}{\sqrt{\omega_{1}}}||p(0)||_{2}\sqrt{t}+||q(0)||_{2} . \label{MtInEq}$$
2. Suppose that $$\sum_{k\ne0}|p_{k}(0)|\ln|k|<\infty.\label{omzeroPcond}$$ Then there is a constant $c>0$ such that for all $t\geqslant1$: $$M(t)\leqslant\frac{\sqrt{2}}{\omega_{1}\pi}|P|\ln(t)+||q(0)||_{2}+c,\quad P=\sum_{k}p_{k}(0).$$
3. For all $\delta>1/2$ there exists at least one initial condition $q(0)=0,p(0)\in l_{2}(\mathbb{Z})$ such that $$\lim_{t\rightarrow\infty}\frac{q_{0}(t)}{\sqrt{t}}\ln^{\delta}t=\frac{\Gamma(\delta)}{\sqrt{2\omega_1}}>0$$ where $\Gamma$ is the gamma function.
From the case 2.a we see that if $\omega_{0}=0$ and initial velocities of the particles are all zero, then $|q_{k}(t)|$ are uniformly bounded.
The assertions 2.c is an attempt to answer the question on the accuracy in the basic inequality (\[MtInEq\]) from 2.a with respect to the rate of growth in $t$.
Asymptotic behavior
-------------------
Next we will formulate theorems concerning asymptotic behavior of $q_{k}(t)$ in several cases.
Define Fourier transform of the sequence $u=\{u_{k}\}\in l_{2}(\mathbb{Z})$: $$\widehat{u}(\lambda)=\sum_{k}u_{k}e^{ik\lambda},\ \lambda\in\mathbb{R}.$$ Note that $\widehat{u}(\cdot)\in L_{2}([0,2\pi])$, i.e. $$\int_{0}^{2\pi}|\widehat{u}(\lambda)|^{2}\ d\lambda=2\pi\sum_{k}|u_{k}|^{2}<\infty.$$
Further on we will use the Fourier transform of the initial conditions: $$Q(\lambda)=\widehat{q(0)}(\lambda),\quad P(\lambda)=\widehat{p(0)}(\lambda).$$
For complex valued functions $f,g$ on $\mathbb{R}$ and constant $c\in\mathbb{C}$ we will write $f(x)\asymp c+ g(x) / \sqrt{x}$, if $f(x)=c+g(x) / \sqrt{x} +\bar{\bar{o}}(1 / \sqrt{x})$ as $x\rightarrow\infty$.
\[asympbehgz\] Suppose that $\omega_{0}>0$ and $Q,P$ are of class $C^{n}(\mathbb{R})$ for some $n\geqslant2$. Then
1. For any fixed $t\geqslant0$ we have $q_{k}(t)=O(k^{-n})$.
2. For any fixed $k\in\mathbb{Z}$ and $t\rightarrow\infty$ we have the following asymptotic formula: $$\begin{aligned}
q_{k}(t)&\asymp\frac{1}{\sqrt{t}}\bigl(C_{1}\cos(\omega_{1}(t))+S_{1}\sin(\omega_{1}(t))\\
&\quad {} +(-1)^{k}C_{2}\cos(\omega_{2}(t))+(-1)^{k}S_{2}\sin(\omega_{2}(t))),\end{aligned}$$ where $$C_{1}=\frac{1}{\omega_{1}}\sqrt{\frac{\omega_{0}}{2\pi}}Q(0),\quad S_{1}=\frac{1}{\omega_{1}\omega_{0}}\sqrt{\frac{\omega_{0}}{2\pi}}P(0)$$ $$C_{2}=\frac{1}{\omega_{1}}\sqrt{\frac{\omega'_{0}}{2\pi}}Q(\pi),\quad S_{2}=\frac{1}{\omega_{1}\omega'_{0}}\sqrt{\frac{\omega'_{0}}{2\pi}}P(\pi),$$ $$\omega_{1}(t)=t\omega_{0}+\frac{\pi}{4},\quad\omega_{2}(t)=t\omega'_{0}-\frac{\pi}{4},\quad\omega'_{0}=\sqrt{\omega_{0}^{2}+4\omega_{1}^{2}}.$$
3. Let $t=\beta|k|,\ \beta>0$ and $k\rightarrow\infty$. Put $$\gamma(\beta)=\beta^{2}\omega_{1}^{2}-1-\beta\omega_{0}.$$
1. If $\gamma(\beta)>0$ then $$q_{k}(t)\asymp\frac{1}{\sqrt{|k|}}\Bigl(\mathcal{F}_{k}^{+}[Q]-i\mathcal{F}_{k}^{-}\Bigl[\frac{P(\lambda)}{\omega(\lambda)}\Bigr]\Bigr)$$ where we introduce the following functionals for a complex valued function $g(\lambda)$ defined on the real line: $$\begin{aligned}
\mathcal{F}_{k}^{\pm}[g]&=c_{+}(g(\mu_{+})e^{i\omega_{+}(k)}\pm g(-\mu_{+})e^{-i\omega_{+}(k)})\\
&\quad {} +c_{-}(g(\mu_{-})e^{i\omega_{-}(k)}\pm g(-\mu_{-})e^{-i\omega_{-}(k)}),
\\
\omega_{\pm}(k)&=k(\mu_{\pm}+\beta\omega(\mu_{\pm}))\pm\frac{\pi}{4}\mathrm{sign}(k),\\
c_{\pm}&=\frac{1}{2}\sqrt{\frac{\beta\omega(\mu_{\pm})}{2\pi\Delta}},\\
\mu_{\pm}&=-\arccos\frac{1}{\beta^{2}\omega_{1}^{2}}(1\pm\Delta),\\
\Delta&=\sqrt{(\beta^{2}\omega_{1}^{2}-1)^{2}-\beta^{2}\omega_{0}^{2}},\\
\omega(\lambda)&=\sqrt{\omega_{0}^{2}+2\omega_{1}^{2}(1-\cos\lambda)}.\end{aligned}$$
2. If $\gamma(\beta)=0$ and $n\geqslant3$ then $q_{k}(t)=O(k^{-3})$.
3. if $\gamma(\beta)<0$ then $q_{k}(t)=O(k^{-n})$ for $n$ defined above.
Recall that a sufficient condition on $z\in l_{2}(\mathbb{Z})$ for $\widehat{z}\in C^{n}(\mathbb{R})$ is $$\sum_{k}|k|^{n}|z_{k}|<\infty.$$ Thus if the following series converge for some $n\geqslant2$: $$\sum_{k}|k|^{n}|q_{k}(0)|<\infty,\quad\mathrm{and}\quad\sum_{k}|k|^{n}|p_{k}(0)|<\infty,$$ then Theorem \[asympbehgz\] holds.
\[asympbehezero\] Suppose that $\omega_{0}=0$ and $Q,P\in C^{n}(\mathbb{R}),\ n\geqslant6$ then
1. For any fixed $t\geqslant0$ we have $q_{k}(t)=O(k^{-n})$.
2. For any fixed $k\in\mathbb{Z}$ and $t\rightarrow\infty$ one has: $$q_{k}(t)\asymp\frac{P(0)}{2\omega_{1}}+\frac{(-1)^{k}}{\sqrt{t}}\Bigl(C\cos\Bigl(2\omega_{1}t-\frac{\pi}{4}\Bigr)+S\sin\Bigl(2\omega_{1}t-\frac{\pi}{4}\Bigr)\Bigr),$$ where $$C=\frac{1}{\sqrt{\pi\omega_{1}}}Q(\pi),\quad S=\frac{1}{2\omega_{1}\sqrt{\pi\omega_{1}}}P(\pi).$$
Remarks
-------
If $\omega_{0}=0$ then it makes sense to consider the displacement variables: $$z_{k}(t)=x_{k+1}(t)-x_{k}(t)-a,\quad u_{k}(t)=\dot{z}_{k}=v_{k+1}(t)-v_{k}(t),\quad k\in\mathbb{Z}.$$ Suppose that $\{z_{k}(0)\}_{k\in\mathbb{Z}}\in l_{2}(\mathbb{Z})$ and $\{u_{k}(0)\}_{k\in\mathbb{Z}}\in l_{2}(\mathbb{Z})$. It is easy to see that $z_{k}(t)$ solves (\[mainEqForQ\]) with $\omega_{0}=0$. Consequently all formulated assertions for $q_{k}(t)$ in the case $\omega_{0}=0$ hold for variables $z_{k}(t)$. It is interesting to note that the quantity $P$ from item 2.b of theorem \[uniBoundTh\] in terms of variables $z,u$ equals $$P=\sum_{k}u_{k}(0)=\lim_{n\rightarrow\infty}(v_{n}(0)-v_{-n}(0)).$$ So if $$\sum_{k\ne0}|u_{k}|\ln|k|<\infty$$ and initial velocities of the “right” particles and “left” particles are equal, i.e. $\lim_{n\rightarrow\infty}(v_{n}(0)-v_{-n}(0))=0$, then from the case 2.b of theorem \[uniBoundTh\] follows uniform boundedness of displacements $z_{k}(t)$.
Proofs
======
Let us introduce the energy (hamiltonian): $$H=H(q,p)=\sum_{k}\frac{p_{k}^{2}}{2}+\frac{\omega_{0}^{2}}{2}\sum_{k}q_{k}^{2}+\frac{\omega_{1}^{2}}{2}\sum_{k}(q_{k}-q_{k-1})^{2}.$$ One can easily check that the energy is conserved under the dynamics (\[mainEqForQ\]). It means that $H(q(t),p(t))=H(q(0),p(0))$ for all $t\geqslant0$ where $q(t),p(t)$ solves (\[mainEqForQ\]). If $\omega_{0}>0$ then from the energy conservation law and the inequality: $$\sup_{k}|q_{k}(t)|\leqslant\frac{\sqrt{2H(q(t),p(t))}}{\omega_{0}}$$ the uniform boundedness of $|q_{k}(t)|$ follows, i.e. $$\sup_{t\geqslant0}\sup_{k\in\mathbb{Z}}|q_{k}(t)|<\infty$$ and so item (1) of Theorem \[uniBoundTh\] is proved.
Let us analyze the Fourier transform of the solution (\[mainEqForQ\]): $$\widehat{q(t)}(\lambda)=\sum_{k}q_{k}(t)e^{ik\lambda}.$$ The inverse transformation is given by the formula: $$q_{k}(t)=\frac{1}{2\pi}\int_{0}^{2\pi}\widehat{q(t)}(\lambda)e^{-ik\lambda}d\lambda,\ k\in\mathbb{Z}.\label{invFourier}$$
\[qQPformulaLemma\] The solution of (\[mainEqForQ\]) can be expressed as $$q_{k}(t)=Q_{k}(t)+P_{k}(t),\label{qQPformula}$$ where $$\begin{aligned}
Q_{k}(t)&=\frac{1}{2\pi}\int_{0}^{2\pi}Q(\lambda)\cos(t\omega(\lambda))e^{-ik\lambda}d\lambda,\\
P_{k}(t)&=\frac{1}{2\pi}\int_{0}^{2\pi}P(\lambda)\frac{\sin(t\omega(\lambda))}{\omega(\lambda)}e^{-ik\lambda}d\lambda,
\\
Q(\lambda)&=\widehat{q(0)}(\lambda),\quad P(\lambda)=\widehat{p(0)}(\lambda),\\
\omega(\lambda)&=\sqrt{\omega_{0}^{2}+2\omega_{1}^{2}(1-\cos(\lambda))}.\end{aligned}$$
Using (\[mainEqForQ\]) we obtain : $$\begin{aligned}
\frac{d^{2}}{dt^{2}}\widehat{q(t)}(\lambda)&=-\omega_{0}^{2}\widehat{q(t)}(\lambda)+\omega_{1}^{2}\sum_{k}q_{k+1}(t)e^{ik\lambda}+\omega_{1}^{2}\sum_{k}q_{k-1}(t)e^{ik\lambda}-2\omega_{1}^{2}\widehat{q(t)}(\lambda)\\
&=-\omega^{2}(\lambda)\widehat{q(t)}(\lambda).\end{aligned}$$ Thus $\widehat{q(t)}(\lambda)$ for fixed $\lambda$ is coordinate of the harmonic oscillator with the frequency $\omega(\lambda)$ and so the solution of equation for $\widehat{q(t)}(\lambda)$ is $$\widehat{q(t)}(\lambda)=\widehat{q(0)}(\lambda)\cos(t\omega(\lambda))+\widehat{p(0)}(\lambda)\frac{\sin(t\omega(\lambda))}{\omega(\lambda))}.$$ From this equality and the formula of the inverse transformation (\[invFourier\]) lemma follows.
Proof of Theorem \[uniBoundTh\]
-------------------------------
The case $\omega_{0}>0$ we have considered above. Now suppose that $\omega_{0}=0$. We will use the representation (\[qQPformula\]) and some upper bounds for $Q_{k},P_{k}$.
At first we will prove the part 2.a of Theorem \[uniBoundTh\]. From the CauchyBunyakovskySchwarz inequality we obtain the inequalities: $$\begin{aligned}
|Q_{k}(t)|&\leqslant\sqrt{\frac{1}{2\pi}\int_{0}^{2\pi}|Q(\lambda)|^{2}d\lambda}\sqrt{\frac{1}{2\pi}\int_{0}^{2\pi}\cos^{2}(t\omega(\lambda))d\lambda}\\
&\leqslant\sqrt{\frac{1}{2\pi}\int_{0}^{2\pi}|Q(\lambda)|^{2}d\lambda}=||q(0)||_{2} ,
\\
|P_{k}(t)|&\leqslant\sqrt{\frac{1}{2\pi}\int_{0}^{2\pi}|P(\lambda)|^{2}d\lambda}\sqrt{\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\sin^{2}(t\omega(\lambda))}{\omega^{2}(\lambda))}d\lambda}\\
&=||p(0)||_{2}\sqrt{\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\sin^{2}(t\omega(\lambda))}{\omega^{2}(\lambda))}d\lambda}.\end{aligned}$$ In the case $\omega_{0}=0$ one has $\omega(\lambda)=\sqrt{2\omega_{1}^{2}(1-\cos(\lambda))}=2\omega_{1}\sin(\lambda/2)$ and $$\begin{aligned}
I&=\int_{0}^{2\pi}\frac{\sin^{2}(t\omega(\lambda))}{\omega^{2}(\lambda)}d\lambda=2\int_{0}^{\pi}\frac{\sin^{2}(2\omega_{1}t\sin(u))}{(2\omega_{1}\sin(u))^{2}}du\\
&=4\int_{0}^{\pi / 2}\frac{\sin^{2}(2\omega_{1}t\sin(u))}{(2\omega_{1}\sin(u))^{2}}du.\end{aligned}$$ Substituting $x=\sin u$ in the last integral we get: $$I=\frac{1}{\omega_{1}^{2}}\int_{0}^{1}\frac{\sin^{2}(2\omega_{1}tx)}{x^{2}}\frac{1}{\sqrt{1-x^{2}}}dx
=\frac{1}{\omega_{1}^{2}}\biggl(\int_{0}^{1 / \sqrt{2}}\ldots dx+\int_{1 / \sqrt{2}}^{1}\ldots dx\biggr). \label{Irepresent}$$ The integrals in the latter formula will be estimated separately. $$\begin{aligned}
\int_{0}^{1 / \sqrt{2}} \frac{\sin^{2}(2\omega_{1}tx)}{x^{2}}\frac{1}{\sqrt{1-x^{2}}}dx&\leqslant\sqrt{2}\int_{0}^{1 / \sqrt{2}}
\frac{\sin^{2}(2\omega_{1}tx)}{x^{2}}dx\\
&=2\omega_{1}t\sqrt{2}\int_{0}^{\sqrt{2}\omega_{1}t}\frac{\sin^{2}(x)}{x^{2}}dx
\\
&\leqslant2\omega_{1}t\sqrt{2}\int_{0}^{\infty}\frac{\sin^{2}(x)}{x^{2}}dx\\
&=2\omega_{1}t\sqrt{2}\frac{\pi}{2}=\pi\omega_{1}t\sqrt{2}.\end{aligned}$$ One can find the value of the last integral in [@GR], p.713, 3.821 (9). For the second integral in (\[Irepresent\]) we have $$\begin{aligned}
\int\limits_{1/ \sqrt{2}}^{1}\frac{\sin^{2}(2\omega_{1}tx)}{x^{2}}\frac{1}{\sqrt{1-x^{2}}}dx&\leqslant
2\omega_{1}t\int\limits_{1 / \sqrt{2}}^{1}\frac{|\sin(2\omega_{1}tx)|}{x}\frac{1}{\sqrt{1-x^{2}}}dx\\
&\leqslant2\omega_{1}t\sqrt{2}\int\limits_{1/ \sqrt{2}}^{1}\frac{1}{\sqrt{1-x^{2}}}dx
=2\omega_{1}t\sqrt{2}\frac{\pi}{4}=\pi\omega_{1}t\frac{1}{\sqrt{2}}.\end{aligned}$$ Thus we obtain the inequality for $P_{k}$: $$|P_{k}(t)|\leqslant\sqrt{\frac{1}{2\pi\omega_{1}^{2}}\left(\pi\omega_{1}t\sqrt{2}+\pi\omega_{1}t\frac{1}{\sqrt{2}}\right)}||p(0)||_{2}\leqslant2\sqrt{\frac{t}{\omega_{1}}}||p(0)||_{2}.$$ This proves the case 2.a of Theorem \[uniBoundTh\].
Next we will check assertion 2.b. Condition (\[omzeroPcond\]) implies $p(0)\in l_{1}(\mathbb{Z})$ and consequently $P(\lambda)=\widehat{p(0)}(\lambda)$ is a bounded continuous function on $\mathbb{R}$. We have the following representation of $P_{k}(t)$: $$P_{k}(t)=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{P(\lambda)-P(0)}{\omega(\lambda)}\sin(t\omega(\lambda))e^{-ik\lambda}d\lambda+\frac{P(0)}{2\pi}\int_{0}^{2\pi}\frac{\sin(t\omega(\lambda))}{\omega(\lambda)}e^{-ik\lambda}d\lambda.\label{PkRepresF}$$ We estimate the first integral using condition (\[omzeroPcond\]): $$\begin{aligned}
I_{k}&=\biggl|\int_{0}^{2\pi}\frac{P(\lambda)-P(0)}{\omega(\lambda)}\sin(t\omega(\lambda))e^{-ik\lambda}d\lambda\biggr|\\
&\leqslant\int_{0}^{2\pi}\Bigl| \frac{P(\lambda)-P(0)}{\omega(\lambda)}\Bigr| d\lambda
\leqslant\sum_{j}|p_{j}(0)|\int_{0}^{2\pi}\Bigl| \frac{e^{ij\lambda}-1}{2\omega_{1}\sin( \lambda / 2)}\Bigr| d\lambda .\end{aligned}$$ On the other hand $$\begin{aligned}
\int_{0}^{2\pi}\Bigl| \frac{e^{ij\lambda}-1}{\sin(\lambda / 2)}\Bigr| d\lambda&=
\int_{0}^{2\pi}\Bigl| \frac{\sin(j\lambda / 2)}{\sin(\lambda / 2)}\Bigr| d\lambda=2\int_{0}^{\pi}\Bigl| \frac{\sin(ju)}{\sin u}\Bigr| d\lambda\\
&=4\int_{0}^{\pi / 2}\Bigl| \frac{\sin(ju)}{\sin u}\Bigr| d\lambda\leqslant^{(1)}
2\pi\int_{0}^{\pi / 2}\Bigl| \frac{\sin(ju)}{u}\Bigr| d\lambda\\
&\leqslant^{(2)}2\pi(\ln|j|+c).\end{aligned}$$ In the inequality $^{(1)}$ we have used the fact that $\sin x\geqslant (2 / \pi) x$ for $x\in[0; \pi / 2]$ and $^{(2)}$ follows from Lemma \[absSinIneq\] below. Thus we have obtained $$I_{k}\leqslant\frac{\pi}{\omega_{1}}\sum_{j\ne0}|p_{j}(0)|(\ln|j|+c)<\infty.$$ Further we will estimate the second integral in (\[PkRepresF\]): $$\begin{aligned}
J_{k}&=\biggl| \int_{0}^{2\pi}\frac{\sin(t\omega(\lambda))}{\omega(\lambda)}e^{-ik\lambda}d\lambda\biggr| \leqslant
2\int_{0}^{\pi}\frac{|\sin(2\omega_{1}t\sin u)|}{2\omega_{1}\sin u}du\\
&=\frac{2}{\omega_{1}}\int_{0}^{1}\frac{|\sin(2\omega_{1}tx)|}{x}\frac{1}{\sqrt{1-x^{2}}}dx
\\
&\leqslant\frac{2}{\omega_{1}}\biggl(\int_{0}^{1 / \sqrt{2}}\ldots dx+\int_{1 / \sqrt{2}}^{1}\ldots dx\biggr) \leqslant\frac{2}{\omega_{1}}(\sqrt{2}\ln t+c)\end{aligned}$$ for some constant $c>0$ not depending on $t\geqslant 1$ and $k$. In the latter inequality we again have applied lemma \[absSinIneq\].
Finally from (\[PkRepresF\]) and due to the bounds for $I_{k},J_{k}$ we obtain: $$|P_{k}(t)|\leqslant\frac{\sqrt{2}}{\omega_{1}\pi}|P(0)|\ln t+c$$ for some constant $c>0$ not depending on $t\geqslant 1$ and $k$. This completes the proof of the part 2.b.
\[absSinIneq\] For all $b>0$ and all $t_0>0$ there is a constant $c>0$ such that for any $t\geqslant t_0$ the following inequality holds: $$\int_{0}^{b}\frac{|\sin(tx)|}{x}dx\leqslant\ln t+c.$$
Substituting $tx=y$ we get: $$\begin{aligned}
\int_{0}^{b}\frac{|\sin(tx)|}{x}dx&=\int_{0}^{tb}\frac{|\sin y|}{y}dy=\int_{0}^{t_0 b}\frac{|\sin y|}{y}dy+\int_{t_0 b}^{tb}\frac{|\sin y|}{y}dy\\
&\leqslant c+\int_{t_0 b}^{tb}\frac{1}{y}dy=\ln -\ln t_0+c.\end{aligned}$$ This proves the assertion.
Finally we will prove part 2.c of Theorem \[uniBoundTh\]. We will construct the required initial condition in two steps. At first step for any $0<\alpha<1/2$ we will find initial conditions (depending on $\alpha$) such that the corresponding solution satisfies $\lim_{t\rightarrow\infty}q_{0}^{(\alpha)}(t)/t^{\alpha}>0$. At the next step we will integrate these initial conditions with an appropriate weight and prove that the resulting function gives us the answer.
Firstly we prove the assertion if $\omega_{1}=1/2$. Consider initial conditions $q^{(\alpha)}(0),p^{(\alpha)}(0)$ with the following Fourier transforms: $$Q^{(\alpha)}(\lambda)=0,\quad P^{(\alpha)}(\lambda)=\frac{a_{\alpha}}{(|\omega(\lambda)|)^{\alpha}}=\frac{a_{\alpha}}{|\sin \lambda / 2|^{\alpha}}$$ where $0<\alpha< 1/2$ and the constant $a_{\alpha}>0$ is chosen so that $$||P^{(\alpha)}(\lambda)||_{L_{2}([0,2\pi])}^{2}=\int_{0}^{2\pi}|P^{(\alpha)}(\lambda)|^{2}d\lambda=1.$$ Exact formula for $a_{\alpha}$ will be given below. It is obvious that $P^{(\alpha)}(\lambda) \! \in \! L_{2}([0,2\pi])$. So the corresponding initial conditions $$q_{k}^{(\alpha)}(0)=0,\quad p_{k}^{(\alpha)}(0)=\frac{1}{2\pi}\int_{0}^{2\pi}P^{(\alpha)}(\lambda)e^{-ik\lambda}d\lambda$$ lie in $l_{2}(\mathbb{Z})$. From (\[qQPformula\]) we have $$q_{0}(t)=q_{0}^{(\alpha)}(t)=P_{0}(t)=\frac{a_{\alpha}}{2\pi}\int_{0}^{2\pi}\frac{\sin(t\sin \lambda / 2)}{|\sin \lambda / 2|^{\alpha+1}}d\lambda.$$
\[qalphatlemma\] For all $t>0$ the following equality holds: $$q_{0}^{(\alpha)}(t)=\varphi(\alpha)t^{\alpha}+R(\alpha,t),\quad\varphi(\alpha)=2a_{\alpha}\frac{\Gamma(1-\alpha)}{\pi\alpha}\cos\frac{\pi\alpha}{2},$$ where for the remainder term $R$ we have $$|R(\alpha,t)|\leqslant a_{\alpha}\Bigl(3+\frac{2}{t}\Bigr),$$ where $\Gamma$ is the gamma function.
From the definition we have $$q_{0}^{(\alpha)}(t)=\frac{a_{\alpha}}{\pi}\int_{0}^{\pi}\frac{\sin(t\sin\lambda)}{\sin^{\alpha+1}\lambda}d\lambda=
\frac{a_{\alpha}}{\pi} \Bigl( I\Bigl[ 0,\frac{\pi}{4}\Bigr]+I\Bigl[\frac{\pi}{4},\frac{3\pi}{4}\Bigr]+I\Bigl[\frac{3\pi}{4},\pi\Bigr]\Bigr),$$ where $$I[a,b]=\int_{a}^{b}\frac{\sin(t\sin\lambda)}{\sin^{\alpha+1}\lambda}d\lambda.$$
The second integral can easily be estimated: $$\Big| I \Bigl[\frac{\pi}{4},\frac{3\pi}{4}\Bigr] \Big| \leqslant\int_{\pi / 4}^{3\pi/4}
\frac{d\lambda}{(\sin\lambda)^{\alpha+1}}\leqslant\frac{\pi}{2}(\sqrt{2})^{\alpha+1}\leqslant\pi2^{-1/4}<4.$$ The third integral $I[ 3\pi / 4,\pi]$ evidently equals to the first one $I[0, \pi / 4]$ (to see this it is sufficient to make the substitution $x=\pi-\lambda$). Let us evaluate the first integral. Substituting $x=\sin\lambda$ we have: $$\begin{aligned}
I\Bigl[0,\frac{\pi}{4}\Bigr]&=\int_{0}^{1/ \sqrt{2}} \frac{\sin(tx)}{x^{\alpha+1}\sqrt{1-x^{2}}}dx \label{Izeropifour} \\
&=\int_{0}^{1 / \sqrt{2}} \frac{\sin(tx)}{x^{\alpha+1}}dx+\int_{0}^{1 / \sqrt{2}} \Bigl(\frac{\sin(tx)}{x^{\alpha+1}\sqrt{1-x^{2}}}-\frac{\sin(tx)}{x^{\alpha+1}}\Bigr) dx. \nonumber
\end{aligned}$$ Due to the mean-value theorem for all $0\leqslant s\leqslant 1/2$ the following inequality holds: $$\Bigl| \frac{1}{\sqrt{1-s}}-1\Bigr| \leqslant s\max_{0\leq\theta\leq s}\frac{1}{2(\sqrt{1-\theta})^{3}}\leqslant s\frac{ 1} {2 \bigl(\sqrt{1- 1/2}\bigr)^{3}}=s\sqrt{2}.$$ Consequently for the second integral in (\[Izeropifour\]) we have the estimates: $$\begin{aligned}
\biggl| \int_{0}^{1 / \sqrt{2}} \Bigl(\frac{\sin(tx)}{x^{\alpha+1}\sqrt{1-x^{2}}}-\frac{\sin(tx)}{x^{\alpha+1}}\Bigr) dx\biggr| &
\leqslant\int_{0}^{1 / \sqrt{2}} \frac{|\sin(tx)|}{x^{\alpha+1}}\Bigl| \frac{1}{\sqrt{1-x^{2}}}-1\Bigr| dx \\
&\leqslant\int_{0}^{1 / \sqrt{2}} \frac{\sqrt{2}x^{2}}{x^{\alpha+1}}dx
\\
&=\sqrt{2}\frac{1}{2-\alpha}\frac{1}{(\sqrt{2})^{2-\alpha}}\leqslant\frac{1}{(\sqrt{2})^{1-\alpha}}<2.\end{aligned}$$ The first integral in (\[Izeropifour\]) can be expressed as: $$\int_{0}^{1 / \sqrt{2}}\frac{\sin(tx)}{x^{\alpha+1}}dx=t^{\alpha}\int_{0}^{t / \sqrt{2}}
\frac{\sin u}{u^{\alpha+1}}du=t^{\alpha}\int_{0}^{+\infty}\frac{\sin u}{u^{\alpha+1}}du-t^{\alpha}\int_{t / \sqrt{2}}^{+\infty}\frac{\sin u}{u^{\alpha+1}}du.$$ In the latter formula the first integral is Bohmer integral (generalized Fresnel integral) which value can be found in [@GR], p.648, 3.712: $$\begin{aligned}
\int_{0}^{\infty}\frac{\sin u}{u^{\alpha+1}}du&=\frac{1}{\alpha(1-\alpha)}\int_{0}^{\infty}\cos y^{1/(1-\alpha)}\,dy\\
&=\frac{1}{\alpha(1-\alpha)}\frac{\Gamma(1-\alpha)\sin(\frac{\pi}{2}(1-\alpha))}{1/(1-\alpha)}=\frac{\Gamma(1-\alpha)}{\alpha}\cos\frac{\pi\alpha}{2}.\end{aligned}$$ Integrating by parts we estimate the remainder term: $$\begin{aligned}
\biggl| \int_{t / \sqrt{2}}^{+\infty} \frac{\sin u}{u^{\alpha+1}}du\biggr| &=
\biggl| \frac{\cos(t / \sqrt{2})}{(t / \sqrt{2})^{\alpha+1}}
-(1+\alpha)\int_{t/ \sqrt{2}}^{\infty}\frac{\cos u}{u^{\alpha+2}}du\biggr| \\
&\leqslant\frac{2}{t^{\alpha+1}}+(1+\alpha)\int_{t / \sqrt{2}}^{\infty}\frac{1}{u^{\alpha+2}}du\leqslant\frac{4}{t^{\alpha+1}}.\end{aligned}$$ These inequalities complete the proof.
For any $0<\varepsilon<1/2$ define a weight function $$w_{\varepsilon}(\alpha)=\frac{1}{\varphi(\alpha)}\ \frac{1}{( 1/2-\alpha)^{1/2-\varepsilon}}$$ where $\varphi(\alpha)$ is defined in Lemma \[qalphatlemma\].
$w_{\varepsilon}(\alpha)$ is absolutely integrable w.r.t. $\alpha$ on $[0, 1/2]$: $$\int_{0}^{1/2}w_{\varepsilon}(\alpha)d\alpha<\infty.$$
Now we need the exact expression of $a_{\alpha}$: $$\begin{aligned}
\frac{1}{a_{\alpha}^{2}}&=\int_{0}^{2\pi}\frac{1}{(\sin\frac{\lambda}{2})^{2\alpha}}d\lambda=4\int_{0}^{\pi / 2}
\frac{1}{\sin^{2\alpha}x}dx\\
&=2\mathrm{B}\Bigl(\frac{1-2\alpha}{2},\frac{1}{2}\Bigr)=2\frac{\Gamma(\frac{1}{2}-\alpha)\Gamma(\frac{1}{2})}{\Gamma(1-\alpha)}\end{aligned}$$ where $B$ is the beta function ([@GR], p.610, 3.621). Hence $$a_{\alpha}=\sqrt{\frac{\Gamma(1-\alpha)}{2\sqrt{\pi}\Gamma(\frac{1}{2}-\alpha)}}.$$ It is well-known that $\Gamma(z)= 1/z+O(1)$ as $z\rightarrow0$. Thus $w_{\varepsilon}(\alpha)$ has the only one singular point on $[0, 1/2]$ at $\alpha=1/2$ and obviously $$w_{\varepsilon}(\alpha)\sim\frac{c}{(1/2-\alpha)^{1-\epsilon}}$$ as $\alpha\rightarrow 1/2$ for some constant $c$. So $w_{\varepsilon}(\alpha)$ is absolutely integrable on $[0, 1/2]$.
Finally, we will construct required initial condition by its Fourier transform which are defined by the following formulas: $$\tilde{Q}^{(\varepsilon)}(\lambda)=0,\quad\tilde{P}^{(\varepsilon)}(\lambda)=\int_{0}^{1/2}w_{\varepsilon}(\alpha)P^{(\alpha)}(\lambda)d\alpha.$$ The latter integral we understand in the following sense: $$\tilde{P}^{(\varepsilon)}(\lambda)=\lim_{\delta\rightarrow0+}\int_{0}^{1/2-\delta}w_{\varepsilon}(\alpha)P^{(\alpha)}(\lambda)d\alpha.\label{limdDef}$$ Since $L_{2}$-norm of $P^{(\alpha)}(\cdot)$ equals to one and $w_{\varepsilon}(\alpha)$ is absolutely integrable, the limit in (\[limdDef\]) exists and, moreover, one has the inequality: $$||\tilde{P}^{(\varepsilon)}(\cdot)||_{L_{2}([0,2\pi])}\leqslant\int_{0}^{1/2}w_{\varepsilon}(\alpha)d\alpha.$$ Thus the corresponding to $\tilde{Q}^{(\varepsilon)}(\lambda),\tilde{P}^{(\varepsilon)}(\lambda)$ initial conditions: $$\tilde{q}_{k}^{(\varepsilon)}(0)=0,\quad\tilde{p}_{k}^{(\varepsilon)}(0)=\frac{1}{2\pi}\int_{0}^{2\pi}\tilde{P}^{(\varepsilon)}(\lambda)e^{-ik\lambda}d\lambda\label{epsInitCond}$$ lie in $l_{2}(\mathbb{Z})$. Denote $\tilde{q}_{k}^{(\varepsilon)}(t),\tilde{p}_{k}^{(\varepsilon)}(t),\ k\in\mathbb{Z}$ the solution of (\[mainEqForQ\]) with initial condition (\[epsInitCond\]). Due to the lemmas \[qQPformulaLemma\] and \[qalphatlemma\] and FubiniTonelli theorem we have: $$\begin{aligned}
\tilde{q}_{0}^{(\varepsilon)}(t)&=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\sin(t\sin\frac{\lambda}{2})}{\sin\frac{\lambda}{2}}\tilde{P}^{(\varepsilon)}(\lambda)d\lambda\\
&=\int_{0}^{1/2}w_{\varepsilon}(\alpha)\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\sin(t\sin\frac{\lambda}{2})}{\sin\frac{\lambda}{2}}P^{(\alpha)}(\lambda)d\lambda d\alpha
\\
&=\int_{0}^{1/2} \! w_{\varepsilon}(\alpha)q_{0}^{(\alpha)}(t)d\alpha= \! \int_{0}^{1/2} \! \! \frac{1}{\varphi(\alpha)}\,
\frac{1}{(1/2-\alpha)^{1/2-\varepsilon}}\bigl(\varphi(\alpha)t^{\alpha} \! + \! R(\alpha,t)\bigr)d\alpha
\\
&=\int_{0}^{1/2}\frac{t^{\alpha}}{(1/2-\alpha)^{1/2-\varepsilon}}d\alpha+\int_{0}^{1/2}\frac{R(\alpha,t)}{\varphi(\alpha)}\
\frac{1}{(1/2-\alpha)^{1/2-\varepsilon}}d\alpha .\end{aligned}$$ The remainder term in the latter formula can be easily estimated: $$\biggl| \int_{0}^{1/2} \frac{R(\alpha,t)}{\varphi(\alpha)}\ \frac{1}{( 1/2 -\alpha)^{1/2-\varepsilon}}d\alpha\biggr| \leqslant c_{1}+\frac{c_{2}}{t}$$ for some nonnegative constants $c_{1},c_{2}$. Let us find the value of the first integral. Put $\delta=\varepsilon+1/2$: $$\begin{aligned}
\int_{0}^{1/2} \frac{t^{\alpha}}{(1/2-\alpha)^{1-\delta}}d\alpha&=\sqrt{t}\int_{0}^{1/2} t^{-u}u^{\delta-1}du=\sqrt{t}\int_{0}^{1/2}e^{-u\ln t}u^{\delta-1}du\\
&=\sqrt{t}(\ln t)^{-\delta}\int_{0}^{(\ln t)/2} e^{-y}y^{\delta-1}dy.\end{aligned}$$ Thus we have proved Theorem \[uniBoundTh\] item 2.c for the case $\omega_1 =\frac{1}{2}$.
Now suppose that $\omega_1$ is an arbitrary positive number. Consider solution with initial condition $\tilde{q}_{k}^{(\varepsilon)}(0), \tilde{p}_{k}^{(\varepsilon)}(0)$ defined in (\[epsInitCond\]). Denote it by $\tilde{q}_{k}^{\omega_1,(\varepsilon)}(t)$. From (\[qQPformula\]) it is easy to see that $$\tilde{q}_{k}^{\omega_1,(\varepsilon)}(t) = \frac{1}{2\omega_1}\tilde{q}_{k}^{1/2,(\varepsilon)}(2\omega_1 t) = \frac{1}{2\omega_1}\tilde{q}_{k}^{(\varepsilon)}(2\omega_1 t).$$ Hence we obtain limiting equalities: $$\lim_{t\rightarrow\infty}\frac{\tilde{q}_{k}^{\omega_1,(\varepsilon)}(t)}{\sqrt{t}}\ln^{\delta}t =
\frac{1}{2\omega_1} \lim_{t\rightarrow\infty}\frac{\tilde{q}_{k}^{(\varepsilon)}(2\omega_1 t)}{\sqrt{t}}\ln^{\delta}t = \frac{1}{\sqrt{2\omega_1}} \Gamma(\delta).$$ This completes the proof of Theorem \[uniBoundTh\].
Proof of Theorem \[asympbehgz\]
-------------------------------
We will use Lemma \[qQPformulaLemma\]. The first part of Theorem \[asympbehgz\] can be easily derived by integrating by parts $n$ times the following integral: $$\int_{0}^{2\pi}f(\lambda)e^{-ik\lambda}d\lambda$$ where $f(\lambda)$ is a corresponding $C^{n}$ smooth $2\pi$-periodic function.
Next we will need the following lemma.
\[egaymp\] Consider the integral: $$E[g](t)=\frac{1}{2\pi}\int_{0}^{2\pi}g(\lambda)\exp(it\omega(\lambda))d\lambda,\quad \omega(\lambda)=\sqrt{\omega_{0}^{2}+2\omega_{1}^{2}(1-\cos\lambda)}$$ where $g(\lambda)\in C^{n}(\mathbb{R}),\ n\geqslant2$ is a complex valued $2\pi$-periodic function. Then, as $t \rightarrow\infty,$ $$E[g](t)\asymp\frac{1}{\sqrt{t}}\left(c_{1}g(0)\exp\Bigl(i\Bigl(t\omega_{0}+\frac{\pi}{4}\Bigr)\Bigr)+c_{2}g(\pi)\exp\Bigl(i\Bigl(t\omega'_{0}-\frac{\pi}{4}\Bigr)
\Bigr) \right),$$ where $$c_{1}=\frac{1}{\omega_{1}}\sqrt{\frac{\omega_{0}}{2\pi}},\quad c_{2}=\frac{1}{\omega_{1}}\sqrt{\frac{\omega'_{0}}{2\pi}}$$ and $\omega_{0}'$ is defined in Theorem \[asympbehgz\].
We will apply stationary phase method (see [@Erdelyi; @Fedoruk]). Let us find critical points of the phase function, i.e. zeros of the $\frac{d}{d\lambda}\omega(\lambda)$: $$\omega'(\lambda)=\frac{d}{d\lambda}\omega(\lambda)=\frac{\omega_{1}^{2}\sin\lambda}{\omega(\lambda)}=0.$$ So the critical points lying on the interval $[0,2\pi]$ are only $0,\pi,2\pi$. Zero and $2\pi$ are the boundary points, but as $g(\lambda)$ is a $2\pi$-periodic function we can replace the interval $[0,2\pi]$ from the definition of $E[g](t)$ by the interval $[- \pi / 2, 3\pi / 2]$: $$E[g](t)=\frac{1}{2\pi}\int_{-\pi / 2}^{3\pi / 2}g(\lambda)\exp(it\omega(\lambda))d\lambda.$$ Thus the critical points of $\omega(\lambda)$ lying on $[-\pi / 2, 3\pi / 2]$ are only $0,\pi$ and we can apply stationary phase method for the internal critical points. Let us find the second derivative of the $\omega(\lambda)$: $$\omega''(\lambda)=\frac{\omega_{1}^{2}\cos\lambda}{\omega(\lambda)}-\frac{\omega_{1}^{4}\sin^{2}\lambda}{\omega^{3}(\lambda)}.$$ Consequently $\omega''(0)=\omega_{1}^{2} / \omega_{0},$ $\omega''(\pi)=\omega_{1}^{2} / \omega'_{0}$. To prove the lemma it remains to apply the formula (2) from [@Erdelyi], p.51 (or [@Fedoruk], p.163).
Further we will use the equality (\[qQPformula\]). For $Q_{k}$ we have: $$\begin{aligned}
Q_{k}(t)&=\frac{1}{2\pi}\int_{0}^{2\pi}Q(\lambda)\cos(t\omega(\lambda))e^{-ik\lambda}d\lambda=\frac{1}{2}(E[Qe^{-ik\lambda}](t)+\overline{E[\overline{Qe^{-ik\lambda}}](t)}),
\\
P_{k}(t)&=\frac{1}{2\pi}\int_{0}^{2\pi}P(\lambda)\frac{\sin(t\omega(\lambda))}{\omega(\lambda)}e^{-ik\lambda}d\lambda=\frac{1}{2i}\left(E[g](t)-\overline{E[\overline{g}](t)}\right),
\\
g&=\frac{Pe^{-ik\lambda}}{\omega(\lambda)}\end{aligned}$$ where $\overline{z}$ denotes the complex conjugate number of $z\in\mathbb{C}$. Applying Lemma \[egaymp\] to these expressions one can easily obtain the part 2 of Theorem \[asympbehgz\].
Let us prove part 3 of Theorem \[asympbehgz\]. We need the following lemma.
\[FgbetakL\] Consider the integral $$F[g](\beta,k)=\frac{1}{2\pi}\int_{0}^{2\pi}g(\lambda)e^{ik(\lambda+\beta\omega(\lambda))}d\lambda$$ where $g$ satisfies the conditions of Lemma \[egaymp\], $\beta>0,\ k\in\mathbb{Z}$. Define the constant: $$\gamma(\beta)=\beta^{2}\omega_{1}^{2}-1-\beta\omega_{0}.$$ The following assertions hold:
1. if $\gamma(\beta)>0$ then as $k\rightarrow\infty$: $$F[g](\beta,k)\asymp\frac{1}{\sqrt{|k|}}(c_{+}g(\mu_{+})e^{i\omega_{+}(k)}+c_{-}g(\mu_{-})e^{i\omega_{-}(k)})\label{fbetakasymp}$$ where $$\begin{aligned}
\omega_{\pm}(k)&=k(\mu_{\pm}+\beta\omega(\mu_{\pm}))\pm\frac{\pi}{4}\mathrm{sign}(k),\\
c_{\pm}&=\sqrt{\frac{\beta\omega(\mu_{\pm})}{2\pi\Delta}},\quad\mu_{\pm}=-\arccos\frac{1}{\beta^{2}\omega_{1}^{2}}(1\pm\Delta),
\\
\Delta&=\sqrt{(\beta^{2}\omega_{1}^{2}-1)^{2}-\beta^{2}\omega_{0}^{2}};\end{aligned}$$
2. if $\gamma(\beta)=0$ and $n\geqslant3$ then $F[g](\beta,k)=O(k^{-3})$;
3. if $\gamma(\beta)<0$ then $F[g](\beta,k)=O(k^{-n})$.
We will again use the stationary phase method. Consider the phase function: $$h(\lambda)=\lambda+\beta\omega(\lambda).$$ Let us find the critical points: $$h'(\lambda)=1+\beta\omega_{1}^{2}\frac{\sin\lambda}{\sqrt{\omega_{0}^{2}+2\omega_{1}^{2}(1-\cos\lambda)}}=0.\label{hdereq}$$ From this equation we have: $$\omega_{0}^{2}+2\omega_{1}^{2}(1-\cos\lambda)=\beta^{2}\omega_{1}^{4}(1-\cos^{2}\lambda).$$ Making the substitution $x=\cos\lambda$ rewrite the last equation: $$\beta^{2}\omega_{1}^{4}x^{2}-2\omega_{1}^{2}x+(\omega_{0}^{2}+2\omega_{1}^{2}-\beta^{2}\omega_{1}^{4})=0.\label{criticalPointsEqX}$$ The discriminant of this equation is: $$\begin{aligned}
D&=4\omega_{1}^{4}(1-\beta^{2}(\omega_{0}^{2}+2\omega_{1}^{2}-\beta^{2}\omega_{1}^{4}))
=4\omega_{1}^{4}\bigl((\beta^{2}\omega_{1}^{2}-1)^{2}-\beta^{2}\omega_{0}^{2}\bigr)\\
&=4\omega_{1}^{4}(\beta^{2}\omega_{1}^{2}-1-\beta\omega_{0})(\beta^{2}\omega_{1}^{2}-1+\beta\omega_{0}).\end{aligned}$$
Suppose that $\gamma(\beta)\geqslant0$. In that case the roots of the equation (\[criticalPointsEqX\]) are: $$x_{\pm}=\frac{1}{\beta^{2}\omega_{1}^{2}}\Bigl(1\pm\sqrt{(\beta^{2}\omega_{1}^{2}-1)^{2}-\beta^{2}\omega_{0}^{2}}\,\Bigr)
=\frac{1}{\beta^{2}\omega_{1}^{2}}(1\pm\Delta).$$ From the condition $\gamma(\beta)\geqslant0$ it follows that $\beta^{2}\omega_{1}^{2}-1>0$ and thus: $$|x_{\pm}|<\frac{1}{\beta^{2}\omega_{1}^{2}}(1+|\beta^{2}\omega_{1}^{2}-1|)=1.$$ Consequently the phase function has only two critical points $\lambda_{\pm}$ on $[0,2\pi]$: $\cos(\lambda_{\pm})=x_{\pm}$ with the additional condition $\sin(\lambda_{\pm})<0$ which follows from (\[hdereq\]): $$\lambda_{\pm}=2\pi-\arccos\frac{1}{\beta^{2}\omega_{1}^{2}}(1\pm\Delta)=2\pi+\mu_{\pm}.$$ Obviously $\lambda_{+}\ne\lambda_{-}$ iff $\gamma(\beta)>0$. Moreover, $\lambda_{\pm}$ are internal points, i.e. $\lambda_{\pm}\in(0,2\pi)$.
To apply the stationary phase method we should find the signs of $h''(\lambda_{\pm})$. Rewrite the derivative of the phase function: $$h'(\lambda)=1+\beta\omega_{1}^{2}\frac{\sin\lambda}{\omega(\lambda)}.$$ So at $\lambda_{\pm}$ we have: $$\omega(\lambda_{\pm})=-\beta\omega_{1}^{2}\sin\lambda_{\pm}.$$ Further we obtain $$h''(\lambda)=\beta\omega_{1}^{2}\frac{\cos\lambda}{\omega(\lambda)}-\beta\omega_{1}^{4}\frac{\sin^{2}\lambda}{\omega^{3}(\lambda)}.$$ Thus $$\begin{aligned}
h''(\lambda_{\pm})&=-\frac{\cos\lambda_{\pm}}{\sin\lambda_{\pm}}+\frac{1}{\beta\omega_{1}^{2}}\frac{1}{\sin\lambda_{\pm}}=-\frac{1}{\sin\lambda_{\pm}}\left(\cos\lambda_{\pm}-\frac{1}{\beta^{2}\omega_{1}^{2}}\right)
\\
&=\pm\Bigl(-\frac{1}{\sin\lambda_{\pm}}\Bigr)\frac{1}{\beta^{2}\omega_{1}^{2}}\sqrt{(\beta^{2}\omega_{1}^{2}-1)^{2}-\beta^{2}\omega_{0}^{2}}=\pm\frac{1}{\beta\omega(\lambda_{\pm})}\Delta.\end{aligned}$$ Since $\omega(\lambda)>0$ we have $\mathrm{sign}(h''(\lambda_{\pm}))=\pm1$ under the condition $\gamma(\beta)>0$. Hence applying (2) from [@Erdelyi], p.51 we get: $$F[g](\beta,k)\asymp\frac{1}{\sqrt{|k|}}\bigl(c_{+}g(\lambda_{+})e^{i\tilde{\omega}_{+}(k)}+c_{-}g(\lambda_{-})e^{i\tilde{\omega}_{-}(k)}\bigr)$$ where $c_{\pm}$ are defined in (\[fbetakasymp\]) (we have used the fact that $\omega(\lambda)=\omega(-\lambda)$). Note that: $$\begin{aligned}
\tilde{\omega}_{+}(k)&=k(\lambda_{\pm}+\beta\omega(\lambda_{\pm}))\pm\frac{\pi}{4}\mathrm{sign}(k)=2\pi k+\omega_{\pm}(k),
\\
g(\lambda_{\pm})&=g(\mu_{\pm})\end{aligned}$$ where $\omega_{\pm}(k)$ is defined in (\[fbetakasymp\]). Since $k\in\mathbb{Z}$ we have $e^{i\tilde{\omega}_{\pm}(k)}=e^{i\omega_{\pm}(k)}$. This completes the prove in the case $\gamma(\beta)>0$.
If $\gamma(\beta)=0$ then $\lambda_{+}=\lambda_{-}$ and $h''(\lambda_{+})=0$, i.e. $\lambda_{+}$ is a degenerate critical point and so $F[g](\beta,k)=O(k^{-3})$ due to [@Erdelyi], p.52 (or [@Fedoruk], p.163).
Consider the case $\gamma(\beta)<0$. We will prove that $h(\lambda)$ has no critical points on $[0,2\pi]$. Rewrite the discriminant using $\gamma$: $$D=4\omega_{1}^{4}\gamma(\gamma+2\beta\omega_{0}).$$ If $\gamma<0$ and $\gamma+2\beta\omega_{0}>0$ then $D<0$ and the equation (\[criticalPointsEqX\]) has no real roots. Suppose that $\gamma+2\beta\omega_{0}\leqslant0$. In that case $$\Delta^{2}=\frac{D}{4\omega_{1}^{4}}=((\gamma+2\beta\omega_{0})^{2}-2\beta\omega_{0}(\gamma+2\beta\omega_{0}))\geqslant(\gamma+2\beta\omega_{0})^{2}.$$ For the roots $x_{\pm}$ we have the following estimates: $$\begin{aligned}
|x_{\pm}| &\geqslant\frac{1}{\beta^{2}\omega_{1}^{2}}|1-\Delta|\geqslant\frac{1}{\beta^{2}\omega_{1}^{2}}|1-|\gamma+2\beta\omega_{0}||=\frac{1}{\beta^{2}\omega_{1}^{2}}|1+\gamma+2\beta\omega_{0}|\\
&=\frac{1}{\beta^{2}\omega_{1}^{2}}(\beta^{2}\omega_{1}^{2}+\beta\omega_{0})>1.\end{aligned}$$ Thus we have proved that $h(\lambda)$ has no critical points on $[0,2\pi]$. Consequently, $F[g](\beta,k)=O(k^{-n})$. This completes the proof.
Let us prove the remainder part of Theorem \[asympbehgz\]. From Lemma \[qQPformulaLemma\] we have $$\begin{aligned}
Q_{k}(t)&=\frac{1}{2\pi}\int_{0}^{2\pi}Q(\lambda)\cos(t\omega(\lambda))e^{-ik\lambda}d\lambda\\
&=\frac{1}{4\pi}\int_{0}^{2\pi}Q(\lambda)e^{it\omega(\lambda)-ik\lambda}d\lambda+\frac{1}{4\pi}\int_{0}^{2\pi}Q(\lambda)e^{-it\omega(\lambda)-ik\lambda}d\lambda.\end{aligned}$$ Put $t=\beta k$ and rewrite $Q_{k}(t)$: $$\begin{aligned}
Q_{k}(t)&=\frac{1}{4\pi}\int_{0}^{2\pi}Q(\lambda)e^{it\omega(\lambda)-ik\lambda}d\lambda+\frac{1}{2}F[Q](\beta,-k)\\
&=\frac{1}{2}(F[Q^{*}](\beta,k)+F[Q](\beta,-k)),\end{aligned}$$ where we have used the following notation: $$f^{*}(\lambda)=f(2\pi-\lambda)=f(-\lambda).$$ Note that $$\omega_{\pm}(-k)=-\omega_{\pm}(k)$$ for all $k\in\mathbb{Z}$. So in the case $\gamma(\beta)>0$, as $k\rightarrow\infty,$ $t=\beta k,$ $\beta>0$, due to Lemma \[FgbetakL\] we get: $$\begin{aligned}
Q_{k}(t)&\asymp\frac{1}{\sqrt{|k|}}\bigl(c_{+}(Q(\mu_{+})e^{i\omega_{+}(k)}+Q(-\mu_{+})e^{-i\omega_{+}(k)})\\
&\quad {} +c_{-}(Q(\mu_{-})e^{i\omega_{-}(k)}+Q(-\mu_{-})e^{-i\omega_{-}(k)})\bigr)
\\
&=\frac{1}{\sqrt{|k|}}\mathcal{F}_{k}^{+}[Q],\end{aligned}$$ where $\mathcal{F}_{k}^{\pm}$ are defined in Theorem \[asympbehgz\]. Similarly one can obtain the expression for $P_{k}(t)$: $$P_{k}(t)=\frac{1}{2i}\bigl(F[g^{*}](\beta,k)-F[g](\beta,-k)\bigr),\quad g=\frac{P(\lambda)}{\omega(\lambda)}.$$ If $\gamma(\beta)>0$ it is easy to see from the latter formula that $$P_{k}(t)\asymp-\frac{i}{\sqrt{|k|}}\mathcal{F}^{-}[g].$$ This completes the proof of Theorem \[asympbehgz\].
Proof of Theorem \[asympbehezero\]
----------------------------------
We will use Lemma \[qQPformulaLemma\]. The fact that $Q_{k}(t)=O(k^{-n})$ as $t$ is fixed easily follows from the integrating by parts the corresponding integral $n$ times. Let us consider the term $P_{k}(t)$: $$P_{k}(t)=\frac{1}{2\pi}\int_{0}^{2\pi}P(\lambda)\frac{\sin(t\omega(\lambda))}{\omega(\lambda)}e^{-ik\lambda}d\lambda.$$ In the case $\omega_{0}=0$ we have $$\omega(\lambda)=2\omega_{1}\sin\frac{\lambda}{2}.$$ Thus the integrand contains two points where the denominator equals to zero. Since $$\frac{\sin(t\omega(\lambda))}{\omega(\lambda)}=\sum_{k=0}^{\infty}\frac{t^{2k+1}\omega^{2k}(\lambda)}{(2k+1)!},$$ $[\sin(t\omega(\lambda))] / \omega(\lambda)$ is $C^{\infty}(\mathbb{R})$ smooth function w.r.t. $\lambda$ with period $2\pi$. Thus $P_{k}(t)=O(k^{-n})$ as $t$ is fixed. Hence the part 1 of Theorem \[asympbehezero\] is proved.
Next we will prove the remainder part of Theorem \[asympbehezero\]. We need the following lemma.
For all fixed $k\in\mathbb{Z}$ the folowing limit holds $$\lim_{t\rightarrow\infty}P_{k}(t)=\frac{P(0)}{2\omega_{1}}. \label{Pklim}$$
We need another expression for $P_{k}$: $$\begin{aligned}
P_{k}(t)&=\frac{1}{2\pi}\int_{0}^{2\pi}P(\lambda)\frac{\sin(t\omega(\lambda))}{\omega(\lambda)}e^{-ik\lambda}d\lambda
\\
&=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{P(\lambda)-P(0)}{\omega(\lambda)}\sin(t\omega(\lambda))e^{-ik\lambda}d\lambda\\
&\quad {} +P(0)\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\sin(t\omega(\lambda))}{\omega(\lambda)}e^{-ik\lambda}d\lambda
\\
&=\tilde{P}_{k}(t)+P(0)I_{k}(t),\end{aligned}$$ where $$\begin{aligned}
\tilde{P}_{k}(t)&=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{P(\lambda)-P(0)}{\omega(\lambda)}\sin(t\omega(\lambda))e^{-ik\lambda}d\lambda,\\
I_{k}(t)&=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\sin(t\omega(\lambda))}{\omega(\lambda)}e^{-ik\lambda}d\lambda .\end{aligned}$$ Since the integrand $[P(\lambda)-P(0)] / \omega(\lambda)$ in $\tilde{P}_{k}(t)$ is absolutely integrable, we have due to RiemannLebesgue theorem $$\lim_{t\rightarrow\infty}\tilde{P}_{k}(t)=0.$$ Using lemma \[besselExpression\] we obtain (\[Pklim\]).
\[besselExpression\] The following equalities hold: $$I_{k}(t)=\int_{0}^{t}J_{2k}(2\omega_{1}s)ds,\quad\lim_{t\rightarrow\infty}I_{k}(t)=\frac{1}{2\omega_{1}}\label{besselEq}$$ where $J_{k}(t)$ is the Bessel function of the first kind.
From the definition we have: $$I_{k}(t)=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\sin(t\omega(\lambda))}{\omega(\lambda)}e^{-ik\lambda}d\lambda=\frac{1}{\pi}\int_{0}^{\pi}\frac{\sin(2\omega_{1}t\sin\lambda)}{2\omega_{1}\sin\lambda}e^{-2ik\lambda}d\lambda.$$ Thus $$\begin{aligned}
\frac{d}{dt}I_{k}(t)&=\frac{1}{\pi}\int_{0}^{\pi}\cos(2\omega_{1}t\sin\lambda)e^{-2ik\lambda}d\lambda
\\
&=\frac{1}{\pi}\int_{0}^{\pi}\cos(2\omega_{1}t\sin\lambda)\cos(2k\lambda)d\lambda \\
&\quad {} -\frac{i}{\pi}\int_{0}^{\pi}\cos(2\omega_{1}t\sin\lambda)\sin(2k\lambda)d\lambda .\end{aligned}$$ The second term equals zero. To see this one should make the substitution $x=\pi-\lambda$. Whence $$\begin{aligned}
\frac{d}{dt}I_{k}(t)&=\frac{1}{2\pi}\int_{0}^{\pi}\cos(2\omega_{1}t\sin\lambda+2k\lambda)d\lambda\\
&\quad {} +\frac{1}{2\pi}\int_{0}^{\pi}\cos(2\omega_{1}t\sin\lambda-2k\lambda)d\lambda
\\
&=\frac{1}{2}(J_{-2k}(2\omega_{1}t)+J_{2k}(2\omega_{1}t))=J_{2k}(2\omega_{1}t).\end{aligned}$$ It proves the first equality in (\[besselEq\]). The second one follows from [@GR], p.1036, 6.511 (1).
\[ozeroCg\] Consider the integral $$C[g](t)=\frac{1}{2\pi}\int_{0}^{2\pi}g(\lambda)\cos(t\omega(\lambda))d\lambda,$$ where $g(\lambda)\in C^{n}(\mathbb{R}),\ n\geqslant6$ is a complex valued $2\pi$-periodic function. Then $$C[g](t)=\frac{1}{\sqrt{t}}\frac{g(\pi)}{\sqrt{\pi\omega_{1}}}\cos\Bigl(2\omega_{1}t-\frac{\pi}{4}\Bigr)+b\frac{\cos(2\omega_{1}t- \pi / 4)}{t\sqrt{t}}+O(t^{-2}),$$ for some complex constant $b\in\mathbb{C}$.
We apply the stationary phase method. The phase function $\omega(\lambda)=2\omega_{1}\sin(\lambda / 2)$ has the only one critical point $\lambda_{0}=\pi$ on the interval $[0,2\pi]$. The contribution to the asymptotics of $C[g](t)$ as $t\rightarrow\infty$ is determined by the boundary points $0,2\pi$ and the critical point $\pi$. The contribution from the boundary points has the order of $t^{-2}$. Indeed the leading term of the asymptotic expansion corresponding to the point $0$ is: $$-\frac{1}{2}\Bigl(\frac{g(0)e^{i\omega(0)}}{it\omega'(0)}+\frac{g(0)e^{-i\omega(0)}}{-it\omega'(0)}\Bigr)=0.$$ Analogously one has for the point $2\pi$. Hence the contribution to the asymptotics of $C[g](t)$ up to the order $t^{-2}$ determined by the stationary point. From [@Erdelyi] we have the formula $$C[g](t)=\frac{1}{2\pi}\frac{\cos(2\omega_{1}t- \pi / 4)}{\sqrt{t}}\Bigl(g(\pi)\sqrt{\frac{4\pi}{\omega_{1}}}+b\frac{1}{t}+O(t^{-2})\Bigr)$$ for some constant $b$. This completes the proof of Lemma \[ozeroCg\].
Let us continue the proof of Theorem \[asympbehezero\]. From Lemmas \[qQPformulaLemma\] and \[ozeroCg\] we have: $$Q_{k}(t)\asymp\frac{1}{\sqrt{t}}\frac{(-1)^{k}Q(\pi)}{\sqrt{\pi\omega_{1}}}\cos\Bigl(2\omega_{1}t-\frac{\pi}{4}\Bigr)\ \mathrm{as}\ t\rightarrow\infty.$$ Using (\[Pklim\]) we obtain $$\int_{t}^{+\infty}\frac{d}{ds}P_{k}(s)ds=\lim_{T\rightarrow\infty}\int_{t}^{T}\frac{d}{ds}P_{k}(s)ds=\lim_{T\rightarrow\infty}P_{k}(T)-P_{k}(t)=\frac{P(0)}{2\omega_{1}}-P_{k}(t).$$ Whence due to Lemma \[ozeroCg\] we get $$\begin{aligned}
P_{k}(t)&=\frac{P(0)}{2\omega_{1}}-\int_{t}^{+\infty}\frac{d}{ds}P_{k}(s)ds=\frac{P(0)}{2\omega_{1}}-\int_{t}^{+\infty}C[P(\lambda)e^{-ik\lambda}](s)ds
\nonumber \\
&=\frac{P(0)}{2\omega_{1}}-\int\limits_{t}^{+\infty}\Bigl(\frac{1}{\sqrt{s}}\frac{(-1)^{k}P(\pi)}{\sqrt{\pi\omega_{1}}}
\cos\Bigl(2\omega_{1}s-\frac{\pi}{4}\Bigr)+b\frac{\cos(2\omega_{1}s- \pi / 4)}{s\sqrt{s}}\Bigr)ds \nonumber \\
&\qquad {} +O(t^{-1}) . \label{PkexpressionINt}\end{aligned}$$
Applying the second mean-value theorem we have that for all $T\geqslant t$ there is a $\tau\in[t,T]$ such that $$\int_{t}^{T} \! \frac{\cos(2\omega_{1}s \! - \! \pi / 4)}{s\sqrt{s}}ds=\frac{1}{t\sqrt{t}}\int_{t}^{\tau} \! \cos\Bigl(2\omega_{1}s-\frac{\pi}{4}\Bigr)d\tau+\frac{1}{T\sqrt{T}}
\int_{\tau}^{T} \! \cos\Bigl(2\omega_{1}s-\frac{\pi}{4}\Bigr)ds.$$ It is clear that $$\sup_{-\infty<a<b<+\infty}\biggl| \int_{a}^{b}\cos\Bigl(2\omega_{1}s-\frac{\pi}{4}\Bigr)ds\biggr| =\frac{1}{\omega_{1}} .$$ Therefore $$\biggl| \int_{t}^{T}\frac{\cos(2\omega_{1}s- \pi / 4)}{s\sqrt{s}}ds\biggr| \leqslant\frac{2}{\omega_{1}}\frac{1}{t\sqrt{t}}.$$ So we obtain $$\biggl| \int_{t}^{+\infty}\frac{\cos\Bigl(2\omega_{1}s- \pi / 4 \Bigr)}{s\sqrt{s}}ds\biggr|=O(t^{-3/2}).\label{costhreehalfestim}$$ Integrating by parts we have $$\begin{aligned}
&\int_{t}^{+\infty}\frac{1}{\sqrt{s}}\cos\Bigl(2\omega_{1}s-\frac{\pi}{4}\Bigr)ds\\
&\quad =-\frac{1}{2\omega_{1}\sqrt{t}}\sin\Bigl(2\omega_{1}t-\frac{\pi}{4}\Bigr)+\frac{1}{4\omega_{1}}\int_{t}^{+\infty}\frac{1}{s\sqrt{s}}
\sin\Bigl(2\omega_{1}s-\frac{\pi}{4}\Bigr)ds=
\\
&\quad =-\frac{1}{2\omega_{1}\sqrt{t}}\sin\Bigl(2\omega_{1}t-\frac{\pi}{4}\Bigr)+O(t^{-3/2}).\end{aligned}$$ In the last equality we have used the same estimate for the remainder term as in (\[costhreehalfestim\]), which can be easily proved in the same way.
Summing up obtained estimates for (\[PkexpressionINt\]) we get $$P_{k}(t)=\frac{P(0)}{2\omega_{1}}+\frac{(-1)^{k}P(\pi)}{2\omega_{1}\sqrt{\pi\omega_{1}t}}\sin\Bigl(2\omega_{1}t-\frac{\pi}{4}\Bigr)+O(t^{-3/2}).$$ This completes the proof of Theorem \[asympbehezero\].
Acknowledgment
--------------
We would like to thank professor Vadim Malyshev for stimulating discussions and numerous remarks.
[10]{} (1945) . Ac. Sci. USSR, Kiev.
\(2017) From the $N$-body problem to Euler equations. (24), 79–95.
\(2017) Phase diagram for one-way traffic flow with local control. , 849–866.
\(2018) Behavior for large time of a two-component chain of harmonic oscillators. (25), 470–491.
\(2016) Long-time asymptotics of solutions to a hamiltonian system on a lattice. (1), 69–85.
\(2003) On the convergence to statistical equilibrium for harmonic crystals. (6), 2596–2620.
\(1975) Time evolution and ergodic properties of harmonic systems. In: [*Dynamical Systems, Theory and Applications*]{}, J. Moser (eds). Lect. Notes Phys., 144–177. Springer, Berlin, Heidelberg.
\(1977) Stationary non-equilibrium states of infinite harmonic systems. , 97–120.
\(1983) Convergence to stationary states for infinite harmonic systems. (1), 123–155.
\(1980) Dynamics and ergodicity of the infinite harmonic crystal. (2), 43–149.
\(1983) Long-time tails and diffusion. (6), 3216–3233.
\(1985) Exact time evolution of a classical harmonic-oscillator chain. (5), 3221–3236.
\(1974) . AMS. (1977) . Springer.
\(2007) Academic Press. (1956) . Dover Pub., New York.
\(1977) . Nauka, Moscow. (In Russian).
[^1]: Mechanics and Mathematics Faculty, Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991, Russia
|
---
abstract: 'We compare six models (including the baryonic model, two dark matter models, two modified Newtonian dynamics models and one modified gravity model) in accounting for the galaxy rotation curves. For the dark matter models, we assume NFW profile and core-modified profile for the dark halo, respectively. For the modified Newtonian dynamics models, we discuss Milgrom’s MOND theory with two different interpolation functions, i.e. the standard and the simple interpolation functions. As for the modified gravity, we focus on Moffat’s MSTG theory. We fit these models to the observed rotation curves of 9 high-surface brightness and 9 low-surface brightness galaxies. We apply the Bayesian Information Criterion and the Akaike Information Criterion to test the goodness-of-fit of each model. It is found that non of the six models can well fit all the galaxy rotation curves. Two galaxies can be best fitted by the baryonic model without involving the nonluminous dark matter. MOND can fit the largest number of galaxies, and only one galaxy can be best fitted by MSTG model. Core-modified model can well fit about one half LSB galaxies but no HSB galaxy, while NFW model can fit only a small fraction of HSB galaxies but no LSB galaxy. This may imply that the oversimplified NFW and Core-modified profiles couldn’t well mimic the postulated dark matter halo.'
author:
- Xin Li
- Li Tang
- 'Hai-Nan Lin'
title: 'Comparing the dark matter models, modified Newtonian dynamics and modified gravity in accounting for the galaxy rotation curves'
---
Introduction
============
It has long been found that rotation curves of spiral galaxies are significantly discrepant from the predictions of Newtonian theory [@Rubin:1978; @Rubin1980; @Bosma:1981; @Sofue:2001]. According to Newton’s law of gravitation, the gravitational force between two point-like particles is inversely proportional to the square of their separation. Therefore, the rotation velocity of a star far away from the galactic center is inversely proportional to the square root of the distance to the galactic center. However, the observations often show an asymptotically flat rotation curve out to the furthest data points [@Walter2008; @Blok:2008]. There are several ways to reconcile this contradiction. The most direct assumption is that there is a large amount of nonluminous matter (dark matter) that has not been detected yet [@Begeman1991; @Persic1996; @Chemin:2011mf]. In fact, the dark matter hypotheses was first proposed to solve the mass missing problem of the sidereal system [@Kapteyn:1922; @Oort:1932]. However, after decades of extensive research, no direct evidence for the existence of dark matter has been found on the particle physics level. This motivates us to search for other explanations of the discrepancy between the Newtonian dynamical mass and the luminous mass.
One possible way is to modify the Newtonian dynamics. In 1983 or so, M. Milgrom published a series of papers to modify the Newtonian dynamics in order to explain the flatness of galaxy rotation curves, which is well known today as the MOND theory [@Milgrom1983a; @Milgrom1983b; @Milgrom1986]. According to MOND, the Newton’s second law no longer holds if the acceleration is small enough. The true dynamics should be $\mu(a/a_0){\bm a}={\bm a}_N$, where ${\bm a}_N$ is the acceleration in Newtonian theory, ${\bm a}$ is the true acceleration, and $a_0$ is the critical acceleration bellow which the Newtonian theory does not hold. The interpolation function $\mu(x)$ is chosen such that $\mu(x)\rightarrow 1$ when $a\gg a_0$, so the Newton’s acceleration law is recovered. In the deep MOND region $a\ll a_0$, $\mu(x)\approx x$, such that the rotation curve keeps flat at large distance from the galactic center. MOND is a non-relativistic theory for a long time until the relativistic form is constructed by Bekenstein [@Bekenstein2004]. With only one universal parameter $a_0$, MOND has made great success in accounting for the rotation curves of spiral galaxies [@Sanders1996; @Sanders:1998gr; @Sander:2007; @Swaters2010; @Iocco:2015iia].
In addition to modify the Newtonian dynamics, it is also possible to modify the Newtonian gravity (MOG). According to MOG, the Newton’s law of gravitation is invalid at galactic scales. There are various MOG theories. Moffat [@Moffat:2005; @Moffat:2006] proposed the scalar-tensor-vector gravity (STVG) and metric-skew-tensor gravity (MSTG) models, in which the gravitational “constant" is no longer a constant, but is running with distance. Carmeli [@Carmeli1998; @Carmeli2000; @Carmeli2002] showed that the flatness of galaxy rotation curves can be naturally explained if the expansion of the universe is took into account, and argued that dark matter may be a intrinsic property of the spacetime. Horava [@Horava2009a; @Horava2009b; @Horava2009c] presented a candidate quantum field theory of gravity in ($3+1$) dimension spacetime, which is known as the Horava-Lifshitz theory. Grumiller [@Grumiller:2010bz] proposed an effective gravity whose potential contains a Rindler term in addition to the well known terms of general relativity. All of these theories can to a large degree reconcile the mass missing problem of galaxy rotation curves.
In this paper, we make a comprehensive comparison between different models in explaining the galaxy rotation curves. We choose 9 high-surface brightness (HSB) and 9 low-surface brightness (LSB) galaxies and fit the observed rotation curves to three different types of models, i.e. the dark matter, MOND and MOG models. To the dark matter models, we choose the NFW profile [@Navarro:1996; @Navarro:1997] and the core-modified profile [@Brownstein:2009zz] for the dark matter halo. To the MOND models, we study Milgrom’s MOND theory with two different interpolation functions. To the MOG models, we focus on the MSTG theory [@Moffat:2005]. We also compare these models to Newton’s theory without adding the nonluminous dark matter. Thus there are six models in total. The best model to each galaxy is picked out using statistical method.
The outline of this paper is arranged as follows: In Section \[sec:models\], we introduce the theoretical models of galaxy structures and rotation velocities. In Section \[sec:results\], we introduce the data of 9 HSB and 9 LSB galaxies that are used in our fitting. We first obtain their surface brightness parameters by fitting to the photometric data, then obtain the model parameters by fitting to the observed rotation curve data. In Section \[sec:comparison\], we make the model comparison, and use the Bayesian Information Criterion (BIC) and the Akaike Information Criterion (AIC) to pick out the best model. Finally, discussion and summary are given in Section \[sec:summary\].
theoretical models {#sec:models}
==================
Structure of galaxies {#sec:model_brightness}
---------------------
The brightness of galaxy is often assumed to be a direct tracer of its mass distribution. The brightness of a HSB galaxy can in general be decomposed into two components, a ellipsoidal bulge and a flat disk. The bulge is usually modeled by an inhomogeneous ellipsoid with 3D spatial brightness [@Tamm:2005] $$l(a)=l_{0}\exp\left[-\left(\frac{a}{ka_{0}}\right)^{1/N}\right],$$ where $l_{0}$ is the central density, $a_{0}$ is the harmonic mean radius of the bulge, $k$ is a normalization factor, $a=\sqrt{R^{2}+z^{2}/q^{2}}$ is the distance to the galactic center, $R$ and $z$ are the cylindrical coordinates, $q$ is the ratio of the minor axis to the major axis, and $N$ characterizes the shape of the profile. Integrating over $z$, we obtain the 2D surface brightness of the bulge, $$I_b(R)=2\int_R^{\infty}\frac{l(a)a}{\sqrt{a^2-R^2}}da.$$ The thickness of the disk is very small compared to the galaxy size. We assume that the disk is infinitely thin, and model its surface brightness by the exponential law [@Vaucouleurs1959; @Freeman1970], $$\label{eq:I_d}
I_{d}(R)=I_{0}\exp\left(-\frac{R}{h}\right),$$ where $I_{0}$ is the central surface brightness in unit of $M_{\odot}~{\rm pc}^{-2}$, and $h$ is the scale length of the disk. We can equivalently convert Eq. (\[eq:I\_d\]) to the logarithmic units using the relation $$\mu~[{\rm mag}/{\rm arcsec}^{2}]=\mathcal{M_{\odot}}+21.572-2.5\log_{10} I~[L_{\odot}/{\rm pc}^{2}],$$ and obtain $$\mu_d(R)=\mu_0+1.086\left(\frac{R}{h}\right),$$ where $\mu_{0}$ is the central surface brightness in unit of ${\rm mag}~{\rm arcsec}^{-2}$, $\mathcal{M_{\odot}}$ and $L_{\odot}$ are the absolute magnitude and luminosity of the sun in a specific color-band. The total surface brightness of the HSB galaxies is given by $I(R)=I_b(R)+I_d(R)$. The free parameters are obtained by fitting $I(R)$ to the observed photometric data. We assume that the mass-to-light ratios of both the bulge and disk are constants.
For LSB galaxies, the surface brightness can be well approximated by the exponential disk as HSB galaxies, while the bulge component is usually negligible [@Blok:1995]. However, the gas in LSB galaxies is much richer than in HSB galaxies, so it is necessary to be considered. The mass profile of gas can be read from the observational data directly using the Groningen Image Processing System (GIPSY) whose home page is at http://www.astro.rug.nl/$\sim$gipsy/.
Models of rotation velocity {#sec:model}
---------------------------
We first consider in the framework of Newtonian theory. The rotation velocity induced by the spheroidal bulge can be obtained by solving the Poisson equation, and lead to the result [@Tamm:2005] $$V^{2}_{b}(R)=4\pi\sigma qG\int^{R}_{0}\frac{l(a)a^{2}}{\sqrt{R^{2}-e^{2}a^{2}}}da,$$ where $G$ is Newton’s gravitational constant, $\sigma$ is the mass-to-light ratio of the bulge, and $e=\sqrt{1-q^{2}}$ is the eccentricity of the bulge. Similarly, the rotation velocity induced by the infinitely thin exponential disk is given by [@Freeman1970] $$\begin{aligned}
V^{2}_{d}(R) &=& \frac{GM}{2R}\left(\frac{R}{h}\right)^{3}\bigg{[}I_{0}\left(\frac{1}{2}\frac{R}{h}\right)K_{0}\left(\frac{1}{2}\frac{R}{h}\right) \nonumber\\[1mm]
&& -I_{1}\left(\frac{1}{2}\frac{R}{h}\right)K_{1}\left(\frac{1}{2}\frac{R}{h}\right)\bigg{]},\end{aligned}$$ where $M=2\pi \tau h^{2}I_{0}$ is in total mass of the disk, $\tau$ is the mass-to-light ratio of the disk, $I_n$ and $K_n$ are the $n$th order modified Bessel functions of the first and second kinds, respectively. The rotation velocity due to the neutral hydrogen (HI) can be calculated from the mass profile of HI directly using GIPSY. We assume that the mass ratio of helium (He) to HI is $1/3$, and ignore other gases. Therefore, the rotation velocity contributes from the gas is given by $$V^{2}_{\rm gas}=\frac{4}{3}V^{2}_{\rm HI}.$$ Then rotation velocity arising from the combined contributions of bulge, disk and gas, can be written as the squared sum of each component, i.e, $$V^{2}_{N}=V^{2}_{b}+V^{2}_{d}+V^{2}_{\rm gas},$$ For the HSB galaxies, the gas component is negligible, hence $V_{\rm gas}=0$. For the LSB galaxies, the bulge component is negligible, hence $V_b=0$.
There are several models for the dark mater halo, such as the NFW profile [@Navarro:1996; @Navarro:1997], the pseudo-isothermal profile [@Jimenez:2003], the Burkert profile [@Burkert:1995], the Einasto profile [@Merritt:2006], the core-modified profile [@Brownstein:2009zz], and so on. All of these profiles can be generalized to the $(\alpha,\beta,\gamma)$-models [@Hemquist1990; @Zhao:1996; @An:2013]. Here we focus on the NFW profile and the core-modified profile. The density of NFW profile takes the form $$\label{eq:NFW}
\rho_{\rm NFW}=\frac{\rho_s r^{3}_{s}}{r(r+r_{s})^{2}},$$ where $\rho_s$ and $r_{s}$ are the characteristic density and scale length, respectively. The mass of dark matter, which is acquired from the volumetric integration of Eq. (\[eq:NFW\]), contributes partly to the rotation curve, $$V_{\rm s}^{2}=4\pi G\rho_{s} \frac{r^{3}_{s}}{r}\left[\ln\left(1+\frac{r}{r_{s}}\right)-\frac{r}{r+r_{s}}\right].$$ Therefore, the rotation velocity in the NFW model is given by $$V^{2}_{\rm NFW}=V^{2}_{N}+V_{\rm s}^{2}.$$ NFW profile is often quantified by the viral radius $R_{\rm vir}$ and viral mass $M_{\rm vir}$ instead of $r_{s}$ and $\rho_{s}$ [@Navarro:1997; @Wu:2008]. The viral radius $R_{\rm vir}$ is the radius within which the mean density of dark matter is 200 times the critical density $\rho_{\rm cr}$, and the viral mass $M_{\rm vir}$ is the mass of dark matter within $R_{\rm vir}$. These quantities are related by $$\rho_{s}=\frac{200\rho_{\rm cr} R_{\rm vir}^{3}}{3r_{s}^{3}}\left[\ln\left(\frac{R_{\rm vir}+r_{s}}{r_{s}}\right)-\frac{R_{\rm vir}}{R_{\rm vir}+r_{s}}\right]^{-1},$$ $$M_{\rm vir}=200\rho_{\rm cr}\frac{4}{3}\pi R_{\rm vir}^{3},$$ where $\rho_{\rm cr}=3H_{0}^{2}/8\pi G$ is the critical density of the universe, and $H_{0}$=70 km s$^{-1}$ Mpc$^{-1}$ is the Hubble constant.
The NFW profile is singular at the galactic center. To avoid the singularity, Brownstein [@Brownstein:2009zz] proposed the so-called core-modified profile. The density of the core-modified profile takes the form $$\label{eq:core}
\rho_{\rm core}=\frac{\rho_{c}r^{3}_{c}}{r^{3}+r^{3}_{c}}.$$ The mass of dark matter within the sphere of radius $r$ is given by $$\label{eq:core mass}
M(r)=\frac{4}{3}\pi \rho_{c}r^{3}_{c}\left[\ln\left(r^{3}+r^{3}_{c}\right)-\ln\left(r^{3}_{c}\right)\right].$$ Thus, the corresponding rotation velocity is given by $$\label{eq:v}
V^{2}_{c}=\frac{4}{3}\pi G\rho_{c}\frac{r^{3}_{c}}{r}\left[\ln\left(r^{3}+r^{3}_{c}\right)-\ln\left(r^{3}_{c}\right)\right].$$ Therefore, the rotation velocity in the core-modified profile is given by $$\label{eq:v core}
V^{2}_{\rm core}=V^{2}_{N}+V^{2}_{c}.$$
According to MOND theory [@Milgrom1983a; @Milgrom1983b], the Newtonian dynamics is invalid when the acceleration is approaching or below the critical acceleration $a_0$. The effective acceleration is related to the Newtonian acceleration by $$\label{eq:g-gn-relation}
\mu(g/a_0)g=g_N,$$ where $g_N\equiv GM/r^2$ is the Newtonian acceleration, $a_0\approx 1.2\times 10^{-10}$ m s$^{-2}$ is the critical acceleration, and $\mu(x)$ is an interpolation function which has the asymptotic behaviors: $\mu(x)=x$ for $x\rightarrow 0$, and $\mu(x)=1$ for $x\rightarrow \infty$. We choose two interpolation functions, the first one is the standard interpolation function initially proposed by Milgrom [@Milgrom1983a] $$\label{eq:interp-function}
\mu_{1}(x)=\frac{x}{\sqrt{1+x^2}}.$$ Combining Eq. (\[eq:g-gn-relation\]) and Eq. (\[eq:interp-function\]), we can solve for $g$, $$g^2=\frac{1}{2}g_N^2\left(1+\sqrt{1+\left(\frac{2a_0}{g_N}\right)^2}\right).$$ Since $V=\sqrt{gR}$ and $V_N=\sqrt{g_NR}$, we obtain the rotation velocity in the MOND theory $$V^{2}_{\rm MOND1}=\sqrt{\frac{V^{4}_{N}}{2}+\sqrt{\frac{V^{8}_{N}}{4}+R^{2}a^{2}_{0}V^{4}_{N}}}.$$ Another widely used interpolation function is the so-called simple interpolation function[@Famaey:2005] $$\label{eq:interp-function2}
\mu_{2}(x)=\frac{x}{1+x}.$$ This interpolation function can give a better fit to the Milky-Way-like HSB galaxies than the standard interpolation function [@Famaey:2005; @Zhao:2006]. The corresponding rotation velocity is given by $$V^{2}_{\rm MOND2}=\frac{V^{2}_{N}+\sqrt{V^{4}_{N}+4RaV^{2}_{N}}}{2}.$$
The MSTG model is presented by Moffat [@Moffat:2005]. The action of MSTG model is the Einstein-Hilbert action $S_{EH}$ added by a mass term $S_M$, a scale field term $S_F$, and a term characterizes the interaction between mass and scale field $S_{FM}$. In the linear weak field approximation, the MSTG acceleration law of test particles reads $$\label{eq:a_MSTG}
a(R)=-\frac{G_{N}M}{R^{2}}\left\{1+\sqrt{\frac{M_0}{M}}\left[1-\exp(-R/r_{0})\left(1+\frac{R}{r_{0}}\right)\right]\right\},$$ where $G_{N}$ is the Newton’s gravitational constant, $M$ is the mass of the particle, $M_0$ and $r_{0}$ are characteristic parameters. The best fitting to a large amount of galaxy rotation curves shown that both $M_0$ and $r_{0}$ are approximately universal constants, i.e. $M_0\approx 9.6\times10^{11}M_{\odot}$ and $r_0\approx 13.92$ kpc [@Brownstein:2006zz]. Eq. (\[eq:a\_MSTG\]) can be regarded as the Newtonian acceleration except that the Newton’s gravitational constant is replaced by the running gravitational “constant" $$G(R)=G_N\left\{1+\sqrt{\frac{M_0}{M}}\left[1-\exp(-R/r_{0})\left(1+\frac{R}{r_{0}}\right)\right]\right\}.$$ Therefore, the rotation velocity in the MSTG model is given by $$V^{2}_{\rm MSTG}=V^{2}_{N}G(R)/G_N.$$
Best-fitting results {#sec:results}
====================
Best-fitting to the surface brightness {#sec:brightness}
--------------------------------------
Our samples consists of 9 HSB galaxies and 9 LSB galaxies taken from published literatures. All the 9 HSB galaxies are taken from Ref.[@Palunas:2000], and the surface brightness are imaged at I-band. As for the LSB galaxies, 8 of them are imaged at R-band, and the rest one (F730-V1) are imaged at V-band. Our samples are the same to that in Kun et al.[@Kun:2016yys]. We fit the photometric data to the models discussed in Section \[sec:model\_brightness\] using the least-$\chi^2$ method. HSB galaxies are fitted by a bulge plus a disk, while LSB galaxies are fitted by a disk only. We list the surface brightness parameters in Table \[tab:HSB\_brightness\] and Table \[tab:LSB\_brightness\]for HSB galaxies and LSB galaxies, respectively.
\[tab:HSB\_brightness\]
\[tab:LSB\_brightness\]
Best-fitting to the galaxy rotation curves {#sec:best-fitting}
------------------------------------------
We fit the observed rotation curve data to the theoretical models discussed in Section \[sec:model\] using the least-$\chi^2$ method. The best-fitting parameters are obtained by minimizing the $\chi^2$, $$\chi^{2}=\sum^{n}_{i=1}\left[\frac{V_{\rm th}(r_{i})-V_{\rm obs}(r_{i})}{\sigma_{i}}\right]^{2},$$ where V$_{\rm th}$ is the theoretical velocity, V$_{\rm obs}$ is observed velocity, and $\sigma$ is the $1\sigma$ error of V$_{\rm obs}$. In the Baryonic model and MOND models, the only two free parameters are the mass-to-light ratios of the bulge ($\sigma$) and disk ($\tau$). For LSB galaxies, $\sigma\equiv 0$, and there is only one free parameter. The critical acceleration in the MOND models is fixed at $a_{0}=1.2\times10^{-13}~{\rm km~s}^{-2}$ [@Begeman1991]. In the dark matter models, there are two additional parameters, i.e. $M_{\rm vir}$ and $R_{\rm vir}$ in the NFW model, and $\rho_c$ and $r_{c}$ in the core-modified model. In the MSTG models, there are also two additional parameters, i.e. the characteristic mass $M_0$ and scale length $r_{0}$. However, we find that theses two parameters couldn’t be well constrained using our galaxy sample. Therefore, we fix them to the values $M_0=9.6\times10^{11}M_{\odot}$ and $r_0=13.92$ kpc, which are obtained from fitting to a large sample of galaxies and taking the average [@Brownstein:2006zz].
The rotation curve data of HSB galaxies are taken from Palunas [@Palunas:2000]. We list the best-fitting parameters in Table \[tab:HSB\_parameter\]. We also list the reduced chi-square $\chi^{2}/{\rm dof}$, where ${\rm dof}=N-p$ is the degree of freedom, $N$ is the number of data points and $p$ is the number of free parameters. For three galaxies (ESO215G39, ESO322G76 and ESO322G77) in NFW model and five galaxies(ESO215G39, ESO322G77, ESO323G25, ESO509G80, ESO569G17) in the core-modified model, the best-fitting scale parameters ($r_s$ and $r_c$) of the dark matter halo overstep the galaxy scale, which is physically unreasonable. The mass-to-light ratio of disk for ESO383G02 in the core-modified model is unphysically small. Therefore, we do not list them in Table \[tab:HSB\_parameter\]. For four HSB galaxies (ESO323G25, ESO383G02, ESO446G01, ESO569G17), the mass-to-light ratio of the bulge couldn’t be well constrained in the NFW model, and we fix it to be zero. The best-fitting curves accompanied by the observed data are plotted in Fig. \[fig:HSB\]. The error bar represent the $1\sigma$ uncertainty.
The rotation curve data of LSB galaxies are taken from different literatures. F579-V1 is taken from Blok [@Blok:1996], UGC128 and UGC1230 are taken from Hulst [@Hulst:1993], and the rest five galaxies are taken from McGaugh [@McGaugh:2001]. We list the best-fitting parameters in Table \[tab:LSB\_parameter\]. For all the LSB galaxies in NFW model and three LSB galaxies (F561-1, F583-1, UGC5750) in core-modified model, the parameters couldn’t be well constrained. The mass-to-light ratio of disk for F568-3 in the core-modified model is unphysically small. Therefore we do not list them here. The best-fitting curves accompanied by the observed data are plotted in Fig. \[fig:LSB\], where the contributions from the gas are also shown with black dashed curves.
\[tab:HSB\_parameter\]
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"}
\[tab:LSB\_parameter\]
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"}
Model comparison {#sec:comparison}
================
To appraise which model is the best, one may adopt the most direct method by comparing the $\chi^{2}$ of each model, thereby reveals that the model whose $\chi^{2}$ is the smallest is the best. However, a model with more parameters in general has smaller $\chi^2$. Because the dark matter models have two more free parameters than other models, the result is not comprehensive. One may prefer to use the reduced-$\chi^2$, i.e. the $\chi^2$ per degree of freedom to measure the goodness of fit. We presented the reduced-$\chi^{2}$ in Table \[tab:HSB\_parameter\] and Table \[tab:LSB\_parameter\] for HSB and LSB galaxies, respectively. However, the reduced-$\chi^2$ is still not comprehensive enough to depict models. Therefore, in this section, we compare models with statistical analysis based on the likelihood function $\mathcal{L}=\exp\left(-\chi^{2}/2\right)$. One of the most used criteria to describe the goodness-of-fit is the Bayesian Information Criterion (BIC) [@Schwarz:1978], $${\rm BIC}=-2{\rm ln}\mathcal{L}_{\rm max}+p{\rm ln}N.$$ where $N$ is the number of data points in the galaxy rotation curve, and $p$ is the number of free parameters. Another widely used criterion is the Akaike Information Criterion (AIC) [@Akaike:1974], $${\rm AIC}=-2{\rm ln}\mathcal{L}_{\rm max}+2p.$$ We list the $\chi^2$, BIC and AIC values for each model in Table \[tab:comparison\]. Models with the smallest value of BIC or AIC highlighted in boldface are the best models. To be more visible, in Fig. \[fig:barplot\] we plot the number of galaxies that can be best fitted by each model.
\[tab:comparison\]
{width="50.00000%"}
According to the BIC criterion, one HSB galaxy (ESO383G02) and one LSB galaxie (F561-1) are best fitted by the Baryonic model. Only one HSB galaxy (ESO446G01) but no LSB galaxy is best fitted by the NFW model. One HSB galaxy (ESO322G76) but no LSB galaxy is best fitted by the MSTG model. Four LSB galaxies (F579-V1, F730-V1, UGC128 and UGC1230) but no HSB galaxy are best fitted by the core-modified profile. Simple MOND model fits well for three HSB galaxies (ESO322G77, ESO445G19 and ESO509G80) and two LSB galaxies (F563-1 and UGC5750). For the rest three HSB and two LSB galaxies, standard MOND model is the best model.
If we apply the AIC criterion, similar conclusions can be arrived. The only difference between BIC and AIC happens in the HSB galaxy ESO323G25, which according to the BIC criterion is best fitted by standard MOND model, while according to the AIC criterion is best fitted by NFW model. In fact, ESO323G25 can be fitted by both models very well.
Discussion and summary {#sec:summary}
======================
In this paper, we have compared six different models (Baryonic model, NFW profile, core-modified profile, standard MOND, simple MOND and MSTG) in account for the rotation curves of 9 HSB and 9 LSB galaxies. We fitted the observed rotation curve data to theoretical models, and used the Bayesian Information Criterion (BIC) and Akaike Information Criterion (AIC) to appraise which model is the best. We found that non of the six models can well fit all the 18 galaxies. Specifically, non of the HSB galaxies can be well fitted by core-modified model, and non of the LSB galaxies can be well fitted by NFW model. Only one or two (depends on either BIC or AIC is applied) HSB galaxies are best accounted by NFW model. This hits that the dark matter halos, if they really exist, in some cases couldn’t be well mimicked by the oversimplified NFW or core-modified profiles. Among the 18 galaxies, only one HSB galaxy can be best fitted by MSTG model, which implies that MSTG is not a universe model. Two galaxies (one HSB galaxy and one LSB galaxy) are best accounted by Baryonic model. For these two galaxies, it is neither necessary to add the dark matter component, nor necessary to modified the Newtonian dynamics or Newtonian gravity. Three or two HSB galaxies are best met by standard MOND model, and three HSB galaxies are best fitted by simple MOND model. In most case, standard MOND and simple MOND fit the data equally well. In summary, we couldn’t arrive a convincing conclusion to prefer one model and exclude the others.
[90]{}
V. C. Rubin, N. Thonnard, and W. K. Ford, ApJ, [**225**]{}: L107 (1978) V. C. Rubin, W. K. Ford, and N. Thonnard, ApJ, [**238**]{}: 471 (1980) A. Bosma, AJ, [**86**]{}: 1825 (1981) Y. Sofue, and V. Rubin, ARA&A, [**39**]{}: 137 (2001) F. Walter, E. Brinks, W. J. G. de Blok et al, AJ, [**136**]{}: 2563 (2008) W. J. G. de Blok, F. Walter, E. Brinks et al, AJ, [**136**]{}: 2648 (2008) K. G. Begeman, A. H. Broeils, and R. H. Sanders, MNRAS, [**249**]{}: 523 (1991) M. Persic, P. Salucci, and F. Stel, MNRAS, [**281**]{}: 27 (1996) L. Chemin, W. J. G. de Blok, and G. A. Mamon, AJ, [**142**]{}: 109 (2011) J. C. Kapteyn, ApJ, [**55**]{}: 302 (1922) J. H. Oort, Bulletin of the Astronomical Institutes of the Netherlands, [**6**]{}: 249 (1932) M. Milgrom, ApJ, [**270**]{}: 365 (1983) M. Milgrom, ApJ, [**270**]{}: 371 (1983) M. Milgrom, ApJ, [**302**]{}: 617 (1986) J. D. Bekenstein, Phys. Rev. D, [**70**]{}: 083509 (2004) R. H. Sanders, ApJ, [**473**]{}: 117 (1996) R. H. Sanders, and M. A. W. Verheijen, ApJ, [**503**]{}: 97 (1998) R. H. Sander, and E. Noordermeer, MNRAS, [**379**]{}: 702 (2007) R. A. Swaters, R. H. Sanders, and S. S. McGaugh, ApJ, [**718**]{}: 380 (2010) F. Iocco, M. Pato, and G. Bertone, Phys. Rev. D, [**92**]{}: 084046 (2015) J. W. Moffat, JCAP, [**0505**]{}: 003 (2005) J. W. Moffat, JCAP, [**0603**]{}: 004 (2006) M. Carmeli, Int. J. Theor. Phys, [**37**]{}: 2621 (1998) M. Carmeli, Int. J. Theor. Phys, [**39**]{}: 1397 (2000) M. Carmeli, [*Cosmological Special Relativity*]{} (Singapore, World Scientific, 2002) P. Horava, PRD, [**79**]{}: 084008 (2009) P. Horava, JHEP, [**0903**]{}: 020 (2009) P. Horava, PRL, [**102**]{}: 161301 (2009) D. Grumiller, PRL, [**105**]{}: 211303 (2010) J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophys. J, [**462**]{}: 563 (1996) J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophys. J, [**490**]{}: 493 (1997) J. R. Brownstein, PhD thesis, arXiv:0908.0040 A. Tamm, and P. Tenjes 2005, A&A, [**433**]{}: 31 (2005) G. de Vaucouleurs, Hdb. d. Phys, [**53**]{}: 311 (1959) K. C. Freeman, ApJ, [**160**]{}: 811 (1970) W. J. G. de Blok, J. M. van der Hulst, and G. D. Bothun, MNRAS, [**274**]{}: 235 (1995) R. Jimenez, L. Verde, and S. P. Oh, MNRAS, [**339**]{}: 243 (2003) A. Burkert, Astrophys. J. Lett, [**447**]{}: L25 (1995) D. Merritt, A.W. Graham, and B. Moore et al, AJ, [**132**]{}: 2685 (2006) L. Hemquist, ApJ, [**356**]{}: 359 (1990) H. Zhao, Mon. Not. Roy. Astron. Soc, [**278**]{}: 488 (1996) J. An, and H. Zhao, MNRAS, [**428**]{}: 2805 (2013) X.-F. Wu, B. Famaey, G. Gentile, H. Perets and H.-S. Zhao, MNRAS,[**386**]{}: 2199 (2008) B. Famaey, J. Binney, MNRAS, [**363**]{}: 603 (2005) H. S. Zhao, and B. Famaey, ApJ, [**638**]{}, L9 (2006) J. R. Brownstein, and J. W. Moffat, ApJ, [**636**]{}: 721 (2006) P. Palunas, and T. B. Williams, Astrophys. J, [**120**]{}: 2884 (2000) E. Kun, G. Szűcs, and Z. Keresztes et al, arXiv:1604.02465 W. J. G. de Blok, S. S. McGaugh, J. M. van der Hulst, MNRAS, [**283**]{}: 18 (1996) J. M. van der Hulst, E. D. Skillman, T. R. Smith et al, AJ, [**106**]{}: 548 (1993) S. S. McGaugh, V. C. Rubin, and W. J. G. de Blok, AJ, [**122**]{}: 2381 (2001) J. H. Kim, PhD thesis (Maryland: University of Maryland, College Park, 2007) G. Schwarz, Ann Statist, 1978, [**6**]{}: 461 (1978) H. Akaike, IEEE Trans. Automatic Control, [**19**]{}: 716 (1974)
|
---
abstract: 'We explore trust in a relatively new area of data science: Automated Machine Learning (AutoML). In AutoML, AI methods are used to generate and optimize machine learning models by automatically engineering features, selecting models, and optimizing hyperparameters. In this paper, we seek to understand what kinds of information influence data scientists’ trust in the models produced by AutoML? We operationalize trust as a willingness to deploy a model produced using automated methods. We report results from three studies – qualitative interviews, a controlled experiment, and a card-sorting task – to understand the information needs of data scientists for establishing trust in AutoML systems. We find that including transparency features in an AutoML tool increased user trust and understandability in the tool; and out of all proposed features, model performance metrics and visualizations are the most important information to data scientists when establishing their trust with an AutoML tool.'
author:
- Jaimie Drozdal
- Gaurav Dass
- Bingsheng Yao
- Changruo Zhao
- Justin Weisz
- Dakuo Wang
- Michael Muller
- Lin Ju
- Hui Su
bibliography:
- 'main.bib'
title: 'Trust in AutoML: Exploring Information Needs for Establishing Trust in Automated Machine Learning Systems'
---
<ccs2012> <concept> <concept\_id>10003120.10003121.10003122.10003334</concept\_id> <concept\_desc>Human-centered computing User studies</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10003120.10003121.10011748</concept\_id> <concept\_desc>Human-centered computing Empirical studies in HCI</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10010147.10010178</concept\_id> <concept\_desc>Computing methodologies Artificial intelligence</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
|
---
abstract: 'We present [*EzGal*]{}, a flexible python program designed to easily generate observable parameters (magnitudes, colors, mass-to-light ratios) for arbitrary input stellar population synthesis (SPS) models. As has been demonstrated by various authors, for many applications the choice of input SPS models can be a significant source of systematic uncertainty. A key strength of [*EzGal*]{} is that it enables simple, direct comparison of different models sets so that the uncertainty introduced by choice of model set can be quantified. Its ability to work with new models will allow [*EzGal*]{} to remain useful as SPS modeling evolves to keep up with the latest research (such as varying IMFS; @cappellari12). [*EzGal*]{} is also capable of generating composite stellar population models (CSPs) for arbitrary input star formation histories and reddening laws, and can be used to interpolate between metallicities for a given model set. To facilitate use, we have created a web interface to run EzGal and quickly generate magnitude and mass-to-light ratio predictions for a variety of star formation histories and model sets. We make many commonly used SPS models available from the web interface including the canonical @BC03 models, an updated version of these models, the Maraston models, the BaSTI models, and finally the FSPS models. We use [*EzGal*]{} to compare magnitude predictions for the model sets as a function of wavelength, age, metallicity, and star formation history. From this comparison we quickly recover the well-known result that the models agree best in the optical for old, solar metallicity models, with differences at the $\sim0.1$ magnitude level. Similarly, the most problematic regime for SPS modeling is for young ages ($\la 2$ Gyrs) and long wavelengths ($\lambda \gtrsim 7500$Å) where thermally pulsating AGB stars are important and scatter between models can vary from 0.3 mags (Sloan i’) to 0.7 mags (K$s$). We find that these differences are not caused by one discrepant model set and should therefore be interpreted as general uncertainties in SPS modeling. Finally we connect our results to a more physically motivated example by generating CSPs with a star formation history matching the global star formation history of the universe. We demonstrate that the wavelength and age dependence of SPS model uncertainty translates into a redshift dependent model uncertainty, highlighting the importance of a quantitative understanding of model differences when comparing observations to models as a function of redshift.'
author:
- 'Conor L. Mancone and Anthony H. Gonzalez'
bibliography:
- 'ms.bib'
title: 'EzGal: A Flexible Interface for Stellar Population Synthesis Models'
---
Introduction\[sec:intro\]
=========================
[$$M_{AB}\left[z,t(z,z_f)\right] = -2.5\log\left[ \frac{ \int_{-\infty}^\infty \nu^{-1} (1+z)F_\nu[\nu(1+z),t(z,z_f)] R(\nu) d\nu } {\int_{-\infty}^\infty \nu^{-1} R(\nu) d\nu} \right] - 48.60\label{eq:mags}$$]{}
Stellar population synthesis (SPS) modeling provides a valuable tool for studying the evolution of a stellar population as a function of time. For this reason there have been multiple efforts to develop software for modeling the evolution of the spectral energy distribution (SED) of a stellar population. Because there are a number of unknowns in SPS modeling, such as details of post-main sequence stellar evolution and the form of the initial mass function, models from different groups yield a range of results due to different input ingredients in their models. A detailed discussion of these uncertainties and their impact can be found in @C09 and @conroy10. The net result is that the choice of model set is itself a source of uncertainty when using SPS models.
The use of SPS models is a central ingredient for a wide range of active research programs, as is evident even from a simple literature search. SPS models are commonly used to perform SED fitting and estimate a diverse set of properties for stellar populations including ages, redshifts, $k$-corrections, and masses (see for example @blanton07 [@taylor11; @ma12; @foto12]). They are used to fit isochrones to color magnitude diagrams and measure ages and metallicities of resolved stellar populations, to measure the strength of spectral features in observed galaxies, to predict the evolution of a stellar population as a function of age, and to predict observables from simulations (for example @jonsson06 [@ata; @mancone10; @kriek11]). Because of the utility and ubiquity of SPS models, it is important to have simplifying methods for comparing the models to observations as well as to each other.
Of the many SPS model sets the most commonly used is that of Bruzual & Chalot (; 2003) which we use as a reference for comparisons because of its wide use. Another commonly used model set is that of Maraston (; 2005) which includes a detailed treatment of thermally pulsating AGB (TP-AGB) stars, which can dominate the infrared light of a young stellar population. An updated treatment of the TP-AGB phase is also incorporated into the latest version of the models [@CB07 commonly referred to as CB07]. More recent models include the work of Percival et al. (; 2009) which include not only a range of metallicities but also $\alpha-$enhanced models. The FSPS models () from @C09 and @conroy10 are unique in their ability to treat the most important SPS inputs (such as IMF or various uncertain phases of stellar evolution) as free parameters allowing the uncertainties introduced by various SPS inputs to be quantitatively measured.
All of these models predict the evolution of the SED of a stellar population as a function of age, given a star formation history, initial mass function (IMF), and metallicity. However the easiest to measure observables are not the SED or age, but rather the magnitude and redshift. Therefore all of these model sets are most useful when they can be easily translated into predictions of magnitude evolution as a function of redshift. This transformation involves assuming a formation redshift (the redshift at which star formation starts), calculating a cosmology-dependent luminosity distance, and projecting the SEDs through filter response curves to calculate magnitudes, $e$-corrections, and $k$-corrections. The $e$-corrections specify the amount of observed magnitude evolution that is due to the aging of a stellar population, while the $k$-corrections specify the amount of evolution due to observing a different part of the SED at different redshifts. Together the $e$-corrections, $k$-corrections, and distance moduli specify the magnitude evolution of a stellar population as a function of redshift.
While these steps are straight-forward, in the past there has not been a simple and consistent tool to do this for all model sets. and both come with code for calculating magnitude evolution as a function of redshift and both come with a number of commonly used filter response curves for user convenience. In contrast, and calculate and distribute the absolute magnitude evolution of the stellar populations for a fixed set of filters.
This lack of directly comparable outputs between different model sets is the reason why we have developed [*EzGal*]{}, a python program that calculates magnitude evolution as a function of redshift from models of the evolution of an SED as a function of age. [*EzGal*]{} comes with a number of the most commonly used filter response curves, and more can be easily added by the user. It includes the latest Vega spectrum from STScI[^1] so that magnitudes can be calculated on both the Vega and AB systems. By using the stellar mass information that comes with all of these model sets [*EzGal*]{} can also calculate mass-to-light ratios in any filter. This requires calculating the absolute magnitude of the Sun in any filter and so the latest solar spectrum from STScIis also included with [*EzGal*]{}. [*EzGal*]{} can interpolate between models, which is useful for generating models with the same metallicity from different model sets. It can also generate CSPs with arbitrary input star formation histories or dust reddening laws. Finally, [*EzGal*]{} can read in SEDs in ASCII format or in the binary ised format that the and models are distributed in. In principle this allows it to work with any model, enabling easy comparison with any new codes in the future. [*EzGal*]{} is designed to be an easy-to-use tool for predicting observables from SPS models and greatly simplifying the task of comparing different SPS model sets.
This paper explains how [*EzGal*]{} works and gives a detailed comparison between commonly used model sets. Section \[sec:procedure\] describes details of how [*EzGal*]{} works and discusses calculating magnitudes (Section \[sec:mags\]), generating composite stellar populations (Section \[sec:csp\]), and calculating masses and mass-to-light ratios (Section \[sec:mls\]). In Section \[sec:compare\] we present a detailed comparison between the model sets. Section \[sec:web\] lists EzGal resources currently available from the internet such as the web interface. Our conclusions are found in Section \[sec:conclusions\].
Program Procedure {#sec:procedure}
=================
Calculating Magnitudes {#sec:mags}
----------------------
[*EzGal*]{} calculates apparent magnitudes, absolute magnitudes, $e$-corrections, and $k$-corrections from the model sets as a function of redshift. Conceptually, these quantities are all easy to calculate and are derived from the rest-frame and observed-frame absolute magnitudes as a function of age and redshift. [*EzGal*]{} uses Equation \[eq:mags\] to calculate observed-frame absolute magnitudes as a function of redshift ($z$) and formation redshift ($z_f$). This equation calculates the absolute AB magnitude as a function of redshift and age, $M_{AB}\left[z,t(z,z_f)\right]$, for an SPS model by projecting the redshifted SED, $F_\nu[\nu(1+z),t(z,z_f)]$, at the given age, $t(z,z_f)$ (with the age determined by redshift and formation redshift), through the filter response curve, $R(\nu)$, and comparing this to the flux of a zero mag AB source. For the purposes of this equation the SED should have units of ergs$^{-1}$Hz$^{-1}$cm$^{-2}$ and should be the observed flux for a galaxy at a distance of 10 pc. The age of the galaxy, $t(z,z_f)$, is given by $t(z,z_f) = T_U(z) - T_U(z_f)$ where $T_U(z)$ is the age of the universe as a function of redshift given the cosmology. By default [*EzGal*]{} assumes a WMAP 7 cosmology [@komatsu11 $\Omega_m=0.272, \Omega_\Lambda=0.728, h=0.704$], although any cosmology can be used. To calculate the rest-frame absolute magnitude, [*EzGal*]{} calculates $M_{AB}[ 0, t(z,z_f) ]$.
[*EzGal*]{} also calculates a number of filter properties using standard STScI definitions, including mean wavelength, pivot wavelength, average wavelength, effective dimensionless gaussian width, effective width, equivalent width, and rectangular width.[^2]$^,$[^3] The conversion from AB to Vega magnitudes is calculated for each filter by using the included Vega spectrum to calculate the AB magnitude of Vega in the filter. The Vega spectrum is described in @bohlin04 and comes from IUE spectrophotometry from 0.12 - 0.17$\mu$m, HST STIS spectroscopy from 0.17 - 1.01$\mu$m, and a Kurucz model atmosphere at longer wavelengths. Finally, the absolute magnitude of the Sun is also calculated by projecting the solar spectrum through the filter response curve in the same way as everything else. The solar spectrum used by EzGal is an observed spectrum of the Sun from 0.12 - 2.5 [@sun] which we have extended using a Kurucz model atmosphere at longer wavelengths. Specifically, we take a model atmosphere with solar metallicity, $T_{eff}$ = 5777K, and $log_g$ = 4.44, normalize it to match the observed solar spectrum from 1.5 - 2.5 , and then use it where the observed spectrum ends.
Calculating Composite Stellar Populations {#sec:csp}
-----------------------------------------
[*EzGal*]{} generates composite stellar population (CSP) models from simple stellar population (SSP) models in the standard way. Conceptually, the SED of a CSP at some age is given by a weighted average of SSPs as a function of age, where the weight for a given SSP is equal to the relative strength of star formation (compared to the total amount of star formation) for the CSP at that time. The effect of dust can also be included if desired. Mathematically, [*EzGal*]{} uses Equation \[eq:csp\] to calculate the evolution of the SED of a CSP as a function of time.
$$F(\lambda,t) = \frac{ \int_{0}^t\Psi(t-t')F_{SSP}(\lambda,t')\Gamma(\lambda,t')dt' }{ \int_0^{T_U}\Psi(t')dt' }\label{eq:csp}$$
In this equation $F(\lambda,t)$ is the flux of the CSP as a function of wavelength and time, $\Psi(t)$ is the star formation rate as a function of time, $F_{SSP}(\lambda,t')$ is the flux of the SSP as a function of wavelength and time, $\Gamma(\lambda,t')$ is the impact of dust as a function of wavelength and time, and $T_U$ is the age of the universe at z=0. The factor of $\int_0^{T_U}\Psi(t')dt'$ normalizes the CSP such that one solar mass of stars is generated over the entire star forming epoch. EzGal can work with arbitrary star formation histories and dust laws. A typical dust law is a @charlot00 dust law with $\Gamma(\lambda) = e^{-\tau(t)(\lambda/5500\mathrm{\AA})^{-0.7}}$ where $\tau(t) = 1.0$ for $t \leq 10^7$yr and $\tau(t) = 0.5$ for $t > 10^7$yr. Equation \[eq:csp\] represents the same general methodology used by and to generate CSPs. uses a different normalization and instead divides by $\int_0^t\Psi(t')dt'$ so that the CSPs have one solar mass of stars at all ages. does not provide any CSPs with their models.
In practice [*EzGal*]{} uses Simpson’s rule to numerically evaluate the top integral in Equation \[eq:csp\]. When performing numeric integration it is often necessary to sub-sample the age grid of the SSPs to properly sample any sharp features in the star formation history or in the evolution of the SEDs. In order to minimize execution time and still ensure high fidelity in the numeric integration, [*EzGal*]{} uses an iterative algorithm to decide how finely to sub-sample the age grid. [*EzGal*]{} performs the integral in Equation \[eq:csp\] at wavelengths of 3000, 8000, and 12000Å with increasingly finer levels of age sub-sampling until the difference between two subsequent integrals drops below some tuneable threshold (in magnitudes).
[cccccccc]{} & 221 & 0.005 - 2.5 & No & 6 & Yes & Yes & No\
& 68 & 0.05 - 3.5 & No & 5 & Yes & No & Yes\
& 221 & 0.005 - 2.5 & No & 6 & Yes & Yes & No\
& 56 & 0.005 - 2 & Yes & 10 & No & No & Yes\
& 189 & 0.01 - 1.5 & No & 22 & Yes & Yes & Yes\
[ccccc]{} GALEX FUV & 1536 & 246 & 17.20 & -2.093\
GALEX NUV & 2300 & 730 & 10.04 & -1.659\
Sloan u’ & 3556 & 558 & 6.37 & -0.916\
ACS WFC F435W & 4318 & 845 & 5.37 & 0.102\
WFC3 F438W & 4325 & 616 & 5.34 & 0.152\
Sloan g’ & 4702 & 1158 & 5.12 & 0.100\
ACS WFC F475W & 4746 & 1359 & 5.10 & 0.096\
WFC3 F475W & 4773 & 1343 & 5.08 & 0.096\
WFC3 F555W & 5308 & 1563 & 4.86 & 0.023\
ACS WFC F555W & 5360 & 1124 & 4.84 & 0.005\
WFC3 F606W & 5887 & 2183 & 4.73 & -0.085\
ACS WFC F606W & 5921 & 1992 & 4.72 & -0.088\
Sloan r’ & 6175 & 1111 & 4.64 & -0.144\
WFC3 F625W & 6241 & 1461 & 4.64 & -0.150\
ACS WFC F625W & 6311 & 1308 & 4.63 & -0.165\
Sloan i’ & 7489 & 1045 & 4.53 & -0.357\
WFC3 F775W & 7647 & 1170 & 4.53 & -0.382\
ACS WFC F775W & 7691 & 1320 & 4.53 & -0.389\
WFC3 F814W & 8026 & 1538 & 4.52 & -0.419\
ACS WFC F814W & 8055 & 1733 & 4.52 & -0.425\
Sloan z’ & 8946 & 1125 & 4.51 & -0.518\
ACS WFC F850lp & 9013 & 1239 & 4.51 & -0.521\
WFC3 F850lp & 9167 & 1181 & 4.52 & -0.522\
WFC3 F105W & 10550 & 2649 & 4.53 & -0.647\
WFC3 F110W & 11534 & 4430 & 4.54 & -0.761\
J & 12469 & 2088 & 4.56 & -0.901\
WFC3 F125W & 12486 & 2845 & 4.56 & -0.903\
WFC3 F140W & 13922 & 3840 & 4.60 & -1.078\
WFC3 F160W & 15370 & 2683 & 4.65 & -1.254\
H & 16448 & 2538 & 4.70 & -1.365\
Ks & 21623 & 2642 & 5.13 & -1.838\
[*WISE*]{} 3.4$\mu$m & 33682 & 6824 & 5.95 & -2.668\
IRAC 3.6$\mu$m & 35569 & 6844 & 6.07 & -2.787\
IRAC 4.5$\mu$m & 45020 & 8707 & 6.57 & -3.260\
[*WISE*]{} 3.6$\mu$m & 46179 & 10508 & 6.62 & -3.307\
IRAC 5.8$\mu$m & 57450 & 12441 & 7.05 & -3.753\
IRAC 8$\mu$m & 79156 & 25592 & 7.67 & -4.394\
We verify our procedure for generating CSPs by comparing magnitude predictions for CSPs generated with [*EzGal*]{} from and models to magnitude predictions for CSPs generated by the code distributed with and . We find differences that are small and negligible: for the differences in magnitude are $<0.005$ mags for short ($\tau=0.1$Gyr) and long ($\tau=1.0$Gyr) dust-free exponentially-decaying bursts, and for the differences are $<0.01$ mags for short bursts and $<0.005$ mags for long bursts. These differences are larger than the maximum error set in our numerical integration (0.001 mags), but errors at these levels can easily be accounted for by small differences in the procedures used by different groups.
Calculating Mass-to-Light Ratios and Masses {#sec:mls}
-------------------------------------------
To calculate rest-frame mass-to-light ratios in any filter, $F$, given redshift and formation redshift, four pieces of information are required: the age as a function of redshift and formation redshift, $t(z,z_f)$, the stellar mass as a function of age, $M_*[t(z,z_f)]$, the rest-frame absolute magnitude of the stellar population as a function of age, $M_F[t(z,z_f)]$, and the absolute magnitude of the Sun in the filter, $M_{\odot,F}$. Again, the conversion from redshift and formation redshift to age requires assuming a cosmology, for which EzGal defaults to a WMAP 7 cosmology [@komatsu11 $\Omega_m=0.272, \Omega_\Lambda=0.728, h=0.704$]. The rest-frame mass-to-light ratio in a given filter as a function of redshift and fromation redshift is then given by:
[$$\frac{M_*}{L_F}(z,z_f) = \frac{M_*[t(z,z_f)]}{ 10^{ -0.4*\{M_F[t(z,z_f)]-M_{\odot,F}\} } }\label{eq:ml}$$]{}
[*EzGal*]{} uses Equation \[eq:ml\] to calculate rest-frame mass-to-light ratios. It uses its own calculation of the absolute magnitude evolution of a stellar population as a function of age, calculates the absolute magnitude of the Sun using the solar spectrum from STScI, and gets stellar masses directly from the model sets (which typically distribute stellar mass as a function of age along with the SED). The resulting mass-to-light ratios depend on the chosen model set, star formation history, and initial mass function. [*EzGal*]{} also calculates an observed-frame mass-to-light ratio as a function of redshift using the observed-frame absolute magnitude of the model and the observed-frame absolute magnitude of the Sun. The latter is calculated by redshifting the solar spectrum to the given redshift and projecting it through the bandpass normally.
For the purposes of estimating the mass of an observed galaxy only two pieces of information are required from the models: the stellar mass as a function of redshift and the apparent magnitude of the model as a function of redshift. With these values in hand the mass of an observed galaxy with an assumed redshift ($z$) and formation redshift ($z_f$) can be calculated as:
[$$M_{*,g}(z,z_f) = M_*(z,z_f)*10^{ -0.4*(m_{g,F} - m_F(z,zf)) }\label{eq:mass},$$]{}
where $M_*(z,z_f)$ is the stellar mass of the model as a function of $z$ and $z_f$, $m_{g,F}$ is the apparent magnitude of the galaxy in a given passband, and $m_F(z,zf)$ is the apparent magnitude of the model in the same passband as a function of $z$ and $z_f$.
Model Comparison {#sec:compare}
================
Model Set Overview
------------------
In this paper we compare results from five different SPS model sets: , , , , and . These model sets include a varying range of metallicities and IMFs, and have different spectral resolutions and age grids. has the highest metallicity model ($Z = 3.5Z_\odot$) while , , and have the lowest ($Z = Z_\odot$/200). has the finest grid in metallicity space with 22 metallicities from $Z = 0.01Z_\odot - 1.5Z_\odot$ and is the only model set that distributes models with all three common IMFs: Salpeter, Chabrier, and Kroupa. Finally, is the only model set herein to publish models with alpha enhanced metallicities. This information is provided as a quick reference for comparing model sets and is summarized in Table \[tbl:models\] which includes the number of ages in each model set, the number of metallicities provided, and the IMFs provided.
To facilitate direct comparisons between model sets we interpolate between the models to generate a new set of models for each model set with the same metallicities. Our new models have metallicities of Z = 0.05, 0.1, 0.2, 0.4, 0.8, 1.0, and 1.5 times $Z_\odot$ or Z = 0.001, 0.002, 0.004, 0.008, 0.016, 0.02, and 0.03. For all of our comparisons below we use these interpolated models. We also choose to restrict our comparisons to models with the same IMF. As there is no IMF that is covered by all five model sets we do all comparisons using a Salpeter IMF, and therefore in the comparisons below the models from are not included. For each of our interpolated models we use EzGal to generate four CSP models. The CSPs are dust-free, exponentially decaying bursts with e-folding timescales of 0.1, 0.5, 1.0, and 10.0 Gyrs.
Filter Set Overview
-------------------
For convenience to [*EzGal*]{} users and to enable a basic model comparison we generate a filter set for use with [*EzGal*]{}. Our filter set includes many commonly used filters: the [*GALEX*]{} FUV and NUV filters, the Sloan filters, all wide [*HST*]{} ACS WFC and WFC3 filters, 2MASS filters, [*Spitzer*]{} IRAC filters, and the [*WISE*]{} 3.4 and 4.6$\mu$m filters. The filters come from a number of sources and all represent total transmission: CCD, telescope, filter, and a basic atmosphere when appropriate. The properties of the filter set are summarized in Table \[tbl:filters\] which has the pivot wavelength and rectangular width for each filter as well as the absolute AB magnitude of the Sun through each filter and the calculated AB to Vega conversion. The latter is in magnitudes such that the Vega magnitude of a galaxy is its AB magnitude plus the listed conversion. This Table is also reproduced on the web for quick reference.
Comparison {#sec:comparison}
----------
We begin our comparison by examining the fidelity of magnitude predictions for the models in Sloan filters. Figure \[fig:simple\_comparison\] shows the predicted i’ band rest-frame absolute magnitude (top) and g’-i’ color as a function of age for SSP models with a solar metallicity and Salpeter IMF. As can be seen from this Figure, differences are typically $\sim$0.1 - 0.2 magnitudes. To better explore how the scatter depends on age and wavelength, we plot the scatter between models as a function of age, wavelength, metallicity and star formation history in Figure \[fig:sloan\]. The top left panel of this Figure illustrates the scatter between the predicted magnitudes of the models (, , , and ) for the Sloan filters u’, g’, r’, i’, z’ as a function of age for an SSP with a Salpeter IMF and a metallicity of $Z=0.001$. The panels to the right show the same thing but for $Z=0.008$, $Z=0.02$, and $Z=0.03$. The bottom row of panels shows the impact of changing star formation histories. All the models in the bottom panel have solar metallicity ($Z=0.02$) and a Salpeter IMF. The first plot on the bottom rows shows the scatter between the model sets for an SSP, the next for a dust-free exponentially decaying burst of star formation with an e-folding time ($\tau$) of 1.0 Gyrs, and the last for a dust-free exponential burst with $\tau = 10.0$ Gyrs. Scatter in this case refers to the standard deviation of the magnitudes predicted by the different models at a given age and through a particular filter.
A number of conclusions can be drawn from Figure \[fig:sloan\]. First, the best-case comparison is for solar metallicities and intermediate to old ages ($\gtrsim$ 4 Gyrs), for which differences between the models are at most 0.1 mags and drop to $\sim$0.05 mags at the oldest ages. For the Sloan i’ and Sloan z’ filters the scatter increases by a factor of $\sim$2 for younger ages ($\lesssim$ 2 Gyrs). This is particularly true for sub-solar metallicities, and the scatter in Sloan i’ and Sloan z’ increases systematically at these young ages when going from metallicities of $Z=0.02$ to $Z=0.008$ and $Z=0.001$, reaching differences as large as $\sim$0.4 mags. For the three bluest Sloan filters the scatter is $\lesssim 0.1$ mags for all ages and metallicities.
The bottom series of panels highlights the impact of an extended star formation history, the effect of which is to smooth out the scatter between models as a function of age. At longer wavelengths when the models differ more at younger ages, this smoothing has a tendency to increase errors at latter times and decrease errors at earlier ones. Therefore in this case model uncertainty for extended star formation histories will be larger at later times if the star formation history includes a substantial presence of young stars (ages $\lesssim$ 3 Gyrs).
Figure \[fig:ir\] is the same as Figure \[fig:sloan\] but now various near-IR bands are plotted: J, H, K$s$, and Spitzer/IRAC 3.6 and 4.5$\mu$m. The first thing to note is that for an older ($\gtrsim$ 3 Gyr) solar metallicity SSP the differences in JHK$s$ are comparable to the differences in the Sloan bands (i.e. Figure \[fig:sloan\]), while the Spitzer/IRAC bands typically have larger errors in this same regime. The scatter between the models now has a stronger age dependency, and for ages $\lesssim$2 Gyrs the model uncertainty increases to 0.3 mags (J) and 0.6 mags (3.6$\mu$m). Metallicity has the opposite impact on the scatter between models in the NIR for young ($\lesssim$ 3 Gyrs) and intermediate to old ($\gtrsim$ 3 Gyrs) stellar populations. For younger ages the scatter increases systematically while going to lower metallicities. This effect is particularly pronounced in the K$s$ band which has a maximum scatter of $\sim$0.35 mags for young stellar populations with solar metallicity, but a maximum scatter of $\sim$0.7 mags for young stellar populations with $Z=0.001$. For older stellar populations the scatter is roughly constant or even decreasing (IRAC 3.6 and 4.5$\mu$m) as the metallicity decreases.
The general trend of increasing scatter towards younger ages is by no means a new discovery but is strongly influenced by uncertainties with the thermally pulsating AGB (TP-AGB) phase (, @marigo08, ). This short lived phase in stellar evolution is poorly understood observationally and theoretically; observationally due to its rarity and theoretically because the properties of a TP-AGB star are strongly dependent upon mass loss, which is not predicted theoretically [@C09]. Unfortunately for stellar modeling, TP-AGB stars can dominate the light of a stellar population at long wavelengths ($\lambda \ga 1\mu$m) for ages $\ga 10^8$yrs. While it is most important in the NIR, it can also impact red optical filters to a smaller extent , and so can readily explain the systematic trend to higher scatters seen as a function of wavelength and age in Figures \[fig:sloan\] and \[fig:ir\]. Moreover, it can exacerbate differences for models with different metallicities because the TP-AGB stars used to calibrate the models typically have unknown metallicities [@C09], creating an additional source of uncertainty. This likely explains the substantially higher scatter seen for young ages, sub-solar metallicities, and long wavelengths.
The differences seen in Figures \[fig:sloan\] and \[fig:ir\] are best viewed as lower limits for the uncertainties introduced by SPS modeling. This is because agreement between the models can simply be caused by similar methodologies used by the various modeling groups, and does not necessarily imply that the models are doing a better job of agreeing with actual stellar populations. For instance, we noted above that for old stellar populations the scatter between models is typically the same or smaller for sub-solar metallicities than for solar metallicities. This fact is not surprising since all the model sets used herein are all compared to or calibrated to match Milky Way globular clusters, which are old and metal poor systems.
Figure \[fig:diffs\] demonstrates that the scatters seen in Figures \[fig:sloan\] and \[fig:ir\] are not driven by just one model set. This Figure shows the differences between the predicted magnitudes of these four models through the Sloan and NIR filters for four different ages and two metallicities. All the models in this Figure are SSPs with a Salpeter IMF. The left panel in Figure \[fig:diffs\] is for models with solar metallicity and the right panel is for models with a metallicity of $Z=0.001$. Each panel is divided up into four plots corresponding to four different ages: 1 Gyr (top left), 2 Gyrs (top right), 6 Gyrs (bottom left), and 10 Gyrs (bottom right). The lines in each plot represent the differences between the predicted absolute magnitude through each filter in each model set minus the predicted absolute magnitude of . In general, the models are distributed throughout the full range of magnitudes covered by the models. This shows that apparent disagreements in Figures \[fig:sloan\] and \[fig:ir\] are not caused by one discrepant model set. Therefore, the scatter seen in Figures \[fig:sloan\] and \[fig:ir\] is representative of the general uncertainties between the SPS models.
Finally, we note that our results are robust against the choice of model sets used for our comparison. For instance it might seem expedient to exclude the models from the above analysis because substantial effort has been put forth to understand the TP-AGB phase since was published. However, excluding this model set from the analaysis makes no appreciable differences in our results, which simply reflects the fact that the models are rarely an outlier in Figure \[fig:diffs\]. Our conclusions also remain unchanged if we instead compare the , , and model sets with a Kroupa IMF. This once again emphasizes that the differences noted in this paper reflect general uncertainties in SPS modeling and are not caused by one discrepant model set.
A Practical Example
-------------------
We perform one final model comparison to demonstrate the utility of [*EzGal*]{} as well as to reinforce the above results and show how they can impact current work. We generate new CSP models for our model sets using a more physically motivated SFH, which is the global SFH of the universe. We use the global star formation rate density as a function of redshift from @gonzalez10 which includes data from @reddy09, @bouwens08, @bouwens07, and @schiminovich05. This gives the relative star formation rate in the universe as a function of redshift from $z=0.3$ to $z=8.5$. We further set the star formation rate to zero at $z=0$ and $z>10$ to prevent our star formation history from having any discontinuities. While the star formation rate is unlikely to turn on suddenly at $z=10$ or turn off at $z=0$, in practice this assumption makes little difference and does not impact our example.
Using [*EzGal*]{} we generate a CSP from this star formation history for a solar metallicity galaxy with a Salpeter IMF for our four comparison models (, , , and ). We then use [*EzGal*]{} to generate apparent magnitude predictions for each CSP model through the Sloan r’, 2MASS H, and IRAC 3.6$\mu$m filters as a function of redshift, assuming a formation redshift of $z_f=10.0$. Finally we calculate the scatter between the predicted magnitudes of the models in the same way as in our previous comparisons. We show the scatter between models as a function of filter and redshift in Figure \[fig:z\_scatter\], as well as the star formation history used to generate the CSP models.
The trends seen in Figure \[fig:z\_scatter\] are caused primarily by two effects: increasing model uncertainty for younger stellar populations and the changing rest-frame wavelengths traced by each filter as a function of redshift. For $3.6\mu$m the model scatter peaks in the $1 \la z \la 3$ range. At higher redshifts the $3.6\mu$m filter traces the rest-frame optical where the models agree well. Strong star formation from $2 \la z \la 5$ guarantees that there is a substantial presence of young stars over this redshift range, and therefore the increasing importance of TP-AGB stars leads to increased uncertainty, as does the fact that the $3.6\mu$m filter traces longer wavelengths where TP-AGB stars are again more important. For $z \la 2$ the star formation rate begins to drop and the stellar populations become steadily older. Since the models agree well for old ages, this causes the model scatter to peak shortly after the star formation rate peaks and then steadily decline to $z=0$.
In the H-band the scatter between models is relatively constant and typically $\la$0.1 mags. This low scatter occurs because the H-band filter always traces regions of parameter space for which the models agree well. At high redshift when the stellar populations are young and the impact of TP-AGB stars is important, the H-band traces the rest-frame optical which is unaffected by TP-AGB stars. At low redshift the steadily dropping star formation rate leads to an increasing mean age, once again minimizing the impact of TP-AGB stars and leading to low uncertainties.
Similarly for Sloan r’ the scatter between models is typically $\la$0.1 mags at low and high redshift. However, there is a strong and sudden peak in the model scatter at $z\sim2$. This same feature is also present at precisely the same redshift and significance in all the Sloan filters and the J band, although we only show Sloan r’ in Figure \[fig:z\_scatter\]. The fact that this peak shows up in a variety of filters at the same redshift means that the underlying uncertainty depends primarily on age, not wavelength. At this redshift the Sloan filters and the J band are all tracing rest-frame wavelengths blueward of the 4000Å break. In contrast both the H band and IRAC 3.6$\mu$m filters trace wavelengths redward of the 4000Å break, and neither has evidence for a similar increase in model scatter. Therefore we conclude that this peak in model scatter is caused by uncertainty in modeling young stellar populations blueward of the 4000Å break.
Most importantly, Figure \[fig:z\_scatter\] illustrates one more reason why it is important to use quantitative methods to estimate the impact of SPS model uncertainties. Observations of galaxies at various redshifts through a given filter will trace stellar populations with a variety of ages and wavelengths. Moreover, the uncertainties in SPS modeling depend sensitively on wavelength and age. The result of these facts is that, in practice, SPS model uncertainty often depends on redshift in hard to predict ways. Therefore, for studies that investigate how stellar populations evolve as a function of redshift it is vital to verify that this redshift dependent model uncertainty is not causing spurious results. This is best done through quantitative comparison of the models to each other or of the observations to many different models, tasks that [*EzGal*]{} is designed for.
EzGal Web Resources {#sec:web}
===================
A number of [*EzGal*]{} resources are available through the [*EzGal*]{} website,[^4] including two different interfaces. The first[^5] allows the user to instantly download magnitude, $k$-correction, $e$-correction, $e+k$-correction, mass-to-light ratio, mass, and solar magnitudes for a given model set and filter as a function of redshift for a set of precalculated formation redshifts and cosmologies. The second interface[^6] allows for arbitrary choice of formation redshift and cosmology and emails the calculated results to you, which typically takes a minute or two.
An up-to-date copy of Table \[tbl:filters\] is maintained on the [*EzGal*]{} website with basic filter information, solar magnitudes, and calculated AB to Vega conversions listed for all filters available through the website. Also distributed with this table is a plot of magnitude, mass-to-light ratio, and $k$-correction evolution as a function of redshift for each filter, a plot of the filter response curve, and a data file giving the filter response curve used by EzGal.
A download page is provided where the source code for [*EzGal*]{} can be downloaded, as well as [*EzGal*]{}-ready model files. This includes the original SSP models distributed with all the model sets discussed in this paper, as well as the interpolated SSPs and generated CSPs that we use for our comparison. Finally we distribute a manual for the [*EzGal*]{} API describing how to use [*EzGal*]{} from within python.
Conclusions {#sec:conclusions}
===========
[*EzGal*]{} provides a convenient framework for transforming SPS models from theoretical quantities to directly observable magnitudes and colors. It includes code for generating composite stellar population models with arbitrary star formation histories and dust extinction. In principle it can work with any model set, providing a simple and consistent framework for comparing the predictions of different model sets and estimating errors introduced by the choice of model set.
We demonstrate the latter property of [*EzGal*]{} by predicting the magnitude evolution for five model sets (, , , , ) as a function of age, filter, metallicity, and star formation history. We compare the predictions between the models and note substantial uncertainty (0.3-0.7 mags) for young stellar populations (ages $\la 2$ Gyrs) at long wavelengths ($\lambda \ga$ 1$\mu$m), a region of well-known uncertainty caused by the contribution of thermally pulsating AGB stars. We note that for old ages, optical filters, and solar metallicities the models agree at the $\sim0.1$ mag level. For old ages at all wavelengths the models agree as well if not better at sub-solar metallicities than at solar metallicities, which likely reflects the fact that the models are all compared to or calibrated with Milky Way globular clusters. These differences are best viewed as lower limits on the uncertainties inherent in SPS modeling because it does not include systematic errors in assumptions or methodologies that are shared by all model sets.
Finally we calculate the scatter between our models for a solar metallicity stellar population with a Salpeter IMF and a star formation history matching the global star formation history of the universe. We conclude that the derived model uncertainty depends upon redshift and filter in hard to predict ways. This highlights the importance of using quantitative methods to estimate model uncertainty, especially when comparing observations to models as a function of redshift.
These results illustrate the utility of [*EzGal*]{} in simplifying the process of working with SPS model sets, making it easy to compare observations to multiple model sets as well as to compare model sets to each other. In turn, this provides a simple method to quantify the uncertainties introduced by choice of SPS model set, as well as to find robust or disparate regions in parameter space (age, wavelength, metallicity, etc).
We hope this will help other researchers both in interpreting data and planning new observations. If you use [*EzGal*]{} in your research we appreciate a citation to this paper. However, it is especially important that you cite the paper that describes the model set you are using, since [*EzGal*]{} is simply an interface for working with already existing models. We were not involved in the creation of any of the model sets discussed in this paper, although the authors have been generous enough to allow us to redistribute their models through our web interface. So when using model sets distributed with [*EzGal*]{} please be sure to cite the appropriate papers. If you download models through our website, you will find lists of references for the model sets on the download pages.
We gratefully acknowledge the authors of all five of the model sets included in this paper for giving us permission to redistribute their work in this way. We especially thank Charlie Conroy, Maurizio Salaris, Santi Cassisi, Stéfane Charlot, Gustavo Bruzual, and Claudia Maraston for their input on this project. We are also grateful to our many collaborators - Adam Stanford, Peter Eisenhardt, Yen-Ting Lin, Greg Snyder, and others who have tested [*EzGal*]{} extensively and provided us with invaluable feedback. Finally, we would like to thank the anonymous referee who’s comments have greatly improved the presentation in this paper. This paper is based upon work supported by the National Science Foundation under grant AST-0708490.
[^1]: http://www.stsci.edu/hst/observatory/cdbs/calspec.html
[^2]: http://www.stsci.edu/hst/wfpc2/documents/handbook/cycle17/ch6\_exposuretime2.html\#480221
[^3]: http://www.stsci.edu/hst/wfc3/documents/handbooks/currentIHB/c06\_uvis06.html\#57
[^4]: http://www.baryons.org/ezgal/
[^5]: http://www.baryons.org/ezgal/model
[^6]: http://www.baryons.org/ezgal/model\_server
|
---
abstract: 'We present a new mode of hydrogen burning on neutron stars (NSs) called diffusive nuclear burning (DNB). In DNB, the burning occurs in the exponentially suppressed tail of hydrogen that extends to the hotter regions of the envelope where protons are readily captured. Diffusive nuclear burning changes the compositional structure of the envelope on timescales $\sim 10^{2-4} \,{\rm yrs}$, much shorter than otherwise expected. This mechanism is applicable to the physics of young pulsars, millisecond radio pulsars (MSPs) and quiescent low mass X-ray binaries (LMXBs).'
author:
- Philip Chang
- Lars Bildsten
title: Diffusive Nuclear Burning in Neutron Star Envelopes
---
Introduction
============
The composition of NS envelopes affects their cooling (Potekhin, Chabrier & Yakovlev 1997) and thermal radiation (Romani 1987). The amount of material needed to change the spectral profile of the thermal radiation from the surface of a NS is miniscule ($\sim
10^{-20} {\ensuremath{M_\odot}}$). Such a small amount of contamination could easily be produced from spallation (Bildsten, Salpeter & Wasserman 1992) of fallback material (Woosley & Weaver 1995) soon after a supernova explosion.
Recent X-ray observations of the thermal spectrum from young radio pulsars have yielded tantalizing clues of their photospheric makeup (see Pavlov, Zavlin & Sanwal 2002 for a review). The thermal emission from young NSs can be fit with two models: magnetic hydrogen atmospheres or blackbody atmospheres. Both atmospheres fit the thermal spectra equally well, however they yield different effective temperatures and different solid angles for the emission area. The model which yields a more reasonable radius for the NS at the preferred distance is then taken as the favored model (Pavlov et al. 2002). For radio pulsars younger than $\sim 10^{4-5}\,{\rm
yrs}$ (e.g. Vela), the radius is reasonable for a magnetic hydrogen atmosphere model (Pavlov et al. 2001). For pulsars older than $\sim
10^{4-5}\,{\rm yrs}$ (e.g. PSR B0656+14), a blackbody model is favored (see Pavlov et al. 2002 and references therein). This suggests a possible evolution of hydrogen to more blackbody like elements on a timescale of $10^{4-5}\,{\rm yrs}$, and these indications have motivated our work.
At the photosphere, the local temperature ($T \sim 10^6\,{\rm K}$) and density ($\rho \sim 1 \,{\rm g}\,{\rm cm}^{-3}$) are too low to allow any significant nuclear evolution over $10^{4-5}\,{\rm yrs}$. However at a depth 1 m underneath the photosphere, the temperature is roughly two orders of magnitude greater ($T \sim 10^8\,{\rm K}$). Consider a NS envelope of hydrogen on carbon (we use C here as our fiducial proton capturing nucleus, results for other nuclei are in Chang & Bildsten 2003; CB03 hereafter) as shown in Figure 1.
{width="40.00000%"}
\[fig:envelope\_diagram\]
In diffusive equilibrium, the separation between the hydrogen and carbon is not strict. Rather a diffusive tail of hydrogen penetrates deep into the carbon layer. Protons easily reach a depth where the temperature is sufficiently high for rapid capture onto C. Since the hydrogen’s diffusive tail is exponentially suppressed, there will be a region where proton captures are peaked, which we call the burning layer. The consumption of the hydrogen by burning will drive the diffusive tail out of equilibrium, which sets up a diffusive current of hydrogen that flows down from the hydrogen layer into the burning layer. We refer to this burning mechanism (first mentioned by Chui & Salpeter 1964 and initially calculated by Rosen 1968) as diffusive nuclear burning (DNB; CB03). Over time, given that there is no interstellar accretion or other processes to refresh the hydrogen layer, the H is depleted.
This scenario presents several competing timescales. The first is the proton capture timescale in the burning layer, $\tau_{\rm nuc}$. The second is the time it takes protons to diffuse into the burning layer, $\tau_{\rm diff}$. In the case where $\tau_{\rm diff} \ll \tau_{\rm
nuc}$, the diffusive tail is always in equilibrium. In the opposite limit, the diffusive tail is modified by the burning. For simplicity we will only discuss the first case here.
The equilibrium structure of the diffusive tail is calculated from hydrostatic balance for each ion, $$\begin{aligned}
\label{eq:hb}
\frac {dP_i} {dr} &=& -n_i \left( A_i m_p g - Z_i e E\right), \\
\frac {dP_e} {dr} &=& -n_e\left(m_e g + e E\right),\end{aligned}$$ where $P_i$, $n_i$, $A_i$, $Z_i$ are the pressure, number density, atomic number and charge of the $i$’th ion species and $E$ is the upward pointing electric field found by demanding charge neutrality, $n_e = \sum n_i Z_i$. The thermal structure is determined from the constant flux equation, $$\frac {dT} {dr} = -\frac {3 \kappa \rho} {16 T^3}T_e^4,
\label{eq:flux}$$ where $\kappa$ is the opacity and $T_e$ is the effective temperature. Given appropriate microphysics, the equilibrium thermal and compositional structure can be calculated (see CB03 for additional details). The microphysics we have chosen in this paper are valid for the non-magnetic case, $B < 10^9 \,{\rm G}$.
{width="55.00000%"}
The local hydrogen burning rate is $m_p n_H/\tau_{\rm nuc}$, hence the total hydrogen burning rate per area on a NS, $\zeta_H$, is $$\zeta_H = \frac {y_H} {\tau_{\rm col}} =
\int \frac {n_H m_p} {\tau_H(n_H, n_C, T)} dz,
\label{eq:col_burning_rate}$$ where $y_H = \int m_p n_H dz$ is the integrated column of hydrogen and $\tau_{\rm col}$ is the characteristic burning time for that column. In Figure 2, we show the equilibrium structure for a fiducial NS model with $T_e = 8 \times 10^5\,{\rm K}$ and the total burning rate per log column, $y = P/g$. The Gaussian peak in the burning rate traces out the burning zone which is centered around a column of $y \approx
10^6\,{\rm g\,cm}^{-2}$, which is about 20 cm below where most of the H resides.
{width="55.00000%"}
We relate the burning rate per column into a total mass burning rate $\dot{M}_{\rm DNB} = 4\pi R_{*}^2 \zeta_H$. In Figure 3, we plot the characteristic burning time $\tau_{\rm col}$ and associated $\dot{M}_{\rm DNB}$ as a function of the total hydrogen column, $y_H$. For a given envelope with an initial column of hydrogen at a fixed core temperature, the evolution follows the curve up to the photosphere at $y_H \sim 1\,{\rm g\,cm}^{-2}$. Because the evolution follows a simple power law, the lifetime for any given envelope is completely determined by the characteristic burning time, $\tau_{\rm
col}$, at the photosphere. The curve for a central temperature of $T_c = 5 \times 10^7\,{\rm K}$ is cut off at large columns since the energy release from proton captures is comparable to the flux leaving the envelope. Hence our assumption of constant flux is violated. What is also notable about Figure 3 is that the curves follow a power law dependence which we derive in CB03.
Future directions for DNB are including the microphysics for magnetic NSs, calculating DNB without the constraint $\tau_{\rm diff} \ll
\tau_{\rm nuc}$ and allowing for an intervening helium layer. DNB allows for young neutron stars to have atmospheres other than hydrogen soon after birth. For instance, NSs with carbon, nitrogen or oxygen photospheres can easily exist after burning off any initial H. DNB may help explain the recent observation of magnetic oxygen lines on 1E1207+56 (see De Luca’s review in this volume, also see Hailey & Mori 2002).
This research was supported by NASA via grant NAG 5-8658 and by the NSF under Grants PHY99-07949 and AST01-96422. L. B. is a Cottrell Scholar of the Research Corporation.
Bildsten, L., Salpeter, E. E. & Wasserman, I. 1992, , 384, 143
Chang, P. & Bildsten, L. 2003, to appear in , astro-ph/0210218
Chiu, H. Y., Salpeter, E. E., 1964, , 12, 413
Hailey, C. J. & Mori, K. 2002, , 578, L133
Pavlov, G. G., Zavlin, V. E., Sanwal, D., Burwitz, V., Garmire, G. P. 2001, , 552, L129
Pavlov, G. G., Zavlin, V. E., & Sanwal, D. 2002, in Proceedings of the 270. Heraeus Seminar on Neutron Stars, Pulsars and Supernova Remnants, ed. W. Becker, H. Lesch and J. Trumper, astro-ph/0206024
Potekhin, A. Y., Chabrier, G., & Yakovlev, D. G. 1997, , 323, 415
Potekhin, A. Y. & Yakovlev, D. G. 2001, , 374, 213
Romani, R. W. 1987, , 313, 718
Rosen, L. C., 1968, , 1, 372
Woosley, S. E. & Weaver, T. A. 1995, , 101, 181
|
---
abstract: 'Riemannian Geometry, Topology and Dynamics permit to introduce partially defined holomorphic functions on the variety of representations of the fundamental group of a manifold. The functions we consider are the complex valued Ray–Singer torsion, the Milnor–Turaev torsion, and the dynamical torsion. They are associated essentially to a closed smooth manifold equipped with a (co)Euler structure and a Riemannian metric in the first case, a smooth triangulation in the second case, and a smooth flow of type described in section \[S:2\] in the third case. In this paper we define these functions, describe some of their properties and calculate them in some case. We conjecture that they are essentially equal and have analytic continuation to rational functions on the variety of representations. We discuss the case of one dimensional representations and other relevant situations when the conjecture is true. As particular cases of our torsions, we recognize familiar rational functions in topology such as the Lefschetz zeta function of a diffeomorphism, the dynamical zeta function of closed trajectories, and the Alexander polynomial of a knot. A numerical invariant derived from Ray–Singer torsion and associated to two homotopic acyclic representations is discussed in the last section.'
address:
- |
Dept. of Mathematics, The Ohio State University,\
231 West 18th Avenue, Columbus, OH 43210, USA.
- |
Department of Mathematics, University of Vienna,\
Nordbergstrasse 15, A-1090, Vienna, Austria.
author:
- Dan Burghelea
- Stefan Haller
title: |
Torsion, as a function on the space of\
representations
---
[^1]
Introduction {#S:intro}
============
For a finitely presented group $\Gamma$ denote by $\operatorname{Rep}(\Gamma;V)$ the algebraic set of all complex representations of $\Gamma$ on the complex vector space $V$. For a closed base pointed manifold $(M,x_0)$ with $\Gamma=\pi_1(M,x_0)$ denote by $\operatorname{Rep}^M(\Gamma;V)$ the algebraic closure of $\operatorname{Rep}^M_0(\Gamma;V)$, the Zariski open set of representations $\rho\in\operatorname{Rep}(\Gamma;V)$ so that $H^*(M;\rho)=0$. The manifold $M$ is called $V$-acyclic iff $\operatorname{Rep}^M(\Gamma;V)$, or equivalently $\operatorname{Rep}^M_0(\Gamma;V)$, is non-empty. If $M$ is $V$-acyclic then the Euler–Poincaré characteristic $\chi(M)$ vanishes. There are plenty of $V$-acyclic manifolds.
If $\dim V=1$ then $\operatorname{Rep}(\Gamma;V)=(\mathbb C\setminus 0)^k\times F$, where $k$ denotes the first Betty number of $M$, and $F$ is a finite Abelian group. If in addition $M$ is $V$-acyclic and $H_1(M;\mathbb Z)$ is torsion free, then $\operatorname{Rep}^M(\Gamma;V)=(\mathbb C\setminus 0)^k$. There are plenty of $V$-acyclic ($\dim V=1$) manifolds $M$ with $H_1(M;\mathbb Z)$ torsion free.
In this paper, to a $V$-acyclic manifold and an Euler or coEuler structure we associate three partially defined holomorphic functions on $\operatorname{Rep}^M(\Gamma;V)$, the complex valued Ray–Singer torsion, the Milnor–Turaev torsion, and the dynamical torsion, and describe some of their properties. They are defined with the help of a Riemannian metric, resp. smooth triangulation resp. a vector field with the properties listed in section \[SS:2.3\], but are independent of these data.
We conjecture that they are essentially equal and have analytic continuation to rational functions on $\operatorname{Rep}^M(\Gamma;V)$ and discuss the cases when we know that this is true. If $\dim V=1$ they are genuine rational functions of $k$ variables.
We calculate them in some cases and recognize familiar rational functions in topology (Lefschetz zeta function of a diffeomorphism, dynamical zeta function of some flows, Alexander polynomial of a knot) as particular cases of our torsions, cf. section \[S:7\].
The results answer the question
- *Is the Ray–Singer torsion the absolute value of a holomorphic function on the space of representations?*[^2]
(for a related result consult [@BK05]) and establish the analytic continuation of the dynamical torsion. Both issues are subtle when formulated inside the field of spectral geometry or of dynamical systems and can hardly be decided using internal technologies in these fields. There are interesting dynamical implications on the growth of the number of instantons and of closed trajectories, some of them improving on a conjecture formulated by S.P. Novikov about the gradients of closed Morse one form, cf. section \[S:8\].
.1in
This paper surveys results from [@BH04], [@BH03'], [@BH05] and reports on additional work in progress on these lines. Its contents is the following.
In section \[S:2\], for the reader’s convenience, we recall some less familiar characteristic forms used in this paper and describe the class of vector fields we use to define the dynamical torsion. These vector fields have finitely many rest points but infinitely many instantons and closed trajectories. However, despite their infiniteness, they can be counted by appropriate counting functions which can be related to the topology and the geometry of the underlying manifold cf. [@BH03]. The dynamical torsion is derived from them.
All torsion functions referred to above involve some additional topological data; the Milnor–Turaev and dynamical torsion involve an Euler structure while the complex Ray–Singer torsion a coEuler structure, a sort of Poincaré dual of the first. In section \[S:3\] we define Euler and coEuler structures and discuss some of their properties. Although they can be defined for arbitrary base pointed manifolds $(M,x_0)$ we present the theory only in the case $\chi(M)=0$ when the base point is irrelevant.
While the complex Ray–Singer torsion and dynamical torsion are new concepts the Milnor–Turaev torsion is not, however our presentation is somehow different from the traditional one. In section \[S:4\] we discuss the algebraic variety of cochain complexes of finite dimensional vector spaces and introduce the Milnor torsion as a rational function on this variety. The Milnor–Turaev torsion is obtained as a pull back by a characteristic map of this rational function.
Section \[S:5\] is about analytic torsions. In section \[SS:5.1\], we recall the familiar Ray–Singer torsion slightly modified with the help of a coEuler structure. This is a positive real valued function defined on $\operatorname{Rep}^M_0(\Gamma;V)$, the variety of the acyclic representations. We show that this function is independent of the Riemannian metric, and that it is the absolute value of a rational function, provided the coEuler structure is integral. In section \[SS:5.3\] we introduce the complex valued Ray–Singer torsion, and show the relation to the first. The complex Ray–Singer torsion, denoted $\mathcal S\mathcal T$, is a meromorphic function on a finite cover of the space of representations and is defined analytically using regularized determinants of elliptic operators but not self adjoint.
The Milnor–Turaev torsion, defined in section \[SS:6.1\], is associated with a smooth manifold, a given Euler structure and a homology orientation and is constructed using a smooth triangulation. Its square is conjecturally equal to the complex Ray–Singer torsion as defined in section \[SS:5.3\], when the coEuler structure for Ray–Singer corresponds, by Poincaré duality map, to the Euler structure for Milnor–Turaev. The conjecture is true in many relevant cases, in particular for $\dim V=1$.
Up to sign the dynamical torsion, introduced in section \[SS:6.2\], is associated to a smooth manifold and a given Euler structure and is constructed using a smooth vector field in the class described in section \[SS:2.3\]. The sign can be fixed with the help of an equivalence class of orderings of the rest points of $X$, cf. section \[SS:6.2\]. A priori the dynamical torsion is only a partially defined holomorphic function on $\operatorname{Rep}^M(\Gamma;V)$ and is defined using the instantons and the closed trajectories of $X$. For a representation $\rho$ the dynamical torsion is expressed as a series which might not be convergent for each $\rho$ but is certainly convergent for $\rho$ in a subset $U$ of $\operatorname{Rep}^M(\Gamma;V)$ with non-empty interior. At present this convergence was established only in the case of rank one representations. The existence of $U$ is guaranteed by the exponential growth property (EG) (cf. section \[SS:2.3\] for the definition) required from the vector field.
The main results, Theorems \[T:1\], \[T:2\] and \[T:3\], establish the relationship between these torsion functions, at least in the case $\dim V=1$, and a few other relevant cases. The same relationship is expected to hold for $V$ of arbitrary dimension.
One can calculate the Milnor–Turaev torsion when $M$ has a structure of mapping torus of a diffeomorphism $\phi$ as the “twisted Lefschetz zeta function” of the diffeomorphism $\phi$, cf. section \[SS:7.1\]. The Alexander polynomial as well as the twisted Alexander polynomials of a knot can also be recovered from these torsions cf. section \[SS:7.3\]. If the vector field has no rest points but admits a closed Lyapunov cohomology class, cf. section \[SS:7.2\], the dynamical torsion can be expressed in terms of closed trajectories only, and the dynamical zeta function of the vector field (including all its twisted versions) can be recovered from the dynamical torsion described here.
In section \[SS:8.1\] we express the phase difference of the Milnor–Turaev torsion of two representations in the same connected component of $\operatorname{Rep}^M_0(\Gamma;V)$ in terms of the Ray–Singer torsion. This invariant is analogous to the Atiyah–Patodi–Singer spectral flow but has not been investigated so far. Section \[S:8\] discusses some progress towards a conjecture of Novikov which came out from the work on dynamical torsion.
Characteristic forms and vector fields {#S:2}
======================================
Euler, Chern–Simons, and Mathai–Quillen form {#SS:2.1}
--------------------------------------------
Let $M$ be smooth closed manifold of dimension $n$. Let $\pi:TM\to M$ denote the tangent bundle, and ${\mathcal O}_M$ the orientation bundle, which is a flat real line bundle over $M$. For a Riemannian metric $g$ denote by $$\operatorname{e}(g)\in\Omega^n(M;{\mathcal O}_M)$$ its Euler form, and for two Riemannian metrics $g_1$ and $g_2$ by $$\operatorname{cs}(g_1,g_2)\in
\Omega^{n-1}(M;{\mathcal O}_M)/d(\Omega^{n-2}(M;{\mathcal O}_M))$$ their Chern–Simons class. The following properties follow from and below. $$\begin{aligned}
d \operatorname{cs}(g_1,g_2) &=& \operatorname{e}(g_2)-\operatorname{e}(g_1)
\label{E:csg:i}
\\
\operatorname{cs}(g_2,g_1) &=& -\operatorname{cs}(g_1,g_2)
\label{E:csg:ii}
\\
\operatorname{cs}(g_1,g_3) &=& \operatorname{cs}(g_1,g_2) + \operatorname{cs}(g_2,g_3)
\label{E:csg:iii}\end{aligned}$$ If the dimension of $M$ is odd both $\operatorname{e}(g)$ and $\operatorname{cs}(g_1, g_2)$ vanish.
Denote by $\xi$ the Euler vector field on $TM$ which assigns to a point $x\in TM$ the vertical vector $-x\in T_xTM$. A Riemannian metric $g$ determines the Levi–Civita connection in the bundle $\pi:TM\to M$. There is a canonic $n$-form $\operatorname{vol}(g)\in\Omega^n(TM;\pi^*{\mathcal O}_M)$, which assigns to an $n$-tuple of vertical vectors *their volume times their orientation* and vanishes when contracted with horizontal vectors and a global angular form, see for instance [@BT82], is the differential form $$A(g)
:=\frac{\Gamma(n/2)}{(2\pi)^{n/2}|\xi|^n}
i_\xi\operatorname{vol}(g)\in\Omega^{n-1}(TM\setminus M;\pi^*{\mathcal O}_M).$$ In [@MQ86] Mathai and Quillen have introduced a differential form $$\Psi(g)\in \Omega^{n-1}(TM\setminus M;\pi^*{\mathcal O}_M).$$ When pulled back to the fibers of $TM\setminus M\to M$ the form $\Psi(g)$ coincides with $A(g)$. If $U\subseteq M$ is an open subset on which the curvature of $g$ vanishes, then $\Psi(g)$ coincides with $A(g)$ on $TU\setminus U$. In general we have the equalities $$\begin{aligned}
d\Psi(g) &=& \pi^*\operatorname{e}(g).
\label{dpsie}
\\
\Psi(g_2)-\Psi(g_1) &=& \pi^*\operatorname{cs}(g_1,g_2) \mod d\Omega^{n-2}(TM \setminus M;\pi^* {\mathcal O}_M).
\label{pg1pg2cs}\end{aligned}$$
Euler and Chern–Simons chains {#SS:2.22}
-----------------------------
For a vector field $X$ with non-degenerate rest points we have the singular $0$-chain $\operatorname{e}(X)\in C_0(M;\mathbb Z)$ defined by $\operatorname{e}(X):=\sum_{x\in\mathcal X}\operatorname{IND}(x)x$, with $\operatorname{IND}(x)$ the Hopf index.
For two vector fields $X_1$ and $X_2$ with non-degenerate rest points we have the singular $1$-chain rel. boundaries $\operatorname{cs}(X_1,X_2)\in C_1(M;\mathbb Z)/\partial C_2(M;\mathbb Z)$ defined from the zero set of a homotopy from $X_1$ to $X_2$ cf. [@BH04]. They are related by the formulas, see [@BH04], $$\begin{aligned}
\partial \operatorname{cs}(X_1,X_2) &=& \operatorname{e}(X_2)-\operatorname{e}(X_1)
\label{E:11}
\\
\operatorname{cs}(X_2,X_1) &=& -\operatorname{cs}(X_1,X_2)
\label{E:12i}
\\
\operatorname{cs}(X_1,X_3) &=& \operatorname{cs}(X_1,X_2) + \operatorname{cs}(X_2,X_3).
\label{E:13}\end{aligned}$$
Kamber–Tondeur one form {#SS:2.2}
-----------------------
Let $E$ be a real or complex vector bundle over $M$. For a connection $\nabla$ and a Hermitian structure $\mu$ on $E$ define a real valued one form $\omega(\nabla,\mu)\in\Omega^1(M;{\mathbb R})$ by $$\label{E:6}
\omega(\nabla,\mu)(Y):=-\frac12\operatorname{tr}\bigl(\mu^{-1}\cdot (\nabla_Y\mu)\bigr),
\quad Y\in TM.$$ Here we consider $\mu$ as an element in $\Omega^0(M;\hom(E,\bar E^*))$, where $\bar E^*$ denotes the dual of the complex conjugate bundle. With respect to the induced connection on $\hom(E,\bar E^*)$ we have $\nabla_Y\mu\in\Omega^1(M;\hom(E,\bar E^*))$ and therefore $\mu^{-1}\cdot\nabla_Y\mu\in\Omega^1(M;\operatorname{end}(E,E))$. Actually the latter one form has values in the endomorphisms of $E$ which are symmetric with respect to $\mu$, and thus the (complex) trace, see , will indeed be real. Since any two Hermitian structures $\mu_1$ and $\mu_2$ are homotopic, the difference $\omega(\nabla,\mu_2)-\omega(\nabla,\mu_1)$ will be exact. If $\nabla$ is flat then $\omega(\nabla,\mu)$ is closed and its cohomology class independent of $\mu$.
Replacing the Hermitian structure by a non-degenerate symmetric bilinear form $b$, we define a complex valued one form $\omega(\nabla,b)\in\Omega^1(M;{\mathbb C})$ by a similar formula $$\label {E:7}
\omega(\nabla,b)(Y):=-\frac12\operatorname{tr}\bigl(b^{-1}\cdot(\nabla_Yb)\bigr),\quad Y\in TM.$$ Here we regard $b$ as an element in $\Omega^0(M;\hom(E,E^*))$. If two non-degenerate symmetric bilinear forms $b_1$ and $b_2$ are homotopic, then $\omega(\nabla,b_2)-\omega(\nabla,b_1)$ is exact. If $\nabla$ is flat, then $\omega(\nabla,b)$ is closed. Note that $\omega(\nabla,b)\in\Omega^1(M;\mathbb C)$ depends holomorphically on $\nabla$.
Vector fields, instantons and closed trajectories {#SS:2.3}
-------------------------------------------------
Consider a vector field $X$ which satisfies the following properties:
- All rest points are of hyperbolic type.
- The vector field has exponential growth at all rest points.
- The vector field is of Lyapunov type.
- The vector field satisfies Morse–Smale condition.
- The vector field has all closed trajectories non-degenerate.
Precisely this means that:
- In the neighborhood of each rest point the differential of $X$ has all eigenvalues with non-trivial real part; the number of eigenvalues with negative real part is called the index and denoted by $\operatorname{ind}(x)$; as a consequence the stable and unstable stable sets are images of one-to-one immersions $i_x^\pm:W^\pm_x\to M$ with $W^\pm_x$ diffeomorphic to $\mathbb R^{n-\operatorname{ind}(x)}$ resp. $\mathbb R^{\operatorname{ind}(x)}$.
- With respect to one and then any Riemannian metric $g$ on $M$, the volume of the disk of radius $r$ in $W^-_x$ (w.r. to the induced Riemannian metric) is $\leq e^{Cr}$, for some constant $C>0$.
- There exists a real valued closed one form $\omega$ so that $\omega(X)_x<0$ for $x$ not a rest point.[^3]
- For any two rest points $x$ and $y$ the mappings $i^-_x$ and $i^+_y$ are transversal and therefore the space of non-parameterized trajectories form $x$ to $y$, $\mathcal T(x,y)$, is a smooth manifold of dimension $\operatorname{ind}(x)-\operatorname{ind}(y)-1$. Instantons are exactly the elements of $\mathcal T(x,y)$ when this is a smooth manifold of dimension zero, i.e. $\operatorname{ind}(x)-\operatorname{ind}(y)-1=0$.
- Any closed trajectory is non-degenerate, i.e. the differential of the return map in normal direction at one and then any point of a closed trajectory does not have non-zero fixed points.
Recall that a trajectory $\theta$ is an equivalence class of parameterized trajectories and two parameterized trajectories $\theta_1$ and $\theta_2$ are equivalent iff $\theta_1(t+c)= \theta_2(t)$ for some real number $c.$ Recall that a closed trajectory $\hat\theta$ is a pair consisting of a trajectory $\theta$ and a positive real number $T$ so that $\theta(t+T)= \theta(t).$
Property (L), (H), (MS) imply that for any real number $R$ the set of instantons $\theta$ from $x$ to $y$ with $-\omega([\theta])\leq R$ is finite and properties (L), (H), (MS), (NCT) imply that for any real number $R$ the set of the closed trajectory $\hat\theta$ with $-\omega([\hat \theta])\leq R$ is finite. Here $[\theta]$ resp $[\hat\theta]$ denote the homotopy class of instantons resp. closed trajectories [^4].
Denote by $\mathcal P _{x,y}$ the set of homotopy classes of paths from $x$ to $y$ and by $\mathcal X_q$ the set of rest points of index $q$. Suppose a collection $\mathcal O=\{\mathcal O_x\mid x\in\mathcal X\}$ of orientations of the unstable manifolds is given and (MS) is satisfied. Then any instanton $\theta$ has a sign $\epsilon(\theta)=\pm1$ and therefore, if (L) is also satisfied, for any two rest points $x\in{\mathcal X}_{q+1}$ and $y\in \mathcal X_{q}$ we have the counting function of instantons $\mathbb I^{X,\mathcal O}_{x,y}: \mathcal P_{x,y}\to\mathbb Z$ defined by $$\label {E:8}
\mathbb I^{X,\mathcal O}_{x,y}(\alpha):=\sum_{\theta\in\alpha}\epsilon(\theta).$$
Under the hypothesis (NCT) any closed trajectory $\hat\theta$ has a sign $\epsilon(\hat\theta)=\pm1$ and a period $p(\hat\theta)\in\{1,2,\dotsc\}$, cf. [@H02]. If (H), (L), (MS), (NCT) are satisfied, as the set of closed trajectories in a fixed homotopy class $\gamma\in[S^1,M]$ is compact, we have the counting function of closed trajectories $\mathbb Z_X:[S^1,M]\to\mathbb Q$ defined by $$\label{E:9}
\mathbb Z_X(\gamma):=\sum_{\hat\theta\in\gamma}\epsilon(\hat\theta)/p(\hat\theta).$$ Here are a few properties about vector fields which satisfy (H) and (L).
\[P:1\] 1. Given a vector field $X$ which satisfies (H) and (L) arbitrary close in the $C^r$-topology for any $r\geq0$ there exists a vector field $Y$ which agrees with $X$ on a neighborhood of the rest points and satisfies (H), (L), (MS) and (NCT).
2\. Given a vector field $X$ which satisfies (H) and (L) arbitrary close in the $C^0$-topology there exists a vector field $Y$ which agrees with $X$ on a neighborhood of the rest points and satisfies (H), (EG), (L), (MS) and (NCT).
3\. If $X$ satisfies (H), (L) and (MS) and a collection $\mathcal O$ of orientations is given then for any $x\in\mathcal X_q$, $z\in\mathcal X_{q-2}$ and $ \gamma\in\mathcal P_{x,z}$ one has[^5] $$\label{E:10}
\sum_{\begin{smallmatrix} y\in{\mathcal X}_{q-1},\alpha\in\mathcal P_{x,y},
\beta\in \mathcal P_{y,z} \\ \alpha* \beta=\gamma\end{smallmatrix}}
\mathbb I^{X,\mathcal O}_{x,y}(\alpha)\cdot
\mathbb I^{X,\mathcal O}_{y,z}(\beta)=0.$$
This proposition is a recollection of some of the main results in [@BH03], see Proposition 3, Theorem 1 and Theorem 5 in there.
Euler and coEuler structures {#S:3}
============================
Although not always necessary in this section as in fact always in this paper $M$ is supposed to be closed connected smooth manifold.
Euler structures {#SS:3.1}
----------------
Euler structures have been introduced by Turaev [@Tu90] for manifolds $M$ with $\chi(M)=0$. If one removes the hypothesis $\chi(M)=0$ the concept of Euler structure can still be considered for any connected base pointed manifold $(M,x_0)$ cf. [@B99] and [@BH04]. Here we will consider only the case $\chi(M)=0$. The set of Euler structures, denoted by ${\mathfrak{Eul}}(M;{\mathbb Z})$, is equipped with a free and transitive action $$m:H_1(M;{\mathbb Z})\times{\mathfrak{Eul}}(M;{\mathbb Z})\to{\mathfrak{Eul}}(M;{\mathbb Z})$$ which makes ${\mathfrak{Eul}}(M;{\mathbb Z})$ an affine version of $H_1(M;{\mathbb Z})$. If ${\mathfrak e}_1,{\mathfrak e}_2$ are two Euler structure we write ${\mathfrak e}_2-{\mathfrak e}_1$ for the unique element in $H_1(M;{\mathbb Z})$ with $m({\mathfrak e}_2-{\mathfrak e}_1,{\mathfrak e}_1)={\mathfrak e}_2$.
To define the set ${\mathfrak{Eul}}(M;{\mathbb Z})$ we consider pairs $(X,c)$ with $X$ a vector field with non-degenerate zeros and $c\in C_1(M;\mathbb Z)$ so that $\partial
c=\operatorname{e}(X)$. We make $(X_1,c_1)$ and $(X_2,c_2)$ equivalent iff $c_2-c_1=\operatorname{cs}(X_1,X_2)$ and write $[X,c]$ for the equivalence class represented by $(X,c)$. The action $m$ is defined by $m([c'],[X, c]):=[X,c'+c]$.
\[O:1\] Suppose $X$ is a vector field with non-degenerate zeros, and assume its zero set $\mathcal X$ is non-empty. Moreover, let ${\mathfrak e}\in{\mathfrak{Eul}}(M;\mathbb Z)$ be an Euler structure and $x_0\in M$. Then there exists a collection of paths $\{\sigma_x\mid x\in\mathcal X\}$ with $\sigma_x(0)=x_0$, $\sigma_x(1)=x$ and such that ${\mathfrak e}=[X,c]$ where $c=\sum_{x\in\mathcal X}\operatorname{IND}(x)\sigma_x$.
A remarkable source of Euler structures is the set of homotopy classes of nowhere vanishing vector fields. Any nowhere vanishing vector field $X$ provides an Euler structure $[X,0]$ which only depends on the homotopy class of $X$. Still assuming $\chi(M)=0$, every Euler structure can be obtained in this way provided $\dim(M)>2$. Be aware, however, that different homotopy classes may give rise to the same Euler structure.
To construct such a homotopy class one can proceed as follows. Represent the Euler structure ${\mathfrak e}$ by a vector field $X$ and a collection of paths $\{\sigma_x\mid x\in\mathcal X\}$ as in Observation \[O:1\]. Since $\dim(M)>2$ we may assume that the interiors of the paths are mutually disjoint. Then the set $\bigcup_{x\in\mathcal
X}\sigma_x$ is contractible. A smooth regular neighborhood of it is the image by a smooth embedding $\varphi:(D^n,0)\to(M,x_0)$. Since $\chi(M)=0$, the restriction of the vector field $X$ to $M\setminus\operatorname{int}(D^n)$ can be extended to a non-vanishing vector field $\tilde X$ on $M$. It is readily checked that $[\tilde X,0]={\mathfrak e}$. For details see [@BH04].
If $M$ dimension larger than 2 an alternative description of ${\mathfrak{Eul}}(M;{\mathbb Z})$ with respect to a base point $x_0$ is ${\mathfrak{Eul}}(M;{\mathbb Z})=\pi_0({\mathfrak X}(M,x_0))$, where ${\mathfrak X}(M,x_0)$ denotes the space of vector fields of class $C^r$, $r\geq 0$, which vanish at $x_0$ and are non-zero elsewhere. We equip this space with the $C^r$-topology and note that the result $\pi_0({\mathfrak X}(M,x_0))$ is the same for all $r$, and since $\chi(M)=0$, canonically identified for different base points.
Let $\tau$ be a smooth triangulation of $M$ and consider the function $f_\tau:M\to{\mathbb R}$ linear on any simplex of the first barycentric subdivision and taking the value $\dim(s)$ on the barycenter $x_s$ of the simplex $s\in\tau$. A smooth vector field $X$ on $M$ with the barycenters as the only rest points all of them hyperbolic and $f_\tau$ strictly decreasing on non-constant trajectories is called an Euler vector field of $\tau$. By an argument of convexity two Euler vector fields are homotopic by a homotopy of Euler vector fields.[^6] Therefore, a triangulation $\tau$, a base point $x_0$ and a collection of paths $\{\sigma_s\mid s\in\tau\}$ with $\sigma_s(0)=x_0$ and $\sigma_s(1)=x_s$ define an Euler structure $[X_\tau,c]$, where $c:=\sum_{s\in\tau}(-1)^{n+\dim(s)}\sigma_s$, $X_\tau$ is any Euler vector field for $\tau$, and this Euler structure does not depend on the choice of $X_\tau$. Clearly, for fixed $\tau$ and $x_0$, every Euler structure can be realized in this way by an appropriate choice of $\{\sigma_s\mid s\in\tau\}$, cf. Observation \[O:1\].
Co-Euler structures {#SS:3.2}
-------------------
Again, suppose $\chi(M)=0$.[^7] Consider pairs $(g,\alpha)$ where $g$ is a Riemannian metric on $M$ and $\alpha\in\Omega^{n-1}(M;\mathcal O_M)$ with $d\alpha=\operatorname{e}(g)$ where $\operatorname{e}(g)\in\Omega^n(M;\mathcal O_M)$ denotes the Euler form of $g$, see section \[SS:2.1\]. We call two pairs $(g_1,\alpha_1)$ and $(g_2,\alpha_2)$ equivalent if $$\operatorname{cs}(g_1,g_2)=\alpha_2-\alpha_1
\in\Omega^{n-1}(M;\mathcal O_M)/
d\Omega^{n-2}(M;\mathcal O_M).$$ We will write ${\mathfrak{Eul}}^*(M;{\mathbb R})$ for the set of equivalence classes and $[g,\alpha]$ for the equivalence class represented by the pair $(g,\alpha)$. Elements of ${\mathfrak{Eul}}^*(M;{\mathbb R})$ are called *coEuler structures.*
There is a natural action $$m^*:H^{n-1}(M;{\mathcal O}_M)\times{\mathfrak{Eul}}^*(M;{\mathbb R})\to{\mathfrak{Eul}}^*(M;{\mathbb R})$$ given by $$m^*([\beta],[g,\alpha]):=[g,\alpha-\beta]$$ for $[\beta]\in H^{n-1}(M;\mathcal O_M)$. This action is obviously free and transitive. In this sense ${\mathfrak{Eul}}^*(M;{\mathbb R})$ is an affine version of $H^{n-1}(M;\mathcal O_M)$. If ${\mathfrak e}^*_1$ and ${\mathfrak e}^*_2$ are two coEuler structures we write ${\mathfrak e}^*_2-{\mathfrak e}^*_1$ for the unique element in $H^{n-1}(M;{\mathcal O}_M)$ with $m^*({\mathfrak e}^*_2-{\mathfrak e}^*_1,{\mathfrak e}^*_1)={\mathfrak e}^*_2$.
\[O:2\] Given a Riemannian metric $g$ on $M$ any coEuler structure can be represented as a pair $(g,\alpha)$ for some $\alpha\in\Omega^{n-1}(M;{\mathcal O}_M)$ with $d\alpha=\operatorname{e}(g)$.
There is a natural map $\operatorname{PD}:{\mathfrak{Eul}}(M;{\mathbb Z})\to{\mathfrak{Eul}}^*(M;{\mathbb R})$ which combined with the Poincaré duality map $D:H_1(M;{\mathbb Z})\to H_1(M;{\mathbb R})\to H^{n-1}(M;{\mathcal O}_M)$, the composition of the coefficient homomorphism for ${\mathbb Z}\to{\mathbb R}$ with the Poincaré duality isomorphism,[^8] makes the diagram below commutative: $$\xymatrix{
H_1(M;{\mathbb Z})\times{\mathfrak{Eul}}(M;{\mathbb Z})
\ar[d]_{D\times\operatorname{PD}}\ar[r]^-{m}
&
{\mathfrak{Eul}}(M;{\mathbb Z})\ar[d]^{\operatorname{PD}}
\\
H^{n-1}(M;{\mathcal O}_M)\times{\mathfrak{Eul}}^*(M;{\mathbb R})
\ar[r]^-{m^*}
&
{\mathfrak{Eul}}^*(M;{\mathbb R})
}$$
There are many ways to define the map $\operatorname{PD}$, cf. [@BH04]. For example, assuming $\chi(M)=0$ and $\dim M>2$ one can proceed as follows. Represent the Euler structure by a nowhere vanishing vector field ${\mathfrak e}=[X,0]$. Choose a Riemannian metric $g$, regard $X$ as mapping $X:M\to TM\setminus M$, set $\alpha:=X^*\Psi(g)$, put $\operatorname{PD}({\mathfrak e}):=[g,\alpha]$ and check that this does indeed only depend on ${\mathfrak e}$.
A coEuler structure ${\mathfrak e}^*\in{\mathfrak{Eul}}^*(M;{\mathbb R})$ is called *integral* if it belongs to the image of $\operatorname{PD}$. Integral coEuler structures constitute a lattice in the affine space ${\mathfrak{Eul}}^*(M;{\mathbb R})$.
If $\dim M$ is odd, then there is a canonical coEuler structure ${\mathfrak e}^*_0\in{\mathfrak{Eul}}^*(M;{\mathbb R})$; it is represented by the pair $[g,0]$, with any $g$ Riemannian metric. In general this coEuler structure is not integral.
Complex representations and cochain complexes {#S:4}
=============================================
Complex representations {#SS:4.1}
-----------------------
Let $\Gamma$ be a finitely presented group with generators $g_1,\dotsc,g_r$ and relations $$R_i(g_1,g_2,\dotsc,g_r)=e,\quad i=1,\dotsc,p,$$ and $V$ be a complex vector space of dimension $N$. Let $\operatorname{Rep}(\Gamma;V)$ be the set of linear representations of $\Gamma$ on $V$, i.e. group homomorphisms $\rho:\Gamma\to\operatorname{GL}_{\mathbb C}(V)$. By identifying $V$ to ${\mathbb C}^N$ this set is, in a natural way, an algebraic set inside the space ${\mathbb C}^{rN^2+1}$ given by $pN^2+1$ equations. Precisely if $A_1,\dotsc,A_r,z$ represent the coordinates in ${\mathbb C}^{rN^2+1}$ with $A:=(a^{ij})$, $a^{ij}\in{\mathbb C}$, so $A\in{\mathbb C}^{N^2}$ and $z\in{\mathbb C}$, then the equations defining $\operatorname{Rep}(\Gamma;V)$ are $$\begin{aligned}
z\cdot\det(A_1)\cdot\det(A_2)\cdots\det(A_r)&=&1
\\
R_i(A_1,\dotsc,A_r)&=&{\text{\rm id}},\qquad i=1,\dotsc,p\end{aligned}$$ with each of the equalities $R_i$ representing $N^2$ polynomial equations.
Suppose $\Gamma=\pi_1(M,x_0)$, $M$ a closed manifold. Denote by $\operatorname{Rep}^M_0(\Gamma;V)$ the set of representations $\rho$ with $H^*(M;\rho)=0$ and notice that they form a Zariski open set in $\operatorname{Rep}(\Gamma;V)$. Denote the closure of this set by $\operatorname{Rep}^M(\Gamma;V)$. This is an algebraic set which depends only on the homotopy type of $M$, and is a union of irreducible components of $\operatorname{Rep}(\Gamma;V)$.
Recall that every representation $\rho\in\operatorname{Rep}(\Gamma;V)$ induces a canonical vector bundle $F_\rho$ equipped with a canonical flat connection $\nabla_\rho$. They are obtained from the trivial bundle $\tilde M\times V\to\tilde M$ and the trivial connection by passing to the $\Gamma$ quotient spaces. Here $\tilde M$ is the canonical universal covering provided by the base point $x_0$. The $\Gamma$-action is the diagonal action of deck transformations on $\tilde M$ and of the action $\rho$ on $V$. The fiber of $F_\rho$ over $x_0$ identifies canonically with $V$. The holonomy representation determines a right $\Gamma$-action on the fiber of $F_\rho$ over $x_0$, i.e. an anti homomorphism $\Gamma\to\operatorname{GL}(V)$. When composed with the inversion in $\operatorname{GL}(V)$ we get back the representation $\rho$. The pair $(F_\rho,\nabla_\rho)$ will be denoted by $\mathbb F_\rho$.
If $\rho_0$ is a representation in the connected component $\operatorname{Rep}_\alpha(\Gamma;V)$ one can identify $\operatorname{Rep}_\alpha(\Gamma;V)$ to the connected component of $\nabla_{\rho_0}$ in the complex analytic space of flat connections of the bundle $F_{\rho_0}$ modulo the group of bundle isomorphisms of $F_{\rho_0}$ which fix the fiber above $x_0$.
An element $a\in H_1(M;{\mathbb Z})$ defines a holomorphic function $${\det}_a:\operatorname{Rep}^M(\Gamma;V)\to\mathbb C_*.$$ The complex number $\det_a(\rho)$ is the evaluation on $a\in H_1(M;{\mathbb Z})$ of $\det(\rho):\Gamma\to{\mathbb C}_*$ which factors through $H_1(M;{\mathbb Z})$. Note that for $a,b\in H_1(M;{\mathbb Z})$ we have $\det_{a+b}=\det_a\det_b$. If $a$ is a torsion element, then $\det_a$ is constant equal to a root of unity of order, the order of $a$.
The space of cochain complexes {#SS:4.3}
------------------------------
Let $k= (k_0,k_1,\dotsc,k_n)$ be a string of non-negative integers. The string is called admissible, and will write $k\geq0$ in this case, if the following requirements are satisfied $$\begin{aligned}
k_0-k_1+k_2\mp\cdots+(-1)^nk_n&=&0
\label{E:14}
\\
k_i-k_{i-1}+ k_{i-2}\mp\cdots+(-1)^ik_0&\geq&0
\quad\text{for any $i\leq n-1$.}
\label{E:15}\end{aligned}$$
Denote by $\mathbb D(k)=\mathbb D(k_0,\dotsc,k_n)$ the collection of cochain complexes of the form $$C=(C^*,d^*):
0\to C^0 \xrightarrow{d^0}
C^1\xrightarrow{d^1}
\cdots
\xrightarrow{d^{n-2}}
C^{n-1}
\xrightarrow{d^{n-1}} C^n\to 0$$ with $C^i:={\mathbb C}^{k_i}$, and by $\mathbb D_{\text{\rm ac}}(k)\subseteq\mathbb D(k)$ the subset of acyclic complexes. Note that $\mathbb D_{\text{\rm ac}}(k)$ is non-empty iff $k\geq0$. The cochain complex $C$ is determined by the collection $\{d^i\}$ of linear maps $d^i:{\mathbb C}^{k_i}\to{\mathbb C}^{k_{i+1}}$. If regarded as the subset of those $\{d^i\}\in\bigoplus_{i=0}^{n-1}L({\mathbb C}^{k_i},{\mathbb C}^{k_{i+1}})$, with $L(V,W)$ the space of linear maps from $V$ to $W$, which satisfy the quadratic equations $d^{i+1}\cdot d^i=0$, the set $\mathbb D(k)$ is an affine algebraic set given by degree two homogeneous polynomials and $\mathbb D_{\text{\rm ac}}(k)$ is a Zariski open set. The map $\pi_0:\mathbb D_{\text{\rm ac}}(k)\to\operatorname{Emb}(C^0,C^1)$ which associates to $C\in\mathbb D_{\text{\rm ac}}(k)$ the linear map $d^0$, is a bundle whose fiber is isomorphic to $\mathbb D_{\text{\rm ac}}(k_1-k_0,k_2,\dotsc,k_n)$.
This can be easily generalized as follows. Consider a string $b=(b_0,\dotsc,b_n)$. We will write $k\geq b$ if $k-b=(k_0-b_0,\dotsc,k_n-b_n)$ is admissible, i.e. $k-b\geq0$. Denote by $\mathbb D_b(k)=\mathbb D_{(b_0,\dotsc,b_n)}(k_0,\dotsc,k_n)$ the subset of cochain complexes $C\in\mathbb D(k)$ with $\dim(H^i(C))=b_i$. Note that $\mathbb D_b(k)$ is non-empty iff $k\geq b$. The obvious map $\pi_0:\mathbb D_b(k)\to L(C^0,C^1;b_0)$, $L(C^0,C^1;b_0)$ the space of linear maps in $L(C^0,C^1)$ whose kernel has dimension $b_0$, is a bundle whose fiber is isomorphic to $\mathbb D_{b_1,\dotsc,b_n}(k_1-k_0+b_0,k_2,\dotsc,k_n)$. Note that $L(C^0, C^1; b_0)$ is the total space of a bundle $\operatorname{Emb}(\underline{{\mathbb C}}^{k_0}/L,\underline{{\mathbb C}}^{k_1})\to\operatorname{Gr}_{b_0}(k_0)$ with $L\to\operatorname{Gr}_{b_0}(k_0)$ the tautological bundle over $\operatorname{Gr}_{b_0}(k_0)$ and $\underline{{\mathbb C}}^{k_0}$ resp. $\underline{{\mathbb C}}^{k_1}$ the trivial bundles over $\operatorname{Gr}_{b_0}(k_0)$ with fibers of dimension $k_0$ resp. $k_1$. As a consequence we have
\[P:2\] 1. $\mathbb D_{\text{\rm ac}}(k)$ and $\mathbb D_b(k)$ are connected smooth quasi affine algebraic sets whose dimension is $$\dim\mathbb D_b(k)=\sum_j(k^j-b^j)\cdot\Bigl(
k^j-\sum_{i\leq j}(-1)^{i+j}(k^i-b^i)
\Bigr).$$
2\. The closures $\hat{\mathbb D}_{\text{\rm ac}}(k)$ and $\hat{\mathbb D}_b(k)$ are irreducible algebraic sets, hence affine algebraic varieties, and $\hat{\mathbb D}_b(k)=\bigsqcup_{k\geq b'\geq b}{\mathbb D}_{b'}(k)$.
For any cochain complex in $C\in\mathbb D_{\text{\rm ac}}(k)$ denote by $B^i:=\operatorname{img}(d^{i-1})\subseteq C^i= {\mathbb C}^{k_i}$ and consider the short exact sequence $0\to B^i \xrightarrow{\textrm{inc}} C^i \xrightarrow {d^i} B^{i+1}\to 0$. Choose a base ${\mathfrak b}_i$ for each $B_i$, and choose lifts $\overline{{\mathfrak b}}_{i+1}$ of ${\mathfrak b}_{i+1}$ in $C^i$ using $d^i$, i.e. $d^i(\overline{{\mathfrak b}}_{i+1})={\mathfrak b}_{i+1}$. Clearly $\{{\mathfrak b}_i,\overline{\mathfrak b}_{i+1}\}$ is a base of $C^i$. Consider the base $\{{\mathfrak b}_i,\overline{{\mathfrak b}}_{i+1}\}$ as a collection of vectors in $C^i={\mathbb C}^{k_i}$ and write them as columns of a matrix $[{\mathfrak b}_i,\overline{{\mathfrak b}}_{i+1}]$. Define the torsion of the acyclic complex $C$, by $$\tau(C):=(-1)^{N+1}\prod^n_{i=0}\det[{\mathfrak b}_i,\overline{{\mathfrak b}}_{i+1}]^{(-1)^i}$$ where $(-1)^N$ is Turaev’s sign, see [@FT99]. The result is independent of the choice of the bases ${\mathfrak b}_i$ and of the lifts $\overline{{\mathfrak b}}_i$ cf. [@M66] [@FT99], and leads to the function $$\tau:\mathbb D_{\text{\rm ac}}(k)\to{\mathbb C}_*.$$ Turaev provided a simple formula for this function, cf. [@Tu01], which permits to recognize $\tau$ as the restriction of a rational function on $\hat{\mathbb D}_{\text{\rm ac}}(k)$.
For $C\in\hat{\mathbb D}_{\text{\rm ac}}(k)$ denote by $(d^i)^t:{\mathbb C}^{k_{i+1}}\to{\mathbb C}^{k_i}$ the transpose of $d^i:{\mathbb C}^{k_i}\to{\mathbb C}^{k_{i+1}}$, and define $P_i=d^{i-1}\cdot(d^{i-1})^t+(d^i)^t\cdot d^i$. Define $\Sigma(k)$ as the subset of cochain complexes in $\hat{\mathbb D}_{\text{\rm ac}}(k)$ where $\ker P\neq0$, and consider $S\tau:\hat{\mathbb D}_{\text{\rm ac}}(k)\setminus\Sigma(k)\to\mathbb C_*$ defined by $$S\tau(C)
:=\Bigl(\prod_{i\ {\text{\rm even}}}(\det P_i)^{i}
\big/\prod_{i\ {\text{\rm odd}}}(\det P_i)^{i}\Bigr)^{-1}.$$ One can verify
Suppose $k=(k_0,\dotsc,k_n)$ is admissible.
1\. $\Sigma(k)$ is a proper subvariety containing the singular set of $\hat{\mathbb D}_{\text{\rm ac}}(k)$.
2\. $S\tau=\tau^2$ and implicitly $S\tau$ has an analytic continuation to ${\mathbb D}_{\text{\rm ac}}(k)$.
In particular $\tau$ defines a square root of $S\tau$. We will not use explicitly $S\tau $ in this writing however it justifies the definition of complex Ray–Singer torsion.
Analytic torsion {#S:5}
================
Let $M$ be a closed manifold, $g$ Riemannian metric and $(g,\alpha)$ a representative of a coEuler structure ${\mathfrak e}^*\in{\mathfrak{Eul}}^*(M;{\mathbb R})$. Suppose $E\to M$ is a complex vector bundle and denote by $\mathcal C(E)$ the space of connections and by $\mathcal F(E)$ the subset of flat connections. $\mathcal C(E)$ is a complex affine (Fréchet) space while $\mathcal F(E)$ a closed complex analytic subset (Stein space) of $\mathcal C(E)$. Let $b$ be a non-degenerate symmetric bilinear form and $\mu$ a Hermitian (fiber metric) structure on $E$. While Hermitian structures always exist, non-degenerate symmetric bilinear forms exist iff the bundle is the complexification of some real vector bundle, and in this case $E\simeq E^*$.
The connection $\nabla\in\mathcal C(E)$ can be interpreted as a first order differential operator $d^\nabla:\Omega^*(M;E)\to\Omega^{*+1}(M;E)$ and $g$ and $b$ resp. $g$ and $\mu$ can be used to define the formal $b$-adjoint resp. $\mu$-adjoint $\delta^\nabla_{q;g,b}$ resp. $\delta^\nabla_{q;g,\mu}:\Omega^{q+1}(M;E)\to\Omega^q(M;E)$ and therefore the Laplacians $$\Delta^\nabla_{q;g,b}
\ \text{resp.}\
\Delta^\nabla_{q;g,\mu}:\Omega^q(M;E)\to\Omega^q(M;E).$$ They are elliptic second order differential operators with principal symbol $\sigma_\xi=|\xi|^2$. Therefore they have a unique well defined zeta regularized determinant (modified determinant) $\det(\Delta^\nabla_{q;g,b})\in{\mathbb C}$ (${\det}'(\Delta^\nabla_{q;g,b})\in{\mathbb C}_*$) resp. $\det(\Delta^\nabla_{q;g,\mu})\in{\mathbb R}_{\geq 0}$ (${\det}'(\Delta^\nabla_{q;g,\mu})\in{\mathbb R}_{>0}$) calculated with respect to a non-zero Agmon angle avoiding the spectrum cf. [@BH05]. Recall that the zeta regularized determinant (modified determinant) is the zeta regularized product of all (non-zero) eigenvalues.
Denote by $$\begin{aligned}
\Sigma(E,g,b)
&:=\bigl\{\nabla\in\mathcal C(E)\bigm|
\ker(\Delta^\nabla_{*;g,b})\neq0\bigr\}
\\
\Sigma(E,g,\mu)
&:=\bigl\{\nabla\in\mathcal C(E)\bigm|
\ker(\Delta^\nabla_{*;g,\mu})\neq0\bigr\}\end{aligned}$$ and by $$\Sigma(E):=\bigl\{\nabla\in\mathcal F(E)
\bigm| H^*(\Omega^*(M;E),d^\nabla)\neq 0\bigr\}.$$ Note that $\Sigma(E,g,\mu)\cap\mathcal F(E)=\Sigma(E)$ for any $\mu$, and $\Sigma(E,g,b)\cap\mathcal F(E)\supseteq\Sigma(E)$. Both, $\Sigma(E)$ and $\Sigma(E,g,b)\cap\mathcal F(E)$, are closed complex analytic subsets of $\mathcal F(E)$, and $\det(\Delta^\nabla_{q;g,\cdots})
={\det}'(\Delta^\nabla_{q;g,\cdots})$ on $\mathcal F(E)\setminus \Sigma(E,g,\cdots)$.
We consider the real analytic functions: $T^{\text{\rm even}}_{g,\mu}:\mathcal C(E)\to\mathbb R_{\geq 0}$, $T^{\text{\rm odd}}_{g,\mu}:\mathcal C(E)\to\mathbb R_{\geq 0}$, $R_{\alpha, \mu}:\mathcal C(E)\to\mathbb R_{>0}$ and the holomorphic functions $T^{\text{\rm even}}_{g,b}:\mathcal C(E)\to\mathbb C$, $T^{\text{\rm odd}}_{g,b}:\mathcal C(E)\to\mathbb C$, $R_{\alpha,b}:\mathcal C(E)\to \mathbb C_*$ defined by: $$\label{E;16'}
\begin{aligned}
T^{\text{\rm even}}_{g,\cdots}(\nabla):=& \prod_{q\ {\text{\rm even}}}(\det\Delta^\nabla_{q;g,\cdots})^q,
\\
T^{\text{\rm odd}}_{g,\cdots}(\nabla) :=& \prod_{q\ {\text{\rm odd}}}(\det\Delta^\nabla_{q;g,\cdots})^q ,
\\
R_{\alpha,\cdots}(\nabla):= &e^{\int_M\omega(\cdots,\nabla)\wedge\alpha}.
\end{aligned}$$ We also write $T'^{\,{\text{\rm even}}}_{g,\cdots}$ resp. $T'^{\,{\text{\rm odd}}}_{g,\cdots}$ for the same formulas with ${\det}'$ instead of $\det$. These functions are discontinuous on $\Sigma(E,g,\cdots)$ and coincide with $T^{\text{\rm even}}_{g,\cdots}$ resp. $T^{\text{\rm odd}}_{g,\cdots}$ on $\mathcal F(E)\setminus\Sigma(E,g,\cdots)$. Here $\cdots$ stands for either $b$ or $\mu$. For the definition of real or complex analytic space/set, holomorphic/meromorphic function/map in infinite dimension the reader can consult [@D] and [@KM], although the definitions used here are rather straightforward.
Let $E_r\to M$ be a smooth real vector bundle equipped with a non-degenerate symmetric positive definite bilinear form $b_r$. Let $\mathcal C(E_r)$ resp. $\mathcal F(E_r)$ the space of connections resp. flat connections in $E_r$. Denote by $E\to M$ the complexification of $E_r$, $E=E_r\otimes\mathbb C$, and by $b$ resp. $\mu$ the complexification of $b_r$ resp. the Hermitian structure extension of $b_r$. We continue to denote by $\mathcal C(E_r)$ resp. $\mathcal F(E_r)$ the subspace of $\mathcal C(E)$ resp. $\mathcal F(E)$ consisting of connections which are complexification of connections resp. flat connections in $E_r$, and by $\nabla$ the complexification of the connection $\nabla\in\mathcal C(E_r)$. If $\nabla\in\mathcal C(E_r)$, then $$\operatorname{Spect}\Delta^\nabla_{q;g,b}=\operatorname{Spect}\Delta^\nabla_{q;g,\mu}\subseteq\mathbb R_{\geq 0}$$ and therefore $$\label{E:16}
\begin{aligned}
T^{{\text{\rm even}}/{\text{\rm odd}}}_{g,b}(\nabla)&=\bigl|T^{{\text{\rm even}}/{\text{\rm odd}}}_{g,b}(\nabla)\bigr|
=T^{{\text{\rm even}}/{\text{\rm odd}}}_{g,\mu}(\nabla),
\\
T'^{\,{\text{\rm even}}/{\text{\rm odd}}}_{g,b}(\nabla)&=\bigl|T'^{\,{\text{\rm even}}/{\text{\rm odd}}}_{g,b}(\nabla)\bigr|
=T'^{\,{\text{\rm even}}/{\text{\rm odd}}}_{g,\mu}(\nabla),
\\
R_{\alpha,b}(\nabla)&=\bigl|R_{\alpha,b}(\nabla)\bigr|=R_{\alpha,\mu}(\nabla).
\end{aligned}$$
Observe that $\Omega^*(M;E)(0)$ the (generalized) eigen space of $\Delta^\nabla_{*;g,b}$ corresponding to the eigen value zero is a finite dimensional vector space of dimension the multiplicity of $0$. The restriction of the symmetric bilinear form induced by $b$ remains non-degenerate and defines for each component $\Omega^q(M;E)(0)$ an equivalence class of bases. Since $d^\nabla$ commutes with $\Delta^\nabla_{*;g,b}$, $\bigl(\Omega^*(M;E)(0),d^\nabla\bigr)$ is a finite dimensional complex. When acyclic, i.e. $\nabla\in\mathcal F(E)\setminus\Sigma(E)$, denote by $$T_{\text{\rm an}}(\nabla,g,b)(0)\in\mathbb C_*$$ the Milnor torsion associated to the equivalence class of bases induced by $b$.
The modified Ray–Singer torsion {#SS:5.1}
-------------------------------
Let $E\to M$ be a complex vector bundle, and let ${\mathfrak e}^*\in{\mathfrak{Eul}}^*(M;\mathbb R)$ be a coEuler structure. Choose a Hermitian structure (fiber metric) $\mu$ on $E$, a Riemannian metric $g$ on $M$ and $\alpha\in\Omega^{n-1}(M;\mathcal O_M)$ so that $[g,\alpha]={\mathfrak e}^*$, see section \[SS:3.2\]. For $\nabla\in\mathcal F(E)\setminus\Sigma(E)$ consider the quantity $$T_{\text{\rm an}}(\nabla,\mu,g,\alpha):=\bigl(T^{{\text{\rm even}}}_{g,\mu}(\nabla)
/T^{{\text{\rm odd}}}_{g,\mu}(\nabla)\bigr)^{-1/2}
\cdot R_{\alpha,\mu}(\nabla)\in\mathbb R_{>0}$$ referred to as the *modified Ray–Singer torsion.* The following proposition is a reformulation of one of the main theorems in [@BZ92], cf. also [@BFK01] and [@BH04].
\[P:3\] If $\nabla\in\mathcal F(E)\setminus\Sigma(E)$, then $T_{\text{\rm an}}(\nabla,\mu,g,\alpha)$ is gauge invariant and independent of $\mu,g,\alpha$.
When applied to $\mathbb F_\rho$ the number $T_{\text{\rm an}}^{{\mathfrak e}^*}(\rho):=T_{\text{\rm an}}(\nabla_\rho,\mu,g,\alpha)$ defines a real analytic function $T_{\text{\rm an}}^{{\mathfrak e}^*}:\operatorname{Rep}^M_0(\Gamma;V)\to{\mathbb R}_{>0}$. It is natural to ask if $T_{\text{\rm an}}^{{\mathfrak e}^*}$ is the absolute value of a holomorphic function.
The answer is no as one can see on the simplest possible example $M= S^1$ equipped with the the canonical coEuler structure ${\mathfrak e}^*_0$. In this case $\operatorname{Rep}^M(\Gamma;\mathbb C)=\mathbb C\setminus 0$, and $T_{\text{\rm an}}^{{\mathfrak e}^*_0}(z)=|\frac{(1-z)}{z^{1/2}}|$, cf. [@BH05]. However, Theorem \[T:2\] in section \[SS:6.1\] below provides the following answer to the question (Q) from the introduction.
If ${\mathfrak e}^*$ is an integral coEuler structure, then $T_{\text{\rm an}}^{{\mathfrak e}^*}$ is the absolute value of a holomorphic function on $\operatorname{Rep}^M_0(\Gamma;V)$ which is the restriction of a rational function on $\operatorname{Rep}^M(\Gamma;V)$. For a general coEuler structure $T^{{\mathfrak e}^*}_{\text{\rm an}}$ still locally is the absolute value of a holomorphic function.
Complex Ray–Singer torsion {#SS:5.3}
--------------------------
Let $E$ be a complex vector bundle equipped with a non-degenerate symmetric bilinear form $b$. Suppose $(g,\alpha)$ is a pair consisting of a Riemannian metric $g$ and a differential form $\alpha\in\Omega^{n-1}(M;\mathcal O_M)$ with $d\alpha=\operatorname{e}(g)$. For any $\nabla\in\mathcal F(E)\setminus\Sigma(E)$ consider the complex number $$\label{E:00}
\mathcal{ST}_{\text{\rm an}}(\nabla,b,g,\alpha)
:=\bigl(T'^{\,{\text{\rm even}}}_{g,b}(\nabla)/T'^{\,{\text{\rm odd}}}_{g,b}(\nabla)\bigr)^{-1}
\cdot R_{\alpha,b}(\nabla)^2
\cdot T_{\text{\rm an}}(\nabla,g,b)(0)^2\in\mathbb C_*$$ referred to as the *complex valued Ray–Singer torsion.*[^9]
It is possible to provide an alternative definition of $\mathcal{ST}_{\text{\rm an}}(\nabla,b,g,\alpha)$. Suppose $R>0$ is a positive real number so that the Laplacians $\Delta^\nabla_{q;g,b}$ have no eigen values of absolute value $R$. In this case denote by $\det^R\Delta^\nabla_{q;g,b}$ the regularized product of all eigen values larger than $R$ w.r. to a non-zero Agmon angle disjoint from the spectrum $T^{R,{\text{\rm even}}}_{g,b}$ resp. $T^{R,{\text{\rm even}}}_{g,b}$ the quantities defined by the formulae with $T^{R,{\text{\rm even}}/{\text{\rm odd}}}(\Delta)$ instead of $T^{'\,{\text{\rm even}}/{\text{\rm odd}}}(\Delta)$. Consider $\Omega^*(M;E)(R)$ to be the sum of generalized eigen spaces of $\Delta^\nabla_{*;g,b}$ corresponding to eigen values smaller in absolute value than $R$. $(\Omega^*(M;E)(R), d^\nabla)$ is a finite dimensional complex. As before $b$ remains non-degenerate and when acyclic (and this is the case iff $(\Omega^*(M;E),d^\nabla)$ is acyclic) denote by $T_{\text{\rm an}}(\nabla,g,b)(R)$ the Milnor torsion associated to the equivalence class of bases induced by $b$. It is easy to check that $$\label{E:000}
\mathcal{ST}_{\text{\rm an}}(\nabla,b,g,\alpha)
=\bigl(T^{R,{\text{\rm even}}}_{g,b}(\nabla)/T^{R,{\text{\rm odd}}}_{g,b}(\nabla)\bigr)^{-1}
\cdot R_{\alpha,b}(\nabla)^2
\cdot T_{\text{\rm an}}(\nabla,g,b)(R)^2$$
\[P:6\] 1. $\mathcal{ST}_{\text{\rm an}}(\nabla,b,g,\alpha)$ is a holomorphic function on $\mathcal F(E)\setminus\Sigma(E)$ and the restriction of a meromorphic function on $\mathcal F(E)$ with poles and zeros in $\Sigma(E)$.
2\. If $b_1$ and $b_2$ are two non-degenerate symmetric bilinear forms which are homotopic, then $\mathcal{ST}_{\text{\rm an}}(\nabla,b_1,g,\alpha)=\mathcal{ST}_{\text{\rm an}}(\nabla,b_2,g,\alpha)$.
3\. If $(g_1,\alpha_1)$ and $(g_2,\alpha_2)$ are two pairs representing the same coEuler structure, then $\mathcal{ST}_{\text{\rm an}}(\nabla,b,g_1,\alpha_1)=\mathcal{ST}_{\text{\rm an}}(\nabla,b,g_2,\alpha_2)$.
4\. We have $\mathcal{ST}_{\text{\rm an}}(\gamma\nabla,\gamma b,g,\alpha)=\mathcal{ST}_{\text{\rm an}}(\nabla,b,g,\alpha)$ for every gauge transformation $\gamma$ of $E$.
5\. $\mathcal{ST}_{\text{\rm an}}(\nabla_1\oplus\nabla_2,b_1\oplus b_2,g,\alpha)=
\mathcal{ST}_{\text{\rm an}}(\nabla_1,b_1,g,\alpha)\cdot\mathcal{ST}_{\text{\rm an}}(\nabla_2,b_2,g,\alpha)$.
To check the first part of this proposition, one shows that for $\nabla_0\in\mathcal F(E)$ one can find $R>0$ and an open neighborhood $U$ of $\nabla_0\in \mathcal F(E)$ such that no eigen value of $\Delta^\nabla_{q;g,b}$, $\nabla\in U$, has absolute value $R$. The function $\bigl(T^{R,{\text{\rm even}}}_{g,b}(\nabla)/T^{R,{\text{\rm odd}}}_{g,b}(\nabla)\bigr)^{-1}$ is holomorphic in $\nabla\in U$. Moreover, on $U$ the function $T_{\text{\rm an}}(\nabla,g,b)(R)^2$ is meromorphic in $\nabla$, and holomorphic when restricted to $U\setminus\Sigma(E)$. The statement thus follows from .
The second and third part of Proposition \[P:6\] are derived from formulas for $d/dt(\mathcal{ST}_{\text{\rm an}}(\nabla,b(t),g,\alpha))$ resp. $d/dt(\mathcal{ST}_{\text{\rm an}}(\nabla,b,g(t),\alpha)$ which are similar to such formulas for Ray–Singer torsion in the case of a Hermitian structure instead of a non-degenerate symmetric bilinear form, cf. [@BH05]. The proof of 4) and 5) require a careful inspection of the definitions. The full arguments are contained in [@BH05].
As a consequence to each homotopy class of non-degenerate symmetric bilinear forms $[b]$ and coEuler structure ${\mathfrak e}^*$ we can associate a meromorphic function on $\mathcal F(E)$. The reader unfamiliar with the basic concepts of complex analytic geometry on Banach/ Frechet manifolds can consult [@D] and [@KM]. Changing the coEuler structure our function changes by multiplication with a non-vanishing holomorphic function as one can see from . Changing the homotopy class $[b]$ is actually more subtle. We expect however that $\mathcal{ST}$ remains unchanged when the coEuler structure is integral.
Denote by $\operatorname{Rep}^{M,E}(\Gamma;V)$ the union of components of $\operatorname{Rep}^M(\Gamma;V)$ which consists of representations equivalent to holonomy representations of flat connections in the bundle $E$. Suppose $E$ admits non-degenerate symmetric bilinear forms and let $[b]$ be a homotopy class of such forms. Let $x_0\in M$ be a base point and denote by $\mathcal G(E)_{x_0,[b]}$ the group of gauge transformations which leave fixed $E_{x_0}$ and the class $[b]$. In view of Proposition \[P:6\], $\mathcal{ST}_{\text{\rm an}}(\nabla, b, g,\alpha)$ defines a meromorphic function $\mathcal{ST}_{\text{\rm an}}^{{\mathfrak e}^*,[b]}$ on $\pi^{-1}(\operatorname{Rep}^{M,E}(\Gamma;V) \subseteq \mathcal F(E)/\mathcal G_{x_0,[b]}$. Note that $\pi:\mathcal F(E)/\mathcal G_{x_0,[b]}\to\operatorname{Rep}(\Gamma;V)$ is an principal holomorphic covering of its image which contains $\operatorname{Rep}^{M,E}(\Gamma;V)$. We expect that the absolute value of this function is the square of modified Ray–Singer torsion. The expectation is true when $(E,b)$ satisfies $P_r$ below.
The pair $(E,b)$ satisfies *Property $P_r$* if it is the complexification of a pair $(E_r,b_r)$ consisting of a real vector bundle $E_r$ and a non-degenerate symmetric positive definite $\mathbb R$-bilinear form $b_r$ and the space of flat connections $\mathcal F(E_r)$ is a real form of the space $\mathcal F(E)$.
We summarize this in the following Theorem.
\[T:1\] With the hypotheses above we have.
1\. If ${\mathfrak e}^*_1$ and ${\mathfrak e}^*_2$ are two coEuler structures then $$\mathcal{ST}_{\text{\rm an}}^{{\mathfrak e}^*_1,[b]}=\mathcal{ST}_{\text{\rm an}}^{{\mathfrak e}^*_2,[b]}\cdot
e^{2([\omega(\nabla,b)],D^{-1}({\mathfrak e}^*_1-{\mathfrak e}^*_2))}$$ with $D:H_1(M;{\mathbb R})\to H^{n-1}(M;\mathcal O_M)$ the Poincaré duality isomorphism.
Suppose that $(E,b)$ satisfies property $(P_r)$. Then:
2\. If ${\mathfrak e}^* $ is integral then $\mathcal{ST}_{\text{\rm an}}^{{\mathfrak e}^*,[b]}$ is independent of $[b]$ and descends to a rational function on $\operatorname{Rep}^{M,E}(\Gamma;V)$ denoted $\mathcal{ST}_{\text{\rm an}}^{{\mathfrak e}^*}$.
3\. We have $$\bigl|\mathcal{ST}_{\text{\rm an}}^{{\mathfrak e}^*,[b]}\bigr|=(T_{\text{\rm an}}^{{\mathfrak e}^*}\cdot\pi)^2.$$
We expect that both 2) and 3) remain true for an arbitrary pair $(E,b)$.
Property 5) in Proposition \[P:6\] shows that up to multiplication with a root of unity the complex Ray–Singer torsion can be defined on all components of $\operatorname{Rep}^M(\Gamma;V)$, since $F=\oplus_kE$ is trivial for sufficiently large $k$.
Milnor–Turaev and dynamical torsion {#S:6}
===================================
Milnor–Turaev torsion {#SS:6.1}
---------------------
Consider a smooth triangulation $\tau$ of $M$, and choose a collection of orientations $\mathcal O$ of the simplices of $\tau$. Let $x_0\in M$ be a base point, and set $\Gamma:=\pi_1(M,x_0)$. Let $V$ be a finite dimensional complex vector space. For a representation $\rho\in\operatorname{Rep}(\Gamma;V)$, consider the chain complex $(C^*_\tau(M;\rho),d^{\mathcal O}_\tau(\rho))$ associated with the triangulation $\tau$ which computes the cohomology $H^*(M;\rho)$.
Denote the set of simplexes of dimension $q$ by $\mathcal X_q$, and set $k_i:=\sharp(\mathcal X_i)\cdot\dim(V)$. Choose a collection of paths $\sigma:=\{\sigma_s\mid s\in\tau\}$ from $x_0$ to the barycenters of $\tau$ as in section \[SS:3.1\]. Choose an ordering $o$ of the barycenters and a framing $\epsilon$ of $V$. Using $\sigma$, $o$ and $\epsilon$ one can identify $C^q_\tau(M;\rho)$ with ${\mathbb C}^{k_q}$. We obtain in this way a map $$t_{\mathcal O,\sigma,o,\epsilon}:\operatorname{Rep}(\Gamma;V)\to\mathbb D(k_0,\dotsc,k_n)$$ which sends $\operatorname{Rep}^M_0(\Gamma;V)$ to $\mathbb D_{\text{\rm ac}}(k_0,\dotsc,k_n)$. A look at the explicit definition of $d^{\mathcal O}_\tau(\rho)$ implies that $t_{\mathcal O,\sigma,o,\epsilon}$ is actually a regular map between two algebraic sets. Change of $\mathcal O,\sigma,o,\epsilon$ changes the map $t_{\mathcal O,\sigma,o,\epsilon}$.
Recall that the triangulation $\tau$ determines Euler vector fields $X_\tau$ which together with $\sigma$ determine an Euler structure ${\mathfrak e}\in{\mathfrak{Eul}}(M;\mathbb Z)$, see section \[SS:3.1\]. Note that the ordering $o$ induces a cohomology orientation $\mathfrak o$ in $H^*(M;{\mathbb R})$. In view of the arguments of [@M66] or [@Tu86] one can conclude (cf. [@BH04]):
\[P:7\] If $\rho\in\operatorname{Rep}^M_0(\Gamma;V)$ different choices of $\tau, \mathcal O, \sigma, o, \epsilon$ provide the same composition $\tau\cdot t_{\mathcal O, \sigma, o, \epsilon}(\rho)$ provided they define the same Euler structure ${\mathfrak e}$ and homology orientation $\mathfrak o$.
In view of Proposition \[P:7\] we obtain a well defined complex valued rational function on $\operatorname{Rep}^M(\Gamma;V)$ called the Milnor–Turaev torsion and denoted from now on by $\mathcal T_{\text{\rm comb}}^{{\mathfrak e},\mathfrak o}$.
\[T:2\] 1. The poles and zeros of $\mathcal T_{\text{\rm comb}}^{{\mathfrak e},\mathfrak o}$ are contained in $\Sigma(M)$, the subvariety of representations $\rho$ with $H^*(M;\rho)\neq 0$.
2\. The absolute value of $\mathcal T_{\text{\rm comb}}^{{\mathfrak e},\mathfrak o}(\rho)$ calculated on $\rho\in\operatorname{Rep}^M_0(\Gamma;V)$ is the modified Ray–Singer torsion $T_{{\text{\rm an}}}^{{\mathfrak e}^*}(\rho)$, where ${\mathfrak e}^*=\operatorname{PD}({\mathfrak e})$.
3\. If ${\mathfrak e}_1$ and ${\mathfrak e}_2$ are two Euler structures then $\mathcal T_{\text{\rm comb}}^{{\mathfrak e}_2,\mathfrak o}
=\mathcal T_{\text{\rm comb}}^{{\mathfrak e}_1,\mathfrak o}
\cdot\det_{{\mathfrak e}_2-{\mathfrak e}_1}$ and $\mathcal T_{\text{\rm comb}}^{{\mathfrak e},-\mathfrak o}=
(-1)^{\dim V}\cdot\mathcal T_{\text{\rm comb}}^{{\mathfrak e},\mathfrak o}$ where $\det_{{\mathfrak e}_2-{\mathfrak e}_1}$ is the regular function on $\operatorname{Rep}^M(\Gamma;V)$ defined in section \[SS:4.1\].
4\. When restricted to $\operatorname{Rep}^{M,E}(\Gamma;V)$, $E$ a complex vector bundle equipped with a non-degenerate symmetric bilinear form $b$ so that $(E,b)$ satisfies Property $P_r$, $(\mathcal T_{\text{\rm comb}}^{{\mathfrak e},\mathfrak o})^2=\mathcal{ST}_{\text{\rm an}}^{{\mathfrak e}^*}$, where ${\mathfrak e}^*=\operatorname{PD}({\mathfrak e})$.
We expect that 4) remains true without any hypothesis. Parts 1) and 3) follow from the definition and the general properties of $\tau$, part 2) can be derived from the work of Bismut–Zhang [@BZ92] cf. also [@BFK01], and part 4) is discussed in [@BH05], Remark 5.11.
Dynamical torsion {#SS:6.2}
-----------------
Let $X$ be a vector field on $M$ satisfying (H), (EG), (L), (MS) and (NCT) from section \[SS:2.3\]. Choose orientations $\mathcal O$ of the unstable manifolds. Let $x_0\in M$ be a base point and set $\Gamma:=\pi_1(M,x_0)$. Let $V$ be a finite dimensional complex vector space. For a representation $\rho\in\operatorname{Rep}(\Gamma;V)$ consider the associated flat bundle $(F_\rho,\nabla_\rho)$, and set $C^q_X(M;\rho):=\Gamma(F_\rho|_{\mathcal X_q})$, where $\mathcal X_q$ denotes the set of zeros of index $q$. Recall that for $x\in\mathcal X$, $y\in\mathcal X$ and every homotopy class $\hat\alpha\in\mathcal P_{x,y}$ parallel transport provides an isomorphism $(\operatorname{pt}^\rho_{\hat\alpha})^{-1}:(F_\rho)_y\to(F_\rho)_x$. For $x\in\mathcal X_q$ and $y\in\mathcal X_{q-1}$ consider the expression: $$\label{E:18}
\delta_X^{\mathcal O}(\rho)_{x,y}:=\sum_{\hat\alpha\in\mathcal P_{x,y}}
\mathbb I_{x,y}^{X,\mathcal O}(\hat\alpha)(\operatorname{pt}^\rho_{\hat\alpha})^{-1}.$$ If the right hand side of is absolutely convergent for all $x$ and $y$ they provide a linear mapping $\delta^{\mathcal O}_X(\rho):C^{q-1}_X(M;\rho)\to
C^q_X(M;\rho)$ which, in view of Proposition \[P:1\](3), makes $\bigl(C^*_X(M;\rho),\delta_X^{\mathcal O}(\rho)\bigr)$ a cochain complex. There is an integration homomorphism $\operatorname{Int}_X^{\mathcal O}(\rho):\bigl(\Omega^*(M;F_\rho),d^{\nabla_\rho}\bigr)\to
\bigl(C^*_X(M;\rho),\delta_X^{\mathcal O}(\rho)\bigr)$ which does not always induce an isomorphism in cohomology.
Recall that for every $\rho\in\operatorname{Rep}(\Gamma;V)$ the composition $\operatorname{tr}\cdot\rho^{-1}:\Gamma\to{\mathbb C}$ factors through conjugacy classes to a function $\operatorname{tr}\cdot\rho^{-1}:[S^1,M]\to{\mathbb C}$. Let us also consider the expression $$\label{E:19}
P_X(\rho):=\sum_{\gamma\in[S^1,M]}\mathbb Z_X(\gamma)(\operatorname{tr}\cdot\rho^{-1})(\gamma).$$ Again, the right hand side of will in general not converge.
\[P:8\]
There exists an open set $U$ in $\operatorname{Rep}^M(\Gamma;V)$, intersecting every irreducible component, s.t. for any representation $\rho\in U$ we have:
a\) The differentials $\delta^{\mathcal O}_X(\rho)$ converge absolutely.
b\) The integration $\operatorname{Int}_X^{\mathcal O}(\rho)$ converges absolutely.
c\) The integration $\operatorname{Int}_X^{\mathcal O}(\rho)$ induces an isomorphism in cohomology.
d\) If in addition $\dim V=1$, then $$\label{E:19'}
\sum_{\sigma\in H_1(M;{\mathbb Z})/\operatorname{Tor}(H_1(M;{\mathbb Z}))}\,\biggl|\sum_{[\gamma]\in\sigma}
\mathbb Z_X(\gamma)(\operatorname{tr}\cdot\rho^{-1})(\gamma)\biggr|$$ converges, cf. . Here the inner (finite) sum is over all $\gamma\in[S^1,M]$ which give rise to $\sigma\in H_1(M;{\mathbb Z})/\operatorname{Tor}(H_1(M;{\mathbb Z}))$.
This Proposition is a consequence of exponential growth property (EG) and requires (for d)) Hutchings–Lee or Pajitnov results. A proof in the case $\dim V=1$ is presented in [@BH03']. The convergence of is derived from the interpretation of this sum as the Laplace transform of a Dirichlet series with a positive abscissa of convergence.
We expect d) to remain true for $V$ of arbitrary dimension.[^10] In this case we make precise by setting $$\label{E:19''}
P_X(\rho):=\sum_{\sigma\in H_1(M;{\mathbb Z})/\operatorname{Tor}(H_1(M;{\mathbb Z}))}\,\sum_{[\gamma]\in\sigma}
\mathbb Z_X(\gamma)(\operatorname{tr}\cdot\rho^{-1})(\gamma).$$
A Lyapunov closed one form $\omega$ for $X$ permits to consider the family of regular functions $P_{X;R}$, $R\in{\mathbb R}$, on the variety $\operatorname{Rep}(\Gamma;V)$ defined by: $$P_{X;R}(\rho):=\sum_{\hat\theta,-\omega(\hat\theta)\leq R}
(\epsilon(\hat\theta)/p(\hat\theta))\operatorname{tr}(\rho(\hat\theta)^{-1}).$$ If converges then $\lim_{R\to \infty}P_{X;R}$ exists for $\rho$ in an open set of representations. We expect that by analytic continuation this can be defined for all representations except ones in a proper algebraic subvariety. This is the case when $\dim V=1$ or, for $V$ of arbitrary dimension, when the vector field $X$ has only finitely many simple closed trajectories. In this case $\lim_{R\to \infty}P_{X;R}$ has an analytic continuation to a rational function on $\operatorname{Rep}(\Gamma;V)$, see section \[S:8\] below.
As in section \[SS:6.1\], we choose a collection of paths $\sigma:=\{\sigma_x\mid x\in\mathcal X\}$ from $x_0$ to the zeros of $X$, an ordering $o'$ of $\mathcal X$, and a framing $\epsilon$ of $V$. Using $\sigma,o,\epsilon$ we can identify $C^q_X(M;\rho)$ with $\mathbb
C^{k_q}$, where $k_q:=\sharp(\mathcal X_q)\cdot\dim(V)$. As in the previous section we obtain in this way a holomorphic map $$t_{\mathcal O,\sigma,o',\epsilon}:
U\to\hat{\mathbb D}_{\text{\rm ac}}(k_0,\dotsc,k_n).$$
An ordering $o'$ of $\mathcal X$ is given by orderings $o'_q$ of $\mathcal X_q$, $q=0,1,\dotsc,n$. Two orderings $o'_1$ and $o'_2$ are equivalent if $o'_{1,q}$ is obtained from $o'_{2,q}$ by a permutation $\pi_q$ so that $\prod_q\operatorname{sgn}(\pi_q)=1$. We call an equivalence class of such orderings a *rest point orientation.* Let us write $\mathfrak o'$ for the rest point orientation determined by $o'$. Moreover, let ${\mathfrak e}$ denote the Euler structure represented by $X$ and $\sigma$, see Observation \[O:1\]. As in the previous section, the composition $\tau\cdot t_{\mathcal
O,\sigma,o',\epsilon}:U\setminus\Sigma\to\mathbb C_*$ is a holomorphic map which only depends on ${\mathfrak e}$ and $\mathfrak o'$, and will be denoted by $\tau_X^{{\mathfrak e},\mathfrak o'}$. Consider the holomorphic map $P_X:U\to\mathbb C$ defined by formula . The *dynamical torsion* is the partially defined holomorphic function $$\mathcal T^{{\mathfrak e},\mathfrak o'}_X:=\tau^{{\mathfrak e},\mathfrak o'}_X\cdot e^{P_X}:
U\setminus\Sigma\to\mathbb C_*.$$
The following result is based on a theorem of Hutchings–Lee and Pajitnov [@H02] cf. [@BH03'].
\[T:3\] If $\dim V=1$ the partially defined holomorphic function $\mathcal T^{{\mathfrak e},\mathfrak o'}_X$ has an analytic continuation to a rational function equal to $\pm\mathcal T_{\text{\rm comb}}^{{\mathfrak e},\mathfrak o}$.
It is hoped that a generalization of Hutchings–Lee and Pajitnov results which will be elaborated in subsequent work [@BH05b] might led to the proof of the above result for $V$ of arbitrary dimension.
Examples {#S:7}
========
Milnor–Turaev torsion for mapping tori and twisted Lefschetz zeta function {#SS:7.1}
--------------------------------------------------------------------------
Let $\Gamma_0$ be a group, $\alpha:\Gamma_0\to \Gamma_0$ an isomorphism and $V$ a complex vector space. Denote by $\Gamma:=\Gamma_0\times_{\alpha}\mathbb Z$ the group whose underlying set is $\Gamma_0\times\mathbb Z$ and group operation $(g',n) * (g'',m) :=(\alpha^m(g')\cdot g'', n+m)$. A representation $\rho:\Gamma\to\operatorname{GL}(V)$ determines a representation $\rho_0(\rho):\Gamma_0\to\operatorname{GL}(V)$ the restriction of $\rho$ to $\Gamma_0\times 0$ and an isomorphism of $V$, $\theta(\rho)\in\operatorname{GL}(V)$.
Let $(X,x_0)$ be a based point compact space with $\pi_1(X,x_0)=\Gamma_0$ and $f:(X,x_0) \to (X,x_0)$ a homotopy equivalence. For any integer $k$ the map $f$ induces the linear isomorphism $f^k: H^k(X; V) \to H^k(X; V)$ and then the standard Lefschetz zeta function $$\zeta_f (z):=\frac{\prod_{k\ {\text{\rm even}}}\det(I-zf^k)}{\prod_{k\ {\text{\rm odd}}}\det(I-zf^k)}.$$
More general if $\rho$ is a representation of $\Gamma$ then $f$ and $\rho=(\rho_0(\rho), \theta(\rho))$ induce the linear isomorphisms $f^k_\rho: H^k(X;\rho_0(\rho)) \to H^k(X; \rho_0(\rho))$ and then the $\rho$-twisted Lefschetz zeta function $$\zeta_f(\rho,z):=\frac{\prod_{k\ {\text{\rm even}}}\det(I-zf^k_\rho)}{\prod_{k\ {\text{\rm odd}}}\det(I-zf^k_\rho)}.$$
Let $N$ be a closed connected manifold and $\varphi:N\to N$ a diffeomorphism. Without loss of generality one can suppose that $y_0\in N$ is a fixed point of $\varphi$. Define the mapping torus $M=N_\varphi$, the manifold obtained from $N\times I$ identifying $(x,1)$ with $(\varphi(x),0)$. Let $x_0=(y_0,0)\in M$ be a base point of $M$. Set $\Gamma_0:=\pi_1(N,n_0)$ and denote by $\alpha:\pi_1(N,y_0)\to \pi_1(N,y_0)$ the isomorphism induced by $\varphi$. We are in the situation considered above with $\Gamma=\pi_1(M,x_0)$. The mapping torus structure on $M$ equips $M$ with a canonical Euler structure ${\mathfrak e}$ and canonical homology orientation $\mathfrak o$. The Euler structure ${\mathfrak e}$ is defined by any vector field $X$ with $\omega(X)<0$ where $\omega:=p^*dt\in\Omega^1(M;\mathbb R)$; all are homotopic. The Wang sequence $$\label{E:Wang}
\cdots\to H^*(M;\mathbb F_\rho)\to H^*(N;i^* (\mathbb F_\rho))
\xrightarrow{\varphi^*_{ \rho} -{\text{\rm id}}}
H^*(N;i^*(\mathbb F_\rho) )
\to H^{*+1}(M;\mathbb F_\rho)\to\cdots$$ implies $H^*(M;\mathbb F_\rho)=0$ iff $\det(I-\varphi^k_\rho)\neq0$ for all $k$. The cohomology orientation is derived from the Wang long exact sequence for the trivial one dimensional real representation. For details see [@BH04]. We have
\[P:11\] With these notations $\mathcal T_{\text{\rm comb}}^{{\mathfrak e},\mathfrak o}(\rho)=\zeta_\varphi(\rho,1)$.
This result is known cf. [@BJ96]. A proof can be also derived easily from [@BH04].
Vector fields without rest points and Lyapunov cohomology class {#SS:7.2}
---------------------------------------------------------------
Let $X$ be a vector field without rest points, and suppose $X$ satisfies (L) and (NCT). As in the previous section $X$ defines an Euler structure ${\mathfrak e}$. Consider the expression . By Theorem \[T:3\] we have:
\[O:5\] With the hypothesis above there exists an open set $U\subseteq\operatorname{Rep}^M(\Gamma;V)$ so that converges, and $e^{P_X}$ is a well defined holomorphic function on $U$. The function $e^{P_X}$ has an analytic continuation to a rational function on $\operatorname{Rep}^M(\Gamma;V)$ equal to $\pm\mathcal T^{{\mathfrak e},\mathfrak o}_{\text{\rm comb}}$. The set $U$ intersects non-trivially each connected component of $\operatorname{Rep}^M(\Gamma;V)$.
The Alexander polynomial {#SS:7.3}
------------------------
If $M$ is obtained by surgery on a framed knot, and $\dim V=1$, then $\operatorname{Rep}(\Gamma;V)={\mathbb C}\setminus0$, and the function $(z-1)^2 \mathcal T_{\text{\rm comb}}^{{\mathfrak e},\mathfrak o}$ equals the Alexander polynomial of the knot, see [@Tu02]. Any twisted Alexander polynomial of the knot can be also recovered from $\mathcal T_{\text{\rm comb}}^{{\mathfrak e},\mathfrak o}$ for $V$ of higher dimension. One expects that passing to higher dimensional representations $\mathcal T_{\text{\rm comb}}^{{\mathfrak e},\mathfrak o}$ captures even more subtle knot invariants.
Applications {#S:8}
============
The invariant $A^{{\mathfrak e}^*}(\rho_1,\rho_2)$ {#SS:8.1}
--------------------------------------------------
Let $M$ be a $V$-acyclic manifold and ${\mathfrak e}^*$ a coEuler structure. Using the modified Ray–Singer torsion we define a ${\mathbb R}/\pi{\mathbb Z}$ valued invariant (which resembles the Atiyah–Patodi–Singer spectral flow) for two representations $\rho_1$, $\rho_2$ in the same component of $\operatorname{Rep}^M_0(\Gamma;V)$.
By a holomorphic path in $\operatorname{Rep}^M_0(\Gamma;V)$ we understand a holomorphic map $\tilde\rho:U\to\operatorname{Rep}^M_0(\Gamma;V)$ where $U$ is an open neighborhood of the segment of real numbers $[1,2]\times\{0\}\subset{\mathbb C}$ in the complex plane. For a coEuler structure ${\mathfrak e}^*$ and a holomorphic path $\tilde\rho$ in $\operatorname{Rep}_0^M(\Gamma;V)$ define $$\label{E:01}
\arg^{{\mathfrak e}^*}(\tilde\rho)
:=\Re\biggl(2/i\int_1^2\frac{{\partial}(T_{{\text{\rm an}}}^{{\mathfrak e}^*}\circ\tilde\rho)}
{T_{{\text{\rm an}}}^{{\mathfrak e}^*}\circ\tilde\rho}\biggr) \mod\pi.$$ Here, for a smooth function $\varphi$ of complex variable $z$, $\partial\varphi$ denotes the complex valued $1$-form $(\partial\varphi/\partial z)dz$ and the integration is along the path $[1,2] \times0\subset U$. Note that
\[O:4\] 1. Suppose $E$ is a complex vector bundle with a non-degenerate bilinear form $b$, and suppose $\tilde\rho$ is a holomorphic path in $\operatorname{Rep}^{M,E}_0(\Gamma;V)$. Then $$\arg^{{\mathfrak e}^*}(\tilde\rho)=\arg\Bigl(\mathcal{ST}_{\text{\rm an}}^{{\mathfrak e}^*,[b]}(\tilde\rho(2))
\big/\mathcal{ST}_{\text{\rm an}}^{{\mathfrak e}^*,[b]}(\tilde\rho(1))\Bigr) \mod\pi.$$ As consequence
2\. If $\tilde\rho'$ and $\tilde\rho''$ are two holomorphic paths in $\operatorname{Rep}^M_0(\Gamma;V)$ with $\tilde\rho'(1)=\tilde\rho''(1)$ and $\tilde\rho'(2)=\tilde\rho''(2)$ then $$\arg^{{\mathfrak e}^*}(\tilde\rho')=\arg^{{\mathfrak e}^*}(\tilde\rho'') \mod\pi.$$
3\. If $\tilde\rho'$, $\tilde\rho''$ and $\tilde\rho'''$ are three holomorphic paths in $\operatorname{Rep}^M_0(\Gamma;V)$ with $\tilde\rho'(1)=\tilde\rho'''(1)$, $\tilde\rho'(2)=\tilde\rho''(1)$ and $\tilde\rho''(2)=\tilde\rho'''(2)$ then $$\arg^{{\mathfrak e}^*}(\tilde\rho''')=\arg^{{\mathfrak e}^*}(\tilde\rho')+\arg^{{\mathfrak e}^*}(\tilde\rho'')\mod\pi.$$
Observation \[O:4\] permits to define a ${\mathbb R}/\pi{\mathbb Z}$ valued numerical invariant $A^{{\mathfrak e}^*}(\rho_1,\rho_2)$ associated to a coEuler structure ${\mathfrak e}^*$ and two representations $\rho_1,\rho_2$ in the same connected component of $\operatorname{Rep}^M_0(\Gamma;V)$. If there exists a holomorphic path with $\tilde\rho(1)=\rho_1$ and $\tilde\rho(2)=\rho_2$ we set $$A^{{\mathfrak e}^*}(\rho_1,\rho_2):=\arg^{{\mathfrak e}^*}(\tilde\rho)\mod\pi.$$ Given any two representations $\rho_1$ and $\rho_2$ in the same component of $\operatorname{Rep}^M_0(\Gamma;V)$ one can always find a finite collection of holomorphic paths $\tilde\rho_i$, $1\leq i\leq k$, in $\operatorname{Rep}^M_0(\Gamma;V)$ so that $\tilde\rho_i(2)=\tilde\rho_{i+1}(1)$ for all $1\leq i<k$, and such that $\tilde\rho_1(1)=\rho_1$ and $\tilde\rho_k(2)=\rho_2$. Then take $$A^{{\mathfrak e}^*}(\rho_1,\rho_2):=\sum_{i=1}^k\arg^{{\mathfrak e}^*}(\tilde\rho_i)\mod\pi.$$
In view of Observation \[O:4\] the invariant is well defined, and if ${\mathfrak e}^*$ is integral it is actually well defined in ${\mathbb R}/2\pi{\mathbb Z}$. This invariant was first introduced when the authors were not fully aware of “the complex Ray–Singer torsion.” The formula is a more or less obvious expression of the phase of a holomorphic function in terms of its absolute value, the Ray–Singer torsion, as positive real valued function. By Theorem \[T:2\] the invariant can be computed with combinatorial topology and by section \[S:7\] quite explicitly in some cases. If the representations $\rho_1, \rho_2$ are unimodular then the coEuler structure is irrelevant. It is interesting to compare this invariant to the Atiyah–Patodi–Singer spectral flow; it is not the same but are related.
Novikov conjecture
------------------
Let $X$ be a smooth vector field which satisfies (H), (L), (MS), (NCT). Suppose $\omega$ is a real valued closed one form so that $\omega(X)_x<0$, $x$ not a rest point (Lyapunov form). Define the functions $I^X_{x,y}:\mathbb R\to\mathbb Z$ and $Z^X:\mathbb R\to \mathbb Q$ by $$\begin{aligned}
I^{X,\mathcal O}_{x,y}(R):= &\sum _{\hat \alpha,\ \omega (\hat \alpha)<R}
\mathbb I^{X,\mathcal O}_{x,y} (\hat \alpha)\\
Z^X(R):= &\sum _{\hat \theta,\ \omega (\hat \theta)<R} \mathbb Z^X(\hat\theta)
\end{aligned}$$
Part (a) of the following conjecture was formulated by Novikov for $X=\operatorname{grad}_g\omega$, $\omega$ a Morse closed one form when this vector field satisfies the above properties.
a\) The function $I^{X,\mathcal O}_{x,y} (R)$ has exponential growth.
b\) The function $Z^X(R)$ has exponential growth.
Recall that a function $f:\mathbb R \to \mathbb R$ is said to have exponential growth iff there exists constants $C_1, C_2$ so that $| f(x) | <C_1 e^{C_2}$.
As a straight forward consequence of Proposition \[P:8\] we have
a\) Suppose $X$ satisfies (H), (MS), (L) and (EG). Then part a) of the conjecture above holds.
b\) Suppose $M$ is $V$-acyclic for some $V$ with $\dim(V)=1$. Moreover, assume $X$ satisfies (H), (MS), (L), (NTC) and (EG). Then part b) of the conjecture above holds.
This result is proved in [@BH05]; The $V$-acyclicity in part b) is not necessary if (EG) is replaced by an apparently stronger assumption (SEG). Prior to our work Pajitnov has considered for vector fields which satisfy (H), (L), (MS), (NCT) an additional property, condition $(\mathfrak C \mathcal Y)$, and has verified part (a) of this conjecture. He has also shown that the vector fields which satisfy (H), (L), (MS), (NCT) and $(\mathfrak C \mathcal Y)$ are actually $C^0$ dense in the space of vector fields which satisfy (H), (L), (MS), (NCT). It is shown in [@BH05] that Pajitnov vector fields satisfy (EG), and in fact (SEG).
A question in dynamics {#SS:7.4}
----------------------
Let $\Gamma$ be a finitely presented group, $V$ a complex vector space and $\operatorname{Rep}(\Gamma;V)$ the variety of complex representations. Consider triples $\underline a:=\{a,\epsilon_-,\epsilon_+\}$ where $a$ is a conjugacy class of $\Gamma$ and $\epsilon_\pm\in\{\pm1\}$. Define the rational function $\operatorname{let}_{\underline a}:\operatorname{Rep}(\Gamma;V)\to\mathbb C$ by $$\operatorname{let}_{\underline a}(\rho)
:=\Bigl(\det\bigl({\text{\rm id}}-(-1)^{\epsilon_-}\rho(\alpha)^{-1}\bigr)
\Bigr)^{(-1)^{\epsilon_-+\epsilon_+}}$$ where $\alpha \in \Gamma$ is a representative of $a.$
Let $(M,x_0)$ be a $V$-acyclic manifold and $\Gamma=\pi_1(M,x_0)$. Note that $[S^1,M]$ identifies with the conjugacy classes of $\Gamma$. Suppose $X$ is a vector field satisfying (L) and (NCT). Every closed trajectory $\hat\theta$ gives rise to a conjugacy class $[\hat\theta]\in[S^1,M]$ and two signs $\epsilon_\pm(\hat\theta)$. These signs are obtained from the differential of the return map in normal direction; $\epsilon_-(\hat\theta)$ is the parity of the number of real eigenvalues larger than $+1$ and $\epsilon_-(\hat\theta)$ is the parity of the number of real eigenvalues smaller than $-1$. For a simple closed trajectory, i.e. of period $p(\hat\theta)=1$, let us consider the triple $\underline{\hat\theta}
:=\bigl([\hat\theta],\epsilon_-(\hat\theta),\epsilon_+(\hat\theta)\bigr)$. This gives a (at most countable) set of triples as in the previous paragraph.
Let $\xi\in H^1(M;\mathbb R)$ be a Lyapunov cohomology class for $X$. Recall that for every $R$ there are only finitely many closed trajectories $\hat\theta$ with $-\xi([\hat\theta])\leq R$. Hence, we get a rational function $\zeta_R^{X,\xi}:\operatorname{Rep}(\Gamma;V)\to\mathbb C$ $$\zeta_R^{X,\xi}:=\prod_{-\xi([\hat\theta])\leq R}\operatorname{let}_{\underline{\hat\theta}}$$ where the product is over all triples $\underline{\hat\theta}$ associated to simple closed trajectories with $-\xi([\hat\theta])\leq R$. It is easy to check that formally we have $$\lim_{R\to\infty}\zeta_R^{X,\xi}=e^{P_X}.$$ It would be interesting to understand in what sense (if any) this can be made precise. We conjecture that there exists an open set with non-empty interior in each component of $\operatorname{Rep}(\Gamma;V)$ on which we have true convergence. In fact there exist vector fields $X$ where the sets of triples are finite in which case the conjecture is obviously true.
[XXXX]{}
M.F. Atiyah, V.K. Patodi and I.M. Singer, *Spectral asymmetry and Riemannian geometry. II,* Math. Proc. Cambridge Philos. Soc. **78**(1975), 405–432.
J.M. Bismut and W. Zhang, *An extension of a theorem by Cheeger and Müller,* Astérisque **205**, Société Mathématique de France, 1992.
R. Bott and L.W. Tu, *Differential forms in algebraic topology.* Graduate texts in Mathematics **82**. Springer Verlag, New York–Berlin, 1982.
M. Braverman and T. Kappeler, *Refined analytic torsion,*\
preprint [math.DG/0505537.]{}
D. Burghelea, *Removing Metric Anomalies from Ray–Singer Torsion,* Lett. Math. Phys. **47**(1999), 149–158.
D. Burghelea, L. Friedlander and T. Kappeler, *Relative Torsion,* Commun. Contemp. Math. **3**(2001), 15–85.
D. Burghelea and S. Haller, *Euler structures, the variety of representations and the Milnor–Turaev torsion,* preprint [math.DG/0310154.]{}
D. Burghelea and S. Haller, *Laplace transform, dynamics and spectral geometry (version 1),* preprint [math.DG/0405037.]{}
D. Burghelea and S. Haller, *Dynamics, Laplace transform, and spectral geometry (version 2),* preprint [math.DG/0508216.]{}
D. Burghelea and S. Haller, *Complex valued Ray–Singer torsion.* Preprint [math.DG/0604484.]{}
D. Burghelea and S. Haller, *Dynamical torsion, a non-commutative version of a theorem of Hutchings–Lee and Pajitnov,* in preparation.
S. Dineen, *Complex Analysis in Locally Convex Spaces.* Mathematics Studies **57**, North-Holland, 1981.
M. Farber, *Singularities of the analytic torsion,* J. Differential Geom. **41**(1995), 528–572.
M. Farber and V. Turaev, *Poincaré–Reidemeister metric, Euler structures and torsion,* J. Reine Angew. Math. **520**(2000), 195–225.
R. Hartshorne, *Algebraic Geometry.* Graduate Texts in Mathematics **52**. Springer Verlag, New York–Heidelberg, 1977.
M. Hutchings, *Reidemeister torsion in generalized Morse Theory,* Forum Math. **14**(2002), 209–244.
B. Jiang, *Estimation of the number of periodic orbits,* Pac. J. Math. **172**(1996), 151–185.
A. Kriegl and P.W. Michor, *The convenient setting of global analysis.* Mathematical Surveys and Monographs **53**, American Mathematical Society, Providence, RI, 1997.
J. Marcsik, *Analytic torsion and closed one forms.* Thesis, Ohio State University, 1995.
V. Mathai and D. Quillen, *Superconnections, Thom Classes, and Equivariant differential forms,* Topology **25**(1986), 85–110.
J. Milnor, *Whitehead torsion,* Bull. Amer. Math. Soc. **72**(1966), 358–426.
W. Müller, private communication.
L. Nicolaescu, *The Reidemeister Torsion of $3$–manifolds.* De Gruyter Studies in Mathematics [**30**]{}, Walter de Gruyter & Co., Berlin, 2003.
D. Quillen, *Determinants of Cauchy–Riemann operators on Riemann surfaces,* Functional Anal. Appl. **19**(1985), 31–34. Translation of Funktsional. Anal. i Prilozhen. **19**(1985), 37–41.
D.B. Ray and I.M. Singer, *R–torsion and the Laplacian on Riemannian manifolds,* Adv. in Math. **7**(1971), 145–210.
N. Steenrod, *The topology of fiber bundles.* Reprint of the 1957 edition. Princeton University Press, Princeton, NJ, 1999.
V. Turaev, *Reidemeister torsion in Knot theory,* Uspekhi Mat. Nauk **41**(1986), 119–182.
V. Turaev, *Euler structures, nonsingular vector fields, and Reidemeister-type torsions,* Math. USSR–Izv. **34**(1990), 627–662.
V. Turaev, *Introduction to combinatorial torsions.* Notes taken by Felix Schlenk. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2001.
V. Turaev, *Torsion of 3-dimensional manifolds.* Progress in Mathematics **208**. Birkhäuser Verlag, Bassel–Boston–Berlin, 2002.
G.W. Whitehead, *Elements of homotopy theory.* Graduate Texts in Mathematics [**61**]{}. Springer Verlag, New York–Berlin, 1978.
[^1]: Part of this work was done while both authors enjoyed the hospitality of the Max Planck Institute for Mathematics in Bonn. A previous version was written while the second author enjoyed the hospitality of the Ohio State University. The second author was partially supported by the *Fonds zur Förderung der wissenschaftlichen Forschung* (Austrian Science Fund), project number [P17108-N04]{}
[^2]: A similar question was considered in [@Q85] and a positive answer provided.
[^3]: This $\omega$ has nothing in common with $\omega(\nabla,b)$ notation used in the previous section.
[^4]: For a closed trajectory the map whose homotopy class is considered is $\hat \theta: \mathbb R/ T\mathbb Z\to M.$
[^5]: It is understood that only finitely many terms from the left side of the equality are not zero. Here $*$ denotes juxtaposition.
[^6]: Any Euler vector field $X$ satisfies (H), (EG), (L) and has no closed trajectory, hence also satisfies (NCT). The counting functions of instantons are exactly the same as the incidence numbers of the triangulation hence take the values $1$, $-1$ or $0$.
[^7]: The hypothesis is not necessary and the theory of coEuler structure can be pursued for an arbitrary base pointed smooth manifold $(M,x_0)$, cf. [@BH04].
[^8]: We will use the same notation D for the Poincaré duality isomorphism $D:H_1(M;{\mathbb R})\to H^{n-1}(M; \mathcal O_M)$.
[^9]: The idea of considering $b$-Laplacians for torsion was brought to the attention of the first author by W. Müller [@M]. The second author came to it independently.
[^10]: Even more, we conjecture that converges absolutely on an open set $U$ as in Proposition \[P:8\].
|
---
abstract: 'Liquid Argon Time Projection Chamber detectors are ideally suited for studying neutrino interactions and probing the parameters that characterize neutrino oscillations. The ability to drift ionization particles over long distances in purified argon and to trigger on abundant scintillation light allows for excellent particle identification and triggering capability. In these proceedings the details of the ArgoNeuT test-beam project will be presented after a brief introduction to the detector technique. ArgoNeuT is a 175 liter detector exposed to Fermilab’s NuMI neutrino beamline. The first neutrino interactions observed in ArgoNeuT will be presented, along with discussion of the various physics analyses to be performed on this data sample.'
author:
- 'M. Soderberg, for the ArgoNeuT Collaboration'
title: 'ArgoNeuT: A Liquid Argon Time Projection Chamber Test in the NuMI Beamline'
---
Introduction
============
Liquid Argon Time Projection Chambers (LArTPCs) are an appealing class of detectors that offers exceptional opportunities for studying neutrino interactions thanks to the bubble-chamber quality images they can provide. The unique combination of position resolution, calorimetry, and scalability provided by LArTPCs make them a possible technology choice for future massive detectors. There is an active program in the U.S. to develop LArTPCs, with the final goal of constructing a massive detector that can be used as a far detector in a long-baseline neutrino oscillation experiment. The Argon Neutrino Teststand, or ArgoNeuT, project is an important early step in this program, and it will be the focus of this document.
LArTPC Technique
================
The LArTPC technique has been around for several decades, with pioneering work done as part of the ICARUS experiment [@Rubbia; @ICARUS]. A wire chamber is placed in highly-purified liquid argon, and an electric field is created within this detector. Neutrino interactions with the argon inside the detector volume produce ionization electrons that drift along the electric field until they reach finely segmented and instrumented anodes ($\textit{i.e.}$ - wireplanes), upon which they produce signals that are utilized for imaging and analyzing the event that occurred, as shown in Fig. $\ref{fig:tpc}$. Applying proper bias voltages to the wireplanes, such that electrons drift undisturbed through the initial planes, allows several complementary views of the same interaction that can be combined into a three-dimensional image of the event[@Grids]. Calorimetric measurements can be extracted from the pulses observed on the wireplanes.
![Schematic diagram of TPC operation. Ionization is drifted along an electric field to multiple planes of readout wires, each of which is instrumented with low-noise preamplifiers and fast digitization electronics. Position of the wire within the plane, and knowledge of drift times, allow event spatial reconstruction.[]{data-label="fig:tpc"}](./pics/figure1.jpg){width="2in"}
This technique allows for very precise imaging, the resolution being dependent on several factors: wire pitch, plane spacing, sampling rate, and electronics S/N levels. The wire pitch is typically on the order of several millimeters, the specific value being chosen to maximize resolution without sacrificing S/N levels. The rapid sampling rate ($\approx$5MHz) characteristic of the readout electronics, combined with the slow drift speed of ionization ($\approx$1.5mm/$\mu$s) at nominal electric field values, equates to an image resolution of fractions of a millimeter along the drift direction (which is the coordinate common to all the wireplanes of the TPC). The technology is further made attractive in that the number of electronics channels required for the detector does not scale directly with the volume of the detector if the drift distance is increased appropriately. This scaling feature, along with the relatively low cost of argon, makes LArTPCs an intriguing option for future massive neutrino detectors.
While LArTPCs are an intriguing detector technology, they are not without their challenges. One of the biggest challenges is producing and maintaining argon that is pure enough to allow the ionization to drift for the required distances. To address this issue new filters that can cleanse the argon to the required purity levels necessary for a LArTPC experiment, and can also be regenerated when they have become saturated, have been developed[@Filter]. These new filters are a necessary step along the path to massive detectors, and they have already been used by several test stands built in the U.S. with the goals of studying detector material effects on argon purity, and looking for cosmic-ray events in a LArTPC [@Yale].
ArgoNeuT
========
ArgoNeuT is a LArTPC that is currently running in the NuMI beamline at Fermilab. The ArgoNeuT project was started in order to gain experience building and operating LArTPCs in a real beam environment, and also to collect a very interesting data sample that will be used to develop simulation and reconstruction code. ArgoNeuT will provide a sample of neutrino interactions in liquid argon for the first time ever in the U.S., and for the first time ever in a low-energy beam. The only previous LArTPC to operate in a neutrino beam was a 50-liter TPC built as part of the ICARUS program that ran in the WANF beam at CERN in the late 1990’s [@50L]. The energy of the NuMI beam (peaking at $\approx$3 GeV) is significantly lower than that of the WANF beam (mean energy of $\approx$24 GeV), making the data accumulated by ArgoNeuT particularly interesting since this is an energy-range relevant to neutrino oscillation physics.
TPC
---
ArgoNeuT’s TPC is a rectangular volume measuring 90cm x 40cm x 48cm, containing an active volume of $\approx$175 liters of liquid, that is positioned inside of a vacuum jacketed cryostat. Figure \[fig:tpc\_pics\] depicts the TPC before and after it was inserted into the inner cryostat volume. The TPC consists of three wireplanes, each with 4mm wire pitch. The innermost induction plane has vertical wire orientation, but is not instrumented with readout electronics and is used primarily for pulse shaping. The middle induction plane has 240 wires oriented at +60$^{\circ}$ with respect to the horizontal beam direction, while the outermost collection plane has 240 wires oriented at -60$^{\circ}$ with respect to the horizontal beam direction. Both the middle induction plane and the collection plane are instrumented with readout electronics. The maximum drift distance in the TPC, from the cathode to the first induction plane is 48cm. The operating cathode voltage of 25kV creates an electric field of 500V/cm, at which the drift speed is 1.55 mm/$\mu$s. The inside of the TPC contains 23 field “rings" that are 1cm in thickness, formed from machined copper-clad G10 sheets, with consecutive rings connected by four 100M$\Omega$ resistors in parallel.
![The ArgoNeuT TPC.[]{data-label="fig:tpc_pics"}](./pics/figure2a.jpg "fig:"){width="3in"} ![The ArgoNeuT TPC.[]{data-label="fig:tpc_pics"}](./pics/figure2b.jpg "fig:"){width="3in"}
Electronics
-----------
A custom electronic readout system has been built for the ArgoNeuT detector. Bias voltage distribution cards (BVDCs), that provide filtered voltage to the wireplanes, are placed directly on the TPC. Each BVDC connects to 24 TPC wires, and sends output signals to ribbon cables that connect to a custom feedthrough circuit board designed by Fermilab. Preamplifier boards, each of which contain 16 FET preamplifiers, reside in a Faraday-cage enclosure surrounding the signal-feedthrough board flange. The input signals to the preamplifier boards are sent through a wide bandwidth filter that removes frequencies outside of the expected range. The amplified signals are sent to digitization boards (ADF2 cards, on loan from the D0 experiment) which sample the waveform at 5MHz (198ns/sample). The DAQ system is triggered by a clock signal associated with the NuMI beam spill, causing each channel to begin recording 2048 ADC samples with 10-bit resolution. The total readout time for a single trigger is $\approx$400$\mu$s, which is significantly longer than the maximum drift time of particles in the TPC (333$\mu$s) allowing for pre/post-sampling of each spill. The pre/post-sampling information is useful for removing spurious tracks that come from outside of the beam window. Figure \[fig:electronic\_pics\] shows several of the components of the TPC readout system.
![Some of the components of the ArgoNeuT readout system. Left: Bias Voltage Distribution Card. Center: Preamplifier board. Right: ADF2 board.[]{data-label="fig:electronic_pics"}](./pics/figure3a.jpg "fig:"){width="1.0in" height="1.2in"} ![Some of the components of the ArgoNeuT readout system. Left: Bias Voltage Distribution Card. Center: Preamplifier board. Right: ADF2 board.[]{data-label="fig:electronic_pics"}](./pics/figure3b.jpg "fig:"){width="1.0in" height="1.2in"} ![Some of the components of the ArgoNeuT readout system. Left: Bias Voltage Distribution Card. Center: Preamplifier board. Right: ADF2 board.[]{data-label="fig:electronic_pics"}](./pics/figure3c.jpg "fig:"){width="1.0in" height="1.2in"}
Cryogenics
----------
The main component of the ArgoNeuT cryogenic system is a 550 liter vacuum-insulated cryostat that houses the TPC and contains feedthrough ports for all of the instrumentation of the experiment. ArgoNeuT uses a self-contained recirculation system to continually pass the liquified argon in the system through the new Fermilab style filters. Boil-off vapor from the inner cryostat is directed vertically up to a 300W Gifford-McMahon cryocooler, where it is condensed and directed back down through one of three return paths to the inner cryostat. Two of these return paths contain filters, while the third is a bypass line that is used during maintenance of the other pathways. The three return lines merge before entering the cryostat, wherein they are guided down to the bottom of the liquid volume and empty through a sintered-metal cap. Figure \[fig:cryo\_pics\] shows the ArgoNeuT detector as it looked in the summer of 2008 during a commissioning run on the surface. In this figure the outer cryostat flange is removed, showing the inner cryostat wrapped in superinsulation.
![ArgoNeuT’s Cryogenic System. The outer cryostat flange is removed.[]{data-label="fig:cryo_pics"}](./pics/figure4.jpg){width="2.5in"}
The experiment is outfitted with numerous safety features to maintain the Oxygen Deficiency Hazard (ODH) rating of the NuMI tunnel. All cryogenic plumbing contains relief valves that are routed to a vent line that extends up the NuMI shaft and out to the surface. In this way any argon gas that is vented, or that might leak from a pipe, is not released into the tunnel but rather is guided out to the surface. The outer cryostat acts as a secondary containment vessel in case the inner vessel leaks, and a “bathtub" acts as tertiary containment in case both cryostats develop leaks. The “bathtub" contains ODH sensors that trigger alarms and mixing fans if a leak is detected. A dedicated process-control system was built for ArgoNeuT that allows remote monitoring of all systems and remote control over important valves.
Location
--------
ArgoNeuT is currently running in the NuMI tunnel at Fermilab, where it is positioned approximately in the center of the beam, and directly upstream of the MINOS near detector. ArgoNeuT’s TPC is too small to contain the majority of muons produced in neutrino interactions from the energetic NuMI beamline. To compensate for the information lost by particles exiting the detector, ArgoNeuT will utilize the MINOS near detector as a range stack to capture the full trajectories of these particles. Since MINOS is magnetized there is the possibility to perform charge identification by matching a track in ArgoNeuT with a track in MINOS.
![Diagram of experiments in the NuMI tunnel. ArgoNeuT is located directly upstream of the MINOS near detector, and directly downstream of the MINERvA detector.[]{data-label="fig:tunnel"}](./pics/figure5.jpg){width="3in"}
ArgoNeuT Physics
================
Despite ArgoNeuT’s small detector volume, it will collect a significant sample of neutrino/antineutrino interactions. As was mentioned previously, this will be the first such sample from a LArTPC operating in the U.S., and the first sample ever in a low-energy (few GeV) neutrino beam. There are several physics analyses that can be carried out with the ArgoNeuT data sample, with one of the most interesting coming from the several thousand charged-current quasi-elastic (CCQE) events that will be collected in antineutrino mode. These events will be recognized by a muon created in the volume of the TPC, possibly accompanied by one or more proton tracks created by a final-state neutron. By utilizing the MINOS near-detector to range-out muons originating from ArgoNeuT their energy can be determined, allowing a first measurement of the CCQE cross-section on argon for neutrinos in the few-GeV range. This result will be particularly interesting for neutrino oscillation physics, since knowledge of the CCQE cross-section is crucial to oscillation analyses. The ability of LArTPCs to see low-energy nuclear fragments created in neutrino interactions, and to determine their impact on the cross-section measurements, may also prove to be very interesting.
Members of the ArgoNeuT collaboration are working on developing a full software environment for analyzing their data. The software being developed will be utilized for everything from simulating neutrino interactions in a LArTPC to reconstructing the interactions starting from the raw TPC data. Such software does not currently exist in the U.S., but is a necessary tool for any future large LArTPC detector where the statistics of the data will be greatly increased. One of the main goals of ArgoNeuT is to utilize this new software to fully develop the $dE/dx$ particle identification technique, and to provide a measurement of the capabilities of the technology to separate electron tracks from photon tracks. This software will be used for future LArTPC detectors, such as the MicroBooNE experiment, so early experience gained from ArgoNeuT will be important.
ArgoNeuT Status
===============
![Neutrino event candidate from ArgoNeuT. The raw data for the instrumented induction and collection planes are displayed.[]{data-label="fig:argoneut_event1"}](./pics/figure6.jpg){width="80mm"}
ArgoNeuT was filled with liquid argon for the first time in its underground location in May 2009. The initial electron lifetime was much lower than anticipated, but after several weeks of recirculating through the closed-loop filtration system the lifetime had recovered significantly. Many neutrino events were recorded during this initial run before the summer 2009 Fermilab shutdown. The raw data from several of these events are displayed in Figs. \[fig:argoneut\_event1\], \[fig:argoneut\_event2\], and \[fig:argoneut\_event3\]
Each event display depicts the information from the instrumented induction and collection plane of ArgoNeuT. The horizontal axis is the wire number within each of the planes, while the vertical axis is the sampling time of the DAQ, which is common to both the induction and collection views. Figure \[fig:argoneut\_event3\] depicts the collection plane view, and also shows the raw pulse information for a particular wire ($\#$140) of the collection plane. There are three clear pulses visible in this wire, with the third pulse containing a double-peak that indicates the presence of two closely spaced tracks.
![Neutrino event candidate from ArgoNeuT.[]{data-label="fig:argoneut_event2"}](./pics/figure7.jpg){width="80mm"}
![Neutrino event candidate from ArgoNeuT. The raw signal from wire $\#$140 of the collection plane is displayed.[]{data-label="fig:argoneut_event3"}](./pics/figure8.jpg){width="80mm"}
The collaboration is currently developing algorithms for analyzing the TPC data. The dark vertical bands visible in the collection view of Figures \[fig:argoneut\_event2\] and \[fig:argoneut\_event3\] are a result of a baseline shift in the ADCs after recording energetic hits. Fourier deconvolution can be performed on the raw data to accurately remove this baseline shift and return the true waveform. Several hit finding methods are being considered to isolate the important sections of the waveforms recorded after each trigger. Extracting overlapping hits, such as those depicted in Figure \[fig:argoneut\_event3\], will provide information about the two-track separation achievable in these detectors.
ArgoNeuT will continue to take data through the Fall of 2009 and into early 2010, primarily in antineutrino mode.
The author would like to acknowledge the support staff at Fermilab for their invaluable contributions in the planning and construction of ArgoNeuT. Also, the Department of Energy and the National Science Foundation.
[9]{} The Liquid-argon time projection chamber: a new concept for Neutrino Detector, C. Rubbia, CERN-EP/77-08 (1977); Design, construction and tests of the ICARUS T600 detector, Nucl. Inst. Meth., A527 329-410 (2004); Design of Grid Ionization Chambers, O. Bunemann, T.E. Cranshaw, and J.A. Harvey; Canadian Journal of Research, 27, 191-206, (1949); A Regenerable Filter for Liquid Argon Purification, A. Curioni *et. al*, Nucl. Inst. Meth., A605 306-311 (2009); The Yale Liquid Argon Time Projection Chamber, A. Curioni, B. Fleming, M. Soderberg, arXiv:0804.0415 (2008); Performance Of A Liquid Argon Time Projection Chamber Exposed To The WANF Neutrino Beam, F. Arneodo *et. al*, Phys. Rev. D.74, 112001 (2006);
|
---
abstract: 'Event-by-event fluctuations of the average transverse momentum of produced particles near mid-rapidity have been measured by the PHENIX Collaboration in $\sqrt{s_{NN}}=200$ GeV Au+Au and p+p collisions at the Relativistic Heavy Ion Collider. The fluctuations are observed to be in excess of the expectation for statistically independent particle emission for all centralities. The excess fluctuations exhibit a dependence on both the centrality of the collision and on the $p_T$ range over which the average is calculated. Both the centrality and $p_T$ dependence can be well reproduced by a simulation of random particle production with the addition of contributions from hard scattering processes.'
author:
- 'S.S. Adler'
- 'S. Afanasiev'
- 'C. Aidala'
- 'N.N. Ajitanand'
- 'Y. Akiba'
- 'J. Alexander'
- 'R. Amirikas'
- 'L. Aphecetche'
- 'S.H. Aronson'
- 'R. Averbeck'
- 'T.C. Awes'
- 'R. Azmoun'
- 'V. Babintsev'
- 'A. Baldisseri'
- 'K.N. Barish'
- 'P.D. Barnes'
- 'B. Bassalleck'
- 'S. Bathe'
- 'S. Batsouli'
- 'V. Baublis'
- 'A. Bazilevsky'
- 'S. Belikov'
- 'Y. Berdnikov'
- 'S. Bhagavatula'
- 'J.G. Boissevain'
- 'H. Borel'
- 'S. Borenstein'
- 'M.L. Brooks'
- 'D.S. Brown'
- 'N. Bruner'
- 'D. Bucher'
- 'H. Buesching'
- 'V. Bumazhnov'
- 'G. Bunce'
- 'J.M. Burward-Hoy'
- 'S. Butsyk'
- 'X. Camard'
- 'J.-S. Chai'
- 'P. Chand'
- 'W.C. Chang'
- 'S. Chernichenko'
- 'C.Y. Chi'
- 'J. Chiba'
- 'M. Chiu'
- 'I.J. Choi'
- 'J. Choi'
- 'R.K. Choudhury'
- 'T. Chujo'
- 'V. Cianciolo'
- 'Y. Cobigo'
- 'B.A. Cole'
- 'P. Constantin'
- 'D.G. d’Enterria'
- 'G. David'
- 'H. Delagrange'
- 'A. Denisov'
- 'A. Deshpande'
- 'E.J. Desmond'
- 'O. Dietzsch'
- 'O. Drapier'
- 'A. Drees'
- 'R. du Rietz'
- 'A. Durum'
- 'D. Dutta'
- 'Y.V. Efremenko'
- 'K. El Chenawi'
- 'A. Enokizono'
- 'H. En’yo'
- 'S. Esumi'
- 'L. Ewell'
- 'D.E. Fields'
- 'F. Fleuret'
- 'S.L. Fokin'
- 'B.D. Fox'
- 'Z. Fraenkel'
- 'J.E. Frantz'
- 'A. Franz'
- 'A.D. Frawley'
- 'S.-Y. Fung'
- 'S. Garpman'
- 'T.K. Ghosh'
- 'A. Glenn'
- 'G. Gogiberidze'
- 'M. Gonin'
- 'J. Gosset'
- 'Y. Goto'
- 'R. Granier de Cassagnac'
- 'N. Grau'
- 'S.V. Greene'
- 'M. Grosse Perdekamp'
- 'W. Guryn'
- 'H.-[Å]{}. Gustafsson'
- 'T. Hachiya'
- 'J.S. Haggerty'
- 'H. Hamagaki'
- 'A.G. Hansen'
- 'E.P. Hartouni'
- 'M. Harvey'
- 'R. Hayano'
- 'X. He'
- 'M. Heffner'
- 'T.K. Hemmick'
- 'J.M. Heuser'
- 'M. Hibino'
- 'J.C. Hill'
- 'W. Holzmann'
- 'K. Homma'
- 'B. Hong'
- 'A. Hoover'
- 'T. Ichihara'
- 'V.V. Ikonnikov'
- 'K. Imai'
- 'D. Isenhower'
- 'M. Ishihara'
- 'M. Issah'
- 'A. Isupov'
- 'B.V. Jacak'
- 'W.Y. Jang'
- 'Y. Jeong'
- 'J. Jia'
- 'O. Jinnouchi'
- 'B.M. Johnson'
- 'S.C. Johnson'
- 'K.S. Joo'
- 'D. Jouan'
- 'S. Kametani'
- 'N. Kamihara'
- 'J.H. Kang'
- 'S.S. Kapoor'
- 'K. Katou'
- 'S. Kelly'
- 'B. Khachaturov'
- 'A. Khanzadeev'
- 'J. Kikuchi'
- 'D.H. Kim'
- 'D.J. Kim'
- 'D.W. Kim'
- 'E. Kim'
- 'G.-B. Kim'
- 'H.J. Kim'
- 'E. Kistenev'
- 'A. Kiyomichi'
- 'K. Kiyoyama'
- 'C. Klein-Boesing'
- 'H. Kobayashi'
- 'L. Kochenda'
- 'V. Kochetkov'
- 'D. Koehler'
- 'T. Kohama'
- 'M. Kopytine'
- 'D. Kotchetkov'
- 'A. Kozlov'
- 'P.J. Kroon'
- 'C.H. Kuberg'
- 'K. Kurita'
- 'Y. Kuroki'
- 'M.J. Kweon'
- 'Y. Kwon'
- 'G.S. Kyle'
- 'R. Lacey'
- 'V. Ladygin'
- 'J.G. Lajoie'
- 'A. Lebedev'
- 'S. Leckey'
- 'D.M. Lee'
- 'S. Lee'
- 'M.J. Leitch'
- 'X.H. Li'
- 'H. Lim'
- 'A. Litvinenko'
- 'M.X. Liu'
- 'Y. Liu'
- 'C.F. Maguire'
- 'Y.I. Makdisi'
- 'A. Malakhov'
- 'V.I. Manko'
- 'Y. Mao'
- 'G. Martinez'
- 'M.D. Marx'
- 'H. Masui'
- 'F. Matathias'
- 'T. Matsumoto'
- 'P.L. McGaughey'
- 'E. Melnikov'
- 'F. Messer'
- 'Y. Miake'
- 'J. Milan'
- 'T.E. Miller'
- 'A. Milov'
- 'S. Mioduszewski'
- 'R.E. Mischke'
- 'G.C. Mishra'
- 'J.T. Mitchell'
- 'A.K. Mohanty'
- 'D.P. Morrison'
- 'J.M. Moss'
- 'F. M[ü]{}hlbacher'
- 'D. Mukhopadhyay'
- 'M. Muniruzzaman'
- 'J. Murata'
- 'S. Nagamiya'
- 'J.L. Nagle'
- 'T. Nakamura'
- 'B.K. Nandi'
- 'M. Nara'
- 'J. Newby'
- 'P. Nilsson'
- 'A.S. Nyanin'
- 'J. Nystrand'
- 'E. O’Brien'
- 'C.A. Ogilvie'
- 'H. Ohnishi'
- 'I.D. Ojha'
- 'K. Okada'
- 'M. Ono'
- 'V. Onuchin'
- 'A. Oskarsson'
- 'I. Otterlund'
- 'K. Oyama'
- 'K. Ozawa'
- 'D. Pal'
- 'A.P.T. Palounek'
- 'V.S. Pantuev'
- 'V. Papavassiliou'
- 'J. Park'
- 'A. Parmar'
- 'S.F. Pate'
- 'T. Peitzmann'
- 'J.-C. Peng'
- 'V. Peresedov'
- 'C. Pinkenburg'
- 'R.P. Pisani'
- 'F. Plasil'
- 'M.L. Purschke'
- 'A.K. Purwar'
- 'J. Rak'
- 'I. Ravinovich'
- 'K.F. Read'
- 'M. Reuter'
- 'K. Reygers'
- 'V. Riabov'
- 'Y. Riabov'
- 'G. Roche'
- 'A. Romana'
- 'M. Rosati'
- 'P. Rosnet'
- 'S.S. Ryu'
- 'M.E. Sadler'
- 'N. Saito'
- 'T. Sakaguchi'
- 'M. Sakai'
- 'S. Sakai'
- 'V. Samsonov'
- 'L. Sanfratello'
- 'R. Santo'
- 'H.D. Sato'
- 'S. Sato'
- 'S. Sawada'
- 'Y. Schutz'
- 'V. Semenov'
- 'R. Seto'
- 'M.R. Shaw'
- 'T.K. Shea'
- 'T.-A. Shibata'
- 'K. Shigaki'
- 'T. Shiina'
- 'C.L. Silva'
- 'D. Silvermyr'
- 'K.S. Sim'
- 'C.P. Singh'
- 'V. Singh'
- 'M. Sivertz'
- 'A. Soldatov'
- 'R.A. Soltz'
- 'W.E. Sondheim'
- 'S.P. Sorensen'
- 'I.V. Sourikova'
- 'F. Staley'
- 'P.W. Stankus'
- 'E. Stenlund'
- 'M. Stepanov'
- 'A. Ster'
- 'S.P. Stoll'
- 'T. Sugitate'
- 'J.P. Sullivan'
- 'E.M. Takagui'
- 'A. Taketani'
- 'M. Tamai'
- 'K.H. Tanaka'
- 'Y. Tanaka'
- 'K. Tanida'
- 'M.J. Tannenbaum'
- 'P. Tarj[á]{}n'
- 'J.D. Tepe'
- 'T.L. Thomas'
- 'J. Tojo'
- 'H. Torii'
- 'R.S. Towell'
- 'I. Tserruya'
- 'H. Tsuruoka'
- 'S.K. Tuli'
- 'H. Tydesj[ö]{}'
- 'N. Tyurin'
- 'H.W. van Hecke'
- 'J. Velkovska'
- 'M. Velkovsky'
- 'L. Villatte'
- 'A.A. Vinogradov'
- 'M.A. Volkov'
- 'E. Vznuzdaev'
- 'X.R. Wang'
- 'Y. Watanabe'
- 'S.N. White'
- 'F.K. Wohn'
- 'C.L. Woody'
- 'W. Xie'
- 'Y. Yang'
- 'A. Yanovich'
- 'S. Yokkaichi'
- 'G.R. Young'
- 'I.E. Yushmanov'
- 'W.A. Zajc'
- 'C. Zhang'
- 'S. Zhou'
- 'S.J. Zhou'
- 'L. Zolin'
title: 'Measurement of Non-Random Event-by-Event Fluctuations of Average Transverse Momentum in $\sqrt{s_{NN}}=200$ GeV Au+Au and p+p Collisions'
---
The measurement of fluctuations in the event-by-event average transverse momentum of produced particles in relativistic heavy ion collisions has been proposed as a probe of phase instabilities near the QCD phase transition [@Heis01; @Step98; @Step99], which could result in classes of events with different properties, such as the effective temperature of the collision. Fluctuation measurements could also provide information about the onset of thermalization in the system [@Gav03]. The resulting phenomena can be observed by measuring deviations of the event-by-event average $p_T$, referred to here as $M_{p_T}$, of produced charged particles from the expectation for statistically independent particle emission [@Stod95; @Shur98] after subtracting contributions from fluctuations arising from physical processes such as elliptic flow and jet production.
Several $M_{p_T}$ fluctuation measurements have been reported in heavy ion collisions [@NA49; @CERES; @PPG005; @STAR], including a study by PHENIX [@PPG005] in $\sqrt{s_{NN}}=130$ GeV Au+Au collisions which set limits on the magnitude of non-random fluctuations in $M_{p_T}$. Recently, STAR has reported fluctuations in excess of the random expectation, within the PHENIX limits, at the same collision energy [@STAR]. For the first results from $\sqrt{s_{NN}}=200$ GeV Au+Au and p+p collisions reported here, upgrades of the PHENIX central arm spectrometers [@PHNIM] have expanded the azimuthal acceptance from $58.5^\circ$ to $180.0^\circ$ within the pseudorapidity range of $|\eta|<0.35$. Pad chamber and calorimeter detectors have also been utilized for improved background rejection. As a result, the sensitivity of the PHENIX spectrometer to the observation of fluctuations in $M_{p_T}$ due to event-by-event fluctuations in the effective temperature [@PPG005; @Korus01] has improved by greater than a factor of two.
Minimum bias events triggered by a coincidence between the Zero Degree Calorimeters (ZDC) and the Beam-Beam Counters (BBC), with a requirement that the collision vertex, which is measured with an r.m.s. resolution of less than 6 mm in central collisions and 8 mm in the most peripheral collisions, be within 5 cm of the nominal origin, are used in this analysis. Event centrality for Au+Au collisions, which is defined using correlations in the BBC and ZDC analog response [@PPG002], is divided into several classes, each containing an average of 244,000 analyzed events. These classes are associated to the estimated average number of participants in the collision, $<N_{part}>$, which is derived using a Glauber model Monte Carlo calculation with the BBC and ZDC detector response taken into account [@Glauber].
Charged particle momenta are reconstructed in the PHENIX central arm spectrometers with a drift chamber and a radially adjacent pixel pad chamber. Non-vertex track background rejection is provided by pixel pad chambers and calorimeters located further outward radially from the collision vertex [@Mit02]. The momentum resolution is $\frac{\delta
p}{p} \simeq 0.7\% \oplus 1.0\% \times p$ (GeV/c).
$M_{p_T}$ is calculated for each event, which contains a number of reconstructed tracks within a specified $p_T$ range, $N_{tracks}$. The $p_T$ range is always given a lower bound of 200 MeV/c and a varying upper bound, $p_{T}^{max}$, from 500 MeV/c to 2.0 GeV/c. There is a minimum $N_{tracks}$ cut of 3 in both Au+Au events (removing 0%, 4.6%, and 29% of events in the 0-50%, 50-60%, and 60-70% centrality ranges, respectively, when $p_{T}^{max}$ = 2.0 GeV/c) and p+p events (removing 59% of the events).
There are several measures by which the magnitude of non-random fluctuations can be quantified, namely $\phi_{p_T}$ [@Gaz92; @Mro98], $\nu_{dynamic}$ [@Pru02], and $F_{p_T}$ [@PPG005]. The calculation of $F_{p_T}$ is based upon the magnitude of the fluctuation, $\omega_{p_T}$, defined as $$\omega_{p_T} = \frac{(<M_{p_T}^2> - <M_{p_T}>^2)^{1/2}}{<M_{p_T}>} = \frac{\sigma_{M_{p_T}}}{<M_{p_T}>}.$$ $F_{p_T}$ is defined as the fractional deviation of $\omega_{p_T}$ from a baseline estimate defined using mixed events, $$F_{p_T} = \frac{(\omega_{(p_T,~data)}-\omega_{(p_T,~mixed)})}{\omega_{(p_T,~mixed)}}.$$
Mixed event $M_{p_T}$ distributions are validated by comparisons to a calculation of $M_{p_T}$ assuming statistically independent particle emission using parameters extracted from the inclusive $p_T$ distributions of the data [@Tan01]. For the 0-5% centrality class, which suffers the most from tracking inefficiency, the effects of two-track resolution, and background contributions, the mixed event $M_{p_T}$ distribution yields a value of $F_{p_T}=0.04\%$ with respect to the calculation. The results of this comparison are included in the estimates of the systematic errors. Further details on the mixed event procedure and a discussion of contributions to the value of $F_{p_T}$ from detector efficiency and resolution effects can be found in the description of the data analysis of $\sqrt{s_{NN}}=130$ GeV Au+Au collisions [@PPG005].
Comparisons of the data and mixed event $M_{p_T}$ distributions for the 0-5% and 30-35% centrality classes are shown in Fig. 1. Any excess fluctuations are small and are difficult to distinguish by eye in a direct overlay of the $M_{p_T}$ distributions. Therefore, the comparison is also shown as residuals of the difference between the data and mixed event distributions in units of standard deviations of the individual data points. The double-peaked shape in the residual distributions is an artifact of the fact that the mixed event distributions, which always have a smaller standard deviation in $M_{p_T}$ than the data, are normalized to minimize the total $\chi^2$ of the residual distribution.
![\[fig:1\] Comparisons between the data and mixed event $M_{p_T}$ distributions for the representative 0-5% and 30-35% centrality classes. Plots a) and c) show direct comparisons of the data (points) and normalized mixed event (solid line) $M_{p_T}$ distributions. Plots b) and d) show the residuals between the data and mixed events in units of standard deviations of the data points from the mixed event points.](residuals){width="1.0\linewidth"}
Figure 2 shows the magnitude of $F_{p_T}$, expressed in percent, as a function of centrality for Au+Au collisions with $p_{T}^{max}$ = 2.0 GeV/c. The error bars are dominated by time-dependent systematic effects during the data taking period due to detector variations, which are minimized using strict time-dependent cuts on the mean and standard deviations of the inclusive $p_T$ and $N_{tracks}$ distributions. Statistical errors are below $F_{p_T}$ = 0.05% for all centralities. The systematic errors are determined by dividing the entire dataset into ten separate subsets for each centrality class and extracting the standard deviation of the $F_{p_T}$ values calculated for each subset. From Fig. 2, a significant non-random fluctuation is seen that appears to peak in mid-central collisions. However, the magnitude of the observed fluctuations are within previously published limits [@PPG005]. In addition, the value of $F_{p_T}$ for the most peripheral Au+Au collisions is consistent with, albeit slightly below, the value measured by the same PHENIX apparatus in minimum bias $\sqrt{s_{NN}}$ = 200 GeV p+p collisions. If the magnitude of $F_{p_T}$ is entirely due to fluctuations in the effective temperature of the system [@Korus01], this measurement corresponds to a fluctuation of $\sigma_{T}/<T>$=1.8% at 0-5% centrality and 3.7% at 20-25% centrality.
![\[fig:2\] $F_{p_T}$ (in percent, $0.2$ GeV/c $<p_T<2.0$ GeV/c) as a function of centrality, which is expressed in terms of the number of participants in the collision, $N_{part}$. The solid squares represent the Au+Au data. The solid triangle represents the minimum bias p+p data point. The open triangle is the result from an analysis of PYTHIA minimum bias p+p events within the PHENIX acceptance. The error bars include statistical and systematic errors and are dominated by the latter. The curves are the results of a Monte Carlo simulation with hard processes modelled using PYTHIA with a constant (dotted curve) and $R_{AA}$-scaled (dashed curve) hard scattering probability factor, and include the estimated contribution due to elliptic flow.](FtCent){width="1.0\linewidth"}
To further understand the source of the non-random fluctuations, $F_{p_T}$ is measured over a varying $p_T$ range for which $M_{p_T}$ is calculated, $0.2$ GeV/c $<p_T<p_{T}^{max}$. Figure 3 shows $F_{p_T}$ plotted as a function of $p_{T}^{max}$ for the 20-25% centrality class. A trend of increasing $F_{p_T}$ for increasing $p_{T}^{max}$ is observed for this and all other centrality classes. The majority of the contribution to $F_{p_T}$ appears to be due to correlations of particles with $p_T>1.0$ GeV/c, where $F_{p_T}$ increases disproportionately to the small increase (only 14%) of $N_{tracks}$ in this region.
![\[fig:3\] $F_{p_T}$ (in percent) of non-random fluctuations as a function of the $p_T$ range over which $M_{p_T}$ is calculated, $0.2$ GeV/c $<p_T<p_{T}^{max}$, for the 20-25% centrality class ($N_{part}$=181.6). The curve is the result of a Monte Carlo simulation with hard-scattering processes modelled using PYTHIA with $S_{prob}(N_{part})$ = 0.075 and $R_{AA}$ = 0.41 [@phenixSupp]. The error bars include statistical and systematic errors and are dominated by the latter. The contribution of elliptic flow is estimated to be negligible at this centrality.](FtPtMax){width="1.0\linewidth"}
The behavior of $F_{p_T}$ as a function of centrality and $p_T$ is similar to trends seen in measurements of elliptic flow [@phenixFlow]. The contribution of elliptic flow to the magnitude of $F_{p_T}$ is investigated using a Monte Carlo simulation whereby events are generated with a Gaussian distribution of $N_{tracks}$ particles determined by a fit to the data and a random reaction plane azimuthal angle, $\Phi$, between 0 and $2\pi$. Independent particles within an event are generated following the inclusive $p_T$ distribution with azimuthal angles, $\phi$, distributed according to collective elliptic flow described by the function $\frac{dN}{d(\phi-\Phi)}=1+2 v_2 cos(2(\phi-\Phi))$. The values of the $v_2$ parameter are linearly parameterized as a function of $p_T$ and centrality using PHENIX measurements of inclusive charged hadrons [@phenixFlow]. Only generated particles that lie within the PHENIX azimuthal acceptance are included in the calculation of $M_{p_T}$. This simulation estimates that the contribution of elliptic flow to $F_{p_T}$ is largely cancelled out by the symmetry of the PHENIX acceptance, and is negligible for central collisions. The estimated elliptic flow contribution to the value of $F_{p_T}$ is less than 0.1% for $N_{part}>150$, increasing to about 0.6% for $N_{part}<100$. Note that $F_{p_T}$ measured for minimum bias p+p collisions, where collective flow is not expected to contribute, is non-zero ($1.9 \pm 0.6\%$), implying that a non-flow contribution may also be present in peripheral Au+Au collisions.
Figure 3 illustrates that a large contribution to the observed non-random fluctuations is due to the correlation of high $p_T$ particles, such as might be expected from correlations due to jet production [@Liu03]. In order to estimate the contribution due to jets, a Monte Carlo simulation is again applied. Events are generated with a Gaussian distribution of $N_{tracks}$ particles as independent particles that follow an $m_T$-exponential fit to the inclusive data $p_T$ distribution. Hard processes are defined to occur at a uniform rate per generated particle, $S_{prob}(N_{part})$, for each centrality class. This is the only parameter that is allowed to vary in the simulation. As Au+Au events are being generated, single $\sqrt{s_{NN}}=200$ GeV p+p hard-scattering events generated by the PYTHIA event generator [@PYTHIA] and filtered by the PHENIX acceptance are embedded into the event. The addition of the PYTHIA events affects the mean and standard deviation of the inclusive $p_T$ spectra by less than 0.1%. The value of $F_{p_T}$ has been extracted from 100,000 PYTHIA events for minimum bias p+p collisions, yielding $F_{p_T}$=2.06% within the PHENIX acceptance, which is consistent with the measured value of $F_{p_T}=1.9 \pm 0.6\%$.
Two scenarios are considered for studies of the centrality-dependence of jet contributions to the value of $F_{p_T}$: 1) with $S_{prob}(N_{part})$ set at a constant rate for all centrality classes, and 2) with $S_{prob}(N_{part})$ scaled for each centrality class by the PHENIX measurement of the suppression of high $p_T$ charged particles, which is characterized by the nuclear modification factor, $R_{AA}$, integrated over $p_T>4.5$ GeV/c [@phenixSupp]. The $p_T$ value at which $R_{AA}$ is extracted has little effect on the simulation results, which change by less than 0.2% for 0-5% centrality if the $R_{AA}$ measurement at $p_T=2.0$ GeV/c is used instead. The latter scenario is intended to model the effect of the suppression of jets due to energy loss in the nuclear medium [@Wang98] on the fluctuation signal. The initial value of $S_{prob}(N_{part})$ for both scenarios is normalized so that the $F_{p_T}$ result from the $R_{AA}$-scaled simulation matches that of the data for the 20-25% centrality class. The results of the simulation as a function of $p_{T}^{max}$, with $S_{prob}(N_{part})$ scaled by $R_{AA}$, are represented by the dashed curve in Fig. 3 for the 20-25% centrality class, The trend of increasing $F_{p_T}$ with increasing $p_{T}^{max}$ observed in the data is reproduced by the simulation reasonably well.
The results of the two hard scattering simulation scenarios are shown in Fig. 2 as a function of centrality. The model curves include the small contribution estimated from the elliptic flow simulation. The dotted curve is the result with $S_{prob}(N_{part})$ fixed for all centralities. The dashed curve is the result with $S_{prob}(N_{part})$ scaled by $R_{AA}$ as a function of centrality. Within this simulation, the decrease of $F_{p_T}$ for the more peripheral events is explained as a decrease in the signal strength relative to number fluctuations from the small and decreasing value of $N_{tracks}$. If $S_{prob}(N_{part})$ remains constant, the value of $F_{p_T}$ decreases only slightly when going from mid-central to central collisions, in contradiction with the large decrease seen in the data over this centrality range. When $S_{prob}(N_{part})$ is scaled by $R_{AA}$ as a function of centrality, the trend in the simulation of decreasing $F_{p_T}$ with increasing centrality is more consistent with the data.
To summarize, the PHENIX experiment has observed a positive non-random fluctuation signal in event-by-event average transverse momentum, measured as a function of centrality and $p_T$ in $\sqrt{s_{NN}}=200$ GeV Au+Au and p+p collisions. The increase of $F_{p_T}$ with increasing $p_T$ implies that the majority of the fluctuations are due to correlated high $p_T$ particles. A Monte Carlo simulation that includes elliptic flow and a PYTHIA-based hard scattering description can consistently describe contributions to the signal as a function of centrality and $p_T$ with a simple implementation of jet suppression.
We thank the staff of the Collider-Accelerator and Physics Departments at BNL for their vital contributions. We acknowledge support from the Department of Energy and NSF (U.S.A.), MEXT and JSPS (Japan), CNPq and FAPESP (Brazil), NSFC (China), CNRS-IN2P3 and CEA (France), BMBF, DAAD, and AvH (Germany), OTKA (Hungary), DAE and DST (India), ISF (Israel), KRF and CHEP (Korea), RMIST, RAS, and RMAE, (Russia), VR and KAW (Sweden), U.S. CRDF for the FSU, US-Hungarian NSF-OTKA-MTA, and US-Israel BSF.
H. Heiselberg, .
M. Stephanov [*et al.*]{}, .
M. Stephanov [*et al.*]{}, .
S. Gavin, nucl-th/0308067, to be published.
L. Stodolsky, .
E. Shuryak, .
NA49 Collaboration, H. Appelshäuser [*et al.*]{}, .
CERES Collaboration, D. Adamová [*et al.*]{}, .
PHENIX Collaboration, K. Adcox [*et al.*]{}, .
STAR Collaboration, preprint nucl-ex/0308033, to be published.
PHENIX Collaboration, K. Adcox [*et al.*]{}, .
R. Korus and S. Mrówczyński, .
PHENIX Collaboration, K. Adcox [*et al.*]{}, .
PHENIX Collaboration, preprint nucl-ex/0307022, to be published.
J. T. Mitchell [*et al.*]{}, .
M. Gaździcki and S. Mrówczyński, .
S. Mrówczyński, .
C. Pruneau [*et al.*]{}, .
M. J. Tannenbaum, .
PHENIX Collaboration, S. S. Adler [*et al.*]{}, preprint nucl-ex/0305013, to be published; PHENIX Collaboration, K. Adcox [*et al.*]{}, ; STAR Collaboration, C. Adler [*et al.*]{} .
Q. Liu and T. Trainor, .
T. Sjöstrand, Computer Physics Commun. [**82**]{}, 74 (1994). Version 5.720 with MSEL=1 for embedded hard scattering events and MSEL=2 for min. bias p+p events, CKIN(3)=0.0, MSTP(32)=4, and MSTP(33)=1.
PHENIX Collaboration, preprint nucl-ex/0308006, to be published.
X. N. Wang, .
|
---
abstract: 'We present ab-initio density functional (DFT) calculations of the vibrational spectra of neutral Magnesium phthalocyanine (MgPc) molecule and of its Raman scattering intensities.'
author:
- |
Jaroslav Tóbik$^{a}$, Erio Tosatti$^{a,b}$\
$^a$International School for Advanced Studies (SISSA),\
and INFM Democritos National Simulation Center, Via Beirut 2, I-34014 Trieste, Italy\
$^b$The Abdus Salam International Centre for Theoretical Physics (ICTP),\
strada Costiera 11, 34100 Trieste, Italy
title: Raman Tensor Calculation for Magnesium Phthalocyanine
---
Introduction
============
Organic materials are attractive for electronic industry, because they promise large versatility based on organic chemistry, low price based on very common elements used as building blocks and interesting mechanical properties. There is quite a vigorous research activity in organic materials for electrical applications - from insulators trough semiconductors to conductors and even superconductors. Here we are concerned with metal phthalocyanines (MPc). Electronic structure calculations [@Liao] and recent experimental observation of the possibility to dope these materials by adding electrons [@Morpurgo] resembles somewhat the situation of fullerenes. In our previous paper we speculated about possible phase diagrams for electron doped MgPc [@Erio-PRL]. While these possibilities are under active experimental considerations, we noticed that for the commonest experimental diagnostic, namely Raman scattering there is no reference calculation of either modes nor Raman intensities (with exception of ZnPc [@Tackley-PCCP1]). In this paper we focused our interest on the vibrational properties of MgPc, which we wish to study ab-initio. While our final aim will be to compute spectra of doped and undoped molecules, this work will be restricted to neutral undoped MgPc, which has not been studied so far.
Technical details.
==================
All our calculations were done using the PWscf software package [@PWSCF], which is a plane-wave basis set DFT implementation. We used the LDA approximation with Slater approximation for exchange and Perdew-Zunger functional for correlations effects. In order to lower the plane wave energy cut-off we used the non-local RRKJ3 ultra–soft pseudopotentials [@RRKJ3]. The kinetic energy cut-off was 35Ry for the wave functions basis set and 280Ry for the charge density basis set. The plane-wave basis set assumed periodic boundary conditions. The unit cell had dimensions 21$\times$21$\times$11[Å]{}, therefore including enough vacuum to represent the isolated molecule.
By energy optimization we found the equilibrium molecular structure with structural parameters as in Table \[tab-struct\]. The molecule has $D_{4h}$ symmetry group and is planar. The electronic structure agrees well with former calculations of Liao et. al.[@Liao]. Some levels near the HOMO–LUMO gap are listed in the table \[tab-electrons\].
We calculated the vibrational spectra of neutral, undoped MgPc by means of the density functional perturbation theory [@Baroni-RMP]. The dynamical matrix was calculated in Cartesian coordinates. In the isolated molecule with $N$ atoms there are $3N-6$ genuine vibrations while 6 modes corresponds to translation–rotational degrees of freedom with zero frequency. In reality due to periodic images of molecules there is a weak virtual interaction which can cause “libration” of the molecule at nonzero, even imaginary frequency. We eliminated these modes by transforming the dynamical matrix to internal coordinates and setting the corresponding dynamical matrix elements to zero. We checked that after this procedure we obtained eigenvectors of correct symmetry. Assignment of irreducible representation to all eigenvectors was done by projection on symmetry adapted bases of all linearly independent atomic displacement. For $N=57$ atoms there are $165$ vibrations $$\Gamma_{vib}=14A_{1g}+13A_{2g}+14B_{1g}+14B_{2g}+13E_g+6A_{1u}+8A_{2u}+7B_{1u}+
7B_2u+28E_u$$ where $E_g$ and $E_u$ are two–fold degenerated modes. The calculated frequencies are summarized in Fig. \[fig-spectrum\].
The Raman intensity is related to the change of molecular polarizability due to the deformation introduced by the vibration. We calculated the polarizability derivatives with respect to deformations in the static limit.
With each vibrational mode $q_i$ is associated a Raman tensor $T^i$ given by $$T^i_{\mu \nu}=\frac{d\alpha_{\mu \nu}}{dq_i}
\label{raman-ten}$$ Here $\mu$, $\nu$ are Cartesian indices, $\alpha_{\mu \nu}$ is the polarizability tensor defined as $\alpha_{\mu\nu}=\frac{\partial ^2U}{\partial E_{\mu} \partial E_{\nu}}$, $U$ is the total energy and $\vec{E}$ an external electric field. Therefore one has to calculate third derivatives of total energy $\frac{\partial^3 U}{\partial E_{\mu} \partial E_{\nu} \partial x_l}$. The energy derivatives with respect to atomic displacements are forces acting on atomic nuclei, and are calculated via the Hellman-Feynman theorem. We extracted the force dependence on electric field by numerical differences. Applied fields were $\pm 1\times 10^{-3}$, $\pm 2\times 10^{-3}$, $\pm 4\times 10^{-3}$ a.u.. Details about application of electric field within periodic boundary conditions are the same as in Ref. [@Andrea-JCP]. There are alternative methods for the Raman tensor calculation using linear response theory [@Deinzer; @Lazzeri]. However numerical derivative method with finite field is computationaly less demanding for the present case, where moreover the implementation is very simple.
From the Raman tensor Eq.(\[raman-ten\]) we calculated for each mode the gas phase (angle averaged) Raman scattering cross section following [@Pederson; @Long-book] in the form: $$\begin{aligned}
\frac{d \sigma _i}{d \Omega}&=& \frac{(2\pi\nu_0)^4}{c^4}\frac{h(n_i+1)}{8\pi\nu_i}\frac{45{\alpha'}_i^2+7{\gamma'}_i^2}{45} \label{ram-cross}\\
\alpha'_i&=&\frac{1}{3}\left(T^i_{xx}+T^i_{yy}+T^i_{zz}\right) \nonumber\\
{\gamma'}_i^2&=&\frac{1}{2}\lbrace
\left( T^i_{xx}-T^i_{yy} \right)^2 + \left( T^i_{xx}-T^i_{zz} \right)^2 +
\left( T^i_{yy}-T^i_{zz} \right)^2 +
6\lbrack \left(T^i_{xy}\right)^2 + \left(T^i_{xz}\right)^2 +\left(T^i_{yz}\right)^2 \rbrack
\rbrace \nonumber \\
n_i&=&{\left[ \exp \left(\frac{h\nu_i}{kT}\right) -1\right]}^{-1}
\nonumber \end{aligned}$$ $\nu_0$ is frequency of incident light, $n_i$ is equilibrium occupation number for the initial vibrational state $i$ at the given temperature $T$. The $\alpha '$ and ${\gamma '}^2$ are isotropic and anisotropic parts of the Raman tensor. Both are invariant under rotations. Formula (\[ram-cross\]) is valid for the most common experimental setup when incident ray, observed ray, and incident light polarization of electric field are perpendicular to each other.
The Raman tensor of a particular mode must belong to the same irreducible representation as that vibration mode. In a coordinate system connected to the molecule as indicated on Fig. (\[fig-mol-scheme\]) all symmetric tensors of second rank are divided according to irreducible representations in the form: $$A_{1g}:\left(\begin{array}{ccc}
a & 0 & 0 \\
0 & a & 0 \\
0 & 0 & b
\end{array} \right), \quad
B_{1g}:\left(\begin{array}{ccc}
c & 0 & 0 \\
0 & -c & 0 \\
0 & 0 & 0
\end{array} \right), \quad
B_{2g}:\left(\begin{array}{ccc}
0 & d & 0 \\
d & 0 & 0 \\
0 & 0 & 0
\end{array} \right), \quad
E_g:\left(\begin{array}{ccc}
0 & 0 & e \\
0 & 0 & f \\
e & f & 0
\end{array} \right)
\label{ram-tensors}$$ Traceless irreducible representations possess only the anisotropic part, while the $A_{1g}$ mode has both parts. Weights in front of $\alpha'$ and $\gamma'$ in formula \[ram-cross\] (45 and 7 respectively) can be changed by observation geometry, thus providing a way to (partially) assign symmetry to vibration eigenmodes (see for example Ref. [@Long-book]). We note that for the $E_g$ representation it is possible to define a single independent parameter $\sqrt{e^2+f^2}$ because of the degeneracy and orthogonality of degenerated modes.
Results and discussion
======================
In Table \[tab-struct\] we give our LDA optimized structure of MgPc. In comparison with data in references [@Liao; @Ruan] we see a reasonable agreement. The difference can be attributed to different density functional (GGA in Ref. [@Liao]). LDA is known to slightly overbind in covalent bonds. Kohn-Sham orbitals energies (Tab. \[tab-electrons\]) are given relative to vacuum zero taken as the SCF electrostatic potential far from the molecule. They are also in good agreement with Ref. [@Liao] as far as we can read off values from fig. 2 of Ref. [@Liao].
Vibrational spectra of various metal-phthalocyanines were experimentally measured [@Tackley-PCCP2] by means of Raman spectroscopy. In the case of ZnPc there is also a DFT calculation of the vibrational and Raman spectra [@Tackley-PCCP1]. In general the most intense peak was found to be in the interval 1500-1550cm$^{-1}$ depending on the central atom.
Our calculated vibration spectrum of MgPc is on Fig.(\[fig-spectrum\]). Above the frequency line for Raman active modes we write the corresponding non-zero component of Raman tensor as defined by Eq.(\[ram-tensors\]). The simulated Raman spectrum according to formula \[ram-cross\] is shown on Fig.(\[fig-raman\]). To simulate spectrum for each intensity and frequency we added Gaussian with the spread $\sigma$=5cm$^{-1}$. We considered the low temperature limit ($n_i=0$) and low incident light frequency limit. Our most intense Raman scattering mode is at 1587cm$^-1$ with symmetry $B_{1g}$. This is similar to ZnPc which is at 1516cm$^{-1}$ with symmetry $B_1$.
The results of the presented calculation should be compared with Raman data of MgPc, as soon as available. It will also serve as a starting point for future work on the electron doped molecules [@Morpurgo].
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to A. Dal Corso for his help and assistance and to A. Morpurgo, M. Craciun, and S. Margadonna for correspondence. This work was partly supported by MIUR COFIN No. 2003028141-007, MIUR COFIN No. 2004023199-003, by FIRB RBAU017S8R operated by INFM, by MIUR FIRB RBAU017S8 R004, and by INFM (Iniziativa trasversale calcolo parallelo).
[99]{} M.S. Liao and S. Scheiner, J. Chem. Phys. [**114**]{}, 9780 (2001). M.F. Craciun, S. Rogge, D.A. Wismeijer, M.J.L. den Boer, T.M. Klapwijk, and A.F. Morpurgo, in [*Proceedings of the 12th International Conference on Scanning and Tunneling Microscopy/Spectroscopy and Related Techniques*]{}, edited by P.M. Koenraad and M. Kemerink, AIP Conference Proceedings [**696**]{}, 489 (2003). M.F. Craciun, S. Rogge, M.J.L. den Boer, S. Margadonna, K. Prassides, Y. Iwasa, and A.F. Morpurgo, arXiv:cond-mat/0401036 v2. E. Tosatti, M. Fabrizio, J. Tobik, G.E. Santoro, Phys. Rev. Lett. [**93**]{}, 117002 (2004). S. Baroni, A. Dal Corso, S. de Gironcoli, and P. Giannozzi, [http://www.pwscf.org]{}. A.M. Rappe, K.M. Rabe, E. Kaxiras, J.D. Joannopoulos, Phys. Rev. B[**41**]{}, 1227 (1990). S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi,Rev. Mod. Phys. [**73**]{}, 515 (2001). J. Tóbik, A. Dal Corso, J. Chem. Phys. [**120**]{}, 9934 (2004). G. Deinzer and D. Strauch, Phys. Rev. B [**66**]{}, 100301 (2002). M. Lazzeri and F. Mauri, Phys. Rev. Lett. [**90**]{}, 036401. D. Porezag and M.R. Pederson, Phys. Rev. B [**54**]{}, 7830 (1996). D.A. Long, [*Raman Spectroscopy*]{} (McGraw-Hill, 1977, ISBN: 0-07-038675-7). C.-Y. Ruan, V. Mastryukov, and M. Fink, J. Chem. Phys [**111**]{}, 3035 (1999). D.R. Tackley, G. Dent, and W.E. Smith, Phys. Chem. Chem. Phys. [**3**]{}, 1419 (2001). D.R. Tackley, G. Dent, and W.E. Smith, Phys. Chem. Chem. Phys. [**2**]{}, 3949 (2000).
level KS-eigenvalue \[eV\] symmetry
----------- ---------------------- ----------
LUMO+1 -2.241 $b_{1u}$
LUMO -3.692 $e_g$
HOMO -5.132 $a_{1u}$
HOMO-1 -6.092 $b_{2g}$
HOMO-2 -6.446 $b_{2u}$
HOMO-3,-4 -6.526 $e_u$
: Kohn-Sham eigenvalues spectra and irreducible representation of wave–functions near the HOMO–LUMO gap. Note that $e_g$ and $e_u$ orbitals are two–fold degenerated.[]{data-label="tab-electrons"}
bond DFT Ref. [@Liao] Experiment Ref. [@Ruan] this work
-------------------------- ------------------ ------------------------- -----------
$Mg-N_1 $ 2.008 1.990 1.991
$N_1-C_1$ 1.377 1.386 1.359
$N_2-C_1$ 1.335 1.317
$C_1-C_2$ 1.465 1.411 1.445
$C_2-C'_2$ 1.415 1.468 1.400
$C_2-C_3$ 1.395 1.400 1.379
$C_3-C_4$ 1.397 1.399 1.381
$C_4-C'_4$ 1.406 1.412 1.392
$C-H$ 1.090 1.121 1.092
$\theta_{C_1-N_1-C'_1} $ 109.7 109.5 109.9
$\theta_{N_2-C_1-N_1} $ 127.5 125.9 127.1
$\theta_{N_1-C_1-C_2} $ 108.6 108.9 108.4
: Structure of MgPc. Bond lengths are in Å, angles in degrees. Notations corresponds to labels on Fig. \[fig-mol-scheme\]
. \[tab-struct\]
|
---
address:
- 'FERMILAB, P.O. Box 500, Batavia, IL 60510, USA'
- 'CERN, TH Division, CH–1211 Geneva 23, Switzerland'
author:
- 'M. CARENA'
- 'C.E.M. WAGNER'
title: ELECTROWEAK BARYOGENESIS AND HIGGS PHYSICS
---
\#1[0= -.025em0-0 .05em0-0 -.025em.0433em0 ]{} \#1;\#2
=0 =\#1
\#1[$^{\scriptstyle #1}$]{} \#1\#2\#3[[*Nucl. Phys.*]{} [**[B\#1]{}**]{} (19\#2) \#3]{} \#1\#2\#3[[*Phys. Lett.*]{} [**[B\#1]{}**]{} (19\#2) \#3]{} \#1\#2\#3[[*Phys. Rev.*]{} [**[D\#1]{}**]{} (19\#2) \#3]{} \#1\#2\#3[[*Phys. Rev. Lett.*]{} [**[\#1]{}**]{} (19\#2) \#3]{} \#1\#2\#3[[*Z. Phys.*]{} [**C\#1**]{} (19\#2) \#3]{} \#1\#2\#3[[*Prog. Theor. Phys.*]{} [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[[*Mod. Phys. Lett.*]{} [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[[*Phys. Rep.*]{} [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[[*Ann. Phys.*]{} [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[[*Rev. Mod. Phys.*]{} [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[[*Helv. Phys. Acta*]{} [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[[*JETP Lett.*]{} [**\#1**]{} (19\#2) \#3]{} 11[m\_[\_[1]{}]{}]{} 22[m\_[\_[2]{}]{}]{} 12[m\_[\_[1,2]{}]{}]{} 11[m\_[\_[1]{}]{}]{} 22[m\_[\_[2]{}]{}]{} 12[m\_[\_[1,2]{}]{}]{} 2[\^[2]{}]{}
FERMILAB-Pub-97/95-T\
CERN-TH/97-74\
hep-ph/9704347
[**Electroweak Baryogenesis and Higgs Physics [^1]**]{}\
\
$^\dagger$ Fermi National Accelarator Laboratory\
P.O. Box 500, Batavia, IL 60510, USA\
$^\ddagger$ Theory Division, CERN\
CH-1211 Geneva 23, Switzerland\
Abstract
Electroweak Baryogenesis is a particularly attractive theoretical scenario, since it relies on physics which can be tested at present high energy collider facilities. Within the Standard Model, it has been shown that the requirement of preserving the baryon number generated at the weak scale leads to strong bounds on the Higgs mass, which are already inconsistent with the present experimental limits. In the Minimal Supersymmetric extension of the Standard Model, we demonstrate that light stop effects can render the electroweak phase transition sufficiently strongly first order, opening the possibility of electroweak baryogenesis for values of the Higgs mass at the LEP2 reach. The generation of the observed baryon asymmetry also requires small chargino masses and new CP-violating phases associated with the stop and Higgsino mass parameters. We discuss the direct experimental tests of this scenario and other relevant phenomenological issues related to it.
April 1997
,
INTRODUCTION
============
One of the fundamental problems of particle physics is to understand the origin of the observed baryon asymmetry of the universe. The mechanism for the generation of baryon number may rely on physics at very high energies, of order of the Planck or Grand Unification scale and hence difficult to test at present high energy colliders. Even in this case, the final result for the baryon asymmetry will always be affected by low energy physics. Indeed, although baryon number is preserved in the Standard Model at the classical level, it is violated through anomalous processes at the quantum level [@anomaly]. As the Universe cools down, unless specific conditions on the baryon and lepton asymmetries generated at high energies are fulfilled [@HerbiG], the anomalous processes will tend to erase the baryon asymmetry generated at high energies. If the baryon asymmetry is completely washed out at temperatures far above the weak scale, the observed baryon number must proceed from physical processes at temperatures close to $T_c$, at which the electroweak phase transition takes place.
Baryogenesis at the electroweak phase transition is a very attractive alternative since it relies only on physics at the weak scale, and it is hence testable in the near future. In principle, the Standard Model (SM) fulfills all the requirements [@baryogenesis] for a successful generation of baryon number [@reviews]. Non-equilibrium processes occur at the first order electroweak phase transition, baryon number is violated by anomalous processes and CP is violated by explicit phases in the CKM matrix. In order to quantitatively estimate the generated baryon number, one should take into account that the baryon number violating processes are effective also after the electroweak phase transition, and are only suppressed by a Boltzman factor $$\Gamma \simeq T^4 \exp\left(-\frac{E_{\rm sph}}{T} \right),
\label{rate}$$ where the sphaleron energy, $E_{\rm sph}$, is equal to the height of the barrier separating two topologically inequivalent vacua [@sphalerons]. The sphaleron energy is given by $$E_{\rm sph}(T) \simeq B \frac{2 M_W(T)}{\alpha_w(T)}$$ with $B \simeq {\cal O}(2)$ being a slowly varying function of the Higgs quartic coupling, $M_W$ the weak gauge boson mass and $\alpha_w$ the weak gauge coupling. If the phase transition were second order, or very weakly first order, the baryon number violating processes would be approximately in equilibrium and no effective baryon number will survive at $T \simlt T_c$. Comparing the rate, Eq. (\[rate\]), with the rate of expansion of the universe one can obtain the condition under which the generated baryon number will be preserved after the electroweak phase transition. This implies a bound on the sphaleron energy, $E_{\rm sph}(T_c)/T_c \simgt 45$ [@first] or, equivalently, $$\frac{v(T_c)}{T_c} \simgt 1. \;\;\;\;\;
\label{orderp}$$
Since $v(T_c)/T_c$ is inversely proportional to the quartic coupling appearing in the low energy Higgs effective potential, the requirement of preserving the generated baryon asymmetry puts an upper bound on the value of the Higgs mass. The actual bound depends on the particle content of the theory at energies of the order of the weak scale. In the case of the Standard Model, the present experimental bounds on the Higgs mass are already too strong, rendering the electroweak phase transition too weakly first order. Hence, within the Standard Model, the generated baryon asymmetry at the electroweak phase transition cannot be preserved [@first], as perturbative [@improvement]$^-$[@pertres2] and non-perturbative [@nonpert; @Jansen] analyses have shown. It is interesting to notice that, even in the absence of experimental bounds, the requirement of a sufficiently strong first order phase transition leads to bounds on the Higgs mass such that the electroweak breaking minimum would become unstable unless new physics were present at scales of the order of the weak scale [@earlyst]$^-$[@CEQ]. We shall review these bounds below. On the other hand, CP-violating processes are suppressed by powers of $m_f/M_V$, where $m_f$ are the light-quark masses and $M_V$ is the mass of the vector bosons. These suppression factors are sufficiently strong to severely restrict the possible baryon number generation [@fs; @huet]. Therefore, if the baryon asymmetry is generated at the electroweak phase transition, it will require the presence of new physics at the electroweak scale.
The most natural extension of the Standard Model, which naturally leads to small values of the Higgs masses is low energy supersymmetry. It is hence highly interesting to test under which conditions baryogenesis is viable within this framework [@early]$^-$[@mariano2]. It was recently shown [@CQW]$^-$[@Delepine] that the phase transition can be sufficiently strongly first order only in a restricted region of parameter space, which strongly constrains the possible values of the lightest stop mass, of the lightest CP-even Higgs mass (which should be at the reach of LEP2) and of the ratio of vacuum expectation values, $\tan\beta$. These results have been confirmed by explicit sphaleron calculations in the Minimal Supersymmetric Standard Model (MSSM) [@MOQ].
On the other hand, the Minimal Supersymmetric Standard Model, contains, on top of the CKM matrix phase, additional sources of CP- violation and can account for the observed baryon asymmetry.[^2] New CP-violating phases can arise from the soft supersymmetry breaking parameters associated with the stop mixing angle. Large values of the mixing angle are, however, strongly restricted in order to preserve a sufficiently strong first order electroweak phase transition [@mariano1]$^-$[@CQW]. Therefore, an acceptable baryon asymmetry requires a delicate balance between the value of the different mass parameters contributing to the left-right stop mixing, and their associated CP-violating phases. Moreover, the CP-violating currents must originate from the variation of the CP-odd phases appearing in the couplings of stops, charginos and neutralinos to the Higgs particles. This variation would be zero if $\tan\beta$ were a constant, implying that the heavy Higgs doublet can not decouple at scales far above $T_c$, or equivalently, the CP-odd Higgs mass should not be much larger than $M_Z$. On the other hand, since the phase transition becomes weaker for lighter CP-odd Higgs bosons, a restricted range for the CP-odd and charged Higgs masses may be obtained from these considerations.
The scenario of Electroweak Baryogenesis (EWB) has crucial implications for Higgs physics and imposes important constraints on the supersymmetry breaking parameters. Most appealing, this scenario makes definite predictions, which may be tested at present or near future colliders.
In section 2 we shall present the improved one-loop finite temperature Higgs effective potential. In section 3 we discuss the Standard Model case, on the light of present experimental constraints on the Higgs mass and the requirement of stability of the physical vacuum. In section 4, we study the strength of the electroweak phase transition within the minimal supersymmetric extension of the standard model, discussing in detail the effect of light stops in expanding the allowed Higgs mass range and analyzing the conditions to avoid color breaking minima. We also discuss the strength of the electroweak phase transition in the cases of large and small values of the CP-odd Higgs mass, and analyse the new sources of CP-violation which may contribute to the generation of baryon asymmetry within the MSSM. In section 5 we study the generation of baryon asymmetry. In section 6 we ellaborate on the experimental tests of this scenario both at LEP2 and the Tevatron, and discuss the predictions for some rare flavor changing neutral current processes within this framework. In section 7 we summarize the prospects for electroweak baryogenesis.
FINITE TEMPERATURE HIGGS EFFECTIVE POTENTIAL
============================================
As we explained above, the requirement of preserving the baryon asymmetry after the phase transition implies that $v(T_c)/T_c$ must be larger than one. To extract the implications of this requirement, a detailed knowledge of the finite temperature effective potential of the Higgs field is needed. The finite temperature effective potential for the neutral component of the Higgs field may be computed at the one-loop level [@DJ], $$\begin{aligned}
V(\phi,T) = V_{\rm tree}(\phi) + V_1(\phi,0) + V_1(\phi,T)\end{aligned}$$ where $V_{\rm tree}(\phi)$, $V_1(\phi,0)$ and $V_1(\phi,T)$ are the tree level, one-loop zero temperature and one-loop finite temperature contributions to the effective potential, respectively. Their expressions are given by $$\begin{aligned}
V_{\rm tree}(\phi)& = &m^2 \phi^2 + \frac{\lambda}{2} \phi^4
\nonumber\\
V_1(\phi,0) & = & \sum_i \frac{n_i}{64 \pi^2}
m_i^4(\phi) \left[ \log\left(
\frac{m_i^2(\phi)}{Q^2}\right) - c_i \right]
\nonumber\\
V_1(\phi,T) & = & \sum_i \frac{n_i}{2 \pi^2 \beta^4}
\int_0^{\infty} dx \; x^2 \log\left( 1 \pm \exp-(x^2 +
\beta^2 m_i^2(\phi))^{1/2}
\right),
\label{effpot}\end{aligned}$$ where $m_i(\phi)$ is the mass of the $i$-field in the background of the field $\phi$, $n_i$ is its total number of degrees of freedom, $c_i = 5/6$ for vector bosons and 3/2 for scalars and fermions. $\beta^{-1}$ is proportional to the temperature and the plus and minus sign in $V_1(\phi,T)$ are associated with fermionic and bosonic particles, respectively. Observe that the contribution of heavy particles to the temperature dependent part of the effective potential is exponentially suppressed. For values of the masses $m_i(\phi) \simlt 2 \; T$, the effective potential admits a high temperature expansion. In this limit, the contribution of bosonic particles to the Higgs effective potential is given by [@DJ] $$V_1^{\rm b}(\phi,T) =
\sum_i n_i \left\{ \frac{m_i^2(\phi)}{24 \beta^2} - \frac{1}{12 \pi}
\frac{m_i^3(\phi)}{\beta} - \frac{1}{64 \pi^2} m_i^4(\phi)
\log(m_i^2(\phi) \beta^2) +... \right\}
\label{eq:cubic}$$ while that of fermions is given by $$V_1^{\rm f}(\phi,T) =
\sum_i n_i \left\{ \frac{m_i^2(\phi)}{ 48 \beta^2} +
\frac{m_i^4(\phi)}{64 \pi^2}
\log\left(m_i^2(\phi) \beta^2\right) +...\right\}.$$ Therefore, in the region of validity of the high temperature expansion, the effective potential reads, $$V(\phi,T) = D (T^2 - T_0^2) \phi^2
- E T \phi^3 + \frac{\lambda}{2} \phi^4 +...$$ where $D$, $E$ and $\lambda$ are temperature dependent functions, $T_0$ is the temperature at which the curvature of the potential vanishes at the origin and we have chosen the normalization such that $<\phi> = v/\sqrt{2}$, with $v(0) \sim 246$ GeV. The minimization of the potential at $T = T_c$, the temperature at which the electroweak symmetry breaking and the electroweak symmetry preserving minima become degenerate, leads to $$\frac{\phi(T_c)}{T_c} = \frac{E}{\lambda}.
\label{eq:vot}$$ Hence, the strength of the phase transition is directly proportional to the coefficient of the cubic term induced by the presence of bosonic particles, like the gauge bosons, with masses $m_B = k_B \phi^2$ (see Eq. (\[eq:cubic\])).
Higher loop corrections are important to define the correct infrared properties of the effective potential, and to reduce its gauge dependence. The most important corrections come from the so-called Daisy graphs [@improvement], which effectively amount to replace $$\sum_i \frac{n_i \; m_i^3(\phi)}{12 \pi} \rightarrow
\sum_i \frac{n_i \; m_i^3(\phi,T)}{12 \pi}$$ in Eq. (\[eq:cubic\]), where $$m_i^2(\phi,T) = m_i^2(\phi) + \Pi(T)
\label{eq:thermalm}$$ and $\Pi(T)$ is the temperature dependent vacuum polarization contribution to the thermal masses.
An important observation is that the strength of the phase transition is inversely proportional to the squared of the Higgs mass. This is due to the fact that, at zero temperature $$m_H^2 = \lambda \; v^2,
\label{mhiggs}$$ and the value of $\lambda$ at the critical temperature is of the order of its zero temperature value. Hence, from Eqs. (\[orderp\]), (\[eq:vot\]) and (\[mhiggs\]), the requirement of preservation of the baryon asymmetry leads to an upper bound on the Higgs mass.
THE STANDARD MODEL CASE
=======================
The Electroweak Phase Transtion
-------------------------------
In the Standard Model, the effect of including thermal masses, Eq. (\[eq:thermalm\]), implies that only the transverse modes of the electroweak gauge bosons will contribute to the cubic term in the effective potential and hence the Daisy improvement leads to a weaker phase transition than the one predicted at the one-loop level. The coefficient of the cubic term is given by $$E_{\rm SM} = \frac{2}{3} \left(\frac{2 M_w^3 + M_Z^3}{\sqrt{2} \pi v^3}
\right)$$ and hence the upper bound on the Higgs mass derived from Eqs. (\[orderp\]) and (\[eq:vot\]) is $$m_H \simlt 40 \; {\rm GeV}.
\label{eq:bound1l}$$
Although the Daisy resummation includes the dominant higher loop corrections to the effective potential, there are additional two-loop effects which have been shown to lead to non-negligible corrections [@twoloop], making the phase transition more strongly first order and increasing slightly the above upper bound on the Higgs mass. Non-perturbative effects have been taken into account through lattice studies [@nonpert]. These simulations have been done both in four dimensions as in the effective three dimensional theory arising at high temperatures. A more involved perturbative resummation has been performed [@pertres2], showing excellent agreement with the lattice results [@Jansen]. In general, the results for the upper bound on the Higgs mass derived from the non-perturbative studies do not differ in a significant way from those ones coming from perturbative analyses. Numerically, the upper bound obtained from the lattice is somewhat higher than the results obtained from the improved one-loop analysis. The result of the one-loop analysis, Eq. (\[eq:bound1l\]), may be hence quoted as a conservative bound on the Higgs mass.
Stability Bounds and Experimental Limits on $m_H$.
--------------------------------------------------
The low values of the Higgs mass required to preserve the baryon asymmetry are clearly in conflict with the current experimental bounds on this quantity. The present LEP bound on the Standard Model Higgs mass reads [@Alephl] $$m^{SM \; exp.}_H \simgt 70 {\rm GeV}$$ and hence, for the mechanism of electroweak baryogenesis to survive, new physics should be present at the weak scale.
Actually, this argument might have been made even in the absence of experimental constraints, by analysing the stability of the physical vacuum [@earlyst]$^-$[@CEQ]. This may be understood by considering the renormalization group improved effective potential for the neutral Higgs at zero temperature, which is approximately given by $$V(\phi) = m^2 \phi^2 + \frac{\lambda(\phi)}{2} \phi^4,
\label{eq:Veff}$$ where $\lambda(\phi)$ means that the quartic coupling must be evaluated at the relevant scale at which the effective potential is analysed. The dominant contributions to the renormalization group equation of the Higgs quartic coupling are $$\frac{d \lambda}{dt} = \frac{3}{8 \pi^2} \left( \lambda^2
+ \lambda h_t^2 - h_t^4 \right) + {\rm electroweak} \;
{\rm corrections},
\label{eq:lambda}$$ where $h_t$ is the top quark Yukawa coupling, $t = \log(Q^2/\Lambda^2)$, with $Q$ the renormalization group scale and $\Lambda$ the cutoff of the effective theory. For large values of $\lambda$, the quartic coupling of the Higgs fields, grows indefinitely with rising energy and an upper bound on $m_H$ follows from the requirement of perturbative consistency of the theory up to a given cutoff scale $\Lambda$ below $M_{\rm Pl}$. The upper bound on $m_H$ depends mildly on the top quark mass through the impact of the top quark Yukawa coupling on the running of the quartic coupling $\lambda$.
On the other hand, the effect of the large values of $h_t$ on the renormalization group evolution of the quartic coupling, may drive $\lambda$ to negative values at large energy scales, thus destabilizing the standard electroweak vacuum. The requirement of vacuum stability in the Standard Model imposes a lower bound on the Higgs boson mass for a given cutoff scale. This bound on $m_H$ is defined as the lower value of $m_H$ for which $\lambda(\phi) \geq 0$ for any value of $\phi$ below the scale $\Lambda$ at which new physics beyond the Standard Model should appear. From Eq. (\[eq:lambda\]), it is clear that the stability condition of the effective potential demands new physics at lower scales for larger values of $m_t$ and lower values of $m_H$.
Fig. 1 [@LEPRep] shows the perturbativity and stability bounds on $m_H$ as a function of the physical top quark mass $M_t$, for different values of the cutoff $\Lambda$ at which new physics is expected. Present experimental results lead to a precise knowledge of the value of the top quark mass, $M_t = 175 \pm 6$ GeV [@TopM]. Hence, as follows from Fig. 1, independently of the experimental bounds on the Higgs mass, the theoretical upper bound on the Higgs mass obtained from the requirement of preserving the baryon asymmetry, $m_H \simeq 40$ GeV, implies an upper bound on the scale of new physics of the order of the electroweak scale.
This new physics will affect the structure of the effective potential at the weak scale, and hence the upper bound on the Higgs mass derived from requiring a sufficiently strong first order phase transition has to be revised.
BEYOND THE STANDARD MODEL: SUPERSYMMETRY
========================================
The arguments presented in section 3 depend strongly on the structure of the effective Higgs potential. Hence, the Higgs mass bounds could be simply avoided by complicating the Higgs structure. Models with two Higgs doublets are among the simplest ones, and hence they have attracted some attention. Two Higgs doublet models, in general, lead to charge breaking minima, unless the vacuum expectation values of both Higgs doublets are alligned in such a way that the electromagnetic symmetry is preserved. Moreover, they generally lead to flavor changing neutral currents which are beyond the present experimental limits. There are several ways to avoid these problems, and models of this type have been analysed in the literature. However, there is no clear motivation for the extension of the Higgs sector within the Standard Model. On the contrary, two Higgs doublets are necessary in the context of supersymmetric theories.
The most appealing extension of the Standard Model is the Minimal Supersymmetric Standard Model (MSSM) [@reviewsu]. Supersymmetry relates bosonic and fermionic degrees of freedom. For each chiral fermion (gauge boson) of the Standard Model, a complex scalar (Majorana fermion) appears in the theory, with equal gauge quantum numbers as the Standard Model field ones. Moreover, supersymmetry implies a relation between the couplings of the bosonic and fermionic degrees of freedom, yielding a cancellation of the quadratic divergencies associated with the radiative corrections to the scalar Higgs masses, and providing a technical explanation of the hierarchy stability from $M_{Pl}$ to the electroweak scale.
Supersymmetry provides a solution to most of the problems affecting the multi-Higgs systems. Two Higgs doublets are naturally required, to cancel the anomalies generated by the superpartners of the Higgs bosons. Moreover, flavor changing neutral currents are naturally suppressed since supersymmetry requires that only one of the Higgs doublets couples to the up-like (down-like) quarks. In addition, the effective Higgs potential is such that the vacuum state is naturally alligned towards a charge conserving minimum in the Higgs sector of these models.
The Higgs spectrum of the Minimal Supersymmetric extension of the Standard Model consists of two CP-even bosons, a CP-odd and a charged Higgs bosons [@Hhunter]. The heaviest CP-even and the charged Higgs masses are of the order of the CP-odd Higgs mass, $m_A$, and for large values of $m_A$ they form a heavy Higgs doublet which decouples at low energies. In this limit, the lightest Higgs doublet contains the three Goldstone modes, as well as a CP-even state. Moreover, supersymmetry relates the Higgs quartic couplings to the weak gauge couplings leading to an upper bound on the lightest CP-even Higgs mass, $$m_h^2 \leq M_Z^2 \cos^2 2\beta + {\rm rad.} \;
{\rm corr.},
\label{eq:htree}$$ where $\tan\beta = v_2/v_1$ and $v_2$ ($v_1$) is the vacuum expectation value of the Higgs field $H_2$ ($H_1$) which couples to the up (down) quarks. The last term in the above equation denotes the radiative corrections, which are induced through supersymmetry breaking effects. The main contributions will be discussed in section 4.1.
Supersymmetry is particularly appealing for the scenario of electroweak baryogenesis, since it naturally provides small values of the Higgs mass, and hence tends to give a relatively strong first order phase transition. Moreover, in a supersymmetric theory the negative contributions of the top quark to the renormalization group evolution of the Higgs quartic couplings are compensated by the effects of its supersymmetric partner, providing a natural solution to the vacuum stability problem. However, since supersymmetry is broken in nature, this argument depends strongly on the supersymmetry breaking scale. Indeed the supersymmetry breaking scale may be identified with the scale of new physics (see section 3). Since the particles which couple more strongly to the Higgs are the top quark and its supersymmetric partners, the relevant scale of new physics, in relation to the stability of the Higgs potential, is given by the stop masses. From Fig. 1 we see that in order to preserve the stability of the effective potential, for a Higgs mass of order of the present experimental bound, the lightest stop mass must be of the order of the weak scale.
Higgs and Stop Masses in the MSSM
---------------------------------
The stop masses have an incidence on the Higgs potential which goes beyond the problem of vacuum stability. The stop radiative corrections affect the value of the parameters appearing in the effective potential of the Higgs field in a way which depends on the exact value of the stop masses [@Higgs1l]. For values of the stop masses close to the top mass, there is an approximate cancellation between the top and stop loop effects and the tree-level relation between $m_h$ and $M_Z$, Eq. (\[eq:htree\]), is recovered. For very large values of the stop masses, instead, the tree-level relation is strongly affected by radiative corrections and the effective theory is similar to a non-supersymmetric two Higgs doublet model.
It is interesting to discuss in some detail the properties of the superpartners of the top quark. The left handed and right handed stops are not mass eigenstates, due to the appearence of effective trilinear couplings between the left- and right-handed stops and the Higgs fields $${\cal L}_{3} = - h_t \left( \epsilon_{ij} A_t H_2^j Q^i U
- \mu^* H_1^{* i} Q^i U \right) + h.c.,$$ where $Q$ is the scalar top-bottom left-handed doublet and $U$ is the charge conjugate of the right handed scalar top, $A_t$ is a soft supersymmetry breaking mass parameter and $\mu$ is the Higgs superpartner (Higgsino) mass term. Once the neutral components of the Higgs doublets acquire vacuum expectation values, a mixing term appears between the left and right handed stops, leading to the following mass matrix $${\cal M}_{st}^2 = \left[
\begin{array}{cc}
m_Q^2 + m_t^2 + D_L & m_t \left(A_t - \mu^*/\tan\beta\right) \\
m_t \left(A_t^* - \mu/\tan\beta\right)
& m_U^2 + m_t^2 +D_R
\end{array}
\right]
\equiv
\left[
\begin{array}{cc}
m^2_{LL} & m^2_{LR} \\
m^2_{LR} & m^2_{RR}
\end{array}
\right] \;\;,
\label{eq:stopmatrix}$$ where $m_Q^2$ and $m_U^2$ are the soft supersymmetry breaking square mass parameters of the left and right handed stops, respectively, $D_L$ and $D_R$ are the (relatively small) D-term contributions to the stop masses, and $m_t = h_t <H_2>$ is the running top quark mass. The stop mass eigenvalues are then given by $$m^2_{\widetilde{T},\tilde{t}}= \frac{m^2_{LL} + m^2_{RR}}{2} \pm
\sqrt{\left(\frac{m^2_{LL} - m^2_{RR}}{2}\right)^2 +|m^2_{LR}|^2}.$$
The lightest CP-even Higgs mass is a monotonically increasing function of the CP-odd Higgs mass $m_A$. As mentioned above, for large values of the CP-odd Higgs mass, $m_A \gg M_Z$, the heavy Higgs doublet decouples and we obtain an upper bound on the lightest CP-even Higgs mass. This value has been computed at the one and two-loop level, and considering a renormalization group resummation [@Higgs1l]. In the large $m_A$ limit, with $m_Q \simeq m_U$, a simple formula is obtained at the two-loop level [@CEQW], $$\begin{aligned}
m_h^2 & = & M_Z^2 \cos^2 2 \beta \left(1 - \frac{3 m_t^2}{4 \pi^2
v^2} t \right)
\nonumber\\
& + & \frac{3m_t^4}{2\pi^2v^2} \left[ \frac{X_t}{2}
+ t + \frac{1}{16\pi^2} \left(
\frac{3 m_t^2}{v^2} - 32 \pi \alpha_3\right)
\left( X_t t + t^2 \right) \right]
\label{Higgs2l}\end{aligned}$$ where $m_t$ and $\alpha_3 = g_3^2/4 \pi$, with $g_3$ the strong gauge coupling, are evaluated at the top quark mass scale, $$t = \log\left(\frac{M_S^2}{m_t^2}\right)
\;\;\;\;\;\;\;\;\;\;\;\;
X_t = \frac{2 |\tilde{A}_t|^2}{M_S^2} \left(
1 - \frac{\left|\tilde{A}_t\right|^2}{12 M_S^2} \right)$$ with $\tilde{A}_t = A_t - \mu^{*}/\tan\beta$, $M_S^2 = (m_Q^2 + m_U^2)/2 + m_t^2$, and we have implicitly assumed that $|m_t \tilde{A}_t| \simlt 0.5 M_S^2$.
The first term in Eq. (\[Higgs2l\]) reproduces the tree-level contribution to the lightest Higgs mass, Eq. (\[eq:htree\]). The tree level value of the lightest Higgs mass increases for larger values of $\tan\beta$ and tends to zero for $\tan\beta$ equal to one. The most important radiative corrections to the Higgs mass value are positive, proportional to $m_t^4$, and increase logarithmically with the supersymmetry breaking scale $M_S$. The stop mixing parameter plays also a very important role in determining the Higgs mass value. A maximum value for the Higgs mass is obtained for $|\tilde{A}_t| \simeq \sqrt{6} M_S$. Such large values of the stop mass mixing parameter are, however, disfavor by model building considerations. As we shall show in section 4.2, large values of $|\tilde{A}_t| \simgt 0.5 M_S$ also make the phase transition more weakly first order.
Since the phase transition becomes stronger for lower values of the Higgs mass, it is interesting to analyse the conditions under which the lightest CP-even Higgs mass becomes close to the present experimental bound $m_h \simgt 70$ GeV. Low values of the Higgs mass $m_h \simlt 85$ GeV are only obtained for $\tan\beta \simlt$ 4. Very low values of $\tan\beta$ are associated with large values of the top quark Yukawa coupling, and for a given value of $m_t$ a lower bound on $\tan\beta$ may be obtained by requiring perturbative consistency of the theory up to scales of order of the grand unification scale. This requirement leads to values of $\tan\beta \simgt 1.2$ for the acceptable experimental range for the top quark mass [@IR]. If $\tan\beta$ is close to one, at least one of the stop masses has to be large, in order to overcome the present experimental limits on $m_h$. For large splittings between the two stop mass eigenvalues, one has to go beyond the approximation of Eq. (\[Higgs2l\]) [@CEQW]. We shall briefly discuss this case in section 4.2.
The Electroweak Phase Transition
--------------------------------
### Lightest stop mass effects on the phase transition.
As we discussed in section 3, for a fixed Higgs mass, the strength of the first order phase transition may be enhanced by increasing the value of the effective cubic term in the finite temperature Higgs potential. This may be achieved by the presence of extra bosonic degrees of freedom [@AndH], with sizeable couplings to the Higgs sector. Within the minimal supersymmetric model, the bosonic particles which couple strongly to the Higgs which acquires vacuum expectation value are the supersymmetric partners of the top quark. Since the cubic term is screened by field independent mass contributions, a relevant enhancement of the cubic term of the effective Higgs potential may only be obtained for small values of the lightest stop mass $m_{\tilde t} \simlt m_t$ [@CQW].
The lightest stop must be mainly right-handed in order to naturally suppress its contribution to the parameter $\Delta\rho$ and hence preserve a good agreement with the precision measurements at LEP. This can be naturally achieved if the soft supersymmetry breaking mass parameter of the left-handed stop, $m_Q$, is much larger than $M_Z$. We shall first discuss the large CP-odd mass limit, $m_A \gg M_Z$. In this case, the heaviest Higgs doublet decouples and the low energy theory contains only one light Higgs boson, $\phi$, with similar properties to the Standard Model one, $$\phi = \cos\beta H_1 + \sin\beta H_2.$$
For moderate left-right stop mixing, from Eq. (\[eq:stopmatrix\]) it follows that the lightest stop mass is then approximately given by \^2 m\_U\^2 + D\_R + m\_t\^2() ( 1 - ) \[eq:lightst\] where $m_t(\phi) = h_t \sin\beta \; \phi$. Hence, the lightest stop contribution to the finite temperature Higgs potential, necessary to overcome the SM constraints on the Higggs mass, strongly depends on the value of $m_U^2$. At finite temperature, however, the field mass receives vacuum polarization contributions which have a strong impact on the size of the induced cubic terms in the effective finite temperature Higgs potential. Indeed, $$m_{\tilde{t}}^2(\phi,T) =
m_{\tilde{t}}^2(\phi,0) + \Pi_R(T)$$ where $\Pi_R(T) \simeq 4 g_3^2
T^2/9
+h_t^2/6[1+\sin^2\beta\left(1-
|\widetilde{A}_t|^2/m_Q^2\right)]T^2$ is the finite temperature self-energy contribution to the right-handed squarks [@CQW; @CE]. The improved one-loop finite temperature effective potential is then given by $$V_{\rm eff}^{\rm MSSM} = m^2(T) \phi^2 -
T \left[ E_{\rm SM} \phi^3 + (2 N_c)
\frac{m_{\tilde{t}}^{3/2}(\phi,T)}{12 \pi} \right]
+ \frac{\lambda(T)}{2} \phi^4 +...
\label{eq:temppot}$$ where $N_c = 3$ is the number of colors, and $E_{\rm SM}$ is the strength of the cubic term in the Standard Model case.
Observe that the heaviest stop leads to a relevant contribution to the zero-temperature effective potential, which can be absorved into a redefinition of the mass and quartic coupling parameters. Large values of $m_Q$ have the effect of increasing the value of the Higgs mass. Indeed, for $m^2_Q \gg m^2_U$ and moderate values of $\tilde{A}_t$, the lightest Higgs mass expression at the one-loop level reads, $$m_h^2 = M_Z^2 \cos^2 2\beta + \frac{3m_t^4}{4 \pi^2 v^2}
\log\left(\frac{m_{\tilde{t}}^2 m_{\tilde{T}}^2}{m_t^4}
\right) + {\cal O}\left(\frac{\tilde{A}_t^2}{m_Q^2}\right)$$ where $m_{\tilde{T}}^2 \simeq m_Q^2 + m_t^2$ is the heaviest stop square mass [^3]. Although larger values of $m_h$ are welcome in order to avoid its experimental bound, since they are associated with an increase of the quartic coupling, they necessarily lead to a weakening of the first order phase transition. Therefore, very large values of $m_Q$, above a few TeV, are disfavored from this point of view. The finite temperature effects of the heaviest stop are, instead, exponentially suppressed.
From Eq. (\[eq:temppot\]) it follows that, in general, as happens with the longitudinal components of the gauge bosons, the lightest stop contribution to the effective potential does not induce a cubic term. This is mainly due to the fact that the effective stop plasma mass squared at $\phi =0$, $$(m^{\rm eff}_{\;\widetilde{t}})^2 = -\widetilde{m}_U^2 + \Pi_R(T)
\label{plasm}$$ with $\widetilde{m}_U^2 \equiv
- m_U^2$, is generally positive and of order of $T^2$. If the right handed stop plasma mass at $\phi = 0$, Eq. (\[plasm\]), vanished, a large value of the effective cubic term would be obtained. Since $v(T_c)/T_c \simeq \sqrt{2} E/\lambda$, an upper bound on $v(T_c)/T_c$ may be obtained from these considerations, namely < ()\_[SM]{} + , \[totalE\] where $m_t = \overline{m}_t(m_t)$ is the on-shell running top quark mass in the $\overline{{\rm MS}}$ scheme. The first term on the right hand side of Eq. (\[totalE\]) is the SM contribution ()\_[SM]{} ()\^2, and the second term is the contribution that would be obtained through the right handed stops in the limit of a vanishing plasma mass. The upper bound on $v(T_c)/T_c$ is almost an order of magnitude larger than the one obtained in the Standard Model, implying that Higgs masses of order $M_Z$ may be consistent with electroweak baryogenesis. Although the exact cancellation of the effective stop mass at $\phi = 0$ is not likely to occur, it is clear that a partial cancellation is necessary to increase the cubic term coefficient considerably [^4]. A small plasma mass can only be obtained through sizeable values of $\widetilde{m}_U$, this means, negative sizeable values of the right-handed stop mass parameter. Moreover, as it is clear from Eq. (\[eq:lightst\]), the trilinear mass term, $\widetilde{A}_t$, must be $|\widetilde{A}_t|^2 \ll m_Q^2$ in order to avoid the suppression of the effective coupling of the lightest stop to the Higgs. Negative values of the right handed stop mass parameter open the window for electroweak baryogenesis. However, they may be associated with the appearence of color breaking minima at zero and finite temperature. It is hence important to discuss briefly the constraints on $\widetilde{m}_U$ which may be obtained from the requirement of avoiding color breaking minima deeper than the physical one.
Color Breaking Minima
---------------------
Let us first analyse the possible existence of color breaking minima in the case of zero stop mixing. In this case, since $m_Q^2 \gg |m_U|^2$ the only fields which may acquire vacuum expectation values are the fields $\phi$ and $U$. At zero temperature, the effective potential is given by V\_[eff]{}(,U) = -m\_\^2 \^2 + \^4 + m\_U\^2 U\^2 + U\^4 + \_t\^2 \^2\^2 U\^2 \[effpphiu\] where $\lambda$ is the radiatively corrected quartic coupling of the Higgs field, with its corresponding dependence on the top/stop spectrum through the one loop radiative corrections, $\widetilde{g}_3^2/3$ is the radiatively corrected quartic self-coupling of the field $U$ and $\widetilde{h}_t^2$ is the bi-bilinear $\phi-U$ coupling. The latter couplings are well approximated by $\widetilde{g}_3 \simeq g_3$ and $\widetilde{h}_t \simeq h_t$. The minimization of this potential leads to three extremes, at: [**(i)**]{} $\phi =0$, $U\neq0$; [**(ii)**]{} $U=0$, $\phi \neq 0$ and [**(iii)**]{} $\phi \neq 0$, $U \neq 0$. The corresponding expressions for the vacuum fields are: $$\label{solutions}
\begin{array}{rllll}
{\bf (i)}& U & = & 0, &
{\displaystyle \phi^2 = \frac{m_{\phi}^2}{\lambda}; }
\\ & & & & \\
{\bf (ii)}& \phi & = & 0, &
{\displaystyle U^2 = \frac{3 \widetilde{m}_U^2}{\widetilde{g}_3^2}; }
\\ & & & & \\
{\bf (iii)} & \phi^2 & = &
{\displaystyle \frac{m_{\phi}^2 - 3 \widetilde{m}_U^2 \widetilde{h}_t^2
\sin^2\beta/\widetilde{g}_3^2}{\lambda - 3 \widetilde{h}_t^4 \sin^4\beta/
\widetilde{g}_3^2}, } &
{\displaystyle U^2 =
\frac{\widetilde{m}_{U}^2 - m_{\phi}^2 \widetilde{h}_t^2
\sin^2\beta/\lambda}{\widetilde{g}_3^2/3 -
\widetilde{h}_t^4 \sin^4\beta/\lambda}. }
\end{array}$$ It is easy to show that the branch (iii) is continuosly connected with branches (i) and (ii). One can also show that the branch (iii) defines a family of saddle point solutions, the true (local) minima being defined by (i) and (ii). Hence, the requirement of absence of a color breaking minimum deeper than the physical one is given by [@CQW] $$\widetilde{m}_U \leq
\left( \frac{m_h^2 \; v^2 \; \widetilde{g}_3^2}{12} \right)^{1/4}.
\label{boundmu}$$ For a Higgs mass $m_h \simeq 70$ GeV, the bound on $\widetilde{m}_U$ is of order 80 GeV. This must be compared with the characteristic value of $\Pi_R(T) \simeq
{\cal O}\left((100 {\rm GeV})^2\right)$, implying that even for values of $\tilde{m}_U$ close to the upper bound on this quantity, a non-negligible screening to the effective cubic term contributions will be present.
It can also be shown that, for $m_Q \gg \widetilde{m}_U$, the bound $\widetilde{m}_U$ derived above, Eq. (\[boundmu\]) is sufficient to assure the stability of the physical ground state for all values of $\widetilde{A}_t$ such that the experimental limits on the lightest stop mass are preserved. As we shall show quantitatively below, and it is clear from our previous discussion, Eq. (\[totalE\]), large values of $\widetilde{A}_t$ induce a large suppression of the potential enhancement in the strength of the first order phase transition through the light top squark, and are hence disfavoured from the point of view of electroweak baryogenesis.
One must also consider the conditions under which the potential may be metastable, but with a lifetime larger than the age of the universe. Even in this case, in general, the constraint -\_U\^2+\_R(T\_c) > 0 \[stability\] must be fulfilled. Indeed, if Eq. (\[stability\]) were not fulfilled, the universe would be driven to a charge and color breaking minimum at $T\geq T_c$ (see Eq.(\[upot\]) below). A more conservative requirement can be obtained demanding that the critical temperature for the transition to the color breaking minimum, $T_c^U$, should be below $T_c$. Due to the strength of the stop coupling to the gluon and squark fields, one should expect the color breaking phase transition to be stronger than the electroweak one.
Let us analyse the finite temperature effective potential for the $U$ field, which is given by [@CQW] $$V_U = \left(-\widetilde{m}_U^2 + \gamma_U T^2 \right) U^2 -
T E_U U^3 + \frac{\lambda_U}{2} U^4,
\label{upot}$$ where $$\begin{aligned}
\gamma_U & \equiv & \frac{\Pi_R(T)}{T^2} \simeq
\frac{4 g_3^2}{9} + \frac{h_t^2}{6}\left[ 1 +
\sin^2\beta (1 - \widetilde{A}_t^2/m_Q^2)
\right] ; \;\;\;\;\;\;\;\;\;\;\;\;\;\;
\lambda_U \simeq \frac{g_3^2}{3}
\nonumber\\
E_U & \simeq & \left[\frac{\sqrt{2} g_3^2}{6 \pi} \left( 1 +
\frac{2}{3\sqrt{3}} \right) \right]
\nonumber\\
& + &
\left\{ \frac{g_3^3}{12\pi}\left(\frac{5}{3\sqrt{3}} + 1\right)
+ \frac{h_t^3 \sin^3\beta (1-\widetilde{A}_t^2/m_Q^2)^{3/2}}{3 \pi}
\right\}.
\label{totalEu}\end{aligned}$$ The contribution to $E_U$ inside the square brackets comes from the transverse gluons, $E_U^g$, while the one inside the curly brackets comes from the squark and Higgs contributions \[Although included in the numerical results, in the above we have not written explicitly the small hypercharge contributions to $E_U$ and $\gamma_U$.\]. We ignore the gluino and left handed squark contributions since they are assumed to be heavy and, hence, their contributions to the finite temperature effective potential is Boltzman suppressed. Observe that we have written the contributions that would be obtained if the field-independent effective thermal mass terms of the squark and Higgs fields were exactly vanishing at the temperature $T_c$. Although for values of $\widetilde{m}_U^2$ which induce a large cubic term in the Higgs potential, $T_c$ is actually close to the temperature at which these masses vanish, an effective screening is always present. This means that the value of $E_U$ given above is somewhat overestimated.
The difference between $T_0^U$, the temperature at which $\mstopeff(\phi = 0) = 0$, and $T_c^U$, is given by $$T_c^U = \frac{T_0^U}{\sqrt{1 - E_U^2/ 2\lambda_U\gamma_U }}.$$ In order to assure a transition from the $SU(2)_L \times U(1)_Y$ symmetric minimum to the physical one at $T = T_c$, we should replace the condition (\[stability\]) by the condition which assures that $T_c^U < T_c$ [@CQW], $$-\widetilde{m}_U^2 + \Pi_R(T) > \widetilde{m}_U^2
\frac{\epsilon}{1-\epsilon} \simeq \widetilde{m}_U^2 \epsilon,
\label{stability2}$$ with $\epsilon = E_U^2/2\lambda_U\gamma_U$, a small number. In the following, we shall require the stability condition, Eq. (\[stability2\]), while using the value of $E_U$ given in Eq. (\[totalEu\]). We shall also show the result that would be obtained if only the gluon contributions to $E_U$, $E_U^{g}$, would be considered. The difference between both procedures is just a reflection of the uncertainties involved in our analysis.
### Strength of the First Order Phase Transition in the Large $m_A$ Limit
Let us first present the results for zero mixing. Fig. 2 shows the order parameter $v(T_c)/T_c$ for the phase transition as a function of the running light stop mass, for $\tan\beta = 2$, $m_Q = 500$ GeV and $M_t = 175$ GeV. These parameters imply a Higgs mass $m_h \simeq 70$ GeV, a result which depends weakly on $\widetilde{m}_U$. We see that for smaller (larger) values of $\mstop$ ($\widetilde{m}_U$), $v(T_c)/T_c$ increases in accordance with the above discussion in this section. The diamond in fig.2 marks the lower bound on the stop mass coming from the bound on color breaking vacua at $T=0$, Eq. (\[boundmu\]). The cross and the star denote the bounds that would be obtained by requiring the condition (\[stability2\]), while using the total and gluon-induced trilinear coefficient, $E_U$ and $E_U^g$, respectively. We see that the light stop effect is maximum for values of $\widetilde{m}_U^2$ such that condition (\[stability2\]) is saturated, which leads to values of $\mstop \simeq 140$ GeV ($\widetilde{m}_U\simeq 90$ GeV) and $v(T_c)/T_c \simeq 1.75$.
The preservation of condition (\[boundmu\]) demands slightly larger stop mass values. The analysis shows that there is a large region of parameter space for which $v(T_c)/T_c \geq$ 1 and is not in conflict with any phenomenological constraint.
Fig. 3 shows the results of $v(T_c)/T_c$ for zero mixing and $m_Q = 500 $ GeV as a function of $\tan\beta$ and for the values of $\widetilde{m}_U$ such that the maximum effect is achieved. We also plot in the figure the corresponding values of the stop and Higgs masses. As in Fig. 2, the solid \[dashed\] line represents the result when the bound (\[boundmu\]) \[the stability bound of Eq. (\[stability2\])\] is saturated. We see that $v(T_c)/T_c$ increases for lower values of $\tan\beta$, a change mainly associated with the decreasing value of the Higgs mass, or equivalently, of the Higgs self-coupling. For values of $\tan\beta \simeq 2.7$, one gets $v(T_c)/T_c \simeq 1$, and hence the value of the Higgs mass yields the upper bound consistent with electroweak baryogenesis. This bound is approximately given by $m_h \simeq 80$ GeV. If the bound on color breaking minima, Eq. (\[boundmu\]), is ignored, then condition (\[stability2\]) yields an upper bound on $m_h$ close to 100 GeV, in accordance with the qualitative discussion presented above (Similar bounds on the Higgs mass, $m_h\! < \!100\!$ GeV, are obtained when two loops corrections are included and condition (\[boundmu\]) is saturated [@inprep]).
Due to the logarithmic dependence of $m_h$ on $m_Q$, larger values of $m_Q$ have the effect of enhancing the Higgs mass values. It turns out that, for zero mixing, the results for $v(T_c)/T_c$ depend on the Higgs mass and on the value of $m_U$, but not on the specific value of $m_Q$. Hence, different values of $m_Q$ have the only effect of shifting (up or down) the preferred values of $\tan\beta$. In particular, the fixed point solution, which corresponds to values of $\tan\beta \simeq 1.6$ for $M_t$ = 175 GeV, leads to values of $m_h \simgt 70$ GeV and $v(T_c)/T_c \simgt 1$, so far $m_Q$ is above 1 TeV and below a few TeV.
The effect of mixing in the stop sector is very important for the present analysis. For fixed values of $m_Q$ and $\tan\beta$, increasing values of $\widetilde{A}_t$ have a negative effect on the strength of the first order phase transition for three reasons. First, large values of $\widetilde{A}_t$ lead to larger values of the Higgs mass $m_h$. Second, as shown in Eq. (\[totalE\]) they suppress the stop enhancement of the cubic term. Finally, there is an indirect effect associated with the constraints on the allowed values for $\widetilde{m}_U$. This has to do with the fact that for larger values of $\widetilde{A}_t$, the phase transition temperature increases, making more difficult an effective suppression of the effective mass $\mstopeff$, Eq. (\[plasm\]). Of course, this third reason is absent if the bound (\[boundmu\]) is ignored. As we have shown above, for zero mixing the bounds (\[orderp\]), (\[boundmu\]) and (\[stability2\]) are only fulfilled for values of the stop mass larger than approximately 140 GeV. Light stops, with masses $\mstop \simlt 100$ GeV, can only be consistent with these constraints for larger values of the mixing mass parameter $\widetilde{A}_t$.
Fig. 4 shows the result for $v(T_c)/T_c$ as a function of $\widetilde{A}_t$ for $\tan\beta = 1.7$, $m_Q$ = 500 GeV, and values of $m_U$ such that the maximal light stop effect is achieved. The same conventions as in Fig. 3 have been used. Due to the constraints on $\widetilde{m}_U$, light stops with $\mstop \simlt M_W$, may only be obtained for values of $\widetilde{A}_t \simgt 0.6\; m_Q$. For these values of $\widetilde{A}_t$, the phase transition temperature is large enough to induce large values of $\mstopeff$, for all values of $\widetilde{m}_U$ allowed by Eq. (\[boundmu\]). In Fig. 4, we have chosen the parameters such that they lead to the maximum value of the mixing parameter $\widetilde{A}_t/m_Q$ consistent with $v(T_c)/T_c \geq 1$ and the Higgs mass bound. For values of $\widetilde{m}_U$ such that the bounds on color breaking minima are preserved, the mixing effects on the stop masses are small, and the lightest stop remains heavier than 100 GeV. If, however, the weaker bound, Eq. (\[stability2\]), were required (thin and thick dashed lines in Fig. 4), light stops, with masses of order $80$–$90$ GeV would not be in conflict with electroweak baryogenesis.
### Sources of CP-violation and the CP-odd Higgs Mass
The new source of CP-violation, beyond the one contained in the Cabibbo-Kobayashi-Maskawa matrix, may be either explicit [@ex1] or spontaneous [@noi] in the Higgs sector (which requires at least two Higgs doublets). In both cases, particle mass matrices acquire a nontrivial space-time dependence when bubbles of the broken phase nucleate and expand during a first-order electroweak phase transition. The crucial observation is that this space-time dependence cannot be rotated away at two adjacent points by the same unitary transformation. This provides sufficiently fast nonequilibrium CP-violating effects inside the wall of a bubble of broken phase expanding in the plasma and may give rise to a nonvanishing baryon asymmetry through the anomalous $(B+L)$-violating transitions [@sp] when particles diffuse to the exterior of the advancing bubble.
As we already mentioned, new CP-violating phases arise through the soft supersymmetry breaking parameters associated with the left-right stop mixing, namely $A_t$ and $\mu$, Eq. (\[eq:stopmatrix\]). The stop induced current is hence proportional to the variation of the phase of $(A_t H_2 - \mu^* H_1) = |A_t H_2 - \mu^* H_1|
\exp(i \phi_{\tilde{A}})$. It is easy to show that $$\partial_{\nu} \phi_{\tilde{A}} \sim
{\rm Im}[A_t \mu] \left(H_2 \partial_{\nu}H_1 - H_1
\partial_{\nu}H_2 \right).$$
The phase of $\mu$ enters also in the chargino sector. If we consider the chargino square mass matrix $${\cal M}_{\rm ch} {\cal M}_{\rm ch}^{\dagger} =
\left[ \begin{array}{cc}
M_2^2 + g^2 H_2^2 & g(M_2 H_1 + \mu^* H_2) \\
g(M_2 H_1 + \mu H_2) & |\mu|^2 + g^2 H_1^2 \end{array}
\right],$$ where $g = 2 M_w/v$ is the SU(2) gauge coupling and $M_2$ is the (assumed) real soft supersymmetry breaking mass of the supersymmetric partners of the weak gauge bosons, the chargino induced CP-violating current is proportional to the variation of the phase of the mixing term, $(M_2 H_1 + \mu^* H_2) =
|M_2 H_1 + \mu^* H_2| \exp(i\phi_{\tilde{\mu}})$. It follows that $$\partial_{\nu} \phi_{\tilde{\mu}} \sim
{\rm Im}[M_2 \mu] \left(H_2 \partial_{\nu}H_1 - H_1
\partial_{\nu}H_2 \right).$$ Defining $H^2 = H_1^2 +H_2^2$, to a good approximation, the currents are proportional to the function $$H_1(z) \partial_z H_2(z) - H_2(z) \partial_z H_1(z)
\equiv H^2(z) \partial_z\beta(z),
\label{current}$$ with $z$ the time component of the four vector $z_{\nu}$. Since the time variation of the Higgs fields in the plasma frame is due to the expansion of the bubble wall through the thermal bath, ignoring the curvature of the bubble wall and assuming that the bubble wall is moving along the $z_3$ axis with velocity $v_w$, any quantity becomes a function of ${\bf z}= z_3 + v_w z$, the coordinate normal to the wall surface. Eq. (\[current\]) should vanish smoothly for values of ${\bf z}$ outside the bubble wall. Since $\partial_z\beta \equiv v_w \partial_{\bf z} \beta$ in Eq. (\[current\]) denotes the derivative of the ratio of vacuum expectation values of the Higgs fields, it will be non-vanishing only if the CP-odd Higgs mass takes values of the order of the critical temperature. Values of the CP-odd Higgs mass $m_A \simlt 200$ GeV are, however, associated with a weaker first order phase transition. Fig. 5 shows the behaviour of the order parameter $v/T$ in the \[$m_A-\tan\beta$\] plane, for $\widetilde{A}_t = 0$, $m_Q = 500$ GeV and values of $\widetilde{m}_U$ close to its upper bound, Eq. (\[boundmu\]). In order to interpret correctly the results of Fig. 5 one should remember that the Higgs mass bounds are somewhat weaker for values of $m_A < 150$ GeV. However, even for values of $m_A$ of order 80 GeV, in the low $\tan\beta$ regime the lower bound on the Higgs mass is of order 60 GeV. Hence, it follows from Fig. 5 that, to obtain a sufficiently strong first order phase transition, $v(T_c)/T_c \simgt 1$, the CP-odd Higgs mass must fulfill the condition $m_A \simgt 150$ GeV.
In order to compute the CP-violating sources, the variation of the angle $\beta$ along the bubble wall should be computed. The Higgs profiles along the wall are likely to follow the path of minimal energy connecting the electroweak symmetry preserving and the symmetry breaking vacua in the Higgs potential. The Higgs potential in the case of low values of the CP-odd Higgs mass may be computed by methods similar to those ones explained in section 2, by preserving the field dependence on both Higgs fields [@mariano2; @CQRVW]. For small values of the fields $H_i$, as those appearing close to the symmetric phase, the Higgs potential for the neutral CP-even components of the Higgs doublets, $H_1$ and $H_2$, may be approximated by $$V(H_1,H_2,T) = m_1^2(T) H_1^2 + m_2^2(T) H_2^2 - 2 m_3^2(T)
H_1 H_2 + {\cal O}(H_i^3) T +...$$ The value of $\beta$ close to the symmetric phase may be easily computed at the temperature $T_0$ at which the curvature at the origin vanishes [^5], $$m_3^4(T_0) = m_1^2(T_0) m_2^2(T_0).$$
Under these conditions, the perturbations of the Higgs fields close to the origin will follow the path such that the value of the potential is minimized along it, namely, $$m_1^2(T_0) v_1^2 + m_2^2(T_0) v_2^2 - 2 m_3^2(T_0) v_1 v_2 = 0$$ or, equivalently, $$\tan^2\beta(T_0,H_1\simeq 0,H_2\simeq 0) =
\frac{m_1^2(T_0)}{m_2^2(T_0)} \simeq
\frac{m_1^2(T_c)}{m_2^2(T_c)}.$$
The exact value of $\beta$ at the critical temperature may be computed by a numerical simulation of the full effective potential. Hence, the variation of $\beta$ along the bubble wall may be approximately given by $$\Delta\beta \simeq \beta(T_c) - \arctan(m_1(T_c)/m_2(T_c)).$$ This quantity tends to zero for large values of $m_A$ like $\Delta\beta \sim H^2/m_A^2$. A numerical estimate [@CQRVW] gives that, for $m_A = 200$ GeV, $\Delta\beta \simeq 0.015$ and hence values of $m_A > 300$ GeV imply a strong suppression of the generated CP-violating sources. We shall fix $m_A = 200$ GeV for most of the following analyis.
GENERATION OF THE BARYON ASYMMETRY
==================================
Baryogenesis is fueled by CP-violating sources which are locally induced by the passage of the wall [@thick; @thicknoi]. These sources do not provide net baryon number. Indeed, in the absence of baryon number violating processes, the generated baryon number will be zero. Since the CP-violation sources are non-zero only inside the wall, it was first thought that a detail analysis of the rate of anomalous processes inside the bubble wall was necessary to estimate the generated baryon number. However, these first analyses ignored the crucial role played by diffusion [@tra]. Indeed, transport effects allow CP-violating charges to efficiently diffuse into the symmetric phase –in front of the advancing bubble wall– where anomalous electroweak baryon number violating processes are unsuppressed. This amounts to greatly enhancing the final baryon asymmetry.
In order to estimate the generated baryon number, a set of coupled classical Boltzmann equations describing particle distribution densities should be solved. These equations take into account particle number changing reactions [@cha] and they allow to trace the crucial role played by diffusion [@tra]. Since the weak anomalous processes affect only the left handed quarks and leptons, the relevant CP-violating sources are those which can lead through particle interactions to a net chiral charge for the Standard Model quarks. The new CP-violating sources we are considering are associated with the parameters $A_t$ and $\mu$, therefore, the relevant currents are the stop, chargino and neutralino ones. Although the masses of the first and second generation squarks, as well as the sbottom ones, are affected by the phase of the $\mu$ parameter, they couple very weakly to the Higgs and hence they play no role in the computation of the CP-violating currents.
The CP-violating sources for left- and right-handed squarks, charginos and neutralinos are converted into sources of chiral quarks via supergauge and top quark Yukawa interactions, respectively. Indeed, the top Yukawa coupling is sufficiently strong, so that the top Yukawa induced processes are in approximate thermal equilibrium. The same happens with the supergauge interactions, if the gauginos are not much heavier than the critical temperature. Moreover, the strong sphaleron processes are the most relevant sources of first and second generation chiral quarks, and hence they must be taken into account while computing the generated baryon number.
The stop, chargino and neutralino currents at finite temperature may be computed by using diagramatic methods [@chou]$^-$[@riotto]. For small values of the mixing mass parameter, $|\tilde{A}_t|/m_Q < 0.5$, and large values of $m_Q \gg T$, the CP-violating stop induced current is naturally suppressed. The current associated with neutral and charged higgsinos is the most relevant one, and it may be written as [@CQRVW]
\[corhiggs\] J\^\_=\^where $\psi$ is the Dirac spinor \[Dirac\] =(
[c]{} \_2\
\_1
) and $\widetilde{H_2}=\widetilde{H}_2^0$ ($\widetilde{H}_2^+$), $\widetilde{H_1}=\widetilde{H}_1^0$ ($\widetilde{H}_1^-$) for neutral (charged) higgsinos.
The vacuum expectation value of the (zero component of the) higgsino current is approximately given by [^6] $$\langle J_{\widetilde{H}}^0(z)\rangle \simeq
{\rm Im}(\mu)\: \left(H_1(z) \partial_z H_2(z) - H_2(z)
\partial_z H_1(z) \right)
\left[ 3 M_2 \; g^2 \; {\cal G}^{\widetilde{W}}_{\widetilde{H}}
\right],
\label{currenth}$$ where $$\begin{aligned}
{\cal G}^{\widetilde{W}}_{\widetilde{H}} & \simeq &
\int_0^\infty dk
\frac{k^2}
{2 \pi^2 \Gamma_{\widetilde{H}}
\omega_{\widetilde{H}} \omega_{\widetilde{W}}}
\left( {\rm Im}(n_{\widetilde{H}}) +
{\rm Im}(n_{\widetilde{W}}) \right)
I_2(\omega_{\widetilde{H}},
\Gamma_{\widetilde{H}},
\omega_{\widetilde{W}},\Gamma_{\widetilde{W}})
\phantom{\frac{1}{2^2}}
\nonumber\\\end{aligned}$$ with $n_{\widetilde{H}(\widetilde{W})} =
1/\left[\exp\left(\omega_{\widetilde{H}(\widetilde{W})}/T
+ i \Gamma_{\widetilde{H}(\widetilde{W})}/T \right)
+ 1 \right]$, where $\omega^2_{\widetilde{H}}
=k^2+ |\mu|^2, \;\;$ $\omega^2_{\widetilde{W}}
=k^2+ M_2^2$, while $\Gamma_{\widetilde{H}}$ and $\Gamma_{\widetilde{W}}$ are the damping rate of charged and neutral Higgsinos and winos, repectively. Since these damping rates are dominated by weak interactions [@weldon; @henning], we shall take $\Gamma_ {\widetilde{H}} \simeq \Gamma_{\widetilde{W}}$ to be of order of $ 5\times 10^{-2} T$. Moreover, the function $I_2$ is given by [@CQRVW] $$I_2(a,b,c,d) = \frac{r_1^2 - 1}{2 \left(r_1^2 + 1 \right)
\left[(a+c)^2 + (b+d)^2 \right]} +
\frac{r_2^2 - 1}{2 \left(r_2^2 + 1 \right)
\left[(a-c)^2 + (b+d)^2 \right]},
\nonumber \\
\;\;$$ where $r_1 = (a+c)/(b+d)$ and $r_2 = (a-c)/(b+d)$. The above expression, Eq. (\[currenth\]), proceeds from an expansion in derivatives of the Higgs field and it is valid only when the mean free path $\Gamma_{\widetilde{W}(\widetilde{H})}^{-1}$ is smaller than the scale of variation of the Higgs background determined by the wall thickness and the wall velocity, $\Gamma_{\widetilde{W}(\widetilde{H})} L_w/v_w \gg 1$.
As mentioned before, the above currents may be used to compute the particle densities, once diffusion and particle changing interaction effects are taken into account. We shall not discuss this in detail here, but we shall limit ourselves to present the most important aspects related to the generation of baryon number.
The particle densities we need to include are the left-handed top doublet $q_L\equiv(t_L+b_L)$, the right-handed top quark $t_R$, the Higgs particle $h\equiv(H_1^0, H_2^0, H_1^-, H_2^+)$, and the superpartners $\widetilde{q}_L$, $\widetilde{t}_R$ and $\widetilde{h}$. The interactions able to change the particle numbers are the top Yukawa interaction with rate $\Gamma_t$, the top quark mass interaction with rate $\Gamma_m$, the Higgs self-interactions in the broken phase with rate $\Gamma_{\cal H}$, the strong sphaleron interactions with rate $\Gamma_{{\rm ss}}$, the weak anomalous interactions with rate $\Gamma_{\rm ws}$ and the gauge interactions. The system may be described by the densities ${\cal Q} = q_L + \widetilde{q}_L$, ${\cal {\cal T}}=t_R+\widetilde{t}_R$ and ${\cal H}=h+\widetilde{h}$. CP-violating interactions with the advancing bubble wall produce source terms $\gamma_{\widetilde{H}}=\partial_z\langle
J_{\widetilde{H}}
^0(z)\rangle$ for Higgsinos and $\gamma_R=\partial_z \langle J_{R}^0(z)\rangle$ for right-handed stops, which tend to push the system out of equilibrium.
If the system is near thermal equilibrium and particles interact weakly, the particle number densities $n_i$ may be expressed as $n_i = k_i \mu_i T^2/6$, where $\mu_i$ is the local chemical potential and $k_i$ are statistical factors of order 2 (1) for light bosons (fermions) in thermal equilibrium, and Boltzmann suppressed for particles heavier than $T$. Assuming that the rates $\Gamma_t$ and $\Gamma_{{\rm ss}}$ are fast so that ${\cal Q}/k_q-
{\cal H}/k_{\cal H}-{\cal T}/k_{\cal T}={\cal O}(1/\Gamma_t)$ and $2{\cal Q}/k_q-{\cal T}/k_{\cal T}+
9({\cal Q}+{\cal T})/k_b={\cal O}(1/\Gamma_{{\rm ss}})$, one can find the equation governing the Higgs density [@newmethod2] $$\label{equation}
v_{\omega}{\cal H}^\prime-\overline{D}
{\cal H}^{\prime\prime}+\overline{\Gamma}{\cal H}-
\widetilde{\gamma}=0,$$ where the derivatives are now with respect to ${\bf z}$, $\overline{D}$ is the effective diffusion constant, $\widetilde{\gamma}$ is an effective source term in the frame of the bubble wall and $\overline{\Gamma}$ is the effective decay constant. An analytical solution to Eq. (\[equation\]) satisfying the boundary conditions ${\cal H}(\pm\infty)=0$ may be found in the symmetric phase (defined by ${\bf z}<0$) using a ${\bf z}$-independent effective diffusion constant and a step function for the effective decay rate $\overline{\Gamma}= \widetilde{\Gamma} \theta({\bf z})$. A more realistic form of $\overline{\Gamma}$ would interpolate smoothly between the symmetric and the broken phase values. We have checked, however, that the result is insensitive to the specific position of the step function inside the bubble wall.
The analytical solution to the diffusion equations for ${\bf z} < 0$ leads to [@CQRVW; @newmethod2] $$\label{higgs1}
{\cal H}({\bf z})={\cal A}\:{\rm e}^{{\bf z}v_{\omega}/\overline{D}},$$ and for ${\bf z} >0$, $$\begin{aligned}
\label{higgs3}
{\cal H}({\bf z}) & = & \left( {\cal B}_{+} -
\frac{1}{\overline{D}(\lambda_+ - \lambda_-)}
\int_0^{{\bf z}} du \widetilde \gamma(u) e^{-\lambda_+ u} \right)
e^{\lambda_{+} {\bf z}}
\nonumber\\
&+& \left( {\cal B}_{-} -
\frac{1}{\overline{D}(\lambda_- - \lambda_+)}
\int_0^{\bf{z}} du \widetilde \gamma(u) e^{-\lambda_- u} \right)
e^{\lambda_{-} {\bf z}}.\end{aligned}$$ where $$\lambda_{\pm} = \frac{ v_{\omega} \pm
\sqrt{v_{\omega}^2 + 4 \widetilde{\Gamma}
\overline{D}}}{2 \overline{D}},$$ and $\widetilde \gamma({\bf z}) = v_{\omega} \partial_{{\bf z}}
J_0({\bf z}) f(k_i)$, $J_0$ being the total CP-violating current resulting from the sum of the right-handed stop and Higgsino contributions and $f(k_i)$ a coefficient depending on the number of degrees of freedom present in the thermal bath and related to the definition of the effective source [@newmethod2]. Imposing the continuity of ${\cal{H}}$ and ${\cal{H}}'$ at the boundaries, we find [@CQRVW] $$\label{higgs2}
{\cal A}= {\cal B}_{+}\left(1-\frac{\lambda_-}{\lambda_+}\right)=
{\cal B}_{-}\left(\frac{\lambda_+}{\lambda_-}-1\right)=
\frac{1}{\overline{D} \; \lambda_{+}} \int_0^{\infty} du\;
\widetilde \gamma(u)
e^{-\lambda_+ u}.$$ From the form of the above equations one can see that CP-violating densities diffuse in a time $t\sim \overline{D}/ v_{\omega}^2$ and the assumptions leading to the analytical form of ${\cal H}({\bf z})$ are valid provided $\Gamma_t,\Gamma_{{\rm ss}}\gg
v_{\omega}^2/\overline{D}$.
The equation governing the baryon asymmetry $n_B$ is given by [@newmethod2] $$\label{bau}
D_q n_B^{\prime\prime}-v_{\omega} n_B^\prime-
\theta(-{\bf z})n_f\Gamma_{{\rm ws}}n_L=0,$$ where $\Gamma_{{\rm ws}}=6\kappa\alpha_w^4T$ is the weak sphaleron rate ($\kappa\simeq 1$) [^7], and $n_L$ is the total number density of left-handed weak doublet fermions, $n_f=3$ is the number of families and we have assumed that the baryon asymmetry gets produced only in the symmetric phase. Expressing $n_L({\bf z})$ in terms of the Higgs number density [@newmethod2] $$n_L=\frac{9k_q k_{\cal T}-8k_b k_{\cal T}
-5 k_b k_q}{k_{\cal H}(k_b+9 k_q+9 k_{\cal T})}\:{\cal H}$$ and making use of Eqs. (\[higgs1\])-(\[bau\]), we find that $$\frac{n_B}{s}=-g(k_i)\frac{{\cal A}\overline{D}\Gamma_{{\rm ws}}}
{v_{\omega}^2 s},$$ where $s=2\pi^2 g_{*s}T^3/45$ is the entropy density ($g_{*s}$ being the effective number of relativistic degrees of freedom) and $g(k_i)$ is a numerical coefficient depending upon the light degrees of freedom present in the thermal bath.
Fig. 6 shows the value of the phase of the parameter $\mu$ needed to obtain the observed baryon asymmetry, $n_B/s \simeq 4 \times 10^{-11}$, within the approximations given above and using a semirealistic approximation for the Higgs profiles [@CQRVW]. The wall velocity is taken to be $v_{\omega} = 0.1$, while the bubble wall width is taken to be $L_{\omega} = 25/T$. Our results, are, however, quite insensitive to the specific choice of $v_{\omega}$ and $L_{\omega}$. It is interesting to note that realistic values of the baryon asymmetry may only be obtained for values of the CP-violating phases of order one, and for a very specific region of the \[$\mu-M_2$\] plane. Values of the phases lower than 0.1 are only consistent with the observed baryon asymmetry for values of $|\mu|$ of order of the gaugino mass parameters. This is due to a resonant behaviour of the induced Higgsino current for $|\mu| \simeq M_2$ (See Eq. (\[currenth\])).
In conclusion, the requirement of a sufficiently strong first order phase transition and of sizeable CP-violating currents may be only satisfied within a very specific region of parameters within the MSSM. The realization of this scenario will imply very specific signatures which may be tested in future runs of existing experimental facilities. We shall expand on this issue in the next section.
EXPERIMENTAL TESTS OF ELECTROWEAK\
BARYOGENESIS
==================================
In the previous sections, we have shown that the scenario of electroweak baryogenesis favors Higgs masses $m_h \simlt 80$ GeV. Slightly heavier Higgs bosons may be consistent with this scenario only if higher-order (or non-perturbative) effects render the phase transition more strongly first order than what is suggested by one-loop analyses. A hint in this direction was obtained in recent works [@JoseR; @JRBC], where it was shown that, when two-loop corrections are included, the requirement of preservation of the generated baryon asymmetry may be fulfilled, for values of the Higgs masses of order 80 GeV even for $m_U \simeq 0$ (see, for comparison, the results of Fig. 2, for which $m_h \simeq 70$ GeV). An ongoing two-loop analysis for $m_U^2 \simlt 0$ shows an interesting extension of the allowed $m_h$ region [@inprep]. As we discussed above, the phase transition may also become moderately stronger assuming that the physical ground state is metastable. In view of all present studies, it may be concluded that a Higgs mass above 95 GeV will put very strong constraints on the scenario of electroweak baryogenesis within the MSSM. Hence, the most direct experimental way of testing this scenario is through the search for the Higgs boson at LEP2.
At LEP2, the ligthest CP-even neutral Higgs bosons may be produced in association with $Z$ via Higgs-strahlung $$e^+ e^- \rightarrow Z^* \rightarrow Z \; h,
\label{eq:rateh}$$ or in association with the neutral CP-odd Higgs scalar, $$e^+ e^- \rightarrow Z^* \rightarrow h \; A.$$ The associated $Ah$ production becomes increasingly important for rising values of $\tan\beta$. However, for values of the CP-odd Higgs mass above 100 GeV it is kinematically forbidden, and hence, it is not relevant for testing the scenario of electroweak baryogenesis, for which $m_A \simgt 150$ GeV.
The Higgs production rate (\[eq:rateh\]) is equal to the Standard Model one times a projection factor. This projection factor, takes into account the component of the lightest CP-even Higgs on the Higgs which acquires vacuum expectation value (which is the one which couples to the $Z$ in the standard way), $$\sigma_{MSSM}(e^+e^- \rightarrow Z \;h) =
\sigma_{SM}(e^+e^- \rightarrow Z \;h) \times
\sin^2(\beta - \alpha).$$ For large values of the CP-odd Higgs mass, the heavy Higgs doublet decouples and $\sin(\beta - \alpha) \rightarrow 1$. Indeed, the Higgs sector of the theory behaves effectively as the Standard Model one. This transition is achieved rather fast, and for CP-odd Higgs masses above 150 GeV, the cross section differs only slightly from the Standard Model one. Hence, in the limit of interest for this discussion, the mass reach for the lightest CP-even Higgs boson within the MSSM is almost indistinguishable from the one of the Standard Model Higgs.
The search for the Higgs boson is performed by taking into account the dominant decay modes of the Higgs into bottom and $\tau$ pairs. Barring the possibility of supersymmetric decay channels, which only appear in very limited regions of parameter space, which will be directly tested through SUSY particle searches, the Higgs decays approximately 90 $\%$ of the time into $b\bar{b}$ pairs and 8 $\%$ of the time into $\tau
\bar{\tau}$ pairs. The $Z$ boson may decay in jets (70 $\%$), charged leptons (10 $\%$) or neutrinos (20 $\%$).
Based on the experimental simulations [@LEPRep], it is possible to derive the exclusion and discovery limits for the lightest CP-even Higgs mass as a function of the luminosity for the expected LEP2 energy range [^8]. The contours are defined at 5$\sigma$ for the discovery and 95$\%$ C.L. for the exclusion limits. The results of the combination of the four experiments for a center of mass energy of $\sqrt{s} = 192$ GeV are shown in Fig. \[fig:lum\]. In Table \[tab:SM\] we summarize the minimum luminosities which are needed per experiment for exclusion and discovery for the largest Higgs mass values that can be realistically reached at center-of-mass energies of 175, 192 and 205 GeV; beyond these maximum mass values the required luminosities increase sharply for the exclusion and discovery of the Higgs particle.
------------------- -------------- ------------------------ -------------- -------------------------
Exclusion Discovery
$\sqrt{s}$\[GeV\] $m_H$\[GeV\] $L_{min}$\[pb$^{-1}$\] $m_H$\[GeV\] $L_{min}$ \[pb$^{-1}$\]
per experiment per experiment
175 83 75 82 150
192 98 150 95 150
205 112 200 108 300
------------------- -------------- ------------------------ -------------- -------------------------
: [*Maximal Higgs masses that can be excluded or discovered with an integrated luminosity $L_{min}$ per experiment at the three representative energy values of 175, 192 and 205 GeV, if the four LEP experiments are combined.*]{}[]{data-label="tab:SM"}
It is important to compare the above results with the currently expected energy range and luminosity of the LEP2 experiment. LEP2 is expected to run at a center of mass energy of $\sqrt{s} \simeq 184$ GeV during the summer of 1997 and to collect a total integrated luminosity of approximately 100 pb$^{-1}$ per experiment. Taken into account the results of the above analysis, LEP2 at $\sqrt{s} \simeq $ 184 GeV will be able to discover a Standard Model-like Higgs boson with a mass up to approximately 85 GeV. In case of negative searches, this will set a lower bound on the lightest CP-even Higgs mass of order 90 GeV for values of $m_A > 150$ GeV. Clearly, the exact range will finally depend on the real performance of the experiments.
Therefore, if the scenario of electroweak baryogenesis is realized in nature, the 1997 run of the LEP2 experiment has excellent prospects for detecting a Higgs. If, however, no signal is found, this will pose very strong constraints on the present scenario. For instance, in order to preserve a sufficiently strong first order phase transition the model will be driven into a corner of $\tilde{A}_t\ll m_Q $. In addition, the enhancement of the phase transition due to higher order effects will be crucial. It is clear that non-perturbative information, as well as a deeper insight on the question of metastability of the physical vacuum will then be necessary to decide the fate of this scenario.
Moreover, the LEP2 experiment is expected to achieve a final center of mass energy of about $\sqrt{s} \simeq 192$ GeV and to collect a total integrated luminosity $L \simgt 100$ pb$^{-1}$ per year and per experiment. As can be inferred from Table 1, this will lead to a discovery limit of order 95 GeV and an exclusion limit of about 100 GeV. This will definitely test the possibility of baryogenesis at the electroweak scale within the MSSM, since larger values of the Higgs mass are unlikely to be consistent with this scenario.
If the Higgs is found at LEP2, the second test will come from the search for the lightest stop at the Tevatron collider (the stop mass is typically too large for this particle to be seen at LEP). The stop can be pair produced at the Tevatron through gluon processes. It can subsequently decay into bottom and chargino with almost one hundred percent branching ratio, unless the chargino mass is very close or above the stop mass. If this is the case, the stops decays through a loop into charm and neutralino. The signal from the tree level decay can be either a single lepton plus missing energy and b- and light quark-jets, or dilepton plus missing energy and b-jets [^9], $$\begin{aligned}
\tilde{t} \rightarrow b \tilde{\chi}^{\pm} \;\;&
\tilde{\chi}^{\pm} \rightarrow \tilde{\chi}^0_1 \; l^{\pm} \; \nu \;\;&
\tilde{\chi}^{\pm} \rightarrow \tilde{\chi}^0_1 \; qq
\nonumber \\
\tilde{t} \rightarrow b \tilde{\chi}^{\pm} \;\;&
\tilde{\chi}^{\pm} \rightarrow \tilde{\chi}^0_1 \; l^{\pm} \; \nu \;\;&
\tilde{\chi}^{\pm} \rightarrow \tilde{\chi}^0_1 \; l^{\pm} \; \nu .\end{aligned}$$ If the above channel is kinematically forbidden, the stop signal will then be missing energy plus two acollinear jets, $$\tilde{t} \rightarrow c \tilde{\chi}^0_1 .$$ At present, the D0 experiment has analysed only 14 pb$^{-1}$ of the 100 pb$^{-1}$ data in the $\tilde{t} \rightarrow \tilde{\chi}^0_1 c$ channel and they are able to search for stop masses up to about 100 GeV, depending on the values of the neutralino mass considered. Studies about the prospects for stop searches at the Run II of the Tevatron [@TeV2000] (main injector phase at 2 TeV center of mass energy and 2 $fb^{-1}$ of intergrated luminosity ) show a maximal mass reach for stops of about 150 GeV in the $\tilde{t} \rightarrow \tilde{\chi}^{\pm}_1 b$ channel and about 120 GeV in the $\tilde{t} \rightarrow \tilde{\chi}^0_1 c$ channel. Forseen upgrades of the Tevatron achieving a total integrated luminosity of 10/25 $fb^{-1}$ will allow to discover a top squark with mass below the top quark, although optimization in the event selection procedure is necessary, specially in the neutralino-charm decay channel. Hence, already the Run II of the Tevatron to begin in 1999, will start testing an important region of the stop mass range consistent with electroweak baryogenesis. The forseen upgrades, if approved, will provide a crucial test of the framework under analysis.
If both particles are found, the last crucial test will come from $B$ physics. The selected parameter space leads to values of the branching ratio ${\rm BR}(b\rightarrow s\gamma)$ different from the Standard Model case [@bsgasusy]. Although the exact value of this branching ratio depends strongly on the value of $m_A$ and the $\mu$ and $A_t$ parameters, the typical difference with respect to the Standard Model prediction is of the order of the present experimental sensitivity and hence in principle testable in the near future. Indeed, for the typical spectrum considered here, due to the relatively low values of the light charged Higgs mass, the branching ratio ${\rm BR}(b \rightarrow s \gamma)$ is somewhat higher than in the SM case, unless it is properly cancelled by the light stop contributions. Figure 8 shows the dependence of ${\rm BR}(b\rightarrow s \gamma)$ on $A_t$ for $M_2 = \mu = 200$ GeV, $\widetilde{m}_U = \widetilde{m}_U^c$, $\tan\beta = 2$, $m_Q = 500$ GeV and $m_A = 300$ GeV [^10]. The solid line represents the leading order result obtained by setting a renormalization scale $Q = m_b$ in the leading order QCD corrections, where $m_b$ is the bottom mass. The dashed line represents the result obtained by setting a normalization scale $Q = 0.5 \; m_b$, which leads to results in agreement with the most recent next to leading order corrections in the Standard Model [@nolbsga]. It is clear from the figure that negative values of Re($A_t \times \mu$) are favored to get consistency with the present experimental range [@bsgaexp], BR$(b\rightarrow s\gamma)^{\rm
exp} = (2.3 \pm 0.6) \times 10^{-4}$. Since negative values of Re($A_t\times \mu$) imply non-negligible mixing in the stop sector, this rare b-decay imposes very strong constraints on the scenario of electroweak baryogenesis. More information may be obtained by the precise measurement of CP-violating asymmetries at B-factories. Indeed, since the light stop couples via superweak interactions to the bottom sector, the large CP-odd phases associated with this scenario will naturally imply a departure from the Standard Model predictions for these CP-violating asymmetries .
CONCLUDING REMARKS
==================
If the scenario of electroweak baryogenesis is realized in nature, it will demand new physics at scales of order of the weak scale. A light Higgs, with mass at the reach of LEP2 will strongly favor this scenario, while new light scalars with relevant couplings to the Higgs field must also be present. These properties are naturally fulfilled within supersymmetric extensions of the Standard Model. The Higgs is naturally light, while the new scalars are provided by the supersymmetric partners of the top quark. Since the stops are charged and colored particles, their large multiplicity helps in enhancing the strength of the first order phase transition, allowing the preservation of the generated baryon number.
The most relevant new CP-violating sources are associated with the supersymmetric partners of the charged and neutral Higgs and weak gauge bosons. These CP-violating sources, which appear through the Higgsino-gaugino mixing terms, must be of order one in order to have a relevant effect in the generation of the baryon asymmetry. Since sizeable phases in the Higgsino mass parameter could lead to unacceptable values for the electric dipole moment of the neutron, one needs to require that the first and second generation squark masses are of the order of a few TeV, or else, an unnatural cancellation between different contributions must take place.
It is interesting to emphazise that the mechanism of electroweak baryogenesis can be consistent with the general framework of unification of couplings. In fact, performing a detailed renormalization group analysis, one can match the specific hierarchy of soft supersymmetry breaking terms at low energies required by the electroweak baryogenesis scenario together with large, perturbative values of the top Yukawa coupling, as those associated with the unification of bottom-tau Yukawa couplings or the top quark infrared fixed point structure. A recent study [@CCOPW] shows that, depending on the scale at which supersymmetry breakdown is transmitted to the observable sector, the above implies very specific constraints on the stop and Higgs mass parameters of the theory at high energies.
Most important, the electroweak baryogenesis explanation can be explicitly tested at present and near future experiments. In the last phase of LEP2, a center of mass energy $\sqrt{s}
\simeq 192$ GeV will be achieved, with a total integrated luminosity of about 100$-$150 pb$^{-1}$ per year and per experiment. A lightest CP-even Higgs, with mass of about 100 GeV is expected to be detectable (or otherwise excluded) providing a definite test of the scenario of EWB within the MSSM. The CP-odd mass within this framework must be sufficiently large in order to avoid weakening the first order phase transition, and it must be sufficiently small to avoid the suppression of the new CP-violating sources. Altogether this implies that the lightest Higgs should be quite Standard model like. If the Higgs is found, the next test of this scenario will come from stop searches at a high luminosity Tevatron facility. Moreover, although this is not required, the charginos and neutralinos might be light, at the reach of LEP2.
Another potential experimental test of this model comes from the rate of flavor changing neutral current processes, like $b \rightarrow s \gamma$. We have shown that, if the CP-odd Higgs mass is not sufficiently large, this rate will be in general above the Standard Model predictions, unless Re$(A_t \times\mu) \simlt 0$. Hence, rare processes put additional constraints on the allowed parameter space. Moreover, light stops and light charginos, with additional CP-violating phases associated with the $\mu$ and $A_t$ parameters, will have a relevant impact on B-physics, which may be testable at B-factories [@bfac].
In summary, the realization of electroweak baryogenesis will not only provide the answer to one of the most interesting open questions of particle physics, but it will imply a rich phenomenology at present and near future colliders.\
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank J.R. Espinosa, A. Riotto, I. Vilja and, in particular, M. Quiros for enjoyable and fruitful collaborations related to this subject. We would also like to thank P. Chankowski, J. Cline, H. Haber, K. Kainulainen, G. Kane, M. Laine, A. Nelson, M. Pietroni, S. Pokorski and M. Shaposhnikov for many pleasant and interesting discussions.
References {#references .unnumbered}
==========
[99]{} G. t’Hooft, ; . H. Dreiner and G. Ross, . A.D. Sakharov, . For recent reviews, see: A.G. Cohen, D.B. Kaplan and A.E. Nelson, [*Annu. Rev. Nucl. Part. Sci.*]{} [**43**]{} (1993) 27; M. Quirós, ; V.A. Rubakov and M.E. Shaposhnikov, Phys. Usp. 39 (1996) 461 \[hep-ph/9603208\]. P. Arnold and L. McLerran, ; and [**D37**]{} (1988) 1020; S.Yu Khlebnikov and M.E. Shaposhnikov, ; F.R. Klinkhamer and N.S. Manton, ; B. Kastening, R.D. Peccei and X. Zhang, ; L. Carson, Xu Li, L. McLerran and R.-T. Wang, ; M. Dine, P. Huet and R. Singleton Jr., . M. Shaposhnikov, [*JETP Lett.*]{} 44 (1986) 465; and [**B299**]{} (1988) 797. M.E. Carrington, ; M. Dine, R.G. Leigh, P. Huet, A. Linde and D. Linde, ; ; P. Arnold, ; J.R. Espinosa, M. Quirós and F. Zwirner, ; W. Buchmüller, Z. Fodor, T. Helbig and D. Walliser, . J. Bagnasco and M. Dine, ; P. Arnold and O. Espinosa, ; Z. Fodor and A. Hebecker, . W. Buchmüller, Z. Fodor and A. Hebecker, ; W. Buchmüller, preprint DESY 96-216 \[hep-ph/9610335\] K. Kajantie, K. Rummukainen and M.E. Shaposhnikov, ; Z. Fodor, J. Hein, K. Jansen, A. Jaster and I. Montvay, ; K. Kajantie, M. Laine, K. Rummukainen and M.E. Shaposhnikov, K. Jansen, Nucl. Phys. (Proc. Supl.) [**B47**]{} (1996) 196. G.R. Farrar and M.E. Shaposhnikov, , ([**E**]{}): [**71**]{} (1993) 210 and . M.B. Gavela. P. Hernández, J. Orloff, O. Pène and C. Quimbay, ; ; P. Huet and E. Sather, . N. Cabibbo, L. Maiani, G. Parisi and R. Petronzio, ; M. Lindner, ; M. Lindner, M. Sher and H.W. Zaglauer, ; M. Sher , ; C. Ford, D.R.T. Jones, P.W. Stephenson and M.B. Einhorn, G. Altarelli and I. Isidori, . J.A. Casas, J.R. Espinosa and M. Quiros, ; . CDF Collaboration, F. Abe [it et al.]{}, ; D0 Collaboration, S. Abachi [*et al.*]{} ; P. Tipton, presented at the XXVIII International Conference on High-Energy Physics, Warsaw, Poland, July 1996. G.F. Giudice, ; S. Myint, . J.R. Espinosa, M. Quirós and F. Zwirner, . A. Brignole, J.R. Espinosa, M. Quirós and F. Zwirner, . M. Carena, M. Quirós and C.E.M. Wagner, . J.R. Espinosa, . M. Laine, ; J. M. Cline, K. Kainulainen, ; G.R. Farrar and M. Losada, preprint RU-96-26 \[hep-ph/9612346\]. D. Delepine, J.M. Gerard, R. Gonzalez Felipe and J. Weyers, . J.M. Moreno, D.H. Oaknin and M. Quirós, to appear in [*Nucl. Phys.*]{} [**B**]{} \[hep-ph/9605387\] and preprint IEM-FT-146/96 \[hep-ph/9612212\]. M. Worah, preprint SLAC-PUB-7417, \[hep-ph/9702423\] L. Dolan and R. Jackiw, . M. Carena, P. Zerwas and the Higgs Physics Working Group, in Vol. 1 of Physics at LEP2, G. Altarelli, T. Sjöstrand and F. Zwirner, eds., Report CERN 96-01, Geneva (1996). G. Cowan, for the ALEPH collaboration, CERN seminar, Feb. 25th, 1997. http://alephwww.cern.ch/ALPUB/seminar/Cowan-172-jam/cowan.html See, e.g., H. P. Nilles, [*Phys. Rep.*]{} [**110**]{} (1984) 1; H.E. Haber and G.L. Kane, [*Phys. Rep.*]{} [**117**]{}(1985) 75. See, e.g., J.F. Gunion, H.E. Haber, G.L. Kane and S.Dawson, [*The Higgs Hunter’s Guide*]{}, Addison-Wesley 1990. Y. Okada, M. Yamaguchi and T. Yanaguida, ; ; J. Ellis, G. Ridolfi and F. Zwirner, ; K. Sasaki, M. Carena and C.E.M. Wagner, ; H.E. Haber and R. Hempfling, ; R. Barbieri, M. Frigeni and F. Caravaglios, ; H.E. Haber and R. Hempfling, . M. Carena, J.R. Espinosa, M. Quirós and C.E.M. Wagner, ; M. Carena, M. Quirós and C.E.M. Wagner, ; H.E. Haber, R. Hempfling and A.H. Hoang, CERN-TH/95-216, preprint \[hep-ph/9609331\]. G.W. Anderson and L.J. Hall, . D. Comelli and J.R. Espinosa, preprint DESY 96-114, FTUV/96-37, IFIC/96-45, IEM-FT-134/96 \[hep-ph/9606438\]. A.G. Cohen, D.B. Kaplan, and A.E. Nelson, . D. Comelli and M. Pietroni, [*Phys. Lett.*]{} [**B306**]{} (1993) 67; D. Comelli and M. Pietroni and A. Riotto, [*Nucl. Phys.*]{} [**B412**]{} (1994) 441; [*Phys. Rev.*]{} [**D50**]{} (1994) 7703; [*Phys. Lett.*]{} [**343**]{} (1995) 207; J.R. Espinosa, J.M. Moreno, M. Quirós, [*Phys. Lett.*]{} [**B319**]{} (1993) 505. V.A. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov, [*Phys. Lett.*]{} [**B155**]{} (1985) 36. M. Carena, M. Quiros, A. Riotto, I. Vilja and C.E.M. Wagner, CERN preprint CERN-TH/96-242, \[hep-ph/9702409\]. L. Mc Lerran [*et al.*]{}, [*Phys. Lett.*]{} [**B256**]{} (1991) 451; M. Dine, P. Huet and R. Singleton Jr., [*Nucl. Phys.*]{} [**B375**]{} (1992) 625; M. Dine [*et al.*]{}, [*Phys. Lett.*]{} [**B257**]{} (1991) 351; A.G. Cohen and A.E. Nelson, [*Phys. Lett.*]{} [**B297**]{} (1992) 111. D. Comelli, M. Pietroni and A. Riotto, [*Phys. Lett.*]{} [**B354**]{} (1995) 91 and [*Phys. Rev.*]{} [**D53**]{} (1996) 4668. A.G. Cohen, D.B. Kaplan, and A.E. Nelson, [*Phys. Lett.*]{} [**336**]{} (1994) 41. M. Joyce, T. Prokopec and N. Turok, [*Phys. Rev. Lett.*]{} [**75**]{} (1995) 1695, (E): [*ibidem*]{} 3375; D. Comelli, M. Pietroni and A. Riotto, [*Astropart. Phys.*]{} [**4**]{} (1995) 71. See, for instance: K. Chou, Z. Su, B. Hao and L. Yu, [*Phys. Rep.*]{} [**118**]{} (1985) 1 and references therein. N.P. Landsmann and Ch.G. van Weert, [*Phys. Rep.*]{} [**145**]{} (1987) 141. A. Riotto, [*Phys. Rev.*]{} [**D53**]{} (1996) 5834. P. Huet and A.E. Nelson, ; . H.A. Weldon, [*Phys. Rev.*]{} [**D26**]{} (1982) 1394; V.V. Klimov, [*Sov. Phys. JETP*]{} [**55**]{} (1982) 199. P.A. Henning, [*Phys. Rep.*]{} [**253**]{} (1995) 235. J. Ambjorn and A. Krasnitz, . P. Arnold, D. Son and L.G. Yaffe, preprint UW-PT-96-19 \[hep-ph/9609481\]. B. de Carlos and J.-R. Espinosa, preprint SUSX-TH-97-005 \[hep-ph/9703212\]. M. Carena, M. Quiros, A. Riotto and C.E.M. Wagner, in preparation. M. Carena, P. Chankowski, M. Olechowski, S. Pokorski and C.E.M. Wagner, preprint CERN-TH-96-241, \[hep-ph/9612261\], to appear in Nucl. Phys. [**B**]{}. L. Alvarez Gaumé, J. Polchinski and M.B. Wise, ; J. Bagger, S. Dimopoulos and E. Masso, ; M. Carena, T.E. Clark, C.E.M. Wagner, W.A. Bardeen and K. Sasaki, ; N. Polonsky, . Report of the TeV2000 study Group, FERMILAB-PUB-96/082. S. Bertolini, F. Borzumati, A. Masiero and G. Ridolfi, ; R. Barbieri and G. Giudice, . C. Greub, T. Hurth and D. Wyler, ; ; K. Chetyrkin, M. Misiak and M. Münz, \[hep-ph/9612313\]; K. Adel and Y.P. Yao, ; C. Greub and T. Hurth \[hep-ph/9703349\]. M.S. Alam et al. (CLEO Collaboration), . See, for example, A. Cohen, D.B. Kaplan, F. Lepeintre and A.E. Nelson, .
[^1]: To appear in [*Perspectives on Higgs Physics II*]{}, ed. G.L. Kane, World Scientific, Singapore
[^2]: An interesting scenario, relying only on the Standard Model phases was recently suggested [@Worah]. However, since it requires a large mixing between the second and third generation squarks, an analysis of the strength of the first order phase transition will be necessary to decide about its viability.
[^3]: The two loop corrections to the Higgs mass, in the limit of $m_Q^2 \gg m_U^2$ are also available [@CEQW]
[^4]: Higher loop corrections are important, making values of $m_U \simgt 0$ possible [@JoseR]
[^5]: Strictly speaking, we are interested in the behaviour of the potential at $T=T_c$. Given the small quantitative difference between $T_0$ and $T_c$, we shall identify both temperatures in the following discussion
[^6]: We display here only the dominant contribution to the current [@CQRVW]
[^7]: The value of $\kappa$ is still subject of debate [@SphalRate; @ASY]
[^8]: As explained above, we are considering scenarios with relatively heavy CP-odd Higgs bosons, for which the limits for the lightest CP-even Higgs coincide with the ones for the SM Higgs
[^9]: we are only considering the case of exact R-Parity conservation
[^10]: We have ignored the effect of the CP-violating phases for these computations
|
---
abstract: 'The phenomenon of spontaneous symmetry breaking is investigated in the dynamic thermalization of a degenerate quantum system. A three-level system interacting with a heat bath is carefully studied to this end. It is shown that the three-level system with degenerate ground states might have different behaviors depending on the details of the interaction with the heat bath when the temperature approaches zero. If we introduce an external field to break the degeneracy of the ground states and let it approach zero after letting the temperature approach zero, then two possibilities will arise: the steady state is a definite one of the degenerate states independent of the initial state, or the steady state is dependent on the initial state in a complicated way. The first possibility corresponds to a spontaneous symmetry breaking of the system and the second one implies that the heat bath could not totally erase the initial information in certain cases.'
author:
- 'Jie-Qiao Liao'
- 'H. Dong'
- 'X. G. Wang'
- 'X. F. Liu'
- 'C. P. Sun'
title: 'Spontaneous symmetry breaking in thermalization and anti-thermalization'
---
*Introduction.—*Conventionally, thermalization is understood as a dynamic process in which an open quantum system (OQS) approaches an equilibrium state with the same temperature $T$ as that of its heat bath [@Breuer]. According to the third law of thermodynamics (the generic version), the entropy of a non-degenerate system will vanish at the absolute zero temperature. That is to say, at the absolute zero temperature, the system will reach a steady pure state. Is it still the case when the system has degenerate ground states? This turns out to be a subtle problem related to the process of thermalization. In quantum physics, thermalization seems to be a much more complex concept than in classical physics. In fact, most recently a new kind of thermalization, called canonical thermalization [@Popescu; @Goldstein; @Gemmer; @Dong], has been proposed.
At zero temperature a system with degenerate ground states might have a finite entropy depending on the degree of degeneracy $d$: $S=k_{B}\ln d$ [@kersonhuang]. Roughly speaking, this can be understood by studying the following two non-commutative limit processes: letting the perturbation introduced to break the degeneracy approach zero and letting the temperature of the heat bath approach zero. Actually, in the thermal equilibrium case (the system is thermalized to reach a thermal equilibrium state), if we first take the second limit and then take the first limit, the OQS will reach a definite pure state and thus have a vanishing entropy. This is a phenomenon of spontaneous symmetry breaking (SSB) [@kersonhuang; @ssb]. On the other hand, if we reverse the order of these two limit processes, the OQS will reach a maximally mixing state of all the degenerate ground states and thus possess a non-vanishing entropy.
![(Color online). The schematic illustration of spontaneous symmetry breaking (SSB) in thermalization. (a) The heat bath usually does not single out a particular state from the degenerate ground states at zero temperature; (b) Breaking the symmetry of the potential will cause a preference of the system to a particular ground state at zero temperature; (c) After the symmetry is recovered at zero temperature, the system will remain in the preferred state.[]{data-label="ssb"}](ssb.eps){width="6"}
Unfortunately, the above discussion about the thermalization of a system with degenerate ground states proves to be overly simplified. In the study of the dynamic thermalization of a simple system, we find that the above mentioned SSB in the thermalization can only happen when bath induced transition is not forbidden by some selection rule. If there is a selection rule to forbid the bath induced transition between the degenerate ground states, the steady thermalized state will depend on the initial state and thus the SSB will not appear. In this case, the OQS enjoys the so called anti-thermalization effect: some information of the initial state is kept after the OQS is thermalized to a steady state.
We will study a three-level system interacting with a heat bath. The two lower (or higher) energy states $|g_{1}\rangle$ and $|g_{2}\rangle$ of this three-level system are degenerate and can be split by applying an external field. The heat bath is modeled as the bath of harmonic oscillators. The dynamic process of the system’s approaching the steady state at zero temperature will be carefully analyzed from the master equation approach. We will prove that if there exists a non-vanishing coupling to the bath for arbitrary two energy levels of the three-level system, then the third law of thermodynamics is valid thanks to the SSB. On the other hand, for the conventional $\Lambda$- and V-type atoms, we will reveal the exotic anti-thermolization effect. This effect happens as a result of the absence of the bath coupling induced quantum transition between $|g_{1}\rangle$ and $%
|g_{2}\rangle$ and the occurrence of the quantum interference between the transition to $|g_{1}\rangle$ and the transition from $|g_{2}\rangle$. At zero temperature, these conclusions coincide with those reached in the context of the spontaneous emission of $V$-type atom in vacuum [@Agarwal].
*SSB in thermalization.—*In general, the Hamiltonian $%
\hat{H}_{S}$ of an OQS to be thermalized can be written as $\hat{H}%
_{S}=\sum_{n,\alpha }E_{n}|n,\alpha \rangle \langle n,\alpha |$, where $%
|n,\alpha \rangle $ $(\alpha =1,2,\cdots,d_{n})$ are degenerate states correspond to the same eigenvalue $E_{n}$ $(n=1,2,3,\cdots)$ and $d_n$ is the degree of degeneracy. Let an external field be applied to break the energy level degeneracy as $E_{n}\rightarrow E_{n}+\Delta _{\alpha }$ and then let this degeneracy split system contact with a heat bath of temperature $T$ for a time longer than the conventional relaxation time. Then the system is supposed to be thermalized to the thermal equilibrium state $\hat{\rho}_{S}\left(\Delta _{\alpha },\beta \right)=\sum_{n,\alpha }%
\exp[-\beta \left( E_{n}+\Delta _{\alpha }\right)]Z^{-1}|n,\alpha \rangle
\langle n,\alpha |$, where $\beta =1/(k_{B}T)$ is the inverse temperature (hereafter, we set $k_{B}=1,\hbar =1$) and $Z=\mathtt{Tr}[\exp (-\beta \hat{H%
}_{S})]$ is the partition function of the OQS.
We observe that for $\hat{\rho}_{S}\left( \Delta _{\alpha },\beta \right) $, there exist the following two limit processes: $$\begin{aligned}
\lim_{\Delta _{\alpha }\rightarrow 0}\lim_{\beta \rightarrow +\infty }\hat{%
\rho}_{S}\left( \Delta _{\alpha },\beta \right) & =|1,1\rangle \langle 1,1|,
\\
\lim_{\beta \rightarrow +\infty }\lim_{\Delta _{\alpha }\rightarrow 0}\hat{%
\rho}_{S}\left( \Delta _{\alpha },\beta \right) & =\frac{1}{d_{1}}%
\sum_{\alpha =1}^{d_{1}}|1,\alpha \rangle \langle 1,\alpha |,\end{aligned}$$ where $|1,1\rangle $ denotes the ground state with vanishing energy in the presence of the external field. Note that taking the two limits in different orders leads to completely different results, the former being a reflection of the SSB phenomenon.
We would like to remark that though both of the two results are correct in the mathematical sense, the former is physically more acceptable than the latter, which is in accordance with the generic version of the third law of thermodynamics. Indeed, in view of the existence of the perturbation breaking the degeneracy, it seems reasonable to let the temperature approach zero first in the calculation. However, things are not so simple. In fact, due to quantum interference effect, even if we let the temperature approach zero first in the calculation, the happening of SSB is not unconditional. This is the main conclusion of this letter.
![(Color online). (a) The schematic diagram of a three-level system immersed in heat bath, which consists of a set of harmonic oscillators. (b) The energy level diagram of the three-level system, arbitrary two energy levels of which couples with the heat bath.[]{data-label="schematicdiagram"}](schematicdiagram.eps){width="6"}
*Dynamic thermalization of a three-level system.—*Generally, the process of dynamic thermalization begins from a factorized initial state $\hat{\rho}(0)=\hat{\rho}_{S}(0)\otimes \hat{\rho}_{B}(\beta )$, where $\hat{\rho}_{B}(\beta )$ is the thermal state of the heat bath. Let the heat bath be modeled as the harmonic oscillator system with the Hamiltonian $\hat{H}_{B}=\sum_{j}\omega _{j}\hat{a}_{j}^{\dag }\hat{a}_{j}$. Then we have $\hat{\rho}_{B}(\beta )=\exp(-\beta \hat{H}_{B})/Z_{B}$ where $%
Z_{B}=\mathtt{Tr}[\exp (-\beta \hat{H}_{B})]$ is the partition function. The time evolution of the total system driven by the coupling $\hat{H}_{I}$ between the system and the heat bath is determined by $\hat{U}(t)=\exp [-i(%
\hat{H} _{S}+\hat{H}_{B}+\hat{H}_{I})]$. The steady state of the system, as $%
t\rightarrow \infty$ at $T=0$, can then be obtained by calculating the reduced density matrix $\hat{\rho}_{S}(t,\beta )=\mathtt{Tr}_{B}[\hat{U}(t)
\hat{\rho }_{S}(0)\otimes \hat{\rho}_{B}\hat{U}^{\dag }(t)]$ of the OQS ($%
\mathtt{Tr} _{B}$ stands for tracing over the heat bath).
To be specific, let us study the dynamic thermalization of a simple three-level system. The Hamiltonian $\hat{H}_{S}=\omega _{2}\hat{\sigma}%
_{ee}+\Delta \hat{\sigma}_{g_{1}g_{1}}$ of the three-level system (as illustrated in Fig. \[schematicdiagram\]) is written in terms of the flip operators $\hat{\sigma}_{\alpha \beta }=|\alpha \rangle \langle \beta |$ ($%
\alpha ,\beta =e,g_{1},g_{2}$), where $\omega_{l}=\omega_{e}-\omega_{g_{l}}$ ($l=1,2$) and $\Delta =\omega _{2}-\omega _{1}$. Here, we choose the eigen-energy of the state $|g_{2}\rangle$ as the energy zero point. The interaction Hamiltonian of the three-level system with its heat bath reads $$\begin{aligned}
\hat{H}_{I}=\sum_{l=1,2}\hat{\sigma}_{eg_{l}}\hat{B}_{l}+\hat{\sigma}
_{g_{1}g_{2}}\hat{B}_{3}+h.c.,\end{aligned}$$ where $\hat{B}_{l}=\sum_{j}\eta _{l}(\omega _{j})\hat{a}_{j}$ $(l=1,2,3)$. For simplicity, we assume $\eta _{l}(\omega_{j})$ to be real below.
Under the Born-Markov approximation, the evolution of the reduced density matrix of the three-level system is governed by the master equation, $$\dot{\hat{\rho}}_{S}=-i[\hat{\rho}_{S},\Delta \hat{\sigma}
_{g_{2}g_{2}}]+\sum_{l=1}^{3}\mathcal{L}_{l}[\hat{\rho}_{S}]+\mathcal{L}%
_{X}[ \hat{\rho}_{S}], \label{mastereqaution}$$ where
$$\begin{aligned}
\mathcal{L}_{l=1,2}[\hat{\rho}_{S}]&=&\frac{\gamma_{l}}{2}\left(\bar{n}(\omega_{l})+1\right)
\left(2\hat{\sigma}_{g_{l}e}\hat{\rho}_{S}\hat{\sigma}_{eg_{l}}-\hat{\sigma}_{ee}\hat{\rho}_{S}-\hat{\rho}_{S}\hat{\sigma}_{ee}\right)
+\frac{\gamma_{l}}{2}\bar{n}(\omega_{l})
\left(2\hat{\sigma}_{eg_{l}}\hat{\rho}_{S}\hat{\sigma}_{g_{l}e}-\hat{\sigma}_{g_{l}g_{l}}\hat{\rho}_{S}-\hat{\rho}_{S}\hat{\sigma}_{g_{l}g_{l}}\right),\nonumber \\
\mathcal{L}_{3}[\hat{\rho}_{S}]&=&\frac{\gamma_{3}}{2}\left(\bar{n}(\Delta)+1\right)
\left(2\hat{\sigma}_{g_{2}g_{1}}\hat{\rho}_{S}\hat{\sigma}_{g_{1}g_{2}}-\hat{\sigma}_{g_{1}g_{1}}\hat{\rho}_{S}-\hat{\rho}_{S}\hat{\sigma}_{g_{1}g_{1}}\right)
+\frac{\gamma_{3}}{2}\bar{n}(\Delta)
\left(2\hat{\sigma}_{g_{1}g_{2}}\hat{\rho}_{S}\hat{\sigma}_{g_{2}g_{1}}-\hat{\sigma}_{g_{2}g_{2}}\hat{\rho}_{S}-\hat{\rho}_{S}\hat{\sigma}_{g_{2}g_{2}}\right),\nonumber \\
\mathcal{L}_{X}[\hat{\rho}_{S}]&=&\left[\frac{\gamma_{12}}{2}\left(\bar{n}(\omega_{1})+1\right)+\frac{\gamma_{21}}{2}\left(\bar{n}(\omega_{2})+1\right)\right]
\left(\hat{\sigma}_{g_{1}e}\hat{\rho}_{S}\hat{\sigma}_{eg_{2}}+\hat{\sigma}_{g_{2}e}\hat{\rho}_{S}\hat{\sigma}_{eg_{1}}\right)+\frac{\gamma_{12}}{2}\bar{n}(\omega_{1})
\left(\hat{\sigma}_{eg_{1}}\hat{\rho}_{S}\hat{\sigma}_{g_{2}e}+\hat{\sigma}_{eg_{2}}\hat{\rho}_{S}\hat{\sigma}_{g_{1}e}\right.\nonumber \\
&&
\left.-\hat{\sigma}_{g_{2}g_{1}}\hat{\rho}_{S}-\hat{\rho}_{S}\hat{\sigma}_{g_{1}g_{2}}\right)+\frac{\gamma_{21}}{2}\bar{n}(\omega_{2})
\left(\hat{\sigma}_{eg_{1}}\hat{\rho}_{S}\hat{\sigma}_{g_{2}e}+\hat{\sigma}_{eg_{2}}\hat{\rho}_{S}\hat{\sigma}_{g_{1}e}
-\hat{\sigma}_{g_{1}g_{2}}\hat{\rho}_{S}-\hat{\rho}_{S}\hat{\sigma}_{g_{2}g_{1}}\right).\end{aligned}$$
Here, the decay rates $\gamma _{l}=2\pi \varrho (\omega _{l})|\eta
_{l}(\omega _{l})|^{2}$ and $\gamma _{lm}=2\pi \varrho (\omega _{l})\eta
_{l}(\omega _{l})\eta _{m}(\omega _{l}),$ for $\omega _{3}=\Delta $ and $%
l\neq m,m=1,2,$ depend on the mode density $\varrho (\omega )$ of the heat bath; $\bar{n}(\omega )=1/[\exp (\beta \omega )-1]$ is the thermal average excitation number for the boson mode of frequency $\omega $ at temperature $T
$. Note that in the master equation (\[mastereqaution\]), we have neglected the Lamb shifts.
The evolution of the density matrix elements governed by the master equation (\[mastereqaution\]) can be described with the optical Bloch equation $\dot{\mathbf{X}}=\mathbf{MX}$, where the state vector $\mathbf{%
X=X(t)}$ and the coefficient matrix $\mathbf{M}$ are respectively defined as $\mathbf{X}=\mathbf{X}_{R}\oplus \mathbf{X}_{S}$ and $\mathbf{M}=\mathbf{R}%
\oplus \mathbf{S}$, with $\mathbf{X}_{S}=(\langle \hat{\sigma}%
_{eg_{1}}\rangle ,\langle \hat{\sigma}_{eg_{2}}\rangle )^{T}$ and
$$\begin{aligned}
\label{mmatrix}
\textbf{X}_{R}&=&((\bar{n}(\omega _{1})+1)\langle \hat{\sigma}
_{ee}\rangle -\bar{n}(\omega _{1})\langle
\hat{\sigma}_{g_{1}g_{1}}\rangle ,(\bar{n}(\omega _{2})+1)\langle
\hat{\sigma}_{ee}\rangle -\bar{n}(\omega _{2})\langle
\hat{\sigma}_{g_{2}g_{2}}\rangle ,(\bar{n}(\Delta )+1)\langle
\hat{\sigma}_{g_{1}g_{1}}\rangle -\bar{n}(\Delta )\langle
\hat{\sigma} _{g_{2}g_{2}}\rangle ,\mathtt{Re}[C], \mathtt{Im}[C])^{T},\nonumber\\
\textbf{R}&=&\left(
\begin{array}{ccccc}
-\gamma_{1}(2\bar{n}(\omega_{1})+1) & -\gamma_{2}(\bar{n}(\omega_{1})+1) & \gamma_{3}\bar{n}(\omega_{1}) &
\gamma_{12}\bar{n}(\omega_{1})(\bar{n}(\omega_{1})+1)+\gamma_{21}\bar{n}(\omega_{2})(2\bar{n}(\omega_{1})+1)
& 0 \\
-\gamma_{1}(\bar{n}(\omega_{2})+1) & -\gamma_{2}(2\bar{n}(\omega_{2})+1) & -\gamma_{3}\bar{n}(\omega_{2}) &
\gamma_{12}\bar{n}(\omega_{1})(2\bar{n}(\omega_{2})+1)+\gamma_{21}\bar{n}(\omega_{2})(\bar{n}(\omega_{2})+1) & 0 \\
\gamma_{1}(\bar{n}(\Delta)+1) & -\gamma_{2}\bar{n}(\Delta) & -\gamma_{3}(2\bar{n}(\Delta)+1) &
\gamma_{12}\bar{n}(\Delta)\bar{n}(\omega_{1})-\gamma_{21}(\bar{n}(\Delta)+1)\bar{n}(\omega_{2}) & 0 \\
\frac{\gamma_{12}}{2} & \frac{\gamma_{21}}{2} & 0 & R_{44} & \Delta \\
0 & 0 & 0 & -\Delta &R_{55} \\
\end{array}
\right),\nonumber\\
\textbf{S}&=&\left(
\begin{array}{cc}
-\frac{1}{2}[\gamma_{1}(2\bar{n}(\omega_{1})+1)+\gamma_{2}(\bar{n}(\omega_{2})+1)
+\gamma_{3}(\bar{n}(\Delta)+1)] & -\frac{\gamma_{21}}{2}\bar{n}(\omega_{2}) \\
i\Delta-\frac{1}{2}[\gamma_{2}(2\bar{n}(\omega_{2})+1)+\gamma_{1}(\bar{n}(\omega_{1})+1)
+\gamma_{3}\bar{n}(\Delta)] & -\frac{\gamma_{12}}{2}\bar{n}(\omega_{1}) \\
\end{array}
\right),\end{aligned}$$
where $C=\langle \hat{\sigma}_{g_{2}g_{1}}\rangle $ and $%
R_{44}=R_{55}=-[\gamma _{1}\bar{n}(\omega _{1})+\gamma _{2}\bar{n}(\omega
_{2})+\gamma _{3}(2\bar{n}(\Delta )+1)]/2$. We can prove $\mathbf{M}$ to be negative-definite or have vanishing determinant. Thus the optical Bloch equation can possess steady state solutions.
We first consider the thermalization of the three-level system at finite temperature $T\neq 0$. In this case, $\det (\mathbf{M})\neq 0$, thus the steady state solution of the optical Bloch equation is $\mathbf{X}=0$, or $$\begin{aligned}
\frac{\langle \hat{\sigma}_{ee}\rangle _{ss}}{\langle \hat{\sigma}
_{g_{l}g_{l}}\rangle _{ss}}& =e^{-\beta \omega _{l}},\hspace{0.5cm}\frac{
\langle \hat{\sigma}_{g_{1}g_{1}}\rangle _{ss}}{\langle \hat{\sigma}
_{g_{2}g_{2}}\rangle _{ss}}=e^{-\beta \Delta }, \notag \\
\langle \hat{\sigma}_{g_{2}g_{1}}\rangle _{ss}& =\langle \hat{\sigma}
_{eg_{1}}\rangle _{ss}=\langle \hat{\sigma}_{eg_{2}}\rangle _{ss}=0
\label{Boltzmannrelation}\end{aligned}$$ for $l=1,2$. Here $\langle \hat{A}\rangle _{ss}=\mathtt{Tr}_{S}[\hat{A}\hat{%
\rho}_{S}(\infty )]$, $\hat{A}$ being the operator concerned. Considering the normalization condition $\langle \hat{\sigma}_{g_{1}g_{1}}\rangle
_{ss}+\langle \hat{\sigma }_{g_{2}g_{2}}\rangle _{ss}+\langle \hat{\sigma}%
_{ee}\rangle _{ss}=1$, from Eq. (\[Boltzmannrelation\]) we then have $$\langle \hat{\sigma}_{g_{l}g_{l}}\rangle _{ss}=\frac{e^{\beta \omega _{l}}}{
1+e^{\beta \omega _{1}}+e^{\beta \omega _{2}}},$$ for $l=1,2$.
Next, we consider the case with vanishing bath temperature. In this case, we have $\bar{n}(\omega _{1})=\bar{n}(\omega _{2})=\bar{n}(\Delta )=0$ and thus $\det (\mathbf{M})=0$. We thus cannot obtain the steady state solution of the optical Bloch equation by simply setting $\dot{\mathbf{X}}=\mathbf{0}$ as the resulted equation $\mathbf{0}=\mathbf{MX}$ would not have a unique solution. Instead, we have to turn to obtain the transient solution first and then consider the long time behavior. One can expect that it cannot be determined without regard to the details of the interaction or the initial state. Indeed, complexity will arise here. Let us focus on the case with $%
\gamma _{3}\neq 0$ in this section, and leave the case with $\gamma _{3}= 0$ to the next section.
When $\gamma _{3}\neq 0$, namely, there exists a bath induced coupling between the states $|g_{1}\rangle $ and $|g_{2}\rangle $, the transient solution, which is not presented here for the technicalities, results in $%
\langle \hat{\sigma}_{ee}\rangle _{ss}=\langle \hat{\sigma}
_{g_{1}g_{1}}\rangle _{ss}=0$, and $\langle \hat{\sigma}_{g_{2}g_{2}}\rangle
_{ss}=1$. This implies that whether the three-level system has degenerate ground states or not, the steady state of the master equation ([mastereqaution]{}) is just the thermal equilibrium state ($\propto \exp
(-\beta \hat{H}_{S})$) of the OQS even at zero temperature. This conforms to the conventional idea. However, when $\gamma _{3}= 0$, in the next section we will see an exotic nature of thermalization as $T\rightarrow 0$.
Now it is easily seen that taking the limits in different orders leads to the following different results: $\lim_{\Delta\rightarrow 0}\lim_{\beta
\rightarrow +\infty }\langle \hat{\sigma}_{g_{1}g_{1}}\rangle
_{ss}=0,\lim_{\Delta \rightarrow 0}\lim_{\beta \rightarrow +\infty }\langle
\hat{\sigma}_{g_{2}g_{2}}\rangle _{ss}=1$, and $\lim_{\beta \rightarrow
+\infty }\lim_{\Delta \rightarrow 0}\langle \hat{\sigma}_{g_{1}g_{1}}\rangle
_{ss}=\lim_{\beta \rightarrow +\infty }\lim_{\Delta \rightarrow 0}\langle
\hat{\sigma}_{g_{2}g_{2}}\rangle _{ss}=1/2 $. Following the first procedure, we will reach the conclusion that the final steady state is the pure state $%
|g_{2}\rangle $ and that there exists SSB effect in the thermalization . But if we adopt the second procedure, we should conclude that the final steady state is the maximally mixing state $(|g_{1}\rangle\langle
g_{1}|+|g_{2}\rangle \langle g_{2}|)/2$. By the way we remark that the SSB can also be seen from the von Neumann entropy $S\equiv -\mathtt{Tr}_{S}\left[%
\hat{\rho}_{S}(\infty) \ln \hat{\rho}_{S}(\infty)\right]$ of the steady state $\hat{\rho}_{S}(\infty)$ of the three-level system. In Fig. [entropy]{}, we plot the von Neumann entropy $S$ as a function of $\Delta $ and $T$. In the figure the character of double values of $S$ at the point $%
(\Delta ,T)=(0,0)$ is clearly illustrated: along the route ($T=0$, $\Delta
\rightarrow 0$), the von Neumann entropy $S\rightarrow 0$ while along the route ($\Delta =0,T\rightarrow 0$) the von Neumann entropy $S\rightarrow 1$.
![(Color online).The von Neumann entropy of the steady state density matrix for the three-level system is plotted versus the temperature $T$ and the energy difference $\Delta $ between the states $|g_{1}\rangle $ and $%
|g_{2}\rangle $. The two red arrows indicate its multi-value feature as both $T$ and $\Delta$ approach zero[]{data-label="entropy"}](entropy.eps){width="7"}
*Anti-thermalization by quantum interference.—*We have shown that when $\gamma_3\neq 0$, in the dynamic thermalization of the three-level system all initial information will finally be erased. As mentioned above, things are not so simple when $\gamma_3=0$. In fact, the steady state of the $\Lambda$-type three-level system immersed in a zero temperature bath will depend on its initial state if the bath coupling between the two lower levels is forbidden. This phenomenon is referred to as anti-thermalization.
When $\gamma_3=0$, the analysis of the $\Lambda$-type three level system immersed in a heat bath can be made in the same way as presented above. The time evolution is described by the master equation (\[mastereqaution\]) with $\gamma _{3}=0$. At zero temperature, we have $\det (\mathbf{M})=0$. Thus the optical Bloch equation is reduced to $$\begin{aligned}
\label{Blocheqforlambdaatom}
\langle \dot{\hat{\sigma}}_{ee}\rangle &=&-(\gamma _{1}+\gamma _{2})\langle
\hat{\sigma}_{ee}\rangle ,\hspace{0.5cm}\langle \dot{\hat{\sigma}}
_{g_{l}g_{l}}\rangle =\gamma _{l}\langle \hat{\sigma}_{ee}\rangle, \notag \\
\mathtt{Re}[\langle \dot{\hat{\sigma}}_{g_{2}g_{1}}\rangle] & =&\frac{1}{2}%
(\gamma _{12}+\gamma _{21})\langle \hat{\sigma}_{ee}\rangle -\Delta \mathtt{%
Im}[\langle \hat{\sigma}_{g_{2}g_{1}}\rangle], \\
\mathtt{Im}\langle \dot{\hat{\sigma}}_{g_{2}g_{1}}\rangle & =&\Delta \mathtt{%
Re}[\langle \hat{\sigma}_{g_{2}g_{1}}\rangle],\hspace{0.1 cm}\langle \dot{%
\hat{\sigma}}_{eg_{l}}\rangle =-\frac{1}{2}(\gamma _{1}+\gamma _{2})\langle
\hat{\sigma}_{eg_{l}}\rangle, \notag\end{aligned}$$ where $l=1,2$.
These equations can be solved straightforwardly to obtain the transient solutions for $\langle \hat{\sigma}_{ee}(t)\rangle ,\langle \hat{\sigma}%
_{g_{l}g_{l}}(t)\rangle $ and $\langle \hat{\sigma}_{g_{2}g_{1}}(t)\rangle $ where $l=1,2$. From these transient solutions it follows that $$\begin{aligned}
\langle \hat{\sigma}_{g_{l}g_{l}}\rangle _{ss}&=&\langle \hat{\sigma}%
_{g_{l}g_{l}}(0)\rangle +\frac{\gamma _{l}}{\gamma _{1}+\gamma _{2}}\langle
\hat{\sigma}_{ee}(0)\rangle, \\
\mathtt{Re}[\langle \hat{\sigma}_{g_{2}g_{1}}\rangle _{ss}]& =&\mathtt{Re}%
[\langle \hat{\sigma}_{g_{2}g_{1}}(0)\rangle] +\frac{(\gamma _{12}+\gamma
_{21})}{2(\gamma _{1}+\gamma _{2})}\langle \hat{\sigma}_{ee}(0)\rangle,
\notag \label{steadystatesolutionlambda}\end{aligned}$$ and $\langle \hat{\sigma}_{ee}\rangle _{ss}=\langle \hat{\sigma}
_{eg_{1}}\rangle _{ss}=\langle \hat{\sigma}_{eg_{2}}\rangle _{ss}=0$ , $%
\mathtt{Im}[\langle \hat{\sigma}_{g_{2}g_{1}}\rangle _{ss}]=\mathtt{Im}%
[\langle \hat{\sigma}_{g_{2}g_{1}}(0)\rangle]$, where $l=1,2$. This is just the steady state solution of equation (\[Blocheqforlambdaatom\]).
Equations (\[steadystatesolutionlambda\]) clearly show that the steady state of the $\Lambda $-type three-level system depends on its initial state. In the steady state, the decaying probabilities from the excited state $|e\rangle$ to the ground states $|g_{1}\rangle$ and $|g_{2}\rangle $ are respectively $\gamma _{1}/(\gamma _{1}+\gamma _{2})$ and $\gamma
_{2}/(\gamma _{1}+\gamma _{2})$. Moreover, as is shown in Eq. ([steadystatesolutionlambda]{}), in the present case the dynamic thermalization will preserve or even increase the off-diagonal elements of the density matrix of the initial state while in the previous case with $\gamma_3\neq 0$, the dynamic thermalization will lead the system to a steady state whose density matrix possesses no off-diagonal elements. It is also noticed that the steady state of the three-level system is independent of the initial off-diagonal elements between $|e\rangle $ and $|g_{l}\rangle $ ($l=1,2$), and when it is initially prepared in the superposition state of the two ground states its final steady state will be the same as the initial state. Finally let us present two simple examples. Take $\gamma
_{1}=\gamma_{2}=\gamma _{12}=\gamma _{21}=\gamma $ and $\Delta =0$, then for the initial state $|\psi (0)\rangle =|e\rangle $ we have the steady state $%
|\psi (\infty )\rangle =(|g_{1}\rangle +|g_{2}\rangle )/\sqrt{2}$ and for the initial state $|\psi (0)\rangle =|g_{1}\rangle $ we have the steady state $|\psi (\infty )\rangle =|g_{1}\rangle $.
*Summary.—*In summary, in this letter the SSB effect in dynamic thermalization is studied through a three-level system immersed in a heat bath inducing cycle transition couplings. Careful calculation is carried out from the master equation approach to examine the thermalization dynamics when the temperature approaches zero. By this investigation it is concluded that when there is no selection rule to forbid any one of the bath induced cycle transition couplings, the canonical thermal state can be reached as a steady state solution of the master equation at zero temperature and if the bath induced transition between the two lower (higher) energy states of $\Lambda $-type ($V$-type) atom is forbidden the anti-thermalization phenomenon will happen due to the quantum interference between the transition from the lower state and the transition to the higher state. In this latter case, the final steady state of the three-level system will depend on its initial state, and thus will preserve some of the initial information. This means that the initial information of the system cannot be completely erased by thermalization and the third law of thermodynamics does not work in the conventional fashion.
The work is supported by National Natural Science Foundation of China and the National Fundamental Research Programs of China under Grant.
[9]{} H.P. Breuer and F. Petruccione, *The Theory of Open Quantum Systems* (Oxford University Press, Oxford, 2002).
S. Popescu, A.J. Short, and A. Winter, Nat. Phys. **2**, 754 (2006).
S. Goldstein, J.L. Lebowitz, R. Tumulka, and N. Zanghì, Phys. Rev. Lett. **96**, 050403 (2006).
J. Gemmer and M. Michel, Europhys. Lett. **73**, 1 (2006).
H. Dong, S. Yang, X.F. Liu, and C.P. Sun, Phys. Rev. A **76**, 044104 (2007).
K. Huang, *Statistical Mechanics* (Wiley, New York, 1987).
P.W. Anderson, Science **177**, 393 (1972); P.W. Higgs, Phys. Rev. Lett. **13**, 508 (1964); Y. Nambu and G. Jona-Lasinio, Phys. Rev. **122**, 345 (1961).
G.S. Agarwal, *Quantum statistical theories of spontaneous emission and their relation to other approaches* (Springer, Berlin, 1974).
|
---
abstract: 'Quantum Lyapunov control uses a feedback control methodology to determine control fields which are applied to control quantum systems in an open-loop way. In this work, we adopt two Lyapunov control schemes to prepare an edge state for a fermionic chain consisted of cold atoms loaded in an optical lattice. Such a chain can be described by the Harper model. Corresponding to the two schemes, state distance and state error Lyapunov functions are considered. The results show that both the schemes are effective to prepare the edge state within a wide range of parameters. We found that the edge state can be prepared with high fidelity even **if** there are moderate fluctuations in on-site or hopping potentials. Both control schemes can be extended to similar chains (3$m+d$, $d$=2) of different lengths. Since regular amplitude control field is easier to apply in practice, amplitude-modulated control fields are used to replace the unmodulated one to prepare the edge state. Such control approaches provide tools to explore edge states for one dimensional topological materials.'
author:
- 'X. L. Zhao$^{1}$,Z. C. Shi$^{1}$, M. Qin'
- 'X.X.Yi'
date:
-
-
title: Edge state preparation in one dimensional lattice by quantum Lyapunov control
---
introduction
============
Topological materials are thought to be the candidates to realize fault-tolerant quantum computation [@NP5378] due to their robustness against perturbations. Usually the topological character is indicated by the emergence of edge states for a bulk system. Recently, there have been a great deal of work to explore topological systems related to such states [@PRL109106402; @PRL108220401; @PRL110075303; @PRL110260405; @PRL110180403; @NP10664; @NN6216; @PRB90165412; @PRL110076401]. In two dimensional systems such as the Bi$_2$Te$_3$ nanoribbon, manipulation of edge state by modulating a gate voltage has been reported in experiments [@NN6216]. Similarly, for topological Bi(111) bilayer nanoribbon, through first-principle simulations, the desirable edge state engineering can be realized by chemical decoration [@PRB90165412]. Besides, in one dimensional lattice systems, by coupling the atomic spin states to a laser-induced periodic Zeeman field, a novel scheme is proposed to manipulate the edge state [@PRL110076401]. This attracts both theoretical and experimental interests. Thus manipulation to the edge state is a way to investigate topological systems.
Compared to traditional solid state systems, cold atoms trapped in optical lattices are excellent simulators to investigate various interesting physical topics such as topological insulators [@Kane; @RMP831057]. One important reason is that such systems with tunable on-site and hopping potentials provide more controllable platforms to investigate quantum signatures of many-body systems. Once such a system possesses a structure of topological material, manipulating the edge state would be feasible in light of quantum control methodology.
Time-dependence control to quantum systems is valid to dynamically realize specific control goals [@nature425937; @PRL92187902; @PRA85022312; @42IEEE; @Auto411987; @PRA80052316; @PRL107177204; @PLA378699]. Among them, quantum Lyapunov control has been investigated widely and applied to realize various kinds of objectives [@42IEEE; @Auto411987; @PRA80052316; @PRL107177204; @PLA378699]. It is used to design an open-loop controller by simulating the evolution for an artificial closed-loop quantum system. Namely it is a ‘closed-loop design and open-loop apply’ control strategy. The dynamics is governed by Schrödinger equation. In this control, after determining the suitable control Hamiltonian, the control fields play an important role. They are designed by making the positive Lyapunov function decreases monotonously. In order to design the control fields, in this work, we employ two quantum Lyapunov control schemes: state distance and state error schemes. These two schemes are so intituled since the Lyapunov functions used in them are based on state distance and state error respectively. Note that the control fields would vanish when the system is asymptotically steered to the target state which locates in a set specified by LaSalle’ invariant principle [@Lasalle].
We prepare the edge state for a topological system composed of atoms loaded in an optical lattice by the quantum Lyapunov control methods mentioned above. Such a lattice can be created by two standing waves formed by two laser beams with different wavelengths [@pra75061603; @nature453895]. Such a system can be described by a Hamiltonian of the Harper model [@Harper]. We assume all the cold atoms are loaded in the lowest band of the optical lattice to make the tight-binding limit available. For the open boundary lattice, the edge states emergence with eigenenergies locate apart from the energy subbands. We choose one of the edge states to prepare. Then a control Hamiltonian is needed. Inspired by lattice shaking technique [@PRL95260404; @PRL95170404; @NATURE9769] which can be applied to quantum simulators in optical lattices with tunable structures, in this work, a trigonometrical modulation control Hamiltonian (time-dependent Hermitian) is introduced with a time varying control field to prepare the edge state.
In order to indicate the validity of both control schemes, the fidelity defined by the scalar product of the controlled and goal state is adopted. We found that despite the existence of fluctuations which perturb the on-site or the hopping potentials, the fidelity can reach a high value at the terminated time. This demonstrates the robustness of both control schemes against fluctuations. To examine the generality of both control schemes, we apply them to chains with different lengths with the same configuration(all of them are of type 3$m+d$, $d$=2 [@Bohm]). High fidelity control results manifest the validity of both control schemes in this work. Taking the controllability for operation in experiments into account, we use amplitude-modulated control fields in place of the unmodulated ones to prepare the edge state. Since the sign for the control field would be inclined to switch while its amplitude changes when fidelity approaches 1, we choose the manipulation function to make the amplitude decrease.
This paper is organized as follows. In Section \[model\], we specify the model, i.e. a controlled chain made of cold atoms loaded in an one dimensional optical lattice. In Section \[app\], we exhibit the general procedure to design the control fields in two quantum Lyapunov control schemes, which is exemplified with a trigonometrical modulation control Hamiltonian. Then we examine the control strategies in large control parameter intervals and with random initial states. In Section \[discussions\], we explore the robustness of the two control schemes against the number and amplitudes of on-site energy and hopping fluctuations, application to chains of 3$m+d$ configuration with different lengths and a feasible modulation for the control fields. Finally, we conclude in Section \[sum\].
fermionic chain made of atoms loaded in optical lattice {#model}
=======================================================
In this work we consider a system made of cold atoms trapped in an one dimensional optical lattice. The lattice of straight line shape can be generated by superimposing two standing waves of laser beams with different wavelengths. First we show the procedure to obtain the fermionic chain from single particle Hamiltonian, which has the following form in the periodically modulated lattice $$\begin{aligned}
\label{Hs}
\hat{H}_s&=&\hat{H}_1+\hat{H}_2, \nonumber \\
\hat{H}_1&=&\frac{p_x^2}{2M}+V_1\sin^2(k_1x), \nonumber \\
\hat{H}_2&=&V_2\cos^2(k_2x+\delta),\end{aligned}$$ where $V_j=s_jE_{rj}$ and $k_j=2\pi/\lambda_j(j=1,2)$ are the lattice depth and wavenumbers respectively. $x$ denotes the positions for the atoms on the chain. $s_j$ and $E_{rj}=h^2/(2M\lambda_j^2)$ denote the height of the lattices and recoil energies respectively. $h$ is the Planck constant and $M$ is the mass of the atoms in the lattice. $\delta$ is an arbitrary phase of the second laser beam. We assume all the atoms are trapped in the lowest band of the optical lattice to make the tight-binding approximation available. Then in virtue of the field operators $\Phi(x)$, the Hamiltonian reads $$\begin{aligned}
\hat{H}_0=\int dx\Phi^{\dagger}(x)\hat{H}_s\Phi(x).\label{Hs2}\end{aligned}$$ In the basis of Wannier functions, the field operator can be expanded as $\Phi(x)=\sum_n\hat{c}_n\omega(x-x_n)$. $\hat{c}_n$ here denotes the annihilation operator for the fermion at site $n$ while the spin freedom is not considered. Substitute this into (\[Hs2\]), omitting constant terms, one gets the Hamiltonian $$\begin{aligned}
\begin{aligned}
\hat{H}_0=&-J\sum_{n=1} (\hat{c}^\dagger_n \hat{c}_{n+1}+\mathrm{H.c.})\\
&+\sum_{n=1} V\cos(2\pi\beta n+\delta) \hat{n}_n,\label{H00}
\end{aligned}\end{aligned}$$ where $\beta$=$k_2/k_1$=$p/q$ ($p$ and $q$ are prime to each other), the hopping amplitude $J$=$\int dx \omega^*(x-x_{n})\hat{H}_1\omega(x-x_{n+1})$ and on-site energy $V=\frac{V_2}{2}\int dx \omega^*(x)\cos(2k_2x)\omega(x)$. $\hat{c}^\dagger_n$ ($\hat{c}_n$) are the creation (annihilation) operators for the atoms on-site $n$, and $\hat{n}_n=\hat{c}^\dagger_n \hat{c}_n$. $J$ is set to be the unit of energy in this work and we set $\hbar$=1. We choose $\beta=1/3$, then the chain has a 3$m+d$ configuration: $m$ here means the number of eigenenergies in a energy subband since the total energy band can be divided into $q$ subbands in Harper model [@PRL109106402; @Bohm] and we choose $d$=2 in this work. ‘3’ reflects the periodic character for the chain which results from the ratio of the two wave vectors of the two lasers. ‘$d$’ here is the remainder for the length of the chain divided by ‘3’ which is a character for the destruction of the translation symmetry of the chain. To get $J$ and $V$ mentioned above, estimations have been obtained by calculating the integrals in terms of maximally localized Wannier functions [@PRA72053606]. They are $J\simeq1.43s_1^{0.98}e^{(-2.07\sqrt{s_1})}E_{r1}$ and $V\simeq\frac{s_2\beta^2}{2}e^{-\frac{\beta^\alpha}{s_1^{\gamma}}}E_{r1}$, where $\alpha$ and $\gamma$ are determined by fitting the numerical evaluation of the integral of $V$. Roughly, Gaussian approximation for the Wannier function can also be used to estimate $J$ and $V$ [@njp11033023]. According to these expressions for $J$ and $V$, one can see that several parameters can be adjusted to yield various ratio of $V/J$. Since we focus on the control procedure, moderate value of $V/J$ is chosen directly in this work.
![The number of isolated states vs. $J_s$ and $\delta$. Different colors show the different total number of isolated states in the total energy band, as labeled in the right side. In this work, we set the parameters $L=62$, $V=1.5J$, $J_s$=0, and $\delta=2\pi/5$ (i.e. the red point in this figure) to exhibit the control process.[]{data-label="deVp"}](deVp.pdf){width="7.7cm"}
In order to determine the controlled chain, now we numerically find out the spectrum for the eigenenergies as a function of the parameters in the Hamiltonian. The edge states for the chain locate near the ends. The hopping strength between the two ends (denoted by $J_s$) would affect the energy spectrum of the chain while the chain can be mapped to a ring mathematically without changing the periodic character in the bulk. Namely different strength of $J_s$ induce different number of isolated states in the total band. In this work, the eigenenergies corresponding to the isolated states locate apart from the three energy subbands (since we have set $\beta=1/3$). Here, in order to define such isolated states, we assume the energy difference between the isolated one and their nearest neighbors is $E_d$ and the maximum energy difference between the neighbor eigenenergies in the corresponding neighbor subband is $E_b$. Here the ‘corresponding neighbor subband’ refers to the one that the nearest neighbor eigenenergy belongs to. If $E_d>\chi E_b$, $\chi=2$, we refer it to be the isolated state in this work. With respect to mentioned above, in Fig. \[deVp\], we numerically explore the number of isolated eigenenergies in the total energy band vs. the distinct hopping strength $J_s$ and the phase $\delta$. It can be seen that there would be more than 2 isolated states appearing in some combinations of $J_s$ and $\delta$. The isolated states include the edge states belonging to topological phases. Besides the edge states, the other isolated ones are just located in the gap since the eigenstates are orthogonal to each other. Numerical simulation shows that the eigenenergies corresponding to the edge states do not always locates in the gap between the subbands but may be larger or smaller than all the other eigenenergies. Here $E_d>2E_b$ is obvious a rough criterion to ascertain an isolated state since the energy difference between the isolated one and its neighbor vary gradually with respect to $J_s$ and $\delta$. So further investigations may be needed on this $\text{rough}$ spectrum.
As mentioned above, if the criterion for an isolated state $E_d>\chi E_b$ changes, one obtains a different spectrum from Fig. \[deVp\]. To further determine a model chain, we next examine the character of an edge state which is chosen as the target. Numerical simulations show that the edge state may appear as the $m+1$ eigenstate in the energy band with respect to $J_s$ and $\delta$. Here the eigenstates have been arrayed according to their corresponding eigenenergies in a small-value to large-value manner. Thus $m+1$ refers to the order for the eigenstate in the array. Since a more local edge state is desirable, we next check the localization for the $m+1$ eigenstate. IPR can be used to indicate the degree of localization for a state [@PRA76042333]. If a state $|\phi_j\rangle=\sum_{n=1}^N\psi_j(n)|n\rangle$ ($N$ is the number of basis $|n\rangle$), IPR can be defined as $I_j=\sum_{n=1}^N|\psi_j(n)|^4$ ($j$=$m$+1 in this work). We can see that if $\psi_j(n)$ are distributed homogeneously over all basis $|n\rangle$, namely $|\psi_j(n)|^2\sim1/N$, then $I_j\sim1/N$. Whereas, if $\psi_j(n)$ are localized over a range $\zeta$, namely $|\psi_j(n)|^2\sim1/\zeta$, then $I_j\sim1/\zeta$. So for large $N$, the larger $I_j$ is, the more degree of localization of state $|\phi_j\rangle$. Then we show $I_{m+1}$ vs. $J_s$ and $\delta$ in Fig. \[maxlo\] to pick a combination of $J_s$ and $\delta$ with high IPR. According to this figure, without loss of generality, we choose $J_s=0$ and $\delta=2\pi/5$ mainly to exhibit the control results. According to the mentioned above, we specify the $m+1$ eigenstate as the target in this work. Then the ring retrogresses to an one dimensional chain.
![The quantity $I_{m+1}$ to measure the localization for the $m$+1 edge states vs. $J_s$ and $\delta$. The other parameters are the same as those in Fig. \[deVp\].[]{data-label="maxlo"}](maxlo1.pdf){width="7.7cm"}
Since the model chain has been specified, we next check the energy spectrum for it. This gives us an intuitive knowledge for the target state. Fig. \[ec\] shows the single-excitation energy spectrum while insets (a) and (b) exhibit the population of the edge state-2($m$+1) and $m$+1-state of Hamiltonian (\[H00\]) when $\beta$=1/3 in the open boundary condition. It can be intuitively seen that the population of the edge states both localize at the ends. They are the eigenstates of the natural Hamiltonian (\[H00\]) with the protected eigenenergies [@PRL108220401]. We next show the procedure to prepare the ‘$m+1$’ edge state by two kinds of Lyapunov control schemes.
![Energy spectrum and the distribution of the edge states for the chain when $J_s$=0, $\delta$= $2\pi/5$ as the red point marks in Fig. \[deVp\]. The edge state as (b) shows is chosen as the target state in this work. $\nu$ in the insets are the index for the sites on the chain.[]{data-label="ec"}](ec.pdf){width="7.7cm"}
prepare the edge state by Lyapunov control {#app}
==========================================
First we briefly exhibit the design procedure in quantum Lyapunov control. Generally, the controlled system is governed by Schrödinger equation $$\begin{aligned}
i\dot{|\varphi\rangle}=(\hat{H}_0+\sum_{n}f_n(t)\hat{H}_n(t))|\varphi\rangle,\label{dv}\end{aligned}$$ where $\hat{H}_0$ describes the natural Hamiltonian and $\hat{H}_n(t)$ are the control Hamiltonians with the corresponding real-valued $f_n(t)$. $f_n(t)$ represent the control fields need to be designed by quantum Lyapunov control method. The system state is $|\psi\rangle=\sum_{k=1}^L \xi_k|k\rangle$, where $|k\rangle$ denotes that the atom on site $k$ is excited while the others not, and $\xi_k$ is the corresponding probability amplitude. $L$ denotes the total number of sites on the chain. The control Hamiltonian $\hat{H}_n(t)$ should not commute with the natural Hamiltonian $\hat{H}_0$, (i.e. $[\hat{H}_0,\hat{H}_n(t)]\neq0$), otherwise its effect can be included in the natural Hamiltonian. Note that the target state $|\varphi_f\rangle$ is usually an eigenstate of the natural Hamiltonian, namely $\hat{H}_{0}|\varphi_f\rangle=\lambda_{f}|\varphi_f\rangle$. Then the Lyapunov function $V_L$ related to the controlled state is constructed to design the control field. Then the control fields are determined by making the first-order time derivative of the Lyapunov function $V_L$ negative. Then assisted by the control Hamiltonians with the corresponding designed control fields, the system would be steered to a LaSalle invariant set asymptotically in which the states satisfy $\dot{V}_{L}=0$.
Usually, there are alternatives of Lyapunov functions that can be used. One of the candidates is based on Hilbert-Schmidt distance between the system state $|\varphi(t)\rangle$ and the target state $|\varphi_f\rangle$ [@PFIS] (We call it Lyapunov-A for short hereafter). It is $$\begin{aligned}
V_{A}=\frac{1}{2}(1-|\langle \varphi_f|\varphi(t)\rangle|^{2}),\label{V1}\end{aligned}$$ where $|\langle \varphi_f|\varphi(t)\rangle|^{2}$ denotes the transition probability from $|\varphi(t)\rangle$ to $|\varphi_f\rangle$. According to the description above, the first-order time derivative for $V_{A}$ is $$\begin{aligned}
\begin{aligned}
\dot{V}_A=&-\sum_{n}f_{An}(t)\cdot|\langle\varphi(t)|\varphi_f\rangle|\times\\
& Im[e^{i\arg\langle\varphi(t)|\varphi_f\rangle}\langle\varphi_f|\hat{H}_{n}|\varphi(t)\rangle],
\label{Vdot}
\end{aligned}\end{aligned}$$ where $Im[\bullet]$ denotes the imaginary part of $\bullet$. Thus there are different kinds of control fields $f_{An}(t)$ that meet the requirement $\dot{V}_{A}\leq0$. For example, a succinct and valid choice is $$\begin{aligned}
\begin{aligned}
f_{An}(t)=T_{n} Im[e^{i\arg\langle\varphi(t)|\varphi_f\rangle}\langle\varphi_f|\hat{H}_{n}|\varphi(t)\rangle],
\end{aligned}\label{fd}\end{aligned}$$ where $T_{n}>0$. When $\langle\varphi(t)|\varphi_f\rangle=0$, the angle $\arg\langle\varphi(t)|\varphi_f\rangle$ is uncertain. Without loss of generality, we artificially set $\arg\langle\varphi(t)|\varphi_f\rangle=0$ in this situation.
Another Lyapunov function based on state error [@Auto411987] can be described by (We call it Lyapunov-B for short hereafter) $$\begin{aligned}
\begin{aligned}
V_{B}&=\frac{1}{2}\langle\varphi(t)-\varphi_f|\varphi(t)-\varphi_f\rangle\\
&=1-Re[\langle\varphi_f|\varphi(t)\rangle].\label{V2}
\end{aligned}\end{aligned}$$ $Re[\bullet]$ denotes the real part of $\bullet$. The first-order time derivative for $V_{B}$ is $$\begin{aligned}
\begin{aligned}
\dot{V}_{B}=&-\lambda_f Im[\langle\varphi_f|\varphi(t)\rangle]\\
&-\sum_nf_{Bn}(t)\cdot Im[\langle\varphi_f|\hat{H}_{n}|\varphi(t)\rangle].\label{dotV2}
\end{aligned}\end{aligned}$$ Distinguishing from the first Lyapunov function, here we would employ a simple $\hat{H}_{c0}=I$ where $I$ is the identity matrix in Hilbert space and $f_{B0}=-\lambda_f$ to cancel the first term in (\[dotV2\]). Therefore, we can choose $$\begin{aligned}
\begin{aligned}
f_{Bn}(t)=& T_{n} Im[\langle\varphi_f|\hat{H}_{n}|\varphi(t)\rangle],\\
& n\neq0,\label{fr}
f_{B0}=-\lambda_f,
\end{aligned}\end{aligned}$$ where $T_n>0$. In the next section, we apply both control schemes to generate edge state for the fermionic chain.
To apply Lyapunov control, we next specify the control Hamiltonian. There may be various control Hamiltonians that can be adopted to prepare the edge state for the chain. Taking available techniques into consideration, a trigonometrical shaking Hamiltonian which is generated by electro-optic phase modulator [@PRL95170404] is employed, i.e. $$\begin{aligned}
\hat{H}_r=V_c\cos^2[k_d(x-b\cos(\omega_{c}t))].\label{sk}\end{aligned}$$ $V_c$ is the constant amplitude and $k_d$ denotes the laser wave vector. $\omega_c$ reflects the shaking frequency. $b$ indicates the shaking depth and should not be large otherwise it may induce heating effect leading to failure of this model. In the second quantization form, the control Hamiltonian reads $$\begin{aligned}
\hat{H}_c=\int dx \Phi^\dag(x)\hat{H}_r\Phi(x).\label{sk2}\end{aligned}$$ As all atoms are loaded in the lowest band of the optical lattice, in terms of Wannier basis similar to (\[Hs2\]), the control Hamiltonian has the matrix element $$\begin{aligned}
\begin{aligned}
&\hat{H}_c(m,n)\\
&=\delta_{m,n}V_{cd}\cos[2\pi \frac{k_d}{k_1}m+2k_db\cos(\omega_{c}t)],\label{skm}
\end{aligned}\end{aligned}$$ where $V_{cd}=\frac{V_c}{2}\int dx\omega^*(x)\cos(2k_dx)\omega(x)$ and $m, n$ are integers indicating the order for atoms on the chain. The parameters $2k_db$ and $\omega_c$ can be modulated in experiments [@PRL95170404]. The total Hamiltonian reads $\hat{H}=\hat{H_0}+f_c(t)\hat{H}_c$. Here $f_{c}(t)$ is the time varying control field determined by quantum Lyapunov method. They can be tuned by changing the voltage on the electro-optic phase modulator. In this work, we denote the control fields as $f_A(t)$ in the Lyapunov-A and $f_B(t)$ in the Lyapunov-B schemes respectively.
![Fidelity at time $t$=3000 (in units of $1/J$) vs. $2k_db$ and $\omega_c$ (in units of $J$) for Lyapunov-A and Lyapunov-B schemes in (a) and (b) respectively. We have set $V=1.5J$, $\beta$=1/3, $\delta=2\pi/5$ in the natural Hamiltonian. $V_{cd}$=5$J$ in the control Hamiltonian (\[skm\]). We have used the laser for the control 2$\pi k_d/k_1\simeq$4.97, the chain of length $L=62$.[]{data-label="DEOPT"}](DEOPT.pdf){width="7.7cm"}
![Evolution of the fidelity for the Lyapunov-A and Lyapunov-B schemes and the control fields. We have chosen $2k_db$=0.2, $\omega_c=115J$ which is available in experimental technology [@PRL95170404; @NATURE9769]. $f_A$ and $f_B$ are the control fields for Lyapunov-A and Lyapunov-B schemes respectively. The other parameters in natural Hamiltonian are same to those in Fig. \[DEOPT\].[]{data-label="FDE"}](fidelityde.pdf){width="7.7cm"}
To exhibit the control effect, we quantify it by the fidelity which is defined as $$\begin{aligned}
\mathcal{F}(t)=|\langle\varphi(t)|\varphi_{edge}\rangle|^{2},\label{Fid}\end{aligned}$$ where $|\varphi_{edge}\rangle$ is the target state as is shown by the inset (b) in Fig. \[ec\]. We examine the fidelity at terminated time $t=3000$ vs. $2k_db$ and $\omega_c$ numerically in Fig. \[DEOPT\] for the two kinds of control schemes. In these simulations, we have set the initial state with equally projection to the basis $|k\rangle$. It can be seen that both schemes can be used to yield high fidelity in a wide range of parameter interval.
To be more concrete, we show the dynamics of the control process in Fig. \[FDE\] for both the control schemes, when $2k_db$=0.2 and $\omega_c=115J$. Numerical simulation shows that the fidelity can reach more than 0.95 at time $t$=3000 while each control field approximately vanishes. To intuitively compare the control results with the target, in Fig. \[pop\], we plot the density distribution at time $t=3000$. It can be seen that the two profiles matches well at the terminated time.
In practice there may be various kinds of initial states. In consideration of this, we tested 100 site-occupation random initial states for both kinds of control schemes to check their validity for preparing the edge state in Fig. \[randDCER\]. It can be seen that both control schemes are effective to complete the control goal.
![The result density distribution at time $t$=3000 as in Fig. \[FDE\] for the two control schemes and the goal match well.[]{data-label="pop"}](pop.pdf){width="7.7cm"}
![The evolution of the fidelity for 100 random initial states in each control scheme. (a) is for the Lyapunov-A scheme while (b) is for Lyapunov-B scheme. The red solid lines represent the average for the blues.[]{data-label="randDCER"}](randDCER.pdf){width="7.7cm"}
Discussions
===========
![Fidelity at time $t$=3000 vs. the number and the maximal strength of the random fluctuations averaged over 30 times for each value. (a) and (b): fidelity for on-site energy and hopping fluctuations in Lyapunov-A scheme; (c) and (d): fidelity for on-site energy and hopping fluctuations in Lyapunov-B scheme. The parameters are same to those in Fig. \[FDE\] where $\eta_o$ and $\eta_h$ are in units of $J$. All the figures have the same color map.[]{data-label="SDEOH"}](SDEOH.pdf){width="8cm"}
In this section we provide discussions for both control schemes with respect to their robustness against fluctuations on the chain, expandability to chains of different lengths and modulation for the control fields.
Inevitably there exist types of fluctuations on the chain which mainly be classified as on-site energy and hopping types. These two kinds of fluctuations can be denoted as $\delta \hat{V}$=$\eta_o\hat{H}_{po}$ and $\delta \hat{J}$=$\eta_h\hat{H}_{ph}$. In matrix form, $\hat{H}_{po}(m,k)=\delta_{m,k}$ and $\hat{H}_{ph}(m,l)=\hat{H}^*_{ph} (l,m)=\delta_{m,l+1}$, here $m, k\in [1,L]$ and $l\in [1,L-1]$ are random integers representing positions for the fluctuations. In the simulation, we set the strength of the fluctuations randomly distributed in the interval: \[0,$\eta_o$($\eta_h$)\]. Namely, there are a number of sites (at random positions on the chain) with strength (distributed randomly within an interval) added to the on-site or hopping potential during the control process whereas the target state is still the edge state of the original chain without fluctuations. We should confirm that the positions and strength of these fluctuations are assumed fixed during the control process even they are random. We examine numerically the robustness of both control schemes for these two kinds of fluctuations by fidelity at time $t$=3000 vs. the number and strength of these fluctuations in Fig. \[SDEOH\] for these two kinds of control schemes. From this figure, it can be seen that, with the increasing of the number and strength of fluctuations, fidelity would decline. This indicates the hindering effect of these fluctuations to the preparing procedures. However these fluctuations would change the original energy spectrum since they destroy the structure of the chain. Thus too many or fluctuation with too large amplitude would destroy this model which needs further discussion. Whereas in a range of the number and strength of these fluctuations, the fidelity can reach a high value which manifests both control schemes are robust against both kinds of fluctuations.
Since chains with different lengths may be the controlled objects, to examine the expandability of both control schemes, we apply them to such 3$m+d$ chains with $d$=2, within the length of $\langle32,35,...,95\rangle$ sites. ‘3’ here reflects the periodic character of the chain, ‘m’ is a integer and ‘d’ is the remainder of the length of the chain divided by 3. In this length scope, the fidelity can reach a high value at time $t$=3000 even with slightly decrease with lengthening of the chain which is shown in Fig. \[length\].
![Fidelity at time $t$=3000 vs. several lengths of the chains with the configuration 3$m+d$ while $d$=2, where the other parameters are same to those in Fig. \[FDE\]. The nail graph with square heads denote the control results for Lyapunov-A scheme while those with circle heads denote the results for Lyapunov-B scheme.[]{data-label="length"}](length.pdf){width="7.7cm"}
![Compare of the results for amplitude-modulated wave control and the original Lyapunov-A scheme. $f_{Ao}$ denotes the amplitude-modulated control field in Lyapunov-A scheme, while $f_{A}$ is the unmodulated one. We have set $A=0.3$ and $\kappa=50$ in (\[mbb\]). The control fields are shut down when the fidelity reach 0.95. The other parameters are same to those in Fig. \[FDE\]. []{data-label="od"}](od.pdf){width="7.7cm"}
![Compare of the results for amplitude-modulated wave control and the original Lyapunov-B scheme. $f_{Bo}$ denotes the amplitude-modulated control field in Lyapunov-B scheme, while $f_{B}$ is the unmodulated one. We have also set $A=0.3$ and $\kappa=50$ in (\[mbb\]). The control fields are shut down when the fidelity reach 0.95. The other parameters are same to those in Fig. \[FDE\].[]{data-label="oe"}](oe.pdf){width="7.7cm"}
Compared to the original designed control fields such as those in Fig. \[FDE\], more regular control fields may be desirable in practice since regular control fields are easier to be generated than the original one. To achieve such a control field, we inspect the property of Lyapunov control. In this control, the sign plays a more important role than the amplitude of the control field since the sign of the control field determines the decreasing trend of the positive Lyapunov function but the amplitude determines the decreasing rate. An amplitude-invariant square waves has been used to replace the unmodulated control field [@PRA86022321] to complete a control process. However when the fidelity approaches 1, the evolution of the controlled system would be more sensitive to the amplitude of control field, i.e. the sign of the control field would switch more frequently between ‘+’ and ‘-’ if its amplitude stays invariant. Indeed in general, control fields with time-dependent envelopes can be used to realize a control goal. We use
$$\begin{aligned}
f_c(t)=\left \{
\begin{array}{rl}
F(t),~~~ f_c(t)>0, \\
-F(t),~~~f_c(t)<0, \\
\end{array}\label{bangbang}
\right.\end{aligned}$$
here $F(t)>0$ is the control field modulated by a time varying envelope. There are various kinds of envelope functions which can be employed to realize a control goal. Considering the sensitivity of the sign of the control field to its amplitude when the state approaches the goal, we choose an envelope for the control field in terms of time $t$ as $$\begin{aligned}
F(t)=\frac{A}{1+\kappa t}, A,\kappa >0.\label{mbb}\end{aligned}$$ By these modulated control fields, the fidelity can reach 0.95 at time less than the results in the unmodulated cases as is shown in Fig. \[od\] and Fig. \[oe\]. Here $A$ and $\kappa$ in (\[mbb\]) can be tuned flexibly. Finally, even the control fields are obtained by closed-loop simulation, whereas in light of Lyapunov control strategy, control fields with the same profile may be used in open-loop control process to prepare the edge state.
summary {#sum}
=======
The fermionic chain discussed in this paper can be created by loading cold atoms in optical lattice. Such an optical lattice can be created by two standing waves formed by laser beams of different wavelengths. This chain can be mapped to a ring with the same periodic structure mathematically. And it has attractive spectrum, in which the number of isolated eigenenergy depends on the hopping and the modulated phase $\delta$ in the Hamiltonian. After specified a 3$m+d$ fermionic chain defined in the text, we present proposals to prepare an edge state by quantum Lyapunov control with state distance (Lyapunov-A) and state error (Lyapunov-B) schemes. By both control schemes, an initial state with equal population on each site can be steered to the edge state with high fidelity in a wide range of control parameters. In the simulation, we have chosen 100 site-occupation random initial states to show the validity of the control. And both schemes are available in the presence of fluctuations in on-site energies and hopping potentials. To reduce the difficulty in realization, the control fields can be replaced with amplitude-modulated ones. This is because the sign plays a more crucial role than the amplitude of the control field to achieve a control goal. Such control methods provide ways to explore edge state for topological materials which possesses novel properties.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
This work is supported by the National Natural Science Foundation of China (Grant No. 11534002 and 61475033).
References
==========
J. Moore, Nature Physics **5**, 378 (2009). Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Phys. Rev. Lett. **109**, 106402 (2012). L. Lang, X. Cai and, S. Chen, Phys. Rev. Lett. **108**, 220401 (2012). S. L. Zhu, Z. D. Wang, Y. H. Chan, and L. M. Duan, Phys. Rev. Lett. **110**, 075303 (2013). F. Grusdt, M. Höning, and M. Fleischhauer, Phys. Rev. Lett. **110**, 260405 (2013). S. Ganeshan, K. Sun, and S. D. Sarma, Phys. Rev. Lett. **110**, 180403 (2013). I. K. Drozdov, A. Alexandradinata, S. Jeon, S. Nadj-Perge, H. Ji, R. J. Cava, B. A. Bernevig, and A. Yazdani, Nature Physics **10**, 664 (2014). F. Xiu, L. He, Y. Wang, L. Cheng, L. T. Chang, M. Lang, G. Huang, X. Kou, Y. Zhou, X. Jiang, Z. Chen, J. Zou, A. Shailos, and K. L. Wang, Nature Nanotechnology **6**, 216 (2011). X. Li, H. Liu, H. Jiang, F. Wang, and J. Feng, Phys. Rev. B **90**, 165412 (2014). X. J. Liu, Z. X. Liu, and M. Cheng, Phys. Rev. Lett. **110**, 076401 (2013). M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. **82**, 3045 (2010). X. L. Qi and S. C. Zhang, Rev. Mod. Phys. **83**, 1057 (2011). O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. H$\ddot{a}$nsch, and I. Bloch, Nature **425**, 937 (2003). M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, Phys. Rev. Lett. **92**, 187902 (2004). M. Bruderer, K. Franke, S. Ragg, W. Belzig, and D. Obreschkow, Phys. Rev. A **85**, 022312 (2012). S. Grivopoulos and B. Bamieh, In Proceedings of the 42nd IEEE conference on decision and control (2003). M. Mirrahimi, P. Rouchon, and G. Turinici, Automatica **41**, 1987 (2005). X. X. Yi, X. L. Huang, Chunfeng Wu, and C. H. Oh, Phys. Rev. A **80**, 052316 (2009). X. Wang, S. Vinjanampathy, F. W. Strauch, and K. Jacobs, Phys. Rev. Lett. **107**, 177204 (2011). S. C. Hou, L. C. Wang, and X. X. Yi, Physics. Letters. A **378**, 699 (2014). J. La Salle and S. Lefschetz, *Stability by Lyapunovs Direct Method with Applications* (Academic Press, New York, 1961). J. E. Lye, L. Fallani, C. Fort, V. Guarrera, M. Modugno, D. S. Wiersma, and M. Inguscio, Phys. Rev. A **75**, 061603(R) (2007). Giacomo Roati, Chiara DErrico, Leonardo Fallani, Marco Fattori, Chiara Fort, Matteo Zaccanti, Giovanni Modugno, Michele Modugno and Massimo Inguscio, Nature **453**, 895, (2008). P. G. Harper, Proc. Phys. Soc. London Sect. A **68**, 874 (1955). A. Eckardt, C. Weiss, and M. Holthaus, Phys. Rev. Lett. **95**, 260404 (2005). N. Gemelke, E. Sarajlic, Y. Bidel, S. Hong, and S. Chu, Phys. Rev. Lett. **95**, 170404 (2005). C. V. Parker, L. C. Ha, and C. Chin, Nature Physics **9**, 769 (2013). A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, and J. Zwanziger, *The Geometric Phase in Quantum Systems* (Springer-Verlag, Berlin, 2003).
F. Gerbier, A. Widera, S. Fölling, O. Mandel, T. Gericke, and I. Bloch, Phys. Rev. A **72**, 053606 (2005).
M. Modugno, New J. Phys. **11**, 033023 (2009). O. Giraud, J. Martin, and B. Georgeot, Phys. Rev. A **76**, 042333 (2007).
P. Vettori, ‘On the convergence of a feedback control strategy for multilevel quantum systems’, Proc. the Fifteenth Int. Symposium MTNS, South Bend, Indiana, USA, (2002).
S. C. Hou, M. A. Khan, X. X. Yi, D. Dong, and I. R. Petersen, Phys. Rev. A **86**, 022321 (2012).
|
---
abstract: 'Symmetry is conventionally described in a contrariety manner that the system is either completely symmetric or completely asymmetric. Using group theoretical approach to overcome this dichotomous problem, we introduce the degree of symmetry (DoS) as a non-negative continuous number ranging from zero to unity. DoS is defined through an average of the fidelity deviations of Hamiltonian or quantum state over its transformation group $G$, and thus is computable by making use of the completeness relations of the irreducible representations of $G$. The monotonicity of DoS can effectively probe the extended group for accidental degeneracy while its multi-valued natures characterize some (spontaneous) symmetry breaking.'
address: |
$^{1}$Beijing Computational Science Research Center, Beijing 100084, China\
$^{2}$Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China\
$^{3}$State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, and University of the Chinese Academy of Sciences, Beijing 100190, China\
$^{4}$Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
author:
- 'Y. N. Fang$^{1,3,4}$, G. H. Dong$^{1,4}$, D. L. Zhou$^{2}$ and C. P. Sun$^{1,4}$'
title: Quantification of Symmetry
---
#### Introduction.
Symmetry is a theme of modern physics, which plays a crucial role in the understanding of fundamental interactions of the microscopic world [@2014; @Sundermeyer; @BOOK] as well as the emergence of macroscopic orders [@1984; @Anderson; @BOOK]. It has become evident that both the elementary particle structure and the emergent phenomena, e.g., superconductivity and Bose-Einstein condensation, are originated from symmetry and its spontaneous breaking [@1956; @Penrose; @PR; @1961; @Nambu; @PR; @1991; @Leggett]. Its applications range from particle physics [@1964; @Englert; @PRL; @1964; @Higgs; @PRL; @2014; @Kibble; @PTRSA; @2004; @Witten; @Nature] to condensed matter physics [@2015; @Navon; @Science; @2014; @Liu; @Nat; @Commun], and even to biological systems [@2015; @Duboc; @Ann; @Rev; @2013; @Saito; @RMP].
Conventionally, symmetry is dealt in a dichotomous fashion that a physical system either possesses or not possesses a symmetry. In the group theoretical approach, the symmetry of a quantum system is usually considered by checking that if the system is invariant or not under some transformations, which sometimes form a symmetry group $G$. The symmetry breaking of the system can be described as a reduction of the symmetry group to its subgroup. Although this conventional approach has succeed in classifying the spectrum structure and even various phases of matters, it is not natural for us because there is not a room for the intermediate circumstance, namely, a continuous measure of symmetry has not been found. Actually, such intermediate issues exist objectively and needs to be properly quantified. For example, a charged particle moving in a central potential possesses SO(3) symmetry. When a static magnetic field is applied, no matter how weak it is, the SO(3) symmetry is said to be broken into SO(2). However, SO(3) symmetry can still be approximately used to simplify the equations describing the dynamics and the energy level structure when the magnetic field is weak enough. Another example is the nuclear system that possesses the isospin SU(2) symmetry and thus its energy spectrum of strong interaction can approximately, but effectively, be classified, although the electromagnetic force could break this SU(2) symmetry.
In this regard, it would be of much interest to present a quantitative description of symmetry and its (spontaneous) breaking in this intermediate circumstance, which could determine the extent of approximation for using a given symmetry in practice. To this end, we, in this letter, introduce a continuous measure of symmetry, i.e., the degree of symmetry (DoS), by considering that symmetry is a relative concept: the particular subset $G$ of all physically-allowed transformations needs to be specified for assigning a symmetry to a physical system. More specifically, for a given set $G$ of transformations on the Hamiltonian or the quantum state $F=H$, or $\rho$, we first define a dual of DoS, the degree $A(G,F)$ of asymmetry (DoAS), by averaging the fidelity deviations (see definition below) over $G$. Generally, the DoAS ranges from zero to unity, and thus the DoS $S(G,F)=1-A(G,F)$ also satisfies $0\leq S(G,F)\leq1$. Evidently, $S(G,F)$ offers symmetry an intermediate description to avoid the dichotomy in the conventional group theoretical analysis.
We will show that, if we chose $G$ as a group, the DoS , bounded with $1/2\leq S(G,F)\leq1$, facilitates a general computable measure of symmetry based on the irreducible representations of $G$. It is potential in identifying various natures of symmetry that are important to emergent phenomena, such as the spontaneous symmetry breaking (SSB). For example, the thermodynamic SSB corresponds to multi-valued natures of DoS at the low temperature, which is similar to the depiction of the spontaneous magnetization [@1987; @Huang; @BOOK]. It is also shown that the multi-level crossing by a proliferation of energy levels brings a peak to the DoS and the extended group can be given to account for the hidden symmetry from accidental degeneracy.
#### Degree of symmetry.
We consider a quantum system with Hamiltonian $H$, and a set $G$ of $n_{G}$ transformations on its Hilbert space $\mathcal{H}$. When $OHO^{-1}=H$ for $O\in G$, we say that $H$ (the quantum system) is symmetric with respect to the transformation $O$. Actually, all symmetric transformations form a group $G'(\subset G)$. It is obvious that, the deviations of $OHO^{-1}$ from $H$ measure the extent of the asymmetry of $H$ with respect to the transformation set. Thus, we use their average over $G$ to define the degree of symmetry breaking (asymmetry) DoAS $$A(G,H)=\frac{1}{4|\tilde{H}|{}^{2}}\overline{|[R(g),H]|^{2}}$$ where $|O|=\sqrt{\mathrm{Tr}\{O^{\dagger}O\}}$ indicates the Frobenius norm [@2015; @Watrous; @BOOK] while $\overline{f(g)}|_{G}\equiv\overline{f(g)}=n_{G}^{-1}\sum_{g\in G}f(g)$ is an average of a (group) function $f(g)$ defined on $G$, and later the subscript $G$ will be occasionally omitted; If $G$ is a group, then $R:\mbox{ }g\rightarrow R(g)\in End(\mathcal{H})$ is a *d*-dimensional representation of $g\in G$. Otherwise, $R(g)$ represents a unitary transformation on $\mathcal{H}$. Here, $|[R(g),H]|^{2}=|R(g)^{\dagger}HR(g)-H|^{2}$ is the fidelity deviation of $H$ under the action of $g$, and $\tilde{H}=H-d^{-1}\mathrm{Tr}\{H\}$ is a re-biased Hamiltonian such that it is invariant under the zero-point energy shifting $H\rightarrow H+\epsilon$ for $\epsilon$ being a real number.
It is easy to prove that $0\leq A(G,H)\leq1$, thus the DoS defined by $S(G,H)=1-A(G,H)$ or $$S(G,H)=\frac{1}{4|\tilde{H}|^{2}}\overline{|\{R(g),\tilde{H}\}|^{2}}\label{DoS}$$ ranges from zero to unity and thus quantifies the extent of the symmetry of $H$ with respect to $G$. The above definition of DoS is evidently reasonable in physics since it possesses the following properties (for the proofs see the supplemental material [@supplemental; @material]): (1) Tighter bound when $G$ forms a transformation group $0\leq A(G,H)\leq1/2\leq S(G,H)\leq1$; (2) Independence of DoS on the basis, i.e, $S(WGW^{\dagger},WHW^{\dagger})=S(G,H)$, where $W$ is a unitary transformation and $WGW^{\dagger}=\{WR(g)W^{\dagger}|g\in G\}$; (3) Scaling invariance, i.e., $S(G,\lambda H)=S(G,H)$; (4) Independence of the choice of the zero-point energy, i.e., $S(G,H+\epsilon)=S(G,H)$; (5) Hierarchy property $n_{G'}S(G_{s},H)\leq n_{G}S(G,H)$ for a subset $G_{s}(\subset G\text{\ensuremath{)}}$ with $n_{G'}$ elements.
When $G$ becomes a group, in the spaces $\mathcal{H}^{(l)}$ of its $l$th irreducible representations with finite dimensions $d_{l}$, the DoS $S=S(G,H)$ is re-expressed as $$S=\frac{1}{2}+\sum_{l}\frac{1}{2d_{l}}(\sum_{\alpha}\frac{\langle l,\alpha|H|l,\alpha\rangle}{|\tilde{H}|}-\frac{\mathrm{Tr}\{H\}d_{l}}{d|\tilde{H}|})^{2},\label{DoS under irr-basis}$$ where $|l,\alpha\rangle$ is a basis vector of $\mathcal{H}^{(l)}$ ($\alpha=1,2,...,d_{l}$), and we have used the completeness relations of irreducible representations [@supplemental; @material]. The bisection point 1/2 from property (1) is also reflected in above equation, since each term contributes non-negatively in the summation over $l$. We point out that, by using Eq.(\[DoS under irr-basis\]), DoS is feasible to be computed based on the measurements with respect to the basis $\{|l,\alpha\rangle\}$. Otherwise, for a continuous group, a straightforward calculation of DoS from Eq.(\[DoS\]) should need to carry out the group integral with the Haar measure, e.g., the sum over SO(3) becomes a Lie group integral [@supplemental; @material].
#### Symmetry breaking.
Let $G$ be a symmetry group of the quantum system with Hamiltonian $H$. A perturbation $H'=\lambda V$ breaks the symmetry into the subgroup $G_{s}(\subset G)$, i.e., $[V,R(g)]\ne0$ for $g\in G-G_{s}$ and $[V,R(g')]=0$ for $g'\in G_{s}$. For the total Hamiltonian $H(\lambda)=H+\lambda V$, we calculate the DoS under the symmetry breaking [@supplemental; @material] $$S(G,H(\lambda))=1-\frac{A(G,V)\lambda^{2}}{\lambda^{2}+\xi\lambda+\eta},\label{DoS symmetry breaking}$$ where $A(G,V)$ is the DoAS of the Hermitian operator $V$, the other two coefficients are defined as $\xi=2\mathrm{Tr}\{\tilde{H}\tilde{V}\}|\tilde{V}|^{-2}$ and $\eta=|\tilde{H}|{}^{2}|\tilde{V}|{}^{-2}$. The above equation exactly reflects the duality between symmetry and asymmetry: the maximal symmetry breaking due to the perturbation corresponds to the minimal symmetry of the considered system. When $|\lambda|$ is increased, there exists a special point $\lambda_{\mathrm{A}}=-2\xi^{-1}\eta$ where the DoS reaches a local minimum $S_{\mathrm{min}}=1-A(G,V)\csc^{2}\varphi$; here $\varphi$ is the angle between $\tilde{H}$ and $\tilde{V}$ [@supplemental; @material].
The following two examples are used to illustrate the above conception on quantifying the extent of symmetry and its breaking. First, let us consider a particle residing on a four-site lattice with the following Hamiltonian $$H=\mbox{\ensuremath{\sum}}_{i}\epsilon|i\rangle\langle i|+\mbox{\ensuremath{\sum}}_{ij}h|i\rangle\langle j|,\label{Four site Hamiltonian H}$$ where $|i\rangle$ ($i=0,1,2,3$) is the single particle state with site $i$ occupied. The site energy $\epsilon$ and the hopping strength $h$ are site-independent for the regular tetrahedron geometry [\[]{}see Fig. 1a[\]]{}, and thus $H$ is symmetric to all transformations from the $T_{d}$ group, which contains (combined) rotations and mirror reflections sending a regular tetrahedron into itself [@1964; @Tinkham; @BOOK]. In this example, we let the symmetry $T_{d}$ break into $C_{3v}$ through the following perturbation $$H'=\lambda[\delta_{0}|0\rangle\langle0|+\delta_{1}\mbox{\ensuremath{\sum}}_{i=1}^{3}(|i\rangle\langle0|+h.c.)],\label{Four site Hamiltonian V}$$ where $\lambda\delta_{0}$ and $\lambda\delta_{1}$ are the deviations of the energy and the coupling related to the 0th site. It is well known that $C_{3v}$ has two one-dimensional irreducible representations $A_{1}$ and $A_{2}$, as well as one two-dimensional irreducible representation $E$ [@1964; @Tinkham; @BOOK], which correspond to the three kinds of energy levels with one or two-fold degeneracies.
The above symmetry breaking from $T_{d}$ to $C_{3v}$ is quantified by the DoS through Eq.(\[DoS symmetry breaking\]) with $G=T_{d}$. Straightforward calculation shows exact results $A(T_{d},V)=(2\gamma^{2}+16)^{-1}(\gamma^{2}+4)$, $\xi=16(\gamma^{2}+8)^{-1}\delta_{1}^{-1}h$, and $\eta=\xi\delta_{1}^{-1}h$. Here, $\gamma=\delta_{1}^{-1}\delta_{0}$ is the ratio between the two parameters in $H'$. As shown in Fig. 1b, the DoS reaches unity when $\lambda=0$, indicating the full $T_{d}$ symmetry that possessed by the original Hamiltonian $H$. The symmetry breaking perturbation $H'$ suppresses the DoS first quadratically in $\lambda$ and then, as $|\lambda|$ further increased to approach the strong perturbing limit ($|\lambda|\rightarrow\infty$), reaches a $\gamma$-dependent asymptotically value $(2\gamma^{2}+16)^{-1}(\gamma^{2}+12)$.
In this model, the special point $\lambda_{\mathrm{A}}=-2\delta_{1}^{-1}h$, where the DoS reaches the local minimum, indicates an avoid level crossing in the energy spectrum. To see this, we rewrite $H(\lambda)$ in terms of the standard basis of irreducible representations by using the projection operator method [@1964; @Tinkham; @BOOK; @2011; @Maze]. The resulting four-dimensional Hilbert space contains two $A_{1}$-representations and one $E$-representation of $C_{3v}$ [@1964; @Tinkham; @BOOK]. The two levels that transform according to the two $A_{1}$-representations are coupled and the corresponding avoid level crossing point $\lambda_{*}$ is related to $\lambda_{\mathrm{A}}$ by $$\lambda_{*}=\frac{6-\gamma}{12+\gamma^{2}}\lambda_{\mathrm{A}}.$$ Especially, for $\delta_{0}\ll\delta_{1}$ the avoid level crossing happens approximately at $\lambda_{\mathrm{A}}/2$ [\[]{}see Fig. 1b,c[\]]{}.
Another example demonstrates the DoS of the breaking of the continuous symmetry. The system we considered is a particle with angular momentum $j$, whose Hamiltonian reads $$H=\epsilon J^{2},\mbox{ }H'=\lambda J_{z},\label{SO 3 model Hamiltonian}$$ where $J_{i}$ ($i=x,y,z$) are components of the angular momentum operator and $J^{2}=J_{x}^{2}+J_{y}^{2}+J_{z}^{2}$. In this model, the O(3) symmetry of $H$ is broken by the perturbation, described by $H'$, to O(2). With $G=\mathrm{O(3)}$, the DoS is calculated as $1-[2\lambda^{2}+\epsilon^{2}j(j+2)]^{-1}\lambda^{2}$ [@supplemental; @material]. Unlike the previous model, here the DoS does not show a local minimum and decays monotonically as $\left|\lambda\right|$ increasing. Comparison with the generic result Eq.(\[DoS symmetry breaking\]) indicates the underlying condition $\mathrm{Tr}\{\tilde{H}\tilde{V}\}=0$, which is fulfilled by the Hamiltonian Eq.(\[SO 3 model Hamiltonian\]).
#### Accidental degeneracy.
Accidental degeneracy of energy levels appears in a quantum system when its parameters are changed to cause a level crossing. It is usually not relevant to the geometric symmetry, but our DoS can reveal the existence of the hidden symmetry. Actually, accidental degeneracy also implies symmetry. The greater the degeneracy, the greater the symmetry.
![(color online). (a) Schematics of the three-site model, with symmetry $D_{3}$ breaking into $Z_{2}$ by the perturbation Eq.(\[perturbation Hamiltonian three site model-1\]). (b) Energy spectrum vs $\lambda/h$ for the three-site model, showing two accidental degeneracies at $\lambda_{01}$ and $\lambda_{02}$ (blue vertical line). (c) DoS vs $\lambda/h$ with respect to $G_{\mathrm{T}}$, showing that the accidental degeneracy at $\lambda_{02}$ is identified with the maximum of the DoS.](Fig_2)
For the general Hamiltonian $H(\lambda)$ defined above, we introduce the additional transformations: the $\mathrm{U}(2)$ operations on the two $\lambda$-dependent energy levels of $H(\lambda)$ (or $\mathrm{U}(N)$ operations for the more general $N$ levels crossing), which will become degenerate as $\lambda$ tuned to $\lambda_{0}$. Because $H(\lambda_{0})$ is proportional to the identity operator in the degenerate subspace and, as a result, commuted with all $\mathrm{U}(2)$ operations, the symmetry group $G$ of $H(\lambda_{0})$ is extended to a larger one $G_{\mathrm{T}}=\langle G,\mathrm{U}(2)\rangle,$ which is generated by elements in $G$ and $\mathrm{U}\text{(2)}$. It is expected that the behavior of DoS could manifest the hidden symmetry that implied by the enlarged group $G_{\mathrm{T}}$: the level crossing at $\lambda_{0}$ could result in a local dip in the DoAS, when the parameter $\lambda$ is tuned close to $\lambda_{0}$. To see this, we expand the Hamiltonian linearly around $\lambda_{0}$, i.e., $H(\lambda)\approx H(\lambda_{0})+\partial_{\lambda}H(\lambda_{0})(\lambda-\lambda_{0})$. Since $[R(g),H(\lambda_{0})]=0$ for $g\in G_{\mathrm{T}}$, the DoAS is written as $$\begin{aligned}
A(G_{\mathrm{T}},H(\lambda)) & \propto & A(G_{\mathrm{T}},\partial_{\lambda}H(\lambda_{0}))(\lambda-\lambda_{0})^{2}.\label{DoAS argument-1}\end{aligned}$$ Thus, by the duality, the accidental degeneracy indeed manifests itself as a local maximum at $\lambda_{0}$ in DoS.
To illustrate the above idea, we consider the following three-site model whose Hamiltonian is of the same form as Eq.(\[Four site Hamiltonian H\]) except that $i\in\{1,2,3\}$. And the perturbation term $$H'=\lambda[|1\rangle\langle1|+|3\rangle\langle3|-(|1\rangle\langle3|+h.c.)]\label{perturbation Hamiltonian three site model-1}$$ breaks the symmetry from $D_{3}$ to $Z_{2}=\{e,\sigma\}$. Here, the transformation $\sigma$ interchanges the basis state $|1\rangle$ with $|3\rangle$. The energy spectrum of $H(\lambda)$ contains two $\Gamma_{1}$ levels $E_{1\pm}=\epsilon+h/2\pm\lambda_{02}$ and one $\Gamma_{2}$ level $E_{2}=\epsilon-h+2\lambda$, where $\Gamma_{i=1,2}$ are two irreducible representations of $Z_{2}$. The spectrum shows two accidental degeneracies between the $\Gamma_{1}$ and the $\Gamma_{2}$ levels at $\lambda_{01}=0$ and $\lambda_{02}=3h/2$, respectively [\[]{}see Fig. 2b[\]]{}.
Indeed, at the accidental degeneracy, $H(\lambda_{02})$ becomes more symmetric since there exists the additional symmetric transformations of $\mathrm{U}(2)$: $R(\omega_{0};\hat{n},\omega)=\exp[i(\omega_{0}-\hat{n}\cdot\vec{s}\omega)]$ with pseudo spin-1/2 operators $\vec{s}$ defined by $s_{x}=(|\psi_{1+}\rangle\langle\psi_{2}|+|\psi_{2}\rangle\langle\psi_{1+}|)/2$ *et al*.. Here, $|\psi_{m}\rangle$ is the eigenstate associated with level $E_{m}$. The extended symmetry group $G_{\mathrm{T}}$ for $H(\lambda_{02})$ is still $\mathrm{U}(2)$ since $Z_{2}\subset\mathrm{U}(2)$ [@supplemental; @material]. Thus, the two-fold degenerate subspace supports a two-dimensional irreducible representation of $G_{\mathrm{T}}$. It is shown that the DoS $S(G_{\mathrm{T}},H(\lambda))$= $1-3[\lambda^{2}-\lambda_{02}\lambda+\lambda_{02}^{2}]^{-1}(\lambda-\lambda_{02})^{2}/8$ reaches the unity when $\lambda=\lambda_{02}$ [\[]{}see Fig. 2c[\]]{}. Therefore, the DoS indeed signals the hidden symmetry. We notice that, without the geometric symmetry, the above $\mathrm{U}(2)$ symmetry defined in the subspace spanned by $|\psi_{1+}\rangle$ and $|\psi_{2}\rangle$ at the accidental degenerate point is similar to the dynamical symmetry $\mathrm{SO}(4)$ of the non-relativistic hydrogen atom [@2011; @Sakurai; @BOOK].
#### DoS of quantum state and spontaneous symmetry breaking.
In emergent phenomena, the symmetry of the system ground state can be different from that of the underlying Hamiltonian or Lagrangian. This difference is roughly regarded as the spontaneous symmetry breaking (SSB) [@1961; @Nambu; @PR; @1964; @Higgs; @PRL; @1984; @Anderson; @BOOK]. For a better depiction of those phenomena, we need to introduce the DoS of quantum state (DoSS) $\rho$, which is analog to the DoS of Hamiltonian by the Eq.(\[DoS\]) $$S(G,\rho)=\frac{1}{4|\rho|^{2}}\overline{|\{R(g),\rho\}|^{2}}\label{DoS for quantum state}$$ where $\rho$ is the density matrix of a quantum state. It possesses the similar properties (1)-(5) except for $S(G,\rho)=S(G,\rho+\epsilon)$, which we need not to require for physics.
![(color online). (a) Energy spectrum of a system with degenerate ground states $\{|G_{\alpha}\rangle\}$ at $\lambda=0$. Upon whether $T\rightarrow0$ (i) before or (ii) after $\lambda\rightarrow0$, the thermal equilibrium state Eq.(\[state in SSB\]) approaches different final states. (b) DoS of quantum state (DoSS) $S(\mbox{O(3)},\rho)$ vs $\lambda/\epsilon$ and $\beta\epsilon$ for the angular momentum model. The multi-valued natures of the DoSS at $T=0$ and $\lambda=0$ are reflected as the two non-commuting limiting processes indicated by (i) and (ii).](Fig_3)
We now use DoSS to characterize the SSB in thermodynamics. We consider the thermalization of a quantum system with degenerate ground states $\{|G_{\alpha}\rangle|\alpha=1,2,...,d_{G}\}$, i.e., $H|G_{\alpha}\rangle=\varepsilon_{0}|G_{\alpha}\rangle$ [@2009; @Liao; @arXiv]. At the zero temperature such system will have a non-vanishing entropy $S=k_{\mathrm{B}}\ln d_{G}$, known as the modified third law of thermodynamics [@1987; @Huang; @BOOK]. By introducing a perturbation $H'=\lambda V$ to break the symmetry so that $|G_{\alpha=0}=G_{0}\rangle$ becomes the unique ground state, the thermodynamic SSB is described as the following two non-commutative limiting processes: (i) $T\rightarrow0$ and then $\lambda\rightarrow0$; (ii) $\lambda\rightarrow0$ and then $T\rightarrow0$. In these two non-commutative limiting processes, the following state $$\rho=\frac{1}{Z}\mbox{\ensuremath{\sum}}_{\alpha\ne0}e^{-\varepsilon_{\alpha}/T}|G_{\alpha}\rangle\langle G_{\alpha}|+\frac{1}{Z}e^{-\varepsilon_{0}/T}|G_{0}\rangle\langle G_{0}|+...\label{state in SSB}$$ will approach to $\rho_{\mathrm{f}1}=|G_{0}\rangle\langle G_{0}|$ and $\rho_{\mathrm{f2}}=d_{G}^{-1}\sum_{\alpha}|G_{\alpha}\rangle\langle G_{\alpha}|$ respectively [\[]{}see Fig. 3a[\]]{}.
To see the quantitative details of such thermodynamic SSB, we use the DoSS defined by Eq.(\[DoS for quantum state\]). Let $G$ be a symmetry group of $H$, and $\{|G_{\alpha}\rangle\}$ span an irreducible representation of $G$. Because $\rho_{\mathrm{f2}}$ is proportional to the identity operator, thus $[R(g),\rho_{\mathrm{f2}}]=0$. This implies that the limiting process (ii) results in a final state with $S(G,\rho_{\mathrm{f2}})=1$. On the other hand, from Schur’s theorem [@1964; @Tinkham; @BOOK], in the limiting process (i) there always exists some $g\in G$ such that $[R(g),\rho_{\mathrm{f1}}]\ne0$ and consequently $S(G,\rho_{\mathrm{f1}})<1$. This in turn implies an SSB since the final state $\rho_{\mathrm{f1}}$ does not retain the full symmetry of the underlying microscopic Hamiltonian $H$.
The above relation between the DoSS and $\rho_{\mathrm{f}1,2}$ suggests the multi-valued natures of DoSS at ($T=0$, $\lambda=0$) upon different limiting processes as an SSB witness $$\lim_{\beta\rightarrow\infty}\lim_{\lambda\rightarrow0}S(G,\rho)=1,\mbox{ }\lim_{\lambda\rightarrow0}\lim_{\beta\rightarrow\infty}S(G,\rho)<1,\label{SSB witness}$$ where $\beta=1/T$ is the inverse temperature.
To illustrate the SSB with an example, let us consider the angular momentum model Eq.(\[SO 3 model Hamiltonian\]) again, which shows a spontaneous breaking of O(3) symmetry. In the subspace with $j=1/2$, the ground state is two-fold degenerate without $H'$, i.e., $|1/2,\pm1/2\rangle$. For a generic thermal state $\rho=Z^{-1}\exp[-\beta H(\lambda)]$, DoSS is shown to be $S(\mbox{O(3)},\rho)=(3+\cosh^{-1}\beta\lambda)/4$ [@supplemental; @material], whose multi-valued natures at ($T=0$, $\lambda=0$) is shown in Fig. 3b. Specifically, when $\beta\epsilon\gg1$ while $\lambda\ne0$ the DoSS is nearly at a constant value 3/4. Then, by tuning the coupling $\lambda$ to zero, the DoSS remains fixed at the same constant value (see (i) in Fig. 3b). In contrast, if one first fix the coupling $\lambda=0$ at the high temperature, the DoSS as a function of $\beta$ from zero to infinity will follow the blue arrowed line (corresponding to the possess (ii)) in Fig. 3b. In the latter case, the DoSS at large $\beta$ is unity. In analog to Eq.(\[SSB witness\]), in this example it is shown that $$\lim_{\beta\rightarrow\infty}\lim_{\lambda\rightarrow0}S(G,\rho)=1,\mbox{ }\lim_{\lambda\rightarrow0}\lim_{\beta\rightarrow\infty}S(G,\rho)=\frac{3}{4}.$$ Here, $G=\mathrm{O}(3)$. On the other hand, the difference in DoSS reflected by above equations is also understood through inspecting on the final state, which is $\rho_{\mathrm{f1}}=\left|1/2,-1/2\left\rangle \right\langle 1/2,-1/2\right|$ or $\rho_{\mathrm{f2}}=2^{-1}\sum_{m}|1/2,m\rangle\langle1/2,m|$ upon limiting processes (i/ii). Clearly, $\rho_{\mathrm{f1}}$ is not invariant under the $\pi$-rotation that represented by the $\sigma_{x}$ operation in the $j=1/2$ subspace, thus results in a DoSS smaller than unity.
#### Conclusion.
In this letter, we introduce a continuous measure, the degree of symmetry (DoS), for the symmetry of quantum system, which largely extrapolates the dichotomous approach of symmetry based on group representation theory. It is shown that the DoS possesses some good properties, such as basis-independent, invariant under the zero-point energy shifting as well as the scaling transformation. Since it can be expressed as an average of physical operators under the basis of irreducible representations for transformation groups, this measure is thus computable and detectable based on some quantum measurements.
In contrast to the previous explorations based on the abstract concepts of fuzz set [@2010; @Garrido; @Symmetry-1] and transform information [@1997; @Vstovsky; @Found; @Phys-1], our introduced DoS can feasibly open many applications in physics. As illustrated in this letter, the DoS is capable of identifying symmetry relevant phenomena and effects, such as the accidental level crossings and the spontaneous symmetry breaking. This, therefore, implies that the DoS could be a useful measure in related future studies, e.g., in characterizing systems near quantum criticality since it is closely related to the multi-level crossings [@2006; @Quan; @PRL].
We thank X. F. Liu, P. Zhang, X. G. Wang, S. X. Yu, and L. P. Yang for helpful discussions. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11421063, 11534002, and 11475254) and the National 973 program (Grant Nos. 2014CB921403, 2012CB922104, and 2014CB921202).
[10]{} K. Sundermeyer, *Symmetries in Fundamental Physics* 2nd Ed., (Springer, Switzerland, 2014).
P. W. Anderson, *Basic Notions of Condensed Matter Physics*, (Benjamin-Cummings Publishing Company, Canada, 1984).
O. Penrose and L. Onsager, Phys. Rev. **104**, 576 (1956).
A. J. Leggett and F. Sols, Found. Phys. **21**, 353 (1991).
Y. Nambu and G. Jona-Lasinio, Phys. Rev. **122**, 345 (1961).
E. Witten, Nature, **429**, 507 (2004).
T. W. B. Kibble, Phil. Trans. R. Soc. A **373**, 33 (2014).
F. Englert and R. Brout, Phys. Rev. Lett. **13**, 321 (1964).
P. W. Higgs, Phys. Rev. Lett. **13**, 508 (1964).
M. Liu, D. A. Powell, I. V. Shadrivov, M. Lapine, and Y. S. Kivshar, Nat. Commun. **5**, 4441 (2014).
N. Navon, A. L. Gaunt, R. P. Smith, and Z. Hadzibabic, Science, **347**, 167 (2015).
Y. Saito and H. Hyuga, Rev. Mod. Phys. **85**, 603, (2013).
V. Duboc, P. Dufourcq, P. Blader, and M. Roussigné, Ann. Rev. Genet. **49**, 26 (2015).
Kerson Huang, *Statistical Mechanics* 2nd Ed., (Wiley, New York, 1987).
J. Watrous, *Theory of Quantum Information* draft Ed., (2015).
see supplemental material at ...
M. Tinkham, *Group Theory and Quantum Mechanics*, (McGraw-Hill, United States of America, 1964).
J. R. Maze, A. Gali, E. Togan, Y. Chu, A. Trifonov, E. Kaxiras, and M. D. Lukin, New J. Phys. **13**, 025025 (2011).
J. J. Sakurai and J. Napolitano, *Modern Quantum Mechanics* 2nd Ed., (Addison-Wesley, Boston, 2011).
J. Q. Liao, H. Dong, X. G. Wang, X. F. Liu, and C. P. Sun, arXiv: 0909.1230v1.
A. Garrido, Symmetry **2**, 707, (2010).
G. V. Vstovsky, Found. Phys. **27**, 10 (1997).
H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, Phys. Rev. Lett. **96**, 140604 (2006).
|
---
abstract: 'We compare our Monte Carlo reaction rates (see Paper II of this series) to previous results that were obtained by using the classical method of computing thermonuclear reaction rates. For each reaction, the comparison is presented using two types of graphs: the first shows the change in reaction rate uncertainties, while the second displays our new results normalized to the previously recommended reaction rate. We find that the rates have changed significantly for almost all reactions considered here. The changes are caused by (i) our new Monte Carlo method of computing reaction rates (see Paper I of this series), and (ii) newly available nuclear physics information (see Paper III of this series).'
address:
- 'Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599-3255, USA; Triangle Universities Nuclear Laboratory, Durham, NC 27708-0308, USA'
- 'Centre de Spectrométrie Nucléaire et de Spectrométrie de Masse (CSNSM), UMR 8609, CNRS/IN2P3 and Université Paris Sud 11, Bâtiment 104, 91405 Orsay Campus, France'
author:
- 'C. Iliadis'
- 'R. Longland'
- 'A. E. Champagne'
- 'A. Coc'
title: |
Charged-Particle Thermonuclear Reaction Rates:\
IV. Comparison to Previous Work
---
,
,
Introduction {#intro}
============
In the final paper of this series, referred to as Paper IV, we compare our Monte Carlo reaction rates to previously derived results. In Paper I we point out that previously reported (or “classical") reaction rates have a different meaning from our Monte Carlo rates. The latter are presented in Paper II and are subject to a straightforward statistical interpretation, while the former are statistically meaningless. Consequently, one has to be very careful when comparing such qualitatively and quantitatively different results. Since we did not want to complicate the discussion in Paper II by comparing “apples with oranges", we avoided so far a comparison between present and previous reaction rates.
On the other hand, we realize that for practical purposes many readers are precisely interested in such a comparison: by how much did a particular reaction rate change? How accurate was a previous reaction rate? Does a given rate used in a particular stellar model need updating or not? How significant is this new Monte Carlo method compared to classical techniques of computing thermonuclear reaction rates? In order to answer these questions, we present a comparison of our new rates with previous results.
Section 2 provides a few useful comments. A brief summary is presented in Sec. 3. Results are presented in graphical form in the Appendix, together with information on how to interpret the figures.
Discussion
==========
The comparison of Monte Carlo with classical reaction rates is presented in graphical form in the Appendix. For each reaction considered, two types of graphs are presented. The first shows the change in reaction rate uncertainties, while the second graph displays the new results normalized to the previously recommended reaction rate. Detailed information on how to interpret the graphs is also given in the Appendix.
Generally, the previous results used for the comparison are adopted from the latest reference that lists recommended reaction rates and, if available, associated uncertainties in [*tabular form*]{}. This reference is provided after each rate comparison figure. Most of the previous reaction rates are adopted from Angulo et al. [@Ang99] or Iliadis et al. [@Ili01]. For some reactions, newer rates have been presented in the literature, but in graphical form only. In such cases we requested (and obtained) from the original authors the numerical reaction rates used to produce the literature plots.
Visual inspection of the graphs reveals that for almost all reactions our new thermonuclear rates differ substantially from previous results. The changes are caused both by our new Monte Carlo method of computing reaction rates (see Paper I of this series), and by newly available nuclear physics information (see Paper III of this series). We did not analyze in detail for each reaction the sources of a rate change at a given temperature. Such a comprehensive analysis would require considerably more efforts in computing time and manpower. Nevertheless, we feel that many readers may find the graphs presented here useful for a variety of purposes.
Summary
=======
Our new Monte Carlo reaction rates (see Paper II) are compared with previous results. For each reaction the comparison is presented in graphical form. It is found that the rates of almost all reactions considered here have changed significantly. The changes are caused by our new Monte Carlo method of computing reaction rates (see Paper I) and by newly available nuclear physics information (see Paper III).
Acknowledgement
===============
The authors would like to thank Dan Bardayan, Shawn Bishop, Barry Davids and Hendrik Schatz for providing us with numerical results for their previous reaction rates. This work was supported in part by the U.S. Department of Energy under Contract No. DE-FG02-97ER41041.
For each reaction, the comparison of present (Monte Carlo) with previous (classical) thermonuclear rates is presented in two graphs. The x-axis shows the entire temperature range of T=0.01-10.0 GK, while the y-axis displays a reaction rate ratio. The meaning of the curves is given below.\
AAAAAAAAAAAAAAAA= $N_A\left<\sigma v\right>_{high}^{MC}$ Present high Monte Carlo reaction rate\
$N_A\left<\sigma v\right>_{med}^{MC}$ Present median Monte Carlo reaction rate\
$N_A\left<\sigma v\right>_{low}^{MC}$ Present low Monte Carlo reaction rate\
$N_A\left<\sigma v\right>_{high}^{clas}$ Previous high classical reaction rate\
$N_A\left<\sigma v\right>_{ad}^{clas}$ Previous adopted classical reaction rate\
$N_A\left<\sigma v\right>_{low}^{clas}$ Previous low classical reaction rate\
\
[*Top graph:*]{}\
\
upper solid line $N_A\left<\sigma v\right>_{high}^{MC}/N_A\left<\sigma v\right>_{med}^{MC}$\
lower solid line $N_A\left<\sigma v\right>_{low}^{MC}/N_A\left<\sigma v\right>_{med}^{MC}$\
upper dashed line $N_A\left<\sigma v\right>_{high}^{clas}/N_A\left<\sigma v\right>_{ad}^{clas}$\
lower dashed line $N_A\left<\sigma v\right>_{low}^{clas}/N_A\left<\sigma v\right>_{ad}^{clas}$\
\
[*Bottom graph:*]{}\
\
upper thin solid line $N_A\left<\sigma v\right>_{high}^{MC}/N_A\left<\sigma v\right>_{ad}^{clas}$\
thick solid line $N_A\left<\sigma v\right>_{med}^{MC}/N_A\left<\sigma v\right>_{ad}^{clas}$\
lower thin solid line $N_A\left<\sigma v\right>_{low}^{MC}/N_A\left<\sigma v\right>_{ad}^{clas}$\
![\[\] Previous reaction rates: Ref. [@CF88]. No numerical reaction rates are presented in more recent work of Ref. [@GO90].](PlotComparec14pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@CF88]. No numerical reaction rates are presented in more recent works of Refs. [@BU07; @GA87; @GO92; @LU04].](PlotComparec14ag.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Goe00]. Rate uncertainties have not been determined in their work.](PlotComparen14ag.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ang99].](PlotComparen15ag.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Fis06].](PlotCompareo15ag.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ang99].](PlotCompareo16pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ang99].](PlotCompareo16ag.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Cha07].](PlotCompareo17pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Cha07].](PlotCompareo17pa.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ang99].](PlotCompareo18pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ang99]. A more recent rate is displayed in Fig. 3 of Ref. [@La08], but numerical rate values are not available.](PlotCompareo18pa.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ang99].](PlotCompareo18ag.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Bar00].](PlotComparef17pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@CO00].](PlotComparef18pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@CO00].](PlotComparef18pa.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@VA98]. Only lower and upper rate limits are shown in Fig. 7 of Ref. [@VA98]; we estimated the previous adopted rate by using the geometric mean of the rate limit values.](PlotComparene19pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ang99].](PlotComparene20pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ang99].](PlotComparene20ag.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotComparene21pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotComparene22pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ang99].](PlotComparene22ag.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Jae01].](PlotComparene22an.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@DAu04]. Rate uncertainties have not been determined previously.](PlotComparena21pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ang99].](PlotComparena22pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotComparena23pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotComparena23pa.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Cag01]. A more recent rate is reported in Tab. IV of Ref. [@He07], but the differences compared to Ref. [@Cag01] are relatively small.](PlotComparemg22pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Her98]. A more recent rate is displayed in Fig. 2 of Ref. [@Vis07], but numerical rate values are not available.](PlotComparemg23pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotComparemg24pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@str08]. Rate uncertainties have not been determined previously.](PlotComparemg24ag.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotComparemg25pgt.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotComparemg25pgg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotComparemg25pgm.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotComparemg26pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Sch97]. Previous rate uncertainties are displayed in Fig. 2 of Ref. [@Sch97], but numerical values for the upper and lower rate limits are not available.](PlotCompareal23pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Her95]. Rate uncertainties have not been determined previously.](PlotCompareal24pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Wre09b].](PlotCompareal25pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotCompareal26gpg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Har00]. The entry for the recommended rate at T=0.5 GK in Tab. 5 of Ref. [@Har00] is presumably a misprint.](PlotCompareal27pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotCompareal27pa.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Guo06]. Rate uncertainties have not been determined previously.](PlotComparesi26pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili99].](PlotComparesi27pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ang99].](PlotComparesi28pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotComparesi29pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotComparesi30pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Her95]. Rate uncertainties have not been determined previously.](PlotComparep27pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Bar07].](PlotComparep29pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotComparep31pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotComparep31pa.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Wre09]; two values given in their Tab. IV, at T=0.04 GK (“high" value) and T=0.25 GK (“low" value), are erroneous.](PlotCompares30pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili99].](PlotCompares31pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotCompares32pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Her95]. Rate uncertainties have not been determined previously.](PlotComparecl31pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Sch05]. Note that the [*stellar*]{} rate is displayed in the original work, but we compare our results to the previous [*laboratory*]{} rate.](PlotComparecl32pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotComparecl35pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotComparecl35pa.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Her95]. Rate uncertainties have not been determined previously.](PlotComparear34pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili99].](PlotComparear35pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotComparear36pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01]. Rate uncertainties have not been determined previously.](PlotComparek35pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili99].](PlotCompareca39pg.eps){height="17.05cm"}
![\[\] Previous reaction rates: Ref. [@Ili01].](PlotCompareca40pg.eps){height="17.05cm"}
[00]{} C. Angulo et al., Nucl. Phys. A 656 (1999) 3. D. W. Bardayan et al., Phys. Rev. C 62 (2000) 055804. D. W. Bardayan et al., Phys. Rev. C 74 (2006) 045804; and D. W. Bardayan, private communication (2009). D. W. Bardayan et al., Phys. Rev. C 76 (2007) 045803; and D. W. Bardayan, private communication (2009). L. Buchmann et al., Phys. Rev. C 75 (2007) 012804(R). J. A. Caggiano et al., Phys. Rev. C 64 (2001) 025802. G. R. Caughlan and W. A. Fowler, At. Data Nucl. Data Tables 40 (1988) 283. A. Chafa et al., Phys. Rev. C 75 (2007) 035810. A. Coc et al., Astron. Astrophys. 357 (2000) 561. J. M. D’Auria et al., Phys. Rev. C 69 (2004) 065804; and S. Bishop, private communication (2009). J. L. Fisker, J. Görres, M. Wiescher and B. Davids, Astrophys. J. 650 (2006) 332; and B. Davids, private communication (2009). M. Gai et al., Phys. Rev. C 36 (1987) 1256. J. Görres et al., Nucl. Phys. A 517 (1990) 329. J. Görres et al., Nucl. Phys. A 548 (1992) 414. J. Görres et al., Phys. Rev. C 62, 055801 (2000). B. Guo et al., Phys. Rev. C 73 (2006) 048801. S. Harissopulos et al., Eur. Phys. J. A 9 (2000) 479. J. J. He et al., Phys. Rev. C 76 (2007) 055802. H. Herndl et al., Phys. Rev. C 52 (1995) 1078. H. Herndl, M. Fantini, C. Iliadis, P. M. Endt and H. Oberhummer, Phys. Rev. C 58 (1998) 1798. C. Iliadis et al., Astrophys. J. 524 (1999) 434. C. Iliadis et al., Astrophys. J. Suppl. 134 (2001) 151. M. Jaeger et al., Phys. Rev. Lett. 87, 202501 (2001). M. La Cognata et al., Phys. Rev. Lett. 101 (2008) 152501. M. Lugaro et al., Astrophys. J. 615 (2004) 934. H. Schatz et al., Phys. Rev. Lett. 79 (1997) 3845. H. Schatz et al., Phys. Rev. C 72 (2005) 065804; and H. Schatz, private communication (2009). E. Strandberg et al., Phys. Rev. C 77 (2008) 055801. G. Vancraeynest et al., Phys. Rev. C 57 (1998) 2711. D. W. Visser et al., Phys. Rev. C 76 (2007) 065803. C. Wrede et al., Phys. Rev. C 79 (2009) 045808. C. Wrede, Phys. Rev. C 79 (2009) 035803.
|
---
author:
-
-
bibliography:
- 'IEEEabrv.bib'
- 'main.bib'
title: Wind speed prediction using multidimensional convolutional neural networks
---
Deep learning, Wind speed prediction, Convolutional neural networks, Feature learning, Short-term forecasting
|
---
abstract: 'We study heat transport in a one-dimensional inhomogeneous quantum spin $1/2$ system. It consists of a finite-size XX spin chain coupled at its ends to semi-infinite XX and XY chains at different temperatures, which play the role of heat and spin reservoirs. After using the Jordan-Wigner transformation we map the original spin Hamiltonian into a fermionic Hamiltonian, which contains normal and pairing terms. We find the expressions for the heat currents and solve the problem with a non-equilibrium Green’s function formalism. We analyze the behavior of the heat currents as functions of the model parameters. When finite magnetic fields are applied at the two reservoirs, the system exhibits rectifying effects in the heat flow.'
address:
- '$^1$ Departamento de Física, FCEyN, Universidad de Buenos Aires, Pabellón 1,Ciudad Universitaria, 1428, Buenos Aires, Argentina.'
- '$^2$ Comisión Nacional de Energ[í]{}a Atómica, Centro Atómico Bariloche and Instituto Balseiro, 8400 S.C. de Bariloche, Argentina.'
author:
- 'Liliana Arrachea$^1$ , Gustavo S. Lozano$^1$ and A. A. Aligia$^2$'
title: '** Thermal transport in one-dimensional spin heterostructures**'
---
Introduction
=============
In the last decade there has been a renewed interest related to the research of thermal transport in one-dimensional magnetic systems [@Solo; @dash] . Most of these studies have been motivated by unusual high values of thermal conductance of some materials, as for example reported in Ref. . From the theoretical side, several works calculating and discussing thermal conductivity in different model Hamiltonians have also appeared [@lista5], [@lista3]. On the other hand, spin Hamiltonians provide the natural scenario to implement quantum computation. This motivated interesting proposals of a variety of physical systems, like arrays of quantum dots, optical lattices and nuclear magnetic resonace experiments which are architectured to effectively behave like one-dimensional spin systems. [@design]
Most of the theoretical studies on heat transport in spin systems are performed within the linear response regime, assuming a very small temperature gradient $\bigtriangledown T$, and using a Kubo formula. Altough this formula is widely used in calculating thermal transport properties it has several conceptual difficulties, particularly if $\bigtriangledown T$ is not small [@mahan; @dash]. The thermal averaging assumes a constant temperature, and (as in all Kubo formulas), the time dependence is governed by the total Hamiltonian $H=H_{0}+H'$, including the perturbation $H' \sim \bigtriangledown T$ , while the thermodynamic averaging is done with $H_{0}$, assuming a fast evolution. For finite $\bigtriangledown T$ the separation of $H$ into an unperturbed part $H_{0}$ and a perturbation $H'$ is in principle ambiguous. On the other hand, real experiments are performed by coupling the system under study to macroscopic systems with well defined temperatures that act as reservoirs. Therefore, it seems desirable to develop alternative approaches designed to treat systems out of equilibrium. Important progress has been achieved by studying the properties of non-equilibrium steady states of XX and XY chains within an algebraic setting which allows one to obtain explicit analytical expressions for different quantities such as entropy production [@pillet]. Recently, the usefulness of Kubo formula for the investigation of heat transport in quantum systems has been discussed [@gemmer1] and the investigation of energy transport through spin systems beyond Kubo formula has been addressed on the basis of master equations and quantum Monte Carlo methods. [@gemmer2] Nevertheless, it is not clear how to extend these ideas to more general situations. In this work we address the problem of thermal transport in one dimensional magnetic systems from a different perspective. We study a problem that to the best of our knowledge has not been considered yet. We study the heat flow through a *spin chain heterostructure*, which we generically define as a set of finite or semi-infinite spin chains attached by their ends. Each piece of the heterostructure can be in principle described by a [*different*]{} Hamiltonian. For definiteness, we shall consider in this work a finite central system connected to two (left and right) semi-infinite chains, with a* finite* temperature difference between them[@karevski; @gemmer2]. This problem is the thermal analog of electronic transport through mesoscopic structures, nanodevices or molecules connected to conducting leads with a finite applied bias voltage, a subject of intense research in recent years. In fact, this type of set-up is the common situation found in the study of charge transport in electronic systems, where a central system is connected to charge reservoirs that also act as thermal baths. This is the basis of the “Landauer” approach, which is one of the most common frameworks to study transport properties of meso- and nano- devices in the last years. [@transporte]
In addition, as it is well known, one-dimensional spin $1/2$ systems can be mapped, via the Jordan-Wigner transformation to fermionic systems. Thus, the model under investigation is equivalent to an electronic heterostructure where very well established techniques, as the Schwinger-Keldysh non equilibrium Green’s functions method can be applied [@Kel].
We focus on a simple device where the central and right parts of the system can be described by XX spin 1/2 chains, and the left part corresponds to an anisotropic XY chain. The Jordan-Wigner transformation maps these models into bilinear fermionic systems, rendering the theoretical study simpler. We show that this simple device presents an interesting physical effect: due to a mechanism reminiscent of Andreev reflections in superconductors, this device could act as thermal diode. This rectifying effect might be useful for applications. We argue that this is a generic feature, which remains valid for anisotropic XYZ models.
The paper is organized as follows: in Section II we present the model, and the non-equilibrium formalism based on Green’s functions. Section III contains the numerical results. Section IV is a summary and discussion.
The formalism
=============
Model
-----
![[ (Color online) Schematic digram showing the set up of the system. The right and central chains correspond to $XX$ models while the left chain is described by an $XY$ model. Left and right chains are taken semi-infinite at temperatures $T_L$ and $T_R$ respectively.]{}[]{data-label="dibujo"}](fig0.eps "fig:"){width="80mm" height="50mm"} [ ]{}
We consider a system of three one-dimensional spin 1/2 chains coupled through their ends as shown in Fig. \[dibujo\]. The system is described by the following Hamiltonian $%
H=H_{L}+H_{C}+H_{R}+H_{coup}$, where $$\begin{aligned}
H_{\alpha } &=&J_{\alpha }\sum_{i=1}^{N_{\alpha }-1}[S_{\alpha
,i}^{x}S_{\alpha ,i+1}^{x}+\lambda _{\alpha }S_{\alpha ,i}^{y}S_{\alpha
,i+1}^{y}] \nonumber \\
&&-B_{\alpha }\sum_{i=1}^{N_{\alpha }}S_{\alpha ,i}^{z}, \nonumber \\
H_{coup} &=&J^{\prime }[S_{L,1}^{x}S_{C,1}^{x}+S_{L,1}^{y}S_{C,1}^{y}]+
\nonumber \\
&&J^{\prime }[S_{R,1}^{x}S_{C,N_{C}}^{x}+S_{R,1}^{y}S_{N_{C},1}^{y}],
\label{ham1}\end{aligned}$$where in the first line, the index $\alpha =L,C,R$ (left, center, right) labels the different chains. Each chain is characterized by its nearest-neighbor exchange $J_{\alpha }$, with anisotropy $\lambda _{\alpha }$ for the ratio of the interaction along the $y$ direction with respect to that along $x,$ and the magnetic field $B_{\alpha }$ applied along the $z$ direction. The isotropic case for a given chain corresponds to $\lambda
_{\alpha }=1$. The Hamiltonian $H_{coup}$ describes the coupling between the central chain, and the left and right ones. In terms of the representation for the spin operators: $S^{\pm }=S^{x}\pm iS^{y}$, the above Hamiltonian reads $$\begin{aligned}
H_{\alpha } &=&\frac{J_{\alpha }}{4}\sum_{l=1}^{N_{\alpha }-1}[(1-\lambda
_{\alpha })(S_{\alpha ,l}^{+}S_{\alpha ,l+1}^{+}+S_{\alpha ,l}^{-}S_{\alpha
,l+1}^{-}) \nonumber \\
&&+(1+\lambda _{\alpha })(S_{\alpha ,l}^{+}S_{\alpha ,l+1}^{-}+S_{\alpha
,l}^{-}S_{\alpha ,l+1}^{+})] \nonumber \\
&&-B_{\alpha }\sum_{l=1}^{N_{\alpha }}S_{\alpha ,l}^{z}, \nonumber \\
H_{coup} &=&\frac{J^{\prime }}{2}%
[S_{L,1}^{+}S_{C,1}^{-}+S_{L,1}^{-}S_{C,1}^{+}]+ \nonumber \\
&&\frac{J^{\prime }}{2}%
[S_{R,1}^{-}S_{C,N_{C}}^{+}+S_{R,1}^{+}S_{N_{C},1}^{-}]. \label{ham2}\end{aligned}$$For the isotropic case, $\lambda _{\alpha }=1$ only the flip-flop terms with products of one raising and and one lowering spin operators survive.
We now introduce the Jordan-Wigner transformation to map the spin 1/2 Hamiltonian into a fermionic Hamiltonian through: $S_{\alpha
,l}^{+}=f_{\alpha ,l}^{\dagger }\exp (i \pi \sum_{j}^{\prime }f_{\alpha
,j}^{\dagger }f_{\alpha ,j})$, where $\sum_{j}^{\prime }$ denotes a sum over all the positions located at the left of the position $\alpha ,l$. Similarly, the other spin operators transform as $S_{\alpha ,l}^{-}=\exp
(-i \pi \sum_{j}^{\prime }f_{\alpha ,j}^{\dagger }f_{\alpha ,j})f_{\alpha ,l}$ and $S_{\alpha ,l}^{z}=f_{\alpha ,l}^{\dagger }f_{\alpha ,l}-1/2$, where the operators $f_{l},f_{l}^{\dagger }$ obey fermionic commutation rules: $%
\{f_{\alpha ,l},f_{\alpha ^{\prime },l^{\prime }}^{\dagger }\}=\delta
_{l,l^{\prime }}\delta _{\alpha ,\alpha ^{\prime }}$, and $\{f_{\alpha
,l}^{\dagger },f_{\alpha ^{\prime },l^{\prime }}^{\dagger }\}=\{f_{\alpha
,l},f_{\alpha ^{\prime },l^{\prime }}\}=0$. Substituting in the Hamiltonian (\[ham2\]), we get $$\begin{aligned}
H_{\alpha } &=&\sum_{l=1}^{N_{\alpha }-1}\{w_{\alpha }[f_{\alpha
,l}^{\dagger }f_{\alpha ,l+1}+f_{\alpha ,l+1}^{\dagger }f_{\alpha ,l}]
\nonumber \\
&&+\Delta _{\alpha }[f_{\alpha ,l}^{\dagger }f_{\alpha ,l+1}^{\dagger
}+f_{\alpha ,l+1}f_{\alpha ,l}]\} \nonumber \\
&&-\mu _{\alpha }\sum_{l=1}^{N_{\alpha }}f_{\alpha ,l}^{\dagger }f_{\alpha
,l}, \nonumber \\
H_{coup} &=&w^{\prime }[f_{L,1}^{\dagger }f_{C,1}+f_{C,1}^{\dagger }f_{L,1}]+
\nonumber \\
&&w^{\prime }[f_{C,N_{C}}^{\dagger }f_{R,1}+f_{R,1}^{\dagger }f_{C,N_{C}}],
\label{ham3}\end{aligned}$$where $w_{\alpha }=J_{\alpha }(1+\lambda _{\alpha })/4$, $\Delta _{\alpha
}=J_{\alpha }(1-\lambda _{\alpha })/4$, $\mu _{\alpha }=B_{\alpha }^{z}$, and $w^{\prime }=J^{\prime }/2$. Therefore, in the language of fermionic operators, the Hamiltonian contains normalterms, with a creation and a destruction operator, as well as anomalousterms, with two creation or two destruction operators. The normal ones are a hopping term between nearest neighbors ($w_{\alpha}$), which is originated in the flip-flop spin terms and a chemical potential ($\mu_{\alpha}$) coupled to the fermionic density, which comes from the magnetic field pointing along the $z$-direction. The anomalous terms ($\Delta_{\alpha}$) are similar to those of a one-dimensional Hamiltonian with a gap function with $p$-wave symmetry, decoupled in the Bardeen-Cooper-Schrieffer (BCS) approximation and are originated by the anisotropy between the $X$ and $Y$ exchange interaction.
Inspired in this analogy, we focus our study on a junction between a chain with isotropic interactions ($XX$ spin chain) and an anisotropic one ($XY$ spin chain), which in the fermionic language is similar to a normal-superconductor junction. Such a situation is realized in a configuration with $\lambda _{R}=\lambda _{C}=0$ and $\lambda _{L}\neq 0$. We also assume that the left and right chains are at temperatures $T_{L}$ and $T_{R}$ respectively and they are both of infinite length( $%
N_{L}\rightarrow \infty $ and $N_{R}\rightarrow \infty $)
Energy balance
--------------
A consistent procedure to define an expression for the heat current from first principles, is to analyze the evolution of the energy stored in a small volume of the system and derive the corresponding equation for the conservation of the energy. [@liliheat] For the present Hamiltonian we choose an elementary volume containing two nearest-neighbor positions of the chain. We place the volume enclosing the sites $l,l+1$ within the central (XX) chain, which in the fermionic language contains only normal terms. We work in units where $\hbar=1$. The equation for the conservation of the energy enclosed by this volume is $$\begin{aligned}
\frac{dE_{l,l+1}}{dt} &=&\frac{J_{C}}{2}\frac{d}{dt}\langle
S_{l}^{+}S_{l+1}^{-}+S_{l}^{-}S_{l+1}^{+}\rangle -B_{C}^{z}\frac{d\langle
S_{l}^{z}\rangle }{dt} \nonumber \\
&=&w_{C}\frac{d}{dt}\langle f_{C,l}^{\dagger }f_{C,l+1}+f_{C,l+1}^{\dagger
}f_{C,l}\rangle -
\nonumber \\
&&B_{C}^{z}\frac{d}{dt}\langle f_{C,l}^{\dagger
}f_{C,l}\rangle \nonumber \\
&=&-iw_{C}\langle \lbrack H,f_{C,l}^{\dagger }f_{C,l+1}+f_{C,l+1}^{\dagger
}f_{C,l}]\rangle \nonumber \\
&&+i\mu _{C}\langle \lbrack H,f_{C,l}^{\dagger
}f_{C,l}]\rangle=J^Q_{l+1,l+2}-J^Q_{l-1,l}, \label{de}\end{aligned}$$where $J^Q_{l,l+1}$ is the heat current flowing from $l$ to $l+1$, which in the present setup coincides with the energy current. Its explicit expression is obtained from the evaluation of the above commutator, which gives $$\begin{aligned}
J^Q &=&J^Q_{l,l+1}=J^Q_{l-1,l} \nonumber \\
&=&i\varepsilon _{l,l+1}^{C}(\varepsilon _{l-1,l}^{C}\langle
f_{C,l-1}^{\dagger }f_{C,l+1}-f_{C,l+1}^{\dagger }f_{C,l-1}\rangle
\nonumber \\
&&+\varepsilon _{l+1,l+1}^{C}\langle f_{C,l+2}^{\dagger
}f_{C,l+1}-f_{C,l+1}^{\dagger }f_{C,l+2}\rangle ), \label{j1}\end{aligned}$$where $\varepsilon _{l,l^{\prime }}^{C}$ denotes the matrix element $%
l,l^{\prime }$ of the Hamiltonian $H_{C}$. In order to evaluate the above current it is convenient to introduce the lesser Green’s functions $$G_{\alpha l,\beta l^{\prime }}^{<}(t,t^{\prime })=i\langle f_{\beta
l^{\prime }}^{\dagger }(t^{\prime })f_{\alpha l}(t)\rangle , \label{gle}$$thus $$\begin{aligned}
J^Q &=&2\mbox{Re}\{\varepsilon
_{l,l+1}^{C}G_{Cl+1,Cl-1}^{<}(t,t)\varepsilon _{l-1,l}^{C} \nonumber \\
&&+\varepsilon _{l+1,l+1}^{C}G_{Cl+1,Cl+2}^{<}(t,t)\varepsilon
_{l+2,l+1}^{C}\}. \label{je}\end{aligned}$$The lesser Green’s functions are one of the basic elements within Keldysh non-equilibrium Green’s function formalism [@Kel]. They are evaluated by solving the equations of motion (Dyson’s equations), which for our model can be written as follows $$\sum_{k}G_{Cl,Ck}^{<}(\omega )[\delta _{k,l^{\prime }}\omega -\varepsilon
_{k,l^{\prime }}^{C}]=0 \label{gle2}$$for coordinates $l,l^{\prime }$ lying within the central chain. We have used the stationary property of the system, as a consequence of which the Green’s functions depend on the difference $t-t^{\prime }$, which allows us to transform: $G_{j,j^{\prime }}^{<}(t-t^{\prime })=\int_{-\infty }^{+\infty
}d\omega /(2\pi )e^{-i\omega (t-t^{\prime })}G_{j,j^{\prime }}^{<}(\omega )$. Thus, using the above equation in Eq. (\[je\]) the energy current can be also expressed in the following way $$\begin{aligned}
J^Q & = & 2\mbox{Re}\{\int_{-\infty }^{+\infty }\frac{d\omega }{2\pi }\omega
[\varepsilon _{l,l+1}^{C}G_{Cl+1,Cl}^{<}(\omega ) \nonumber \\
& & +\varepsilon _{l+1,l+1}^{C}G_{Cl+1,Cl+1}^{<}(\omega ) ]\}. \label{jqq1}\end{aligned}$$ However $\mbox{Re}
\{G_{j+1,j+1}^{<}(\omega )\}=0$. Thus, the heat current reduces to $$J^Q=2\mbox{Re}\{\int_{-\infty }^{+\infty }\frac{d\omega }{2\pi }\omega
\varepsilon _{l,l+1}^{C}G_{Cl+1,Cl}^{<}(\omega ) \}. \label{jqq}$$ Similarly, if we evaluate the heat current through the contacts $L-C$ and $C-R$, we find $$\begin{aligned}
J^Q &=&2w^{\prime }\mbox{Re}\{\int_{-\infty }^{+\infty }\frac{d\omega }{%
2\pi }\omega G_{L1,C1}^{<}(\omega )\}, \nonumber \\
&=&2w^{\prime }\mbox{Re}\{\int_{-\infty }^{+\infty }\frac{d\omega }{2\pi }%
\omega G_{CN,R1}^{<}(\omega )\}. \label{jq2}\end{aligned}$$
Solving Dyson’s equations
-------------------------
In order to evaluate $G^{<}$ and the heat current we follow a treatment close to that presented in Ref. . We define the retarded normal and Gorkov Green’s functions $$\begin{aligned}
G_{\alpha l,\beta l^{\prime }}^{R}(t,t^{\prime }) &=&-i\Theta (t-t^{\prime
})\langle \{f_{\alpha ,l}(t),f_{\beta ,l^{\prime }}^{\dagger }(t^{\prime
})\}\rangle , \nonumber \\
F_{\alpha l,\beta l^{\prime }}^{R}(t,t^{\prime }) &=&-i\Theta (t-t^{\prime
})\langle \{f_{\alpha ,l}^{\dagger }(t),f_{\beta ,l^{\prime }}^{\dagger
}(t^{\prime })\}\rangle . \label{gf}\end{aligned}$$ Before writing down the Dyson’s equations satisfied the these fullGreen’s function let us define the following freeparticle $\hat{g}_{\alpha }^{0}(\omega )$ and hole $\hat{\overline{g}}_{\alpha }^{0}(\omega )$ Green’s functions $$\begin{aligned}
&&[\hat{g}_{\alpha }^{0}(\omega )]_{\alpha l,\alpha l^{\prime }}^{-1}=\delta
_{l,l^{\prime }}(\omega + i \eta) -\varepsilon _{l,l^{\prime }}^{\alpha } \nonumber \\
&&[\hat{\overline{g}}_{\alpha }^{0}(\omega )]_{\alpha l,\alpha l^{\prime
}}^{-1}=\delta _{l,l^{\prime }}(\omega+ i \eta) +\varepsilon _{l,l^{\prime }}^{\alpha
}, \label{g0}\end{aligned}$$with $\eta= 0^+$.
From now on we will work in Fourier space and we will not write explicitly the $\omega$ dependence of Green’s functions unless necessary. For the left chain, we also define the functions $\hat{G}_{L}^{0}$, $\hat{\overline{G}}_{L}^{0}$ containing the paring term contribution, through the relations $$\begin{aligned}
&&[\hat{G}_{L}^{0}]^{-1}=[g_{L}^{0}]^{-1}-\hat{\Delta}_{L}%
\hat{\overline{g}}_{L}^{0}\hat{\Delta}_{L}, \nonumber \\
&&[\hat{\overline{G}}_{L}^{0}]^{-1}=[\overline{g}_{L}^{0}]^{-1}-\hat{\Delta}_{L}
\hat{g}_{L}^{0}\hat{\Delta}_{L}, \end{aligned}$$ We also introduce $$\begin{aligned}
&& \hat{g}^{0}=\sum_{\alpha =L,C,R}\hat{g}_{\alpha }^{0} ,\;\;
\hat{\overline{g}}^{0}=\sum_{\alpha =L,C,R}\hat{\overline{g}}_{\alpha }^{0},\nonumber \\
&&\hat{G}^{0}=\hat{G}_{L}^{0}+\hat{g}_{C}^{0}+%
\hat{g}_{R}^{0}, \nonumber \\ &&
\hat{\overline{G}}^{0}=\hat{\overline{G}}_{L}^{0}+\hat{%
\overline{g}}_{C}^{0}+\hat{\overline{g}}_{R}^{0},
\nonumber \\
&&\hat{F}_{L}^{0}=\hat{\overline{G}}^{0}\hat{\Delta}_{L}%
\hat{g}_{L}^{0}, \nonumber \\
&&\hat{\overline{F}}_{L}^{0}=\hat{G}^{0}\hat{\Delta}_{L}%
\hat{\overline{g}}_{L}^{0}.\end{aligned}$$Here $\varepsilon _{l,l^{\prime }}^{\alpha }$ and $\Delta _{\alpha l,\alpha l^{\prime
}}^{ }$ are matrices defined on the coordinates of the chain $\alpha
=L,C,$ or $R$, containing respectively, the normal and anomalous elements of the Hamiltonian. In the case we are studying only $\hat{\Delta}_{L}$ is non-vanishing.
To obtain these Green’s functions, we write the following Dyson’s equation which relate them with Green’s functions of the disconnected" chains $\hat{g}_{\alpha }^{0}$ and $\hat{\overline{g}}%
_{\alpha }^{0}$ (see below) and the matrix elements of the contacts $$\begin{aligned}
&&\{[\hat{g}_{L}^{0}]^{-1}+[\hat{g}_{C}^{0}]^{-1}+[\hat{g}%
_{R}^{0}]^{-1}- \nonumber \\ &&
[\hat{W}]\}\hat{G}^{R}-\hat{\Delta}_{L}\hat{F}%
^{R}=\hat{1}, \label{dyret1} \\
&&\{[\hat{\overline{g}}_{L}^{0}]^{-1}+[\hat{\overline{g}}%
_{C}^{0}]^{-1}+[\hat{\overline{g}}_{R}^{0}]^{-1}+
\nonumber \\
&&[\hat{W}]\}\hat{F}^{R}-\hat{\Delta}_{L}\hat{G}%
^{R}=\hat{0}. \label{dyret2}\end{aligned}$$ The matrix $\hat{W}=\hat{W}_{L}+\hat{W}_{R}$ contains the matrix elements of $H_{cont}$ describing the connections between the central and left parts and between the central and right parts.
The above equations can be rewritten in a more convenient form by recourse to the following procedure. From Eq. (\[dyret2\]) $$\hat{F}^{R}=\hat{\overline{g}}^{0}[\hat{\Delta}_{L}\hat{G}%
^{R}-\hat{W}\hat{F}^{R}], \label{f}$$$$\hat{G}^{R}=\hat{g}^{0}+\hat{g}^{0}[\hat{\Delta}%
_{L}\hat{F}^{R}+\hat{W}\hat{G}^{R}], \label{g}$$
Substituting (\[f\]) in (\[dyret1\]) and (\[g\]) in (\[dyret2\]) one obtains $$\begin{aligned}
& &\hat{G}^{R} =\hat{G}^{0}(1+\hat{W}\hat{%
G}^{R})+\hat{\overline{F}}_{L}^{0}\hat{W}\hat{F}^{R}, \label{dyret3} \\
& &\hat{F}^{R} =\hat{F}_{L}^{0}(1+%
\hat{W}\hat{G}^{R})-\hat{\overline{G}}^{0}\hat{W}\hat{F}%
^{R}, \label{dyret4}\end{aligned}$$ Let us now consider Eq. (\[dyret3\]) for the following particular coordinates $$\begin{aligned}
G_{Cl,Cl^{\prime }}^{R}&=&g_{C,l,l^{\prime }}^{0}+g_{C,l,1}^{0}w^{\prime }G_{L1,Cl^{\prime }}^{R}
\nonumber \\
&&+g_{C,l,N}^{0}w^{\prime }G_{R1,Cl^{\prime }}^{R},
\label{con1} \\
G_{L1,Cl^{\prime }}^{R} &=&G_{L,1,1}^{0}w^{\prime
}G_{C1,Cl^{\prime }}^{R} \nonumber \\
&&+\overline{F}_{L,1,1}^{0}w^{\prime }F_{C1,Cl^{\prime
}}^{R}, \label{con2} \\
G_{R1,Cl^{\prime }}^{R} &=&g_{R,1,1}^{0}w^{\prime
}G_{CN,Cl^{\prime }}^{R}. \label{con3}\end{aligned}$$Substituting Eqs. (\[con2\]) and (\[con3\]) in Eq. (\[con1\]) it is easy to see that the Dyson’s equation for the two indices corresponding to coordinates of $C$ can be written as follows $$\begin{aligned}
&&\{[\hat{g}_{C}^{0}]^{-1}-\hat{\Sigma}^{R,gg}\}\hat{G}%
_{C}^{R}
+\hat{\Sigma}^{R,gf}\hat{F}_{C}^{R}=\hat{1},
\label{dyret5} \\
&&\{[\hat{\overline{g}}_{L}^{0}]^{-1}-\hat{\Sigma}^{R,ff}\}%
\hat{F}_{C}^{R} +\hat{\Sigma}^{R,fg}\hat{G}_{C}^{R}=\hat{0},
\label{dyret6}\end{aligned}$$ where the matrices of the above equations have sizes $N_{C}\times N_{C}$ and elements corresponding to the coordinates of the central chain. The self-energy matrices are $$\begin{aligned}
\Sigma _{l,l^{\prime }}^{R,ff} &=&\delta _{l,l^{\prime }}|w^{\prime
}|^{2}[\delta _{l,1}\overline{G}_{L,1,1}^{0}+\delta _{l,N_{C}}%
\overline{g}_{R,1,1}^{0} \nonumber \\
\Sigma _{l,l^{\prime }}^{R,gg} &=&\delta _{l,l^{\prime }}|w^{\prime
}|^{2}[\delta _{l,1}G_{L,1,1}^{0}+\delta
_{l,N_{C}}g_{R,1,1}^{0}] \nonumber \\
\Sigma _{l,l^{\prime }}^{R,gf} &=&\delta _{l,l^{\prime }}|w^{\prime
}|^{2}\delta _{l,1}\overline{F}_{L,1,1}^{0}, \nonumber \\
\Sigma _{l,l^{\prime }}^{R,fg} &=&\delta _{l,l^{\prime }}|w^{\prime
}|^{2}\delta _{l,1}F_{L,1,1}^{0}. \label{sigmas}\end{aligned}$$The explicit expressions for these functions imply the evaluation of all the functions appearing in the right hand sides of (\[sigmas\]). Notice that these functions have been defined from manipulations of the Dyson’s equations corresponding to $H_{L}$ or $H_{R}$ *isolated* from the central chain. We indicate a procedure for the calculation of these functions in appendix A. Note also that since the right and left parts of the system are held at two different but constant temperatures, these Green’s functions can be calculated at equilibrium. The advantage of the above representation becomes clear by writing ([dyret6]{}) as $$\begin{aligned}
&&\hat{F}_{C}^{R}=\hat{\overline{g}}_{C}\hat{\Sigma}%
^{R,fg}\hat{G}^{R}, \nonumber \\
&&[\hat{\overline{g}}_{C}]^{-1}=[\hat{\overline{g}}_{C}^{0}]^{-1}-\hat{\Sigma}^{R,ff}, \label{fc}\end{aligned}$$and substituting it in (\[dyret5\]). The result leads to the solution of the retarded normal Green’s function within $C$ $$\begin{aligned}
\hat{G}_{C}^{R} &=&\{[\hat{g}_{C}^{0}]^{-1}-\hat{\Sigma}%
_{\mbox{{\it eff}}}^{-1}, \nonumber \\
\hat{\Sigma}^{R}_{\mbox{{\it eff}}} &=&\hat{\Sigma}^{R,gg}+\hat{\Sigma}%
^{R,gf}\hat{\overline{g}}_{C}\hat{\Sigma}^{R,fg}.
\label{gc}\end{aligned}$$
The results obtained so far correspond to the retarded Green’s functions and self energies. The lesser Green’s function with coordinates within $C$ can be easily obtained by recourse to Langreth rules [@langr; @hern]. In particular one obtains [@lilisup]$$\hat{G}_{C}^{<}=\hat{G}_{C}^{R}\hat{\Sigma}^{<}_{\mbox{{\it eff}}}\hat{G}_{C}^{A}, \label{gcles}$$where the advanced Green’s function is obtained from the retarded one, by means of the relation $\hat{G}_{C}^{A}(\omega )=[\hat{G}_{C}^{R}(\omega
)]^{\dagger }$ and the lesser component of the self-energy is $$\begin{aligned}
& &\hat{\Sigma}^{<}_{\mbox{{\it eff}}} =\hat{\Sigma}^{<,gg}+
\hat{\Lambda}^{R}\hat{\Sigma}^{<,fg}+
\nonumber \\ & &\hat{\Sigma}%
^{<,gf}\hat{\Lambda}^{A} +
\hat{\Lambda}^{R}\hat{\Sigma}^{<,ff}\hat{\Lambda}%
^{A}, \label{sigefles}\end{aligned}$$with $\hat{\Lambda}^{R}(\omega )=\hat{\Sigma}_{\alpha }^{R,gf}(\omega )\hat{%
\overline{g}}_{C}(\omega )$ and $\hat{\Lambda}^{A}(\omega )=[\hat{\Lambda}%
^{R}(\omega )]^{\dagger }$. The self-energies have components $$\Sigma
_{l,l^{\prime }}^{<,\nu ,\nu ^{\prime }}(\omega )=i\delta _{l,l^{\prime
}}[\delta _{l,1}\Gamma _{L}^{\nu ,\nu ^{\prime }}(\omega )f_{L}(\omega
)+\delta _{l,N}\Gamma _{R}^{\nu ,\nu ^{\prime }}(\omega )f_{R}(\omega )]
\nonumber$$ being $\Gamma _{L}^{\nu ,\nu ^{\prime }}(\omega )=-2\mbox{Im}[\Sigma ^{R,\nu
\nu ^{\prime }}(\omega )_{1,1}]$ and $\Gamma _{R}^{\nu ,\nu ^{\prime
}}(\omega )=-2\mbox{Im}[\Sigma ^{R,\nu \nu ^{\prime }}(\omega )_{N,N}]$ with $\nu ,\nu ^{\prime }=g,f$. The Fermi functions $f_{\alpha }(\omega )$, with $%
\alpha =L,R$ depend on the temperatures $T_{L}$ and $T_{R}$ of the left and right chains respectively: $f_{\alpha }(\omega )=1/(1+e^{\omega /T_{\alpha
}})$, in units where $k_B=1$.
Finally, the lesser counterparts of Eqs. (\[con2\]) and (\[con3\]), which correspond to Green’s functions with one of the coordinates in the central ($C$) chain and the other one in the left ($L$) or right ($R$) chain, can be calculated by recourse again to Langreth rules [langr,hern]{} $$\begin{aligned}
G_{L1,Cl^{\prime }}^{<} &=&G_{L,1,1}^{0,<}w^{\prime
}G_{C1,Cl^{\prime }}^{A} +G_{L,1,1}^{0,R}w^{\prime }G_{C1,Cl^{\prime }}^{<}
\nonumber \\
&&+\overline{F}_{L,1,1}^{0,<}w^{\prime }F_{C1,Cl^{\prime
}}^{A} +\overline{F}_{L,1,1}^{0,R}w^{\prime }F_{C1,Cl^{\prime
}}^{<}, \label{conles2} \\
G_{R1,Cl^{\prime }}^{<} &=&g_{R,1,1}^{0,<}w^{\prime
}G_{CN,Cl^{\prime }}^{A} +g_{R,1,1}^{0,R}w^{\prime }G_{CN,Cl^{\prime }}^{<}.
\label{conles3}\end{aligned}$$
Heat currents and transmission functions
----------------------------------------
We focus on the expression for the heat current evaluated in the contact between the central chain $C$ and $R$ given in Eq. (\[jq2\]). Using Eq. (\[conles3\]) one obtains $$\begin{aligned}
J^Q &=&-2 |w' |^2 \int_{-\infty }^{+\infty }\frac{d\omega }{2\pi }\omega \mbox{Re}%
[G_{CN,CN}^{<}(\omega )g_{R,1,1}^{0,A}(\omega ) \nonumber \\
&&+G_{CN,CN}^{R}(\omega )g_{R,1,1}^{0,<}(\omega )]. \label{jal2}\end{aligned}$$Using (\[gcles\]) and after some algebra (see Ref. ), it is found $$\begin{aligned}
J^Q &=&\int_{-\infty }^{+\infty }\frac{d\omega }{2\pi }\omega \lbrack
f_{L}(\omega )-f_{R}(\omega )][T^{n}(\omega )-T^{a}(\omega )], \nonumber \\
T^{n}(\omega ) &=&\Gamma _{R}^{gg}(\omega )|G_{C,N,1}^{R}(\omega
)|^{2}\Gamma _{L, {\mbox{\it eff}}}^{gg}(\omega ), \nonumber \\
T^{a}(\omega ) &=&\Gamma _{R}^{gg}(\omega )
| \overline{\Lambda }_{N,N}(\omega )|^{2} \Gamma _{R}^{ff}(\omega ),
\label{jqal3}\end{aligned}$$where $$\begin{aligned}
\!\!\!\!\!\!&& \Gamma_{L,\mbox{{\it eff}}}^{gg} =\Gamma _{L}^{gg}+2\mbox{Re}%
[\Gamma _{L}^{gf}\Lambda _{1,N}^{A}]
+|\Lambda _{1,1}^{R}|^{2}\Gamma _{L}^{ff}+|\Lambda
_{1,N}^{R}|^{2}\Gamma _{R}^{ff}, \nonumber \\
&&\overline{\Lambda }^R_{N,N} =G_{C,N,1}^{R}\Lambda^R_{1,N}. \label{gamlam}\end{aligned}$$
The difference of Fermi functions in the expression of $J^Q$, reflects the fact that the existence of a non-vanishing heat current through the central system depends on the existence of a difference of temperatures between the left and right chains. The details of the model are enclosed in the behavior of the “normal” and “anomalous” transmission functions $T^{n}(\omega )$ and $T^{a}(\omega )$, which are analogous to those defined in Ref. in the context of particle transport in a setup with normal and superconducting wires. The first function has, in fact, the structure of a transmission. Notice that it depends on the densities of states of the right and left chains through the functions $\Gamma_R$ and $\Gamma_{L}^{eff}$, and one the particle propagator between the first and last points of the central chain. Instead, $T^{a}(\omega )$ actually has the structure of a reflection process. Notice that it depends on the density of states for particles and holes of the right reservoir and on a multiparticle propagator $\overline{\Lambda }^R_{N,N}$ at the last point of the central chain. Typical plots for these functions are shown in Fig. \[tt\]. These functions do not depend on the temperatures $T_L$ and $T_R$ and are non-vanishing only within a finite range of energies of a width that is set by the largest exchange parameter between the left, right and central chains. These functions are symmetric with respect to $\omega=0$ for $B_L=B_R=0$ (see Fig. \[tt\]). This symmetry is broken for finite $B_{\alpha}$, since the effect of a finite magnetic field in one of the side chains is to shift the corresponding function as $\Gamma^{\nu \nu'}_{\alpha}(\omega) \rightarrow
\Gamma^{\nu \nu'}_{\alpha}(\omega-B_{\alpha})$.
In the language of fermionic systems, two different kinds of processes take place in a normal-superconductor junction. For energies higher than the gap, the transport is due to the tunneling of normal single particle high-energy excitations. This mechanism contributes to the electronic transmission function $T^n(\omega)$. Instead, for low energies, below the gap, the transport is due to the mechanism known as “Andreev reflection”, which implies the combination of two fermions of the normal side into a Cooper pair within the superconducting one, leaving a hole that is reflected back from the junction into the normal side. Because of this mechanism, $T^n(\omega) \sim T^a(\omega) \sim 1 $ for energies within the superconducting gap, i.e. $|\omega| \leq \Delta_L$. The effective conversion of electrons into Cooper pairs taking place in the mechanism of Andreev reflection helps to partice transport. Mathematically, this is reflected by the fact that the total particle transmission function is $T^n(\omega)+T^a(\omega)$. [@lilisup; @btk] Instead, in the case of heat transport, $T^a(\omega)$ and $T^n(\omega)$ contribute with opposite sign, as explicitly shown in Eq. (\[jqal3\]), i.e. the mechanism of Andreev reflection, plays a negative role regarding the heat transport. The consequence is a vanishing heat transport due to excitations within the energy window defined by the superconducting gap.
In the original language of interacting spins, the above picture translates as follows. Low energy spin excitations traveling from the isotropic chain via flip-flop processes in the $z$ direction meet an energy gap at the other side of junction due to the anisotropic interaction which tends to favor flip-flop processes in a different direction. This favors the simultaneous raising or lowering of two spins at two neighboring positions of the chain and causes multiscattering processes in which a portion of the incident spin wave packet manages to twist and propagate into the other side, at the same time that a portion becomes reflected and propagates back.
We can describe the behavior of $J^Q$ for low $T$ and small temperature gradients $\delta T$ as follows. Writing $T_R=T$ and $T_L=T_R+\delta T$ we can approximate the difference of Fermi functions in Eq. (\[jqal3\]) as $$f_L(\omega)-f_R(\omega)\sim \frac{\partial f_R(\omega)}{\partial T} \delta T .$$ On the other hand, from Fig. \[tt\], we can write,
$$\begin{aligned}
T^n(\omega)-T^a(\omega) \sim &0 & \omega \leq 2\Delta_L \\
T^n(\omega)-T^a(\omega)\sim & 1 &2\Delta_L \leq \omega \leq 2J\end{aligned}$$
leading to, $$\begin{aligned}
\label{expo}
J^{Q} &=& \frac{\delta T }{\pi} \int_{2\Delta_L}^{+2J}d\omega \omega
\frac{\partial f_R(\omega)}{\partial T} \end{aligned}$$ For low enough temperatures, $T\ll2\Delta_L \ll 2J$, this expression can be further approximated as, $$\begin{aligned}
\label{expo2}
J^{Q} &\sim& \frac{\delta T }{\pi} \frac{\partial }{\partial T}\int_{2\Delta_L}^{\infty}d\omega \omega e^{-\beta \omega}
\\
& \sim & \frac{4}{\pi} \delta T \frac{
\Delta_L^2}{T} e^{-\frac{2\Delta_L}{T}}\;\;\;\; \end{aligned}$$
Therefore, for $T<\Delta_L$, the heat current is exponentially small.
On the other hand, for $\Delta_L= 0$, the behavior of the $J^Q$ is fully due to normal tunneling. For low $T$ we can perform a Sommerfeld expansion on the Fermi function to get $$\begin{aligned}
\label{linear}
J^{Q} &=& \frac{2}{T^2} \delta T \frac{\partial }{\partial \beta} (
\int_{0}^{+\infty }\frac{d\omega }{2\pi }\omega f(\omega ) T^{n}(\omega ) )\nonumber \\
& \sim & \frac{ \pi}{3} T \delta T , \;\;\;\; T \ll J_{\alpha},\;\; \Delta_L = 0.\end{aligned}$$
Results
=======
In this section we discuss the behavior of the heat current as a function of the different ingredients of the spin system. For simplicity, we consider identical exchange parameters along the left, central and right chains: $J_L=J_C=J_R =J'=J$. Without loss of generality we set $J=1$. Thus, we focus on a spin heterostructure with a single junction between a semi-infinite XX and a semi-infinite XY chain, which in the fermionic language translates to a single S-N junction. For this particular configuration, our results do not depend on the length of the central chain.
![[ (Color online) Top left: Conductance, defined as the ratio of heat current to temperature gradient between right and left chains. Top right: specific heat of the central system. Bottom: ratio of the above magnitudes. Parameters are $T_L=T+\protect\delta_T, T_R=T$ where $\protect\delta_T=0.005$, all chemical potentials set to zero, and (from top to bottom) $\Delta_L=0,0.25,0.5$. ]{}[]{data-label="conductividad"}](fig2n.eps "fig:"){width="70mm" height="90mm"} [ ]{}
As discussed in the previous subsection, the structure of the expression (\[jqal3\]) for the heat current clearly reflects the fact that for small temperature differences, we obtain a behavior of the form $$J^Q=-K \delta T,$$ where the coefficient $K$ can be interpreted as a [*thermal conductance*]{}. It is tempting to relate this coefficient with the conductivity $\kappa $ evaluated in several works on the basis of linear response theory. If we assume that the relation between the two coefficients is similar to the one between electrical conductance and electrical conductivity, $K$ and $\kappa$ should differ just by a geometrical factor. However, to the best of our knowledge, a rigorous relation between these two coefficients has not been presented in the literature. Nevertheless, the behavior of $K$ as a function of $T$ shown in the left upper panel of Fig. \[conductividad\] for the case of two connected XX chains (see the plot corresponding to $\Delta_L=0$) is similar to the one reported in the literature for homogeneous and isotropic chains [@lista2], [@lista5],[@lista3]. In this case, the anomalous component is zero and $K$ increases linearly in $T$ for low temperature \[see Eq. (\[linear\])\], as discussed at the end of the previous section. The conductance reaches a maximum at $T \sim J$. and decreases at higher temperatures, as a consequence of the finite bandwith (energy window) for the spin excitations amenable to cross the central chain transporting energy from one side to the other one. As expected, for a fixed temperature $K $ decreases for increasing values of $\Delta_L $. In agreement with the behavior discussed in the previous section, $K$ is exponentially small for $T<\Delta_L $ \[see Eq. (\[expo\])\], For higher temperatures, the high energy excitations are allowed to perform tunneling above the energy gap, with the concomitant increase of $K$. As in the case with $\Delta_L=0$, the maximum is achieved at $T\sim J$.
We also evaluate the specific heat for the [*equilibrium*]{} central system in contact to the side chains at the same temperature $T$ as follows:
$$C(T) = - \frac{2}{N} \sum_{l=1}^N \int_{-\infty}^{+\infty}\frac{d \omega}{2 \pi}
\frac{\partial f(\omega)}{\partial T} \omega
\mbox{Im}[G^R_{C,l,l}(\omega) ]$$
In a normal metallic system as described by Drude model, this quantity is related to the thermal conductivity through $\kappa = v l C/3$, being $v$ the Fermi velocity of the electrons and $l$ their mean free path [@aschcroft-mermin]. We plot this quantity in the right upper panel of Fig. \[conductividad\]. This physical quantity is almost insensitive to the opening of the energy gap and the different plots, corresponding to different values of $\Delta_L$ almost coincide within the scale of the figure. From the lower panel of Fig. \[conductividad\] we see that while for high temperatures there is a linear relation between $K$ and $C$, this is not the case at lower temperatures where Andreev type processes are relevant.
As stressed before, our calculation is not restricted to small temperature gradients. We show in Fig. \[curr2\] a plot of the heat current for several values of the anisotropy parameter $\Delta_L$ as a function of the temperature of the left chain while the temperature of the right chain $T$ is set fixed to zero. The figure clearly shows the suppression of the current as a consequence of the Andreev reflection phenomena mentioned before. In fact for $T < \Delta_L$ the current is exponentially small, while it grows for higher temperatures.
Finally, in Fig. \[asy3\] we illustrate the behavior of the heat current when finite different magnetic fields are applied at the two side chains. The effect of applying magnetic fields at both sides of the junction leads to a interesting effect which me name “thermal diode effect”. As discussed in the previous section, a finite magnetic field originates a shift in the arguments of the functions $\Gamma_{\alpha}^{\nu \nu'}(\omega)$, which leads to asymmetries in the transmission functions $T^n(\omega)$ and $T^a(\omega)$. For $\Delta_L=0$, only the normal transmission function and the functions $\Gamma^{gg}_{\alpha}(\omega)$ are non-vanishing. Furthermore, these functions are identical and gapless for $\alpha=L,R$. Therefore, the heat flow is perfectly antisymmetrical (the sign of the current is reversed preserving the absolute value) under the simultaneous change $T_L \leftrightarrow T_R$ and $B_L \rightarrow B_R$. Instead, for a finite $\Delta_L$, the situation changes. A gap opens for the excitations of the left chain and the functions $ \Gamma^{\nu \nu'}_{L}(\omega) =0$ vanish for $|\omega| < \Delta_L $, while the functions $\Gamma^{g g}_R(\omega)=\Gamma^{ff}_R(-\omega)$ remain finite. The consequence is an asymmetry in the behavior of the transmission functions under the change $B_L \rightarrow B_R$. The result is an effect of thermal rectification. That is, the magnitude of the current $J^Q$ when ($T_L=T$,$\mu_L=\mu$) and ($T_R=T'$,$\mu_R=\mu'$) is different to $J^{Q'}$ when ($T_L=T'$,$\mu_L=\mu'$) and ($T_R=T$,$\mu_R=\mu $), which means that the device is more likely to conduct heat when the temperature difference is applied in one direction than in the other. We display the phenomena for two different values of $\Delta_L$. We show the current when ($\mu_L=0.3, T_L=T, \mu_R=0, T_R=0$) with dots and the current when ($\mu_R=0.3, T_R=T, \mu_L=0, T_L=0$) with a full line. When the value of $\Delta_L=0.2$ both currents are rather large and similar but when $\Delta_L=0.75$ the currents are smaller and clearly different.
Summary and discussion
======================
We have presented a theoretical framework to study heat transport in one-dimensional spin heterostructures.
In the present work we have focussed on a simple system composed of a junction between an anisotropic (XY) and an isotropic (XX) chain under the effect of an inhomogeneous magnetic field along the $z$ direction. Using the Jordan-Wigner transformation to map the problem into a fermionic system and using the non equilibrium Keldysh-Schwinger formalism we have obtained exact expression for the heat current in terms of Green’s functions of the “disconnected” spin chain components. The resulting expressions can be evaluated numerically in a simple way. In the limits $\Delta_L \ll T \ll J$ and $T \ll \Delta_L \ll J$ explicit analytic expressions can also be given. We have studied the heat transport as a function of the different parameters of the model and we have shown that when different magnetic fields are applied at the end chains, a rectifying effect in the heat current occurs. This effect might be of interest for applications. Its origin can be traced back to the appearance of paring terms induced by the anisotropy parameter, which are in turn responsible for an Andreev reflection type mechanism.
In this work we have analyzed a simple model. However this methodology can be straightforwardly extended to more complex structures with many junctions and disorder. Our treatment relies on the Jordan-Wigner transformation which maps the original spin Hamiltonians into fermionic ones. In the case we have considered, the latter are bilinear. In more generic models, although we expect the rectifying effect still to be present, the technical analysis could be more complicated. For instance, in the isotropic Heisenberg model, which in addition to the exchange interaction along $x$ and $y$ directions, contains an additional exchange term along the $z$ direction, the Jordan-Wigner transformation translates such a term into a many-body fermionic interaction, which does not enable a straightforward analytical solution of the problem, as in the case we considered here. Nevertheless, the Green’s function formalism offers a framework for the construction of systematic approximations to treat those terms. Numerical methods could also be useful to deal with models containing many-body terms. [@feiguin] As in electronic systems, many-body terms are expected to introduce further inelastic scattering processes, which could add further ingredients in addition to the transport mechanisms we have discussed here. We hope to report on some of these issues in future work.
Acknowledgments {#acknowledgments .unnumbered}
===============
This investigation was sponsored by PIP 5254 and PIP 112-200801-00466 of CONICET, PICT 2006/483 of the ANPCyT and projects X123 and X403 from UBA. We are partially supported by CONICET. G.S.L and L.A thanks C.Batista, D.Cabra, and D.Karevski for interesting comments.
Green’s functions for an open chain with $p$-wave superconductivity
===================================================================
In this appendix we show a derivation of the Green’s functions $G_{L,1,1}^{0}(\omega )$, $\overline{G}_{L,1,1}^{0}(\omega )$, $%
F_{L,1,1}^{0}(\omega )$ and $\overline{F}_{L,1,1}^{0}(\omega )$, entering Eq. (\[sigmas\]), which correspond to the end of a half infinite chain with $p$-wave superconductivity in the BCS approximation. Making the superconducting parameter $\Delta_L =0$, the first two Green’s functions give the corresponding result for the normal chain $g_{R,1,1}^{0}(\omega )$ and $\overline{g}_{R,1,1}^{0}(\omega )$ respectively.
The Green’s functions of the open chain can be solved considering a ring of $N$ sites, periodic except for the fact that the energy at one site (which we label as site 0) is increased by an energy $A$, and then taking the limit $%
N,A\rightarrow +\infty $. The Hamiltonian is
$$\begin{aligned}
H &=&\sum_{l=0}^{N-1}\{w[f_{l}^{\dagger }f_{l+1}+f_{l+1}^{\dagger }f_{l}]
\nonumber \\
&&+\Delta \lbrack f_{l}^{\dagger }f_{l+1}^{\dagger }+f_{l+1}f_{l}]\}
\nonumber \\
&&-\mu \sum_{l=0}^{N-1}f_{l}^{\dagger }f_{l}+Af_{0}^{\dagger }f_{0}.
\label{ah}\end{aligned}$$
We have solved the problem using two different methods: i) solving the equations of motion in Fourier space, and ii) solving a Dyson’s equation that relates the above Green’s functions to those of the periodic chain ($A=0$) which can be obtained easily using Bloch theorem. Both results of course coincide, but the latter method involves a simpler algebra. We define a matrix $$\tilde{G} =
\left(
\begin{array}{cc}
G_{L,1,1}^{0}(\omega ) & F_{L,1,1}^{0}(\omega )\\
\overline{F}_{L,1,1}^{0}(\omega ) &\overline{G}_{L,1,1}^{0}(\omega )
\end{array}
\right),$$ with the Green’s functions for $A\neq 0$, and a corresponding matrix $\tilde{g}$ for $A = 0$. These matrices are equivalent to the ones obtained by using Nambu’s representation for the Hamiltonian and the Green’s functions. [@cuevas] From the equations of motion of these Green’s functions, one obtains
$$\tilde{G}=\tilde{g}+\tilde{g}\tilde{A}\tilde{G}\text{,} \label{adys}$$
where $\tilde{A}$, is proportional to $A$.
Solving Eq. (\[adys\]) for $\tilde{G}$ and taking the limit $A \rightarrow +\infty$ the following expressions result
$$\begin{aligned}
G_{L,1,1}^{0} &=&h_0(\omega )-\frac{h_1^{2}(\omega )}{%
h_0(\omega )}-\frac{h_2^{2}(\omega )}{h_0^{\ast }(-\omega )%
}. \label{aaa} \\
F_{L,1,1}^{0}&=&h_2(\omega )\left( \frac{h_1^{\ast }(-\omega )}
{h_0^{\ast }(-\omega )}-\frac{h_1(\omega )}{h_0(\omega )}
\right) . \label{aba}\end{aligned}$$
$$\overline{G}_{L,1,1}^{0}(\omega)=-(G_{L,1,1}^{0})^*(-\omega)
\,\,\,
\overline{F}%
_{L,1,1}^{0}=-(F_{L,1,1}^{0})^*(-\omega)$$
The $h$-functions entering the second members of Eqs. (\[aaa\]) and (\[aba\]) are Green’s functions of the periodic chain and can be calculated easily in Fourier space. The result is
$$\begin{aligned}
h_0(\omega ) &=&\frac{1}{N} \sum_{k} \frac{\omega +\epsilon _{k}}{\omega ^{2}-\epsilon _{k}^{2}-\Delta _{k}^{2}}, \nonumber \\
h_1(\omega ) &=&\frac{1}{N} \sum_{k} \frac{(\omega +\epsilon
_{k})\cos k}{\omega ^{2}-\epsilon _{k}^{2}-\Delta _{k}^{2}}, \nonumber \\
h_2(\omega ) &=&\frac{1}{N} \sum_{k} \frac{2\Delta \sin ^{2}k}
{\omega ^{2}-\epsilon _{k}^{2}-\Delta _{k}^{2}}, \label{agp}\end{aligned}$$
where in the second members $\omega $ includes an infinitesimally small imaginary part, $\epsilon _{k}=2w\cos k-\mu $, and $\Delta _{k}=2\Delta \sin
k$.
For $N\rightarrow +\infty $, the sums can be replaced by integrals. Decomposing the integrands into a sum of simple fractions with denominators linear in $\cos k$ and numerators independent of $k$, the integrals can be evaluated analytically using [@gra]
$$I(b)=\frac{1}{\pi } \int_{0}^{\pi } \frac{dk}{\cos k+b}=\frac{1}{\sqrt{%
b^{2}-1}}, \label{ain}$$
where the sign of the root is determined by the sign of the imaginary part of the second member.
Defining
$$\begin{aligned}
\tilde{\omega} &=&\frac{\omega }{2w},\text{ }\tilde{\Delta}=\frac{\Delta }{w},
\text{ }\tilde{\mu}=\frac{\mu }{2w},\text{ }d=1-\tilde{\Delta}^{2},
\nonumber \\
r &=&\left[ (\tilde{\omega}^{2}-\tilde{\Delta}^{2})d+(\tilde{\Delta}
\tilde{\mu})^{2}\right] ^{1/2}/d. \nonumber \\
b_{1} &=&\tilde{\mu}/d-r,\text{ }b_{2}=\tilde{\mu}/d+r, \nonumber \\
d_1 &=&\left[ 4wrd\right] ^{-1}, \label{aux}\end{aligned}$$
the result takes the form
$$\begin{aligned}
h_0(\omega ) &=&d_1 [(r-\tilde{\omega}-\tilde{\mu}\tilde{\Delta}^{2}/b)I(b_{1}) \nonumber \\
&&+(r+\tilde{\omega}+\tilde{\mu}\tilde{\Delta}^{2}/b)I(b_{2})], \nonumber \\
h_1(\omega ) &=&-d_1 \{2r-[(r-\tilde{\omega}-\tilde{\mu}\tilde{\Delta}^{2}/b)b_{1}I(b_{1})
\nonumber \\
&&+(r+\tilde{\omega}+\tilde{\mu}\tilde{\Delta}^{2}/b)b_{2}I(b_{2})]\},
\nonumber \\
h_2(\omega ) &=&d_1 \tilde{\Delta}\{2r+[I(b_{1})]^{-1}-[I(b_{2})]^{-1}\}. \label{agp2}\end{aligned}$$
[99]{}
For a recent review see for instance, A. V.Sologubenko, T. Lorenz, H. R. Ott, A. Freimuth, J. Low Temp. Phys. **147**, 387 (2007). See also, X. Zotos, P. Prelovsek, in *“Interacting Electrons in Low Dimensions”, book series “Physics and Chemistry of Materials with Low-Dimensional Structures”*, Kluwer Academic Publishers (2003), Abhishek Dhar, Advances In Physics, [**57**]{}, 457 (2008).
A. V. Sologubenko, E. Felder, K. Giannò, H. R. Ott, A. Vietkine, and A. Revcolevschi, Phys. Rev. **B 62**, R6108 (2000); A. V. Sologubenko, K. Giannò, H. R. Ott, A. Vietkine, and A. Revcolevschi, ibid. **64**, 054412 (2001); A. V. Sologubenko, H. R. Ott, G. Dhalenne, and A. Revcolevschi, Europhys. Lett. **62** 540 (2003). For some recent works on spin chain and ladders see for instance, F.Heidrich-Meisner, A. Honecker, and W. Brenig, Phys. Rev. **B 71**, 184415 (2005); A. V. Rozhkov and A. L. Chernyshev , Phys. Rev. Lett. **94**, 087201 (2005); P. Jung, R. W. Helmes, and A. Rosch, Phys. Rev. Lett. **96**, 067202 (2006); K. Louis, P. Prelovsek and X. Zotos, Phys. Rev. **B 74**, 235118 (2006); P. Jung and A. Rosch, Phys. Rev. **B 75**, 245104 (2007); E. Boulat, P. Mehta, N. Andrei, E. Shimshoni, and A. Rosch, Phys. Rev. **B 76**, 214411 (2007); A. V. Savin, G. P. Tsironis, and X. Zotos, Phys. Rev. **B 75**, 214305 (2007)
See also, H. Castella, X. Zotos, and P. Prelovsek, Phys. Rev. Lett. [**74**]{}, 972 (1995); S. Fujimoto and N. Kawakami, J. Phys. [ **A 31**]{}, 465 (1998); X. Zotos, Phys. Rev. Lett. [**82**]{}, 1764 (1999);T. Prosen and D. K. Campbell, Phys. Rev. Lett. [**84**]{}, 2857 (2000); J. V. Alvarez and C. Gros, Phys.Rev. Lett [**88**]{}, 077203 (2002); E. Shimshoni, N.Andrei, and A. Rosch, Phys. Rev. [**B 68**]{}, 104401 (2003); K. Saito, Phys. Rev. B [**67**]{}, 064410 (2003); F.Heidrich-Meisner, A. Honecker, D.C. Cabra, and W. Brenig, Phys. Rev. B [**66**]{}, 140406(R) (2002), Phys. Rev. [**B 68**]{}, 134436 (2003), Phys. Rev. Lett. [**92**]{}, 069703 (2004) E. Orignac, R.Chitra, R.Citro, Phys. Rev. B [**67**]{}, 134426 (2003); X. Zotos, Phys. Rev. Lett. [**92**]{}, 067202 (2004).
H. M. Pastawski, P. R. Levstein, and G. Usaj, Phys. Rev. Lett. [**75**]{}, 4310 (1995); D. Loss and D. P. DiVincenzo, Phys. Rev. A [**57**]{}, 120 (1998); T. S. Cubitt and J. I. Cirac, Phys. Rev. Lett. [**100**]{}, 180406 (2008) and references therein.
G.D. Mahan, *Many Particle Physics* (Kluver/Plenum, New York, 2000).
W.Aschbacher and C.A.Pillet, J.Stat.Phys. [**112**]{}, 1153 (2003), Y.Ogata, Phys. Rev. [**66**]{}, 066123 (2002), Y.Ogata, Phys. Rev. [**66**]{}, 016135 (2002).
M. Michel, G. Mahler, and J. Gemmer, Phys. Rev. Lett. [**95**]{}, 180602 (2005); J. Gemmer, R. Steinigeweg, and M. Michel, Phys. Rev. B [**73**]{}, 104302 (2006).
M. Michel, O. Hess, H. Wichterich, and J. Gemmer, Phys. Rev. B [**77**]{}, 104303 (2008).
For other type of baths see D.Karevski and T.Platini, unpublished.
S. Datta, *Electronic Transport in Mesoscopic Systems* (Cambridge University press, 1995);Y. Imry, *Introduction to Mesoscopic Physics*, (Oxford University Press, 1997).
J. Rammer and H. Smith, Rev. Mod. Phys. [**58**]{}, 323 (1986).
L. Arrachea, M. Moskalets and L. Martin-Moreno, Phys. Rev. [**B 75**]{}, 245420 (2007); L. Arrachea and M. Moskalets, cond-mat/0903.1153.
L. Arrachea, Phys. Rev. B, in press, cond-mat/0811.1648.
D.C. Langreth, in *Linear and Non-Linear Transport in Solids*, J.T. Devreese, E. Van Dofen (Eds.), (Plenum Press, New York, 1076).
A. Hernández, V. M. Apel, F. A. Pinheiro, and C. H. Lewenkopf, Physica A **385**, 148 (2007).
G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys. Rev. B [**25**]{}, 4515 (1982). N.W.Ashcroft and N. D. Mermin, “Solid State Physics”, Holt, Rinehart and Winston. 1976.
J. C. Cuevas, A. Martín-Rodero, and A. Levy Yeyati, Phys. Rev. B, [**54**]{}, 7366 (1996).
I.S. Gradshtein and I.M. Ryzhik, *Tables of Integrals, Series and Products* (Academic Press, New York, 1980)
A. E. Feiguin, S. R. White, and D. J. Scalapino, Phys. Rev. B. [**75**]{}, 024505 (2007); A. E. Feiguin, S. R. White, D. J. Scalapino, and I. Affleck; Phys. Rev. Lett. [**101**]{}, 217001 (2008);
|
---
author:
- Keunwoo Peter Yu
- |
Yi Yang\
ASAPP Inc.\
New York, NY 10007\
[{yyang+peter}@asapp.com]{}
bibliography:
- 'cite-strings.bib'
- 'cites.bib'
- 'cite-definitions.bib'
title: |
One Model to Recognize Them All:\
Marginal Distillation from NER Models with Different Tag Sets
---
|
---
author:
- 'Wei Fan,'
- 'Angelos Fotopoulos,'
- Stephan Stieberger
- 'and Tomasz R. Taylor'
bibliography:
- 'celestial\_amp.bib'
title: On Sugawara construction on Celestial Sphere
---
Introduction {#sec:intro}
============
The null infinity of $D\!=\!4$ asymptotically flat spacetime is the product of a conformal two–sphere (*celestial sphere*) ${\cal C S}^2$ with a null line. It was realized some time ago, that for asymptotically flat space-times the Poincare group can be extended to the local BMS group. The local BMS algebra, is an infinite dimensional extension of the Poincare algebra. It contains the Virasoro algebra generators, which generate the conformal group, and in addition supertranslations. This is very suggestive of a conformal field theory living on the null infinity of Minkowski space-time.
Indeed, scattering amplitudes in $D\!=\!4$ Minkowski spacetime can be recast, via a Mellin transform, into conformal correlation functions (*celestial amplitudes*) on the celestial sphere [@Pasterski:2016qvg; @Pasterski:2017kqt; @Pasterski:2017ylz; @Schreiber:2017jsr; @Cheung:2016iub; @Banerjee:2019prz][^1]. The theory describing the dynamics of celestial amplitudes is expected to be a novel conformal field theory on ${\cal CS}^2$. The celestial amplitudes, correspond to a subset of the correlators on the celestial conformal field theory (CCFT). One of the main motivations, for studying such a theory, is the proposal that CCFT is a holographic description of 4d physics in Minkowski spacetime [@Cheung:2016iub; @strominger_lectures_2017; @Pasterski:2019msg]. This originates from early studies of flat holography [@deboer_holographic_2003] and the BMS algebra [@bondi_gravitational_1962; @sachs_gravitational_1962; @barnich_symmetries_2010] of asymptotic symmetries. Recently, the study of CCFT and its properties has led to several advances along various aspects of the proposed theory [@Schreiber:2017jsr; @Stieberger:2018edy; @Kapec:2016jld; @fan_soft_2019; @Fotopoulos:2019tpe; @Pate:2019lpp; @Fotopoulos:2019vac; @distler_double-soft_2019; @Nandan:2019jas; @Guevara:2019ypd; @Adamo:2019ipt; @Laddha:2020kvp; @Cardona:2019ywu], but a lot remains in order to make the CCFT a solid proposal for flat space-time holography.
In CCFT, a particle is described [@Pasterski:2016qvg; @Pasterski:2017kqt] by conformal primary wave function. These functions are labeled by the conformal weights $(h,\bar{h})$ and position $(z,\bar{z})$. The coordinates $(z,\bar{z})$ on the celestial sphere are related to the asymptotic direction of the four-momentum in 4d Minkowski spacetime. The scaling dimension $\Delta$ and spin helicity $J$ can be obtained from the conformal weights via $\Delta = h+ \bar{h}, J=h-\bar{h}$. In CCFT, each particle corresponds to a conformal field operator with ${\rm Re}(\Delta)=1$, i.e. $\Delta=1+i\lambda, \lambda\in\mathbb{R}$ [@Pasterski:2017kqt]. For massless particles, conformal field operators are Mellin transforms of plane wave functions in the 4d Minkowski spacetime [@Pasterski:2017kqt; @Cheung:2016iub].
At this stage, many details of the CCFT are under investigation. From current studies nontrivial and elegant features have already been discovered, especially its algebraic structure extracted from celestial operator spectra and celestial amplitudes. At the classical level, the Ward identities of conserved currents and energy-momentum tensor have been shown to correspond to soft theorem of gluons and gravitons [@strominger_asymptotic_2014; @strominger_bms_2014; @Cheung:2016iub; @Conde:2016rom; @Mao:2017tey; @Himwich:2019dug; @Banerjee:2019tam; @Puhm:2019zbl]. For gluons, conserved currents of the CCFT correspond to the conformal soft limit [@donnay_conformally_2019; @Pate:2019mfs; @Law:2019glh] of conformal operators, i.e. $\Delta=1~(\lambda=0)$.[^2] A similar picture holds between conformally soft gravitons and the energy-momentum tensor [@Cheung:2016iub; @Kapec:2016jld]. The studies of the conformal soft limit of celestial amplitudes allowed making a direct connection between Ward identities of currents on ${\cal CS}^2$ and the low energy theorems of gluons [@fan_soft_2019] and gravitons [@fan_soft_2019; @Pate:2019lpp; @Fotopoulos:2019tpe]. Moreover, the collinear singularities of gluon and graviton amplitudes, correspond to the case where two operators of the CCFT approach each other. The study of collinear singularities of celestial amplitudes was used in [@fan_soft_2019; @Fotopoulos:2019vac] to derive the operator product expansion (OPE) of celestial operators. Similar work has appeared recently for massive states [@Law:2020tsg].
In this work we want in particular to elaborate on the energy-momentum tensor $T(z)$ which generates the Virasoro algebra on the ${\cal CS}^2$. The energy-momentum tensor $T(z)$ is a $\Delta=2$ conformal field operator that can be constructed through a shadow transformation from the $\Delta=0$ operator of the graviton [@Kapec:2016jld; @Cheung:2016iub]. The OPE of this energy-momentum tensor with the conformal field operators of gluons and gravitons was derived in [@Fotopoulos:2019tpe; @Fotopoulos:2019vac] and indeed it was found that these conformal operators transforms as Virasoro primary fields. In addition, OPEs of all the BMS generators, superrotations and supertranslations, were derived [@Fotopoulos:2019vac][^3]. The study of these OPEs allowed us to derive the BMS algebra [@Barnich:2017ubf] of asymptotic symmetries. An alternative proposal for the energy-momentum tensor of a pure gluon theory appeared in [@He:2015zea]. It was shown that positive helicity soft gluons correspond to holomorphic conserved currents on the ${\cal CS}^2$ which generate a $D\!=\!2$ Kac-Moody algebra. In standard CFT, in the presence of a Kac-Moody algebra, we can use the Sugawara construction [@Sugawara:1967rw] to build the energy-momentum tensor. It is natural to ask ourselves if it is possible to extend this construction on the CCFT. Some initial attempts in this direction using double soft theorems of gluons appeared in [@McLoughlin:2016uwa; @Nande:2017dba]. These results showed several interesting features and short-comings of the Sugawara construction. In the current paper we will approach this problem from the point of view of celestial amplitudes. We will construct the Sugawara tensor from the double conformal soft limit of gluons. Furthermore, we will derive its OPE with conformal operators of gluons and discuss its properties depending on whether the operators are soft or hard. We will conclude that the Sugawara energy-momentum tensor can only capture the dynamics of the soft sector of the theory confirming earlier observations in [@Cheung:2016iub]. In the setup of an Einstein-Yang-Mills (EYM) theory, we will present an alternative approach, based on a pair of gluon conformal operators. This is reminiscent of the double copy or Kawai–Lewellen–Tye (KLT) construction of gravity amplitudes from gauge theory amplitudes. This energy momentum tensor captures the dynamics of soft and hard operators alike. We will see that we can extend this construction to include the supertranslation generators as well.
This article is organized as follows. In Section \[sec:Prelim\], we review the notation, useful formulas of the CCFT and the Mellin transform of 4d gluon amplitudes which generates celestial amplitudes. We also review the Kac-Moody algebra on ${\cal CS}^2$ and the proposed Sugawara energy-momentum tensor based on conformally soft gluon operators. In Sections \[sec:Sugawara\_ope\] and \[sec:coincide\_result\], we compute the celestial amplitudes with a Sugawara energy-momentum tensor insertion. We discuss in details the case of $SU(N)$ and general gauge groups studying the Mellin transform of MHV gluon amplitudes [@Parke:1986gb] under the double conformal soft limit. We derive the OPE of the energy momentum tensor with conserved currents. Our results agree with standard Kac-Moody current algebra expectations. In section \[sec:gengroup\] we re-derive the OPEs of the Sugawara tensor applying the OPE between currents and then taking the conformal soft limits. Our results are consistent. Finally, in section \[sec:shadow\] we derive an energy momentum tensor which is inspired by the double copy or KLT construction of gravity amplitudes from gauge amplitudes. This energy-momentum tensor captures the conformal properties of both hard and soft operators.
Remarks on gluon operator products and Sugawara construction {#sec:Prelim}
============================================================
The $D=4$ momentum of a massless particle is parametrized by coordinates $(z,\bar{z})$ of the celestial sphere as $$\label{eq:pz}
p^{\mu}=\epsilon \omega q^{\mu}, \quad q^{\mu}=\frac{1}{2}\left(1+|z|^{2}, z+\bar{z},-i(z-\bar{z}), 1-|z|^{2}\right),$$ with $\omega$ the light-cone energy and $\epsilon=\pm$ indicating outgoing/incoming particles. The asymptotic direction along which the particle propagates is given by the null vector $q^\mu({z,\bar{z}})$. This vector is parametrised by the coordinates $(z,\bar{z})$ on the celestial sphere. In working on the celestial sphere we will need to transform plane wave solutions to conformal primary wave functions [@Pasterski:2017kqt]. For gluons, a conformal primary wave function with conformal dimension $(h,\bar{h})$ is given by the following expression: $$\label{eq:primary_wave}
A_{\mu}^{\Delta, J}=g(\Delta) V_{\mu J}^{\Delta}+\text {pure gauge term},$$ where $g(\Delta)=\tfrac{(\Delta-1)}{\Gamma(\Delta+1)}$ is the normalization constant, $\Delta=h+\bar{h}$ is the scaling dimension and $J=h-\bar{h}=\pm 1$ is the spin helicity. The function $V_{\mu J}^{\Delta}$ is the Mellin transform of the 4d plane wave function $$\label{eq:mellin}
V_{\mu}^{\Delta, J}\left(X^{\mu}, z, \bar{z}\right) \equiv \partial_{J} q_{\mu} \int_{0}^{\infty} d \omega\ \omega^{\Delta-1} e^{\mp i \omega q \cdot X-\varepsilon \omega}\ , \qquad J=\pm 1\ ,$$ where $\partial_J=\partial_z$ for $J=+1$ and $\partial_J=\partial_{{\bar{z}}}$ for $J=-1$. The polarization vectors are $\partial_{z} q^{\mu}=\epsilon_{+}^{\mu}(p)$ and $\partial_{\bar{z}} q^{\mu}=\epsilon_{-}^{\mu}(p)$. The scaling dimensions are $\Delta=1+i\lambda, \lambda\in\mathbb{R}$, [@Pasterski:2017kqt]. In a similar manner for gravitons the conformal primary wave function is
\[confprimMellin2\] H\^[,]{}\_ (X\^, z, [|[z]{}]{})\_J q\_\_J q\_\_0\^d \^[-1]{} e\^[i q X -]{} ,=2 . where $J=+1$ for $\ell=+2$ and $J=-1$ for $\ell=-2$. The conformal (quasi-primary) wave functions can be written as \[confprimexpr3\] G\^[,]{}\_= f()H\^[,]{}\_ + with the normalization constant $ f(\D)=
{1\over 2}{\D(\D-1) \over \Gamma(\D+2)}$. The presence of these normalization factors makes it clear that, as mentioned in the introduction, fields with spin 1 become pure gauge when $\D=1$ while fields with spin 2 become pure diffeomorphisms for $\D=0,1$.
In this work we will study $D=4$ tree–level gluon amplitudes[^4] $A_{n}$ and their celestial sphere representation $\Ac_n$. For a generic gauge group $G$ the $D=4$ gluon scattering amplitudes can be expressed as a sum over partial subamplitudes as follows[^5] $$\label{eq:4d_amp}
A_{n}(\{\om_i, q_i,J_i\}) = \sum_{\sigma \in S_{n-1}} \operatorname{Tr}\left(T^{a_1}T^{a_{\sigma(2)}}\ldots T^{a_{\sigma(n)}}\right) A^{\sigma}_{J_1J_2\ldots J_{n}}\left(\{\omega_{i}, z_{i}, \bar{z}_{i}\}\right)\ ,$$ with $T^a$ gauge generators in the fundamental representation of the gauge group $G$, the spin helicities denoted by $J_i=\pm1,i=1,2,\ldots,n$ and $A^{\sigma}_{J_1J_2\ldots J_{n}}$ is the partial subamplitude for a given permutation $\sigma$ expressed in celestial coordinates $\{\omega_{i}, z_{i}, \bar{z}_{i}\}$. The CCFT amplitudes are identified with the space–time amplitudes transformed from the plane wave basis into the conformal basis (\[eq:primary\_wave\],\[confprimexpr3\]) by using properly normalized Mellin transform[@Pasterski:2016qvg; @Pasterski:2017ylz; @Schreiber:2017jsr; @Stieberger:2018edy]. Concretely, the gluon partial subamplitudes $A^{\sigma}_{J_1J_2\ldots J_{n}}$ give rise to the celestial gluon amplitude: $$\begin{aligned}
\label{eq:mellin_partial_amp}
\mathcal{A}^{\sigma}_{J_{1} \ldots J_{n}}\left\{(\Delta_{i}, z_{i}, \bar{z}_{i}\}\right)&=\left(\prod_{i=1}^{n} g\left(\Delta_{i}\right) \int_0^\infty d \omega_{i}\ \omega_{i}^{\Delta_1-1}\right) A^{\sigma}_{J_{1} \ldots J_{n}}\left(\{\omega_{i}, z_{i}, \bar{z}_{i}\}\right)\delta^{4}\left(\sum_{i=1}^{n}\epsilon_{i}\omega_{i} q_{i}\right)\ .
\end{aligned}$$ In Eq.(\[eq:mellin\_partial\_amp\]) $\epsilon_i= +1$ or $-1$ depending whether the particles are incoming or outgoing, respectively. The full CCFT correlator is identified with the S–matrix element transformed from the plane wave basis into conformal basis: $$\label{eq:celestial_amp}
\left\langle\mathcal{O}_{\Delta_{1} J_{1}}^{a_{1}} \mathcal{O}_{\Delta_{2} J_{2}}^{a_{2}}\ldots \mathcal{O}_{\Delta_{n} J_{n}}^{a_{n}}\right\rangle =\sum_{\sigma \in S_{n-1}} \mathcal{A}_{J_{1} J_{2} \cdots J_{n}}^{\sigma} \operatorname{Tr}\left(T^{a_{1}} T^{a_{\sigma(2)}} \ldots T^{a_{\sigma(n)}}\right):=
\Ac_n\left(\{\Delta_{i}, z_{i}, \bar{z}_{i},J_i\}\ri),$$ where $\mathcal{O}_{\Delta_{i} J_{i}}^{a_{i}}$ is the conformal field operator which corresponds to a gluon conformal primary wave function (\[eq:primary\_wave\]).
On the celestial sphere, the limit $z\to w$ of coinciding positions for two operators, corresponds to $q^\mu(z)\to q^\mu(w)$ for the 4d gluon particles. This limits corresponds to the collinear momentum limit $p^\mu(z)\parallel p^\mu(w)$. It is well known that gauge and gravity amplitudes have collinear singularities and based on the discussion above, they give rise to the OPE singularities of the holographic CCFT. In [@fan_soft_2019; @Fotopoulos:2019vac] (see also [@Pate:2019lpp]), using collinear limits of the 4d gluon amplitude, it was shown that the CCFT has the following OPEs for gluon conformal primaries \[opepp\] \^[a]{}\_[\_1, +]{}(z,[|[z]{}]{})\^[b]{}\_[\_2,+]{}(w,|w) = f\^[abc]{}\^[c]{}\_[(\_1+\_2-1),+]{}(w,|w)+ , with C\_[(+,+)+]{}(\_1,\_2)=1-\[ope1\] , and \^[a]{}\_[\_1 ,+]{}(z,[|[z]{}]{})&&\^[b]{}\_[\_2,-]{}(w,|w)=\
&& f\^[abc]{}\^[c]{}\_[(\_1+\_2-1),-]{}(w,|w) \[opemp\]\
&&+ f\^[abc]{}\^[c]{}\_[(\_1+\_2-1),+]{}(w,|w)\
&& +C\_[(+-)–]{}(\_1,\_2)\^[ab]{}[O]{}\_[(\_1+\_2),-2]{}(w,|w)\
&&+ C\_[(+-)++]{}(\_1,\_2)\^[ab]{}[O]{}\_[(\_1+\_2),+2]{}(w,|w) + ,with: C\_[(+,-)-]{}(\_1,\_2)&=&\[ope3\] ,\
C\_[(+,-)+]{}(\_1,\_2)&=&\[ope4\] ,\
C\_[(+-)–]{}(\_1,\_2)&=& - ,\
C\_[(+-)++]{}(\_1,\_2) &=& - .The subleading terms [^6] of the mixed helicity OPE are associated with corrections of the EYM theory and won’t be important in the pure YM case. Nevertheless, we present these terms here since they will be important in the shadow construction of the energy momentum tensor of section \[sec:shadow\].
In this work we will explore further the properties of a class of correlators, which involve the conformally soft gluon operators $\l \to 0$ [@Donnay:2018neh; @fan_soft_2019]. On the CCFT side these lead to $\D=1$ conserved currents: $$\label{eq:conserved_current}
j^{a}(z)=\mathcal{O}_{\Delta=1,J=+}^{a}(z, \bar{z}), \quad \bar{j}^{a}(\bar{z})=\mathcal{O}_{\Delta=1,J=-}^{a}(z, \bar{z}).$$ The conserved currents suggest an emerging infinite dimensional symmetry algebra, commonly known as Kac-Moody current algebra. The soft limit $\D_1 \to1$ (\[opepp\]) leads to the following OPE on gluon conformal primaries: \[eq:JOope\] j\^a(z) O\^b\_[,+]{}(w) \~[f\^[a b c]{} O\^c\_[,+]{}(w) z-w]{} . In the case of same helicity gluons the consecutive soft limit is equivalent to the double soft limit. Taking the consecutive soft limit $\D_1,\D_2\to 1$ in (\[opepp\]), we are led to the holomorphic current algebra: \[eq:JJope\] j\^a(z) j\^b(w) \~[f\^[a b c]{} j\^c(w) z-w]{} . On the other hand, the soft limit $\D_1$ on the mixed OPE (\[opemp\]) leads to the following result \[eq:JOmope\] j\^a(z) O\^[b]{}\_[,-]{}(w)\~[f\^[a b c]{} O\^c\_[,-]{}(w) z-w]{} and similar results for $\bar{j}^a({\bar{z}})$. Now taking the consecutive double soft limit $\D_1,\D_2 \to 1$ of the mixed helicity gluon OPE we see that the result depends on the order of limits. Specifically taking the soft limit of the positive helicity gluon always first we get \[eq:jbjope\] j\^a(z) |[j]{}\^[b]{}([|[w]{}]{})\~[f\^[a b c]{} |[j]{}\^c([|[w]{}]{}) z-w]{} , and taking the negative one first followed by the positive we get: \[eq:bjjope\] |[j]{}\^a([|[z]{}]{}) j\^[b]{}(w)\~[f\^[a b c]{} j\^c(w) [|[z]{}]{}-[|[w]{}]{}]{} . The order of the soft limits is crucial when we have opposite helicity states or equivalently opposite spin operators.
The relations above imply that the antiholomorphic currents $\bar{j}^a({\bar{z}})$ transform in the adjoint representation of the Kac-Moody symmetry generated by the holomorphic currents $j^a(z)$ and vice-versa. As explained in more details in [@He:2015zea] a symmetric limit which realizes both the holomorphic and antiholomorphic Kac-Moody non-Abelian algebras is not possible. This seems to be related to 3d Chern-Simons theory on a manifold with boundary. The theory naively has two gauge connections $A_z $ and $A_{{\bar{z}}}$, which generate Kac-Moody symmetries, but in the non-Abelian case, boundary conditions eliminate one of them leaving only one copy. In [@Cheung:2016iub] this idea was further explored. The 4d Minkowski spacetime is written as a foliation of $AdS_3$ slices. There, it was demonstrated that indeed the soft sector of the theory leads to a CS theory on the $AdS_3$ slices. For the non-Abelian case boundary conditions allow either positive or negative helicity gluons. The $AdS_3/CFT_2$ correspondence implies only a single copy of a Kac-Moody algebra for the soft gluon sector. We conclude that in the CCFT we need to consider correlators where either the positive or the negative helicity gluons are conformally soft, but not both. When discussing the energy momentum tensor, we chose to study the realisation of the holomorphic Kac-Moody algebra generated by $j^a(z)$ in (\[eq:conserved\_current\]).
It is known that for two dimensional $CFT_2$ the energy momentum tensor for affine current algebras is given by the Sugawara construction. As mentioned before we expect only one copy of the Kac-Moody algebra and therefore only one Sugawara energy momentum tensor. The soft sector of positive helicity gauge bosons forms a sub-$CFT_2$ of the full CCFT. Hard particles are sources of soft radiation[^7]. On the CCFT side, correlation functions factorize into a hard and a soft part. The soft part is expected to be described by a current algebra and its conformal properties encoded in the Ward identities of the Sugawara energy momentum tensor. In this paper we will construct the Sugawara energy momentum tensor using the double conformal soft limit of celestial amplitudes like . We will consider gluon amplitudes and study the limit where the Sugawara tensor becomes collinear with conformally soft positive helicity gluons, the holomorphic current algebra currents $j^a(z)$.
The Sugawara construction [@Sugawara:1967rw; @DiFrancesco:1997nk] gives an expression of the energy momentum tensor in terms of gauge currents $$\label{eq:sugawara_0}
T^S(w)=\fc{1}{2k+C_2}\ \sum_a :J^a(w)J^a(w):\ ,$$ where $k$ is the level of the affine current algebra and the quadratic Casimir[^8] of the adjoint representation is $C_2=\delta^{ab} f^{acd}f^{bcd}$, which is twice the dual Coxeter number $h(g)$, i.e. $C_2=2h(g)$. Usually, for free fields the normal ordering is achieved by subtracting the corresponding two–point correlator \[eq:sugawaranormal\] T\^S(w\_1)= \_[w\_2w\_1]{} {\_a J\^a(w\_1)J\^a(w\_2)-} where $\dim g=\sum_a\delta_{aa}$ is the dimension of the underlying gauge group [@Polchinski:1998rr]. We assume roots of length–squared two. A more general normal ordering can explicitly be imposed by the following contour integral: $$\label{eq:Sugawara_1}
T^S(w_1)=\fc{1}{2k+C_2}\ \fc{1}{2\pi i}\oint_{w_1} \fc{dw_2}{w_2-w_1}\ \sum_a J^a(w_2)J^a(w_1)\ .$$ This more general definition of normal ordering takes into account all possible singular terms and is appropriate in case of fields which are not free [@DiFrancesco:1997nk].
In the following sections we will construct the Sugawara energy momentum tensor first implementing the normal ordering prescription (\[eq:sugawaranormal\]) $$\label{eq:sugawara_def}
T^S(z) =\gamma\sum_{a} j^{a}(z) j^{a}(z)=\gamma \lim\limits_{\Delta,\Delta'\to 1} \lim\limits_{z' \to z} \sum_a \mathcal{O}^a_{\Delta, +}(z, \bar{z})\mathcal{O}^a_{\Delta', +} (z', \bar{z}')\ ,$$ where $\gamma$ is a normalization constant depending on the details of the Yang-Mills theory. For simplicity, we will ignore this normalization constant and determine its value in the end of the computation. We expect that this value will help determine a discrepancy regarding the level $k$ observed in [@Cheung:2016iub]. The relation above is to be considered always as an insertion in a celestial CFT correlator. We will demonstrate that in the case of celestial correlators of MHV amplitudes, (\[eq:sugawara\_def\]) is indeed the energy momentum tensor for the sub-CFT of currents $j^a(z)$. The expected OPE of $T^S(z)$ with a conserved current $j^a(w)$ should be $$\label{eq:tj-ope}
T^S(z)j^a(w) = \frac{1}{(z-w)^{2}} j^a(w) +\frac{1}{z-w} \partial_{w} j^a(w) +\ldots\ .$$ The collinear limit of the Sugawara tensor with negative and positive helicity hard states will also be discussed. We will see that conformal invariance of the full CCFT including the hard sources, necessitates additional contributions to the energy momentum tensor beyond the Sugawara construction. Our discussion in section \[sec:shadow\] extends our construction to a double copy (or KLT) type energy momentum tensor, where the conformally soft graviton in [@Fotopoulos:2019tpe] is described as a pair of conformally soft gluons. This provides an alternative definition of the energy momentum tensor which includes both soft and hard modes on equal footing. It is nevertheless distinct to the Sugawara tensor, since it does not include a bilinear of the dimension one currents $j^a(z)$.
Gluon amplitudes, gauge current insertion and operator products {#sec:Sugawara_ope}
===============================================================
In this Section we shall discuss gluon amplitudes [(\[eq:4d\_amp\])]{} with insertion of a pair of gauge currents [(\[eq:sugawara\_def\])]{}. From the CCFT theory point of view the Sugawara construction (\[eq:sugawara\_def\]) corresponds to performing the double conformal soft limit of two positive helicity gluons taken to be collinear at the same time. In order to study the OPEs of this tensor with primaries we start at the $D=4$ tree–level $n+2$–point gluon MHV amplitude $A_{n+2}(\{\om_i, q_i,J_i\})$. We shall construct the Sugawara energy–momentum $T^S(z)$ and derive its OPE with conserved currents $j(z)$ in the celestial amplitude . In the latter we use the conformal primary operators $\mathcal{O}^{a_{n+1}}_{\Delta_{n+1}+}$ and $\mathcal{O}^{a_{n+2}}_{\Delta_{n+2}+}$ of the last two gluons to construct $T^S(z)$. Obviously, the result does not depend on this choice. Their color indices will be contracted $a_{n+2}=a_{n+1}=a$ and their positions will approach each other by taking the limit $z_{n+1},z_{n+2}\to z$. In the next step, in section \[sec:coincide\_result\] we shall take the conformal soft limit $\Delta_{n+1}, \Delta_{n+2}\to 1$ to get $T^S(z)\sim j^a(z)j^a(z)$. The OPE with the primaries $j(z)$ is extracted by taking the coinciding limit $z \to z_j, j=3,4,\ldots,n$ of $T^S(z)$ with primaries $\mathcal{O}^{a_j}_{\Delta_j+}$. Finally, since we are interested only in the soft sector, at the end we will take the conformal soft limit $\Delta_j\to 1$ to get the OPE $T^S(z)j^{a_j}(z_j)$.
In order to study the OPEs with primaries we shall focus on the MHV case. Hence, we will restrict [(\[eq:4d\_amp\])]{} to the $D=4$ tree–level $n+2$–point gluon MHV amplitude [@Parke:1986gb] $A_{n+2}(\{\om_i, q_i,J_i\})$ with spin helicities $J_1=J_2=-1,J_i=+1,i=3,4,\ldots,n+2$ and their corresponding partial subamplitudes $A^{\sigma}_{J_1J_2\ldots J_{n+2}}$ in celestial sphere representation $$\begin{aligned}
\label{eq:partial_amp}
A^{\sigma}_{J_1J_2\ldots J_{n+2}}\left(\{\omega_{i}, z_{i}, \bar{z}_{i}\}\right) &= \frac{\langle12\rangle^4}{\langle1\sigma(2)\rangle\langle\sigma(2)\sigma(3)\rangle\ldots\langle\sigma(n+2)1\rangle} \nonumber\\
&= \frac{\omega_{1} \omega_{2}}{\omega_{3} \omega_{4} \ldots \omega_{n+2}} \frac{z_{12}^{4}}{z_{1\sigma(2)} z_{\sigma(2)\sigma(3)}\ldots z_{\sigma(n+2)1}}\ ,
\end{aligned}$$ with $z_{jk}=z_j-z_k$.
To summarize, in the following two subsections we shall demonstrate the following relation $$\begin{aligned}
\lim_{z_{n+1}\ra z_j} A_{n+2}(\{g_{n+2}^+,g_1,&\ldots,g_n,g_{n+1}^+\})
=-\fc{\tilde C_2(G)}{\omega_{n+1}\omega_{n+2}}\nonumber\\
&\times\lf(\fc{1}{(z_{n+1}-z_{j})^2}+
\fc{\tilde\partial_{z_j}}{z_{n+1}-z_{j}}\ri) A_{n}(\{g_1,\ldots,g_{n}\}),\ j=1,\ldots,n\ ,\label{OPEn}\end{aligned}$$ with the full $n$ gluon amplitude $A_{n}(\{g_1,\ldots,g_{n}\})$ and the quadratic Casimir $$\tilde C_2(G)=2 C_2(G).$$ The relation [(\[OPEn\])]{} holds for any gauge group $G$. Above we have introduced the derivative \[tderi\] \_[z\_j]{}= -4 , which singles out the two gluons $j_1,j_2$ with negative helicity. Of course, for [(\[eq:partial\_amp\])]{} we have: $j_1=1,j_2=2$. For $j\neq j_1,j_2$ we get $\tilde\partial_{z_j}=\partial_{z_j}$ and the relation [(\[OPEn\])]{} takes the form of [(\[eq:mellin\_0\])]{} further used in section \[sec:coincide\_result\].
Gluon amplitudes and operator products for $\mathbf{SU(N)}$
-----------------------------------------------------------
Let us first discuss[^9] the gauge group $G=SU(N)$. For simplicity, we also include the photon and choose the gauge group[^10] to be $U(N)$. The fundamental representation of $U(N)$ satisfies the following useful relations: $$\label{eq:group_formula}
\left(T^{a}\right)_{k}^{j}\left(T^{a}\right)_{l}^{s}=\delta_{l}^{j} \delta_{k}^{s}, \quad
\left[T^{a}, T^{b}\right]=i\sqrt{2} f^{a b c} T^{c}\equiv i\tilde f^{abc}T^c.$$
In the following computation, we firstly analyze the $D=4$ MHV amplitudes and then perform the Mellin transform in section \[sec:coincide\_result\]. From the partial amplitude , various poles of the OPE obviously come from the denominators $z_{n+1,n+2}$, $z_{n+2,j}$ and $z_{n+1,j}$. The poles $z_{n+1,n+2}$ are to be subtracted under the normal ordering of $T^S(z)$ (\[eq:sugawaranormal\]). We will show though that since our tree amplitudes imply that the level $k$ of the Kac-Moody is zero, such a subtraction wont be necessary. All such $z_{n+1,n+2}$ poles will drop automatically.
Among all the permutations, the double poles arise in the following 6 kinds of ordering $$\label{eq:pole_double_ordering}
\begin{array}{ccc}
A(\ldots,j,n+1,n+2,\ldots)&A(\ldots,n+1,j,n+2,\ldots)&A(\ldots,n+1,n+2,j,\ldots),\\
A(\ldots,j,n+2,n+1,\ldots)&A(\ldots,n+2,j,n+1,\ldots)&A(\ldots,n+2,n+1,j,\ldots),
\end{array}$$ while the single poles arise in the following 12 possible orderings $$\label{eq:pole_single_ordering}
\begin{array}{ccc}
A(\ldots,j,n+1,\ldots,n+2,\ldots)&A(\ldots,n+1,\ldots,j,n+2,\ldots)&A(\ldots,n+1,n+2,\ldots,j,\ldots) \nonumber\\
A(\ldots,n+1,j,\ldots,n+2,\ldots)&A(\ldots,n+1,\ldots,n+2,j,\ldots)&A(\ldots,n+2,n+1,\ldots,j\ldots) \nonumber\\
A(\ldots,j,\ldots,n+2,n+1,\ldots)&A(\ldots,n+2,j,\ldots,n+1,\ldots)&A(\ldots,n+2,\ldots,n+1,j\ldots) \nonumber\\
A(\ldots,j,\ldots,n+1,n+2,\ldots)&A(\ldots,j,n+2,\ldots,n+1,\ldots)&A(\ldots,n+2,\ldots,j,n+1\ldots).
\end{array}$$
\
\
\[sec:sub\_double\]
Now let’s analyze the terms that contribute to double poles. Consider the two kinds of ordering $(\sigma(i),j,n+1,n+2,\sigma(i+1))$ and $(\sigma(i),j,n+2,n+1,\sigma(i+1))$. Using the formula , it is straightforward to get the contraction of the color indices for $U(N)$ $$\begin{aligned}
\label{eq:color_1}
& \operatorname{Tr}(\ldots T^{a_{\sigma(i)}}T^{a_j}T^{a}T^{a}T^{a_{\sigma(i+1)}}\ldots)=(N)\operatorname{Tr}(\ldots T^{a_{\sigma(i)}}T^{a_j}T^{a_{\sigma(i+1)}}\ldots).
\end{aligned}$$ hese two ordering have the same group factor, which means their partial amplitudes can be combined together. To get the double poles, we first take the limit $z_{n+2}\to z_{n+1}$, then take the limit $z_{n+1}\to z_j$. It is easy to get the following poles from the diverging parts of the partial amplitudes $$\label{eq:doublepole_1}
\frac{1}{z_{j,n+1}z_{n+1,n+2}z_{n+2,\sigma(i+1)}} + \frac{1}{z_{j,n+2}z_{n+2,n+1}z_{n+1,\sigma(i+1)}} =\frac{1}{z_{j,n+1}^2 z_{j,\sigma(i+1)}} + \frac{2}{z_{j,n+1}z_{j,\sigma(i+1)}^2} ,$$ where a single pole also arises in addition to the double pole. Combining the group factor and the remaining parts of the partial amplitudes, we get an MHV amplitude of $n$ gluons $$\begin{aligned}
\label{eq:doubleorder_1}
\lf(\frac{1}{z_{j,n+1}^2} + \frac{2}{z_{j,n+1}z_{j,\sigma(i+1)}}\ri)&N\operatorname{Tr}(\ldots T^{a_{\sigma(i)}}T^{a_j}T^{a_{\sigma(i+1)}}\ldots)\ \lf(\ldots\frac{1}{z_{\sigma(i),j}z_{j,\sigma(i+1)}}\ldots\ri)\nonumber\\
&=\frac{N}{\omega_{n+1}\omega_{n+2}}\ \lf(\frac{1}{z_{j,n+1}^2} + \frac{2}{z_{j,n+1}z_{j,\sigma(i+1)}}\ri)\ A_{n}(\{\omega_i, q_i,J_i\}).
\end{aligned}$$
For the other two orderings $(\sigma(i),n+1,n+2,j,\sigma(i+1))$ and $(\sigma(i),n+2,n+1,j,\sigma(i+1))$, it is easy to see that the contraction of color indices is the same as . Then it is straightforward to get the following poles from the diverging parts of their partial amplitudes $$\label{eq:doublepole_2}
\frac{1}{z_{\sigma(i),n+1}z_{n+1,n+2}z_{n+2,j}} + \frac{1}{z_{\sigma(i),n+2}z_{n+2,n+1}z_{n+1,j}} =\frac{1}{z_{n+1,j}^2 z_{\sigma(i),j}} + \frac{2}{z_{n+1,j}z_{\sigma(i),j}^2}\ .$$ So the result is also an MHV amplitude of $n$ gluons $$\begin{aligned}
\label{eq:doubleorder_2}
\lf(\frac{1}{z_{n+1,j}^2} + \frac{2}{z_{n+1,j}z_{\sigma(i),j}}\ri)&N\operatorname{Tr}(\ldots T^{a_{\sigma(i)}}T^{a_j}T^{a_{\sigma(i+1)}}\ldots)\ \lf(\ldots\frac{1}{z_{\sigma(i),j}z_{j,\sigma(i+1)}}\ldots\ri)\nonumber\\
&=\frac{N}{\omega_{n+1}\omega_{n+2}}\lf(\frac{1}{z_{n+1,j}^2} + \frac{2}{z_{n+1,j}z_{\sigma(i),j}}\ri)\ A_{n}(\{\omega_i,q_i,J_i\}).
\end{aligned}$$
For the remaining two orderings $(\sigma(i),n+1,j,n+2,\sigma(i+1))$ and $(\sigma(i),n+2,j,n+1,\sigma(i+1))$, the contraction of color indices has a single trace term $\operatorname{Tr}(T^{a_j})=0$, which is zero because it is a gluon. So the contribution is zero.
Combining all the above results, we get the following result from all possible orderings in $$\label{eq:doubleorder}
\frac{2N}{\omega_{n+1}\omega_{n+2}}\ \lf(\frac{1}{z_{n+1,j}^2} + \frac{1}{z_{n+1,j}}\partial_j\ri)\ A_{n}(\{\omega_i, q_i,J_i\}),$$ which contains a double pole and a single pole with a derivative acting on the $j$-th gluon.
Now let us see what would happen if the gauge group is chosen to be $SU(N)$, in which case the following formula is used $$\label{eq:su_n}
(T^a)^j_k (T^a)^s_l= \delta^j_l \delta^s_k-\frac{1}{N} \delta^j_k \delta^s_l\ .$$ The extra term in this formula leads to an extra term $-(1/N)\operatorname{Tr}(\ldots T^{a_{\sigma(i)}}T^{a_j}T^{a_{\sigma(n+1)}}\ldots)$ in the color index contractions of all six kinds of ordering. When combining the partial amplitudes, this extra term cancels out, leaving the complete result the same as . This is of course the $U(1)$ decoupling identity of standard YM, as expected.
\
\
\[sec:sub\_single\]
Now let us analyze the terms that only contribute to single poles. Consider the four orderings $(\sigma(i),j,n+1,\ldots,n+2,\sigma(i+1))$, $(\sigma(i),n+1,j,\ldots,n+2,\sigma(i+1))$, $(\sigma(i),n+2,\ldots,j,n+1,\sigma(i+1))$ and $(\sigma(i),n+2,\ldots,n+1,j,\sigma(i+1))$. Using , the contraction of color indices contains double traces $$\begin{aligned}
\label{eq:single_1}
\operatorname{Tr}(\ldots T^{a_{\sigma(i)}}T^{a_j}T^{a}\ldots &T^{a}T^{a_{\sigma(i+1)}}\ldots) = \operatorname{Tr}(\ldots T^{a_{\sigma(i)}}T^{a}\ldots T^{a}T^{a_j}T^{a_{\sigma(i+1)}}\ldots)\nonumber\\
&=\operatorname{Tr}(\ldots)\operatorname{Tr}(\ldots T^{a_{\sigma(i)}}T^{a_j}T^{a_{\sigma(i+1)}}\ldots)\nonumber\\
\operatorname{Tr}(\ldots T^{a_{\sigma(i)}}T^{a}T^{a_j}\ldots &T^{a}T^{a_{\sigma(i+1)}}\ldots) = \operatorname{Tr}(\ldots T^{a_{\sigma(i)}}T^{a}\ldots T^{a_j}T^{a}T^{a_{\sigma(i+1)}}\ldots)\nonumber\\
&= \operatorname{Tr}(T^{a_j}\ldots)\operatorname{Tr}(\ldots T^{a_{\sigma(i)}}T^{a_{\sigma(i+1)}}\ldots)\ .
\end{aligned}$$ Combining the corresponding partial amplitudes, they cancel out in the coinciding limit $z_{n+2}=z_{n+1}$ and $z_{n+1}=z_{j}$, so the complete result is zero.
Now let’s see what would happen if the gauge group is chosen to be $SU(N)$, where the contraction of color indices contains extra single traces $$\begin{aligned}
\label{eq:single_su}
\operatorname{Tr}(\ldots T^{a_{\sigma(i)}}T^{a_j}T^{a}\ldots &T^{a}T^{a_{\sigma(i+1)}}\ldots) =\operatorname{Tr}(\ldots T^{a_{\sigma(i)}}T^{a}T^{a_j}\ldots T^{a}T^{a_{\sigma(i+1)}}\ldots)\nonumber\\
&= (-\frac{1}{N})\operatorname{Tr}(\ldots T^{a_{\sigma(i)}}T^{a_j}\ldots T^{a_{\sigma(i+1)}}\ldots)\nonumber\\
\operatorname{Tr}(\ldots T^{a_{\sigma(i)}}T^{a}\ldots T^{a}&T^{a_j}T^{a_{\sigma(i+1)}}\ldots) = \operatorname{Tr}(\ldots T^{a_{\sigma(i)}}T^{a}\ldots T^{a_j}T^{a}T^{a_{\sigma(i+1)}}\ldots)\nonumber\\
&= (-\frac{1}{N})\operatorname{Tr}(\ldots T^{a_{\sigma(i)}}\ldots T^{a_{j}}T^{a_{\sigma(i+1)}}\ldots).
\end{aligned}$$ Again the combination of partial amplitudes for these extra terms cancels out in the coinciding limit $z_{n+2}=z_{n+1}$ and $z_{n+1}=z_{j}$. So the result is the same as the case of $U(N)$, which is zero. Similarly, for the four kinds of ordering $(\sigma(i),j,n+2,\ldots,n+1,\sigma(i+1))$, $(\sigma(i),n+2,j,\ldots,n+1,\sigma(i+1))$, $(\sigma(i),n+1,\ldots,j,n+2,\sigma(i+1))$ and $(\sigma(i),n+1,\ldots,n+2,j,\sigma(i+1))$, the complete result is zero. The remaining four kinds of ordering $(\sigma(i),n+2,n+1,\ldots,j,\sigma(i+1))$, $(\sigma(i),n+1,n+2,\ldots,j,\sigma(i+1))$, $(\sigma(i),j,\ldots,n+2,n+1,\sigma(i+1))$ and $(\sigma(i),j,\ldots,n+1,n+2,\sigma(i+1))$ is also zero.
Eventually, after combining all permutations for the singular terms of the 4d MHV amplitude we get the result [(\[OPEn\])]{} for $j=3,4,\ldots,n$ and $\tilde C_2(G)=2N$ for $G=SU(N)$. We can generalize the above manipulations to negative helicity gluons $j=1, \, 2.$ Due to the cyclic structure of the denominator for the partial amplitude , the poles in and are the same for $j=1,\, 2.$ The only difference is in the single pole term, where the numerator $z_{12}^4$ would modify the derivative term of as [(\[tderi\])]{}.
Gluon color sums and operator products for general gauge group {#sec:colorsums}
--------------------------------------------------------------
\#1[\#1]{}
In this subsection we worked out color sums with insertion of a pair of gauge currents. This generalizes the previous discussion for the case of general gauge group $G$. We will need to develop several gauge group identities, some of which are novel and potentially useful for scattering amplitude computations in general. We start with the color decomposition of an $n+2$–point gluon amplitude $$\begin{aligned}
\label{startDDM}
A_{n+2}(\{p_i,J_j\})
&=\sum_{\sigma \in S_n}\f^{a_{n+2} b_{\sigma(1)} x_{1}} \f^{x_{1} b_{\sigma(2)} x_{2}}\ldots \f^{x_{n-1} b_{\sigma(n)} a_{n+1}}\nonumber\\
&\times A_{n+2}\left(n+2, \sigma\left(1,2,\ldots,n\right), n+1\right)\ ,\end{aligned}$$ with the partial subamplitudes $A_{n+2}(\ldots)$. The color decomposition is w.r.t. a $n!$–dimensional basis of subamplitudes $A_{n+2}\left(n+2, \sigma\left(1,2,\ldots,n\right), n+1\right)$ subject to the DDM representation [@DelDuca:1999rs], with the structure constants $$\f^{abc}=-i\ \operatorname{tr}(T^a[T^b,T^c])\ ,$$ with $T^a$ generators in the fundamental representation. Note, that [(\[startDDM\])]{} is just an other representation of the color sum [(\[eq:4d\_amp\])]{} for $n\ra n+2$.
We are interested in the pair of gluons $g_{n+1},g_{n+2}$ of positive helicity and in [(\[startDDM\])]{} we shall consider their double soft limits[^11] $p_{n+1},p_{n+2}\ra0$ $$\begin{aligned}
A_{n+2}\left((n+2)^+, \sigma\left(1,2,\ldots,n\right), (n+1)^+\right)&\ra \fc{\vev{n+1,{\sigma}(1)}}{\vev{n+1,n+2}\vev{n+2,{\sigma}(1)}}\ \nonumber\\
&\times \fc{\vev{{\sigma}(n),{\sigma}(1)}}{\vev{{\sigma}(n),n+1}\vev{n+1,{\sigma}(1)}}
\ A_{n}\left(\sigma\left(1,2,\ldots,n\right)\right)\ ,\label{doublesoft}\end{aligned}$$ and perform the sum over the color indices $a_{n+1}$ and $a_{n+2}$, i.e.: \[Need\] \_[a\_[n+1]{},a\_[n+2]{}]{} \_[a\_[n+1]{}a\_[n+2]{}]{} \^[a\_[n+2]{} b\_[(1)]{} x\_[1]{}]{} \^[x\_[1]{} b\_[(2)]{} x\_[2]{}]{}…\^[x\_[n-1]{} b\_[(n)]{} a\_[n+1]{}]{} . Actually, the object [(\[Need\])]{} is interesting on its own, since it appears in (planar) one–loop gluon amplitudes. It yields the group trace $\operatorname{Tr}(T_A^{b_{\sigma(1)}} \ldots T_A^{b_{\sigma(n)}})$ with the gauge group generators $T_A^a$ in the adjoint representation with $(T_A^a)_{bc}=-\tilde f^{abc}$. E.g. for $n=3$ we have [@vanRitbergen:1998pn]: \[Groupv\] \_a \^[ab\_[1]{}x\_1]{} \^[x\_1b\_[2]{}x\_2]{} \^[x\_2b\_[3]{}a]{}=- C\_2(G) \^[b\_[1]{}b\_[2]{}b\_[3]{}]{} , with the invariant $\tilde C_2(G)\equiv \tilde C_2$ referring to the adjoint representation of the gauge group $G$, cf. also footnote \[grouptheory\]. For general $n\geq 4$ we decompose [(\[Need\])]{} into combinations of symmetric tensors and fewer numbers of structure constants. This way for $n=4$ we have [@vanRitbergen:1998pn]: \[Groupvi\] \^[ab\_[1]{}x\_1]{}\^[x\_1b\_[2]{}x\_2]{} \^[x\_2b\_[3]{}x\_3]{}\^[x\_3b\_[4]{}a]{}= d\_A\^[b\_[1]{}b\_[2]{}b\_[3]{}b\_[4]{}]{}+ C\_2(G) {\^[b\_[1]{}b\_[4]{}a]{}\^[ab\_[2]{}b\_[3]{}]{}-\^[b\_[1]{}b\_[2]{}a]{}\^[ab\_[3]{}b\_[4]{}]{}} , with the symmetric invariant tensor $d_A$ given as trace over symmetrized products of gauge group generators $T^a_A$: d\_A:=d\_A\^[b\_1b\_2b\_3b\_4]{}=\_[S\_4]{}(T\_A\^[a\_[(1)]{}]{}T\_A\^[a\_[(2)]{}]{}T\_A\^[a\_[(3)]{}]{}T\_A\^[a\_[(4)]{}]{}) . Furthermore, for $n=5$ we derive: $$\begin{aligned}
\label{Groupvii}
&\f^{ab_{1}x_1} \f^{x_1b_{2}x_2} \f^{x_2b_{3}x_3}\f^{x_3b_{4}x_4}\f^{x_4b_{5}a}=\nonumber\\
&-\fc{1}{12}\tilde{C}_2(G) \lf\{\f^{b_1b_2a}\f^{ab_3c}\f^{cb_4b_5}+\f^{b_2b_3a}\f^{ab_4c}\f^{cb_5b_1}+
\f^{b_5b_1a}\f^{ab_2c}\f^{cb_3b_4}-\f^{b_1b_4a}\f^{ab_3c}\f^{cb_2b_5}\ri\}\nonumber\\[2mm]
&-\h\ \lf\{ \f^{b_1b_5a}\tilde d_A^{ab_2b_3b_4}+\f^{b_3b_2a}\tilde d_A^{ab_1b_4b_5}+\f^{b_4b_2a}\tilde d_A^{ab_1b_3b_5}+\f^{b_4b_3a}\tilde d_A^{ab_1b_2b_5}\ri\}\ .\end{aligned}$$ For general $n$ we have the relation of the following structure $$\begin{aligned}
\f^{a b_{\sigma(1)} x_{1}} \f^{x_{1} b_{\sigma(2)} x_{2}}\ldots \f^{x_{n-1} b_{\sigma(n)} a}
&=\tilde d_A^{b_{\sigma(1)}b_{\sigma(2)}\ldots b_{{\sigma}(n)}}+\ldots+\nonumber\\
&+\tilde C_2(G)\ \lf\{\tilde f^{n-2} \ldots\ri\}\end{aligned}$$
Let us now introduce celestial coordinates. With $\vev{ij}=(\omega_i\omega_j)^{1/2}z_{ij}$ the split factors in [(\[doublesoft\])]{} can be expressed in terms of celestial coordinates as: $$\begin{aligned}
A_{n+2}\Big((n+2)^+, \sigma\left(1,2,\ldots,n\right),& (n+1)^+\Big)\ra \fc{1}{\omega_{n+1}\omega_{n+2}}\ \lf(\fc{1}{z_{n+1}-z_{n+2}}+\fc{1}{z_{n+2}-z_{{\sigma}(1)}}\ri)\nonumber\\
&\times\lf(\fc{1}{z_{{\sigma}(n)}-z_{n+1}}+\fc{1}{z_{n+1}-z_{{\sigma}(1)}}\ri)
A_{n}\left(\sigma\left(1,2,\ldots,n\right)\right)\ .\label{Doublesoft}\end{aligned}$$ With these preparations we may compute the color sum [(\[startDDM\])]{} supplemented by [(\[Need\])]{} and [(\[Doublesoft\])]{}. One important observation is that all the terms $\tfrac{1}{z_{n+1}-z_{n+2}}$ cancel in the color sum. Therefore, we may safely take the limit $z_{n+2}\ra z_{n+1}$. In the sequel the following universal functions will specify the color sum. There is the function \[universal\] Z\_0=\_[i=1]{}\^n , universal to all color orderings and an other function $$\begin{aligned}
Z_{i_1,i_2,\ldots,i_n}&=\fc{1}{(z_{n+1}-z_{i_1})(z_{n+1}- z_{i_2})}+
\fc{1}{(z_{n+1}-z_{i_2})(z_{n+1}- z_{i_3})}\nonumber\\
&+\ldots+\fc{1}{(z_{n+1}-z_{i_n})(z_{n+1}- z_{i_1})}\ ,\label{rationalZ}\end{aligned}$$ which sums over all neighbours of a given color ordering $(i_1,i_2,\ldots,i_n)$.
\
\
For $n=3$ with [(\[Groupv\])]{} in the limit $z_5\ra z_4$ we find: $$\begin{aligned}
A_{5}(\{g_{5}^+,g_1,g_2,g_3,g_{4}^+\}) &=-\tilde C_2(G)\ \f^{b_1b_2b_3}\ A_3(1,2,3)\nonumber\\
&\times\fc{1}{\omega_{4}\omega_{5}}\fc{1}{ z_{14} z_{24} z_{34}}\
\lf\{ \fc{\ z_{12}\ z_{13}}{z_{14}}
+\fc{ z_{13}z_{23}}{ z_{34}} -\fc{ z_{12}\ z_{23}}{ z_{24}} \ri\}\ .\end{aligned}$$ Then, e.g. for $ z_4\ra z_1$ we have the expansion series: $$\begin{aligned}
\label{OPEi}
\lim_{ z_4\ra z_1}A_{5}(\{g_{5}^+,g_1,g_2,g_3,g_{4}^+\})
&=-\tilde C_2(G)\ \f^{b_1b_2b_3}\ A_3(1,2,3)\cr
&\times\fc{1}{\omega_{4}\omega_{5}}\lf\{\fc{1}{ z_{14}^2}+\fc{1}{ z_{14}}\lf(\fc{1}{ z_{12}}+\fc{1}{ z_{13}}\ri)\ri\}\ .\end{aligned}$$ In the following we shall rewrite the subleading piece $\tfrac{1}{z_{14}}\lf(\ldots\ri)$ of [(\[OPEi\])]{}. The three–point amplitudes are MHV amplitudes [(\[eq:partial\_amp\])]{}. Therefore, the latter assume the generic form $$A_3(1,i_2,i_3)\sim \fc{z_{j_1j_2}^4}{z_{1i_2}z_{i_2i_3}z_{i_31}}\ ,$$ with $j_1,j_2$ denoting those two gluons of negative helicity. After inspecting the rational terms in [(\[OPEi\])]{} we deduce that the terms in the bracket can be represented as derivative w.r.t. $z_1$ on the corresponding amplitude: $$\begin{aligned}
\fc{1}{ z_{14}}\lf(\fc{1}{ z_{12}}+\fc{1}{ z_{13}}\ri)\ A_3(1,i_2,i_3)&=
-\fc{1}{ z_{14}}\ \lf(\fc{\p}{\p z_1} -4\ \fc{\delta^{1,j_1}-\delta^{1,j_2}}{z_{j_1j_2}}\ri)\ A_3(1,i_2,i_3)\nonumber\\
&\equiv \fc{1}{ z_{41}}\ \tilde\p_{z_1}A_3(1,i_2,i_3)\ ,\label{receipt}\end{aligned}$$ with the derivative [(\[tderi\])]{} singling out the two gluons $j_1,j_2$ with negative helicity. Eventually, the limit [(\[OPEi\])]{} gives rise to the following Ward identity: \_[ z\_4z\_1]{}A\_[5]{}({g\_[5]{}\^+,g\_1,g\_2,g\_3,g\_[4]{}\^+}) =- (+) A\_3({g\_1,g\_2,g\_3}) , with the full three gluon amplitude: \[ThreeAmp\] A\_3({g\_1,g\_2,g\_3})=\^[b\_1b\_2b\_3]{} A\_3(1,2,3) . Similar Ward identities can be derived for the other two cases $z_4\ra z_2,z_3$. To this end, we get [(\[OPEn\])]{} for $n=3$ with the amplitude [(\[ThreeAmp\])]{}.
\
\
Next, for $n=4$ with [(\[Groupvi\])]{} in the limit $z_6\ra z_5$ we determine: $$\begin{aligned}
A_{6}(\{g_{6}^+,g_1,g_2,g_3,g_4,g_{5}^+\})&=\fc{1}{\omega_{5}\omega_{6}}\lf\{ \lf[\ \fc{\tilde C_2(G)}{3}\lf(-c_s-c_u\ri)+2d_A\ri] (Z_0-Z_{1234})\ A_4(1,2,3,4)\ri.\nonumber\\
&+\lf[\ \fc{\tilde C_2(G)}{3}\lf(-c_t+c_u\ri)+2d_A\ri] (Z_0-Z_{1324})\ A_4(1,3,2,4)\nonumber\\
&+\lf.\lf[\ \fc{\tilde C_2(G)}{3}\lf(c_s+c_t\ri)+2d_A\ri] (Z_0-Z_{1243})\ A_4(1,2,4,3)\ri\},\label{lastvi}\end{aligned}$$ with the color factors $$\begin{aligned}
\label{FourPointColor}
c_s=\f^{b_1b_2a}\f^{b_3b_4a}\,, \hskip 0.8cm
c_t=\f^{b_1b_3a}\f^{b_2b_4a}\,, \hskip 0.8cm
c_u=\f^{b_4b_1a}\f^{b_2b_3a}\ ,\end{aligned}$$ obeying the Jacobi relation $c_t+c_u=c_s$. Note, that the following Kleiss–Kuijf (KK) relation holds [@Kleiss:1988ne]: \[KK\] A\_4(1,2,3,4)+A\_4(1,2,4,3)+ A\_4(1,3,2,4)=0 . As a consequence any universal term cancels in the above color sum [(\[lastvi\])]{}.
Let us consider the limit $ z_5\ra z_1$, for which we have: \[limits\] Z\_0 , Z\_{
(+),&=1234,\
(+),&=1324,\
(+),&=1243 .
. Again in the same way [(\[receipt\])]{} as in the previous $n=3$ case we are able to rewrite the subleading pieces $\tfrac{1}{z_{15}}\lf(\ldots\ri)$ given in [(\[limits\])]{} and entering [(\[lastvi\])]{}. To this end as a consequence of [(\[KK\])]{} up to the next leading order only the terms multiplying the color factors [(\[FourPointColor\])]{} contribute in the color sum [(\[lastvi\])]{}. The same conclusions can be drawn for the other limits $z_5\ra z_2,z_3,z_4$ resulting in the following Ward identity [(\[OPEn\])]{} with $n=4$ and the full four gluon amplitude: $$\begin{aligned}
A_4(\{g_1,g_2,g_3,g_4\})&=\sum_{\sigma\in S_2}
\f^{b_1b_{\sigma(2)}a}\f^{ab_{\sigma(3)}b_4}\ A_4(1,{\sigma}(2),{\sigma}(3),4)\nonumber\\
&=c_s\ A_4(1,2,3,4)+c_t\ A_4(1,3,2,4)\ .\end{aligned}$$
\
Next, for $n=5$ with [(\[Groupvii\])]{} in the limit $z_7\ra z_6$ we derive: $$\begin{array}{lcl}
&&A_{7}(\{g_{7}^+,g_1,g_2,g_3,g_4,g_5,g_{6}^+\})=-\fc{1}{\omega_{6}\omega_{7}}\nonumber \\[2mm]
&\times&\lf\{\lf[\fc{\tilde C_2(G)}{6}\lf(-c_1-c_2-c_5+c_6\ri)+x_1+x_2+x_3+x_5+x_6+x_8\ri] (Z_0-Z_{12345})\ A_5(1,2,3,4,5)\ri.\\[3mm]
&+&\lf[\fc{\tilde C_2(G)}{6}\lf(c_1-c_7+c_{11}+c_{14}\ri)-x_1+x_2+x_3+x_5+x_6+x_8\ri] (Z_0-Z_{12354})\ A_5(1,2,3,5,4)\\[3mm]
&+&\lf[\fc{\tilde C_2(G)}{6}\lf(c_5+c_9-c_{11}-c_{12}\ri)+x_1+x_2+x_3-x_5+x_6+x_8\ri] (Z_0-Z_{12435})\ A_5(1,2,4,3,5)\\[3mm]
&+&\lf[\fc{\tilde C_2(G)}{6}\lf(c_2+c_{12}+c_{15}-c_{10}\ri)+x_1-x_2+x_3-x_5+x_6+x_8\ri] (Z_0-Z_{12453})\ A_5(1,2,4,5,3)\\[3mm]
&+&\lf[\fc{\tilde C_2(G)}{6}\lf(c_2+c_{11}+c_{14}-c_{15}\ri)+x_1+x_2+x_3+x_5+x_6+x_8\ri] (Z_0-Z_{13245})\ A_5(1,3,2,4,5)\\[3mm]
&+&\lf[\fc{\tilde C_2(G)}{6}\lf(c_6+c_{7}+c_{15}-c_{5}\ri)-x_1+x_2+x_3+x_5+x_6-x_8\ri] (Z_0-Z_{13254})\ A_5(1,3,2,5,4)\\[3mm]
&+&\lf[\fc{\tilde C_2(G)}{6}\lf(c_5+c_{12}-c_{9}-c_{11}\ri)+x_1+x_2+x_3+x_5-x_6-x_8\ri] (Z_0-Z_{13425})\ A_5(1,3,4,2,5)\\[3mm]
&+&\lf[\fc{\tilde C_2(G)}{6}\lf(c_1+c_{3}+c_{9}-c_{2}\ri)+x_1+x_2-x_3+x_5-x_6-x_8\ri] (Z_0-Z_{13452})\ A_5(1,3,4,5,2)\\[3mm]
&+&\lf[\fc{\tilde C_2(G)}{6}\lf(c_2+c_{11}+c_{15}-c_{14}\ri)+x_1+x_2+x_3-x_5-x_6+x_8\ri] (Z_0-Z_{14235})\ A_5(1,4,2,3,5)\\[3mm]
&+&\lf[\fc{\tilde C_2(G)}{6}\lf(c_5+c_{9}+c_{10}+c_{14}\ri)+x_1-x_2+x_3-x_5-x_6+x_8\ri] (Z_0-Z_{14253})\ A_5(1,4,2,5,3)
\end{array}$$ \[lastvii\]
[lcl]{} &+&(Z\_0-Z\_[14325]{}) A\_5(1,4,3,2,5)\
&+&.(Z\_0-Z\_[14352]{}) A\_5(1,4,3,5,2) },
with the color factors [@Bern:2008qj] $$\begin{aligned}
&&
c_{1\phantom{0}} = \f^{b_1 b_2 a}\f^{a b_3 c}\f^{c b_4 b_5}\,, \hskip 0.8cm
c_{2\phantom{1}} = \f^{b_2 b_3 a}\f^{a b_4 c}\f^{c b_5 b_1}\,, \hskip 0.8cm
c_{3\phantom{1}} = \f^{b_3 b_4 a}\f^{a b_5 c}\f^{c b_1 b_2}\,, \nonumber \\&&
c_{4\phantom{1}} = \f^{b_4 b_5 a}\f^{a b_1 c}\f^{c b_2 b_3}\,, \hskip 0.8cm
c_{5\phantom{1}} = \f^{b_5 b_1 a}\f^{a b_2 c}\f^{c b_3 b_4}\,, \hskip 0.8cm
c_{6\phantom{1}} = \f^{b_1 b_4 a}\f^{a b_3 c}\f^{c b_2 b_5}\,, \nonumber \\&&
c_{7\phantom{1}} = \f^{b_3 b_2 a}\f^{a b_5 c}\f^{c b_1 b_4}\,, \hskip 0.8cm
c_{8\phantom{1}} = \f^{b_2 b_5 a}\f^{a b_1 c}\f^{c b_4 b_3}\,, \hskip 0.8cm
c_{9\phantom{1}} = \f^{b_1 b_3 a}\f^{a b_4 c}\f^{c b_2 b_5}\,, \nonumber \\&&
c_{10} = \f^{b_4 b_2 a}\f^{a b_5 c}\f^{c b_1 b_3}\,, \hskip 0.8cm
c_{11} = \f^{b_5 b_1 a}\f^{a b_3 c}\f^{c b_4 b_2}\,, \hskip 0.8cm
c_{12} = \f^{b_1 b_2 a}\f^{a b_4 c}\f^{c b_3 b_5}\,, \nonumber \\ &&
c_{13} = \f^{b_3 b_5 a}\f^{a b_1 c}\f^{c b_2 b_4}\,, \hskip 0.8cm
c_{14} = \f^{b_1 b_4 a}\f^{a b_2 c}\f^{c b_3 b_5}\,, \hskip 0.8cm
c_{15} = \f^{b_1 b_3 a}\f^{a b_2 c}\f^{c b_4 b_5}\,. \hskip 1.5 cm
\label{FivePointColor}\end{aligned}$$ fulfilling various Jacobi relations leaving the set of six independent $\{c_1,c_6,c_9,c_{12},c_{14},c_{15}\}$, and the ten tensors $$\begin{aligned}
&&
x_{1\phantom{0}}= \f^{b_4 b_5 c}d_A^{c b_1 b_2b_3}\,, \hskip 1cm
x_{2\phantom{0}}= \f^{b_3 b_5 c}d_A^{c b_1 b_2b_4}\,, \hskip 1cm
x_{3\phantom{0}}= \f^{b_2 b_5 c}d_A^{c b_1 b_3b_4}\,, \nonumber \\&&
x_{4\phantom{0}}= \f^{b_1 b_5 c}d_A^{c b_2 b_3b_4}\,, \hskip 1cm
x_{5\phantom{0}}= \f^{b_3 b_4 c}d_A^{c b_1 b_2b_5}\,, \hskip 1cm
x_{6\phantom{0}}= \f^{b_2 b_4 c}d_A^{c b_1 b_3b_5}\,, \nonumber \\&&
x_{7\phantom{0}}= \f^{b_1 b_4 c}d_A^{c b_2 b_3b_5}\,, \hskip 1cm
x_{8\phantom{0}}= \f^{b_2 b_3 c}d_A^{c b_1 b_4b_5}\,, \hskip 1cm
x_{9\phantom{0}}= \f^{b_1 b_3 c}d_A^{c b_2 b_4b_5}\,, \nonumber \\&&
x_{10\phantom{0}}= \f^{b_1 b_2 c}d_A^{c b_3 b_4b_5}\ ,
\label{FivePointTensor}\end{aligned}$$ which fulfill the relations $$\begin{aligned}
&&
x_{4\phantom{0}}= -x_1-x_2-x_3\,, \hskip 2cm
x_{7\phantom{0}}= x_1-x_5-x_6\,, \nonumber \\&&
x_{9\phantom{0}}= x_2+x_5-x_8\,, \hskip 2cm
x_{10\phantom{0}}= x_3+x_6+x_8\ ,
\label{TensorRel}\end{aligned}$$ leaving six independent combinations $\{x_1,x_2,x_3,x_5,x_6,x_8\}$.
Again, for the limits $ z_6\ra z_j,\ j=1,\ldots,5$ the leading term $\tfrac{1}{z_{j6}^2}$ can easily be extracted from [(\[lastvii\])]{} by taking into account Jacobi and KK relations. To determine the next leading piece $\tfrac{1}{z_{j6}}$ arising from [(\[rationalZ\])]{} we can proceed in the same way as in the previous case $n=4$ which has lead to the receipt [(\[receipt\])]{}. To this end we find [(\[OPEn\])]{} for $n=5$ with the full five gluon amplitude: $$\begin{aligned}
A_5(\{g_1,g_2,g_3,g_4,g_5\})&=\sum_{\sigma\in S_3}
\f^{b_1b_{\sigma(2)}a}\f^{ab_{\sigma(3)}c}\f^{cb_{\sigma(4)}b_5}\ A_5(1,{\sigma}(2),{\sigma}(3),{\sigma}(4),5)\nonumber\\
&=c_1\ A_5(1,2,3,4,5)+c_{12}\ A_5(1,2,4,3,5)+c_{15}\ A_5(1,3,2,4,5)\nonumber\\
&+c_9\ A_5(1,3,4,2,5)+c_{14}\ A_5(1,4,2,3,5)+c_{6}\ A_5(1,4,3,2,5)\ .\end{aligned}$$
Finally, for generic $n$ we compute the color sum [(\[startDDM\])]{} supplemented by [(\[Need\])]{} and [(\[Doublesoft\])]{} and consider the limit $z_{n+2}\ra z_{n+1}$. From the consideration above it is evident, that for general $n$ we obtain [(\[OPEn\])]{} with the full $n$ gluon amplitude: $$\begin{aligned}
A_{n}(\{g_1,\ldots,g_{n}\})
&=\sum_{\sigma \in S_{n-2}}\f^{b_1b_{\sigma(2)} x_{1}} \f^{x_{1} b_{\sigma(3)} x_{2}}\ldots \f^{x_{n-5} b_{\sigma(n-3)} b_{n-2}}\nonumber\\
&\times A_{n}\left(1, \sigma\left(1,2,\ldots,n-1\right), n\right)\ .\end{aligned}$$ This completes the general proof of equation (\[OPEn\]). In the following section we will perform the Mellin transform of this amplitude and derive the OPE of the Sugawara energy-momentum tensor with primaries.
Mellin transform and the Sugawara energy-momentum tensor {#sec:coincide_result}
========================================================
In this section we will use the results we derived for gauge theory amplitudes to derive the OPE of the Sugawara energy-momentum tensor with the operators of our theory. We will split the discussion into two parts. One part regarding the conformally soft gluons and in the second part we will discuss the hard states. We will see that although the Sugawara energy-momentum tensor has the right OPE to generate conformal transformations for soft operators, it is not so for the hard ones and a modified tensor will be necessary. The correction needed will not be discussed in this work, although in section \[sec:shadow\] we will propose an alternative construction, and it is an interesting open question.
Sugawara energy-momentum tensor and conformally soft gluons {#sec:Sugasoft}
-----------------------------------------------------------
We start with the OPE result [(\[OPEn\])]{} for generic gauge groups $G$. After Mellin transforming the latter we extract the OPE of the energy–momentum tensor $T^S(z_{n+1})$ with the currents $j^{a_j}(z_j)$ from the celestial amplitude . We follow the steps explained in the beginning of section \[sec:Sugawara\_ope\]. The Mellin transform leads to $$\begin{aligned}
\lim_{z_{n+1}\to z_j}\langle&\mathcal{O}_{\Delta_{1} J_{1}}^{a_{1}}(z_1,\bar{z}_1)\ldots \mathcal{O}_{\Delta_{j}+}^{a_{j}}(z_{j},{\bar{z}}_j)\ldots\mathcal{O}_{\Delta_{n+1} +}^{a}(z_{n+1},{\bar{z}}_{n+1}) \mathcal{O}_{\Delta_{n+2}+}^{a}(z_{n+1}, {\bar{z}}_{n+1})\rangle\nonumber \\
&=\lim_{z_{n+1}\to z_j}\left(\prod_{i=1}^{n+2} g\left(\Delta_{i}\right) \int\limits_0^\infty d \omega_{i}\ \omega_{i}^{i \lambda_{i}}\right)\ \times\delta^{4}\left(\sum_{i=1}^{n}\epsilon_{i}\omega_{i} q_{i}+\epsilon_{n+1}\omega'_{n+1}q_{n+1}\right) \nonumber\\
&\times \frac{\tilde C_2(G)}{\omega_{n+1}\omega_{n+2}}\lf(\frac{1}{z_{n+1,j}^2} + \frac{\partial_j}{z_{n+1,j}}\ri)\ A_n(\{-,-,+,\ldots,+\})\ ,
\label{eq:mellin_0}
\end{aligned}$$ where in the coinciding limit[^12] we can define the total energy $\omega'_{n+1}=\omega_{n+1}+\omega_{n+2}$ of the collinear pair.
Consider first the Mellin integral of the double pole part $$\begin{aligned}
\label{eq:mellin_d0}
\frac{\tilde C_2(G)}{z_{n+1,j}^2}&\lim_{z_{n+1}\to z_j}\left(\prod_{i=1}^{n+2} g\left(\Delta_{i}\right) \int_0^\infty d \omega_{i}\ \omega_{i}^{i \lambda_{i}}\right)
\frac{1}{\omega_{n+1}\omega_{n+2}}\nonumber\\
&\times\delta^{4}\left({\sum_{i=1\atop i\neq j}^n}\epsilon_{i}\omega_{i} q_{i}+\epsilon_j\omega'_jq_j\right)A_n(\{-,-,+,\ldots,+\})\ ,
\end{aligned}$$ where in the coinciding limit $z_{n+1}=z_j$ we can further define $\omega'_{j}=\omega_j+\omega'_{n+1}$. The integral of the collinear states becomes $$\begin{aligned}
\label{eq:int_d0}
&\int_0^\infty d\omega_j d\omega_{n+1} d\omega_{n+2}\, \omega_j^{i\lambda_j}\omega_{n+1}^{i\lambda_{n+1}}\omega_{n+2}^{i\lambda_{n+2}} \frac{1}{\omega_j \omega_{n+1}\omega_{n+2}}\ldots \nonumber\\
&=\int_0^\infty d\omega'_j\int_0^{\omega'_j} d\omega'_{n+1} \int_0^{\omega'_{n+1}} d\omega_{n+1}\, \omega_{n+1}^{-1+i\lambda_{n+1}} (\omega'_{n+1} - \omega_{n+1})^{-1+i\lambda_{n+2}}(\omega'_j - \omega'_{n+1})^{-1+i\lambda_{j}}\ldots \nonumber\\
&=B(i\lambda_{n+1},i\lambda_{n+2})B(i\lambda'_{n+1},i\lambda_{j})\int_0^\infty d\omega'_j {\omega'}_j^{-1+i\lambda'_j} \ldots,
\end{aligned}$$ where we use the new variables $\lambda'_{n+1}=\lambda_{n+1}+\lambda_{n+2}$ and $\lambda'_j=\lambda_j+\lambda'_{n+1}$. Combing with the normalization factors $g(\lambda_j), g(\lambda_{n+1}), g(\lambda_{n+2})$ and taking the conformal soft limit $\Delta_{n+1}=\Delta_{n+2}=1$, we obtain the double pole of the OPE $$\label{eq:ope_d0}
\frac{\tilde C_2(G)}{z_{n+1,j}^2} \ \langle\mathcal{O}_{\Delta_{1} J_{1}}^{a_{1}}\ldots \mathcal{O}_{\Delta_{j}+}^{a_{j}}(z_{j})\ldots\rangle\ .$$
Next, for the Mellin integral of the single pole part, we can move the derivative out of the integral by adding an extra term: $$\begin{aligned}
\label{eq:mellin_single}
&\frac{\tilde C_2(G)}{z_{n+1,j}}\lim_{z_{n+1}\to z_j}\partial_j\left\{\left(\prod_{i=1}^{n+2} g\left(\lambda_{i}\right) \int d \omega_{i} \omega_{i}^{i \lambda_{i}}\right)\delta^{4}\left(\sum_{i=1}^{n}\epsilon_{i}\omega_{i} q_{i}+\epsilon_n\omega'_{n+1}q_{n+1}\right)\frac{ A_n(\{\ldots\})}{\omega_{n+1}\omega_{n+2}}\right\} \nonumber\\
&-\frac{\tilde C_2(G)}{z_{n+1,j}}\lim_{z_{n+1}\to z_j}\left(\prod_{i=1}^{n+2} g\left(\lambda_{i}\right) \int d \omega_{i} \omega_{i}^{i \lambda_{i}}\right)\frac{A_n(\{\ldots\})}{\omega_{n+1}\omega_{n+2}} \left\{\partial_j\delta^{4}\left(\sum_{i=1}^{n}\epsilon_{i}\omega_{i} q_{i}+\epsilon_{n+1}\omega'_{n+1}q_{n+1}\right)\right\}.
\end{aligned}$$ For the second term in , the coinciding limit $z_{n+1}=z_j$ and the derivative $\partial_j$ on the delta function do not commute. An explicit computation shows that $$\label{eq:delta_derivative}
\lim_{z_{n+1}\to z_j}\partial_j\delta^{4}\left(\sum_{i=1}^{n}\epsilon_{i}\omega_{i} q_{i}+\epsilon_{n+1}\omega'_{n+1}q_{n+1}\right) =
\frac{\omega_j}{\omega'_j}\partial_j\delta^{4}\left(\sum_{i=1\atop i\neq j}^n\epsilon_{i}\omega_{i} q_{i}+\epsilon_j\omega'_jq_j\right),$$ where on the right hand side the coinciding limit $z_{n+1}=z_j$ was used and we defined $\omega'_j=\omega_j+\omega'_{n+1}$. For the first term in , using the delta function expanded around $z_{n+1}=z_j+z_{n+1,j}$ $$\begin{aligned}
\label{eq:delta_expand}
&\delta^{4}(\sum_{i=1}^{n}\epsilon_{i}\omega_{i} q_{i}+\epsilon_{n+1}\omega'_{n+1}q_{n+1}) \nonumber\\
&= \delta^{4}(\sum_{i=1\atop i\neq j}^n\epsilon_{i}\omega_{i} q_{i}+\epsilon_j\omega'_jq_j) + z_{n+1,j}
\partial_{n+1}\delta^{4}(\sum_{i=1}^{n}\epsilon_{i}\omega_{i} q_{i}+\epsilon_n\omega'_{n+1}q_{n+1})|_{z_{n+1}=z_j}\nonumber\\
&= \delta^{4}(\sum_{i=1\atop i\neq j}^n\epsilon_{i}\omega_{i} q_{i}+\epsilon_j\omega'_jq_j) + z_{n+1,j}\frac{\omega'_{n+1}}{\omega'_j}\partial_j\delta^{4}(\sum_{i=1\atop i\neq j}^n\epsilon_{i}\omega_{i} q_{i}+\epsilon_j\omega'_jq_j)
\end{aligned}$$ we get: $$\begin{aligned}
\label{eq:mellin_single_1st}
\frac{\tilde C_2(G)}{z_{n+1,j}}\lim_{z_{n+1}\to z_j}\partial_j\int&\left\{ \delta^{4}(\sum_{i=1\atop i\neq j}^n\epsilon_{i}\omega_{i} q_{i}+\epsilon_j\omega'_jq_j) + z_{n+1,j}\frac{\omega'_{n+1}}{\omega'_j}\partial_j\delta^{4}(\sum_{i=1\atop i\neq j}^n\epsilon_{i}\omega_{i} q_{i}+\epsilon_j\omega'_jq_j) (\ldots)\right\} \nonumber\\
&=\frac{\tilde C_2(G)}{z_{n+1,j}}\partial_j\int\delta^{4}(\sum_{i=1\atop i\neq j}^n\epsilon_{i}\omega_{i} q_{i}+\epsilon_j\omega'_jq_j) (\ldots)\nonumber\\
& -\frac{\tilde C_2(G)}{z_{n+1,j}}\int\frac{\omega'_{n+1}}{\omega'_j}\partial_j\delta^{4}(\sum_{i=1\atop i\neq j}^n\epsilon_{i}\omega_{i} q_{i}+\epsilon_j\omega'_jq_j) (\ldots) +\mathcal{O}(z_{n+1,j}^0).\end{aligned}$$ Combining the two terms eq. and eq. , the Mellin transform becomes $$\begin{aligned}
\label{eq:int_single}
&\frac{\tilde C_2(G)}{z_{n+1,j}}\partial_j\left\{\left(\prod_{i=1}^{n+2} g\left(\lambda_{i}\right) \int d \omega_{i} \omega_{i}^{i \lambda_{i}}\right)\frac{1}{\omega_{n+1}\omega_{n+2}}\delta^{4}(\sum_{i=1\atop i\neq j}^n\epsilon_{i}\omega_{i} q_{i}+\epsilon_j\omega'_jq_j)\ A_n(\{\ldots\})\ \right\} \nonumber\\
&-\frac{\tilde C_2(G)}{z_{n+1,j}}\left(\prod_{i=1}^{n+2} g\left(\lambda_{i}\right) \int d \omega_{i} \omega_{i}^{i \lambda_{i}}\right)\frac{1}{\omega_{n+1}\omega_{n+2}}\ A_n(\{\ldots\})\ \left\{\partial_j\delta^{4}(\sum_{i=1\atop i\neq j}^n\epsilon_{i}\omega_{i} q_{i}+\epsilon_j\omega'_jq_j)\right\}.
\end{aligned}$$ The energy integral over $\omega_{n+1},\omega_{n+2}$ is the same as the integral of the double pole part , so this part of the OPE is $$\begin{aligned}
\label{eq:ope_s0}
\frac{\tilde C_2(G)}{z_{n+1,j}} \bigg(\partial_j \langle\mathcal{O}_{\Delta_{1} J_{1}}^{a_{1}}\ldots \mathcal{O}_{\Delta_{j}+}^{a_{j}}(z_{j})\ldots\rangle-\big(g(\lambda_j)\int_0^\infty d\omega_j \omega_j^{i\lambda_j}\ldots\big)\ A_n(\{\ldots\})\ \partial_j\delta^{4}(\sum_{i=1}^{n}\epsilon_{i}\omega_{i} q_{i})\bigg),
\end{aligned}$$ where the integration variable is changed from $\omega'_j$ to $\omega_j$ for convenience.
As discussed in section \[sec:Prelim\], the Sugawara energy-momentum tensor is expected to describe the conformal properties of the soft sector of the theory, which corresponds to the conformal soft limit [@fan_soft_2019]. Therefore as explained earlier, we need to take the conformal soft limit $\Delta_j=1$ of the $j$-th conformal field operator. In the second term of , the derivative $\partial_j$ on the delta function contributes an extra $\omega_j$ to the Mellin integral $$\label{eq:delta_j}
\partial_j\delta^{4}(\sum_{i=1}^{n}\epsilon_{i}\omega_{i} q_{i}) = \epsilon_j \omega_j (\partial_jq_j) \frac{\delta^{4}(\sum_{i=1}^{n}\epsilon_{i}\omega_{i} q_{i})}{\sum_{i=1}^{n}\epsilon_{i}\omega_{i} q_{i}}.$$ According to the analysis [@fan_soft_2019], this extra $\omega_j$ gives a Mellin integral without a pole $1/\lambda_j$ so under the $\lambda_j \to 0$ conformal soft limit vanishes due to $g(\lambda_j)\to0$.
Combining the above results, we obtain: $$\begin{aligned}
\lim_{z_{n+1}\to z_j}&\langle\mathcal{O}^{a_1}_{\Delta_{1} ,J_1} \ldots j^{a_{j}}(z_{j})\ldots \mathcal{O}^{a_n}_{\Delta_{n} ,J_n} j^{a}(z_{n+1}) j^{a}(z_{n+1})\rangle =
\nonumber \\
&=\tilde C_2(G)\ \lf(\frac{1}{z_{n+1,j}^2} + \frac{\partial_j}{z_{n+1,j}} \ri) \langle\mathcal{O}^{a_1}_{\Delta_{1} ,J_1} \ldots j^{a_{j}}(z_{j})\ldots \mathcal{O}^{a_n}_{\Delta_{n} ,J_n} \rangle\ , \qquad j=3,\ldots n\ .\label{eq:ope_f0}
\end{aligned}$$ We see that the overall constant of the OPE is $\tilde C_2(G)$. Therefore, in (\[eq:sugawara\_def\]) we need to choose a normalization $\gamma={1\over \tilde C_2(G)}$. We can define therefore, $$\label{eq:sugawara_norm}
T^S(z)=\frac{j^a(z)j^a(z)}{ \tilde C_2(G)}\ ,$$ which agrees with its general definition (\[eq:Sugawara\_1\]) [@DiFrancesco:1997nk] for level $k=0$. We have shown that indeed the OPE of $T^S(z)$ with a current $j^a(w)$ is given by: $$\label{eq:sugawara_ope}
T^S(z)j^a(w)=\frac{h}{(z-w)^2}j^a(w) + \frac{\partial_wj^a(w)}{z-w},\quad h=1 .$$ In section \[sec:gengroup\] we discuss the Sugawara construction and its OPE with the currents $j^a(z)$ for a general group, using directly the soft theorem for conformally soft states [@fan_soft_2019]. It agrees with our result from the previous subsections, namely (\[eq:sugawara\_ope\]).
We close this subsection, with a comment for the case of the negative helicity gluons $j=1,2$ in eq. . For $j=1,2$, the respective Mellin integral in eq. and eq. will give the following result for the $j$-th operator $$\begin{aligned}
\label{eq:tbarj}
g(\lambda_{j})\int_0^\infty d\omega_j\ {\omega}_j^{1+i\lambda_j} \ldots\overset{\lambda_j=0}{=} 0\ \ \ ,\ \ \ j=1,2\ .\end{aligned}$$ It is zero under the conformal soft limit of the $j$-th operator, because there is no $1/\lambda_j$ pole [@fan_soft_2019; @Pate:2019mfs; @Nandan:2019jas] that can cancel the $\lambda_j$ factor in $g(\lambda_j)$. Hence, for the case of MHV amplitudes $A_n(-,-,+,+,\ldots,+)$, the conformal soft limit of the negative helicity states gives zero. This was observed also in [@Pate:2019mfs], which is consistent with the vanishing of the MHV amplitude (\[eq:partial\_amp\]) in momentum space under the soft limit of a negative helicity gluon. In total we can write as $$\begin{aligned}
\lim_{z_j\rightarrow z_{n+1}} &\langle\mathcal{O}_{\Delta_{1} J_{1}}^{a_{1}}\ldots j^{a_{j}}(z_{j})\ldots \mathcal{O}_{\Delta_{n} J_{n}}^{a_{n}} T^S(z_{n+1})\rangle\nonumber\\
&=\begin{cases}
\displaystyle{\left( \frac{1}{(z_{n+1}\!-\!z_j)^2} \!+\! \frac{\partial_j}{(z_{n+1}-z_j)} \right) \langle\mathcal{O}_{\Delta_{1} J_{1}}^{a_{1}}\ldots j^{a_{j}}(z_{j})\ldots\mathcal{O}_{\Delta_{n} J_{n}}^{a_{n}} \rangle\ ,}\ &\ j=3,\ldots,n,\\
\displaystyle{\ 0\ ,}\ &\ j=1,2\ ,
\end{cases}\end{aligned}$$ with $\Delta_j\ra1$.
Therefore we cannot extract any OPE of $T^S(z)$ with $\bar{j}({\bar{w}})$. Naively, we expect this OPE to be regular since the operators $\bar{j}({\bar{w}})$ are antiholomorphic with weights $h=0$ and ${\bar{h}}=1$. Also this assertion, although discussed here only for MHV amplitudes, makes a connection with our earlier discussion regarding the CS interpretation of the theory. Only one set of currents can survive in the soft limit, holomorphic or antiholomorphic. We have made the choice which leads to a holomorphic Kac-Moody algebra and the antiholomorphic currents $\bar{j}({\bar{w}})$ are expected to decouple. Indeed, as shown above, this is the case for MHV amplitudes. But unfortunately this does not hold for $NMHV$ amplitudes, see appendix \[sec:appendixB\]. So we must restrict our discussion solely on correlators which involve only one type of soft gluons, positive or negative ones. In appendix \[sec:appendixC\] we discuss the role of the shadow transform, which allows soft negative helicity states to be expressed as positive helicity ones. Therefore allowing us to have a purely holomorphic correlator, alas in an apparently non-local formulation.
As a final remark, had we chosen to work in a $\overline{ MHV}$ basis we would be led to an antiholomorphic Kac-Moody algebra. For an antiholomorphic Kac-Moody algebra, we will get the $\bar{T}^S(\bar{z})\bar{j}^a(\bar{w})$-OPE as: $$\label{eq:sugawara_Cope}
\bar{T}^S(\bar{z})\bar{j}^a(\bar{w})=\frac{\bar{h}}{(\bar{z}-\bar{w})^2}\bar{j}^a(\bar{w}) + \frac{\partial_{\bar{w}}\bar{j}^a(\bar{w})}{\bar{z}-\bar{w}},\quad \bar{h}=1\ .$$ Similar conclusions as for the holomorphic sector apply in this case.
Comments on the OPE of the Sugawara tensor for hard operators {#sec:comments}
-------------------------------------------------------------
On the other hand, hard operators $O^a_{\D,-}$ as well as $O^a_{\D, +}$ act as color sources for soft modes. The complete theory requires an energy-momentum tensor for the hard states as well. We can try to examine the collinear limit of the Sugawara tensor with a hard operator. Looking carefully at (\[OPEn\]) we see that the single poles for negative helicity states have a modified partial derivative (\[tderi\]). This already poses an issue with the negative helicity gluons. Also, in the Mellin transform derivation we encounter (\[eq:ope\_s0\]). We see that only for the soft limit $\lambda_j \to 0$ we can recover the simple partial derivative of the celestial amplitude. Finally, the double poles for hard operators pose a problem as well. These states have weights $h=i{\lambda\over 2}$ and $\bar{h}= 1+i {\lambda\over 2}$. But the Sugawara energy momentum tensor will give always weights proportional to the eigenvalues of the quadratic Casimir operator on the states. In all situations completing the Mellin integrals and taking the double conformal soft limit for gluons $n+1,n+2$, results in the double poles of (\[eq:sugawara\_ope\]). From the analysis above we conclude, that when the Sugawara energy momentum tensor acts on hard negative or hard positive helicity states we do not derive the desired OPE.
One might wonder if we can consider a subsector of the theory where the Sugawara decouples completely from hard and soft negative helicity states. We know that all positive helicity states can be taken soft under consecutive soft limits. In the previous section we concluded though, that soft limits of negative helicity gluons, lead to vanishing MHV amplitudes. It is also known that a pure plus helicity amplitude does not exist in the gauge theory side and therefore in the CCFT we cannot have a correlator with only $j^a(z)$ operators [^13]. We conclude that we need to modify the Sugawara tensor to account for the proper conformal properties of hard operators. As suggested in [@Cheung:2016iub; @McLoughlin:2016uwa] one should include an additional term in the definition of the full energy-momentum tensor: \[eq:Tful\] T(z)= T\^S(z) +T’(z) . This remains an open problem. So at this stage, in order to have non-vanishing correlators with $j^a(z)$ operator insertions, we need correlators with heavy states of the CCFT. As suggested in [@He:2015zea; @Cheung:2016iub; @Nande:2017dba] we can treat external, heavy negative and positive helicity states as Wilson lines. In section \[sec:shadowKLT\] we will make an alternative proposal based on our analysis for the Einstein-Yang-Mills theory, which works for soft and hard operators alike.
As an example we can consider massive particles as sources of soft gauge radiation [@Nande:2017dba]. Massive particles are described by time-like Wilson lines which source soft gauge bosons. Unlike massless particles whose wave function localizes on the ${\cal C S}^2$ at null infinity, massive particles’ trajectories do not asymptote to the celestial sphere ${\cal C S}^2$ at null infinity. Massless particles i.e gluons correspond to local operators on ${\cal C S}^2$, but massive particles correspond to smeared operators and involve non-local integrals of local operators on ${\cal C S}^2$.
To make our point we restrict to the case of QED to avoid a heavy notation with colour matrices and traces. For QED, consider as in [@Nande:2017dba], the CCFT operator $O(p)$, where $p$ the four-momentum, describes massive states. We will assume that this operator factorizes into two parts. One part $\hat{O}(p)$ is neutral under large gauge transformations and decouples from soft radiation. The second part ${\cal W}_{Q}(p)$, where $Q$ is the charge of the massive state, is a Wilson line, a smeared operator on ${\cal C S}^2$, that transforms under large gauge transformations and describes the coupling of massive states to soft radiation. Correlation functions on the CCFT are expected to factorize [@Feige:2014wja; @Nande:2017dba]: \[eq:factor\] j\_1,j\_2…j\_n O\_1,O\_2…O\_m= \_[hard]{}\_[soft]{} . The Sugawara tensor is expected to be the energy momentum tensor of the sub-CFT of conformally soft operators and Wilson lines of the CCFT. It is very interesting to extend our present discussion in the non-Abelian case and include Wilson operators in our correlators, staring from the hard-soft-collinear factorization of scattering amplitudes in QCD [@Feige:2014wja].
The general gauge group using OPE and conformal soft limits of currents {#sec:gengroup}
=======================================================================
In this section we discuss the Sugawara energy–momentum tensor for general gauge groups by using directly the conformal soft limit result of [@fan_soft_2019] and subsequently the collinear limit. This is different sequence of operations compared to the one of the previous section. It can be applied only for the OPE of the energy momentum tensor with a soft gauge boson. We still consider the MHV case and follow closely the discussion in chapter 15 of [@DiFrancesco:1997nk]. We write the Sugawara energy momentum tensor in the the form (\[eq:Sugawara\_1\]). We will consider the following expression $$\begin{aligned}
\label{eq:geng_1}
{\gamma\over 2 \pi i} & \oint_{z_{n+2}} \ {d z_{n+1}\over z_{n+1}-z_{n+2}}\ \langle \mathcal{O}_{\Delta_{1} -}^{b_{1}}(z_1,\bar{z}_1) \ldots j^{b_n}(z_n) j^a(z_{n+1}) j^a(z_{n+2}) \rangle \nonumber \\
=&\lim_{\D_{n+1}\D_{n+2} \to 1} \ \lim_{\D_{n} \to 1} {\gamma\over 2 \pi i} \oint_{z_{n+2}} \ {d z_{n+1}\over z_{n+1}-z_{n+2}}\ \langle \mathcal{O}_{\Delta_{1} -}^{b_{1}}(z_1,\bar{z}_1) \mathcal{O}_{\Delta_{2} -}^{b_{2}}(z_2,\bar{z}_2) \mathcal{O}_{\Delta_{3} +}^{b_{3}}(z_3,\bar{z}_3) \ldots \nonumber \\
&\mathcal{O}_{\Delta_{n} +}^{b_{n}}(z_n,\bar{z}_n)\mathcal{O}_{\Delta_{n+1} +}^{a}(z_{n+1},\bar{z}_{n+1}) \mathcal{O}_{\Delta_{n+2} +}^{a}(z_{n+2},\bar{z}_{n+2}) \rangle \ ,\end{aligned}$$ where $\gamma$ a normalization constant to be determined soon. Now we apply the soft limit iteratively following closely the derivation in equations (15.51-15.56) of [@DiFrancesco:1997nk]. We use the collinear and soft limits in (\[eq:JJope\]). First, we take the soft limit for the operator $\mathcal{O}_{\Delta_{n} +}^{b_{n}}(z_{n},\bar{z}_{n})$ $$\begin{aligned}
\label{eq:geng_2}
=&\sum_{i=1}^{n-1} \frac{\tilde{f}^{b_{n} b_i c}}{z_{n,i}} \left\langle \mathcal{O}_{\D_1}^{b_{1}}(z_1,\bar{z}_1) \ldots\mathcal{O}_{\D_i}^{c}(z_i,\bar{z}_i) \ldots \ri.\nonumber\\
&\lf.\ldots\mathcal{O}^{b_{n-1}}(z_{n-1},\bar{z}_{n-1})\mathcal{O}_{\Delta_{n+1} +}^{a}(z_{n+1},\bar{z}_{n+1}) \mathcal{O}_{\Delta_{n+2} +}^{a}(z_{n+1},\bar{z}_{n+1})\right\rangle \nonumber \\
&+ \frac{\tilde{f}^{b_{n} a c}}{z_{n,n+1}} \left\langle\mathcal{O}_{\Delta_{1}}^{b_{1}}(z_1,\bar{z}_1)\ldots\mathcal{O}_{\Delta_{n+1},+}^{c}(z_{n+1},\bar{z}_{n+1})\mathcal{O}_{\Delta_{n+2} +}^{a}(z_{n+2},\bar{z}_{n+2})\right\rangle \nonumber \\
&+\frac{\tilde{f}^{ b_{n} a c}}{z_{n,n+2}} \left\langle\mathcal{O}_{\Delta_{1}}^{b_{1}}(z_1,\bar{z}_1)\ldots\mathcal{O}_{\Delta_{n+1},+}^{a}(z_{n+1},\bar{z}_{n+1})\mathcal{O}_{\Delta_{n+2} +}^{c}(z_{n+2},\bar{z}_{n+2})\right\rangle \ .\end{aligned}$$ Then we consider the collinear limit for the operator $\mathcal{O}_{\Delta_{n} +}^{b_{n}}(z_n,\bar{z}_n)$ as it approaches the two operators at $z_{n+1}$ and $z_{n+2}$ The first line above does not contribute to the OPE of interest since it is finite as $z_n\to z_{n+1}$. The last two lines are inserted in the contour integral (\[eq:geng\_1\]). Now we apply the conformal soft limit on the operator $\mathcal{O}^{a}_{\Delta_{n+1} ,+}\to J^a(z_{n+1})$ and use the OPE (\[eq:JJope\]) with $\mathcal{O}^b_{\Delta_{n+2} ,+}$ . $$\begin{aligned}
\label{eq:geng_2b}
& {\gamma\over 2 \pi i} \oint_{z_{n+2}} \ {d z_{n+1}\over z_{n+1,n+2}}\ \Big( \frac{\tilde{f}^{b_{n} a c}\tilde{f}^{c a d}}{z_{n,n+1}z_{n+1,n+2}} \left\langle\mathcal{O}_{\Delta_{1}}^{b_{1}}(z_1,\bar{z}_1)\ldots \mathcal{O}_{\Delta_{n+2} +}^{d}(z_{n+2},\bar{z}_{n+2})\right\rangle \nonumber \\
& +\frac{\tilde{f}^{b_{n} a c}\tilde{f}^{ac d}}{z_{n,n+2}z_{n+1,n+2}} \left\langle\mathcal{O}_{\Delta_{1}}^{b_{1}}(z_1,\bar{z}_1)\ldots \mathcal{O}_{\Delta_{n+2} +}^{d}(z_{n+2},\bar{z}_{n+2})\right\rangle \Big) .\end{aligned}$$ The second term gives only regular terms in the contour integral. Only the first one is relevant. We use the integration formula: \[eq:integral\] [12 i]{} \_[w]{} [d x (x-w)\^n]{} [F(w)(z-x)\^m]{}= [(n+m-2)!(n-1)! (m-1)!]{} [F(w)(z-w)\^[n+m-1]{}]{} . Finally, we derive the following expression $$\begin{aligned}
\label{eq:geng_3}
\gamma \ {-\tilde{C}_2\over (z_n-z_{n+2})^2} \left\langle\mathcal{O}_{\Delta_{1}}^{b_{1}}(z_1,\bar{z}_1)\ldots \mathcal{O}^{b_n}_{\Delta_{n+2} ,+}(z_{n+2},{\bar{z}}_{n+2})\right\rangle,\end{aligned}$$ where the overall minus sign comes from the formula $\tilde{f}^{b_n ac}\tilde{f}^{c ad}=-\tilde{C}_2\delta^{b_nd}$ and will be thrown away because it reflects only the antisymmetric property of the structure constant. At this point we need to expand the correlator for $z_{n+2}$ around $z_n$. We can use a similar method as in [@Fotopoulos:2019tpe] or use the delta function expansion in (\[eq:delta\_expand\]) leading to (\[eq:ope\_s0\]). Finally, lest consider the conformal soft limit of $\mathcal{O}_{\Delta_{n+2} ,+}$ and follow the discussion that leads to (\[eq:ope\_f0\]). In this situation, for the MHV case discussed in the previous section, we can simply Taylor expand the Mellin transform of the partial amplitude (\[eq:partial\_amp\]) At the end we arrive at $$\begin{aligned}
\label{eq:geng_4}
\left\langle\mathcal{O}_{\Delta_{1}}^{b_{1}}(z_1,\bar{z}_1)\ldots T(z_{n+2}) j^{b_n}(z_{n})\right\rangle\sim \gamma \ \tilde{C}_2\ \lf({1\over z_{n+2,n}^2} + \frac{\partial_n}{z_{n+2,n}} \ri) \left\langle\mathcal{O}_{\Delta_{1}}^{b_{1}}\ldots j^{b_n}(z_{n})\right\rangle\ ,\end{aligned}$$ where the choice $\gamma={1\over \tilde{C}_2}$ gives the correct normalization for a level $k=0$ Sugawara energy momentum tensor. This concludes the derivation of the OPE for the Sugawara tensor using an approach with the soft limits first and collinear after.
Energy–momentum tensor from shadow transform and double copy {#sec:shadow}
============================================================
As discussed in the introduction, the set of BMS algebra generators consists of superrotations and supertranslations. The energy momentum tensors $T(z), \bar{T}({\bar{z}})$ encode the superrotation generators and the supertranslation field $P(z,{\bar{z}}) $ encodes the supertranslation generators. In this section we will follow an alternative approach to the Sugawara construction of the energy-momentum tensor. We will follow an observation from [@Pate:2019lpp] to construct the energy-momentum tensor using a pair of dimension zero, opposite helicity gauge bosons. Inspired by this relation, we will propose a similar construction for the supertranslation field $P(z,{\bar{z}}) $.
A double copy construction of the energy momentum tensor {#sec:shadowKLT}
--------------------------------------------------------
In the following we shall consider a pair of dimension zero gauge boson operators with a shadow transform of one of the gauge bosons. In general the shadow transform of an operator of the CCFT is given by the relation [@Osborn:2012vt]: \[eq:shadow\] (z,[|[z]{}]{})=\^a\_[2-,-J]{}(z,[|[z]{}]{})=[ K\_[,J]{}]{} \^a\_[,J]{}(w,[|[w]{}]{}) . where $K_{\D,J}=\D+J-1$[^14]. Following (\[eq:sugawara\_0\]) we can introduce a modified energy momentum tensor on the celestial sphere by choosing a pair of dimension zero gauge boson operators and considering the following expression $$\label{Start}
T(w_1)\sim \lim_{w_2\to w_1}\sum_a \delta^{ab}\ \cO^{a}_{0,+}(w_2,\bar w_2)\tilde\cO^b_{2,+}(w_1,\bar w_1)\ ,$$ based on (\[eq:sugawaranormal\]) for level $k=0$. Here, the first gluon operator $\cO^{a}_{0,+}$ has spin one and vanishing dimension $\Delta_2=0$ (with $h_2=\tfrac{1}{2}+\tfrac{i\lambda_2}{2}, \bar h_2=-\tfrac{1}{2}+\tfrac{i\lambda_2}{2}$, $\lambda_2 \to 0$), while the second operator $\tilde\cO^b_{2,+}$ with $\tilde\Delta_1=2$ arises from a shadow operation \[ShadowJ\] \^b\_[2,+]{}(w\_1,|w\_1)\~d\^2z\_1 (z\_1-w\_1)\^[-3]{}(|z\_1-|w\_1)\^[-1]{} \^b\_[0,-]{}(z\_1,z\_1) , of a gluon operator of negative spin one and vanishing dimension $\Delta_1=0$ (with $h_1=-\tfrac{1}{2}+\tfrac{i\lambda_1}{2}, \bar h_1=\tfrac{1}{2}+\tfrac{i\lambda_1}{2}$, $\lambda_1\to 0$). We have ignored the normalization factors of the shadow transform since they are not important for our arguments below and can be absorbed in an overall normalization for the energy momentum tensor. To proceed we use the OPE of two gluon states of opposite spins, which can be found in [(\[opemp\])]{} $$\begin{aligned}
\cO^{a}_{\D_2,+}(w_2,\bar w_2)\cO^b_{\D_1,-}(z_1,\bar z_1)&=\fc{\Delta_1-1}{\Delta_2(\Delta_1+\Delta_2-2)}\sum_c\frac{\tilde{f}^{abc}}{w_2-z_1} \cO^{c}_{(\D_1+\D_2-1),-}(z_1,\bar z_1)
\nonumber \\
&-2 \delta^{ab}\ \frac{\bar w_2-\bar z_1}{w_2-z_1}\ \fc{(\Delta_1-1)(\Delta_1+1)(\Delta_2-1)}{\Delta_2(\Delta_1+\Delta_2)(\Delta_1+\Delta_2-1)}\ \cO_{(\Delta_1+\Delta_2),-2}(z_1,\bar z_1)\nonumber \\
&+ \tilde{f}^{abc}\ \Lambda(\D_1,\D_2)\ ({\bar{w}}_2-{\bar{z}}_1)\ \cO^c_{\D_1+\D_2+1,-}(z_1,{\bar{z}}_1)+ \nonumber \\
&+\delta^{ab}\ M(\D_1,\D_2)\ ({\bar{w}}_2-{\bar{z}}_1)^2\ \cO_{\D_1+\D_2+2, -2}(z_1,{\bar{z}}_1)+\ldots\ ,
\label{OPEgg}\end{aligned}$$ where $\Lambda(\D_1,\D_2), M(\D_1,\D_2)$ are constants which depend on the details of the $D=4$ theory from which their OPE has been derived. Above we have included possible single–pole or finite terms [@Pate:2019lpp]. After inserting (\[ShadowJ\]) into (\[Start\]) and using [(\[OPEgg\])]{} we arrive at: $$\begin{aligned}
&\lim_{\Delta_1,\Delta_2\rightarrow 0} \ \lim_{w_1\to w_2}
\int d^2z_1\ (z_1-w_1)^{-3}(\bar z_1-\bar w_1)^{-1}\ \delta^{aa}\nonumber \\
&\times \left\{ \fc{2}{\Delta_2(\Delta_1+\Delta_2)} \ \frac{\bar w_2-\bar z_1}{w_2-z_1}\ \cO_{\D_1+\D_2,-2}(z_1,\bar z_1)+ M(\D_1,\D_2) ({\bar{w}}_2-{\bar{z}}_1)^2\ \cO_{\D_1+\D_2+2, -2}(z_1,{\bar{z}}_1)\right\}\label{Eq16}
$$ Note, that the first and third term of the OPE (\[OPEgg\]) cancel after performing the color sum in (\[Start\]). The first term of the equation above can be related to the energy–momentum tensor [@Fotopoulos:2019tpe]: T(w)\~d\^2z (z-w)\^[-4]{} \_[0,-2]{}(z,|z) . In fact, after taking the limit $w_1\to w_2$ we obtain $$\begin{aligned}
&2 \dim g\ \lim_{\Delta_1,\Delta_2\rightarrow 0}
\fc{1}{\Delta_2(\Delta_1+\Delta_2)}\ \int d^2z_1\ (z_1-w_1)^{-4}\ \cO_{0,-2}(z_1,\bar z_1)\nonumber\\
&=2 \dim g\ \lim_{\Delta_1,\Delta_2\rightarrow 0}
\fc{1}{\Delta_2(\Delta_1+\Delta_2)}\ T(w_1)\ ,\end{aligned}$$ with the dimension $\dim g=\delta_{aa}$ of the underlying gauge group. In total we have the following relation $$\label{Total}
T(w_1)=\fc{1}{\tilde{C}_2(G)}\ \lim_{\Delta_1,\Delta_2\rightarrow 0}[\Delta_2(\Delta_1+\Delta_2)]\lim_{w_2 \to w_1} \sum_a \ \cO^{a}_{\Delta_2,+}(w_2,\bar w_2)\tilde\cO^a_{2-\Delta_1,+}(w_1,\bar w_1),$$ which assumes the desired form [(\[Start\])]{}. The latter takes the Sugawara form (\[eq:sugawaranormal\]) upon replacing the factor $\tfrac{1}{2\dim g}$ by $\tfrac{1}{2k+\tilde{C}_2(G)}$ for $k=0$.
Having fixed the normalization constant we can consider the regular terms of the OPE. The limit at $w_2\to w_1$ gives $$\begin{aligned}
\label{eq:regope_1}
\lim_{\Delta_1,\Delta_2\rightarrow 0}&[\Delta_2(\Delta_1+\Delta_2)]\ \delta^{aa} M(\D_1, \D_2)\nonumber \\
&\times \lim_{w_2 \to w_1} \int d^2z_1\ (z_1-w_1)^{-3}(\bar z_1-\bar w_1)^{-1} ({\bar{w}}_2-{\bar{z}}_1)^2\ \cO_{\D_1+\D_2+2, -2}(z_1,{\bar{z}}_1) \nonumber \\
&=\lim_{\Delta_1,\Delta_2\rightarrow 0}[\Delta_2(\Delta_1+\Delta_2)]\ \delta^{aa} \ M(\D_1, \D_2) \int d^2z_1\ (z_1-w_1)^{-3} ({\bar{z}}_1-{\bar{w}}_1)\ \cO_{2, -2}(z_1,{\bar{z}}_1).\end{aligned}$$ The action of this hard operator on primaries will lead to the following potentially singular terms $$\begin{aligned}
\label{eq:regope_2}
&\cO_{2, -2}(z_1,{\bar{z}}_1)\cO^a_{\D, \pm}(w,{\bar{w}}) \sim \rho(\D)\ {z_1-w\over {\bar{z}}_1-{\bar{w}}}\ \cO^a_{\D+2, \pm}(w,{\bar{w}})\end{aligned}$$ Naively, after integration in (\[eq:regope\_1\]) we get, that close to the operator insertion the integral behaves as $$\begin{aligned}
\label{eq:regope_3}
\int d^2z_1\ { ({\bar{z}}_1-{\bar{w}}_1)\over (z_1-w_1)^{3}}\cO_{2, -2}(z_1,{\bar{z}}_1)\cO^a_{\D, \pm}(w,{\bar{w}})\sim
\rho(\D) {{\bar{w}}_1-{\bar{w}}\over w_1-w} \cO^a_{\D+2, \pm}(w,{\bar{w}})\end{aligned}$$ where we have used standard conformal integrals (cf. [@Dolan:2011dv]) \[eq:confint\] d\^2z\_1 [ ([|[z]{}]{}\_1-[|[w]{}]{}\_1)(z\_1-w\_1)\^[3]{}]{} = to extract the singular part of this integral. Equation (\[eq:regope\_3\]) leads to singular behaviour, either pole type if we consider separately the holomorphic limit $w\to w_1$ or of a singular angular distribution if both $w\to w_1$ and ${\bar{w}}\to {\bar{w}}_1$. If we consider strictly EYM theory this term does not exist. As explained in [@Pate:2019lpp], the subleading term in the OPE (\[OPEgg\]) originates from higher derivative bulk interactions of the form $R F^2$. So in the pure EYM case, these are absent and the final result is given by (\[Total\]) and we have demonstrated the desired result.
Nevertheless, we are interested in the energy-momentum tensor for more general theories with higher derivative corrections i.e. $R F^2$ due to quantum, stringy or other effects. The important point is that such corrections will not contribute to the proposal (\[Start\]). We will demonstrate, that the constant $M(\D_1,\D_2)$ has at most single poles under the double soft limit $\D_1,\D_2 \to0$. Then in (\[eq:regope\_1\]) the last term will drop in the double soft limit $\D_1,\D_2\to 0$. The prefactor $\D_2(\D_1+\D_2)$ goes to zero quadratically but $M(\D_1,\D_2)$ has a single pole. To see this we need to follow the discussion of Appendix A in [@Pate:2019lpp]. The term of interest in the OPE (\[OPEgg\]) stems from $RF^2$ higher derivative corrections of EYM .
In general the cubic vertex has the form $$\begin{aligned}
\label{eq:cubvert}
V=\partial^m \Phi_1(x) \Phi_2(x) \Phi_k(x)
\end{aligned}$$ where the fields $\Phi$ can be $A_\mu, h_{\mu \nu}$ but Lorentz indices are suppressed and the total number of derivatives $m$ distributed among all three fields $\Phi_1. \Phi_2, \Phi_k$. The net dimension of the vertex is $d_V= 3+m$. A Mellin transform analysis of the collinear limit in a celestial amplitude, leads to the following result [@Pate:2019lpp] $$\begin{aligned}
\label{eq:mellincub}
\cA\sim \sum_{\a, \b} B(\D_1+m+\a-1,\D_2+\b-1)\int\limits_0^\infty d\omega_P\ \omega_P^{\D_1+\D_2+m-3} A_{\a,\b}(z_1,{\bar{z}}_1, z_2,{\bar{z}}_2, \omega_P,\dots),\end{aligned}$$ where $\omega_P=\omega_1+\omega_2$ and in our case $m=4$. In this case the operator $\Phi_k$ has dimensions $\D_k= \D_1+\D_2 +m-2\rightarrow 2$ which is the dimension of $\cO_{2,-2}$ in (\[eq:regope\_1\]). The remaining Mellin transform is a celestial amplitude with a hard operator insertion $\cO_{2,-2}$ and no poles are expected unlike for soft operators with dimension one. The labels $\a,\b$ determine the different powers of the energy factors in the collinear splitting functions [@fan_soft_2019; @Pate:2019lpp; @Fotopoulos:2019vac] \[eq:split\] Spit\_[s\_1,s\_2]{}\^s(p\_1,p\_2)= [1z\_[12]{}]{}[ \_1\^[m+]{} \_2\^\_P\^[+]{}]{} [1\_1 \_2]{} , with $s_i$ the helicities of the collinear states and $\a,\b\geq -1$ in YM and $\a,\b\geq -2$ in GR. In higher derivative theories with couplings $RF^2$ etc. they are always $\a,\b \geq -m$.
As it is clear from the derivation of (\[eq:mellincub\]) \[eq:M\] M(\_1,\_2)= c B(\_1+m+-1,\_2+-1) , where $c$ is a numerical constant independent of the dimensions $\D_1,\D_2$. We notice that in the limit $\D_1 , \D_2\to 0$, at most single poles can appear in the prefactor $ B(\D_1+m+\a-1,\D_1+\b-1)\to B(3+\a,\b-1)$. Actually, our result is more general, since the Beta function has at most single poles and could be applied to arbitrary higher derivative corrections.
This concludes the proof that the prefactor $M(\D_1,\D_2)$ in (\[eq:regope\_1\]) has at most a single pole as $\D_1,\D_2 \to 0$. In (\[eq:regope\_1\]) we see that automatically the limit leads to zero since we have a double zero from the overall prefactor. We conclude that the result (\[Total\]) holds for more general extensions of EYM.
The Sugawara inspired relation (\[Total\]) gives a gauge gravity relation established as a relation between a pair of gauge boson operators $\cO^{a}_{0,+}, \cO^b_{0,-}$ and a graviton operator $\cO_{0,-2}$ on the celestial sphere. Notice that this is not the usual Sugawara construction since the operators $\cO^{a}_{0,\pm}$ are not dimension one and do not generate a Kac-Moody symmetry. Having said this, it seems that our construction is more like the KLT or double–copy equivalent in CCFT. Note, that the well-known KLT relations express gravitational amplitudes as sums over squares of gauge amplitudes supplemented by a momentum dependent kernel. The latter accounts for disentangling monodromy relations on the string world–sheet. On the other hand, (\[Total\]) gives a direct relation between a graviton and a pair of gauge bosons on the celestial sphere without any additional momentum dependent factors.
A double copy construction for supertranslations {#sec:KLTP}
------------------------------------------------
In [@donnay_conformally_2019; @Fotopoulos:2019vac], it was shown that supertranslations are generated by the operator $P$ \[pdef\] P(z,|z)\_[[|[z]{}]{}]{} [O]{}\_[1,+2]{} ,where its OPE with spin one primaries is given by the relation: \[poope1\] P(z,[|[z]{}]{})\_[,J]{}(w,[|[w]{}]{})= \_[+1,J]{}(w,[|[w]{}]{}) + , J=1 . The presence of $(\D-1)$ factors in the above OPE coefficients implies that the products $P(z)j^a(w)$, $P(z)\bar j^a({\bar{w}})$ are regular. Similar relations hold for the antiholomorphic operator $\bar{P}$. An important property is that supertranslations shift the dimension of the usual fields up $\D \to \D+1$ or equivalently $(h,{\bar{h}})\ra (h+{{1\over 2}},{\bar{h}}+{{1\over 2}})$. This may be checked by applying in particular the momentum operator which generates translations along the light-cone direction (see [@Stieberger:2018onx; @Fotopoulos:2019vac]): \[eq:P00\] P\_[-[[12]{}]{}, -[[12]{}]{}]{}=P\_0+P\_3=e\^[(\_h+ \_[|[h]{}]{})/2]{} .
Proceeding one step ahead one can try a kind of double copy construction of the operator $P(z,{\bar{z}})$ of (\[pdef\]) as well. We will show that: \[eq:Psugawara\] [O]{}\_[1,+2]{}(w\_1,[|[w]{}]{}\_1)\~\_[\_2 0 ,\_1 1]{}\_[w\_2 w\_1]{} \_2 \_a \^a\_[\_2, +]{}(w\_2,[|[w]{}]{}\_2) \^a\_[2-\_1,+]{} (w\_1,[|[w]{}]{}\_1) . To prove the equivalence above, we use once more the OPE (\[OPEgg\]). $$\begin{aligned}
\label{eq:ggOPEb}
&\lim_{\Delta_1\ra 1,\Delta_2\rightarrow 0} \ \lim_{w_1\to w_2}
\int d^2z_1\ (z_1-w_1)^{-2}\ \\
&\times \delta^{aa} \left\{ \fc{4}{\Delta_2} \ \frac{\bar w_2-\bar z_1}{w_2-z_1}\ \cO_{\D_1+\D_2,-2}(z_1,\bar z_1)
+ M(\D_1,\D_2)\ ({\bar{w}}_2-{\bar{z}}_1)^2\ \cO_{\D_1+\D_2+2, -2}(z_1,{\bar{z}}_1)\right\}\ . \nonumber
$$ In the equation above we need to chose a specific order of limits $\D_2\to 0$ first and $\D_1\to 1$ last. The first term is the one we need for our purpose. For EYM there are no higher derivative terms and the leading term is all we need. As in the case of the energy-momentum tensor, for general theories beyond EYM, we need to analyze the potential implications of the subleading operator $ \cO_{\D_1+\D_2+2, -2}\to \cO_{3,-2}$. The coefficient $M(\D_1,\D_2)$ of the subleading term can have a single pole as $\D_2\to 0$ following the collinear limits of celestial amplitudes (\[eq:mellincub\]). So naively this term can create additional contributions in the proposal (\[eq:Psugawara\]). Nevertheless, this operator has the following OPEs with primary operators: $$\begin{aligned}
\label{eq:regope_2b}
&\cO_{3, -2}(z_1,{\bar{z}}_1)\cO^a_{\D, \pm}(w,{\bar{w}}) \sim \rho(\D)\ {z_1-w\over {\bar{z}}_1-{\bar{w}}}\ \cO^a_{\D+3, \pm}(w,{\bar{w}})\ .\end{aligned}$$ Applying the integration in (\[eq:ggOPEb\]) the integrant near the operator insertion behaves as follows $$\begin{aligned}
\label{eq:regope_3b}
\int d^2z_1\ & (z_1-w_1)^{-2} ({\bar{z}}_1-{\bar{w}}_1)^2\ \cO_{3, -2}(z_1,{\bar{z}}_1)\cO^a_{\D, \pm}(w,{\bar{w}})\nonumber\\
&\sim \rho(\D) \int d^2z_1\ { ({\bar{z}}_1-{\bar{w}}_1)^2 \over (z_1-w_1)^{2}} {z_1-w \over {\bar{z}}_1-{\bar{w}}}\ \cO^a_{\D+3, \pm}(w,{\bar{w}})\ .\end{aligned}$$ Unlike (\[eq:regope\_3\]) no poles can emerge from this expression. So finally, we derive \[eq:relation3\] &&\_[\_2 0 ,\_1 1]{}\_[w\_2 w\_1]{} \_2 \_a \^a\_[\_2, +]{}(w\_2,[|[w]{}]{}\_2) \^a\_[2-\_1,+]{} (w\_1,[|[w]{}]{}\_1)\
&&= \_[\_2 0 ,\_1 1]{}\_[w\_2 w\_1]{} [[|[w]{}]{}\_2-[|[z]{}]{}w\_2-z]{} \_[\_1+\_2, -2]{}(z,[|[z]{}]{})\
&&= d\^2z [[|[w]{}]{}\_1-[|[z]{}]{}(w\_1-z)\^3]{} \_[1, -2]{}(z,[|[z]{}]{})\
&& =(w\_1,[|[w]{}]{}\_1)= \_[1,+2]{}(w\_1,[|[w]{}]{}\_1)=\_[1,+2]{}(w\_1,[|[w]{}]{}\_1) , where in the last step we have used the relation for dimension one operators $\tilde{\mathcal{O}}^a_{1, +} (w_1)= \mathcal{O}^a_{1, +} (w_1)$ [@Pasterski:2017kqt]. For completeness we give the OPE of the operators $ \cO_{1,+2}(w)$ with spin one primaries $J=\pm 1$, \[eq:relation 2\] [O]{}\_[1,+2]{}(z,[|[z]{}]{}) \_[\_i,J]{}(w,|w)\~\_[\_[i]{}+1,J]{}(w,[|[w]{}]{}) from [@Fotopoulos:2019vac]. Then applying this on (\[pdef\]) we derive the OPE of $P(z,{\bar{z}})$ with primaries \[eq:relation 3\] P(z,[|[z]{}]{}) \_[\_i, J]{}(w,|w)\~ \_[\_[i]{}+1,+]{}(w,[|[w]{}]{}) One mode of this field is the operator $P_{-{{1\over 2}},-{{1\over 2}}}$ in (\[eq:P00\]).
Conclusions
===========
From the study of scattering amplitudes in four–dimensional Minkowski space–time some striking relations between gravity and gauge amplitudes have emerged. For a review see [@Bern:2019prr]. These observations suggest a deeper connection between gauge and gravity theories and indicate the existence of some gauge structure in quantum gravity. However, the origin of these relations is yet poorly understood in four–dimensional Minkowski space–time. The Mellin transform of gauge and gravitational states and amplitudes to celestial sphere gives a new way of looking at quantum field theory and quantum gravity and might shed light on the underlying symmetries of these amplitude relations. In particular, it seems feasible that the manifestation of double–copy–constructions may have a simpler emergence when considered within the underlying conformal field theory on the celestial sphere.
In this work we discussed the energy-momentum tensor of the pure gauge sector of the CCFT. For the pure gauge theory, it has been suggested [@He:2015zea; @Cheung:2016iub], that a particular subsector of the CCFT, the one of soft operators, can be described by a current algebra, a Kac-Moody algebra. In this work, we used the Sugawara method to construct the energy-momentum tensor $T^S(z)$ from the celestial amplitude of gluons. From the analysis of the soft and collinear limits of gluon amplitudes we extracted the OPEs of the Sugawara energy momentum tensor with primary fields of the CCFT. The OPE of the holomorphic Sugawara energy-momentum tensor has the expected form for soft holomorphic operators $j^a(z)$ which correspond to soft positive helicity gluons. For antiholomorphic soft operators and hard operators, the OPE is not as expected. A modification will be necessary. We discussed these shortcomings and suggested potential resolutions on how to decouple the sub-CFT that describes the positive helicity soft sector from the rest of the theory. We also developed several gauge group identities, some of which are novel and potentially useful for scattering amplitude computations in general.
Subsequently we used CCFT OPEs for EYM theory to construct from a pair of gluon operators the energy momentum tensor and the supertranslation operator of the BMS algebra. This method bears resemblance to the double-copy method that relates gauge and gravity amplitudes. The energy momentum tensor we constructed has the correct action on both the soft and hard operators of the theory. It is a generalization of the Sugawara method, although the Kac-Moody current algebra origin of this construction is not so clear.
There are several open questions which deserve further study. In section \[sec:comments\] we discussed the importance of massive states in relation to the soft sub-sector of the theory. Massive states should correspond to Wilson lines on the CCFT and it is an interesting question how to implement them in the celestial amplitudes picture. It is important to investigate correlators of soft operators with Wilson lines and extract the OPE with the Sugawara energy momentum tensor. Finally, the BMS algebra on the CCFT language was discussed recently in [@Fotopoulos:2019vac]. It would be interesting to compute the algebra using the Sugawara energy-momentum tensor and see if we can have a BMS type of symmetry for the soft sub-sector of the theory.\
We are grateful to Bin Zhu for collaboration in related topics and correspondence. StSt and TT are grateful to Monica Pate, Ana-Maria Raclariu and Andy Strominger for useful conversations. This material is based in part upon work supported by the National Science Foundation under Grant Number PHY–1913328. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Solution of the $\mathbf{n}$-particle momentum-conservating delta function {#sec:appendixA}
==========================================================================
We use [@fan_soft_2019] and give an expression of the $n$-particle momentum-conservating delta functions which appear in the amplitudes (\[eq:mellin\_partial\_amp\]) in terms of energies $\omega_i$ and celestial coordinates $z_i,{\bar{z}}_i$. For the $n$-particle ($n\ge 5)$ momentum-conservating delta function, we choose to use the first four energies $\omega_1, \omega_2, \omega_3,\omega_4$ to localize the solution. This choice is arbitrary and any other choice also works. Define the following cross-ratios of celestial coordinates: $$\label{eq:cross_ratio}
t_i = \frac{z_{12}z_{3i}}{z_{13}z_{2i}}, \quad i=4,5,\ldots, n.$$ Then the $n$-point momentum delta function is solved as $$\label{eq:delta_solution}
\delta^4\big(\sum_{i=1}^{N} \epsilon_i \omega_i q_i\big) = \frac{i}{4}\frac{(1-t_4)(1-\bar{t}_4)}{t_4-\bar{t}_4} \frac{1}{|z_{14}|^2|z_{23}|^2} \prod_{i=1}^{4}\delta(\omega_i-\omega_i^\star).$$ The solutions for the four chosen energies are $$\label{eq:energy_sol}
\omega_i^\star=f_{i5}\omega_5+f_{i6}\omega_6+\ldots+f_{in}\omega_n\ ,$$ where $f_{ij}, i=1,2,3,4, j=5,6,\ldots,n$ are functions of cross-ratios: $$\begin{aligned}
f_{1j}&=t_4 \Big|\frac{z_{24}}{z_{12}}\Big|^2 \frac{(1-t_4)(1-\bar{t}_4)}{t_4-\bar{t}_4}\epsilon_1\epsilon_j\frac{t_j-
\bar{t}_j}{(1-t_j)(1-\bar{t}_j)}\Big|\frac{z_{1j}}{z_{14}}\Big|^2 - \epsilon_1\epsilon_j t_j\Big|\frac{z_{2j}}{z_{12}}\Big|^2\ , \\
f_{2j}&= - \frac{1-t_4}{t_4} \Big|\frac{z_{34}}{z_{23}}\Big|^2 \frac{(1-t_4)(1-\bar{t}_4)}{t_4-\bar{t}_4}\frac{\epsilon_1\epsilon_j}{\epsilon_1
\epsilon_2}\frac{t_j-\bar{t}_j}{(1-t_j)(1-\bar{t}_j)}\Big|\frac{z_{1j}}{z_{14}}\Big|^2 + \frac{\epsilon_1\epsilon_j}{\epsilon_1\epsilon_2} \frac{1-t_j}{t_j}\Big|\frac{z_{3j}}{z_{23}}\Big|^2\ ,\\
f_{3j}&= (1-t_4)\Big|\frac{z_{24}}{z_{23}}\Big|^2 \frac{(1-t_4)(1-\bar{t}_4)}{t_4-\bar{t}_4}\frac{\epsilon_1\epsilon_j}{\epsilon_1
\epsilon_3}\frac{t_j-\bar{t}_j}{(1-t_j)(1-\bar{t}_j)}\Big|\frac{z_{1j}}{z_{14}}\Big|^2 - \frac{\epsilon_1\epsilon_j}{\epsilon_1\epsilon_3} (1-t_j)\Big|\frac{z_{2j}}{z_{23}}\Big|^2\ , \\
f_{4j}&= - \frac{(1-t_4)(1-\bar{t}_4)}{t_4-\bar{t}_4}\frac{\epsilon_1\epsilon_j}{\epsilon_1
\epsilon_4}\frac{t_j-\bar{t}_j}{(1-t_j)(1-\bar{t}_j)}\Big|\frac{z_{1j}}{z_{14}}\Big|^2.
\end{aligned}$$
Seven–gluon NMHV amplitude and $\mathbf{T^S\bar{j}}$ OPE {#sec:appendixB}
========================================================
In this appendix we will compute the mixed $T^S(z)\bar{j}(\bar{w})$-OPE in the seven–gluon NMHV amplitude $A^{7}(-,-,-,+,+,+,+)$. We use the last two operators to define the Sugawara energy momentum tensor $T^S(z_{7})\sim\lim_{z_{6}\to z_{7}} j^a(z_{6})j^a(z_{7})$, and extract its OPE with the first operator $T^S(z_{7})\bar{j}(\bar{z}_1)$.
The explicit form of the subamplitude $A^{7}(-,-,-,+,+,+,+)$ was obtained by the BCFW method in reference [@Britto:2004ap], which reads as: $$\label{eq:7nmhv_1}
\begin{array}{l}
A\left(1^{-}, 2^{-}, 3^{-}, 4^{+}, 5^{+}, 6^{+}, 7^{+}\right)= \frac{\langle 1|2+3| 4]^{3}}{t_{2}^{[3]}\langle 56\rangle\langle 67\rangle\langle 71\rangle[23][34]\langle 5|4+3| 2]} \\
-\frac{1}{\langle 34\rangle\langle 45\rangle\langle 6|7+1| 2]}\left(\frac{\langle 3|(4+5)(6+7)| 1\rangle^{3}}{t_{3}^{[3]} t_{6}^{[3]}\langle 67\rangle\langle 71\rangle\langle 5|4+3| 2]}+\frac{\langle 3|2+1| 7]^{3}}{t_{7}^{[3]}\langle 65\rangle[71][12]}\right)\ .
\end{array}$$ Taking limit of $z_6\to z_7$ for defining the Sugawara $T(z_7)$ and the limit of $z_7\to z_1$ for extracting the mixed OPE, this seven–gluon NMHV amplitude has the following leading order poles in terms of celestial coordinates $$\begin{aligned}
\label{eq:7nmhv_ope_1}
\lim_{z_6\to z_7\to z_1}A^{7}&(---++++)=-
\frac{\omega_{1}\omega_4\omega_5}{\omega_2\omega_3\omega_6\omega_7(\omega_1+\omega_6 +\omega_{7})^2} \frac{\bar{z}_{45}^3}{\bar{z}_{12}\bar{z}_{23}\bar{z}_{34}\bar{z}_{51}}(\frac{1}{ z_{67}z_{71}}) \nonumber\\
&-
\frac{\omega_{2}\omega_3(\omega_6 +\omega_{7})^2}{\omega_1\omega_4\omega_5\omega_6\omega_7(\omega_1+\omega_6 +\omega_{7})^2} \frac{z_{23}^3}{z_{12}z_{34}z_{45}z_{51}}(\frac{1}{ z_{67}\bar{z}_{71}}) \nonumber\\
&- \frac{\omega_{2}\omega_3\omega_7 }{\omega_1^2(\omega_1+\omega_{7})\omega_4 \omega_5\omega_6} \frac{z_{23}^3}{z_{12}z_{34}z_{45}z_{51}}(\frac{1}{ z_{71}\bar{z}_{71}}).
\end{aligned}$$ The Mellin integral of the first term in the above equation is $$\begin{aligned}
\label{eq:7nmhv_int_0}
&g(\lambda_1)g(\lambda_6)g(\lambda_{7})\int_0^\infty d\omega_1 d\omega_6 d\omega_7\, \omega_1^{i\lambda_1}\omega_6^{i\lambda_6}\omega_7^{i\lambda_7} \frac{\omega_1}{ \omega_6\omega_7 (\omega_1+\omega_6 +\omega_7)^2}\ldots \nonumber\\
&=g(\lambda_1)g(\lambda_6)g(\lambda_7)\int_0^\infty d\omega'_1\int_0^{\omega'_1} d\omega'_6 \int_0^{\omega'_6} d\omega_{6}\, \omega_6^{-1+i\lambda_6} (\omega'_6 - \omega_6)^{-1+i\lambda_7}(\omega'_1 - \omega'_6)^{1+i\lambda_{1}}\omega\prime_1^{-2}\ldots \nonumber\\
&=g(\lambda_1)g(\lambda_6)g(\lambda_7)B(i\lambda_6,i\lambda_7)B(i\lambda'_6,2+i\lambda_{1})\int_0^\infty d\omega'_1 {\omega'}_1^{-1+i\lambda'_1} \ldots,
\end{aligned}$$ where the integral is performed with change of variable $\omega'_6=\omega_6+\omega_7$, $\omega'_1=\omega_1+\omega'_6$ and we have defined new quantities $\lambda'_6=\lambda_6+\lambda_7$, $\lambda'_1=\lambda_1+\lambda'_6$. In the conformal soft limit $ \lambda_{6},\lambda_7 \to 0$, the integral is nonzero $$\begin{aligned}
\label{eq:7nmhv_int_1}
&g(\lambda_1)g(\lambda_6)g(\lambda_7)B(i\lambda_6,i\lambda_7)B(i\lambda'_6,2+i\lambda_{1})\int_0^\infty d\omega'_1 {\omega'}_1^{-1+i\lambda'_1} \ldots \nonumber\\
&= \frac{i\lambda_6\Gamma(i\lambda_6)}{\Gamma(2+i\lambda_6)} \frac{i\lambda_7 \Gamma(i\lambda_7)}{\Gamma(2+i\lambda_7)} \frac{i\lambda_{1}}{\Gamma(2+i\lambda'_{1})} \int_0^\infty d\omega'_1 {\omega'}_1^{1+i\lambda'_1} \ldots =g(\lambda_{1}) \int_0^\infty d\omega_1 {\omega}_1^{1+i\lambda_1} \ldots,
\end{aligned}$$ where in the last step we have relabelled $\omega'_1$ as $\omega_1$. Similarly, the Mellin integral of the second term in eq. is $$\begin{aligned}
\label{eq:7nmhv_int_2}
&g(\lambda_1)g(\lambda_6)g(\lambda_{7})\int_0^\infty d\omega_1 d\omega_6 d\omega_7\, \omega_1^{i\lambda_1}\omega_6^{i\lambda_6}\omega_7^{i\lambda_7} \frac{ (\omega_6 +\omega_7)^2}{ \omega_1\omega_6\omega_7 (\omega_1+\omega_6 +\omega_7)^2}\ldots \nonumber\\
&=g(\lambda_1)g(\lambda_6)g(\lambda_7)\int_0^\infty d\omega'_1\int_0^{\omega'_1} d\omega'_6 \int_0^{\omega'_6} d\omega_{6}\, \omega_6^{-1+i\lambda_6} (\omega'_6 - \omega_6)^{-1+i\lambda_7}(\omega'_1 - \omega'_6)^{-1+i\lambda_{1}}\omega\prime_6^2 \omega\prime_1^{-2}\ldots \nonumber\\
&=g(\lambda_1)g(\lambda_6)g(\lambda_7)B(i\lambda_6,i\lambda_7)B(2+i\lambda'_6,i\lambda_{1})\int_0^\infty d\omega'_1 {\omega'}_1^{-1+i\lambda'_1} \ldots.
\end{aligned}$$ In the conformal soft limit $ \lambda_{6},\lambda_7 \to 0$, the integral is zero $$\begin{aligned}
\label{eq:7nmhv_int_3}
&g(\lambda_1)g(\lambda_6)g(\lambda_7)B(i\lambda_6,i\lambda_7)B(2+i\lambda'_6,i\lambda_{1})\int_0^\infty d\omega'_1 {\omega'}_1^{-1+i\lambda'_1} \ldots \nonumber\\
&= i\lambda'_6(1+i\lambda'_6) \frac{i\lambda_6\Gamma(i\lambda_6)}{\Gamma(2+i\lambda_6)} \frac{i\lambda_7 \Gamma(i\lambda_7)}{\Gamma(2+i\lambda_7)} \frac{i\lambda_{1} \Gamma(i\lambda_{1})}{\Gamma(2+i\lambda_{1})\Gamma(2+i\lambda'_{1})} \int_0^\infty d\omega'_1 {\omega'}_1^{-1+i\lambda'_1} \ldots \nonumber\\
&=0.
\end{aligned}$$ Finally, the Mellin integral of the third term in eq. is $$\begin{aligned}
\label{eq:7nmhv_int_4}
&g(\lambda_1)g(\lambda_6)g(\lambda_{7})\int_0^\infty d\omega_1 d\omega_6 d\omega_7\, \omega_1^{i\lambda_1}\omega_6^{i\lambda_6}\omega_7^{i\lambda_7} \frac{ \omega_7}{ \omega_1^2\omega_6 (\omega_1+\omega_7)}\ldots \nonumber\\
&=g(\lambda_1)g(\lambda_7)\int_0^\infty d\omega_1 d\omega_7\, \omega_7^{1+i\lambda_7} \omega_1^{-2+i\lambda_1}(\omega_1+ \omega_7)^{-1}\ldots \nonumber\\
&=g(\lambda_1)g(\lambda_7)B(-1+i\lambda_1,2+i\lambda_7)\int_0^\infty d\omega'_1 {\omega'}_1^{-1+i(\lambda_1+\lambda_7)} \ldots,
\end{aligned}$$ where in the first line we took the conformal soft limit of $\lambda_6\to 0$ and used the following formula [@Nandan:2019jas] $$\lim_{\lambda_6=0}g(i\lambda_6)\int d\omega_6 \omega_6^{-1+i\lambda_6} = \lim_{\lambda_6=0}\int d\omega_6 i\lambda_{6}\omega_6^{-1+i\lambda_6}=1.$$ In the conformal soft limit $\lambda_7 \to 0$, the integral is zero $$\begin{aligned}
\label{eq:7nmhv_int_5}
&g(\lambda_1)g(\lambda_7)B(-1+i\lambda_1,2+i\lambda_7)
= \frac{i\lambda_7\Gamma(2+i\lambda_7)}{\Gamma(2+i\lambda_7)}g(\lambda_1) \frac{\Gamma(-1+i\lambda_1)}{\Gamma(1++i\lambda_1+i\lambda_7)}
&\overset{\lambda_7=0}{=}0.
\end{aligned}$$
Combining the above three Mellin integrals, the final result is $$\begin{aligned}
\label{eq:7nmhv_ope_nontrivial}
\mathcal{A}^{7}(---++++)\overset{z_7\to z_6\to z_1}{=} -\lf(\frac{1}{z_{67}z_{71}}\ri) \mathcal{A}^5(1^-2^-3^-4^+5^+).
\end{aligned}$$ Adding the contribution for $z_6\leftrightarrow z_7$ we derive $$\begin{aligned}
\label{eq:7nmhv_ope_nontrivial2}
\mathcal{A}^{7}(---++++)\overset{z_7, z_6\to z_1}{=} -\lf(\frac{1}{z^2_{71}}\ri) \mathcal{A}^5(1^-2^-3^-4^+5^+).
\end{aligned}$$ Obviously $T(z_7)\bar{j}_(\bar{z}_1)$-OPE has a double pole and it is not zero.
Sugawara OPE with soft shadow operators {#sec:appendixC}
=======================================
In this appendix we discuss the role of the Sugawara energy momentum tensor in correlators with insertions of the soft ($\D\to1$) shadow of spin one conformal primary operators. The shadow of an operator is given by (\[eq:shadow\]). For the case of spin one and dimension one operator this becomes \[eq:shadowD=1\] \^a(z)=-[12]{} |[j]{}\^a(w) In [@Pasterski:2017kqt] the conformal primary wave functions for dimension one operators were shown to be equivalent to their shadow transforms. Nevertheless, in [@donnay_conformally_2019] it was shown that at the subleading order of the $\D\to1$ limit the two conformal primary wave functions differ by a logarithmic mode. We will leave the operator $\widetilde{j}^a_+(z)$ as a distinct operator from $j^a(z)$.
In order to discuss the algebra of the shadow currents with the holomorphic currents $j^a(z)$, we will need the conformal soft theorems (or OPEs) of both $j^a(z)$ and $\bar{j}^a({\bar{z}})$. At this point it is important to distinguish two different situations depending on the order of the consecutive soft limits. This is important in the case of opposite helicity gluons only. The OPEs in (\[eq:jbjope\]) correspond to the case where positive helicity gluons are taken soft before negative ones and vice versa for (\[eq:bjjope\]). The action of the shadow currents on hard primary operators has no ambiguity and agrees with (\[eq:JOope\],\[eq:JOmope\]).
Let us discuss case 1. Using the OPEs in (\[eq:jbjope\]) we derive \[eq:jshjalg1\] && j\^a(z) |[j]{}\^[b]{}([|[w]{}]{})\~[f\^[a b c]{} |[j]{}\^c([|[w]{}]{}) z-w]{}, \^a(z) |[j]{}\^[b]{}([|[w]{}]{})\~[f\^[a b c]{} |[j]{}\^c([|[w]{}]{}) z-w]{}\
&&j\^a(z) j\^[b]{}(w)\~[f\^[a b c]{} j\^c(w) z-w]{} , j\^a(z)\^[b]{}(w)\~reg, \^a(z)\^[b]{}(w)\~reg Since we are discussing the OPE of the Sugawara tensor with a soft operator we can use the method of section \[sec:gengroup\]. It is straight forward to apply the derivation there and derive the following OPE \[eq:Tsj1ope\] T\^S(z) \^a(w)\~reg This implies that the shadow currents $\widetilde{j}^a$ are inert under the conformal transformations generated by the Sugawara energy momentum tensor. This again leads to the necessity to modify the energy momentum tensor to account for the conformal transformation properties of the $\widetilde{j}^a(z)$ holomorphic currents. Nevertheless, this is more promising than considering correlators with antiholomorphic currents, since we found that contrary to expectations the holomorphic Sugawara energy momentum tensor acts on those currents which have weights $(0,1)$ and should be normally inert. Of course the result above does not apply to the case of MHV amplitudes, since conformal soft limit of negative helicity gluons leads to a vanishing correlator. For the case of $N^kMHV$ though Mellin plus shadow transform lead to \[eq:NkMHVcorr\] A\_n(g\_1\^-, g\_2\^-, …g\_ k\^-, g\_[k+1]{}\^+,…g\_n\^+) (z\_1)…(z\_k) j(z\_[k+1]{}) …j(z\_n)which is non vanishing generally and purely holomorphic.\
For case 2, similarly we derive \[eq:jshjalg2\] &&j\^a(z) j\^[b]{}(w)\~[f\^[a b c]{} j\^c(w) z-w]{} , \^[a]{}(w) j\^b(z)\~[f\^[a b c]{} j\^c(w) z-w]{} , \^a(z)\^[b]{}(w)\~reg\
&&|[j]{}\^a([|[z]{}]{}) j\^[b]{}(w)\~[f\^[a b c]{} j\^c(w) [|[z]{}]{}-[|[w]{}]{}]{}, \^b(z)|[j]{}\^[a]{}([|[w]{}]{}) \~[f\^[a b c]{} |[j]{}\^c(|[w]{}) z-w]{}This leads to the surprising conclusion that there is a Kac-Moody algebra of $j^a(z)$ and $\widetilde{j}^a(z)$ which closes only on the $j^a(z)$[^15]. Repeating the previous steps we find \[eq:Tsj2ope\] T\^S(z) \^a(w)\~\_2 We see that if we identify $\widetilde{j}^a(z)\equiv j^a(z)$ we have agreement with the conformal properties of the operators as currents with weights $(1,0)$. This is very interesting and consistent with the identification of dimension one states in [@Pasterski:2017kqt]. Nevertheless, it implies an one-to-two relation between gauge amplitudes and CCFT correlators since in this way any negative helicity gluon is mapped to a positive one. It is plausible that this discrepancy lies in the detailed analysis of the conformal primary wave functions for the dimension one primary and its shadow. In [@donnay_conformally_2019] there is a subtle difference between the two operators due to a dimension one logarithmic operator. So it is more sensible to have the relation $\widetilde{j}^a(z)\simeq j^a(z)$ modulo subleading in the limit $\D\to 1$ logarithmic operators. In that sense the Sugawara energy momentum tensor captures the leading conformal properties of the shadow operators. We leave this interesting question for future work.
[^1]: In theories with gravity the Mellin transform from 4d to the ${\cal C S}^2$ is sensitive to UV divergencies. String amplitudes are known for their soft UV properties. Celestial amplitudes for string theories were discussed in [@Stieberger:2018edy; @Pate:2019mfs]. The role of UV constraints on scattering amplitudes was discussed recently in [@Carrasco:2019qwr].
[^2]: Any operator with $\D\neq1, (\lambda\neq 0)$ is called a hard operator for the purposes of this work
[^3]: A study of representations of the BMS algebra on ${\cal CS}^2$ was initiated in [@Banerjee:2020kaa].
[^4]: Here we use capital $A$ for gluon amplitudes, with $A_n$ representing the full amplitude and $A^\sigma$ representing the partial amplitude. We choose to use the calligraphic $\mathcal{A}$ for Mellin/celestial amplitude. This is different convention from reference [@fan_soft_2019], where the calligraphic $\mathcal{M}$ is used for gluon partial amplitudes.
[^5]: In the amplitudes community, color generators $T^a$ differ from the mathematics definition $t^a$ by a factor of $\sqrt 2$ absorbed into each generator, i.e. $T^a=\sqrt2 t^a$. As a consequence, for the Lie algebra $g$ the commutation relation $[t^a,t^b]=i f^{abc}t^c$ implies $[T^a,T^b]=i \tilde f^{abc}T^c$ with the following dictionary for the structure constants $\tilde f^{abc}=\sqrt 2 f^{abc}$. See also footnote \[grouptheory\] for further details.
[^6]: In our units, the gravitational and gauge coupling constants $\kappa=2$ and $g_{\rm YM}=1$, respectively.
[^7]: As explained in [@He:2015zea; @Cheung:2016iub] hard sources can be described by Wilson lines in the 4d-gauge theory along the spirit of jet physics [@Feige:2014wja]. We will discuss this further in section \[sec:comments\].
[^8]: As a consequence we have $\tilde C_2 =2C_2$, with $\tilde C_2$ referring to the structure constants $\tilde f^{abc}$ of the generators $T^a$, i.e. $\tilde C_2=\delta^{ab} \tilde f^{acd}\tilde f^{bcd}$ \[grouptheory\].
[^9]: A similar construction restricted to $SU(N)$ gauge group and considering double soft limit of gluons appears in [@McLoughlin:2016uwa]. However, in the latter reference the Mellin representation, which will be determined in section \[sec:coincide\_result\] is not addressed. Moreover, our analysis which is based on the CCFT formulation, will be extended to arbitrary gauge groups in subsection \[sec:colorsums\] and section \[sec:gengroup\], respectively.
[^10]: The final result for $SU(N)$ is the same as $U(N)$. We will explain this point later.
[^11]: Note, that in the MHV case the double–soft limit [(\[doublesoft\])]{} gives rise to an exact equation: $$A_{n+2}\left((n+2)^+, \sigma\left(1,2,\ldots,n\right), (n+1)^+\right)=\fc{1}{\vev{n+1,n+2}}\ \fc{\vev{{\sigma}(n),{\sigma}(1)}}{\vev{n+2,{\sigma}(1)}\ \vev{{\sigma}(n),n+1}} A_{n}\left(\sigma\left(1,2,\ldots,n\right)\right)\ .$$ The latter just describes splitting off all dependence on the gluons $g_{n+1},g_{n+2}$ from the remaining amplitude. This is the method discussed in the previous subsection.
[^12]: Without losing generality, we assume that collinear particles are either incoming or outgoing, i.e. $\epsilon_{j}=\epsilon_{n+1}=\epsilon_{n+2}$. In fact, we can assume that all positive helicity particles are outgoing as in reference [@Fotopoulos:2019tpe].
[^13]: See though appendix \[sec:appendixC\] for a formulation which includes shadow operators $\widetilde{j}^a(z)$.
[^14]: It seems there is a clash in the literature concerning the normalization factor for the shadow transform. In [@Dolan:2011dv] the normalization constant is $K_{\D,J}= {\Gamma(2-\D+J)\over \Gamma(\D+J-1)}$ unlike the one of [@donnay_conformally_2019; @Pasterski:2017kqt] which we use in the main text. For $J=-1$ and as $\D\ra 0$, the normalization of [@Dolan:2011dv] behaves as $K_{\D,J}\sim \D$ and goes to zero. It is not clear why this discrepancy occurs, but in this case the only modification will be that in (\[Total\]) we will need only the $(\D_1+\D_2)$ factor in the definition of the energy momentum tensor. The rest of our analysis leads though to the same conclusions.
[^15]: This is plausibly the manifestation of a degeneracy in the algebra.
|
---
abstract: 'Numerical issues arising in computations of viscous flows in corners formed by a liquid-fluid free surface and a solid boundary are considered. It is shown that on the solid a Dirichlet boundary condition, which removes multivaluedness of velocity in the ‘moving contact-line problem’ and gives rise to a logarithmic singularity of pressure, requires a certain modification of the standard finite-element method. This modification appears to be insufficient above a certain critical value of the corner angle where the numerical solution becomes mesh-dependent. As shown, this is due to an eigensolution, which exists for all angles and becomes dominant for the supercritical ones. A method of incorporating the eigensolution into the numerical method is described that makes numerical results mesh-independent again. Some implications of the unavoidable finiteness of the mesh size in practical applications of the finite-element method in the context of the present problem are discussed.'
author:
- 'James E. Sprittles and Yulii D. Shikhmurzaev'
bibliography:
- 'manuscript\_arxiv.bib'
title: 'Viscous flows in corner regions: Singularities and hidden eigensolutions'
---
Introduction
============
The ability of a numerical scheme to accurately approximate a physical problem in a domain containing corners is critical for the description of a number of phenomena, ranging from electromagnetic wave propagation in a waveguide [@juntunen00] to die-swell effects in polymer extrusion [@georgiou90]. Often, an analysis of such problems reveals singular behaviour of variables as the corner is approached, which requires special numerical treatment. Such problems are well known and have been thoroughly investigated in the setting of fracture mechanics, where one considers the propagation of a crack into a material [@duflot06], and have also been studied in some fluid dynamics problems [@wilson06].
Our interest here is the viscous flow in a corner formed between a liquid-fluid free surface and a solid boundary. Although a free surface is generally bent, to leading order as the corner is approached, it is often possible for the purpose of a local analysis to consider the flow domain as having a wedge shape. This is the case, for example, in dynamic wetting flows [@shik07], where the free surface and the solid boundary form what is referred to as the ‘contact angle’ and the liquid-fluid-solid ‘contact line’ moves with respect to the solid surface. This differs from the situation considered in some previous investigations on flow in a corner where the contact line is stationary with respect to the solid and motion is generated by disturbances in the far field [@georgiou89; @georgiou90].
It is well known that the classical fluid-mechanical approach when applied to dynamic wetting problems fails to provide an adequate description of the flow [@huh71]. The conventional remedy to the problem is to relax the no-slip boundary condition on the solid surface and allow for ‘slip’ between the liquid and solid. A number of different forms for this slip behaviour have been examined in the literature (for a recent review see Ch. 3 of [@shik07]). Broadly, these split into conditions which (i) relate tangential stress to the slip velocity, such as the Navier condition [@navier23], or (ii) explicitly prescribe the velocity along the solid surface.
In this paper, we consider numerical problems arising in the second case where the velocity along the solid surface is *a priori* prescribed in a form which ensures that a solution exists, and that in the far field the usual no-slip condition is restored. This approach is appealing to some users due to its mathematical simplicity, and our goal here is to show numerical pitfalls one comes across in its numerical implementation and give a method of overcoming them which provides a framework for modelling this class of problems. A number of functions prescribing the fluid velocity on the solid surface have been proposed in the literature [@dussan76; @zhou90; @somalinga00] and here we consider just one of these which captures all the main features of the problem.
Problem formulation {#pf}
===================
The problem is most easily formulated in a polar coordinate system $(r,\theta)$ in a frame moving with the contact line (now referred to in our two-dimensional domain as the corner point). The wedge is formed by a solid surface at $\theta=0$ which moves at speed $U$ parallel to itself, a flat free surface at $\theta=\alpha$ and a ‘far field’ boundary which is placed at an arc of a sufficiently large radius $r=R$.
The liquid is Newtonian and incompressible, with density $\rho$ and viscosity $\mu$. Near the corner the flow is characterized by a small length scale so that the Reynolds number $\hbox{\it
Re\/}$ based on this scale is small. Then as $\hbox{\it Re\/}\to0$, to leading order in $\hbox{\it
Re\/}$ we have the Stokes flow[^1]. The non-dimensional Stokes equations for the bulk pressure $p$ and the radial and azimuthal components of velocity $(u,v)$ take the form: $$\label{contin} \frac{1}{r}\frac{\partial(ru)}{\partial r}
+\frac{1}{r}\frac{\partial v}{\partial\theta}=0,
\qquad\qquad(0<r<R,\ 0<\theta<\alpha),$$
$$\label{motion_prim}
\frac{\partial p}{\partial r}=\Delta u
-\frac{u}{r^2}-\frac{2}{r^2}\frac{\partial v}{\partial\theta},
\qquad \frac{1}{r}\frac{\partial p}{\partial\theta}=\Delta v
-\frac{v}{r^2}+\frac{2}{r^2}\frac{\partial u}{\partial\theta},$$
where $$\Delta=\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial
}{\partial r} +\frac{1}{r^2}\frac{\partial^2}{\partial\theta^2}.$$
On the solid surface, for a solution not to have multivalued velocity at the corner point [@dussan74; @shik07], we replace the no-slip condition ($u=1$) with a prescribed velocity that has free-slip at the corner point and attains no-slip in the far field, that is: $$\label{form} u = 0,\quad\hbox{at}\quad r=0\qquad\hbox{and}\qquad u\rightarrow1,\quad \hbox{as}\quad
r\rightarrow \infty.$$ Following [@somalinga00], we use an exponential form for this function and, to complete the boundary conditions on the solid surface, it is combined with the usual impermeability condition for the component of velocity normal to the surface: $$\label{vect_ss} u = 1-\exp(-r/s),\quad v=0, \qquad\qquad(0<r<R,\
\theta=0).$$ The region in which the velocity deviates from no-slip is characterized by the value of $s$, which is a (non-dimensional) ‘slip length’.
On the free surface, we have the standard boundary conditions of zero tangential stress and impermeability: $$\label{vect_fs} \frac{\partial u}{\partial\theta}=0,\quad
v=0,\qquad\qquad(0<r<R,\ \theta=\alpha).$$
In the far field, we assume that the flow is fully developed and apply ‘soft’ conditions: $$\label{vect_far_field} \frac{\partial u}{\partial r}=\frac{\partial
v}{\partial r}=0, \qquad\qquad(r=R,\ 0<\theta<\alpha),$$ which imply that the influence of slip has attenuated; these conditions are satisfied by the (multivalued at the corner point) solution obtained using the no-slip condition all along the solid surface [@moffatt64].
Equations (\[contin\])–(\[vect\_far\_field\]) fully specify the problem of interest.
Local asymptotics {#ana}
=================
Consider the leading-order asymptotics for the solution of (\[contin\])–(\[vect\_far\_field\]) as $r\to0$ [@shik07] that we will later need to use in the numerical code and to provide a test of accuracy of the numerical results presented in the next section. After introducing the stream function $\psi$ by $$\label{streamfunction} u = \frac{1}{r}\frac{\partial \psi}{\partial \theta}, \quad v = -
\frac{\partial \psi}{\partial r},$$ equations (\[contin\])–(\[motion\_prim\]) are reduced to a biharmonic equation $\Delta^2\psi=0$ with boundary conditions (\[vect\_ss\])–(\[vect\_fs\]) taking the form $$\label{polar_slip} \frac{\partial\psi}{\partial\theta} =
r\left(1-\exp\left(-r/s\right)\right), \quad\psi=0,
\qquad\qquad(\theta=0,\ 0<r<R),$$ $$\label{polar_free_slip}
\frac{\partial^{2}\psi}{\partial\theta^{2}}=0, \quad \psi=0,
\qquad\qquad(\theta=\alpha,\ 0<r<R),$$ where, for definiteness, we assign the value zero to the streamline coinciding with the wedge’s boundary.
Condition (\[polar\_slip\]) is the only inhomogeneous boundary condition in the problem, i.e. the condition that drives the flow. To leading order as $r\to0$, it has the form $$\label{u_form}
\left.{\dfrac{\partial^{} \psi}{\partial {\theta}^{}}}\right|_{\theta=0}=ar^{2}+O\left(r^{3}\right),$$ where $a=1/s$. An alternative prescribed velocity that satisfies (\[form\]) and (\[u\_form\]) known in the literature [@dussan76] is given by $u=(r/s)/(1+r/s)$ and the asymptotic analyses throughout this paper are equally valid for this function as well. The form (\[u\_form\]) suggests looking for the leading-order term of the local asymptotics in the form $\psi=r^2F(\theta)$, which is a particular case from a known family of separable solutions of the biharmonic equation of the form $\psi=r^\lambda F(\theta)$. After substituting $\psi=r^2F(\theta)$ into $\Delta^2\psi=0$, one arrives at $$\label{stream_2} \psi = r^{2}
\left(B_{1}+B_{2}\theta+B_{3}\sin2\theta+B_{4}\cos2\theta\right),$$ where the constants of integration $B_i~(i=1,\dots,4)$, found from (\[polar\_slip\])–(\[polar\_free\_slip\]), are given by[^2]: $$\label{constants}
B_1=-B_4=\frac{a\alpha\sin2\alpha}{2\alpha\cos2\alpha-\sin2\alpha},\qquad B_2=-B_1/\alpha,\qquad
B_3=B_1\cot2\alpha.$$
The pressure field obtained from (\[motion\_prim\]) using (\[streamfunction\]) and (\[stream\_2\]) has the form $$\label{pressure}
p=4B_{2}\ln r + p_{0}.$$ where $p_{0}$ is a constant which sets the pressure level.
It is immediately obvious from (\[constants\]) that the coefficients are singular when $2\alpha\cos2\alpha-\sin2\alpha=0$, which occurs at a critical value $\alpha_{c}$ determined by $\tan(2\alpha_{c})=2\alpha_{c}$. In the range of interest, i.e. for $0<\alpha_{c}<180^\circ$, we have $\alpha_{c}\approx128.7^\circ$.
It is noteworthy, that in the limit $r\to0$, the velocity scales [*linearly*]{} with $r$ whilst the pressure is [*logarithmically*]{} singular at the corner and is independent of the angular coordinate $\theta$.
Numerical results {#num}
=================
From a numerical viewpoint, the steady fixed-boundary problem considered in this paper is complicated only by the presence of a singularity in the pressure which, according to (\[pressure\]), is logarithmic as $r\rightarrow 0$. The simplicity of the rest of the problem and the availability of asymptotic results, which not only give the behaviour of the velocity and pressure near the corner, but also provide the coefficients, make this a perfect testing ground for a numerical method’s ability to approximate flows in corner regions formed by boundaries on which different types of boundary conditions are applied.
In the standard implementation of the finite-element, as well as finite-difference, algorithm, one assigns an [*a priori*]{} unknown finite value to the pressure at the corner point. If such a code attempts to approximate a solution where the pressure at the corner point is singular, like the one whose local asymptotics we considered earlier, the nodal value of the pressure, as well as the pressure at the neighbouring nodes, will vary as one refines the mesh. In other words, an attempt to approximate a singular analytic solution using regular numerical representations of the unknown function on each element will lead to a numerical ‘solution’ that is mesh-dependent and hence, strictly speaking, it is not a solution to the original problem formulated in terms of PDEs that ‘do not know’ about any mesh.
In order to achieve a uniformly valid solution in the framework of a finite-element method, one approach is to use singular elements, i.e. to redefine the pressure interpolation in the elements that contain the corner point node in such a way that the pressure is allowed to be infinite at the corner point and behave as described by the local asymptotics. A simple implementation of this idea is described in [@wilson06].
In the present work, the problem formulated in Section \[pf\] has been considered using part of a finite-element-based numerical platform which has been developed to simulate a range of microfluidic capillary flows and has already been used to obtain new results for the flow of liquids over surfaces of varying wettability [@sprittles07; @sprittles09]. The idea here, is to modify the finite-element’s basis function $\Phi_p$ associated with the pressure at the corner point. In the standard triangular Taylor-Hood element with six velocity nodes and three pressure nodes, $\Phi_p$ is linear and takes the value 1 at the corner point and 0 at all other pressure nodes. Now, instead of using $\Phi_p$ and determining the coefficient in front of it, we will be using and determining the coefficient in front of a singular basis function $\Phi_s=\Phi_p \ln r$. We denote the computed coefficient of $\Phi_s$ as $B_p$. Instead of $\Phi_p$ one could use other functions to pre-multiply the logarithm, for example $\sin(\pi\Phi_p/2)$; such functions have been tried and it was found that they do not offer noticeable advantages over $\Phi_p$.
It is pointed out in [@wilson06] that the usual Gaussian quadrature is not well suited to the integration of a singular function, such as $\ln r$, and a special Simpson quadrature routine has been suggested to provide a very accurate approximation of the integrals. This routine has also been incorporated into our numerical platform; however, we found that using Gaussian quadrature with enough integration points provided an accurate enough estimation of the integral and was significantly quicker and easier to implement. Sixteen Gauss integration points were found to be more than sufficient.
At the critical angle $\alpha_{c}\approx127.8^\circ$, more complex asymptotic analysis should be considered to resolve the singular behaviour and the results should be incorporated into the code in a way similar to what is being described. However, since this paper is concerned with the general numerical approximation of corner flows which contain singularities and their numerical treatment, we shall consider angles away from this critical value, i.e. the range where, as one might expect at this stage, our asymptotics of Section \[ana\] can be used. First, we shall consider subcritical angles $\alpha<\alpha_{c}$, in particular $\alpha=75^\circ$ as a representative case. We take $s=0.1$ and $R=10$ in all the simulations that we present as an investigation into the variation of these parameters would be about the physical problem rather than its numerics and would detract from the main emphasis of this paper.
Numerical approximation at subcritical corner angles {#75}
----------------------------------------------------
In Fig. \[F:75velocity\], we show the streamlines generated by the exponential slip model. As expected, the prescribed velocity on the solid draws fluid near the solid out of the corner, thus reducing the pressure there which then sucks in fluid from the far field. In the same figure, the components of the radial velocity along the interfaces are compared to the asymptotic predictions; the agreement between the calculated velocity along the liquid-fluid interface and the asymptotic prediction is visibly excellent (agreement along the liquid-solid interface near the corner is guaranteed as the velocity is prescribed, so we give it here just to show the range in which the velocity varies).
![Left: streamlines in increments of $\psi=0.05$, with the $\theta=0,\alpha$ interfaces corresponding to $\psi=0$. Right: comparison of the computed velocity with the analytic prediction (dashed line) along the liquid-solid (1) and liquid-fluid (2) interfaces.[]{data-label="F:75velocity"}](streamlines_75 "fig:") ![Left: streamlines in increments of $\psi=0.05$, with the $\theta=0,\alpha$ interfaces corresponding to $\psi=0$. Right: comparison of the computed velocity with the analytic prediction (dashed line) along the liquid-solid (1) and liquid-fluid (2) interfaces.[]{data-label="F:75velocity"}](velocity_75 "fig:")
The plot of pressure along the two interfaces in Fig. \[F:75pressure\] shows that the pressure is indeed $\theta$-independent, and it is almost graphically indistinguishable from the asymptotic result. To confirm the mesh-independence of the result, we also consider how well the singular behaviour of pressure is captured as the mesh is resolved over ten orders of magnitude using ten different meshes, characterized by the width of the smallest element $r_1$. To do so, we show both the coefficient $B_p$ in front of the basis function $\Phi_s$ (curve 1 in Fig. \[F:75pressure\]) and the appropriate gradient determined from the pressures $p_1,\ p_2$ at the two pressure nodes on the solid surface, for simplicity, closest to the corner point at radial distances $r_1,\ r_2$, which is given by $(p_{2}-p_{1})/(\ln r_{2} - \ln r_{1})$ (curve 1g). This second method allows us to compare the code with the singular elements to one without (curve 2g in the plot) where $B_p$ is not explicitly calculated.
The clearest conclusions are drawn from the results of the second method (curves 1g and 2g). Here, the plot shows that, by using the singular element, the local gradient converges to the asymptotic value of $B_{p}=7.23$ (the dashed line in the figure): without these singular elements the code converges to the incorrect value as the mesh is resolved. This is strong support for the inclusion of singular elements in dynamic wetting codes. Without them, the behaviour of pressure is wrong not only in the element adjacent to the corner point, but, by continuity, also in a neighbourhood of this element.
![Left: comparison of computed pressure along the $\theta=0,\alpha$ interfaces, which are graphically indistinguishable, as the corner is approached with the analytic pressure (dashed line). Right: convergence of the pressure coefficient of $\ln r$ with (1) and without (2) singular elements as the mesh is resolved. $B_{p}$ is calculated from the singular element’s coefficient (1) and from the local gradient (1g) and (2g). The analytic value is 7.23 and $r_{1}$ is the size of the smallest element. []{data-label="F:75pressure"}](pressure_75 "fig:") ![Left: comparison of computed pressure along the $\theta=0,\alpha$ interfaces, which are graphically indistinguishable, as the corner is approached with the analytic pressure (dashed line). Right: convergence of the pressure coefficient of $\ln r$ with (1) and without (2) singular elements as the mesh is resolved. $B_{p}$ is calculated from the singular element’s coefficient (1) and from the local gradient (1g) and (2g). The analytic value is 7.23 and $r_{1}$ is the size of the smallest element. []{data-label="F:75pressure"}](bp_coeff_convergence_75 "fig:")
For $\alpha<\alpha_{c}$ the asymptotics and numerics are in excellent agreement, and the special treatment of the corner singularity was a success. Now we consider a supercritical angle $\alpha=175^\circ>\alpha_{c}$.
Numerical approximation at supercritical corner angles
------------------------------------------------------
The streamlines in Fig. \[F:175velocity\] are as one may intuitively expect, but when we compare the numerical and asymptotic results for the velocity along the liquid-fluid interface there is no agreement. In fact, the asymptotic result predicts that the flow should be [*up*]{} the liquid-fluid interface, which is clearly not the case in the computed solution.
![Left: streamlines in increments of $\psi=0.1$ with the $\theta=0,\alpha$ interfaces corresponding to $\psi=0$. Right: comparison of the velocity along the liquid-fluid interface with the analytic prediction (dashed line).[]{data-label="F:175velocity"}](streamlines_175 "fig:") ![Left: streamlines in increments of $\psi=0.1$ with the $\theta=0,\alpha$ interfaces corresponding to $\psi=0$. Right: comparison of the velocity along the liquid-fluid interface with the analytic prediction (dashed line).[]{data-label="F:175velocity"}](velocity_175 "fig:")
The computed pressure along the liquid-solid interface is given as curve $1$ in Fig. \[F:175pressure\]. It is not only that the pressure strongly deviates from the analytic result (dashed line) as the corner point is approached; one can see that there also appear huge oscillations: the pressure decrease in the element adjacent to the special corner elements (the line to the left of the point $10^{-6}$ in the plot) is followed by a steep increase in the element comprising the corner point (not shown in the semilogarithmic plot). Such mesh-dependence of the numerical result indicates that the obtained solution cannot be regarded as a valid approximation of the solution to the original set of partial differential equations. This conclusion is re-enforced when we study the value of $B_{p}$, the coefficient to the logarithm in the singular element, as we refine the mesh: there is no convergence. A similar trend is observed if we study $B_p$ using the local gradient method. The same conclusions may be drawn for all supercritical angles.
![Left: computed pressure in the vicinity of the corner along the liquid-solid (1) and liquid-fluid (2) interfaces compared to the analytic prediction (dashed line). Right: pressure coefficient compared to the analytic value 1.122 (dashed line) plotted against $r_{1}$ the size of the smallest element. []{data-label="F:175pressure"}](pressure_175 "fig:") ![Left: computed pressure in the vicinity of the corner along the liquid-solid (1) and liquid-fluid (2) interfaces compared to the analytic prediction (dashed line). Right: pressure coefficient compared to the analytic value 1.122 (dashed line) plotted against $r_{1}$ the size of the smallest element. []{data-label="F:175pressure"}](bp_coeff_convergence_175 "fig:")
Thus, the standard FEM coupled with the local-asymptotics-based approximation of the pressure does not allow one to obtain an acceptable numerical approximation for solutions of the Stokes equations in a corner with a combination of Dirichlet and Neumann boundary conditions on the interfaces for all angles greater than $127.8^\circ$.
The robustness of the obtained numerical ‘solution’ suggests that there is a fundamental numerical problem. It should be emphasized that we arrived at this difficulty just by varying the wedge angle in a code that produces excellent results for smaller wedge angles, and then cannot provide any mesh-independent solution after a critical angle. An immediate (tentative) explanation for this situation is that, besides the analytical solution whose asymptotics has been considered in Section \[ana\] and incorporated into the code, there exists a ‘local eigensolution’, i.e. a solution satisfying zero boundary conditions on the sides of the wedge, that becomes dominant for the supercritical angles. We will now examine this conjecture.
Asymptotics of an eigensolution {#eigen}
-------------------------------
The near-field asymptotics of the eigensolution to the biharmonic equation satisfying conditions $$\label{eigen_bcs}
\psi = {\dfrac{\partial^{} \psi}{\partial {\theta}^{}}} = 0,\quad\hbox{on}\quad\theta=0, \qquad\hbox{and}\qquad \psi =
{\dfrac{\partial^{2} \psi}{\partial {\theta}^{2}}} = 0,\quad\hbox{on}\quad\theta=\alpha,$$ is given by $$\psi_{e} = r^{\lambda}\left[A_{1}\sin\left(\lambda\theta\right) +
A_{2}\cos\left(\lambda\theta\right) + A_{3}\sin\left(\left(\lambda-2\right)\theta\right) +
A_{4}\cos\left(\left(\lambda-2\right)\theta\right)\right].$$ Using (\[eigen\_bcs\]) and noting that $\lambda\neq 1,2$, we find that $\lambda$ is determined by the equation: $$\label{lambda_eqn}
2\sin\left(\lambda\alpha\right)\cos\left(\left(\lambda-2\right)\alpha\right)=\lambda\sin\left(2\alpha\right).$$ Defining the degree of freedom by $A\equiv A_{1}$, the boundary conditions (\[eigen\_bcs\]) give: $$\label{coeffs}
A_{2}=-A_{4} =
-A\frac{\lambda(\lambda-2)^{-1}\sin\left(\left(\lambda-2\right)\alpha\right)-\sin\left(\lambda\alpha\right)}
{\cos\left(\left(\lambda-2\right)\alpha\right)-\cos\left(\lambda\alpha\right)},\qquad
A_{3} = -A\frac{\lambda}{\lambda-2}.$$ and the pressure has the form: $$\label{p_eigen}
p_{e} =
4A(\lambda-1)r^{\lambda-2}\left[a_{3}\cos((\lambda-2)\theta)-a_{4}\sin((\lambda-2)\theta)\right]+p_{1}$$ where $a_i=A_i/A,~i=3,4$ and $p_{1}$ is a constant setting the pressure level. Promisingly, equation (\[lambda\_eqn\]) has roots $\lambda\in(1,2)$ for $\alpha\in (\alpha_{c},180^\circ)$, with $\lambda\to2$ as $\alpha\to\alpha_c$, $\lambda\to3/2$ as $\alpha\to180^\circ$ and $\lambda$ as a function of $\alpha$ varying monotonically between these limiting values.
This eigensolution has been derived in a number of other works considering the flow in a corner formed between an impermeable no-slip boundary and an impermeable, sometimes free, zero-tangential stress boundary, e.g. in [@anderson93]; these are sometimes referred to as ‘stick-slip phenomena’ [@richardson70; @georgiou89]. The difference between our problem and the aforementioned flows with static corner points is that, unlike these situations where the flow is driven by the far field, here the fluid motion is generated by the movement of the solid and analytically this behaviour is captured in the asymptotics of Section \[ana\]. The eigensolution comes on top of this solution and, in the near field, it ‘does not know’ about the motion of the solid, although, ultimately, it is the solid’s motion that generates the flow in the far field that gives rise to this solution. The eigensolution exists in the range of subcritical angles as well, but there it is regular in all variables and therefore causes no problem for numerical computations; it is only for $\alpha>\alpha_c$ that the eigensolution becomes both singular and dominant.
For $\alpha<\alpha_c$ the pressure at the corner point can be referred to as single-valued: the coefficient in front of the logarithm is independent of $\theta$. In contrast, the solution for $\alpha>\alpha_c$ is manifestly multivalued as predicted by the eigensolution (\[p\_eigen\]) and as seen numerically: if one takes a vicinity of the corner point, then, no matter how small this vicinity is, there will be points which are equidistant from the corner point with an arbitrarily large pressure difference. Here, being interested in the numerical side of the problem, we set aside physical arguments that might arise in connection with the obtained solution.
The existence of an eigensolution and its dominance for $\alpha>\alpha_c$ suggests that our initial attempt at computing the flow at large angles were flawed because, given the asymptotics of Section \[ana\], we assumed that the pressure scaled as $\ln r$ whereas in fact the most singular term (i) has order $r^{-k}$ where $k=2-\lambda\in(0,1)$ and (ii) is dependent on $\theta$. This suggests a generalisation of our approach: we need to incorporate the new singular behaviour into the special elements adjacent to the corner point. Due to the presence of the eigensolution, we now have an unknown constant $A$ in our asymptotics which will prevent us from comparing *a priori* determined analytic curves with our numerical results. However, once $A$ is determined numerically, we may use it to extrapolate the asymptotic behaviour outside the singular element in which it is calculated, i.e. we may then compare, a now semi-analytic, asymptotic prediction to the computed solution *globally*.
An alternative method that we used to verify our singular element solution and do not describe in detail here is to analytically remove the eigensolution, which is the cause of numerical difficulties, prior to computation and then superimpose it back on after. This approach that has been shown to be essential for the simulation of flows using the Navier slip condition, a Robin-type boundary condition, on the solid surface is described elsewhere [@sprittles09a]. In more complex problems, where the corner is just one element, this method of removing the eigensolution everywhere is overly complex and a local method should be used which removes the eigensolution near to the corner; this method is also described in [@sprittles09a] using examples of full scale dynamic wetting simulations.
### Modified singular elements
In the limit as $r\to0$, the first two terms in the expansion of pressure are $p=
A_{p}r^{\lambda-2}g(\theta)+B_{p}\ln r$. For $\alpha>\alpha_c$ both terms are singular. We will begin by using only the leading order term in $r$ in our singular elements and will return to the two-term approximation later. Taking the leading term, our new singular elements have a basis function of the form: $$\Phi_s = \left\{
\begin{array}{ll}
\Phi_p \ln r , & \hbox{$\alpha<\alpha_c$,} \\
\Phi_p r^{\lambda-2} g(\theta), & \hbox{$\alpha>\alpha_c$,}
\end{array}
\right.$$ where the unknown coefficients are $B_{p}$ and $A_{p}$, respectively.
In Fig. \[F:175pressure\_sing\] with $\alpha=175^\circ$ and hence, from (\[lambda\_eqn\]), $\lambda=1.529$, we see that the implementation of the new singular elements solves our previous problems by (i) removing the oscillations in pressure as the corner point is approached, and (ii) converging as the mesh is refined.
The value of $A=A_{p}$, which is the coefficient of the singular basis functions and is determined by the finite element method, may be used after computation with the supplementary asymptotics of Section \[eigen\] to produce a fully determined asymptotic solution for the velocity and pressure. Then we may compare the analytic results of Section \[eigen\] with our numerical results globally. It should be pointed out that using the value of $A$ to extrapolate the analytic behaviour of the eigensolution well outside the first elements provides a quick check to see if $A$ is in the correct range, this value does not in any way actually determine the velocity field outside the first elements. The comparison with pressure in Fig. \[F:175pressure\_sing\] shows good agreement between numerics and asymptotics; however, very close to the corner point along the free surface the numerical solution is not as smooth as the asymptotic result. This is no surprise given the huge gradients in pressure which are being approximated by linear basis functions both in the radial and angular directions.
![Left: computed pressure distributions in the vicinity of the corner along the liquid-solid (1) and liquid-fluid (2) interfaces compared with the semi-analytic result (dashed lines). Right: convergence of pressure coefficient $A_p$ to 0.092, plotted against $r_{1}$ the size of the smallest element. []{data-label="F:175pressure_sing"}](pressure_175_sing "fig:") ![Left: computed pressure distributions in the vicinity of the corner along the liquid-solid (1) and liquid-fluid (2) interfaces compared with the semi-analytic result (dashed lines). Right: convergence of pressure coefficient $A_p$ to 0.092, plotted against $r_{1}$ the size of the smallest element. []{data-label="F:175pressure_sing"}](ap_coeff_convergence_175 "fig:")
In Fig. \[F:175velocity\_sing\], we compare the velocity along the interfaces of the computed numerical solution to the asymptotic result, showing in particular how the full asymptotic solution is a superposition of the eigensolution $u_{e}$ and the supplementary solution $u_{a}$. We see that, although the supplementary asymptotics predicts that flow will be reversed near the contact line, this is blown away by the strength of the eigensolution, which restores what one would intuitively think is the correct direction for the flow. The agreement we see in this figure is sufficient when we consider that it is determined by the coefficient of the singular pressure which only plays a role in elements adjacent to the corner point.
![Comparison of the computed velocity along the liquid-fluid interface $u$ with the semi-analytic result which is shown decomposed into a contribution from the supplementary asymptotics $u_a$ and the eigensolution contribution $u_e$ with A=0.092.[]{data-label="F:175velocity_sing"}](velocity_175_sing)
### Numerics incorporating two-term asymptotics
For an angle of $\alpha=175^\circ$ we have completely resolved the situation. However, we have observed for smaller angles, roughly $\alpha_c<\alpha<150^\circ$, that, in terms of mesh refinement, convergence is very slow, i.e. a mesh-independent regime is only realised for exceptionally well resolved meshes which are well outside the scope of most numerical platforms where the corner is just one part of a larger problem. In this range of angles, although asymptotically in the limit $r\rightarrow0$ the logarithmic pressure behaviour ($p\sim\ln r$) is overshadowed by the eigensolution, where $p\sim r^{\lambda-2}g(\theta)$, in reality, unavoidable finiteness of the resolution of the mesh means that one could be sufficiently far away from the corner point for the logarithm to be still dominant in the numerics. To see if this is indeed the case, we consider the relative size of the two pressure terms at the edge opposite the corner point in the first elements. Taking this element to have size $10^{-n}$, we see that the two singular functions are comparable, at a critical value $\lambda_{c}$, at the edge of the first element when $$\ln 10^{-n} = 10^{-n(\lambda_{c}-2)},$$ that is when $$\label{lambda_critical}
\lambda_{c} =2-\frac{ \ln |-n\ln10| }{n\ln10}.$$ Taking $n=6$, from (\[lambda\_critical\]) we have that $\lambda_{c} = 1.81$, which means, using (\[lambda\_eqn\]), that the logarithmic behaviour dominates a numerical scheme, with the stated spatial resolution, in the range of angles $\alpha_{c}<\alpha<143^\circ$. Thus, we have seen both numerically and analytically, that even for an extremely well resolved mesh, we should expect that there is a range of angles in which the logarithmic solution cannot be neglected; approximately for $\alpha<150^\circ$. This is a very serious issue which must be resolved as almost all numerical schemes will not be able to afford the required resolution to accurately capture the singular behaviour.
The solution to this issue is to include the second term in the asymptotic expansion of pressure, so that $p= A_{p}\Phi_p r^{\lambda-2} g(\theta)+B_{p}\Phi_p\ln r$ for $\alpha>\alpha_c$. The coefficient to this logarithm $B_{p}$ is known exactly from the supplementary asymptotics (\[pressure\]) and we have already shown in Section \[75\] that this value is numerically reproduced. Therefore, the simplest solution is to prescribe the value of $B_{p}$ and see if this improves the speed of convergence of $A_{p}$.
In Fig. \[F:175\_spedup\] we show how nicely this method works with, as one would expect, the most improvement occurring for smaller angles where the logarithmic pressure is strongest. For example, for $\alpha=150^\circ$, where the converged value is $A_{p} = 1.007$, if we neglect the logarithm, then we require $\simeq 5000$ elements to get within $5\%$ of this value, but, if we include the asymptotic logarithmic behaviour then we only need $\simeq 2500$ elements to attain the same accuracy. Computationally this is a terrific saving and, given the simplicity of its implementation, we conclude that the logarithmic asymptotic behaviour should be included for all angles $\alpha>\alpha_c$.
![A comparison of the convergence of the pressure coefficient $A_{p}$ with mesh size $r_{1}$. With (dashed line) and without (full line) the supplementary asymptotics prescribed, for a range of different angles.[]{data-label="F:175_spedup"}](p_coeff_convergence_lower)
Conclusion
==========
We have shown that accurate numerical calculation of viscous flows near corners of the flow domain in the framework of the finite-element method requires special treatment of the elements adjacent to the corners. In the case of zero-stress/prescribed velocity boundary conditions on the sides of the corner, the range of corner angles is split into two distinct regions. For the angles below the critical angle $\alpha_c\approx128.7$, it is sufficient to use a logarithmic basis function for the pressure, whereas for supercritical angles there appears a ‘hidden’ eigensolution which considerably complicates numerics. In order to obtain an acceptable (i.e. mesh-independent) solution, one has to incorporate this eigensolution into the code by altering the basis functions for the pressure in the elements adjacent to the corner point. Close to the critical angle it becomes necessary to use a two-term asymptotics in the numerical algorithm, including the leading terms of both the eigensolution and the solution of the inhomogeneous problem, as the finiteness of the mesh size could result in the algebraically and logarithmically singular terms having comparable values.
An issue that is now opened up for numerical and analytic investigation is how to generalize the developed methods for a three-dimensional case, i.e. in a situation where both the contact angle and the direction of velocity of the solid vary along a contact line. The first of these aspects becomes particularly challenging when the angle varies from subcritical to supercritical.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors kindly acknowledge the financial support of Kodak European Research and the EPSRC via a Mathematics CASE award.
[^1]: The analysis remains valid for the full Navier-Stokes problem since it considers the $r\to0$ limit, and here we consider the Stokes equations in the whole region as a convenient way to illustrate the idea.
[^2]: Here, we correct a typographical error in $B_1$ on p. 126 of [@shik06] and on p. 153 of [@shik07].
|
---
abstract: 'Game semantics aim at describing the interactive behaviour of proofs by interpreting formulas as games on which proofs induce strategies. In this article, we introduce a game semantics for a fragment of first order propositional logic. One of the main difficulties that has to be faced when constructing such semantics is to make them precise by characterizing definable strategies – that is strategies which actually behave like a proof. This characterization is usually done by restricting to the model to strategies satisfying subtle combinatory conditions such as innocence, whose preservation under composition is often difficult to show. Here, we present an original methodology to achieve this task which requires to combine tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of definable strategies by the means of generators and relations: those strategies can be generated from a finite set of “atomic” strategies and that the equality between strategies generated in such a way admits a finite axiomatization. These generators satisfy laws which are a variation of bialgebras laws, thus bridging algebra and denotational semantics in a clean and unexpected way.'
author:
- 'Samuel Mimram[^1]'
bibliography:
- 'these.bib'
title: 'Presentation of a Game Semantics for First-Order Propositional Logic'
---
Denotational semantics were introduced to provide useful abstract invariants of proofs and programs modulo cut-elimination or reduction. In particular, game semantics, introduced in the nineties, have been very successful in capturing precisely the interactive behaviour of programs. In these semantics, every type is interpreted as a *game*, that is as a set of *moves* that can be played during the game, together with the rules of the game, formalized by a partial order on the moves of the game indicating the dependencies between the moves. Every move in these games is to be played by one of the two players, called *Proponent* and *Opponent*, who should be thought respectively as the program and its environment. Interactions between these two players are sequences of moves respecting the partial order of the game, called *plays*. Every program is characterized by the set of such interactions that it can have with its environment during an execution and thus defines a *strategy* reflecting the interactive behaviour of the program inside the game specified by the type of the program.
In particular, the notion of *pointer game*, introduced by Hyland and Ong [@hyland-ong:full-abstraction-pcf] and independently by Nickau [@nickau:hsf], gave a fully abstract model of PCF – a simply-typed $\lambda$-calculus extended with recursion, conditional branching and arithmetical constants. It has revealed that PCF programs generate strategies with partial memory, called *innocent* because they react to Opponent moves according to their own *view* of the play. Thus innocence – together with another condition called *well-bracketing* – is in their setting a characterization of *definable* strategies, that is strategies which are the interpretation of a PCF term. This seminal work has lead to an extremely successful series of semantics: by relaxing in various ways the innocence constraint on strategies, it became suddenly possible to characterize the behaviour of PCF programs extended with imperative features like states, references, etc.
Unfortunately, these constraints are very specific to game semantics and remain difficult to link with other areas of computer science or algebra. Moreover, the conditions used to characterize definable strategies are very subtle and combinatorial and are thus sometimes difficult to work with. In particular, showing that these conditions are preserved under composition of strategies usually requires a fairly large amount of work.
#### Generating instead of restricting.
In this paper, we introduce a game semantics for a fragment of first-order propositional logic and describe a monoidal category ${\mathbf{Games}}$ of games and strategies in which the proofs can be interpreted. Instead of characterizing definable strategies of the model by restricting the strategies we consider to strategies satisfying particular conditions, we show that we can equivalently use here a kind of converse approach: we explain how to *generate* definable strategies by giving a *presentation* of those strategies, we show that a finite set of definable strategies can be used to generate all definable strategies by composition and tensoring and finitely axiomatize the equality between strategies obtained this way.
We we mean precisely by a presentation is a generalization of the usual notion of presentation of a monoid (or a group, …) presentation to monoidal categories. For example, consider the bicyclic monoid $B$ whose set of elements is $\mathbb{N}\times\mathbb{N}$ and whose multiplication $*$ is defined by $$(m_1,n_1)*(m_2,n_2)=(m_1-n_1+\max(n_1,m_2),n_2-m_2+\max(n_1,m_2))$$ This monoid admits the presentation $\pangle{\;p,q\;|\;pq=1\;}$, where $p$ and $q$ are two *generators* and $pq=1$ is an equation between two elements of the free monoid $M$ on $\{p,q\}$. This means that $B$ is isomorphic to the free monoid $M$ on two generators $p$ and $q$ quotiented by the smallest congruence $\equiv$ (multiplication) such that $pq\equiv 1$, where $1$ is the unit of the free monoid. More explicitly, the morphism of monoids $\varphi:M\to B$ defined by $\varphi(p)=(1,0)$ and $\varphi(q)=(0,1)$ is surjective and induces an injective functor from $M/\equiv$ to $B$: two words $w$ and $w'$ have the same image under $\varphi$ if and only if $w\equiv w'$.
Similarly, we give in this paper a finite set of typed generators from which we can generate a free monoidal category $\mathcal{G}$ by composing and tensoring generators. We moreover give a finite set of typed equations between morphisms of $\mathcal{G}$ and write $\equiv$ for the smallest congruence (composition and tensoring) on morphisms. Then we show that the category $\mathcal{G}/\equiv$, which is the category $\mathcal{G}$ whose morphisms are quotiented by the congruence $\equiv$, is equivalent to the category ${\mathbf{Games}}$ of definable strategies. As a by-product we obtain the fact that the composite of two definable strategies is well-defined which was not obvious from the definition we gave.
#### Strategies as refinements of the game.
Game semantics has revealed that proofs in logics describe particular strategies to explore formulas. A formula $A$ is a syntactic tree expressing in which order its connectives must be introduced in cut-free proofs of $A$: from the root to leaves. In this sense, it can be seen as the rules of a game whose moves correspond to introduction rules of connectives in logics. For instance, consider a formula $A$ of the form $$\label{eq:formula-ffe}
{\forall{x}.} P\quad\Rightarrow\quad{\forall{y}.}{\exists{z}.} Q$$ where $P$ and $Q$ are propositional formulas which may contain free variables. When searching for a proof of $A$, the $\forall y$ connective must be introduced before the $\exists z$ connective and the $\forall x$ connective can be introduced independently. The game – whose moves are first-order connectives – associated to this formula is therefore a partial order on the first-order connectives of the formula which can be depicted as the following diagram (to be read from the bottom to the top) $$\label{eq:game-ffe}
\xymatrix{
&\exists z\\
\forall x&\ar@{-}[u]\forall y\\
}$$ Existential connectives should be thought as Proponent moves (the strategy gives a witness for which the formula holds) and the universal connectives as Opponent moves (the strategy receives a term from its environment, for which it has to show that the formula holds).
Informally, in a first-order propositional logic, the formula (\[eq:formula-ffe\]) can have proofs of the three following shapes $$\inferrule{
\inferrule{
\inferrule{
\inferrule{\vdots}
{P[t/x]\vdash Q[t'/z]}
}
{P[t/x]\vdash{\exists{z}.} Q}
}
{P[t/x]\vdash {\forall{y}.}{\exists{z}.} Q}
}{{\forall{x}.} P\vdash {\forall{y}.}{\exists{z}.} Q}
\qquad
\inferrule{
\inferrule{
\inferrule{
\inferrule{\vdots}
{P[t/x]\vdash Q[t'/z]}
}
{P[t/x]\vdash{\exists{z}.} Q}
}
{{\forall{x}.} P\vdash {\exists{z}.} Q}
}{{\forall{x}.} P\vdash {\forall{y}.} {\exists{z}.} Q}
\qquad
\inferrule{
\inferrule{
\inferrule{
\inferrule{\vdots}
{P[t/x]\vdash Q[t'/z]}
}
{{\forall{x}.} P\vdash Q[t'/z]}
}
{{\forall{x}.} P\vdash {\exists{z}.} Q}
}{{\forall{x}.} P\vdash {\forall{y}.} {\exists{z}.} Q}$$ Here $P[t/x]$ denotes the formula $P$ where every occurrence of the free variable $x$ has been replaced by the term $t$. These proofs introduce the connectives in the orders depicted respectively below $$\xymatrix{
\ar@{-}[d]\forall x\\
\ar@{-}[d]\forall y\\
\exists z\\
}
\qquad
\xymatrix{
\ar@{-}[d]\forall y\\
\ar@{-}[d]\forall x\\
\exists z\\
}
\qquad
\xymatrix{
\ar@{-}[d]\forall y\\
\ar@{-}[d]\exists z\\
\forall x\\
}$$ It should be noted that they are all refinements of the partial order (\[eq:game-ffe\]) corresponding to the formula, in the sense that they have more dependencies between moves: *proofs add causal dependencies between connectives.*
To understand exactly what dependencies which are added by proofs we are interested in, we shall examine precisely proofs of the formula $$\label{eq:formula-ee}
{\exists{x}.} P\quad\Rightarrow\quad{\exists{y}.} Q$$ which induces the following game $$\xymatrix{
\exists x&\exists y\\
}$$ By permuting the use of introduction rules, a proof of the formula (\[eq:formula-ee\]) $$\inferrule{
\inferrule{
\inferrule{\vdots}
{P\vdash Q[t/y]}
}
{P\vdash {\exists{y}.} Q}
}
{{\exists{x}.} P\vdash {\exists{y}.} Q}$$ might be reorganized as the proof $$\inferrule{
\inferrule{
\inferrule{\vdots}
{P\vdash Q[t/y]}
}
{{\exists{x}.} P\vdash Q[t/y]}
}
{{\exists{x}.} P\vdash {\exists{y}.} Q}$$ if and only if the term $t$ used in the introduction rule of the $\exists y$ connective does not have $x$ as free variable. If the variable $x$ is free in $t$ then the rule introducing $\exists y$ can only be done after the rule introducing the $\exists x$ connective. This will be reflected by a causal dependency in the strategy corresponding to the proof, depicted by an oriented wire: $${\cpdfinput{dep_ex.ps}}
$$
We thus build a monoidal category ${\mathbf{Games}}$ of games and strategies. Its objects are *games*, that is total orders on a set whose elements (the *moves*) are polarized (they are either Proponent or Opponent moves). Its morphisms $\sigma:A\to B$ between two objects $A$ and $B$ are the partial orders $\leq_\sigma$ on the moves of $A$ (with polarities inverted) and $B$ which are compatible with both the partial orders of $A$ and of $B$, does not create cycle with those partial orders.
The logic we have chosen to model here (the fragment of first-order propositional logic without connectives) is deliberately very simple in order to simplify our presentation of the category ${\mathbf{Games}}$. We believe however that the techniques used here are very general and could extend to more expressive logics.
Presentations of monoidal categories
====================================
Monoidal categories
-------------------
A *monoidal category* $(\mathcal{C},\otimes,I)$ is a category $\mathcal{C}$ together with a functor $$\otimes:\mathcal{C}\times\mathcal{C}\to\mathcal{C}$$ and natural isomorphisms $$\alpha_{A,B,C}:(A\otimes B)\otimes C\to A\otimes(B\otimes C)
\tcomma\quad
\lambda_A:I\otimes A\to A
\qtand
\rho_A:A\otimes I\to A$$ satisfying coherence axioms [@maclane:cwm]. A symmetric monoidal category $\mathcal{C}$ is a monoidal category $\mathcal{C}$ together with a natural isomorphism $$\gamma_{A,B}:A\otimes B\to B\otimes A$$ satisfying coherence axioms and such that $\gamma_{B,A}\circ\gamma_{A,B}=\id_{A\otimes B}$. A monoidal category $\mathcal{C}$ is strictly monoidal when the natural isomorphisms $\alpha$, $\lambda$ and $\rho$ are identities. To simplify our presentation, in the rest of this paper we only consider strict monoidal categories. Formally, it can be shown that it is not restrictive, using MacLane’s coherence theorem [@maclane:cwm]: every monoidal category is monoidally equivalent to a strict one.
A (strict) *monoidal functor* $F:\mathcal{C}\to\mathcal{D}$ between two strict monoidal categories $\mathcal{C}$ and $\mathcal{D}$ if a functor $F$ between the underlying categories $\mathcal{C}$ and $\mathcal{D}$ such that $F(A\otimes B)=F(A)\otimes F(B)$ for every objects $A$ and $B$ of $\mathcal{C}$, and $F(I)=I$. A monoidal functor $F$ between two strict symmetric monoidal categories $\mathcal{C}$ and $\mathcal{D}$ is *symmetric* when it transports the symmetry of $\mathcal{C}$ to the symmetry of $\mathcal{D}$, that is when $F(\gamma_A)=\gamma_{F(A)}$.
A *monoidal natural transformation* $\theta:F\to G$ between two monoidal functors $F,G:\mathcal{C}\to\mathcal{D}$ is a natural transformation between the underlying functors $F$ and $G$ such that $\theta_{A\otimes
B}=\theta_A\otimes\theta_B$ for every objects $A$ and $B$ of $\mathcal{C}$, and $\theta_I=\id_I$. A monoidal natural transformation $\theta:F\to G$ between two strict symmetric monoidal functors is said to be *symmetric*.
Two monoidal categories $\mathcal{C}$ and $\mathcal{D}$ are *monoidally equivalent* when there exists a pair of monoidal functors $F:\mathcal{C}\to\mathcal{D}$ and $G:\mathcal{D}\to\mathcal{C}$ and two invertible monoidal natural transformations $\eta:\mathrm{Id}_\mathcal{C}\to GF$ and $\varepsilon:FG\to\mathrm{Id}_\mathcal{D}$.
Monoidal theories
-----------------
A *monoidal theory* $\mathcal{T}$ is a strict monoidal category whose objects are the natural integers such that the tensor product on objects is given by addition of integers. By an integer $n$, we mean here the finite ordinal ${\underline{n}}=\{0,1,\ldots,n-1\}$ and the addition is given by ${\underline{m}}+{\underline{n}}={\underline{m+n}}$. A *symmetric monoidal theory* is a monoidal theory where the category is moreover required to be symmetric. An algebra $F$ of a monoidal theory $\mathcal{T}$ in a strict monoidal category $\mathcal{C}$ is a strict monoidal functor from $\mathcal{T}$ to $\mathcal{C}$. Every monoidal theory $\mathcal{T}$ and strict monoidal category $\mathcal{C}$ give rise to a category ${\mathbf{Alg}_{\mathcal{T}}^{\mathcal{C}}}$ of algebras of $\mathcal{T}$ in $\mathcal{C}$ and monoidal natural transformations between them. Examples of such categories are given in Section \[section:presentation-rel\]. Monoidal theories and symmetric monoidal theories are sometimes called respectively PRO and PROP, these terms were introduced by Mac Lane in [@maclane:ca] as abbreviations for respectively “category with products” and “category with products and permutations”.
Monoidal theories generalize equational theories: in this setting, operations are typed, and can moreover have multiple outputs as well as multiple inputs.
Presentations of monoidal categories {#subsection:moncat-presentation}
------------------------------------
In this section, we recall the notion of *presentation* of a monoidal category by the means of 2-dimensional generators and relations.
Suppose that we are given a set $E_1$ whose elements are called *atomic types*. We write $E_1^*$ for the free monoid on the set $E_1$ and $i_1:E_1\to
E_1^*$ for the corresponding injection; the product of this monoid is written $\otimes$ and its unit is written $I$. The elements of $E_1^*$ are called *types*. Suppose moreover that we are given a set $E_2$, whose elements are called *generators*, together with two functions $s_1,t_1:E_2\to E_1^*$ which to every generator associate a type called respectively its *source* and *target*. We call a *signature* such a 4-uple $(E_1,s_1,t_1,E_2)$: $$\xymatrix@C=10ex@R=10ex{
E_1\ar[d]_{i_1}&\ar@<-0.7ex>[dl]_{s_1}\ar@<0.7ex>[dl]^{t_1}E_2\\
E_1^*&\\
}$$
In particular, every strict monoidal category $\mathcal{C}$ generates a signature by taking $E_1$ to be the objects of the category $\mathcal{C}$, $E_2$ its morphisms, such that for every morphism $f:A\to B$, we have $s_1(f)=i_1(A)$ and $t_1(f)=i_1(B)$. Conversely, every signature $(E_1,s_1,t_1,E_2)$ generates a free strict monoidal category $\mathcal{E}$ described as follows. If we write $E_2^*$ for the morphisms of this category and $i_2:E_2\to E_2^*$ for the injection of the generators into this category, we get a diagram $$\xymatrix@C=10ex@R=10ex{
E_1\ar[d]_{i_1}&\ar@<-0.7ex>[dl]_{s_1}\ar@<0.7ex>[dl]^{t_1}E_2\ar[d]_{i_2}\\
E_1^*&\ar@<-0.7ex>[l]_{\overline{s_1}}\ar@<0.7ex>[l]^{\overline{t_1}}E_2^*\\
}$$ in $\Set$ together with a structure of monoidal category on the graph $$\xymatrix@C=10ex@R=10ex{
E_1^*&\ar@<-0.7ex>[l]_{\overline{s_1}}\ar@<0.7ex>[l]^{\overline{t_1}}E_2^*\\
}$$ where the morphisms $\overline{s_1},\overline{t_1}:E_2^*\to E_1^*$ are the morphisms (unique by universality of $E_2^*$) such that $s_1=\overline{s_1}\circ
i_2$ and $t_1=\overline{t_1}\circ i_2$. More explicitly, the category $\mathcal{E}$ has $E_1^*$ as objects and its set $E_2^*$ of morphisms is the smallest set such that
1. there is a morphism $f:A\to B$ in $E_2^*$ for every element $f$ of $E_2$ such that $s_1(f)=A$ and $t_1(f)=B$ (this is the image by $i_2$ of $f$),
2. there is a morphism $\id_A:A\to A$ in $E_2^*$ for every element $A$ of $E_1^*$,
3. for every morphisms $f:A\to B$ and $g:B\to C$ in $E_2^*$ there is a morphism $g\circ f:A\to C$ in $E_2^*$,
4. for every morphisms $f:A\to B$ and $g:C\to D$ in $E_2^*$ there is a morphism $f\otimes g:A\otimes C\to B\otimes D$ in $E_2^*$,
quotiented by equalities imposing that
1. composition is associative and admits identities as neutral element,
2. the tensor product is associative and admits $\id_I$ as neutral element,
3. identities form a monoidal natural transformation $\id:\mathrm{Id}\to\mathrm{Id}$: for every objects $A$ and $B$, $$\id_A\otimes\id_B\qeq \id_{A\otimes B}$$
4. tensor product and composition are compatible in the sense that for every morphisms $f:A\to B$, $g:B\to C$, $f':A'\to B'$ and $g':B'\to C'$, $$(g\circ f)\otimes(g'\circ f')
\qeq
(g\otimes g')\circ (f\otimes f')$$
The *size* ${\left|f\right|}$ of a morphism $f:A\to B$ in $\mathcal{E}$ is defined inductively by $$\begin{array}{c}
{\left|\id\right|}=0
\qquad
{\left|f\right|}=1
\text{ if $f$ is a generator}
\\
{\left|f_1\otimes f_2\right|}
=
{\left|f_1\right|}+{\left|f_2\right|}
\qquad
{\left|f_2\circ f_1\right|}
=
{\left|f_1\right|}+{\left|f_2\right|}
\end{array}$$ In particular, a morphism is of size $0$ if and only if it is an identity.
This construction is a particular case of Street’s 2-computads [@street:limit-indexed-by-functors] and Burroni’s polygraphs [@burroni:higher-word] who made precise the sense in which the generated monoidal category is free on the signature. In particular, the following notion of equational theory is a specialization of the definition of a 3-polygraph to the case where there is only one 0-cell.
A **monoidal equational theory** is a 7-uple $${\mathfrak{E}}=(E_1,s_1,t_1,E_2,s_2,t_2,E_3)$$ where $(E_1,s_1,t_1,E_2)$ is a signature together with a set $E_3$ of *equations* and two morphisms $s_2,t_2:E_3\to E_2^*$, as pictured in the diagram $$\xymatrix@C=10ex@R=10ex{
E_1\ar[d]_{i_1}&\ar@<-0.7ex>[dl]_{s_1}\ar@<0.7ex>[dl]^{t_1}E_2\ar[d]_{i_2}&\ar@<-0.7ex>[dl]_{s_2}\ar@<0.7ex>[dl]^{t_2}E_3\\
E_1^*&\ar@<-0.7ex>[l]_{\overline{s_1}}\ar@<0.7ex>[l]^{\overline{t_1}}E_2^*\\
}$$ such that $$\overline{s_1}\circ s_2=\overline{s_1}\circ t_2
\qtand
\overline{t_1}\circ s_2=\overline{t_1}\circ t_2
\tdot$$
Every equational theory defines a monoidal category $\mathcal{E}/\equiv$ obtained from the monoidal category $\mathcal{E}$ generated by the signature $(E_1,s_1,t_1,E_2)$ by quotienting the morphisms by the congruence $\equiv$ generated by the equations of the equational theory ${\mathfrak{E}}$: it is the smallest congruence (both composition and tensoring) such that $s_2(e)\equiv t_2(e)$ for every element $e\in E_3$. We say that a monoidal equational theory ${\mathfrak{E}}$ is a *presentation* of a strict monoidal category $\mathcal{M}$ when $\mathcal{M}$ is monoidally equivalent to the category $\mathcal{E}$ generated by ${\mathfrak{E}}$. We sometimes informally say that an equational theory $${\mathfrak{E}}=(E_1,s_1,t_1,E_2,s_2,t_2,E_3)$$ has a *generator* $$f\qcolon A\to B$$ to mean that $f$ is an element of $E_2$ such that $s_1(f)=A$ and $t_1(f)=B$. We also say that the equational theory has an *equation* $$f\qeq g$$ to mean that there exists an element $e$ of $E_2$ such that $s_2(e)=f$ and $t_2(e)=g$.
We say that two equational theories are *equivalent* when they generate monoidally equivalent categories. A generator $f$ in an equational theory ${\mathfrak{E}}$ is *superfluous* when the equational theory ${\mathfrak{E'}}$ obtained from ${\mathfrak{E}}$ by removing the generator $f$ and all equations involving $f$, is equivalent to ${\mathfrak{E}}$. Similarly, an equation $e$ is *superfluous* when the equational theory ${\mathfrak{E'}}$ obtained from ${\mathfrak{E}}$ by removing the equation $e$ is equivalent to ${\mathfrak{E}}$. An equational theory is *minimal* when it does not contain any superfluous generator or equation.
An equational presentation $(E_1,s_1,t_1,E_2,s_2,t_2,E_3)$ where $E_1$ is reduced to a set with only one object $\{1\}$ generates a monoidal category which is a monoidal theory.
Presented categories as models
------------------------------
Suppose that a strict monoidal category $\mathcal{M}$ is presented by an equational theory ${\mathfrak{E}}$. We write $\mathcal{E}/\equiv$ for the category generated by ${\mathfrak{E}}$. The proof that ${\mathfrak{E}}$ presents $\mathcal{M}$ can generally be decomposed in three parts:
1. *$\mathcal{M}$ is a model of the equational theory ${\mathfrak{E}}$:* there exists a functor ${\widetilde{-}}$ from the category $\mathcal{E}/\equiv$ to $\mathcal{M}$. This amounts to check that there exists a functor $F:\mathcal{E}\to\mathcal{M}$ such that for every morphisms $f,g:A\to B$ in $\mathcal{E}$, $f\equiv g$ implies $Ff=Fg$.
2. *$\mathcal{M}$ is a fully-complete model of the equational theory ${\mathfrak{E}}$:* the functor ${\widetilde{-}}$ is full.
3. *$\mathcal{M}$ is the initial model of the equational theory ${\mathfrak{E}}$:* the functor ${\widetilde{-}}$ is faithful.
We say that a morphism $f:A\to B$ of $\mathcal{E}/\equiv$ *represents* the morphism ${\widetilde{f}}:{\widetilde{A}}\to{\widetilde{B}}$ of $\mathcal{M}$.
Usually, the first point is a straightforward verification and the second point is easy to show. Proving that the functor ${\widetilde{-}}$ is faithful often requires more work. In this paper, we use the methodology introduced by Lafont in [@lafont:boolean-circuits]. We first define *canonical forms* which are (not necessarily unique) canonical representatives of the equivalence classes of morphisms of $\mathcal{E}$ under the congruence $\equiv$ generated by the equations of ${\mathfrak{E}}$ – proving that every morphism is equal to a canonical form can be done by induction on the size of the morphisms. Then we show that the functor ${\widetilde{-}}$ is faithful by showing that all the canonical forms which have the same image under ${\widetilde{-}}$ are equal.
It should be noted that this is not the only technique to prove that an equational theory presents a monoidal category. In particular, Joyal and Street have used topological methods [@joyal-street:geometry-tensor-calculus] by giving a geometrical construction of the category generated by a signature, in which morphisms are equivalence classes under continuous deformation of progressive plane diagrams (their construction is detailed a bit more in Section \[subsection:string-diagrams\]). Their work is for example extended by Baez and Langford in [@baez-langford:two-tangles] to give a presentation of the 2-category of 2-tangles in 4 dimensions. The other general methodology the author is aware of, is given by Lack in [@lack:composing-props], by constructing elaborate monoidal theories from simpler monoidal theories. Namely, a monoidal theory can be seen as a monad in a particular span category and monoidal theories can therefore be composed, given a distributive law between their corresponding monads. We chose not to use those methods because, even though they can be very helpful to build intuitions, they are difficult to formalize and even more to mechanize – we believe indeed that some of the tedious proofs given in this paper could be somewhat automated.
String diagrams {#subsection:string-diagrams}
---------------
*String diagrams* provide a convenient way to represent the morphisms in the category generated by a presentation. Given an object $M$ in a category $\mathcal{C}$, a morphism $\mu:M\otimes M\to M$ can be drawn graphically as a device with two inputs and one output of type $M$ as follows: $${\cpdfinput{mult_m_label.ps}}
\qquad\text{or simply as}\qquad
{\cpdfinput{mult_m.ps}}$$ when it is clear from the context which morphism of type $M\otimes M\to M$ we are picturing (we sometimes even omit the source and target of the morphisms). Similarly, the identity $\id_M:M\to M$ can be pictured as $${\cpdfinput{id_m.ps}}$$ The tensor $f\otimes g$ of two morphisms $f:A\to B$ and $g:C\to D$ is obtained by putting the diagram corresponding to $f$ above the diagram corresponding to $g$: $${\cpdfinput{f_x_g.ps}}$$ So, for instance, the morphism $\mu\otimes M:M\otimes M\otimes M\to M\otimes M$ can be drawn diagrammatically as $${\cpdfinput{mult_x_id_m.ps}}$$ Finally, the composite $g\circ f$ of two morphisms $f:A\to B$ and $g:B\to C$ can be drawn diagrammatically by putting the diagram corresponding to $g$ at the right of the diagram corresponding to $f$ and by “linking the wires”. $${\cpdfinput{f_o_g.ps}}$$ Thus, the diagram corresponding to the morphism $\mu\circ(\mu\otimes M):M\otimes
M\to M$ is $${\cpdfinput{mult_assoc_l_m.ps}}$$ The associativity law for monoids (see Section \[subsection:monoids\]) $$\mu\circ(\mu\otimes M)
\qeq
\mu\circ(M\otimes\mu)$$ can therefore be represented graphically as $${\cpdfinput{mult_assoc_l_m.ps}}
\qeq
{\cpdfinput{mult_assoc_r_m.ps}}$$
Suppose that $(E_1,s_1,t_1,E_2)$ is a signature. Every element $f$ of $E_2$ such that $$s_1(f)=A_1\otimes\cdots\otimes A_m
\qtand
t_1(f)=B_1\otimes\cdots\otimes B_n$$ where the $A_i$ and $B_i$ are elements of $E_1$, can be represented by a diagram $${\cpdfinput{signature_f.ps}}$$ Bigger diagrams can be constructed from these diagrams by composing and tensoring them, as explained above. Joyal and Street have shown in details in [@joyal-street:geometry-tensor-calculus] that the category of those diagrams, modulo continuous deformations, is precisely the free category generated by a signature (which they call a tensor scheme). For example, the equality $$(M\otimes\mu)\circ(\mu\otimes M\otimes M)
\qeq
(\mu\otimes M\otimes M)\circ(M\otimes\mu)$$ in the category $\mathcal{C}$ given in the example above; this can be shown by continuously deforming the diagram on the left-hand side below into the diagram on the right-hand side: $${\cpdfinput{mu_x_mu_r.ps}}
\qeq
{\cpdfinput{mu_x_mu_l.ps}}$$ All the equalities, given in Section \[subsection:moncat-presentation\], satisfied by the monoidal category generated by a signature have a similar geometrical interpretation.
Some algebraic structures
=========================
In this section, we recall the categorical formulation of some well-known algebraic structures (monoids, bialgebras, …). It should be noted that we give those definitions in the setting of a monoidal category which is *not* required to be symmetric. We suppose that $(\mathcal{C},\otimes,I)$ is a strict monoidal category, fixed throughout the section.
Symmetric objects
-----------------
A *symmetric object* of $\mathcal{C}$ is an object $S$ together with a morphism $$\gamma:S\otimes S\to S\otimes S$$ called *symmetry* and pictured as $${\cpdfinput{sym_s.ps}}$$ such that the diagrams $$\xymatrix{
S\otimes S\otimes S\ar[d]_{S\otimes\gamma}\ar[r]^{\gamma\otimes S}&S\otimes S\otimes S\ar[r]^{S\otimes\gamma}&S\otimes S\otimes S\ar[d]^{\gamma\otimes S}\\
S\otimes S\otimes S\ar[r]_{\gamma\otimes S}&S\otimes S\otimes S\ar[r]_{S\otimes\gamma}&S\otimes S\otimes S\\
}
\qtand
\xymatrix{
&S\otimes S\ar[dr]^{\gamma}&\\
S\otimes S\ar[ur]^{\gamma}\ar[rr]_{S\otimes S}&&S\otimes S\\
}$$ commute. Graphically, $${\cpdfinput{yang_baxter_r.ps}}
\qeq
{\cpdfinput{yang_baxter_l.ps}}$$ and $${\cpdfinput{sym_sym.ps}}
\qeq
{\cpdfinput{id_x_id.ps}}$$ These equations are called the Yang-Baxter equations.
When the monoidal category $\mathcal{C}$ is symmetric, every object $S$ has a symmetry $\gamma=\gamma_{S,S}$ induced by the symmetry of the category.
Monoids {#subsection:monoids}
-------
A *monoid* $(M,\mu,\eta)$ in $\mathcal{C}$ is an object $M$ together with two morphisms $$\mu : M\otimes M\to M
\qtand
\eta : I\to M$$ called respectively *multiplication* and *unit* and pictured respectively as $${\cpdfinput{mult_m.ps}}
\qtand
{\cpdfinput{unit_m.ps}}$$ such that the diagrams $$\vxym{
M\otimes M\otimes M\ar[d]_{M\otimes\mu}\ar[r]^-{\mu\otimes M}&M\otimes M\ar[d]^{\mu}\\
M\otimes M\ar[r]_{\mu}&M
}
\qtand
\vxym{
\ar[dr]_{M}I\otimes M\ar[r]^{\eta\otimes M}&M\otimes M\ar[d]_{\mu}&\ar[l]_{M\otimes\eta}M\otimes I\ar[dl]^{M}\\
&M&
}$$ commute. Graphically, $${\cpdfinput{mult_assoc_l.ps}}
\qeq
{\cpdfinput{mult_assoc_r.ps}}$$ and $$\label{eq:monoid_unit}
{\cpdfinput{mult_unit_l.ps}}
\qeq
{\cpdfinput{mult_unit_c.ps}}
\qeq
{\cpdfinput{mult_unit_r.ps}}$$
A *symmetric monoid* is a monoid which admits a symmetry $\gamma:M\otimes
M\to M\otimes M$ which is compatible with the operations of the monoid in the sense that it makes the diagrams $$\label{eq:monoid-nat}
\begin{array}{c}
\xymatrix{
\ar[d]_{\mu\otimes M}M\otimes M\otimes M\ar[r]^{M\otimes \gamma}&M\otimes M\otimes M\ar[r]^{\gamma\otimes M}&M\otimes M\otimes M\ar[d]^{M\otimes\mu}\\
M\otimes M\ar[rr]_{\gamma}&&M\otimes M\\
}
\\[4ex]
\xymatrix{
\ar[d]_{M\otimes \mu}M\otimes M\otimes M\ar[r]^{\gamma\otimes M}&M\otimes M\otimes M\ar[r]^{M\otimes\gamma}&M\otimes M\otimes M\ar[d]^{\mu\otimes M}\\
M\otimes M\ar[rr]_{\gamma}&&M\otimes M\\
}
\\[4ex]
\xymatrix{
&M\otimes M\ar[dr]^{\gamma}&\\
M\ar[ur]^{\eta\otimes M}\ar[rr]_{\eta\otimes M}&&M\otimes M\\
}
\qquad
\xymatrix{
&M\otimes M\ar[dr]^{\gamma}&\\
M\ar[ur]^{M\otimes\eta}\ar[rr]_{M\otimes\eta}&&M\otimes M\\
}
\\
\end{array}$$ commute. Graphically, $$\begin{array}{cc}
{\cpdfinput{mult_sym_rnat_r.ps}}
=
{\cpdfinput{mult_sym_rnat_l.ps}}
&
{\cpdfinput{mult_sym_lnat_r.ps}}
=
{\cpdfinput{mult_sym_lnat_l.ps}}
\\[8ex]
{\cpdfinput{eta_sym_rnat_l.ps}}
=
{\cpdfinput{eta_sym_rnat_r.ps}}
&
{\cpdfinput{eta_sym_lnat_l.ps}}
=
{\cpdfinput{eta_sym_lnat_r.ps}}
\end{array}$$ A *commutative monoid* is a symmetric monoid such that the diagram $$\xymatrix{
&M\otimes M\ar[dr]^{\mu}&\\
M\otimes M\ar[ur]^{\gamma}\ar[rr]_{\mu}&&M
}$$ commutes. Graphically, $$\label{eq:monoid_mult_comm}
{\cpdfinput{mult_comm.ps}}
\qeq
{\cpdfinput{mult.ps}}$$ A commutative monoid in a symmetric monoidal category is a commutative monoid whose symmetry corresponds to the symmetry of the category: $\gamma=\gamma_{M,M}$. In this case, the equations (\[eq:monoid-nat\]) can always be deduced from the naturality of the symmetry of the monoidal category.
A *comonoid* $(M,\delta,\varepsilon)$ in $\mathcal{C}$ is an object $M$ together with two morphisms $$\delta:M\to M\otimes M
\qtand
\varepsilon:M\to I$$ respectively drawn as $${\cpdfinput{comult_m.ps}}
\qqtand
{\cpdfinput{counit_m.ps}}$$ satisfying dual coherence diagrams. An similarly, the notions symmetric comonoid, cocommutative comonoid and cocommutative comonoid can be defined by duality.
An equational theory of monoids {#subsection:eq-th-monoid}
-------------------------------
The definition of a monoid can be reformulated internally using the notion of equational theory.
\[definition:e-t-monoid\] The *equational theory of monoids* ${\mathfrak{M}}$ has only one object $1$ and two generators $\mu:2\to 1$ and $\eta:0\to 1$ subject to the equations $$\mu\circ(\mu\otimes\id_1)
=
\mu\circ(\id_1\otimes\mu)
\qtand
\mu\circ(\eta\otimes\id_1)
=
\id_1
=
\mu\circ(\id_1\otimes\eta)$$
We write $\mathcal{M}$ for the monoidal category generated by the equational theory ${\mathfrak{M}}$. It can easily be seen that a monoid $M$ in a strict monoidal category $\mathcal{C}$ is essentially the same as a functor from $\mathcal{M}$ to $\mathcal{C}$. More precisely,
The category ${\mathbf{Alg}_{\mathcal{M}}^{\mathcal{C}}}$ of algebras of the monoidal theory $\mathcal{M}$ in $\mathcal{C}$ is equivalent to the category of monoids in $\mathcal{C}$.
Similarly, all the algebraic structures introduced in this section can be defined using algebraic theories.
The presentations given here are not necessarily minimal. For example, in the theory of commutative monoids the equation on the right-hand side of (\[eq:monoid\_unit\]) is derivable from the equation (\[eq:monoid\_mult\_comm\]), the equation on the left-hand side of (\[eq:monoid\_unit\]) and one of the equations (\[eq:monoid-nat\]): $${\cpdfinput{mult_unit_r.ps}}
=
{\cpdfinput{mult_unit_l_sym.ps}}
=
{\cpdfinput{mult_unit_l.ps}}
=
{\cpdfinput{mult_unit_c.ps}}$$ A minimal presentation of this equational theory with three generators and seven equations is given in [@massol:minimality]. However, not all the equational theories introduced in this paper have a known presentation which is proved to be minimal.
Bialgebras
----------
A *bialgebra* $(B,\mu,\eta,\delta,\varepsilon,\gamma)$ in $\mathcal{C}$ is an object $B$ together with four morphisms $$\begin{array}{r@{\quad:\quad}l}
\mu&B\otimes B\to B\\
\eta&I\to B\\
\delta&B\to B\otimes B\\
\varepsilon&B\to I\\
\gamma&B\otimes B\to B\otimes B\\
\end{array}$$ respectively drawn as $${\cpdfinput{mult_b.ps}}
\quad
{\cpdfinput{unit_b.ps}}
\quad
{\cpdfinput{comult_b.ps}}
\quad
{\cpdfinput{counit_b.ps}}
\qtand
{\cpdfinput{sym_b.ps}}$$ such that $\gamma:B\otimes B\to B\otimes B$ is a symmetry for $B$, $(B,\mu,\eta,\gamma)$ is a symmetric monoid and $(B,\delta,\varepsilon,\gamma)$ is a symmetric comonoid. Those two structures should be coherent, in the sense that the diagrams $$\begin{array}{cc}
\xymatrix{
B\otimes B\ar[d]_{\delta\otimes\delta}\ar[r]^-{\mu}&B\ar[r]^-{\delta}&B\otimes B\\
B\otimes B\otimes B\otimes B\ar[rr]_{B\otimes\gamma\otimes B}&&\ar[u]_{\mu\otimes\mu}B\otimes B\otimes B\otimes B\\
}&
\xymatrix{
&B\ar[dr]^{\varepsilon}&\\
I\ar[ur]^{\eta}\ar[rr]_{I}&&I
}\\
\xymatrix{
&B\ar[dr]^{\varepsilon}&\\
B\otimes B\ar[ur]^{\mu}\ar[rr]_{\varepsilon\otimes\varepsilon}&&I\otimes I=I
}&
\xymatrix{
&B\ar[dr]^{\delta}&\\
I=I\otimes I\ar[ur]^{\eta}\ar[rr]_{\eta\otimes\eta}&&B\otimes B
}
\end{array}$$ should commute. Graphically, $$\begin{array}{r@{\qeq}l@{\qquad}r@{\qeq}l}
{\cpdfinput{hopf_l.ps}}&{\cpdfinput{hopf_r.ps}}
&
{\cpdfinput{unit_counit.ps}}&
\\
{\cpdfinput{counit_mult.ps}}&{\cpdfinput{counit_x_counit.ps}}
&
{\cpdfinput{comult_unit.ps}}&{\cpdfinput{unit_x_unit.ps}}
\end{array}$$
A morphism of bialgebras of $\mathcal{C}$ $$f:(A,\mu_A,\eta_A,\delta_A,\varepsilon_A,\gamma_1)\to(B,\mu_B,\eta_B,\delta_B,\varepsilon_B,\gamma_B)$$ is a morphism $f:A\to B$ of $\mathcal{C}$ preserving the structure of bialgebra, that is $$(f\otimes f)\circ\mu_A=\mu_B\circ f
\qqcomma
f\circ\eta_A=\eta_B
\qqcomma
\text{\etc}$$ The symmetric bialgebra is *commutative* (*cocommutative*) when the induced symmetric monoid $(B,\mu,\eta,\gamma)$ (symmetric comonoid $(B,\delta,\varepsilon,\gamma)$) is commutative (cocommutative), and *bicommutative* when it is both commutative and cocommutative.
A *qualitative bialgebra* is a bialgebra $(B,\mu,\eta,\delta,\varepsilon,\gamma)$ such that the diagram $$\xymatrix{
&B\otimes B\ar[dr]^{\mu}&\\
B\ar[ur]^{\delta}\ar[rr]_{B}&&B
}$$ commutes. Graphically, $${\cpdfinput{rel_l.ps}}
\qeq
{\cpdfinput{rel_r.ps}}$$
A bialgebra $B$ in a symmetric monoidal category is a bialgebra whose symmetry morphism $\gamma$ corresponds with the symmetry of the category $\gamma_{B,B}$.
Similarly to what has been explained for monoids in Section \[subsection:eq-th-monoid\], an equational theory of bialgebras, can be defined. We write ${\mathfrak{B}}$ for the equational theory of bicommutative bialgebras and ${\mathfrak{R}}$ for the equational theory of bicommutative qualitative bialgebras.
Dual objects
------------
An object $L$ of $\mathcal{C}$ is said to be *left dual* to an object $R$ when there exists two morphism $$\eta:I\to R\otimes L
\qtand
\varepsilon:L\otimes R\to I$$ called respectively the *unit* and the *counit* of the duality and respectively pictured as $${\cpdfinput{adj_unit_lr.ps}}
\qtand
{\cpdfinput{adj_counit_lr.ps}}$$ making the diagrams $$\vxym{
&L\otimes R\otimes L\ar[dr]^{L\otimes\varepsilon}&\\
L\ar[ur]^{\eta\otimes L}\ar[rr]_L&&L\\
}
\qtand
\vxym{
&R\otimes L\otimes R\ar[dr]^{\varepsilon\otimes R}&\\
R\ar[ur]^{R\otimes\eta}\ar[rr]_{R}&&R\\
}$$ commute. Graphically, $${\cpdfinput{zig_zag_l.ps}}
=
{\cpdfinput{id_L.ps}}
\qtand
{\cpdfinput{zig_zag_r.ps}}
=
{\cpdfinput{id_R.ps}}$$ We write ${\mathfrak{D}}$ for the equational theory associated to dual objects and $\mathcal{D}$ for the generated monoidal category.
If $\mathcal{C}$ is category, two dual objects in the monoidal category $\mathrm{End}(\mathcal{C})$ of endofunctors of $\mathcal{C}$, with tensor product given on objects by composition of functors, are adjoint endofunctors of $\mathcal{C}$. More generally, the theory of adjoint functors in a 2-category is given in [@street:free-adj], the definition of ${\mathfrak{D}}$ is a specialization of this construction to the case where there is only one 0-cell.
A presentation of relations {#section:presentation-rel}
===========================
We now introduce a presentation for the category $\Rel$ of finite ordinals and relations. This result is mentioned in Examples 6 and 7 of [@hyland-power:symmetric-monoidal-sketches] and is proved in three different ways in [@lafont:equational-reasoning-diagrams], [@pirashvili:bialg-prop] and [@lack:composing-props]. The proof we give here has the advantage of being simple to check and can be extended to give a presentation of the category of games and strategies, see Section \[subsection:walking-inno\].
The simplicial category
-----------------------
The simplicial category $\Delta$ is the strict monoidal category whose objects are the finite ordinals and whose morphisms $f:{\underline{m}}\to{\underline{n}}$ are the monotone functions from ${\underline{m}}$ to ${\underline{n}}$.
It has been known for a long time that this category is closely related to the notion of monoid, see [@maclane:cwm] or [@lafont:boolean-circuits] for example. This result can be formulated as follows:
\[property:delta-presentation\] The monoidal category $\Delta$ is presented by the equational theory of monoids ${\mathfrak{M}}$.
In this sense, the simplicial category $\Delta$ impersonates the notion of monoid.
Dually, the monoidal category $\Delta^\op$, which is isomorphic to the category of finite ordinals and (weakly) monotonic functions $f:{\underline{m}}\to{\underline{n}}$ such that $f(0)=0$, impersonates the notion of comonoid:
The monoidal category $\Delta^\op$ is presented by the equational theory of comonoids.
In the next Section, we show how to extend these results to the monoidal category of multirelations.
Multirelations
--------------
A *multirelation* $R$ between two sets $A$ and $B$ is a function from $A\times B\to\N$. It can be equivalently be seen as a multiset whose elements are in $A\times B$, or as a matrix over $\N$, or as a span $$\xymatrix@C=2ex@R=2ex{
&\ar[dl]_sR\ar[dr]^t&\\
A&&B\\
}$$ in the category $\Set$ – for the latest case, the multiset representation can be recovered from the span by $$R(a,b)\qeq\left|\setof{e\in R\tq s(e)=a\tand t(e)=b}\right|$$ for every element $(a,b)\in A\times B$. If $R_1:A\to B$ and $R_2:B\to C$ are two multirelations, their composition is defined by $$R_2\circ R_1
\qeq
(a,c)\mapsto\sum_{b\in B}R_1(a,b)\times R_2(b,c)
\tdot$$ Again, this corresponds to the usual composition of matrices if we see $R_1$ and $R_2$ as matrices over $\N$, and as the span obtained by computing the pullback $$\xymatrix@C=2ex@R=2ex{
&&\ar[dl]R_2\circ R_1\ar[dr]&&\\
&\ar[dl]_{s_1}R_1\ar[dr]^{t_1}&&\ar[dl]_{s_2}R_2\ar[dr]^{t_2}&\\
A&&B&&C\\
}$$ if we see $R_1$ and $R_2$ as spans in $\Set$.
The cardinal ${\left|R\right|}$ of a multirelation $R:A\to B$ is defined by $$|R|\qeq\sum_{(a,b)\in A\times B}R(a,b)
\tdot$$
We write ${\mathbf{MRel}}$ for the monoidal theory of multirelations: its objects are finite ordinals and morphisms are multirelations between them. It is a strict symmetric monoidal category with the tensor product $\otimes$ defined on two morphisms $R_1:{\underline{m_1}}\to{\underline{n_1}}$ and $R_2:{\underline{m_2}}\to{\underline{n_2}}$ by $$R_1\otimes R_2=R_1\cup R_2:{\underline{m_1}}+{\underline{m_2}}\to{\underline{n_1}}+{\underline{n_2}}$$ and the morphisms $$R^\gamma_{{\underline{m}},{\underline{n}}}=
({\underline{m}}\times{\underline{n}})\cup({\underline{n}}\times{\underline{m}}):
{\underline{m}}+{\underline{n}}\to{\underline{n}}+{\underline{m}}$$ as symmetry. In particular, the following multirelations are morphisms in ${\mathbf{MRel}}$: $$\begin{array}{r@{\quad:\quad}l}
R^\mu=(i,j)\mapsto 1&{\underline{2}}\to{\underline{1}}\\
R^\eta=(i,j)\mapsto 1&{\underline{0}}\to{\underline{1}}\\
R^\delta=(i,j)\mapsto 1&{\underline{1}}\to{\underline{2}}\\
R^\varepsilon=(i,j)\mapsto 1&{\underline{1}}\to{\underline{0}}\\
R^\gamma=(i,j)\mapsto
\begin{cases}
0&\text{if $i=j$,}\\
1&\text{otherwise.}
\end{cases}&{\underline{2}}\to{\underline{2}}
\end{array}$$
We now show that multirelations are presented by the equational theory ${\mathfrak{B}}$ bicommutative bialgebras. We write $\mathcal{B}$ for the monoidal category generated by ${\mathfrak{B}}$.
\[property:fmr-bialg\] In ${\mathbf{MRel}}$, $(1,R^\mu,R^\eta,R^\delta,R^\varepsilon)$ is a bicommutative bialgebra.
For every morphism $\phi:m\to n$ in $\mathcal{B}$, where $m>0$, we define a morphism $S\phi:m+1\to n$ by $$S\phi\qeq \phi\circ(\gamma\otimes \id_{m-1})$$ We introduce the following notation which is defined inductively by $${\cpdfinput{gsym.ps}}
\quad\text{is either}\quad
{\cpdfinput{gsym_id.ps}}
\qtor
{\cpdfinput{gsym_sym.ps}}$$ These morphisms are called *stairs*: a stair is therefore either $\id_1$ or $S\phi'$ where $\phi'$ is a stairs. The *length* of a stairs is defined as $0$ if its of the first form and the length of the stairs plus one if it is of the second form.
We define the following notion of *canonical form* inductively: $\phi$ is either $$\label{eq:bialg-nf-eta}
{\cpdfinput{bialg_nf_eta.ps}}$$ or there exists a canonical form $\phi'$ such that $\phi$ is either $$D_i\phi'\qeq{\cpdfinput{bialg_nf_mu.ps}}
\qqtor
E\phi'\qeq{\cpdfinput{bialg_nf_eps.ps}}$$ In the latter case we write respectively $\phi$ as $D_i\phi'$ (where the index $i$ is the length of the stairs) or as $E\phi'$.
Showing that identities are equal to canonical forms require the slightly more general following lemma.
\[lemma:bialg-nf-id\] Any morphism $f=\eta\otimes\cdots\otimes\eta\otimes\id_n:m\to m$ is equal to a canonical form.
By induction on $n$. The result is immediate when $n=0$. Otherwise, we have the equalities of Figure \[fig:bialg-nf-id-proof\] in Appendix which show that $f$ is equal to a morphism of the form $D_i(E\phi)$, where $\phi$ is equal to a canonical form by induction hypothesis.
\[lemma:bialg-nf-comm\] For every morphism $\phi:m\to n$, where $m>0$, for all indices $i$ and $j$ such that $0\leq i\leq n$ and $0\leq j\leq n$, we have $$D_j(D_i\phi)\qeq D_i(D_j\phi)$$
The proof is done by examining separately the cases $j<i$, $i\neq j$ and $j>i$ and showing the result for each case using in particular the derivable equalities shown in Figure \[fig:bialg-stairs-yb\] in Appendix.
From this we deduce that
Every multirelation $R:m\to n$ is represented by a canonical form and two canonical forms representing $R$ are equal.
This is proved by induction on $m$ and on the cardinal ${\left|R\right|}$ of $R$.
1. If $m=0$ then $R$ is represented by a unique normal form which is of the form (\[eq:bialg-nf-eta\]).
2. If $m>0$ and for every $j<n$, $R(0,j)=0$ then $R$ is of the form $R=R^\varepsilon\otimes R'$ and $R$ is necessarily represented by a canonical form $E\phi'$ where $\phi'$ is a canonical form representing $R':(m-1)\to n$, which exists by induction hypothesis.
3. Otherwise, $R$ is necessarily represented by a canonical form of the form $D_k\phi'$, where $k$ is such that $R(0,k)>0$ and $\phi'$ is a canonical form represented by the relation $R':m\to n$ defined by $$R'(i,j)=
\begin{cases}
R(i,j)-1&\text{if $i=0$ and $j=k$,}\\
R(i,j)&\text{else.}\\
\end{cases}$$ and such a canonical form exists by induction hypothesis.
By Lemma \[lemma:bialg-nf-comm\], two canonical forms $\phi_1$ and $\phi_2$ representing $R$, obtained by choosing different values for $k$ in case 3 during the construction of the canonical form are equal.
\[lemma:bialg-nf-init\] Every morphism $f:m\to n$ in $\mathcal{B}$ is equal to a canonical form.
The proof is done by induction on the size ${\left|f\right|}$ of $f$.
- If ${\left|f\right|}=0$ then $m=n$ and $f=\id_m$ which is equal to a canonical form by Lemma \[lemma:bialg-nf-id\].
- If ${\left|f\right|}>0$ then $f$ is of the form $f=h\circ g$ where ${\left|h\right|}=1$ and ${\left|g\right|}={\left|f\right|}-1$. By induction hypothesis, $g$ is equal to a canonical form $\phi$. Since $h$ is of size $1$, it is of the form $h=\id_{m_2}\circ h'\circ\id_{m_1}$ where $h$ is either $\mu$, $\eta$, $\delta$, $\varepsilon$ or $\gamma$. We show the result by case analysis. For the lack of space, we only detail the case where $h'=\mu$. There are four cases to handle which are shown in Figures \[fig:bialg-nf-init-proof1\], \[fig:bialg-nf-init-proof2\], \[fig:bialg-nf-init-proof3\] and \[fig:bialg-nf-init-proof4\].
The category ${\mathbf{MRel}}$ of multirelations is presented by the equational theory ${\mathfrak{B}}$ of bicommutative bialgebras.
Relations
---------
The monoidal category $\Rel$ has finite ordinals as objects and relations as morphisms. This category can be obtained from ${\mathbf{MRel}}$ by quotienting the morphisms by the equivalence relation $\sim$ on multirelations defined as follows. Two multirelations $R_1,R_2:m\to n$ are such that $R_1\sim R_2$ whenever $$\label{eq:multirel-rel}
\forall i<m, \forall j<n,\quad R_1(i,j)\neq 0\tiff R_2(i,j)\neq 0$$ This induces a full monoidal functor $F$ from ${\mathbf{MRel}}$ to $\Rel$. We still write $R^\mu$, $R^\eta$, $R^\delta$, $R^\varepsilon$ and $R^\gamma$ for the images by this functor of the corresponding multirelations. We denote $\mathcal{R}$ for the monoidal category generated by the equational theory ${\mathfrak{R}}$ of qualitative bicommutative bialgebras.
\[lemma:rel-nf-rel\] For every morphism $\phi:m\to n$ in $\mathcal{R}$, where $m>0$, for every index $i$ such that $0\leq i\leq n$, we have $$D_i(D_i\phi)\qeq D_i\phi$$
See Figure \[fig:rel-nf-rel-proof\] in Appendix.
From this Lemma, we deduce that:
The category $\Rel$ of relations is presented by the equational theory ${\mathfrak{R}}$ of qualitative bicommutative bialgebras.
Since $\Rel$ can be obtained from ${\mathbf{MRel}}$ by quotienting morphisms, by Lemma \[property:fmr-bialg\], $(1,R^\mu,R^\eta,R^\delta,R^\varepsilon,R^\gamma)$ is still a bialgebra in $\Rel$ and moreover it satisfies the additional equation making it a qualitative bialgebra. Therefore $\Rel$ is a model of the equational theory ${\mathfrak{R}}$. Moreover, ${\mathfrak{R}}$ is a complete axiomatization of $\Rel$. In order to show this, we use the same notion of canonical form as in the previous Section: we have to show that two canonical forms representing the same relation are equal. This amounts to check that two canonical forms representing two multirelations $R_1$ and $R_2$, which are equivalent by the relation (\[eq:multirel-rel\]), are equal. This can easily be done using Lemmas \[lemma:bialg-nf-comm\] and \[lemma:rel-nf-rel\].
A game semantics for first-order propositional logic {#section:games-strategies}
====================================================
First-order propositional logic
-------------------------------
Suppose that we are given a fixed first-order language $\mathcal{L}$, that is
1. a set of proposition symbols $P,Q,\ldots$ with given arities,
2. a set of function symbols $f,g,\ldots$ with given arities,
3. a set of first-order variables $x,y,\ldots$.
*Terms* $t$ and *formulas* $A$ are respectively generated by the following grammars: $$t\qgramdef x\gramor f(t,\ldots,t)
\qquad\qquad
A\qgramdef P(t,\ldots,t)\gramor{\forall{x}.}{A}\gramor{\exists{x}.}{A}$$ We suppose that application of propositions and functions always respect arities. Moreover, we suppose here that there are proposition and function symbols of any arity (this is needed for the definability result of Proposition \[prop:definability\]). Formulas are considered modulo renaming of variables. Substitution $A[t/x]$ of a free variable $x$ by a term $t$ in a formula $A$ is defined as usual, avoiding capture of variables. We consider the logic associated to these formulas, where proofs are generated by the following inference rules: $$\begin{array}{c@{\qquad}c}
\inferrule{A[t/x]\vdash B}{{\forall{x}.} A\vdash B}{{\text{($\forall$-L)}}}
&
\inferrule{A\vdash B}{A\vdash {\forall{x}.} B}{{\text{($\forall$-R)}}}
\\
&
\text{(with $x$ not free in $A$)}
\\[2ex]
\inferrule{A\vdash B}{{\exists{x}.} A\vdash B}{{\text{($\exists$-L)}}}
&
\inferrule{A\vdash B[t/x]}{A\vdash {\exists{x}.} B}{{\text{($\exists$-R)}}}
\\
\text{(with $x$ not free in $B$)}
&
\\[2ex]
\inferrule{\null}{P(t_1,\ldots,t_n)\vdash P(t_1,\ldots,t_n)}{{\text{(Ax)}}}
&
\inferrule{A\vdash B\\B\vdash C}{A\vdash C}{{\text{(Cut)}}}
\\[2ex]
\end{array}$$
Games and strategies {#subsection:games-strategies}
--------------------
A *game* $A=({M_{A}},\lambda_A,\leq_A)$ consists of a set of moves ${M_{A}}$, a polarization function $\lambda_A:{M_{A}}\to\{-1,+1\}$ which to every move $m$ associates its polarity, and a partial order $\leq_A$ on moves such that every move $m\in{M_{A}}$ defines a finite downward closed set $$m\!\downarrow\qeq\setof{n\in{M_{A}}\tq n\leq_A m}
\tdot$$
A move $m$ is said to be a Proponent move when $\lambda_A(m)=+1$ and an Opponent move else. Suppose that $A$ and $B$ are two games. Their tensor product $A\otimes B$ is defined by $${M_{A\otimes B}}={M_{A}}\uplus{M_{B}}
\tcomma\quad
\lambda_{A\otimes B}=\lambda_A+\lambda_B
\qtand
\leq_{A\otimes B}=\leq_A\cup\leq_B
\tdot$$ The opposite game $A^*$ of the game $A$ is defined by $$A^*=({M_{A}},-\lambda_A,\leq_A)
\tdot$$ Finally, the arrow game $A\llimp B$ is defined by $$A\llimp B\qeq A^*\otimes B
\tdot$$ A game $A$ is *filiform* when the associated partial order is total.
Two partial orders $\leq$ and $\leq'$ on a set $M$ are *compatible* when their relational union $\leq\cup\leq'$ is still an order (is acyclic).
A *strategy* $\sigma$ on a game $A$ is a partial order $\leq_\sigma$ on the moves of $A$ which is compatible with the order of the game and is moreover such that for every moves $m,n\in{M_{A}}$, $$\label{eq:strategy-polarisation}
m<_\sigma n\qtimpl\lambda_A(m)=-1\tand\lambda_A(n)=+1
\tdot$$
The *size* ${\left|A\right|}$ of a game $A$ is the cardinal of ${M_{A}}$ and the *size* ${\left|\sigma\right|}$ of a strategy $\sigma:A$ is the cardinal of the set $$\setof{(m,n)\in{M_{A}}\times{M_{A}}\tq m<_\sigma n}
\tdot$$
If $\sigma:A\llimp B$ and $\tau:B\llimp C$ are two strategies, their composite $\tau\circ\sigma:A\llimp C$ is the partial order $\leq_{\tau\circ\sigma}$ on the moves of $A\llimp C$, defined as the restriction to the set of moves of $A\llimp
C$ of the transitive closure of the union $\leq_\sigma\cup\leq_\tau$ of the partial orders $\leq_\sigma$ and $\leq_\tau$ considered as relations. The identity strategy $\id_A:A\llimp A$ on a game $A$ is the strategy such that for every move $m$ of $A$ we have $m_L\leq_{\id_A}m_R$ if $\lambda(m)=-1$ and $m_R\leq_{\id_A}m_L$ if $\lambda(m)=+1$ when $m_L$ ($m_R$) is the instance of the move $m$ in the left-hand side (right-hand side) copy of $A$. It can easily be checked that for every strategy $\sigma:A\to B$ we have $\id_B\circ\sigma=\sigma=\sigma\circ\id_A$.
Since the composition of strategies is defined in the category of relations, we still have to check that the composite of two strategies $\sigma$ and $\tau$ is actually a strategy. Preservation of the polarization condition (\[eq:strategy-polarisation\]) by composition is easily checked. However, proving that the relation $\leq_{\tau\circ\sigma}$ corresponding to the composite strategy is acyclic is more difficult: a direct proof of this property is combinatorial and a bit lengthy. For now, we define the category ${\mathbf{Games}}$ as the smallest category whose objects are filiform games, whose morphisms between two games $A$ and $B$ contain the strategies on the game $A\llimp B$ and is moreover closed under composition. We will deduce in Corollary \[corol:composition\] that strategies are in fact the only morphisms of this category from our presentation of the category.
If $A$ and $B$ are two games, the game $A{\varolessthan}{}B$ (to be read $A$ *before* $B$) is the game defined by $$M_{A{\varolessthan}{}B}=M_A\uplus M_B
\qcomma
\lambda_{A{\varolessthan}{}B}=\lambda_A+\lambda_B$$ and $\leq_{A{\varolessthan}{}B}$ is the transitive closure of the relation $$\leq_A\cup\leq_B\cup\;\setof{(a,b)\tq a\in M_A\tand b\in M_B}$$ This operation is extended as a bifunctor on strategies as follows. If $\sigma:A\to B$ and $\tau:C\to D$ are two strategies, the strategy $\sigma{\varolessthan}{}\tau:A{\varolessthan}{}C\to B{\varolessthan}{}D$ is defined as the transitive closure of the relation $$\leq_{\sigma{\varolessthan}{}\tau}
\qeq
\leq_\sigma\cup\leq_\tau$$ This bifunctor induces a monoidal structure $({\mathbf{Games}},{\varolessthan}{},I)$ on the category ${\mathbf{Games}}$, where $I$ denotes the empty game.
We write $O$ for a game with only one Opponent move and $P$ for a game with only one Proponent move. It can be easily remarked that filiform games $A$ are generated by the following grammar $$A\qqgramdef I\gramor O{\varolessthan}{}A\gramor P{\varolessthan}{}A$$ A game $X_1{\varolessthan}{}\cdots{\varolessthan}{}X_n{\varolessthan}{}I$ where the $X_i$ are either $O$ or $P$ is represented graphically as $$\xymatrix@R=1ex{
X_1\\
\vdots\\
X_n\\
}$$ A strategy $\sigma:A\to B$ is represented graphically by drawing a line from a move $m$ to a move $n$ whenever $m\leq_\sigma n$. For example, the strategy $\mu^P:P{\varolessthan}{}P\to P$ $${\cpdfinput{mult_P.ps}}$$ is the strategy on $(O{\varolessthan}{}O)\otimes P$ in which both Opponent move of the left-hand game justify the Proponent move of the right-hand game. When a move does not justify (or is not justified by) any other move, we draw a line ended by a small circle. For example, the strategy $\varepsilon^P:P\to I$ drawn as $${\cpdfinput{counit_P.ps}}$$ is the unique strategy from $P$ to $I$.
With these conventions, we introduce notations for some morphisms which are depicted in Figure \[fig:inno-gen\] (perhaps a bit confusingly, the tensor product $\otimes$ on this figure is the ${\varolessthan}{}$ tensor).
$$\begin{array}{r@{\qcolon}l@{\quad}r@{\qcolon}l@{\quad}rcl}
\mu^O&O\otimes O\to O&\mu^P&P\otimes P\to P&\eta^{OP}&\colon&I\to O\otimes P\\
\eta^O&I\to O&\eta^P&I\to P\\
\delta^O&O\to O\otimes O&\delta^P&P\to P\otimes P&\varepsilon^{OP}&\colon&P\otimes O\to I\\
\varepsilon^O&O\to I&\varepsilon^P&P\to I\\
\gamma^O&O\otimes O\to O\otimes O&\gamma^P&P\otimes P\to P\otimes P&\gamma^{OP}&\colon&P\otimes O\to O\otimes P\\
\end{array}$$ respectively drawn as $$\begin{array}{c@{\qquad}c@{\qquad}c}
{\cpdfinput{mult_O.ps}}&{\cpdfinput{mult_P.ps}}&{\cpdfinput{unit_OP.ps}}\\
{\cpdfinput{unit_O.ps}}&{\cpdfinput{unit_P.ps}}\\
{\cpdfinput{comult_O.ps}}&{\cpdfinput{comult_P.ps}}&{\cpdfinput{counit_OP.ps}}\\
{\cpdfinput{counit_O.ps}}&{\cpdfinput{counit_P.ps}}\\
{\cpdfinput{sym_O.ps}}&{\cpdfinput{sym_P.ps}}&{\cpdfinput{sym_OP.ps}}\\
\end{array}$$
A game semantics for proofs
---------------------------
A formula $A$ is interpreted as a game ${\llbracket{A}\rrbracket}$ by $${\llbracket{P}\rrbracket}=I
\qquad
{\llbracket{{\forall{x}.} A}\rrbracket}=O{\varolessthan}{}{\llbracket{A}\rrbracket}
\qquad
{\llbracket{{\exists{x}.} A}\rrbracket}=P{\varolessthan}{}{\llbracket{A}\rrbracket}$$
A proof $\pi:A\vdash B$ is interpreted as the strategy $\sigma:A\llimp B$. The corresponding partial order $\leq_\sigma$ is defined as follows. For every Proponent move $P$ interpreting a quantifier which is introduced by a rule $$\inferrule{A[t/x]\vdash B}{{\forall{x}.} A\vdash B}{{\text{($\forall$-L)}}}
\qqtor
\inferrule{A\vdash B[t/x]}{A\vdash {\exists{x}.} B}{{\text{($\exists$-R)}}}$$ every Opponent move $O$ interpreting an universal quantification $\forall x$ on the right-hand side of a sequent, or an existential quantification $\exists x$ on the left-hand side of a sequent, is such that $O\leq_\sigma P$ whenever the variable $x$ is free in the term $t$. The partial order interpreting a proof $\pi$ can easily be shown to be a strategy.
For example, a proof $$\inferrule{
\inferrule{
\inferrule{
\inferrule{\null}
{P\vdash Q[t/z]}{{\text{(Ax)}}}
}
{P\vdash{\exists{z}.} Q}{{\text{($\exists$-R)}}}
}
{{\exists{y}.} P\vdash{\exists{z}.} Q}{{\text{($\exists$-L)}}}
}
{{\exists{x}.}{{\exists{y}.} P}\vdash{\exists{z}.} Q}{{\text{($\exists$-L)}}}$$ is interpreted respectively by the strategies $${\cpdfinput{strat_ex_xy.ps}}
\quad
{\cpdfinput{strat_ex_x.ps}}
\quad
{\cpdfinput{strat_ex_y.ps}}
\qtand
{\cpdfinput{strat_ex_.ps}}$$ when the free variables of $t$ are $\{x,y\}$, $\{x\}$, $\{y\}$ and $\emptyset$.
The following Proposition shows that our game semantics contains only definable strategies.
\[prop:definability\] For every strategy $\sigma:A\to B$ in ${\mathbf{Games}}$, there exists two propositions $P$ and $Q$ such that $A={\llbracket{\sqcap_1\ldots\sqcap_k P}\rrbracket}$, $B={\llbracket{\sqcap_1\ldots\sqcap_l Q}\rrbracket}$ and there exists a proof $\pi:\sqcap_1\ldots\sqcap_k P\vdash \sqcap_1'\ldots\sqcap_l' Q$ such that ${\llbracket{\pi}\rrbracket}=\sigma$, where $\sqcap_i$ and $\sqcap_i'$ is either $\forall$ or $\exists$.
An equational theory of strategies {#subsection:walking-inno}
----------------------------------
\[definition:innocent-strategies\] The *equational theory of strategies* is the equational theory ${\mathfrak{G}}$ with two types $O$ and $P$ and 13 generators depicted in Figure \[fig:inno-gen\] such that
- $(O,\mu^O,\eta^O,\delta^O,\varepsilon^O,\gamma^O)$ is a bicommutative qualitative bialgebra,
- the Proponent structure is adjoint to the Opponent structure in the sense that the equations of Figure \[fig:PO-adj\] hold.
We write $\mathcal{G}$ for the monoidal category generated by ${\mathfrak{G}}$.
The generators $\mu^P$, $\eta^P$, $\delta^P$, $\varepsilon^P$, $\gamma^P$ and $\gamma^{OP}$ are superfluous in this presentation. However, removing them would seriously complicate the proofs.
With the notations of \[definition:innocent-strategies\], we have:
- $(P,\mu^P,\eta^P,\delta^P,\varepsilon^P,\gamma^P)$ is a qualitative bicommutative bialgebra,
- the Yang-Baxter equalities $${\cpdfinput{yang_baxter_xyz_r.ps}}
\qeq
{\cpdfinput{yang_baxter_xyz_l.ps}}$$ hold whenever $(X,Y,Z)$ is either $(O,O,O)$, $(P,O,O)$, $(P,P,O)$ or $(P,P,P)$,
- the equalities $${\cpdfinput{mult_sym_rnat_P_l.ps}}
=
{\cpdfinput{mult_sym_rnat_P_r.ps}}$$ and $${\cpdfinput{mult_sym_lnat_O_l.ps}}
=
{\cpdfinput{mult_sym_lnat_O_r.ps}}$$ hold (and dually for comultiplications),
- the equalities $${\cpdfinput{eta_sym_rnat_P_l.ps}}
=
{\cpdfinput{eta_sym_rnat_P_r.ps}}$$ and $${\cpdfinput{eta_sym_lnat_O_l.ps}}
=
{\cpdfinput{eta_sym_lnat_O_r.ps}}$$ hold (and dually for counits),
- the equalities $${\cpdfinput{adj_counit_O_r.ps}}
=
{\cpdfinput{adj_counit_O_l.ps}}$$ and $${\cpdfinput{adj_counit_P_r.ps}}
=
{\cpdfinput{adj_counit_P_l.ps}}$$ hold (and dually for the counit of duality).
In the category ${\mathbf{Games}}$ with the monoidal structure induced by ${\varolessthan}{}$, the games $O$ and $P$ together with the morphisms introduced at the end of Section \[subsection:games-strategies\] induce a strategy structure in the sense of Definition \[definition:innocent-strategies\].
We extend the proofs of Section \[section:presentation-rel\] to show that ${\mathfrak{G}}$ is a presentation of the category ${\mathbf{Games}}$.
*Stairs* are defined inductively by $${\cpdfinput{gsym.ps}}$$ is either $${\cpdfinput{gsym_id_O.ps}}
\qtor
{\cpdfinput{gsym_id_P.ps}}$$ or $${\cpdfinput{gsym_sym_O.ps}}
\tor
{\cpdfinput{gsym_sym_P.ps}}
\tor
{\cpdfinput{gsym_sym_OP.ps}}$$ A *canonical form* is either of the form $\phi=\psi\circ\theta$ $$\label{eq:inno-nf-eta}
\text{$\phi$ is}
\qquad
{\cpdfinput{theta_psi.ps}}$$ where a morphism of the form $\theta$ is defined inductively by $$\label{eq:inno-nf-theta}
\text{$\theta$ is either void or}
\qquad
{\cpdfinput{theta_nf_adj.ps}}$$ where $\theta'$ is of the form (\[eq:inno-nf-theta\]), and $\psi$ is defined inductively by $$\label{eq:inno-nf-psi}
\text{$\psi$ is either void or}
\qquad
{\cpdfinput{psi_nf_eta.ps}}
\qtor
{\cpdfinput{psi_nf_id.ps}}$$ where $X$ is either $P$ or $O$ and $\psi'$ is of the form (\[eq:inno-nf-psi\]), or there exists a canonical form $\phi'$ such that $\phi$ is either $$D_i^X\phi'\qeq{\cpdfinput{nf_mu.ps}}
\qqtor
E^X\phi'\qeq{\cpdfinput{nf_eps.ps}}$$ or $$A_i\phi'\qeq{\cpdfinput{nf_adj.ps}}$$ where $X$ is either $P$ or $O$. In the latter case, we write respectively $\phi$ as $D_i^X\phi'$ (where $i$ is the length of the stairs), or as $E^X\phi'$ or as $A_i\phi'$ (where $i$ is the length of the stairs).
\[lemma:inno-nf-comm\] For any morphism $\phi$, we have $$\begin{array}{r@{\qeq}l@{\qquad\qquad}r@{\qeq}l}
D_j^X(D_i^X\phi)&D_i^X(D_j^X\phi)&A_i(A_j\phi)&A_j(A_i\phi)\\
D_i^X(D_i^X\phi)&D_i^X\phi&D_i^O(A_j\phi)&A_j(D_i^O\phi)\\
\end{array}
$$ whenever both members of the equalities are defined, where $X$ is either $P$ or $O$.
Every strategy $\sigma:A\to B$ is represented by a canonical form and two canonical forms representing the same strategy are equal.
This is proved by induction on the respective sizes ${\left|A\right|}$ and ${\left|\sigma\right|}$ of $A$ and $\sigma$.
1. If ${\left|A\right|}=0$ then $\sigma$ is necessarily represented by canonical form of the form (\[eq:inno-nf-eta\]), which is unique.
2. If $A=X{\varolessthan}{}A'$, where $X$ is either $P$ or $O$ and ${M_{X}}=\{m\}$, and for every move $n\in{M_{A\llimp B}}$ we have $m\not<_\sigma n$ then $\sigma$ is necessarily represented by a canonical form $E^X\phi'$ where $\phi'$ is a canonical form representing the restriction of $\sigma$ to $A'\llimp B$.
3. Otherwise, $A$ is of the form $A=X{\varolessthan}{}A'$, where $X$ is either $P$ or $O$ and ${M_{X}}=\{m\}$. A canonical form $\phi$ of $\sigma$ is necessarily of one of the two following forms.
- $\phi=D^X_i\phi'$ where $n$ is the $i$-th move of $B$ and is such that $m<_\sigma n$, and $\phi'$ is a canonical form representing either the strategy $\sigma$ or the strategy $\sigma'$ which is the same strategy as $\sigma$ excepting that $m\not<_{\sigma'}n$ – for the construction part of the lemma we obviously chose the second possibility in order for the induction to work.
- $\phi=A_i\phi'$ where $n$ is the $i$-th move of $A$ and is such that $m<_\sigma n$, and $\phi'$ is a canonical form representing the strategy $\sigma'$ which is the same strategy as $\sigma$ excepting that $m\not<_{\sigma'}n$.
By Lemma \[lemma:inno-nf-comm\], two canonical forms $\phi_1$ and $\phi_2$ representing $\sigma$, obtained by choosing different values for $n$ in case 3 are equal.
Every morphism $f:A\to B$ of $\mathcal{G}$ is equal to a canonical form.
The proof is similar to the proof of Lemma \[lemma:bialg-nf-init\].
The category ${\mathbf{Games}}{}$ is presented by the equational theory ${\mathfrak{G}}$.
As a direct consequence of this Theorem, we deduce that
\[corol:composition\] The composite of two strategies is a strategy.
In particular, acyclicity is preserved by composition.
Conclusion
==========
We have constructed a game semantics for the fragment of first-order propositional logic without connectives and given a presentation of the category ${\mathbf{Games}}$ of games and definable strategies. Our methodology has proved very useful to ensure that the composition of strategies was well-defined.
We consider this work much more as a starting point to bridge semantics and algebra than as a final result. The methodology presented here seem to be very general and many tracks remain to be explored.
First, we would like to extend the presentation to a game semantics for richer logic systems like first-order propositional logic with conjunction and disjunction. Whilst we do not expect many technical complications, this case is much more difficult to grasp and manipulate since a presentation of such a semantics would be a 4-polygraph (one dimension is added since games would be trees instead of lines) and corresponding diagrams now live in a 3-dimensional space.
It would be interesting to know whether it is possible to orient the equalities in the presentations in order to obtain strongly normalizing rewriting systems for the algebraic structures described in the paper. Such rewriting systems are given in [@lafont:boolean-circuits] – for monoids and commutative monoids for example – but finding a strongly normalizing rewriting system presenting the theory of bialgebras is still an open problem.
Finally, many of the proofs given here are repetitive and we believe that many of them could be (at least partly) automated or mechanically checked. However, finding a good representation of diagrams, in order for a program to be able to manipulate them, is a difficult task that we should address in subsequent works.
Acknowledgements {#acknowledgements .unnumbered}
----------------
I would like to thank my PhD supervisor Paul-André Melliès as well as Yves Lafont, Martin Hyland and Albert Burroni for the lively discussion we had, in which I learned so many things.
Figures
=======
$$\begin{array}{r@{=}l}
{\cpdfinput{eta_id_1.ps}}
&
{\cpdfinput{eta_id_2.ps}}
=
{\cpdfinput{eta_id_3.ps}}
=
{\cpdfinput{eta_id_4.ps}}
\\
&
{\cpdfinput{eta_id_5.ps}}
\end{array}$$
$$\begin{array}{r@{\qeq}l}
{\cpdfinput{gsym_yb_l.ps}}
&
{\cpdfinput{gsym_yb_r.ps}}
\end{array}$$
$g$ is of the form $D_i\phi'$ and $f$ is equal to $${\cpdfinput{bialg_nf_mu_mu_case1_1.ps}}
\qeq
{\cpdfinput{bialg_nf_mu_mu_case1_2.ps}}$$ which is of the form $D_j\phi$ where $\phi$ is equal to a canonical form by induction hypothesis.
$g$ is of the form $D_i\phi'$ and and $f$ is equal to $$\begin{array}{cccc}
&{\cpdfinput{bialg_nf_mu_mu_case2_1.ps}}
&=&
{\cpdfinput{bialg_nf_mu_mu_case2_2.ps}}
\\
=&
{\cpdfinput{bialg_nf_mu_mu_case2_3.ps}}
&=&
{\cpdfinput{bialg_nf_mu_mu_case2_4.ps}}
\\
\end{array}$$ which is of the form $D_j(D_k\phi)$ where $\phi$ is equal to a canonical form by induction hypothesis.
$g$ is of the form $D_i\phi'$ and and $f$ is equal to $${\cpdfinput{bialg_nf_mu_mu_case3_1.ps}}$$ which is of the form $D_j\phi$ where $\phi$ is equal to a canonical form by induction hypothesis.
$g$ is of the form $E\phi'$ and and $f$ is equal to $${\cpdfinput{bialg_nf_eta_mu_case1_1.ps}}$$ which is of the form $E\phi$ where $\phi$ is equal to a canonical form by induction hypothesis.
$$\begin{array}{cccc}
&{\cpdfinput{rel_nf_mu_mu_1.ps}}
&=&
{\cpdfinput{rel_nf_mu_mu_2.ps}}
\\
=&
{\cpdfinput{rel_nf_mu_mu_3.ps}}
&=&
{\cpdfinput{rel_nf_mu_mu_4.ps}}
\\
=&
{\cpdfinput{bialg_nf_mu.ps}}
\end{array}$$
$$\begin{array}{c}
{\cpdfinput{mult_P.ps}}
=
{\cpdfinput{comult_O_adj.ps}}
\\[4ex]
{\cpdfinput{comult_P.ps}}
=
{\cpdfinput{mult_O_adj.ps}}
\\[4ex]
{\cpdfinput{unit_P.ps}}
=
{\cpdfinput{counit_O_adj.ps}}
\qquad
{\cpdfinput{counit_P.ps}}
=
{\cpdfinput{unit_P_adj.ps}}
\\[4ex]
{\cpdfinput{sym_P.ps}}
=
{\cpdfinput{sym_O_adj.ps}}
\\[18ex]
{\cpdfinput{sym_OP.ps}}
=
{\cpdfinput{sym_O_adj_OP.ps}}
\\
\end{array}$$
[^1]: This work has been supported by the ANR Invariants algébriques des systèmes informatiques . Physical address: Équipe PPS, CNRS and Université Paris 7, 2 place Jussieu, case 7017, 75251 Paris cedex 05, France. Email address: .
|
---
abstract: 'The very low statistics of cosmic rays above the knee region make their study possible only through the detection of the extensive air showers (EAS) produced by their interaction with the constituents of the atmosphere. The Pierre Auger Observatory located in Argentina is the largest high energy cosmic-ray detection array in the world, composed of fluorescence telescopes, particle detectors on the ground and radio antennas. The Auger Engineering Radio Array (AERA) is composed of 153 autonomous radio stations that sample the radio emission of the extensive air showers in the 30 MHz to 80 MHz frequency range. It covers a surface of 17 km$^2$, has a 2$\pi$ sensitivity to arrival directions of ultra-high energy cosmic rays (UHECR) and provides a duty cycle close to 100%. The electric field emitted by the secondary particles of an air shower is highly correlated to the primary cosmic ray characteristics like energy and mass, and the emission mechanisms are meanwhile well understood. In this contribution, recent progress on the reconstruction of the mass composition and energy measurements with AERA will be presented.'
author:
- Florian Gaté
- for the Pierre Auger Collaboration
title: Radio detection of cosmic rays with the Auger Engineering Radio Array
---
Introduction
============
The Pierre Auger Observatory aims at solving the mysteries of the origin of the ultra-high energy cosmic rays (UHECR). Lots of progress have been done in the last 10 years: a clear cutoff is observed in the energy spectrum around $4\times 10^{19}$ eV [@spectrumpao2010; @Abbasi:2007sv; @AbuZayyad:2012ru], as well as a trend towards a heavier composition at the highest energies, the proton-air cross section is measured at $10^{18}$ eV, exotic scenarii are disfavored as no photon- or neutrino-like showers were detected... For a recent review of the Auger results, see [@Ghia:2015kfz]. The Pierre Auger Collaboration will upgrade its instruments to increase the mass-discrimination capabilities. One way to achieve this goal is to exploit the radio signal emitted by air showers while developing in the atmosphere.
The AERA experiment
===================
The Pierre Auger Observatory [@ThePierreAuger:2015rma], the largest UHECR observatory in the world, hosts a large surface detector (SD) of 1660 Cherenkov tanks spread over 3000 km$^2$, a fluorescence detector (FD) of 27 telescopes at 4 sites around the SD, the AMIGA instrument with buried scintillators for the measurement of the muonic component of air showers, and the Auger Engineering Radio Array (AERA).
AERA consists today of 153 autonomous radio stations that detect the very fast (10-30 ns) transient radio signal emitted by the secondary electrons and positrons of air showers. The electric field is detected by antennas in the band 30-80 MHz (log-periodic dipoles — LPD — in the case of the first 24 stations in 2011 and butterfly dipoles afterwards) and sampled by ADCs running at 180 MHz or 200 MHz depending on the electronics in use. [[FIG.]{}]{} \[aeramap\] shows the current setup.
![Map of the AERA antennas (triangles) together with the other instruments deployed at the Pierre Auger Observatory (water Cherenkov tanks, fluorescence telescopes and AMIGA muon counters).[]{data-label="aeramap"}](aera){width="80mm"}
The stations are powered with solar panels and are used in both self-trigger and external-trigger mode. External triggers come from the SD and the FD. We present here the results obtained using the external-trigger data. They are calibrated in time by the usage of GPS receivers together with a beacon emission that allow to correct for clock drifts. We also use the radio transients emitted by airplanes. The overall timing accuracy is better than 2 ns [@Aab:2015omo].
The amplitude calibration has been studied in detail with the LPD antennas using a known calibrated source carried by weather balloons and, more recently, a remotely piloted drone. The final amplitude accuracy is 9.3% accounting for all kinds of systematic uncertainties [@krausevienna]. This calibration procedure will be applied on the butterfly antennas.
Electric-field emission mechanisms
==================================
Secondary charged particles created during air-shower development emit coherent radiation that extends from frequencies of some kHz up to some hundreds of MHz. The emission extends up to the GHz domain but it is incoherent, less interesting and abandoned today. Electrons and positrons experience multiple diffusion, which is a random process, and the Lorentz force due to the geomagnetic field which is systematic. This results in a net current $\mathbf{j}$ perpendicular to the shower axis and leads to a net electric field. This is called the geomagnetic effect and is the dominant source of radio waves from air showers. This electric field is linearly polarized in the direction of $\mathbf{a}\times\mathbf{B}$ where $\mathbf{a}$ is the shower axis direction and $\mathbf{B}$ is the geomagnetic field direction. An observer detects the same polarization, independently of its position with respect to the shower axis.
The second mechanism of coherent emission is the excess of electrons over positrons. Indeed, positrons annihilate very quickly in air and, in addition to the shower electrons, more electrons are extracted from the medium through Compton, Bhabha and Moeller diffusions. This excess of electrons leads to a net and coherent electric field which is radially polarized with respect to the shower axis. This means that the corresponding polarization depends on the observer position with respect to the shower axis. Its amplitude is one order of magnitude smaller than the geomagnetic electric field.
Both mechanisms have been detected in the data of various experiments [@marinicrc2011; @radioemissionprd; @belletoile:hal-01138851], in particular thanks to the strong progress made in the simulation codes.
These two electric fields interfere and the final electric field has a complex structure that cannot be described by a simple 1D lateral distribution function (LDF). We can use for instance a 2D function taking into account both mechanisms [@Nelles:2014gma], based on the code CoREAS [@coreas2013] as a model of ground distribution of the electric field. This method has been used in AERA to extract the primary energy from the radio signal. We also used the code SELFAS [@selfas2011] to construct the 2D ground distribution of the electric field in order to extract the shower [$X_\mathrm{max}$]{} (see section \[xmax\]).
Primary cosmic ray characteristics from the radio signal
========================================================
The electric field emitted by air showers is now well understood. We can therefore use the simulation to correlate the data and the model to extract the primary cosmic-ray characteristics: its primary energy and its nature through the [$X_\mathrm{max}$]{} of the shower it created.
Energy
------
A strong correlation has been established by AERA between the shower energy measured by the FD or the SD and a radio observable, having units of energy. This observable can be interpreted as the fraction of the primary energy that is radiated in the AERA frequency band 30-80 MHz. The data recorded by the radio stations are first corrected for the antenna and electronics responses [@Abreu:2012pi]. After this step, we obtain the three components of the electric field as a function of time (together with the corresponding Poynting vector) for each station participating in an event. We integrate this Poynting vector over time to get the energy fluence (energy per unit area) for each station. This means that we have the energy fluence at each location of a station, i.e. the measured 2D ground distribution of the energy fluence. We fit these points with the 2D LDF. Once the best 2D LDF is obtained, we integrate it over ground coordinates to get the energy radiated in the 30-80 MHz band [@Aab:2015vta; @Aab:2016eeq]. We rescale the obtained radiated energy $E^\mathrm{Auger}_{30-80~\mathrm{MHz}}$ by the factor $1/\sin^2\alpha$ to take into account the angular distance $\alpha$ from the geomagnetic field. The corrected radiated energy is quadratically correlated to the primary energy $E_\mathrm{CR}$, as expected in the case of a coherent mechanism. The correlation is shown in [[FIG.]{}]{} \[fig:nrj\].
![Corrected radio energy as a function of the primary energy.[]{data-label="fig:nrj"}](energyPRL.pdf){width="65mm"}
From the measurement of the radiated energy in the 30-80 MHz band, we can invert the formula to extract the primary energy: $$\frac{E_\mathrm{CR}}{10^{18}~\mathrm{eV}} = \left(\frac{E^\mathrm{Auger}_{30-80~\mathrm{MHz}}}{10^7~\mathrm{eV}}\cdot\frac{1}{A \sin^2\alpha}\right)^{1/B}$$ with $A=1.58\,\pm\, 0.07$ and $B=1.98\,\pm\, 0.04$. For a primary cosmic-ray energy of $10^{18}$ eV and an angular separation from the geomagnetic field of $90^\circ$, we have a radiated energy in 30-80 MHz of $15.8\pm 0.7~\text{(stat)}\pm 6.7~\text{(sys)}$ MeV.
[$X_\mathrm{max}$]{} estimation {#xmax}
-------------------------------
In this section, we explain how we perform the measurement of the shower [$X_\mathrm{max}$]{} using directly the simulation (here with SELFAS), not using the 2D LDF as in the previous section. The basic idea is to exploit the fact that the shape of the ground distribution of the electric field depends strongly on the distance between the shower core and the point of maximum emission on the shower axis. This distance in turn strongly depends on the mass of the primary cosmic ray.
At fixed energy and zenith angle, light nuclei are more likely to interact at lower altitudes than heavier nuclei. The electric field emission is highly beamed towards the direction of propagation of the shower producing statistically a narrower LDF in the case of light nuclei compared to heavier nuclei. Thus, the topology of the electric field at the ground level is highly correlated to the mass of the primary, as depicted in [[FIG.]{}]{} \[fig:topology\].
![Dependence of the footprint of the radio emission on the mass of the primary. Two LDF are simulated with SELFAS: iron nucleus (left) and proton (right). The vertical axis represents the atmospheric depth and the horizontal lines account for the typical [$X_\mathrm{max}$]{} distributions for proton-induced showers and iron-induced showers.[]{data-label="fig:topology"}](topology.PNG){width="80mm"}
The reconstruction method is based on a comparison of the amplitude of the detected electric field to its simulation. To reconstruct one detected shower, we use a set of simulated events having the measured arrival direction, using an arbitrary energy of $10^{18}$ eV and with realistic [$X_\mathrm{max}$]{} depths for showers initiated by protons and iron nuclei at this energy. The total set is composed of a higher fraction of protons to account for the larger width of the distribution of the first interaction depths for the light nuclei. The center of the simulated LDFs is moved to several positions on the surface of the AERA array. For each tested position, the agreement between the data and the simulation is tested with a $\chi^2$ test. This first step permits us to reconstruct the position of the shower core. During this step the simulated amplitudes are multiplied by a scaling factor defined as the mean amplitude ratio between the data and the simulation. As the electric field amplitude depends linearly on the primary energy, the value of the scaling factor at the core position allows us to reconstruct the primary energy. The $\chi^2$ values obtained with each simulated LDF at the reconstructed core position allows the reconstruction of the detected shower [$X_\mathrm{max}$]{} as the minimum of the function $\chi^2 = f({X_\mathrm{max}})$ as shown in [[FIG.]{}]{} \[fig:chi2xmax\].
![Chi-square for the fit of a detected LDF at AERA and the simulated LDFs as a function of their respective [$X_\mathrm{max}$]{} depths. The values are fitted by a square function represented by the curve. The [$X_\mathrm{max}$]{} depth reconstructed by the radio method for this event is given by the minimum of the quadratic function (i.e. the value that gives the best agreement), highlighted on the plot by the vertical line.[]{data-label="fig:chi2xmax"}](chi2Xmax.PNG){width="80mm"}
The method is applied on a high quality set of multi-hybrid (radio, FD and SD) showers passing the official FD quality cuts; we also require that at least 5 AERA radio stations participate in the event and that the shower has a zenith angle smaller than $55^\circ$. The correlation between the [$X_\mathrm{max}$]{} reconstructed by the radio method and the FD signal is shown in [[FIG.]{}]{} \[fig:xmax\].
![Reconstructed [$X_\mathrm{max}$]{} with the radio method as a function of the FD measurements, the middle line accounts for a one-to-one correlation and the others account for a deviation of $\pm 50$ [$\text{g}/\mathrm{cm}^2$]{}.[]{data-label="fig:xmax"}](Xmax_phingdas.pdf){width="80mm"}
The reconstructed values are in good agreement with the FD measurements and the mean deviation is compatible with zero with a dispersion of $25$ [$\text{g}/\mathrm{cm}^2$]{}. It is important to note that these results are obtained using a realistic simulation of the atmosphere in SELFAS. To do so, we used the Global Data Assimilation System (GDAS) [@gdas]. It gives information (pressure, temperature and air humidity) that allows the computation of the air density as a function of the altitude on a 3 hour basis, in the neighbourhood of the AERA site. For each detected event, the data sets necessary to perform the radio reconstruction are simulated using the air density and refractivity profiles matching the actual experimental conditions at the moment of the detection. The use of a stationary and fixed atmosphere description such as the US Standard model leads to a systematic shift of $17$ [$\text{g}/\mathrm{cm}^2$]{} between the FD measurements and the radio method together with a much larger dispersion. Thus the simulation of the atmosphere matching the experimental conditions is now mandatory to reach the FD precision. Using this method with a realistic description of the atmosphere, we found the same shower core than the one obtained with the SD and FD: the average difference is smaller than $3$ m and the dispersion is smaller than $10$ m. The energy is also well reconstructed as the mean relative difference and dispersion to the SD reconstruction are $3$% and $25$%, respectively.
Conclusion
==========
The latest results provided by AERA show that the radio signal contains the information needed to reconstruct all characteristics of the primary cosmic ray. The primary energy is estimated using the measured radiated energy in the 30-80 MHz band in use in AERA, from a 2D LDF based on CoREAS simulations. The estimation is unbiased and the resolution is of the order of $17\%$. This radiated energy estimator is the same at any experiment site as it is normalized to the geomagnetic field strength. It is hardly dependent on environmental conditions, contrarily to the fluorescence method because the atmosphere is transparent to radio waves in our frequency range of interest. Finally, this is a reliable method as it is based on the classical emission of the electromagnetic part of the shower, which is well understood.
A pure radio method (without the need of other detectors), using only the electric field measurements, has been developed using SELFAS simulations. It has been tested on multi-hybrid (radio, SD, FD) events detected in Auger. The energy and core position are accurately reconstructed and the shower [$X_\mathrm{max}$]{} is in excellent agreement with the FD estimation. This result is obtained when using a realistic description of the atmosphere provided by the GDAS. This is not the case if we use the canonical US Standard model for the atmosphere which is not accurate enough for the level of precision we demand.
We are currently working on providing the composition of ultra-high energy cosmic rays using the radio signal.
[17]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
(), ****, (), .
(), ****, (), .
(), ****, (), .
(), ****, ().
(), ****, (), .
(), ****, P01018 (), .
, in ** (, ).
, in **, Vol. 1, 291 (, ).
, ****, 052002 ().
, , , , , , , ****, () , ****, (), .
, in ** (, ), .
, ****, (), .
(), ****, (), .
(), ****, (), .
(), ****, (), .
, **.
|
---
abstract: 'Almost perfect nonlinear (APN) functions play an important role in the design of block ciphers as they offer the strongest resistance against differential cryptanalysis. Despite more than 25 years of research, only a limited number of APN functions are known. In this paper, we show that a recent construction by Taniguchi provides at least $\frac{\varphi(m)}{2}\left\lceil \frac{2^m+1}{3m} \right\rceil$ inequivalent APN functions on the finite field with ${2^{2m}}$ elements. This is a great improvement of previous results: for even $m$, the best known lower bound has been $\frac{\varphi(m)}{2}\left(\lfloor \frac{m}{4}\rfloor +1\right)$, for odd $m$, there has been no such lower bound at all. Moreover, we determine the automorphism group of Taniguchi’s APN functions.'
author:
- 'Christian Kaspers[^1] and Yue Zhou[^2]'
bibliography:
- '../../Draft/bib.bib'
title: The number of almost perfect nonlinear functions grows exponentially
---
#### Keywords
vectorial Boolean function, APN function, CCZ-equivalence, dimensional dual hyperoval
Introduction {#sec:introduction}
============
A function $f:{\mathbb{F}}_{2^n} \rightarrow {\mathbb{F}}_{2^n}$ is called *almost perfect nonlinear* (APN) if the equation $$f(x+a)+f(x)=b$$ has exactly $0$ or $2$ solutions for any $b\in {\mathbb{F}}_{2^n}$ and any nonzero $a\in {\mathbb{F}}_{2^n}$. APN functions were introduced in [-@nyberg1994] by @nyberg1994. She defined them as the mappings with the highest resistance to differential cryptanalysis, which is one of the most important cryptanalyst tools for block ciphers and was introduced in [-@biham1991] by @biham1991. Moreover, APN functions are strongly connected with finite geometry. In particular, quadratic APN functions are equivalent to a special type of dimensional dual hyperovals. We refer to the work by @yoshiara2008 [@edel2010; @dempwolff2014] for more details.
Since their introduction, APN functions have been studied intensively. For an extended overview of these functions, we refer to the survey by @pott2016. For a long time, only very few APN functions were known, all of which power functions of the form $x \mapsto x^d$. In [-@edel2006], @edel2006 reported the first two examples of non-power APN functions on ${\mathbb{F}}_{2^{10}}$ and ${\mathbb{F}}_{2^{12}}$. Since then, quite a few infinite families of non-power APN functions have been found. A recent list of them was given by @budaghyan2019 [Table 3].
Except for some sporadic examples, every known non-power APN function is equivalent to a quadratic APN function, that can be written in the form $\sum_{0\leq i<j\le n-1}\alpha_{i,j} x^{2^i+2^j}+\sum_{0\leq i \le n-1}\beta_i x^{2^i} +\gamma$ with $\alpha_{i,j},\beta_i,\gamma\in {\mathbb{F}}_{2^n}$ for $i,j = 0, \dots, n-1$ and not all $\alpha_{i,j}=0$. By equivalent we mean there exists a CCZ-equivalence transformation between functions over ${\mathbb{F}}_{2^n}$. This equivalence relation was introduced in [-@carletcharpinzinoviev1998] by @carletcharpinzinoviev1998, it preserves the APN property.
When $n$ is odd, many known APN functions are also permutations on ${\mathbb{F}}_{2^n}$. The most fascinating problem regarding APN functions is to find APN permutations on ${\mathbb{F}}_{2^{n}}$ where $n$ is even. So far, only one such function is known: it was found by @dillon2010 on ${\mathbb{F}}_{2^6}$. This sporadic example is also equivalent to a quadratic APN function.
Another very basic and natural question concerning APN functions is the following.
\[question\] How many inequivalent APN functions on ${\mathbb{F}}_{2^n}$ exist for a given $n$?
Despite its simplicity, this question has not been satisfactorily answered yet. By checking the known APN functions, see , we first notice that all the power APN functions only provide very few inequivalent examples. Little is known, however, about the number of inequivalent non-power APN functions as it is, in general, a very hard problem to prove the non-equivalence of two functions. Only for small dimensions, this problem can be solved computationally, for larger dimensions, one has to solve it theoretically. Studying a special family of non-power APN functions introduced by Pott and the second author [@zhou2013], the present authors [@kasperszhou2020] recently presented a first benchmark to answer for certain fields: they showed that there are at least $\frac{1}{2}\varphi(m)\left(\lfloor m/4\rfloor +1\right)$ inequivalent APN functions on ${\mathbb{F}_{2^{2m}}}$ with $m$ even, where $\varphi$ is Euler’s totient function.
In this paper, we considerably improve this lower bound and extend it to ${\mathbb{F}_{2^{2m}}}$ for any $m \geq 2$. We investigate a family of APN functions defined on ${\mathbb{F}}_{2^{2m}}$ for any $m\geq 2$ that has been found by @taniguchi2019. By completely determining the equivalence of members among this family, we show that the number of inequivalent APN functions on ${\mathbb{F}}_{2^{2m}}$ is at least $$\frac{\varphi(m)}{2}\left\lceil \frac{2^m+1}{3m} \right\rceil.$$ As a corollary, our results enables us to determine the automorphism group of the Taniguchi APN functions.
The paper is organized as follows. In , we introduce all necessary definitions and notations. In , we give an overview of the known classes of APN functions and introduce the constructions by @taniguchi2019 and Pott and the second author [@zhou2013]. Afterwards, we solve the equivalence problem for the Taniguchi APN functions and present their automorphism group in . In , we use these results to establish the aforementioned lower bound on the total number of inequivalent APN functions on ${\mathbb{F}}_{2^{2m}}$. To conclude, we point out several open problems regarding APN functions in .
Preliminaries {#sec:Preliminaries}
=============
In this section, we present all the definitions and basic results needed to follow the paper. Denote by ${\mathbb{F}}_{2}^n$ the $n$-dimensional vector space over the finite field ${\mathbb{F}}_2$ with two elements. A function from ${\mathbb{F}}_2^n$ to ${\mathbb{F}}_2^m$ is called a *vectorial Boolean function* if $m \ge 2$ or simply a *Boolean function* if $m=1$. In this paper, we will only consider vectorial Boolean functions from ${\mathbb{F}}_2^n$ to ${\mathbb{F}}_2^n$, we say functions *on* ${\mathbb{F}}_2^n$. In most cases, we identify the $n$-dimensional vector space ${\mathbb{F}}_2^n$ over ${\mathbb{F}}_{2}$ with the finite field ${\mathbb{F}}_{2^n}$ with $2^n$ elements. This will allow us to use finite field operations and notations. Note that any function on the finite field ${\mathbb{F}}_{2^n}$ can be written as a univariate polynomial mapping of degree at most $2^n-1$. Furthermore, denote by ${\mathbb{F}}_{2^n}^*$ the multiplicative group of ${\mathbb{F}}_{2^n}$. We start by recalling the definition of APN functions from .
\[def:APN\] A function $f \colon {\mathbb{F}}_{2^n} \to {\mathbb{F}}_{2^n}$ is called *almost perfect nonlinear* (APN) if the equation $$f(x+a) + f(x) = b$$ has exactly $0$ or $2$ solutions for any $b \in {\mathbb{F}}_{2^n}$ and any nonzero $a \in {\mathbb{F}}_{2^n}$.
There are several equivalent definitions of almost perfect nonlinear functions. We refer to @budaghyan2014 and @pott2016 for an extended overview of these functions. In this paper, we will only consider *quadratic* APN functions. We define this term using the coordinate function representation of a function on ${\mathbb{F}}_2^n$.
Let $f \colon {\mathbb{F}}_2^n \to {\mathbb{F}}_2^n$, where $$f(x_1,\dots,x_n) =
\begin{pmatrix}
f_1(x_1, \dots, x_n)\\\vdots\\ f_n(x_1, \dots, x_n)
\end{pmatrix}$$ for Boolean coordinate functions $f_1, \dots, f_n\colon {\mathbb{F}}_2^n \to {\mathbb{F}}_2$. The maximal degree of the coordinate functions $f_1, \dots, f_n$ is called the *algebraic degree* of $f$. We call a function of algebraic degree $2$ *quadratic*, and a function of algebraic degree $1$ *affine*. If $f$ is affine and has no constant term, we call $f$ *linear*.
In polynomial mapping representation, any quadratic function $f$ on ${\mathbb{F}}_{2^n}$ can be written in the form $$f(x) = \sum_{\substack{i,j = 0 \\ i < j}}^{n-1}\alpha_{i,j} x^{2^i+2^j} + \sum_{i = 0}^{n-1}\beta_i x^{2^i} + \gamma,$$ and any affine function $f \colon {\mathbb{F}}_{2^n} \to {\mathbb{F}}_{2^n}$ can be written as $$f(x) = \sum_{i=0}^{n-1}\beta_i x^{2^i} + \gamma.$$ If $f$ is affine and $\gamma = 0$, then $f$ is linear. Similar terms are used to describe polynomials over ${\mathbb{F}}_{2^n}$. Denote by ${\mathbb{F}}_{2^n}[X]$ the univariate polynomial ring over ${\mathbb{F}}_{2^n}$. A polynomial of the form $$P(X) = \sum_{i\ge 0}\alpha_i X^{2^i}$$ is called a *linearized polynomial*. Note that there is a one-to-one correspondence between linear functions on ${\mathbb{F}}_2^n$ and linearized polynomials in ${\mathbb{F}}_{2^n}[X] / (X^{2^n}-X)$. In the same way as for univariate polynomials, we define a linearized polynomial in the multivariate polynomial ring ${\mathbb{F}}_{2^n}[X_1,\dots, X_r]$ as a polynomial of the form $$P(X_1,\dots,X_r) = \sum_{j=1}^{r} \left(\sum_{i\ge 0}\alpha_{i,j} X_j^{2^i}\right).$$
We will use such polynomials to study the equivalence of APN functions. In this paper, we are interested in *inequivalent* APN functions. There are several notions of equivalence between vectorial Boolean functions that preserve the APN property. We list them in the following definition.
\[def:equivalence\] Two functions $f,g \colon {\mathbb{F}}_{2^n} \to {\mathbb{F}}_{2^n}$ are called
- *Carlet-Charpin-Zinoviev equivalent* (CCZ-equivalent), if there is an affine permutation $C$ on ${\mathbb{F}}_{2^n} \times {\mathbb{F}}_{2^n}$ such that $$C(G_f) = G_g,$$ where $G_f = \{(x,f(x)) : x \in {\mathbb{F}}_{2^n}\}$ is the graph of $f$,
- *extended affine equivalent* (EA-equivalent) if there exist three affine functions $A_1,A_2,A_3 \colon {\mathbb{F}}_{2^n} \to {\mathbb{F}}_{2^n}$, where $A_1$ and $A_2$ are permutations, such that $$f(A_1(x)) = A_2(g(x)) + A_3(x),$$
- *affine equivalent* if they are extended affine equivalent and $A_3(x) = 0$,
- *linearly equivalent* if they are affine equivalent and $A_1, A_2$ are linear.
CCZ-equivalence is the most general known notion of equivalence that preserves the APN property. Obviously, linear equivalence implies affine equivalence, and affine equivalence implies EA-equivalence. Moreover, it is well known that EA-equivalence implies CCZ-equivalence but, in general, the converse is not true. For quadratic APN functions, however, @yoshiara2012 proved that also the converse holds.
\[prop:yoshiara\] Let $f$ and $g$ be quadratic APN functions on a finite field ${\mathbb{F}}_{2^n}$ with $n \ge 2$. Then $f$ is CCZ-equivalent to $g$ if and only if $f$ is EA-equivalent to $g$.
In this paper, will allow us to prove the CCZ-inequivalence of certain quadratic APN functions by showing that they are EA-inequivalent.
We will often consider functions on vector spaces of even dimension $n = 2m$. Such functions can be represented in a *bivariate* description as a map on ${\mathbb{F}_{2^m}}^2 = {\mathbb{F}_{2^m}}\times {\mathbb{F}_{2^m}}$ with two coordinate functions. In this case, we will describe EA-equivalence as follows: Two functions $f,g \colon {\mathbb{F}_{2^m}}^2 \to {\mathbb{F}_{2^m}}^2$, where $$\begin{aligned}
f(x,y) = (f_1(x,y), f_2(x,y)) &&\text{and} &&g(x,y) = (g_1(x,y), g_2(x,y))
\end{aligned}$$ for coordinate functions $f_1,f_2,g_1,g_2 \colon {\mathbb{F}_{2^m}}^2 \to {\mathbb{F}_{2^m}}$, are EA-equivalent, if there exist affine functions $L,N,M \colon {\mathbb{F}_{2^m}}^2 \to {\mathbb{F}_{2^m}}^2$, where $L$ and $N$ are bijective, such that $$f(L(x,y)) = N(g(x,y)) + M(x,y).$$ Write $$\begin{aligned}
L(x,y) = (L_A(x,y), L_B(x,y)) && \text{and} && M(x,y) = (M_A(x,y), M_B(x,y))
\end{aligned}$$ for affine functions $L_A, L_B, M_A, M_B \colon {\mathbb{F}_{2^m}}^2 \to {\mathbb{F}_{2^m}}$ and $$N(x,y) = \left(N_1(x) + N_3(y),\ N_2(x) + N_4(y)\right)$$ for affine functions $N_1, \dots, N_4 \colon {\mathbb{F}_{2^m}}\to {\mathbb{F}_{2^m}}$. In terms of these newly defined functions, $f$ and $g$ are EA-equivalent if both $$\begin{aligned}
\label{eq:LinEquiv_1}
f_1(L_A(x,y), L_B(x,y)) &= N_1(g_1(x,y)) + N_3(g_2(x,y)) + M_A(x,y),\\
\label{eq:LinEquiv_2}
f_2(L_A(x,y), L_B(x,y)) &= N_2(g_1(x,y)) + N_4(g_2(x,y)) + M_B(x,y)
\end{aligned}$$ hold. They are affine equivalent if $M(x,y) = 0$, and they are linearly equivalent if $M(x,y) = 0$ and the functions $L$ and $N$ are linear.
When studying EA-equivalence, the constants of the affine functions $L_A, L_B$, $M_A, M_B$, $N_1,\dots,N_4$ can be omitted as they only lead to a shift in the input and in the output. Hence, we will usually consider these functions as linear functions and describe them as linearized polynomials in the respective polynomial ring. Equations \[eq:LinEquiv\_1\] and \[eq:LinEquiv\_2\] will form the general framework in the proof of our main theorem.
We will not only solve equivalence problems in this paper, but we will also present the size of the automorphism group of several vectorial Boolean functions.
\[def:automorphism\_group\] Let $f$ be a vectorial Boolean function on ${\mathbb{F}}_{2^n}$. We define the *automorphism group of $f$ under CCZ-equivalence* as the group of affine permutations on ${\mathbb{F}}_{2^n} \times {\mathbb{F}}_{2^n}$ that preserve the graph of $f$. We denote this automorphism group by ${\textnormal{Aut}}(f)$. We analogously define the *automorphism group ${\textnormal{Aut}}_{EA}(f)$ of $f$ under EA-equivalence* and the *automorphism group ${\textnormal{Aut}}_L(f)$ of $f$ under linear equivalence* as the groups of the respective equivalence mappings on ${\mathbb{F}}_{2^n} \times {\mathbb{F}}_{2^n}$.
Regarding the automorphism groups of APN functions, we need the following two lemmas. The first one follows from @yoshiara2012’s [@yoshiara2012] proof of .
\[lem:CCZ\_EA\] Let $f$ be a quadratic APN function on the finite field ${\mathbb{F}}_{2^n}$, where $n \ge 2$. Then $${\textnormal{Aut}}(f) = {\textnormal{Aut}}_{EA}(f).$$
The next result follows from the definitions of the different notions of equivalence in .
\[lem:AutomorphismGroup\] Denote by $({\mathbb{F}}_{2^n},+)$ the additive group of the finite field ${\mathbb{F}}_{2^n}$. Let $f$ be a function on ${\mathbb{F}}_{2^n}$. Then $${\textnormal{Aut}}_{EA}(f) = \left({\mathbb{F}}_{2^n},+\right) \rtimes {\textnormal{Aut}}_L(f).$$
Known classes of APN functions {#sec:known_classes}
==============================
In this section, we give a short overview over the currently known APN functions. In , we present the known APN power functions.
[XXXl]{} &Exponents $d$ &Conditions &Reference
------------------------------------------------------------------------
\
Gold functions &$2^i+1$ &$\gcd(i,n) = 1,\ i \le \lfloor \frac{n}{2} \rfloor$ &[@gold1968; @nyberg1994]
------------------------------------------------------------------------
\
Kasami functions &$2^{2i}-2^i+1$ &$\gcd(i,n) = 1,\ i \le \lfloor \frac{n}{2} \rfloor$ &[@janwa1993; @kasami1971]
------------------------------------------------------------------------
\
Welch function &$2^k+3$ &$n = 2k+1$ &[@dobbertin1999_welch]
------------------------------------------------------------------------
\
Niho function &$2^k+2^{\frac{k}{2}}-1$, $k$ even & $n=2k+1$ &[@dobbertin1999_niho]
------------------------------------------------------------------------
\
&$2^k+2^{\frac{3k+1}{2}}-1$, $k$ odd & $n=2k+1$ &
------------------------------------------------------------------------
\
Inverse function &$2^{2k}-1$ &$n=2k+1$ &[@beth1994; @nyberg1994]
------------------------------------------------------------------------
\
Dobbertin function &$2^{4k} + 2^{3k} + 2^{2k} + 2^{k}-1$ &$n=5k$ &[@dobbertin2001]
------------------------------------------------------------------------
\
This list is conjectured to be complete. APN power functions and their equivalence relations are very well studied. It is well known that the classes in are in general CCZ-inequivalent. Moreover, it is, for example, known that Gold functions are inequivalent for different values of $i$.
As far as non-power APN functions are concerned, the situation becomes much less clear than for power functions. Several infinite families of non-power APN functions have been found, but only for few of them their equivalence relations are known. This includes equivalence relations both between functions from different classes as well as between functions coming from the same class. A current list of known families of APN functions that are CCZ-inequivalent to power functions was recently given by @budaghyan2019 [Table 3]. This list contains nine distinct classes, all of which are quadratic.
In the present paper, we study the family (F12) from this list. It was introduced by @taniguchi2019 who used a criterion developed by @carlet2011 to prove the APN property of his functions. In , we restate @taniguchi2019’s [@taniguchi2019] construction in bivariate representation. Its univariate form can be found in the list by @budaghyan2019.
\[th:TaniguchiAPN\] Let $m \ge 2$ and $k$ be positive integers such that $\gcd(k,m) = 1$. Let $\alpha, \beta \in {\mathbb{F}_{2^m}}$ and $\beta \ne 0$. Then the function $f_{k,\alpha,\beta}: {\mathbb{F}_{2^{2m}}}\to {\mathbb{F}_{2^{2m}}}$, where $$f_{k,\alpha,\beta}(x,y) = \left(x^{2^{2k}(2^k+1)} + \alpha x^{2^{2k}} y^{2^k} + \beta y^{2^k+1},\ xy\right)$$ is APN if and only if the polynomial $X^{2^k+1} + \alpha X + \beta \in {\mathbb{F}_{2^m}}[X]$ has no root.
We remark that the Taniguchi APN functions from are quadratic. In the following lemma we specify the case $\alpha = 0$.
\[lem:alpha=0\] A Taniguchi function $f_{k,0,\beta}$ on ${\mathbb{F}_{2^{2m}}}$ is APN if and only if $m$ is even and $\beta$ is a non-cube in ${\mathbb{F}_{2^m}}^*$.
According to , the function $f_{k,0,\beta}$ is APN if and only if the polynomial $P(X) \in {\mathbb{F}_{2^m}}[X]$, where $P(X) = X^{2^k+1} + \beta$, has no root. Recall that $m$ and $k$ are coprime. Hence, $$\gcd(2^k+1,2^m-1) =
\begin{cases}
1, &\text{if $m$ is odd},\\
3, &\text{if $m$ is even}.
\end{cases}$$ Consequently, if $m$ is odd, $P(X)$ is a permutation polynomial and, thus, always has a root. If $m$ is even, however, then $P(X)$ has a root if and only if $\beta$ is a cube.
The following lemma provides insight on the total number of Taniguchi APN functions for given $m$ and $k$—without considering equivalence—by giving the number of admissible $\beta \in {\mathbb{F}_{2^m}}^*$. This result is due to @bluher2004 [Theorem 5.6] who proved it in a more general setting. In the specific form of the present paper, the result was also obtained by @hellesethkholosha2008.
\[lem:number\_of\_beta\] Let $k,m$ be coprime integers such that $0 < k < m$. The number of $\beta \in {\mathbb{F}_{2^m}}^*$ such that the polynomial $X^{2^k+1} + X + \beta$ has no roots in ${\mathbb{F}_{2^m}}$ is $\frac{2^m-1}{3}$ if $m$ is even and $\frac{2^m+1}{3}$ if $m$ is odd.
In , we present another family of APN functions, which is closely related to @taniguchi2019’s [@taniguchi2019] construction from . It was introduced by Pott and the second author [@zhou2013], and @anbar2019 showed that the conditions on the parameters are not only sufficient but also necessary. The equivalence problem of these APN functions was recently solved by the present authors [@kasperszhou2020].
\[th:ZhouPottAPN\] Let $m$ be an even integer and let $k,s$ be integers, $0 \le k,s \le m$, such that $\gcd(k,m)=1$. Let $\alpha \in {\mathbb{F}_{2^m}}^*$. The function $g_{k,s,\alpha}:{\mathbb{F}_{2^{2m}}}\to {\mathbb{F}_{2^{2m}}}$ defined as $$g_{k,s,\alpha}(x,y) = \left(x^{2^k+1} + \alpha y^{2^s(2^k+1)},\ xy\right)$$ is APN if and only if $s$ is even and $\alpha$ is a non-cube.
In the following and , we restate two results by the present authors [@kasperszhou2020] about the equivalence between Pott-Zhou APN functions that we will need to study the equivalence relations between Taniguchi APN functions in .
\[lem:ZP\_trivial\_equivalence\] Let $m \ge 2$ be an even integer. Let $k,\ell$ be integers coprime to $m$ such that $0 < k,\ell < m$, and let $s,t$ be even integers with $0 \le s,t \le m$. Let $\alpha, \alpha' \in {\mathbb{F}_{2^m}}^*$ be non-cubes. The two APN functions $g_{k,s,\alpha},g_{\ell,t,\alpha'}$ on ${\mathbb{F}_{2^{2m}}}$ from are linearly equivalent
1. \[item:non-cubics\] if $k = \ell$ and $s=t$, no matter which non-cubes $\alpha$ and $\alpha'$ we choose,
2. \[item:k=-l,s=-t\] if $k \equiv \pm \ell \pmod{m}$ and $s \equiv \pm t\pmod{m}$.
\[th:ZP\_equivalence\] Let $m \ge 4$ be an even integer. Let $k,\ell$ be integers coprime to $m$ such that $0 < k,\ell < \frac{m}{2}$, let $s,t$ be even integers with $0 \le s,t \le \frac{m}{2}$, and let $\alpha,\alpha' \in {\mathbb{F}_{2^m}}^*$ be non-cubes. Two Pott-Zhou APN functions $g_{k,s,\alpha}, g_{\ell,t,\alpha'}$ on ${\mathbb{F}_{2^{2m}}}$ from , are CCZ-equivalent if and only if $k = \ell$ and $s = t$.
On the equivalence of Taniguchi APN functions {#sec:Taniguchi_equivalence}
=============================================
In this section, we study the equivalence problem of the Taniguchi APN functions on ${\mathbb{F}_{2^{2m}}}$, which were introduced in . We will answer the question for which values of the parameters $k,\alpha,\beta$ two Taniguchi APN functions $f_{k,\alpha,\beta}$ are CCZ-inequivalent.
As we have pointed out before, Taniguchi APN functions are quadratic. Hence, by and , two Taniguchi APN functions are CCZ-equivalent if and only if they are EA-equivalent, and their automorphism groups under CCZ- and EA-equivalence are the same. We begin by studying the case $\alpha = 0$. Recall from that $f_{k,0,\beta}$ is APN if and only if $m$ is even and $\beta$ is a non-cube.
\[prop:alpha=0\_ZhouPott\] Let $m \ge 2$ be an even integer, and let $0 < k < \frac{m}{2}$ such that $k$ and $m$ are coprime. Let $\beta, \gamma \in {\mathbb{F}_{2^m}}^*$ be non-cubes. The Taniguchi APN function $f_{k,0,\beta}$ on ${\mathbb{F}_{2^{2m}}}$ from is linearly equivalent to the Pott-Zhou APN function $g_{k,2k,\gamma}$ on ${\mathbb{F}_{2^{2m}}}$ from .
If $\beta$ is a non-cube in ${\mathbb{F}_{2^{2m}}}^*$, then $\frac{1}{\beta}$ is as well. From (a), we know that the Pott-Zhou APN function $g_{k,2k,\gamma}$ is linearly equivalent to $g_{k,2k,\frac{1}{\beta}}$. We will show that $f_{k,0,\beta}$ is linearly equivalent to $g_{k,2k,\frac{1}{\beta}}$.
By \[eq:LinEquiv\_1\] and \[eq:LinEquiv\_2\] and the explanations below, the two functions $f_{k,0,\beta}$ and $g_{k,2k,\frac{1}{\beta}}$ are linearly equivalent if there exist bijective mappings $L,N$ on ${\mathbb{F}}_{2^m}^2$, represented by linearized polynomials $L_A(X,Y), L_B(X,Y) \in {\mathbb{F}_{2^m}}[X,Y]$ and $N_1(X),\dots,N_4(X) \in {\mathbb{F}_{2^m}}[X]$, respectively, such that the two equations $$\begin{aligned}
L_A(x,y)^{2^{2k}(2^k+1)} + \beta L_B(x,y)^{(2^k+1)} &= N_1(x^{2^k+1} + \tfrac{1}{\beta} y^{2^{2k}(2^k+1)}) + N_3(xy),\\
L_A(x,y)L_B(x,y) &= N_2(x^{2^k+1}+ \tfrac{1}{\beta} y^{2^{2k}(2^k+1)}) + N_4(xy)
\end{aligned}$$ hold for all $x,y \in {\mathbb{F}_{2^m}}$. The functions $f_{k,0,\beta}$ and $g_{k,2k,\frac{1}{\beta}}$ are linearly equivalent by $$\begin{aligned}
L_A(X,Y) &= Y,& L_B(X,Y)&=X,& N_1(X) &= \tfrac{1}{\beta}X,& N_2(X)=N_3(X)&= 0, &N_4(X) &= X.
\end{aligned}$$ Consequently, $f_{k,0,\beta}$ is linearly equivalent to $g_{k,2k,\gamma}$.
From , we immediately obtain the following results.
\[cor:alpha=0\] Let $m \ge 4$.
1. \[item:alpha=0\_a\] Two Taniguchi APN functions $f_{k,0,\beta}$ and $f_{-k,0,\beta}$ on ${\mathbb{F}_{2^{2m}}}$ are CCZ-equivalent.
2. \[item:alpha=0\_b\] Two Taniguchi APN functions $f_{k,0,\beta}$ and $f_{\ell,0,\beta'}$ on ${\mathbb{F}_{2^{2m}}}$ where $0< k,\ell < \frac{m}{2}$ are CCZ-equivalent if and only if $k = \ell$.
Statement \[item:alpha=0\_a\] follows from in combination with (b). Statement \[item:alpha=0\_b\] follows from in combination with .
We remark that for $m = 2$, all Taniguchi APN functions, no matter if $\alpha$ is zero or not, are CCZ-equivalent to the Gold APN function $x \mapsto x^3$. From now on, we focus on the case $\alpha \ne 0$. In the following , we summarize several results about polynomials of the shape $X^{2^k+1} + X + \beta$ that we need to solve the equivalence problem of the Taniguchi APN functions.
\[lem:polynomial\_transformation\] Let $m \ge 2$, and let $\alpha, \beta \in {\mathbb{F}_{2^m}}^*$. The statement the polynomial $X^{2^k+1} + \alpha X + \beta \in {\mathbb{F}_{2^m}}[X]$ has no roots is equivalent to the following statements:
1. $X^{2^k+1} + X + \frac{\beta}{\alpha^{2^{-k}+1}} \in {\mathbb{F}_{2^m}}[X]$ has no roots,
2. $X^{2^k+1} + X + \beta^{2^i} \in {\mathbb{F}_{2^m}}[X]$, where $i \in \{0, \dots, m-1\}$, has no roots,
3. $X^{2^{-k}+1} + X + \beta \in {\mathbb{F}_{2^m}}[X]$ has no roots.
Let $P(X) = X^{2^k+1} + \alpha X + \beta$ such that $P(X)$ has no root in ${\mathbb{F}_{2^m}}$.
1. If we substitute $X$ by $\alpha^{2^{-k}}X$ in $P(X)$, we obtain $\alpha^{2^{-k}+1}X^{2^k+1} + \alpha^{2^{-k}+1}X + \beta$. Factoring out $\alpha^{2^{-k}+1}$ gives the result.
2. Transform the polynomial $P(X)$ into $X^{2^k+1} + X + \beta^{2^i}$ by applying the automorphism $x \mapsto x^{2^i}$ on the coefficients of $P(X)$.
3. Let $P'(X) = X^{2^{-k}+1} + X + \beta$. Then $P'(X)$ can be transformed into $P(X)$ by the substitution $X \mapsto (X+1)^{2^k}$.
We now focus on the equivalence relations between Taniguchi APN functions.
\[prop:trivial\_equivalences\] Let $m \ge 2$ be an integer. Let $k$ be an integer coprime to $m$ such that $0 < k <m$, and let $\alpha, \beta \in {\mathbb{F}_{2^m}}^*$. Then, the following pairs of Taniguchi APN functions on ${\mathbb{F}_{2^{2m}}}$ from are linearly equivalent:
1. \[item:alpha\_1\] $f_{k,\alpha,\beta}$ and $f_{k,1,\frac{\beta}{\alpha^{2^{-k}+1}}}$,
2. \[item:beta\_frobenius\] $f_{k,1,\beta^{2^i}}$ and $f_{k,1,\beta}$ for $i \in \{0,\dots,m-1\}$,
3. \[item:-k\_k\] $f_{-k,1,\beta}$ and $f_{k,1,\beta}$.
It follows from that all the functions in are APN.
By \[eq:LinEquiv\_1\] and \[eq:LinEquiv\_2\] and the explanations below, two Taniguchi APN functions $f_{k,\alpha,\beta}$ and $f_{\ell,\alpha',\beta'}$ are linearly equivalent if there exist invertible mappings $L,N$ on ${\mathbb{F}}_{2^m}^2$, represented by linearized polynomials $L_A(X,Y), L_B(X,Y) \in {\mathbb{F}_{2^m}}[X,Y]$ and $N_1(X),\dots,N_4(X) \in {\mathbb{F}_{2^m}}[X]$, respectively, such that the two equations $$\begin{aligned}
L_A(x,y)^{2^{2k}(2^k+1)} + \alpha &L_A(x,y)^{2^{2k}} L_B(x,y)^{2^k} +\beta L_B(x,y)^{(2^k+1)} \\ &= N_1(x^{(2^\ell+1)2^{2\ell}} + \alpha' x^{2^{2\ell}} y^{2^\ell} + \beta' y^{2^\ell+1}) + N_3(xy),\\
L_A(x,y)L_B(x,y) &= N_2(x^{(2^\ell+1)2^{2\ell}} + \alpha' x^{2^{2\ell}} y^{2^\ell} + \beta' y^{2^\ell+1}) + N_4(xy)
\end{aligned}$$ hold for all $x,y \in {\mathbb{F}_{2^m}}$. We will give such polynomials for (a)–(c). As we have $N_2(X) = N_3(X) = 0$ in all three cases, we will not restate these polynomials in every case.
1. The functions $f_{k,\alpha,\beta}$ and $f_{k,1,\frac{\beta}{\alpha^{2^{-k}+1}}}$ are linearly equivalent by $$\begin{aligned}
L_A(X,Y) &= X,& L_B(X,Y)&=\tfrac{1}{\alpha^{2^{-k}}}Y,& N_1(X) &=X, &N_4(X) &= \tfrac{1}{\alpha^{2^{-k}}}X.
\end{aligned}$$
2. The functions $f_{k,1,\beta^{2i}}$ and $f_{k,1,\beta}$ are linearly equivalent by $$\begin{aligned}
L_A(X,Y) &= X^{2^i},& L_B(X,Y)&=Y^{2^i},& N_1(X) &= X^{2^i},& N_4(X) &= X^{2^i}.
\end{aligned}$$
3. We first show that $f_{-k,1,\beta}$ and $f_{k,\frac{1}{\beta},\frac{1}{\beta}}$ are equivalent. This can be seen choosing $$\begin{aligned}
L_A(X,Y) &= Y^{2^{3k}},& L_B(X,Y)&=X^{2^{3k}},& N_1(X) &= \beta X,& N_4(X) &= X^{2^{3k}}.
\end{aligned}$$ Using \[item:alpha\_1\], it follows that $f_{k,\frac{1}{\beta},\frac{1}{\beta}}$ is linearly equivalent to $f_{k,1,\beta^{2^{-k}}}$, which, by \[item:beta\_frobenius\], is linearly equivalent to $f_{k,1,\beta}$.
Next, we present our main theorem. We remark that it only holds for $m \ge 3$ as for $m = 2$, all Taniguchi APN functions are CCZ-equivalent to the Gold APN function $x \mapsto x^3$. According to , for $m \ge 3$, every Taniguchi APN function $f_{k,\alpha,\beta}$, where $\alpha \ne 0$, is linearly equivalent to a Taniguchi APN function $f_{\ell,1,\beta'}$, where $0 < \ell < \frac{m}{2}$. Hence, we will only consider functions $f_{k,1,\beta}$ where $0 < k < \frac{m}{2}$ in our theorem. Note that the structure of the proof of is similar to the structure of the proof of by the present authors [@kasperszhou2020]. To keep the paper self-contained we will restate some parts that also appear in [@kasperszhou2020].
\[th:Taniguchi\_equivalence\] Let $m \ge 3$ be an integer, and let $k,\ell$ be integers, $0 < k,\ell < \frac{m}{2}$, coprime to $m$. Let $\beta, \beta' \in {\mathbb{F}_{2^m}}^*$ such that the polynomials $X^{2^k+1} + X + \beta$ and $X^{2^k+1} + X + \beta'$ have no roots in ${\mathbb{F}_{2^m}}$. Two Taniguchi APN functions $f_{k,1,\beta}, f_{\ell,1,\beta'}$ on ${\mathbb{F}_{2^{2m}}}$, where $$f_{k,1,\beta} = (x^{2^{2k}(2^k+1)} + x^{2^{2k}}y^{2^k} + \beta y^{2^k+1},\ xy)$$ and $$f_{\ell,1,\beta'} = (x^{2^{2\ell}(2^\ell+1)} + x^{2^{2\ell}}y^{2^\ell} + \beta' y^{2^\ell+1},\ xy),$$ are CCZ-equivalent if and only if $k = \ell$ and $\beta' = \beta^{2^i}$ for some $i \in \{0,\dots,m-1\}$.
We have shown in that $f_{k,1,\beta}$ and $f_{k,1,\beta^{2^i}}$ are linearly equivalent and thereby CCZ-equivalent. We will now show the converse: if $f_{k,1,\beta}$ and $f_{\ell,1,\beta'}$ are CCZ-equivalent, then $k=\ell$ and $\beta' = \beta^{2^i}$ for some $i \in \{0,\dots,m-1\}$.
For $m = 3$ and $m = 4$, the result can be easily confirmed. If $m=3$, then $k=1$ and, according to , there are three distinct $\beta \in {\mathbb{F}}_{2^3}^*$ such that $X^3+X+\beta$ has no root in ${\mathbb{F}}_{2^3}$. Clearly, if $\beta$ meets this condition, then $\beta^2$ and $\beta^4$ do as well. Consequently, for $m = 4$, all three Taniguchi APN functions belong to the same equivalence class. If $m=4$, then $k=1$ and there are five distinct $\beta \in {\mathbb{F}}_{2^4}^*$ such that $X^3+X+\beta$ has no root, namely $1$ and $\beta, \beta^2, \beta^4, \beta^8$ for some $\beta \ne 1$. Hence, for $m=4$, there exist two equivalence classes: $f_{1,1,1}$ of size $1$ and $f_{1,1,\beta}$, where $\beta \ne 1$, of size $4$. The existence of these two classes was also observed by @taniguchi2019 who computed the $\Gamma$-ranks for these functions.
For the remainder of the proof, let $m\ge 5$. Assume $f_{k,1,\beta}$ and $f_{\ell,1,\beta'}$ are CCZ-equivalent. By , this implies that the functions are also EA-equivalent. Hence, analogously to the proof of , there exist linearized polynomials $L_A(X,Y), L_B(X,Y), M_A(X,Y), M_B(X,Y) \in {\mathbb{F}_{2^m}}[X,Y]$ and $N_1(X), \dots, N_4(X) \in {\mathbb{F}_{2^m}}[X]$, where $$L(X,Y) = (L_A(X,Y), L_B(X,Y))$$ and $$N(X,Y) = (N_1(X) + N_3(Y),\ N_2(X) + N_4(Y))$$ are invertible, such that the equations $$\begin{aligned}
\begin{split}
\label{eq:equiv1}
L_A(x,y)^{2^{2k}(2^k+1)} \,+ &\, L_A(x,y)^{2^{2k}} L_B(x,y)^{2^k} +\beta L_B(x,y)^{2^k+1} \\
&= N_1(x^{(2^\ell+1)2^{2\ell}} + x^{2^{2\ell}} y^{2^\ell} + \beta' y^{2^\ell+1}) + N_3(xy) + M_A(x,y),
\end{split}\\
\label{eq:equiv2}
L_A(x,y)L_B(x,y) &= N_2(x^{(2^\ell+1)2^{2\ell}} + x^{2^{2\ell}} y^{2^\ell} + \beta' y^{2^\ell+1}) + N_4(xy) + M_B(x,y)
\end{aligned}$$ hold for all $x,y \in {\mathbb{F}_{2^m}}$. We write $L_A(X,Y) = L_1(X) + L_3(Y)$ and $L_B(X,Y) = L_2(X) + L_4(Y)$ for linearized polynomials $L_1(X), \dots, L_4(X) \in {\mathbb{F}}_{2^m}[X]$. Hence, $$L(X,Y) = \left(L_1(X)+L_3(Y),\ L_2(X) + L_4(Y)\right).$$ Write $$\begin{aligned}
L_1(X) = \sum_{i=0}^{m-1} a_i X^{2^i},&& L_2(X) = \sum_{i=0}^{m-1} b_i X^{2^i},&& L_3(Y) = \sum_{i=0}^{m-1} \overline{a}_i Y^{2^i},&& L_4(Y) = \sum_{i=0}^{m-1} \overline{b}_i Y^{2^i}.
\end{aligned}$$ Analogously, define linearized polynomials $M_1(X), \dots, M_4(X) \in {\mathbb{F}}_{2^m}[X]$ such that $$M(X,Y) = (M_1(X) + M_3(Y), M_2(X) + M_4(Y)).$$ For the remainder of the proof, let $x,y \in {\mathbb{F}_{2^m}}$. We first prove the following claim.
#### Claim.
*If $f_{k,1,\beta}$ and $f_{\ell,1,\beta'}$ are EA-equivalent, then $k=\ell$ and each of the linearized polynomials $L_1(X), L_2(X), L_3(Y), L_4(Y)$ is a monomial or zero.*\
We will prove the result for $y=0$ and obtain statements for $L_1(X)$ and $L_2(X)$. Using the same approach with $x=0$, identical statements can be obtained for $L_3(Y)$ and $L_4(Y)$. Let $y=0$. Then it follows from \[eq:equiv1\] and $\cref{eq:equiv2}$ that $$\begin{aligned}
\label{eq:y=0_1}
L_1(x)^{2^{2k}(2^k+1)} + L_1(x)^{2^{2k}} L_2(x)^{2^k} +\beta L_2(x)^{2^k+1} &= N_1(x^{(2^\ell+1)2^{2\ell}}) + M_1(x),\\
\label{eq:y=0_2}
L_1(x)L_2(x) &= N_2(x^{(2^\ell+1)2^{2\ell}}) + M_2(x)
\end{aligned}$$ for all $x \in {\mathbb{F}_{2^m}}$. Write $$\begin{aligned}
N_1(X) = \sum_{i=0}^{m-1} c_i X^{2^i}&& \text{and} &&N_2(X) = \sum_{i=0}^{m-1} d_i X^{2^{i-2\ell}}.
\end{aligned}$$ Note that, for convenience, we shift the summation index of $N_2(X)$.
As $L(X,Y)$ has to be invertible, it is not possible that both $L_1(X)$ and $L_2(X)$ are zero. First, suppose $L_1(X) \ne 0$ and $L_2(X) = 0$. For the case $L_1(X) = 0$ and $L_2(X) \ne 0$, an identical result can be obtained by symmetry. If $L_1(X) \ne 0$ and $L_2(X) = 0$, then \[eq:y=0\_2\] does not provide any information as the left-hand side is zero, and \[eq:y=0\_1\] becomes $$\label{eq:L2=0-Gold}
L_1(x)^{2^{2k}(2^k+1)} = N_1(x^{(2^\ell+1)2^{2\ell}}) + M_1(x).$$
From \[eq:L2=0-Gold\], it follows that the Gold APN functions $x \mapsto x^{2^k+1}$ and $x \mapsto x^{2^\ell+1}$ on ${\mathbb{F}_{2^m}}$ have to be EA-equivalent. It is well known that this implies $k = \ell$. The present authors [@kasperszhou2020 Theorem 4.1] moreover showed that if $m \ge 5$, the equivalence mappings between equivalent Gold APN functions are linearized monomials. In our case, this means the polynomial $L_1(X)$ is a linearized monomial. In summary, we obtain $$\begin{aligned}
\label{eq:L1L2_onezero1}
L_1(X) &= a_u X^{2^u}& \text{and}&& L_2(X) &= 0
\end{aligned}$$ for some $u \in \{0, \dots, m-1\}$ and $a_u \in {\mathbb{F}_{2^m}}^*$. If we consider the case $L_1(X) = 0$ and $L_2(X) \ne 0$, we analogously obtain $$\begin{aligned}
\label{eq:L1L2_onezero2}
L_1(X) &= 0& \text{and}&& L_2(X) &=b_u X^{2^u}
\end{aligned}$$ for some $u \in \{0, \dots, m-1\}$ and $b_u \in {\mathbb{F}_{2^m}}^*$. In both cases, $M_1(X)=M_2(X)=0$.
Now, let both $L_1(X), L_2(X) \ne 0$. Then \[eq:y=0\_2\] becomes $$\label{eq:y=0_L1L2nonzero}
\sum_{i=0}^{m-1}a_ib_i x^{2^{i+1}} + \sum_{\substack{i,j=0,\\j \ne i}}^{m-1}a_ib_j x^{2^i+2^j} = \sum_{i=0}^{m-1}d_ix^{(2^\ell+1)2^i} + M_2(x).$$ Note that the first sum on the left-hand side of \[eq:y=0\_L1L2nonzero\] is linearized. Hence, set $M_2(X) = \sum_{i=0}^{m-1}a_ib_i X^{2^{i+1}}$. We rewrite \[eq:y=0\_L1L2nonzero\] as $$\sum_{0 \le i < j \le m-1} (a_ib_j + a_jb_i)x^{2^i+2^j} = \sum_{i=0}^{m-1}d_ix^{2^i+ 2^{i+\ell}}$$ which implies that the equations $$\begin{aligned}
\label{eq:Coefficients1}
a_i b_{i+\ell} + a_{i+\ell} b_i &= d_i \quad\text{for all } i,\\
\label{eq:Coefficients2}
a_i b_j + a_j b_i &=0 \quad\ \text{for } j \ne i, i\pm \ell,
\end{aligned}$$ where the subscripts are calculated modulo $m$, have to hold. We separate the proof into two cases: first, the case that $d_i = 0$ for all $i = 0, \dots, m-1$, and, second, the case that $d_u \ne 0$ for some $u \in \{0,\dots,m-1\}$.
##### Case 1.
In this case, we show that if $d_i=0$ for all $i = 0, \dots, m-1$, similarly to \[eq:L2=0-Gold\], the problem can be reduced to the equivalence problem of Gold APN functions that has been studied by the present authors [@kasperszhou2020 Theorem 4.1]. Assume $d_i = 0$ for all $i = 0, \dots, m-1$, which means $N_2(X) = 0$. In this case, \[eq:Coefficients1\] and \[eq:Coefficients2\] combine to $$\label{eq:Coefficients3}
a_i b_j + a_j b_i =0 \quad \text{for } j \ne i.$$
As $L_1(X)$ and $L_2(X)$ are both nonzero, each polynomial has at least one nonzero coefficient. Assume $a_u$ and $b_{u'}$ are nonzero, where $u,u' \in \{0, \dots, m-1\}$. If $u = u'$, the corresponding term in \[eq:y=0\_2\], that is $a_u b_u X^{2^u+1}$, is linearized and only contributes to $M_2(X)$. If $u \ne u'$, then, by \[eq:Coefficients3\], $$a_u b_{u'} + a_{u'} b_u = 0.$$ Consequently, $a_{u'}$ and $b_u$ have to be nonzero as well, and $a_u, a_{u'}, b_u, b_{u'}$ have to meet the condition $\frac{a_u}{b_u} = \frac{a_{u'}}{b_{u'}}$. Define $\Delta = \frac{a_u}{b_u}$ and note that $\Delta \ne 0$. It follows that $(a_j, b_j)$ satisfies either $$\begin{aligned}
\label{eq:TypeI_TypeII}
a_j = b_j &= 0 &&\text{or} && \frac{a_j}{b_j} = \Delta
\end{aligned}$$ for all $j = 0, \dots, m-1$. Consequently, $b_j = \delta a_j$, where $\delta = \frac{1}{\Delta}$, for all $j=0, \dots, m-1$, and $L_2(X)$ is a multiple of $L_1(X)$, namely $$\label{eq:L2_multiple_L1}
L_2(X) = \delta L_1(X).$$
We plug $L_1(X)$ and $L_2(X)$ into \[eq:y=0\_1\] and obtain $$\label{eq:PolynomialInL1}
L_1(x)^{2^{2k}(2^k+1)} + \delta^{2^k} L_1(x)^{2^k(2^k+1)} + \beta\delta^{2^k+1} L_1(x)^{2^k+1} = N_1(x^{(2^\ell + 1)2^{2\ell}}) + M_1(x).$$ Define a polynomial $T(X) \in {\mathbb{F}_{2^m}}[X]$ as $$T(X) = X^{2^{2k}} + \delta^{2^k} X^{2^k} + \beta \delta^{2^k+1} X$$ and rewrite the left-hand side of \[eq:PolynomialInL1\] as $$T(L_1(x)^{2^k+1}).$$
We show that $T(X)$ is a permutation polynomial. Since $T(X)$ is linearized, it is sufficient to show that $T(X)$ has no nonzero roots. If $T(X)$ had a nonzero root, it would also be a root of the polynomial $$T'(X) = X^{2^{2k}-1} + \delta^{2^k}X^{2^k-1} + \beta \delta^{2^k+1}.$$ Substitute $X^{2^k-1}$ by $Z$. Note that this substitution is one-to-one since $\gcd(2^k-1,2^m-1) = 2^{\gcd(k,m)} - 1 = 1$. We obtain $$T'(Z) = Z^{2^k+1} + \delta^{2^k} Z + \beta \delta^{2^k+1}.$$ By , the polynomial $T'(Z)$ has no root if and only if $P(X) = X^{2^k+1} + X + \beta$ has no root. This holds by the definition of $\beta$.
Hence, we denote by $T^{-1}(X)$ the inverse of $T(X)$ and rewrite \[eq:PolynomialInL1\] as $$\label{eq:GoldCase}
L_1(x)^{2^k+1} = T^{-1}(N_1(x^{(2^\ell+1)2^{2\ell}})) + T^{-1}(M_1(x)).$$ Since $T^{-1}(X)$ is also linearized, \[eq:GoldCase\] describes the equivalence problem of two Gold APN functions as in the case that exactly one of $L_1(X)$ and $L_2(X)$ is zero. By [@kasperszhou2020 Theorem 4.1], it follows that $L_1(X)$ is a monomial. Because of \[eq:L2\_multiple\_L1\], the polynomials $L_1(X)$ and $L_2(X)$ are monomials of the same degree: $$\begin{aligned}
\label{eq:L1L2_samedegree}
L_1(X) &= a_u X^{2^u}& \text{and}&& L_2(X) &= b_u X^{2^u}.
\end{aligned}$$ Moreover, $M_2(X)=a_u b_u X^{2^{u+1}}$ and $M_1(X)=0$.
##### Case 2.
Consider \[eq:Coefficients1\] and \[eq:Coefficients2\] again and assume $d_u \ne 0$ for some $u \in \{0,\dots,m-1\}$ which means $N_2(X) \ne 0$. We will show that in this case, similarly to Case 1, the polynomials $L_1(X)$ and $L_2(X)$ need to be monomials. In contrast to Case 1, however, now $L_1(X)$ and $L_2(X)$ will have different degrees.
If $d_u \ne 0$, then, by \[eq:Coefficients1\], $a_u$ and $b_u$ cannot be zero at the same time. We will separate the proof of Case 2 into two subcases: first, Case 2.1, where both $a_u$ and $b_u$ are nonzero, and second, Case 2.2, where exactly one of $a_u$ and $b_u$ is nonzero. Both these cases will be separated into several subcases again.
##### Case 2.1.
Assume $a_u \ne 0$ and $b_u \ne 0$. It follows from \[eq:Coefficients2\] that all pairs $(a_j,b_j)$, where $j \ne u, u \pm \ell$, satisfy \[eq:TypeI\_TypeII\]. We will first show that the only possible nonzero coefficients are $a_j, b_j$ for $j = u, u \pm \ell, u \pm 2\ell$.
By way of contradiction, assume there exists $\ell' \ne 0, \pm\ell, \pm 2\ell$ such that $a_{u+\ell'}$ and $b_{u+\ell'}$ are nonzero. By \[eq:TypeI\_TypeII\], this implies $\frac{a_{u+\ell'}}{b_{u+\ell'}} = \Delta$. Since $u+\ell' \pm \ell \ne u \pm \ell$, it follows from \[eq:Coefficients1\] with $i =u + \ell'$ that both $(a_{u+\ell},b_{u+\ell})$ and $(a_{u-\ell},b_{u-\ell})$ also have to satisfy one of the equations in \[eq:TypeI\_TypeII\]. Hence, \[eq:TypeI\_TypeII\] holds for all $j=0, \dots, m-1$ which means that $L_2(X)$ is a multiple of $L_1(X)$. However, now \[eq:y=0\_2\] implies $N_2(X) = 0$. This is a contradiction.
Hence, we assume $a_j=b_j = 0$ for $j \ne u, u \pm \ell, u \pm 2\ell$ for the remainder of Case 2.1. We separate its proof into two subcases, both will lead to contradictions.
##### Case 2.1.1.
Suppose $a_{u \pm 2\ell}=b_{u \pm 2 \ell}=0$. In this case, we obtain only one equation from \[eq:Coefficients1\], namely $$a_{u-\ell}b_{u+\ell} + a_{u+\ell}b_{u-\ell} = 0.$$ Hence, either
1. \[item:Case212\_1\] $a_{u-\ell} = a_{u+\ell} = 0$ or $b_{u-\ell} = b_{u+\ell} = 0$, meaning that one of $L_1(X)$ and $L_2(X)$ is a monomial and the other one has at most three nonzero coefficients, or
2. \[item:Case212\_2\] $a_{u-\ell} = b_{u-\ell} = 0$ or $a_{u+\ell} = b_{u+\ell} = 0$, meaning that both $L_1(X)$ and $L_2(X)$ have at most two nonzero coefficients, or
3. \[item:Case212\_3\] $a_{u \pm \ell},b_{u \pm \ell} \ne 0$ and $\frac{a_{u-\ell}}{b_{u-\ell}} = \frac{a_{u+\ell}}{b_{u+\ell}}$, meaning that both $L_1(X)$ and $L_2(X)$ are trinomials.
We will consider each of these three subcases.
##### Subcase \[item:Case212\_1\].
Assume $b_{u-\ell} = b_{u+\ell} = 0$. The case $a_{u-\ell} = a_{u+\ell} = 0$ follows by symmetry. We consider polynomials $$\begin{aligned}
L_1(X) &= a_{u-\ell}X^{2^{u-\ell}} + a_u X^{2^u} + a_{u+\ell}X^{2^{u+\ell}} & \text{and}&& L_2(X) &= b_u X^{2^u}
\end{aligned}$$ which we plug into the left-hand side of \[eq:y=0\_1\]. We obtain $$\begin{aligned}
\nonumber
L_1(x)^{2^{2k}(2^k+1)} &=
a_{u-\ell}^{2^{2k}(2^k+1)} x^{2^{u-\ell+2k}(2^k+1)} +
a_{u}^{2^{2k}(2^k+1)} x^{2^{u+2k}(2^k+1)}
\\&\nonumber\quad+
a_{u+\ell}^{2^{2k}(2^k+1)} x^{2^{u+\ell+2k}(2^k+1)} +
a_{u-\ell}^{2^{3k}}a_u^{2^{2k}} x^{2^{u+2k}(2^{k-\ell}+1)}
\\&\label{eq:Case212_Subcase1_1}\quad+
a_{u}^{2^{3k}}a_{u+\ell}^{2^{2k}} x^{2^{u+\ell+2k}(2^{k-\ell}+1)} +
a_{u+\ell}^{2^{3k}}a_{u-\ell}^{2^{2k}} x^{2^{u-\ell+2k}(2^{k+2\ell}+1)}
\\&\nonumber\quad+
a_{u-\ell}^{2^{3k}}a_{u+\ell}^{2^{2k}} x^{2^{u+\ell+2k}(2^{k-2\ell}+1)} +
a_{u}^{2^{3k}}a_{u-\ell}^{2^{2k}} x^{2^{u-\ell+2k}(2^{k+\ell}+1)}
\\&\nonumber\quad+
a_{u+\ell}^{2^{3k}}a_{u}^{2^{2k}} x^{2^{u+2k}(2^{k+\ell}+1)}
\end{aligned}$$ and $$\begin{split}
\label{eq:Case212_Subcase1_2}
L_1(x)^{2^{2k}} L_2(x)^{2^k} &=
a_{u-\ell}^{2^{2k}} b_u^{2^k} x^{2^{u+k}(2^{k-\ell}+1)} +
a_{u}^{2^{2k}} b_u^{2^k} x^{2^{u+k}(2^k+1)}
\\&\quad+
a_{u+\ell}^{2^{2k}} b_u^{2^k} x^{2^{u+k}(2^{k+\ell}+1)}
\end{split}$$ and $$\label{eq:Case212_Subcase1_3}
\beta L_2(x)^{2^k+1} = \beta b_u^{2^k+1} x^{2^u(2^k+1)}.$$ Recall that the right-hand side of \[eq:y=0\_1\] is $$\sum_{i=0}^{m-1}c_i x^{2^{i+2\ell}(2^\ell+1)} + M_1(x).$$
We will show that not all of the first three terms of \[eq:Case212\_Subcase1\_1\], that all contain the factor $x^{2^k+1}$, can be canceled simultaneously. First, as $0 < \ell < \frac{m}{2}$, the terms cannot cancel each other. Second, if $\ell = \frac{m}{2}-k$, the exponent of $x$ in the sixth term can be written as $2^{u-\frac{m}{2}+2k}(2^k+1)$, but by the same reasoning as above, the sixth term cannot cancel any of the first three terms. Third, if $m$ is odd and $k < \frac{m}{4}$, it is possible that $\ell = 2k$. In this case, the term in \[eq:Case212\_Subcase1\_3\], the first term of \[eq:Case212\_Subcase1\_2\] and the first term of \[eq:Case212\_Subcase1\_1\] all contain the factor $x^{2^{u}(2^k+1)}$ and could potentially cancel each other, but the second and third term of \[eq:Case212\_Subcase1\_1\] cannot be canceled. Analogously, the third term of \[eq:Case212\_Subcase1\_1\] could be canceled if $m$ is odd and $ \frac{m}{4}<k<\frac{m}{2}$ and $\ell = -2k$ but the first and second term would remain. Fourth, if $\ell = k$, the first and the second term of \[eq:Case212\_Subcase1\_1\] could be canceled by the second term of \[eq:Case212\_Subcase1\_2\] and the seventh term of \[eq:Case212\_Subcase1\_1\], respectively. However, the third term would remain. In summary, for arbitrary $k$ and $\ell$, the third term of \[eq:Case212\_Subcase1\_1\] can never be canceled.
We now compare the left-hand side and the right-hand side of \[eq:y=0\_1\]: The summands on the left-hand side that contain the factor $x^{2^i(2^k+1)}$ can only be represented on the right-hand side, if $k = \ell$. Hence, assume $k=\ell$. Now, the fourth and the fifth summand of \[eq:Case212\_Subcase1\_1\] as well as the first summand of \[eq:Case212\_Subcase1\_2\] become linearized. Consequently, $$M_1(X) = a_{u-k}^{2^{2k}} b_u^{2^k} X^{2^{u+k+1}} + a_{u-k}^{2^{3k}}a_u^{2^{2k}} X^{2^{u+2k+1}} +
a_{u}^{2^{3k}}a_{u+k}^{2^{2k}} X^{2^{u+3k+1}}.$$ Next, consider the eighth and the ninth term of \[eq:Case212\_Subcase1\_1\] where the eighth term can be summarized with the third term of \[eq:Case212\_Subcase1\_2\]: $$\begin{aligned}
a_{u+k}^{2^{3k}}a_{u}^{2^{2k}} x^{2^{u+2k}(2^{2k}+1)},&&
(a_{u}^{2^{3k}}a_{u-k}^{2^{2k}} + a_{u+k}^{2^{2k}} b_u^{2^k}) x^{2^{u+k}(2^{2k}+1)}.
\end{aligned}$$ As $m \ge 5$ and $\gcd(k,m) = 1$, we have $2k \not\equiv \pm k \pmod{m}$. Hence, these terms cannot be represented in the form $c_i x^{2^{i+2k}(2^k+1)}$ on the right-hand side of \[eq:y=0\_1\] which means that their coefficients have to be zero. As $a_u \ne 0$, it follows that $a_{u+k} = 0$ which then implies $a_{u-k} = 0$. Hence, $L_1(X)$ and $L_2(X)$ are monomials of the same degree. As this implies $N_2(X)=0$, it contradicts the assumption of Case 2.
##### Subcase \[item:Case212\_2\].
Assume $a_{u-\ell} = b_{u-\ell} = 0$. The case $a_{u+\ell} = b_{u+\ell} = 0$ follows by symmetry. In our case $$\begin{aligned}
L_1(X) &= a_u X^{2^u} + a_{u+\ell}X^{2^{u+\ell}} & \text{and}&& L_2(X) &= b_u X^{2^u} + b_{u+\ell}X^{2^{u+\ell}}.
\end{aligned}$$ On the left-hand side of \[eq:y=0\_1\], we obtain $$\begin{split}
L_1(x)^{2^{2k}(2^k+1)} &=
a_{u}^{2^{2k}(2^k+1)} x^{2^{u+2k}(2^k+1)} +
a_{u+\ell}^{2^{2k}(2^k+1)} x^{2^{u+\ell+2k}(2^k+1)}
\\&\quad+
a_{u}^{2^{3k}}a_{u+\ell}^{2^{2k}} x^{2^{u+\ell+2k}(2^{k-\ell}+1)} +
a_{u+\ell}^{2^{3k}}a_{u}^{2^{2k}} x^{2^{u+2k}(2^{k+\ell}+1)}
\end{split}$$ and $$\begin{split}
L_1(x)^{2^{2k}} L_2(x)^{2^k} &=
a_{u}^{2^{2k}} b_u^{2^k} x^{2^{u+k}(2^k+1)} +
a_{u+\ell}^{2^{2k}} b_{u+\ell}^{2^k} x^{2^{u+\ell+k}(2^k+1)}
\\&\quad+
a_{u}^{2^{2k}} b_{u+\ell}^{2^k} x^{2^{u+\ell+k}(2^{k-\ell}+1)} +
a_{u+\ell}^{2^{2k}} b_u^{2^k} x^{2^{u+k}(2^{k+\ell}+1)}
\end{split}$$ and $$\begin{split}
\beta L_2(x)^{2^k+1} &=
\beta b_u^{2^k+1} x^{2^u(2^k+1)} +
\beta b_{u+\ell}^{2^k+1} x^{2^{u+\ell}(2^k+1)}
\\&\quad+
\beta b_u^{2^k} b_{u+\ell} x^{2^{u+\ell}(2^{k-\ell}+1)} +
\beta b_{u+\ell}^{2^k} b_{u} x^{2^{u}(2^{k+\ell}+1)}.
\end{split}$$
By similar reasoning as in Subcase \[item:Case212\_1\], not all summands containing the factor $x^{2^k+1}$ can be canceled simultaneously. Consequently, we need $k=\ell$ for these terms to be represented on the right-hand side of \[eq:y=0\_1\]. If $k=\ell$, the following terms, which cannot be canceled, occur on the left-hand side of \[eq:y=0\_1\]: $$\begin{aligned}
a_{u+k}^{2^{3k}}a_{u}^{2^{2k}} x^{2^{u+2k}(2^{2k}+1)},&&
a_{u+k}^{2^{2k}} b_u^{2^k} x^{2^{u+k}(2^{2k}+1)},&&
\beta b_{u+k}^{2^{2k}} b_u x^{2^u(2^{2k}+1)}.
\end{aligned}$$ As they cannot be represented in the form $c_i x^{2^{i+2k}(2^k+1)}$ on the right-hand side of \[eq:y=0\_1\], their coefficients need to be zero. Hence $a_{u+k} = b_{u+k} = 0$, which means $L_1(X)$ and $L_2(X)$ are monomials of the same degree. As in Subcase \[item:Case212\_1\], this is a contradiction.
##### Subcase \[item:Case212\_3\].
Now, $$\begin{aligned}
L_1(X) &= a_{u-\ell}X^{2^{u-\ell}} + a_u X^{2^u} + a_{u+\ell}X^{2^{u+\ell}}\\ \text{and } L_2(X) &= b_{u-\ell}X^{2^{u-\ell}} + b_u X^{2^u} + b_{u+\ell}X^{2^{u+\ell}},
\end{aligned}$$ where all coefficients are nonzero and $\frac{a_{u-\ell}}{b_{u-\ell}} = \frac{a_{u+\ell}}{b_{u+\ell}}$. We plug these polynomials into the left-hand side of \[eq:y=0\_1\]. By similar reasoning as in Subcases (i) and (ii), not all terms containing the factor $x^{2^k+1}$ can be canceled. Hence, $k=\ell$. Now, the left-hand side contains the following two summands that cannot be canceled: $$\begin{aligned}
a_{u+k}^{2^{3k}}a_{u}^{2^{2k}} x^{2^{u+2k}(2^{2k}+1)},&&
\beta b_{u}^{2^{2k}} b_{u-k} x^{2^{u-k}(2^{2k}+1)}.
\end{aligned}$$ As none of them can be represented on the right-hand side of \[eq:y=0\_1\], their coefficients need to be zero, which means that $a_{u+k} = b_{u-k} = 0$. This contradicts our assumption.
##### Case 2.1.2.
Suppose that not all of $a_{u \pm 2 \ell}, b_{u \pm 2 \ell}$ are zero. Recall that all pairs $(a_j, b_j)$ where $j \ne u, u\pm \ell$ have to satisfy \[eq:TypeI\_TypeII\]. We consider the case that $a_{u+2\ell}$ and $b_{u+2\ell}$ are nonzero. One can obtain an almost identical result by symmetry when assuming that $a_{u-2\ell}$ and $b_{u-2\ell}$ are nonzero.
If $a_{u+2\ell}, b_{u+2\ell} \ne 0$, then, by \[eq:TypeI\_TypeII\], $\frac{a_{u+2\ell}}{b_{u+2\ell}} = \Delta$. It follows from \[eq:Coefficients2\] that also $(a_{u-2\ell},b_{u-2\ell})$ and $(a_{u-\ell}, b_{u-\ell})$ have to satisfy \[eq:TypeI\_TypeII\]. However, \[eq:Coefficients2\] does not provide any restriction on the values of $a_{u+\ell}$ and $b_{u+\ell}$. If $(a_{u+\ell}, b_{u+\ell})$ satisfies \[eq:TypeI\_TypeII\], then all $(a_j,b_j)$ do and we know from the beginning of Case 2.1 that this implies $N_2(X) = 0$. As before, this is a contradiction. If $(a_{u+\ell}, b_{u+\ell})$ does not satisfy \[eq:TypeI\_TypeII\], then it follows from \[eq:Coefficients2\] that $a_j=b_j=0$ for $j =u-\ell, u-2\ell$. Hence, $$\begin{aligned}
L_1(X) &= a_u X^{2^u} + a_{u+\ell}X^{2^{u+\ell}} + a_{u+2\ell}X^{2^{u+2\ell}}\\
\text{and } L_2(X) &= b_u X^{2^u} + b_{u+\ell}X^{2^{u+\ell}} + b_{u+2\ell}X^{2^{u+2\ell}}.
\end{aligned}$$
As $\frac{a_u}{b_u} = \frac{a_{u+2\ell}}{b_{u+2\ell}}$, this case is similar to Case 2.1.1, Subcase \[item:Case212\_3\], when we substitute $u$ by $u+\ell$, with the only difference that now, one of the middle coefficients $a_{u+\ell}, b_{u+\ell}$ can be zero. However, the arguments used in the previous case leading to the conclusion $k=\ell$ still hold. If $k = \ell$, the left-hand side of \[eq:y=0\_1\] contains the following terms that cannot be canceled: $$\begin{aligned}
a_{u+2k}^{2^{3k}}a_{u}^{2^{2k}} x^{2^{u+2k}(2^{3k}+1)},&&
a_{u+2k}^{2^{2k}}b_{u}^{2^k} x^{2^{u+k}(2^{3k}+1)},&&
\beta b_{u+2k}^{2^k} b_{u} x^{2^u(2^{3k}+1)}.
\end{aligned}$$ They cannot be represented on the right-hand side of \[eq:y=0\_1\], hence, their coefficients need to be zero. This contradicts our assumption that $a_u, a_{u+2k}, b_u, b_{u+2k}$ are nonzero.
##### Case 2.2.
Assume, exactly one of $a_u$ and $b_u$ is nonzero. We show the case $a_u \ne 0$ and $b_u = 0$. The case $a_u = 0$ and $b_u \ne 0$ can be proved analogously. So, assume $a_u \ne 0$ and $b_u = 0$. From \[eq:Coefficients1\] with $i = u$, we obtain the equation $$a_u b_{u+\ell} = d_u.$$ As $d_u \ne 0$, it follows that $b_{u+\ell} \ne 0$. From \[eq:Coefficients2\] with $i = u$, we obtain $$a_u b_j = 0 \quad \text{for } j \ne u, u\pm \ell.$$ Consequently, $b_j = 0$ for $j \ne u \pm \ell$. Now, it follows from \[eq:Coefficients2\] with $i = u + \ell$ that $$a_j b_{u+\ell} = 0 \quad \text{for } j \ne u-\ell,u, u + \ell, u + 2\ell.$$ Consequently, $a_j = 0$ for $j \ne u-\ell,u, u + \ell, u + 2\ell$. We will separate the proof of Case 2.2 into two subcases: in Case 2.2.1, we consider $b_{u-\ell} \ne 0$, in Case 2.2.2, we suppose $b_{u-\ell} = 0$.
##### Case 2.2.1.
Assume $b_{u-\ell} \ne 0$. From \[eq:Coefficients2\] with $i = u-\ell$ and $j = u+2\ell$, we obtain $$a_{u+2\ell} b_{u-\ell} = 0,$$ which implies $a_{u+2\ell} = 0$, and $$a_{u-\ell}b_{u+\ell} + a_{u+\ell} b_{u-\ell} = 0,$$ which, recalling that $b_{u+\ell}$ is nonzero, implies either $a_{u-\ell} = a_{u+\ell} = 0$ or $a_{u-\ell}, a_{u+\ell} \ne 0$ and $\frac{a_{u-\ell}}{b_{u-\ell}} = \frac{a_{u+\ell}}{b_{u+\ell}}$. We separate these two subcases:
##### Subcase (i).
Assume $a_{u-\ell} = a_{u+\ell} = 0$. Then $$\begin{aligned}
L_1(X) &= a_u X^{2^u} &\text{and} &&L_2(X) &= b_{u-\ell}X^{2^{u-\ell}} + b_{u+\ell}X^{2^{u+\ell}}.
\end{aligned}$$ We plug these polynomials into the left-hand side of \[eq:y=0\_1\] and obtain $$L_1(x)^{2^{2k}(2^k+1)} = a_{u}^{2^{2k}(2^k+1)} x^{2^{u+2k}(2^k+1)}$$ and $$L_1(x)^{2^{2k}} L_2(x)^{2^k} =
a_{u}^{2^{2k}} b_{u-\ell}^{2^k} x^{2^{u-\ell+k}(2^{k+\ell}+1)} +
a_{u}^{2^{2k}} b_{u+\ell}^{2^k} x^{2^{u+\ell+k}(2^{k-\ell}+1)}$$ and $$\begin{split}
\label{eq:Case221_Subcase1_1}
\beta L_2(x)^{2^k+1} &=
\beta b_{u-\ell}^{2^k+1} x^{2^{u-\ell}(2^k+1)} +
\beta b_{u+\ell}^{2^k+1} x^{2^{u+\ell}(2^k+1)}
\\&\quad +
\beta b_{u-\ell}^{2^k} b_{u+\ell} x^{2^{u+\ell}(2^{k-2\ell}+1)} +
\beta b_{u+\ell}^{2^k} b_{u-\ell} x^{2^{u-\ell}(2^{k+2\ell}+1)}.
\end{split}$$ As in previous cases, if $k \ne \ell$, not all terms containing the factor $x^{2^k+1}$ can be canceled simultaneously. Thus, we need $k = \ell$. However, if $k = \ell$, the left-hand side \[eq:y=0\_1\] contains the term $$a_u^{2^{2k}}b_{u-k}^{2^k} x^{2^u(2^{2k}+1)}$$ that cannot be represented in the form $c_i x^{2^{i+2k}(2^k+1)}$ on the right-hand side of \[eq:y=0\_1\]. Hence, its coefficient needs to be zero which contradicts our assumption.
##### Subcase (ii).
Assume $a_{u-\ell}, a_{u+\ell} \ne 0$ and $\frac{a_{u-\ell}}{b_{u-\ell}} = \frac{a_{u+\ell}}{b_{u+\ell}}$. Then $$\begin{aligned}
L_1(X) &= a_{u-\ell} X^{2^{u-\ell}} + a_u X^{2^u} +a_{u+\ell} X^{2^{u+\ell}} &\text{and} &&L_2(X) &= b_{u-\ell}X^{2^{u-\ell}} + b_{u+\ell}X^{2^{u+\ell}}.
\end{aligned}$$ We plug these polynomials into the left-hand side of \[eq:y=0\_1\]. Then $L_1(x)^{2^{2k}(2^k+1)}$ is as in \[eq:Case212\_Subcase1\_1\] and $\beta L_2(x)^{2^k+1}$ is as in \[eq:Case221\_Subcase1\_1\]. Moreover, $$\begin{split}
\label{eq:Case221_Subcase2_1}
L_1(x)^{2^{2k}} L_2(x)^{2^k} &=
a_{u-\ell}^{2^{2k}} b_{u-\ell}^{2^k} x^{2^{u-\ell+k}(2^k+1)} +
a_{u+\ell}^{2^{2k}} b_{u+\ell}^{2^k} x^{2^{u+\ell+k}(2^k+1)}
\\&\quad+
a_{u-\ell}^{2^{2k}} b_{u+\ell}^{2^k} x^{2^{u+\ell+k}(2^{k-2\ell}+1)} +
a_{u}^{2^{2k}} b_{u-\ell}^{2^k} x^{2^{u-\ell+k}(2^{k+\ell}+1)}
\\&\quad+
a_{u}^{2^{2k}} b_{u+\ell}^{2^k} x^{2^{u+\ell+k}(2^{k-\ell}+1)} +
a_{u+\ell}^{2^{2k}} b_{u-\ell}^{2^k} x^{2^{u-\ell+k}(2^{k+2\ell}+1)}.
\end{split}$$ By the same reasoning as in Subcase (i), it follows that $k=\ell$. However, if $k = \ell$, then the fourth term of \[eq:Case221\_Subcase2\_1\] cannot be canceled by any other terms on the left-hand side of \[eq:y=0\_1\], neither can it be represented on the right-hand side of \[eq:y=0\_1\]. This implies $b_{u-\ell} = 0$ which contradicts our assumption.
##### Case 2.2.2.
Assume $b_{u-\ell} = 0$. From \[eq:Coefficients2\] with $i = u+\ell$ and $j = u-\ell$, it follows that $$a_{u-\ell}b_{u+\ell} = 0$$ which, recalling that $b_{u+\ell} \ne 0$, implies $a_{u-\ell} = 0$. Then $$\begin{aligned}
L_1(X) &= a_u X^{2^u} + a_{u+\ell} X^{2^{u+\ell}} + a_{u+2\ell} X^{2^{u+2\ell}} &\text{and} &&L_2(X) &= b_{u+\ell}X^{2^{u+\ell}}.
\end{aligned}$$ Plugging these polynomials into \[eq:y=0\_1\], the expressions $L_1(x)^{2^{2k}(2^k+1)}$, $L_1(x)^{2^{2k}} L_2(x)^{2^k}$ and $\beta L_2(x)^{2^k+1}$ are as in \[eq:Case212\_Subcase1\_1\], \[eq:Case212\_Subcase1\_2\] and \[eq:Case212\_Subcase1\_3\], respectively, where we substitute $u$ by $u+\ell$. By the same reasoning as in Case 2.1.1, Subcase (i), it follows that $k=\ell$. If $k=\ell$, analogously to Case 2.1.1, Subcase (i), the following terms occur on the left-hand side of \[eq:y=0\_1\]: $$\begin{aligned}
a_{u+2k}^{2^{3k}}a_{u}^{2^{2k}} x^{2^{u+2k}(2^{3k}+1)},&&
(a_{u+k}^{2^{3k}}a_{u}^{2^{2k}} + a_{u+2k}^{2^{2k}} b_{u+k}^{2^k}) x^{2^{u+2k}(2^{2k}+1)}.
\end{aligned}$$ As neither of them can be represented on the right-hand side of \[eq:y=0\_1\], their coefficients need to be zero. As $a_u \ne 0$, it follows that $a_{u+2k} = 0$, and, consequently, $a_{u+k}=0$. Hence, $L_1(X)$ and $L_2(X)$ are monomials of the form $$\begin{aligned}
\label{eq:L1L2_differentdegree1}
L_1(X) &= a_u X^{2^u} &\text{and} && L_2(X) = b_{u+k} X^{2^{u+k}},
\end{aligned}$$ and $M_1(X) = a_u^{2^{2k}}b_{u+k}^{2^k} X^{2^{u+2k+1}}$.
Note that if we consider Case 2.2 with $a_u = 0$ and $b_u \ne 0$, we obtain $$\begin{aligned}
\label{eq:L1L2_differentdegree2}
L_1(X) &= a_{u+k} X^{2^{u+k}} &\text{and} && L_2(X) = b_{u} X^{2^{u}}
\end{aligned}$$ and $M_1(X)=a_{u+k}^{2^{2k}} b_u^{2^k} X^{2^{u+2k+1}}$ from Case 2.2.2. This concludes the **proof of our Claim**.\
We summarize the results we have obtained so far. If the Taniguchi APN functions $f_{k,1,\beta}$ and $f_{\ell,1,\beta'}$ are EA-equivalent, then $k = \ell$ and $L_1(X)$ and $L_2(X)$ meet the following conditions: either, one of the polynomials $L_1(X)$ and $L_2(X)$ is zero and the other one is a monomial, see \[eq:L1L2\_onezero1\] and \[eq:L1L2\_onezero2\], or both $L_1(X)$ and $L_2(X)$ are monomials, either of the same degree or of degrees $u$ and $u+k$, see \[eq:L1L2\_samedegree\], \[eq:L1L2\_differentdegree1\] and \[eq:L1L2\_differentdegree2\]. Vice versa, the same statements hold for $L_3(Y)$ and $L_4(Y)$.
It remains to be shown that the EA-equivalence of $f_{k,1,\beta}$ and $f_{k,1,\beta'}$ implies $\beta' = \beta^{2^i}$ for some $i \in \{0,\dots,m-1\}$. Combining the results on $L_1(X),L_2(X),L_3(Y),L_4(Y)$ mentioned above, it is clear that the polynomials $L_A(X,Y)$ and $L_B(X,Y)$ have to be of one of the following forms:
1. \[item:a\] $L_A(X,Y) = a_u X^{2^u} + \overline{a}_w Y^{2^w}$ and $L_B(X,Y) = b_u X^{2^u} + \overline{b}_w Y^{2^w}$,
2. \[item:b\] $L_A(X,Y) = a_u X^{2^u} + \overline{a}_w Y^{2^w}$ and $L_B(X,Y) = b_u X^{2^u} + \overline{b}_{w+k} Y^{2^{w+k}}$,
3. \[item:c\] $L_A(X,Y) = a_u X^{2^u} + \overline{a}_{w+k} Y^{2^{w+k}}$ and $L_B(X,Y) = b_u X^{2^u} + \overline{b}_w Y^{2^w}$,
4. \[item:d\] $L_A(X,Y) = a_u X^{2^u} + \overline{a}_w Y^{2^w}$ and $L_B(X,Y) = b_{u+k} X^{2^{u+k}} + \overline{b}_w Y^{2^w}$,
5. \[item:e\] $L_A(X,Y) = a_{u+k} X^{2^{u+k}} + \overline{a}_w Y^{2^w}$ and $L_B(X,Y) = b_u X^{2^u} + \overline{b}_w Y^{2^w}$,
6. \[item:f\] $L_A(X,Y) = a_u X^{2^u} + \overline{a}_w Y^{2^w}$ and $L_B(X,Y) = b_{u+k} X^{2^{u+k}} + \overline{b}_{w+k} Y^{2^{w+k}}$,
7. \[item:g\] $L_A(X,Y) = a_u X^{2^u} + \overline{a}_{w+k} Y^{2^{w+k}}$ and $L_B(X,Y) = b_{u+k} X^{2^{u+k}} + \overline{b}_w Y^{2^w}$,
8. \[item:h\] $L_A(X,Y) = a_{u+k} X^{2^{u+k}} + \overline{a}_{w} Y^{2^{w}}$ and $L_B(X,Y) = b_{u} X^{2^{u}} + \overline{b}_{w+k} Y^{2^{w+k}}$,
9. \[item:i\] $L_A(X,Y) = a_{u+k} X^{2^{u+k}} + \overline{a}_{w+k} Y^{2^{w+k}}$ and $L_B(X,Y) = b_{u} X^{2^{u}} + \overline{b}_w Y^{2^w}$.
Note that, as $L(X,Y) = (L_A(X,Y), L_B(X,Y))$ has to be a permutation polynomial, it is neither possible that $L_A(X,Y)$ or $L_B(X,Y)$ is zero nor that both $L_A(X,Y)$ and $L_B(X,Y)$ depend only on $X$ or only on $Y$. We will show that all cases listed above lead to the conclusion that $L_A(X,Y)$ and $L_B(X,Y)$ need to be monomials of the same degree of the shape $$\begin{aligned}
\label{eq:LALB_monomials}
L_A(X,Y) &= a_u X^{2^u}& \text{and}&& L_B(X,Y) &= b_u Y^{2^u}.
\end{aligned}$$
We rewrite \[eq:equiv1\] and \[eq:equiv2\] considering $k = \ell$: $$\begin{aligned}
\begin{split}
\label{eq:equiv1_k=l}
L_A(x,y)^{2^{2k}(2^k+1)} + &L_A(x,y)^{2^{2k}} L_B(x,y)^{2^k} +\beta L_B(x,y)^{2^k+1} \\
&= N_1(x^{2^{2k}(2^k+1)} + x^{2^{2k}} y^{2^k} + \beta' y^{2^k+1}) + N_3(xy) + M_A(x,y),
\end{split}\\
\label{eq:equiv2_k=l}
L_A(x,y)L_B(x,y) &= N_2(x^{2^{2k}(2^k+1)} + x^{2^{2k}} y^{2^k} + \beta' y^{2^k+1}) + N_4(xy) + M_B(x,y).
\end{aligned}$$
We will plug all the possible combinations \[item:a\]–\[item:i\] into these equations. We begin with \[item:b\]. By proceeding analogously, the cases \[item:c\]–\[item:e\] lead to the same result. If we plug the polynomials of \[item:b\] into the left-hand side of \[eq:equiv2\_k=l\], we obtain $$\begin{split}
\label{eq:b_equiv2}
L_A(x,y)L_B(x,y) &=
a_u b_u x^{2^{u+1}} + \overline{a}_w \overline{b}_{w+k} y^{2^w(2^k+1)}
\\&\quad+
a_u \overline{b}_{w+k} x^{2^u}y^{2^{w+k}} + \overline{a}_w b_u x^{2^u} y^{2^w}.
\end{split}$$ Note that the first term of \[eq:b\_equiv2\] is linearized. As there is no term containing the factor $x^{2^k+1}$, we need $N_2(X) = 0$ on the right-hand side of \[eq:equiv2\_k=l\]. This implies, first, that the coefficient $\overline{a}_w \overline{b}_{w+k}$ of the second summand of \[eq:b\_equiv2\] has to be zero, and second, that the third and the fourth summand of \[eq:b\_equiv2\] cannot be represented simultaneously on the right-hand side of \[eq:equiv2\_k=l\]. The coefficient of the second summand of \[eq:b\_equiv2\] is zero if $\overline{a}_{w}$ or $\overline{b}_{w+k}$ is zero. We separate the proof into two cases:
**Case 1.** Assume $\overline{a}_w = 0$. Note that this implies $a_u \ne 0$ and $\overline{b}_{w+k} \ne 0$ as otherwise $L(X,Y)$ would not be a permutation polynomial. If $\overline{a}_w = 0$, then \[eq:equiv2\_k=l\] holds only if $u=w+k$. Set $u = w+k$ and plug $L_A(x,y)$ and $L_B(x,y)$ into the left-hand side of \[eq:equiv1\_k=l\]. We obtain $$\label{eq:b_equiv1_1}
L_A(x,y)^{2^{2k}(2^k+1)} =
a_u^{2^{2k}(2^k+1)} x^{2^{u+2k}(2^k+1)}$$ and $$\label{eq:b_equiv1_2}
L_A(x,y)^{2^{2k}} L_B(x,y)^{2^k} =
a_u^{2^{2k}} b_u^{2^k} x^{2^{u+k}(2^k+1)} + a_u^{2^{2k}} \overline{b}_u^{2^k} x^{2^{u+2k}} y^{2^{u+k}}$$ and $$\begin{split}
\label{eq:b_equiv1_3}
\beta L_B(x,y)^{2^k+1} &=
\beta b_u^{2^k+1} x^{2^u(2^k+1)} + \beta \overline{b}_u^{2^k+1} y^{2^u(2^k+1)}
\\&\quad +
\beta b_u^{2^k} \overline{b}_u x^{2^{u+k}} y^{2^u} + \beta \overline{b}_u^{2^k} b_u x^{2^u} y^{2^{u+k}}.
\end{split}$$ The fourth summand of \[eq:b\_equiv1\_3\] cannot be canceled by any other summand of \[eq:b\_equiv1\_1\]–\[eq:b\_equiv1\_3\] and it cannot be represented on the right-hand side of \[eq:equiv1\_k=l\]. As $\beta, \overline{b}_u \ne 0$, it follows that $b_u = 0$. Consequently, $L_A(X,Y)$ and $L_B(X,Y)$ are monomials of the same degree as in \[eq:LALB\_monomials\].
**Case 2.** Assume $\overline{b}_{w+k} = 0$. By the same reasoning as above, this implies $b_u \ne 0$ and $\overline{a}_u \ne 0$. Now, \[eq:equiv2\_k=l\] holds for $u=w$. Set $u = w$ and plug $L_A(x,y)$ and $L_B(x,y)$ into the left-hand side of \[eq:equiv1\_k=l\]. The summand $L_A(x,y)^{2^{2k}} L_B(x,y)^{2^k}$ contains the term $$\overline{a}_u^{2^{2k}} b_u^{2^k} x^{2^{u+k}} y^{2^{u+2k}},$$ that has a nonzero coefficient and cannot be canceled by the other terms on the left-hand side of \[eq:equiv1\_k=l\]. However, it cannot be represented on the right-hand side of \[eq:equiv1\_k=l\]. This is a contradiction.
We next study \[item:f\]. By symmetry, the same result also holds for \[item:i\]. Moreover, an analogous approach gives identical results for \[item:g\] and \[item:h\]. If we plug $L_A(X,Y)$ and $L_B(X,Y)$ of \[item:f\] into \[eq:equiv2\_k=l\], we obtain $$\begin{split}
\label{eq:f_equiv2}
L_A(x,y)L_B(x,y) &=
a_u b_{u+k} x^{2^u(2^k+1)} + \overline{a}_w \overline{b}_{w+k} y^{2^w(2^k+1)}
\\&\quad +
a_u \overline{b}_{w+k} x^{2^u}y^{2^{w+k}} + \overline{a}_w b_{u+k} x^{2^{u+k}} y^{2^w}.
\end{split}$$ If all coefficients are nonzero, we need $u = w+2k$ to represent the first and the second summand of \[eq:f\_equiv2\] on the right-hand side of \[eq:equiv2\_k=l\]. Then, however, the fourth term of \[eq:f\_equiv2\] cannot be represented on the right-hand side of \[eq:equiv2\_k=l\], which is a contradiction.
Now assume one of the coefficients is zero. We show the case $b_{u+k} = 0$. If we assume $a_u = 0$ instead, we end up with the same contradiction as in Case 2 of the study of \[item:b\]. By symmetry, analogous results can be obtained when assuming $\overline{a}_w = 0$ or $\overline{b}_{w+k} = 0$. If $b_{u+k} = 0$, it follows that $a_{u}$ and $\overline{b}_{w+k}$ are nonzero as otherwise $L(X,Y)$ would not be a permutation polynomial. Moreover, as the first term of \[eq:f\_equiv2\] vanishes, we need $N_2(X)=0$. Then, also the second term of \[eq:f\_equiv2\] cannot be represented on the right-hand side of \[eq:equiv2\_k=l\] and $\overline{a}_w \overline{b}_{w+k}$ has to be zero. As $\overline{b}_{w+k} \ne 0$, we need $\overline{a}_{w} = 0$ for the second coefficient to be zero. Moreover, we need $u=w+k$ to represent the third summand of \[eq:f\_equiv2\] on the right-hand side of \[eq:equiv2\_k=l\]. Consequently, $L_A(X,Y)$ and $L_B(X,Y)$ are monomials as in \[eq:LALB\_monomials\].
Finally, we study \[item:a\]. If we plug $L_A(X,Y)$ and $L_B(X,Y)$ of \[item:a\] into \[eq:equiv2\_k=l\], we obtain $$\label{eq:a_equiv2}
L_A(x,y)L_B(x,y) =
a_u b_{u} x^{2^{u+1}} + \overline{a}_{w} \overline{b}_{w} y^{2^{w+1}} +
(a_u \overline{b}_{w} + \overline{a}_w b_u) x^{2^u}y^{2^w}.$$ We separate two cases: in the first case, the third term of \[eq:a\_equiv2\] vanishes, in the second case, its coefficient is nonzero.
**Case 1.** We first show, that the third term of \[eq:a\_equiv1\_2\] can only vanish if all coefficients are nonzero. Suppose $a_u = 0$. Then $\overline{a}_w b_u$ has to be zero as well. However, this is not possible, as $a_u = 0$ implies that $\overline{a}_w$ and $b_u$ are nonzero. By symmetry, the same result is obtained if we assume that any other coefficient is zero.
Consequently, assume all coefficients are nonzero and $\frac{a_u}{b_u} = \frac{\overline{a}_w}{\overline{b}_w}$. Then \[eq:a\_equiv2\] does not provide any information, as the left-hand side is a linearized polynomial. We plug $L_A(X,Y)$ and $L_B(X,Y)$ into the left-hand side of \[eq:equiv1\_k=l\] and obtain $$\begin{split}
\label{eq:a_equiv1_1}
L_A(x,y)^{2^{2k}(2^k+1)} &=
a_u^{2^{2k}(2^k+1)} x^{2^{u+2k}(2^k+1)} +
\overline{a}_{w}^{2^{2k}(2^k+1)} y^{2^{w+2k}(2^k+1)}
\\&\quad +
a_u^{2^{3k}} \overline{a}_w^{2^{2k}} x^{2^{u+3k}} y^{2^{w+2k}} +
\overline{a}_w^{2^{3k}} a_u^{2^{2k}} x^{2^{u+2k}} y^{2^{w+3k}}
\end{split}$$ and $$\begin{split}
\label{eq:a_equiv1_2}
L_A(x,y)^{2^{2k}} L_B(x,y)^{2^k} &=
a_u^{2^{2k}} b_u^{2^k} x^{2^{u+k}(2^k+1)} +
\overline{a}_w^{2^{2k}} \overline{b}_w^{2^k} y^{2^{w+k}(2^k+1)}
\\& \quad +
a_u^{2^{2k}} \overline{b}_w^{2^k} x^{2^{u+2k}} y^{2^{w+k}} +
\overline{a}_w^{2^{2k}} b_u^{2^k} x^{2^{u+k}} y^{2^{w+2k}}
\end{split}$$ and $$\begin{split}
\label{eq:a_equiv1_3}
\beta L_B(x,y)^{2^k+1} &=
\beta b_u^{2^k+1} x^{2^u(2^k+1)} + \beta \overline{b}_w^{2^k+1} y^{2^w(2^k+1)} +
\\&\quad +
\beta b_u^{2^k} \overline{b}_w x^{2^{u+k}} y^{2^w} + \beta \overline{b}_w^{2^k} b_u x^{2^u} y^{2^{w+k}}.
\end{split}$$ No matter how we choose $u$ and $w$, the third and the fourth summand of \[eq:a\_equiv1\_1\] cannot be canceled by the terms of \[eq:a\_equiv1\_1\]–\[eq:a\_equiv1\_3\] and they cannot be represented simultaneously on the right-hand side of \[eq:equiv1\_k=l\]. Hence, at least one of the coefficients needs be zero which is a contradiction.
**Case 2.** Assume $a_u \overline{b}_{w} + \overline{a}_w b_u \ne 0$. As there are no terms on the left-hand side of \[eq:equiv2\_k=l\] containing the factors $x^{2^k+1}$ and $y^{2^k+1}$, it follows that $N_2(X)=0$, and we need $u=w$ to represent the third summand of \[eq:a\_equiv2\] on the right-hand side of \[eq:equiv2\_k=l\]. We plug $L_A(X,Y)$ and $L_B(X,Y)$ into \[eq:equiv1\_k=l\] and obtain the same expressions as in \[eq:a\_equiv1\_1\]–\[eq:a\_equiv1\_3\] with $u=w$. Analogously to Case 1, the third and the fourth term of \[eq:a\_equiv1\_1\] cannot be represented on the right-hand side of \[eq:equiv1\_k=l\] at the same time. Hence, $a_u\overline{a}_w$ has to be zero. Assuming $\overline{a}_w = 0$, we obtain, by similar reasoning as in the previous cases, that $L_A(X,Y)$ and $L_B(X,Y)$ have to be monomials of the same degree as in \[eq:LALB\_monomials\]. Assuming $a_u = 0$, we obtain the same contradiction as in the study of \[item:b\], Case 2.
In summary, the only possible choice of $L_A(X,Y)$ and $L_B(X,Y)$ that can satisfy \[eq:equiv1\_k=l\] and \[eq:equiv2\_k=l\] is $L_A(X,Y) = a_u X^{2^u}$ and $L_B(x,y)=\overline{b}_u Y^{2^u}$. If we plug these monomials into \[eq:equiv1\_k=l\], we obtain $$\begin{split}
\label{eq:monomials_in_equiv1}
&a_u^{2^{2k}(2^k+1)} x^{2^{u+2k}(2^k+1)} + a_u^{2^{2k}} \overline{b}_u^{2^k} x^{2^{u+2k}} y^{2^{u+k}} + \beta \overline{b}_u^{2^k+1} y^{2^u(2^k+1)}
\\&\qquad = N_1(x^{2^{2k}(2^k+1)} + x^{2^{2k}} y^{2^k} + \beta' y^{(2^k+1)}) + N_3(xy) + M_A(x,y).
\end{split}$$ Obviously, $N_3(X) = 0$ and $M_A(X,Y) = 0$ and $N_1(X)$ has to be a monomial of degree $u$, the same degree as $L_A(X,Y)$ and $L_B(X,Y)$. Write $N_1(X) = c_u X^{2^u}$. Then \[eq:monomials\_in\_equiv1\] becomes $$\begin{split}
&a_u^{2^{2k}(2^k+1)} x^{2^{u+2k}(2^k+1)} + a_u^{2^{2k}} \overline{b}_u^{2^k} x^{2^{u+2k}} y^{2^{u+k}} + \beta \overline{b}_u^{2^k+1} y^{2^u(2^k+1)}
\\&\qquad = c_ux^{2^{u+2k}(2^k+1)} + c_ux^{2^{u+2k}} y^{u+2^k} + c_u\beta'^{2^u} y^{2^u(2^k+1)}
\end{split}$$ and the coefficients have to meet the following conditions: $$\begin{aligned}
\label{eq:Coefficients_final}
a_u^{2^{2k}(2^k+1)} &= c_u, & a_u^{2^{2k}} \overline{b}_u^{2^k} &= c_u, & \beta \overline{b}_u^{2^k+1} &= c_u \beta'^{2^u}.
\end{aligned}$$
The first two equations of \[eq:Coefficients\_final\] imply $\overline{b}_u = a_u^{2^{2k}}$ and $c_u = \overline{b}_u^{2^k+1}$. Combining the later result with the third equation of \[eq:Coefficients\_final\], it follows that $\beta = \beta'^{2^u}$.
From the proof of , we can deduce the order of the automorphism group of the Taniguchi APN functions. Note that only holds for $m \ge 4$. For $m=2$, the unique Taniguchi APN function $f_{1,1,1}$ on ${\mathbb{F}}_{2^4}$ is CCZ-equivalent to the Gold APN function $x \mapsto x^3$. Its automorphism group has order $5760$. If $m = 3$, the unique Taniguchi APN function $f_{1,1,\beta}$ on ${\mathbb{F}}_{2^6}$ is CCZ-equivalent to the APN function $x \mapsto x^3 + ux^{24} + x^{10}$, where $u$ is primitive in ${\mathbb{F}}_{2^6}$, that was first given by @browning2009. In this case, $|{\textnormal{Aut}}(f_{1,1,\beta})| = 896$.
\[th:Taniguchi-automorphismgroup\] Let $m \ge 4$, and let $f_{k,\alpha,\beta}$ be a Taniguchi APN function from on ${\mathbb{F}_{2^{2m}}}$. Define $\beta' = \frac{\beta}{\alpha^{2^{-k}+1}}$. Then $$|{\textnormal{Aut}}_L(f_{k,\alpha,\beta})| =
\begin{cases}
3m(2^m-1) &\text{if } \alpha = 0 \text{ and } m = 4, \rule[-1em]{0em}{1em}\\
\frac{3}{2}m(2^m-1) &\text{if } \alpha = 0 \text{ and } m \ge 5,\rule[-1em]{0em}{1em}\\
\dfrac{m(2^m-1)}{\min \{u : \beta'^{2^u} = \beta'\}} &\text{if } \alpha \ne 0
\end{cases}$$ and $$|{\textnormal{Aut}}(f_{k,\alpha,\beta})| =
\begin{cases}
3m2^{2m}(2^m-1) &\text{if } \alpha = 0 \text{ and } m =4,\rule[-1em]{0em}{1em}\\
3m2^{2m-1}(2^m-1) &\text{if } \alpha = 0 \text{ and } m \ge 5,\rule[-1em]{0em}{1em}\\
\dfrac{m 2^{2m}(2^m-1)}{\min \{u : \beta'^{2^u} = \beta'\}} &\text{if } \alpha \ne 0.
\end{cases}
\vspace{.5em}$$
We determine $|{\textnormal{Aut}}_L(f_{k,\alpha,\beta})|$, then $|{\textnormal{Aut}}(f_{k,\alpha,\beta})|$ follows from and . If $\alpha = 0$, according to , a Taniguchi APN function $f_{k,0,\beta}$ is linearly equivalent to the Pott-Zhou APN function $g_{k,2k,\beta}$ whose automorphism group was determined by the present authors [@kasperszhou2020 Theorem 5.2].
If $\alpha \ne 0$, we know from (a) that $f_{k,\alpha,\beta}$ is linearly equivalent to $f_{k,1,\beta'}$. We study the case $\alpha = 1$. For $m=4$ the results can be confirmed computationally with `Magma` [@magma]. Assume $m \ge 5$. Then the proof of holds. We count the number of equivalence mappings that map $f_{k,1,\beta'}$ on itself. Therefore, we consider the conditions given in \[eq:Coefficients\_final\] which the coefficients of the linearized monomials $L_A(X,Y)$, $L_B(X,Y)$ and $N_1(X)$ have to meet. We have shown that \[eq:Coefficients\_final\] implies $$\begin{aligned}
\overline{b}_u &= a_u^{2^{2k}}, &c_u &= \overline{b}_u^{2^k+1},& \text{and}&& \beta'^{2^u-1} = 1.
\end{aligned}$$ The number of $u$ such that $\beta'^{2^u-1} = 1$ is given by $$\frac{m}{\min \{u : \beta'^{2^u} = \beta'\}}.$$ Next, we have $2^m-1$ choices for $a_u$. By choosing $a_u$, the coefficients $\overline{b}_u$ and $\overline{c_u}$ are uniquely determined.
From , we easily deduce the following result about the inequivalence of Taniguchi and Pott-Zhou APN functions. Recall that Pott-Zhou APN functions only exist on ${\mathbb{F}_{2^{2m}}}$ where $m$ is even and that we have already solved the case $\alpha = 0$ in .
\[cor:inequivalence\_Taniguchi\_ZhouPott\] Let $m \ge 4$ be even. Let $f_{k,\alpha,\beta}$, where $\alpha \ne 0$, be a Taniguchi APN function from on ${\mathbb{F}_{2^{2m}}}$, and let $g_{\ell,s,\gamma}$ be a Pott-Zhou APN function from on ${\mathbb{F}_{2^{2m}}}$. Then $f_{k,\alpha,\beta}$ and $g_{\ell,s,\gamma}$ are CCZ-inequivalent.
The order of the automorphism group of a vectorial Boolean function is invariant under CCZ-equivalence. For a Taniguchi APN function $f_{k, \alpha, \beta}$ on ${\mathbb{F}_{2^{2m}}}$, we determined the order of the automorphism group ${\textnormal{Aut}}(f_{k, \alpha, \beta})$ in . For a Pott-Zhou APN function $g_{\ell,s,\gamma}$ on ${\mathbb{F}_{2^{2m}}}$, the present authors [@kasperszhou2020 Theorem 5.2] showed that $$|{\textnormal{Aut}}(g_{\ell,s,\gamma})| =
\begin{cases}
3m2^{2m}(2^m-1) &\text{if } s \in \{0, \frac{m}{2}\},\\
3m2^{2m-1}(2^m-1) &\text{otherwise}.\\
\end{cases}$$ As clearly $\frac{m}{\min \{u : \beta'^{2^u} = \beta'\}} \le m$, it follows that $\frac{m}{\min \{u : \beta'^{2^u} = \beta'\}} < \frac{3}{2}m < 3m$. Hence, the automorphism groups of $f_{k,\alpha,\beta}$ and $g_{\ell,s,\gamma}$ are of different order which implies that the functions are CCZ-inequivalent.
On the total number of CCZ-inequivalent Taniguchi APN functions on ${\mathbb{F}_{2^{2m}}}$ {#sec:Taniguchi_number}
==========================================================================================
The results from allow us now to determine the number of CCZ-inequivalent Taniguchi APN functions on ${\mathbb{F}_{2^{2m}}}$ for any $m$. This will be done in by counting the number of parameters $k$, $\alpha$ and $\beta$ that lead to inequivalent functions. Recall from that every Taniguchi APN function $f_{k,\alpha,\beta}$ where $\alpha \ne 0$ is CCZ-equivalent to a function $f_{k,1,\beta'}$ for some $\beta' \in {\mathbb{F}_{2^m}}^*$. Hence, we only need to consider functions with $\alpha = 0$ or $\alpha = 1$. As we know from that $f_{k,0,\beta}$ is equivalent to a Pott-Zhou APN function, whose equivalence problem was solved by the present authors [@kasperszhou2020], we focus on $\alpha = 1$ first.
Recall from that two Taniguchi APN functions $f_{k,1,\beta}$ and $f_{k,1,\beta'}$ on ${\mathbb{F}_{2^{2m}}}$ are CCZ-equivalent if and only if $\beta' = \beta^{2^i}$ for some $i \in \{0, \dots, m-1\}$. Consequently, to obtain the exact number of $\beta$ providing inequivalent functions for fixed $k$, we need to determine the number of orbits of $\beta$ such that $X^{2^k+1} + X + \beta$ has no root in ${\mathbb{F}_{2^m}}$ under the action of the Galois group ${\textnormal{Gal}}({\mathbb{F}_{2^m}}/ {\mathbb{F}}_2)$. We will do this in with the help of the following series of technical lemmas.
\[lem:3k\] If $k>1$ is an integer with $\gcd(k,3) = 1$, then $3k$ does not divide $2^{k}+1$.
Assume, by way of contradiction, that $3k \mid 2^{k}+1$. By the Chinese Remainder Theorem, $2^{k} \equiv -1 \pmod{3}$ which means that $k$ is odd.
Let $k=p_1^{t_1}\cdots p_s^{t_s}$, where $p_1, \dots, p_s$ are prime numbers such that $3<p_1<p_2<\cdots<p_s$ and $t_i\geq 1$ for $i = 1, \dots, s$. For convenience, we set $p=p_1$ and $t=t_1$ in the remainder of this proof.
By the Chinese Remainder Theorem, it also follows that $2^k\equiv -1 \pmod{p^t}$. Denote by $\varphi(x)$ the Euler’s totient function of $x$. Since $2^{2^k} \equiv 1 \pmod{p^t}$ and the unit group of the integer ring ${\mathbb{Z}}_{p^t}$ has order $\varphi(p^t)$, it follows that $\mathrm{ord}_{p^{t}}(2) \mid \gcd(2k, \varphi(p^{t}))$. Note that $\varphi(p^t) = (p-1)p^{t-1}$. As $p-1 < p_i$ for all $i \in \{1,\dots, s\}$, the number $p-1$ is not divisible by any of the $p_i$. Recalling that $k = p^{t} p_2^{t_2}\cdots p_s^{t_s}$, it follows that $\gcd(2k,\varphi(p^t))=2p^{t-1}$. Consequently, $2^{2p^{t-1}}-1\equiv 0 \pmod{p^t}$. Thus $2^{2p^{t-1}}-1\equiv 4^{p^{t-1}}-1\equiv 0 \pmod{p}$. As $4^{p}=4\pmod{p}$, we obtain $4-1\equiv 0\pmod{p}$ which means $p=3$. This is a contradiction to the assumption $3 < p$.
\[lem:zeros\_3\] Suppose that $k$ and $m$ are positive integers satisfying $\gcd(k,m)=1$. Write $m=rp$ for an integer $r$ and a prime $p$. For $\beta\in {\mathbb{F}}_{2^r}$, suppose that the polynomial $P(X)=X^{2^k+1}+X+\beta$ has no root in ${\mathbb{F}}_{2^r}$.
1. \[item:zeros\_a\] If $p\neq 3$, then $P(X)$ has no root in ${\mathbb{F}_{2^m}}$.
2. \[item:zeros\_b\] If $p= 3$, then $P(X)$ has exactly three roots in ${\mathbb{F}_{2^m}}$.
Set $\sigma(x)=x^{2^{r}}$ for $x$ in any extension of ${\mathbb{F}}_{2^r}$.
We show \[item:zeros\_a\] first. Suppose that $P(X)$ has at least one root $x_0 \in {\mathbb{F}_{2^m}}$. Then $x_0, \sigma(x_0), \dots, \sigma^{p-1}(x_0)$ have to be $p$ distinct roots of $P(X)$ in ${\mathbb{F}_{2^m}}$ because $\sigma(P(x_0))= \sigma(x_0)^{2^k+1}+\sigma(x_0)+\beta=0$ and $p$ is prime. @hellesethkholosha2008 [Theorem 1] showed that if $P(X)$ has more than one root, then $P(X)$ has exactly three roots in ${\mathbb{F}_{2^m}}$ which contradicts the assumption that $p\neq3$.
We next prove \[item:zeros\_b\]. Now $m=3r$. If $P(X)$ has at least one root in ${\mathbb{F}_{2^m}}$, by the proof of \[item:zeros\_a\], it has exactly three roots in ${\mathbb{F}_{2^m}}$ and we are done. Assume, by way of contradiction, that $P(X)$ has no root in ${\mathbb{F}_{2^m}}$. First, if $k=1$, then $P(X)$ has degree $3$ and is irreducible over ${\mathbb{F}}_{2^r}$. Therefore, $P(X)$ splits over ${\mathbb{F}_{2^m}}$ which contradicts our assumption.
From now on, assume $k>1$. Write $P(X) = P_1(X)P_2(X) \cdots P_s(X)$ for irreducible polynomials $P_1(X), \dots, P_s(X) \in {\mathbb{F}_{2^m}}$. Since $\deg(P(X)) = 2^k+1$ is odd, there exists a polynomial $P_j(X)$, where $j \in \{1,\dots, s\}$, of odd degree. Denote by $J_{\textnormal{odd}}$ the set of all $j \in \{1,\dots,s\}$ such that $\deg(P_j(X))$ is odd, and let $j^* \in J_{\textnormal{odd}}$ such that $\deg(P_{j^*}(X)) \le \deg(P_j(X))$ for all $j \in J_{\textnormal{odd}}$. Set $\ell = \deg(P_{j^*}(X))$ and note that $\ell > 1$ and $\ell$ is odd. Then $P_{j^*}(X)$ splits over ${\mathbb{F}}_{2^{m\ell}}$, which is an extension of ${\mathbb{F}_{2^m}}$ with $[{\mathbb{F}}_{2^{m\ell}}:{\mathbb{F}_{2^m}}]=\ell$. Consequently, $P(X)$ has a root in ${\mathbb{F}}_{2^{m\ell}}$, and there is no root of $P(X)$ in any proper subfield of ${\mathbb{F}}_{2^{m\ell}}$ containing ${\mathbb{F}_{2^m}}$.
Define $h = \gcd(m\ell, k)$. As $m$ and $k$ are coprime, this implies $h = \gcd(\ell, k)$ and, in particular, $h \mid \ell$. Then ${\mathbb{F}}_{2^h}={\mathbb{F}}_{2^{m\ell}}\cap {\mathbb{F}}_{2^k}$. As $\ell$ is odd, according to @bluher2004 [Theorem 5.6], $P(X)$ has exactly $2^{h}+1$ roots in ${\mathbb{F}}_{2^{m\ell}}$. If $h = 1$, then the roots of $P(X)$ in ${\mathbb{F}}_{2^{m\ell}}$ are also elements of ${\mathbb{F}_{2^m}}$ as $m=3r$. This contradicts our assumption. Hence, assume $h > 1$. We may regard $\sigma$ as an element in ${\textnormal{Gal}}({\mathbb{F}}_{2^{m\ell}}/{\mathbb{F}}_{2^r})$. If $3\nmid \ell$, then it is clear that $x_0$, $\sigma(x_0)$, $\dots, \sigma^{3\ell}(x_0)$ are pairwise distinct for any root $x_0$ of $P(X)$ in ${\mathbb{F}}_{2^{m\ell}}$. If $3\mid \ell$, then $x_0$, $\sigma(x_0)$, $\dots, \sigma^{3\ell}(x_0)$ are still pairwise distinct for any root $x_0$ of $P(X)$ in ${\mathbb{F}}_{2^{m\ell}}$. The reason is as follows. Suppose that $\sigma^{j}(x_0)=x_0$ for some $j<3\ell$ with $j\mid 3\ell$. This means $[{\mathbb{F}}_{2^r}(x_0): {\mathbb{F}}_{2^r}]=j$. Thus, $$[{\mathbb{F}_{2^m}}(x_0):{\mathbb{F}_{2^m}}] =
\begin{cases}
j &\text{if } 3\nmid j,\\
j/3 &\text{if } 3\mid j.
\end{cases}$$ For the first case, $3\nmid j$, as ${\mathbb{F}}_{2^{m\ell}}={\mathbb{F}_{2^m}}(x_0)$ by definition, we get $j=\ell$ which is a contradiction to the assumption that $3\mid \ell$. For the second case, $3\mid j$, we get $\ell = [{\mathbb{F}}_{2^{m\ell}}:{\mathbb{F}_{2^m}}] = [{\mathbb{F}_{2^m}}(x_0):{\mathbb{F}_{2^m}}] = j/3$ which contradicts the assumption $j<3\ell$.
Therefore, $3\ell$ divides $2^h+1$, in particular, as $h \mid \ell$, we obtain $3h \mid 2^h+1$. By , this is only possible if $\gcd(h,3) > 1$ which implies $\gcd(m,k) > 1$. This is a contradiction.
For any two relatively prime positive integers $k$ and $m$, define $$\label{eq:Phi}
\Phi(m) = \{\beta\in {\mathbb{F}}_{2^m}: X^{2^k+1}+X+\beta \text{ has no roots in }{\mathbb{F}}_{2^m}\}$$ and $$M(m) =|\Phi(m)|$$ and $$\label{eq:N(m)_definition}
N(m) = \left|\{\beta\in \Phi(m): \beta\notin{\mathbb{F}}_{2^{m'}} \text{ with } m'< m \text{ and } m'\mid m\}\right|.$$ According to , $$\label{eq:M(m)}
M(m)=\frac{2^m+(-1)^{m+1}}{3}.$$ In the following , we determine the exact value of $N(m)$.
\[lem:N(m)\] Suppose that $m=3^{n_0}\prod_{i=1}^t p_i^{n_i}$ where $n_0$ is a non-negative integer, $p_1, \dots, p_t$ are distinct prime numbers, and $n_1, \dots, n_t$ are positive integers. If $t = 0$, that means $m = 3^{n_0}$ and, in particular, includes the case $m=1$, then $$N(m)=\frac{2^{m}+1}{3}.$$ If $t \ge 1$, then $$\begin{split}
N(m)& = \frac{1}{3} \Bigg( 2^m -\sum_{i=1}^t 2^{\frac{m}{p_i}} + \sum_{\substack{i,j=1,\\j \ne i}}^t 2^{\frac{m}{p_i p_j}}- \dots\\
\label{eq:N(m)}
&\quad \dots + (-1)^\ell \sum_{\substack{i_1,\dots,i_\ell = 1\\\text{pairwise distinct}}}^t 2^{\frac{m}{p_{i_1} \cdots p_{i_\ell}}}+ \dots + (-1)^t2^{\frac{m}{p_1 p_2 \cdots p_t}}-\varepsilon\Bigg),
\end{split}$$ where $$\varepsilon=
\begin{cases}
2 & \text{if } t=1
\text{ and } m \equiv 2 \pmod{4},\\
0 &\text{otherwise}.
\end{cases}$$
By definition, to determine $N(m)$, we have to exclude each element in $\Phi(m)\cap {\mathbb{F}}_{2^{m'}}$ from $\Phi(m)$ for every proper subfield ${\mathbb{F}}_{2^{m'}}$ of ${\mathbb{F}}_{2^m}$. We first consider the case $t=0$: If $p_0 = 1$, which means $m=1$, then $X^{2^k+1} + X + \beta$ has no root in ${\mathbb{F}}_2$ if and only if $\beta = 1$. Hence, $N(1) = 1$. If $p_0 \ge 1$, by , $$\Phi(m)\cap {\mathbb{F}}_{2^{m'}}=
\begin{cases}
\emptyset & \text{if } 3m' \mid m,\\
\Phi(m') & \text{if } 3m' \nmid m.
\end{cases}$$ Hence, we get $N(3^{p_0})=M(3^{p_0})$ and, by \[eq:M(m)\], $M(3^{p_0}) = \frac{2^m+1}{3}$. From now on, assume $t \ge 1$. Then, by the inclusion-exclusion principle, $$\begin{split}
\label{eq:N(m)_ex}
N(m)& =M(m)-\sum_{i=1}^t M\left(\frac{m}{p_i}\right) + \sum_{\substack{i,j=1,\\j \ne i}}^t M\left(\frac{m}{p_ip_j}\right) - \cdots\\
&\quad \dots +(-1)^\ell\sum_{\substack{i_1,\cdots,i_\ell = 1 \\ \text{pairwise distinct}}}^t M\left(\frac{m}{p_{i_1}\cdots p_{i_\ell}}\right) + \cdots + (-1)^t M\left(\frac{m}{p_1\cdots p_t}\right).
\end{split}$$
If $m$ is odd, then $m'$ is odd for all $m' \mid m$. If $4 \mid m$, then $m'$ is even for all $m' = \frac{m}{p_{i_1} \cdots p_{i_\ell}}$ that occur in \[eq:N(m)\_ex\]. Consequently, in these two cases, by , we have $M(m')=\frac{2^{m'}+(-1)^{m+1}}{3}$ for any $m' = \frac{m}{p_{i_1} \cdots p_{i_\ell}}$ occuring in . Plugging $M(m')$ into , we obtain $$\begin{split}
\label{eq:N1}
N(m)=&\frac{1}{3}\Bigg( 2^m - \sum_{i=1}^t 2^{\frac{m}{p_i}} + \sum_{\substack{i,j=1,\\j \ne i}}^t 2^{\frac{m}{p_i p_j}} - \cdots + (-1)^t 2^{\frac{m}{p_1p_2\cdots p_t}} \Bigg)\\
&\quad + \frac{(-1)^{m+1}}{3}\left(1-\binom{t}{1}+\binom{t}{2}-\cdots+(-1)^t \right).
\end{split}$$ Note that the last sum of \[eq:N1\] equals zero which can be seen by using the binomial identity $$(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$$ with $x=1$ and $y=-1$ (or vice versa).
If $m \equiv 2 \pmod{4}$, we set $p_1=2$ and $n_1=1$. By \[eq:M(m)\], $$\label{eq:M(m')}
M(m') =
\begin{cases}
\frac{2^{m'}+1}{3} &\text{if } m' = \frac{m}{2p_{i_2} \cdots p_{i_\ell}},\\
\frac{2^{m'}-1}{3} &\text{if } m' = \frac{m}{p_{i_1} \cdots p_{i_\ell}} \text{ and } i_1, \dots, i_\ell \ne 1.
\end{cases}$$ Plugging \[eq:M(m’)\] into \[eq:N(m)\_ex\], we obtain $$\begin{split}
\label{eq:N(m)_m2mod4}
N(m)=&\frac{1}{3}\Bigg( 2^m - \sum_{i=1}^t 2^{\frac{m}{p_i}} + \sum_{\substack{i,j=1,\\j \ne i}}^t 2^{\frac{m}{p_i p_j}} - \cdots + (-1)^t 2^{\frac{m}{p_1p_2\cdots p_t}} \Bigg)\\
&\quad + \frac{1}{3} \sum_{i=0}^t (-1)^i \left(\binom{t-1}{i-1} - \binom{t-1}{i}\right).
\end{split}$$
We show where the last sum of \[eq:N(m)\_m2mod4\] is coming from and which values it can take. If $t=1$, then $m=3^{n_0} \cdot 2$. Note that $m$ is even and $\frac{m}{2}$ is odd. Hence, in this case, $N(m) = M(m) - M(\frac{m}{2}) = 2^m - 2^\frac{m}{2} -2$, and the last sum of \[eq:N(m)\_m2mod4\] equals $-2$. Now assume $t > 1$. Consider the sum $$\label{eq:sum}
\sum_{\substack{i_1,\cdots,i_\ell = 1 \\ \text{pairwise distinct}}}^t M\left(\frac{m}{p_{i_1}\cdots p_{i_\ell}}\right)$$ from \[eq:N(m)\] for some $\ell \in \{1, \dots, t\}$. This sum consists of $\binom{t}{\ell}$ terms. Assume $p_{i_1} < p_{i_2} < \dots < p_{i_\ell}$. If $i_1 = 1$, which means $p_{i_1} = 2$, then $\frac{m}{2 p_{i_2}\cdots p_{i_\ell}}$ is odd. In this case, we have $\binom{t-1}{\ell-1}$ possibilities to choose $p_{i_2}, \dots, p_{i_\ell}$. On the contrary, if $i_1 \ne 1$, then $\frac{m}{p_{i_1}\cdots p_{i_\ell}}$ is even, and we have $\binom{t-1}{\ell}$ possibilities to choose $p_{i_1}, \dots, p_{i_\ell}$. Combining these results with \[eq:M(m’)\], we have $\binom{t-1}{\ell-1}$ terms of the form $(2^{m'} + 1)$ and $\binom{t-1}{\ell}$ terms of the form $(2^{m'} - 1)$ in the sum from \[eq:sum\]. Note that, by similar reasoning as in the case $m$ odd or $4 \mid m$, this sum is zero if $t > 1$.
Consider $\Phi(m)$ as in \[eq:Phi\]. We have shown in that if $X^{2^k+1} + X + \beta$ has no root in ${\mathbb{F}_{2^m}}$, then neither has $X^{2^k+1} + X + \beta^{2^i}$ for all $i \in \{0,\dots,m-1\}$. Consequently, $\Phi(m)$ decomposes into orbits of $\beta \in {\mathbb{F}_{2^m}}^*$ under the action of the Galois group ${\textnormal{Gal}}({\mathbb{F}_{2^m}}/ {\mathbb{F}}_2)$. In , we count this number of orbits.
\[prop:beta\_number\] Let $\Phi(m)$ as in \[eq:Phi\], and define $$B(m) = \left\{ \{ \beta^{2^i} : i \in \{0,\dots,m-1\} \} : \beta \in \Phi(m)\right\}$$ as the set of orbits of $\beta \in {\mathbb{F}_{2^m}}^*$ such that $X^{2^k+1} + X + \beta$ has no root in ${\mathbb{F}_{2^m}}$ under the action of the Galois group ${\textnormal{Gal}}({\mathbb{F}_{2^m}}/ {\mathbb{F}}_2)$. Moreover, define $b(m) = |B(m)|$. Then $$b(m) = \sum_{m' \mid m,\ 3 \nmid \frac{m}{m'}} \frac{N(m')}{m'},$$ where $N(m')$ is defined as in \[eq:N(m)\_definition\] and can be calculated as in .
For any subfield ${\mathbb{F}}_{2^{m'}}$ of ${\mathbb{F}_{2^m}}$, we count the number of orbits of $\beta \in \Phi(m) \cap {\mathbb{F}}_{2^{m'} }^*$ under the action of ${\textnormal{Gal}}({\mathbb{F}}_{2^{m'}}/{\mathbb{F}}_2)$ that have full length $m'$. This number is given by $\frac{N(m')}{m'}$. It follows from that we only need to consider the orbits in ${\mathbb{F}}_{2^{m'}}$ with $3 \nmid [{\mathbb{F}_{2^m}}: {\mathbb{F}}_{2^{m'}}]$. Adding all these numbers gives $b(m)$.
With the help of , we can eventually determine the number of CCZ-inequivalent Taniguchi APN functions on ${\mathbb{F}_{2^{2m}}}$ in . We give a nice lower bound on this number in .
\[th:numberTaniguchi\] Let $m \ge 3$, and denote by $n(m)$ the number of CCZ-inequivalent Taniguchi APN functions $f_{k,\alpha,\beta}$ from on ${\mathbb{F}_{2^{2m}}}$. Then $$n(m) =
\begin{cases}
\dfrac{\varphi(m)b(m)}{2} &\text{if $m$ is odd},\rule[-1.2em]{0em}{1em}\\
\dfrac{\varphi(m)(b(m)+1)}{2} &\text{if $m$ is even},
\end{cases}$$ where $\varphi$ denotes Euler’s totient function and $b(m)$ is as in .
Let $m \ge 3$. We count the number of CCZ-inequivalent Taniguchi APN functions $f_{k,1,\beta}$ first: According to , for $0 < k,\ell <\frac{m}{2}$ two functions $f_{k,1,\beta}$ and $f_{\ell,1,\beta'}$ are CCZ-equivalent if and only if $k = \ell$ and $\beta = \beta'^{2^i}$ for some $i \in \{0,\dots,m-1\}$. We count the number of pairs $(k,\beta)$ that lead to inequivalent APN functions: As $0 < k < \frac{m}{2}$ and $\gcd(k,m)=1$, we have $\frac{\varphi(m)}{2}$ choices for $k$. The number of admissible $\beta \in {\mathbb{F}_{2^m}}^*$ equals $b(m)$ from .
From , it follows that if $m$ is even, then for every valid choice of $k$ there is additionally exactly one equivalence family of Taniguchi APN functions $f_{k,0,\beta}$. As before, we have $\frac{\varphi(m)}{2}$ choices for $k$.
Note that shows that the number of APN functions on ${\mathbb{F}_{2^{2m}}}$ increases exponentially in $m$.
\[cor:bound\] Let $m \ge 3$, and define $n(m)$ as the number of CCZ-inequivalent Taniguchi APN functions from on ${\mathbb{F}_{2^{2m}}}$. Then $$n(m) \ge \frac{\varphi(m)}{2}\left\lceil\frac{2^m+1}{3m}\right\rceil,$$ where $\varphi$ denotes Euler’s totient function.
Define $B(m)$ and $b(m)$ as in . The value of $b(m)$ is minimal if all the orbits in $B(m)$ have full length $m$. By , this implies $$b(m) \ge
\begin{cases}
\left\lceil\frac{2^m-1}{3m}\right\rceil &\text{if $m$ is even} \rule[-1.2em]{0em}{1em},\\
\left\lceil\frac{2^m+1}{3m}\right\rceil &\text{if $m$ is odd},
\end{cases}$$ and it is easy to see that $\left\lceil\frac{2^m-1}{3m}\right\rceil = \left\lceil\frac{2^m+1}{3m}\right\rceil$ for all $m \ge 3$.
In , we list the exact number of CCZ-inequivalent Taniguchi APN functions obtained from for certain values of $m$. Recall that for $m=2$, there is only one unique Taniguchi APN function. We moreover compare these numbers to the lower bound that we have established in . It can be seen that the bound is very close to the actual number of Taniguchi APN functions.
[\*[16]{}[r]{}]{} m &2 &3 &4 &5 &6 &7 &8 &9 &10 &11 &12 &13 &14 &15 &16
------------------------------------------------------------------------
\
\# &1& 1& 3& 6& 5& 21& 26& 57& 74& 315& 234& 1266& 1185& 2916 &5492
------------------------------------------------------------------------
\
bound&1& 1& 2& 6& 4& 21& 22& 57& 70& 315& 228& 1266& 1173& 2916 &5464
------------------------------------------------------------------------
\
[\*[8]{}[r]{}]{} m &17 &18 &19 &20 &25 &50 &100
------------------------------------------------------------------------
\
\# & 20568& 14595& 82791& 69988& 4473950& $\approx 7.5 \cdot 10^{13}$& $\approx 8.5 \cdot 10^{28}$
------------------------------------------------------------------------
\
bound & 20568& 14565& 82791& 69908& 4473930& $\approx 7.5 \cdot 10^{13}$& $\approx 8.5 \cdot 10^{28}$
------------------------------------------------------------------------
\
Conclusion and open questions {#sec:conclusion}
=============================
In the present paper, we establish a new lower bound on the total number of CCZ-inequivalent APN functions on the finite field ${\mathbb{F}_{2^{2m}}}$. We show that the number of APN functions on ${\mathbb{F}_{2^{2m}}}$ grows exponentially in $m$. For even $m$, our result presents a great improvement of the lower bound previously given by the present authors [@kasperszhou2020]. For odd $m$, this is the first such lower bound.
Our result now shifts the focus on the following open problems concerning APN functions:
- Establish a lower bound on the total number of CCZ-inequivalent APN functions on the finite field ${\mathbb{F}}_{2^n}$ with $n$ odd.
- As it is confirmed now that there are very many quadratic APN functions on ${\mathbb{F}_{2^{2m}}}$, the efforts of finding new constructions of APN functions should focus on the search for non-quadratic ones.
- It was shown by @anbar2019 that Taniguchi APN functions have the classical Walsh spectrum. It would be interesting to find more APN functions with non-classical Walsh spectra.
[^1]: Institute for Algebra and Geometry, Otto von Guericke University Magdeburg, 39106 Magdeburg, Germany (email: [](mailto:christian.kaspers@ovgu.de))
[^2]: Department of Mathematics, National University of Defense Technology, 410073 Changsha, China (email: [](mailto:yue.zhou.ovgu@gmail.com))
|
---
abstract: |
In this paper, we discuss the development of a sublinear sparse Fourier algorithm for high-dimensional data. In Adaptive Sublinear Time Fourier Algorithm" by D. Lawlor, Y. Wang and A. Christlieb (2013) [@lawlor2013adaptive], an efficient algorithm with empirically $O(k\log k)$ runtime and $O(k)$ sampling complexity for the one-dimensional sparse FFT was developed for signals of bandwidth $N$, where $k$ is the number of significant modes such that $k\ll N$.
In this work we develop an efficient algorithm for sparse FFT for higher dimensional signals, extending some of the ideas in [@lawlor2013adaptive]. Note a higher dimensional signal can always be unwrapped into a one dimensional signal, but when the dimension gets large, unwrapping a higher dimensional signal into a one dimensional array is far too expensive to be realistic. Our approach here introduces two new concepts: partial unwrapping” and tilting”. These two ideas allow us to efficiently compute the sparse FFT of higher dimensional signals.
author:
- 'Bosu Choi, Andrew Christlieb, and Yang Wang'
title: |
MULTI-DIMENSIONAL SUBLINEAR\
SPARSE FOURIER ALGORITHM
---
INTRODUCTION {#INTRODUCTION}
============
As the size and dimensionality of data sets in science and engineering grow larger and larger, it is necessary to develop efficient tools to analyze them [@greene2015understanding; @LSST]. One of the best known and most frequently-used tools is the Fast Fourier Transform (FFT). However, in the case that the bandwidth $N$ of frequencies is large, the sampling size becomes large, as dictated by the Shannon-Nyquist sampling theorem. Specifically, the runtime complexity is $O(N{\log}N)$ and the number of samples is $O(N)$. This issue is only exacerbated in the $d$-dimensional setting, where the runtime complexity is $O (N^d{\log}N^d )$ and the number of samples is $O(N^d)$ if we assume the dimension is $d$ and the bandwidth in each dimension is $N$. Due to this curse of dimensionality”, many higher dimensional problems of interest are beyond current computational capabilities of the traditional FFT. Moreover, in the sparse setting where the number of significant frequencies $k$ is small, it is computationally wasteful to compute all $N^d$ coefficients. In such a setting we refer to the problem as being sparse”. For sparse problems, the idea of sublinear sparse Fourier transforms was introduced [@gilbert2002near; @gilbert2005improved; @hassanieh2012nearly; @hassanieh2012simple; @iwen2010combinatorial; @lawlor2013adaptive; @christlieb2016multiscale]. These methods greatly reduce the runtime and sampling complexity of the FFT in the sparse setting. The methods were primarily designed for the one dimensional setting.
The first sparse Fourier algorithm was proposed in [@gilbert2002near]. It introduced a randomized algorithm with $O(k^2{\log}^c N)$ runtime and $O(k^2{\log}^c N)$ samples where $c$ is a positive number that varies depending on the trade-off between efficiency and accuracy. An algorithm with improved runtime $O(k{\log}^c N)$ and samples $O(k{\log}^c N)$ was given in [@gilbert2005improved]. The algorithms given in [@hassanieh2012nearly] and [@hassanieh2012simple] achieved $O(k{\log} N{\log} {N}/{k})$ runtime and gave empirical results. The algorithms in [@gilbert2002near; @gilbert2005improved; @hassanieh2012nearly; @hassanieh2012simple] are all randomized. The first deterministic algorithm using a combinatorial approach was introduced in [@iwen2010combinatorial]. In [@lawlor2013adaptive], another deterministic algorithm was given whose procedure recognizes frequencies in a similar manner to [@hassanieh2012nearly]. The two methods in [@lawlor2013adaptive; @hassanieh2012nearly] were published at the same time and both use the idea of working with two sets of samples, one at $O(k)$ points and the second at the same $O(k)$ points plus a small shift. The ratio of the FFT of the two sets of points, plus extra machinery, lead to fast deterministic algorithms. The first deterministic algorithm [@iwen2010combinatorial] has $O(k^2{\log}^4 N)$ runtime and sampling complexity, and the second one [@lawlor2013adaptive] has $O(k{\log}k)$ runtime and $O(k)$ sampling complexity. Later, [@christlieb2016multiscale] introduced modified methods for noisy data with $O(k{\log}k{\log}N/k)$ runtime. Our method, discussed throughout this paper builds on the method presented in [@lawlor2013adaptive].
The methods introduced in the previous paragraph are for one-dimensional data. In [@ghazi2013sample], practical algorithms for data in two dimensions were given for the first time. In this paper, we develop algorithms designed for higher dimensional data, which is effective even for dimensions in the hundreds and thousands. To achieve our goal, our approach must address the worst case scenario presented in [@ghazi2013sample]. We can find a variety of data sets in multiple dimensions that we want to analyze. A relatively low-dimensional example is MRI data, which is three dimensional. However, when we designed the method in this paper, we had much higher dimensional problems in mind, such as some astrophysical data, e.g., the Sloan Digital Sky Survey and Large Synoptic Survey Telescope [@SDSS; @LSST]. They produce tera- or peta-bytes of imaging and spectroscopic data in very high dimensions. Due to the computational effort of a multi-dimensional FFT, spectral analysis of this high dimensional data necessitates a multidimensional sparse fast Fourier transform. Further, given the massive size of data sets in some current and future problems in science and engineering, it is anticipated that the develop of such an efficient algorithm will play an important role in the analysis of these types of data.
It is not straightforward to extend one dimensional sparse Fourier transform algorithms to multiple dimensions. We face several obstacles. First, we do not have an efficient FFT for multidimensional problems much higher than three. Using projections onto lower-dimensional spaces solves this problem. However, like all projection methods for sparse FFT, one needs to match frequencies from one projection with those from another projection. This [*registration*]{} problem is one of the big challenges in the one dimensional sparse FFT. An equally difficult challenge is that different frequencies may be projected into the same frequency ([*the collision problem*]{}). All projection methods for sparse FFT primarily aim to overcome these two challenges. In higher dimensional sparse FFT, these problems become even more challenging as now we are dealing with frequency vectors, not just scalar frequencies.
As a first step to our goal of a high dimensional sparse FFT, this paper addresses the case for continuous data without noise in a high dimensional setting. In a later paper we shall present an adaptation of the algorithm for noisy data. We introduce effective methods to address the registration and the collision problems. In particular, we introduce a novel [*partial unwrapping*]{} technique that is shown to be highly effective in reducing the registration and collision complexity while maintains the sublinear runtime efficiency. We shall show that empirically we can achieve $O(dk{\log}k)$ computational complexity and $O(dk)$ sampling size for randomly generated test data. In Section \[5\], we present as examples computational results for sparse FFT where the dimensions are 100 and 1000 respectively. For comparison, the traditional $d$-dimensional FFT requires $O(N^d{\log}N^d)$ time complexity and $O(N^d)$ sampling complexity, which is impossible to implement on any computers today.
PRELIMINARIES {#PRELIMINARIES}
=============
Review of the One-Dimensional Sublinear Sparse Fourier Algorithms {#2.1}
-----------------------------------------------------------------
The one-dimensional sublinear sparse Fourier algorithm inspiring our method was developed in [@lawlor2013adaptive]. We briefly introduce the idea and notation of the algorithm before developing the multidimensional ones throughout this paper. We assume a function $f:[0,1)\rightarrow {\mathbb C}$ with sparsity $k$ as the following, $$\label{(1)}
f(t) ~=~ \sum_{j=1}^{k}a_je^{2\pi i w_j t}$$ with bandwidth $N$, i.e., frequency $w_j$ belongs to $[-N/2, N/2)\cap {\mathbb Z}$ and corresponding nonzero coefficient $a_j$ is in ${\mathbb C}$ for all $j$. We can consider it as a periodic function over ${\mathbb R}$ instead of $[0,1)$. The goal of the algorithm is to recover all coefficients $a_j$ and frequencies $w_j$ so that we can reconstruct the function $f$. This algorithm is called the phase-shift" method since it uses equi-spaced samples from the function and those at positions shifted by a small positive number $\epsilon$. To verify that the algorithm correctly finds the frequencies in the bandwidth $N$, $\epsilon$ should be strictly no bigger than $1/N$. We denote a sequence of samples shifted by $\epsilon$ with sampling rate $1/p$, where $p$ is a prime number, as $$\label{(2)}
{\mathbf f}_{p, \epsilon} ~=~ \Big( f(0 + \epsilon), f(\frac{1}{p} + \epsilon), f(\frac{2}{p} +\epsilon), f(\frac{3}{p}+ \epsilon), \cdots, f(\frac{p-1}{p} + \epsilon)\Big).$$ We skip much of the details here. In a nutshell, by choosing $p$ slightly larger than $k$ is enough to make the algorithm work. In [@lawlor2013adaptive] $p$ is set to be roughly $5k$, which is much smaller than the Nyquist rate $N$. Discrete Fourier transform (DFT) is then applied to the sample sequence ${\mathbf f}_{p, \epsilon}$, and the $h$-th element of its result is the following $$\label{(3)}
{\mathcal F}({\mathbf f}_{p,\epsilon})[h] ~=~ p \sum_{w_j = h (\bmod p)} a_j e^{2 \pi i \epsilon w_j}$$ where $h = 0, 1, 2, \dots, p-1$. If there is only one frequency $w_j$ congruent to $h$ modulo $p$, $$\label{(4)}
{\mathcal F}({\mathbf f}_{p,\epsilon})[h] ~=~ pa_je^{2 \pi i \epsilon w_j}.$$ By putting $0$ instead of $\epsilon$, we can get unshifted samples ${\mathbf f}_{p,0}$ and applying the DFT gives $$\label{(5)}
{\mathcal F}({\mathbf f}_{p,0})[h] ~=~ pa_j.$$ This process so far is visualized in the Figure \[1D\]. As long as there is no collision of frequencies with modulo $p$, we can find frequencies and their corresponding coefficients by the following computation $$\begin{aligned}
w_j &~=~ \frac{1}{2\pi \epsilon}{\rm Arg}\Big( \frac{{\mathcal F}({\mathbf f}_{p,\epsilon})[h]} {{\mathcal F}({\mathbf f}_{p,0})[h]} \Big) ,\nonumber\\
a_j &~=~ \frac{1}{p}{\mathcal F}({\mathbf f}_{p,0})[h],\label{(7)}\end{aligned}$$ where the function $\lq\lq {\rm Arg}"$ gives us the argument falling into $[-\pi, \pi).$ Note that $w_j$ should be the only frequency congruent to $h$ modulo $p$, i.e., $w_j$ has no collision with other frequencies modulo $p$. The test to determine whether collision occurs or not is $$\label{(8)}
\frac{\vert{\mathcal F}({\mathbf f}_{p,\epsilon})[h]\vert}{\vert{\mathcal F}({\mathbf f}_{p,0})[h]\vert}~=~ 1.$$ The equality above holds when there is no collision. If there is a collision, the equality does not hold for almost all $\epsilon$, i.e., the test fails to predict a collision for finite number of $\epsilon$ [@lawlor2013adaptive]. Further, it is also shown in the same paper that for any $\epsilon=\frac{a}{b}$ with $a, b$ coprime and $b\geq N$, equality (\[(8)\]) does not hold unless there is no collision. In practical implementations, we choose $\epsilon$ to be ${1}/{cN}$ for some positive integer $c \geq 1$ and allow some small difference $\tau$ between the left and right sides of (\[(8)\]) where $\tau$ is very small positive number.
![Process of 1D sublinear sparse Fourier algorithm[]{data-label="1D"}](1d_graph_3.png){width="99.00000%"}
\[fig1\]
The above process is one loop of the algorithm with a prime number $p$. To explain it from a different view, we can imagine that there are $p$ bins. Then we sort all frequencies into these bins according to their remainder modulo $p$. If there are more than one frequencies in one bin, then a collision happened. If there is only one frequency, then there is no collision. To determine whether a collision occurs, we use the above test. In the case where the test fails, i.e., the ratio is not $1$, we need to use another prime number $p'$. Thus we re-sort the frequencies into $p'$ bins by their remainder modulo $p'$. Even if two frequencies collide modulo $p$, it is likely that they do not collide modulo $p'$. Particularly, the Chinese Remainder Theorem guarantees that with a finite set of prime numbers, $\{p_{\ell}\}$, any frequency within the bandwidth $N$ can be uniquely identified, given $\prod_{\ell} p_{\ell} \geq N$. Algorithmically, for each loop, we choose a different prime number $p'$ and repeat equations (\[(2)\])-(\[(8)\]) with $p$ replaced by $p'$. In this way we can recover all $a_j$ and $w_j$ in sublinear time $O(k{\log} k)$ using $O(k)$ samples. The overall code is shown in Algorithm \[Algorithm1\] referred from [@lawlor2013adaptive].
\
[**Input:**]{}[$f, c, k, N, \epsilon$]{}\
[**Output:**]{}[$R$]{} $R \gets \emptyset$ $i \gets 1$ $ k^{\ast} \gets k - |R|$ $p \gets {\it{i} th}$ prime number $\geq c k^{\ast}$ $g({{t}})=\sum_{({w},a_{w})\in R} a_{w} e^{2 \pi i {w} {t}}$
${f}_{p,\epsilon}[h] = f(\frac{h}{p}) - g(\frac{h}{p}) $ ${f}_{p,0}[h] = f(\frac{h}{p}+\epsilon) - g(\frac{h}{p}+\epsilon)$ $\mathcal{F}({f}_{p, \epsilon})=FFT({f}_{p, \epsilon})$ $\mathcal{F}({f}_{p, 0})=FFT({f}_{p, 0})$ $\mathcal{F}^{sort}({f}_{p, 0})=SORT(\mathcal{F}({f}_{p, 0}))$
$\tilde{w}=\frac{1}{2\pi\epsilon}\rm Arg\Big( \frac{\mathcal{F}^{sort}({f}_{p, \epsilon})[h]}{\mathcal{F}^{sort}({f}_{p, 0})[h]} \Big)$ $a=\frac{1}{p}\mathcal{F}^{sort}({f}_{p, 0})[h]$ $R \gets R\cup ({\tilde{w}}, a)$ $i \gets i+1$
Multidimensioanl Problem Setting and Worst Case Scenario {#2.2}
--------------------------------------------------------
In this section, the multidimensional problem is introduced. Let us consider a function $f: {\mathbb R^d} \rightarrow {\mathbb C}$ such that $$\label{(9)}
f({\mathbf t})~=~\sum_{j=1}^{k}a_j e^{2\pi i{{\mathbf w}_j}\cdot{{\mathbf t}}},$$ where ${\mathbf w}_j \in ([-N/2, N/2)\cap{\mathbb Z})^d$ and $a_j\in{\mathbb C}$. That is, from (\[(1)\]), $t$ is replaced by the $d$-dimensional phase or time vector ${\mathbf t}$, frequency $w_j$ is replaced by the frequency vector ${{\mathbf w}_j}$ and thus the operator between ${{\mathbf w}_j}$ and ${\mathbf t}$ is a dot product instead of simple scalar multiplication. We can see that this is a natural extension of the one-dimensional sparse problem. As in the 1D setting, if we find $a_j$ and ${\mathbf w}_j$, we recover the function $f$.
However, since our time and frequency domain have changed, we cannot apply the previous algorithm directly. If we project the frequencies onto a line, then we can apply the former algorithm so that we can retain sublinear time complexity. Since the operator between frequency and time vectors is a dot product, we can convert projection of frequencies to that of time. For example, we consider the projection onto the first axis, that is, we put the last $d-1$ elements of time vectors as $0$. If the projection is one-to-one, i.e., there is no collision, then we can apply the algorithm in Section \[2.1\] to this projected function to recover the first element of frequency vectors. If there is a collision on the first axis, then we can try another projection onto $i$th axis, $i=2, 3, \cdots, d$, until there are no collisions. We introduce in latter sections how to recover the corresponding remaining $d-1$ elements by extending the test to determine the occurrence of a collision in Section \[2.1\]. Furthermore, to reduce the chance of a collision through projections, we use an unwrapping method” which unwraps frequencies onto a lower dimension guaranteeing a one-to-one projection. There is both a full unwrapping” and a partial unwrapping” method, which are explained in later sections.
We shall call projections onto any one of the coordinate axes a [*parallel projection*]{}. The worst case is where there is a collision for every parallel projection. This obviously happens when a subset of frequency vectors form the vertices of a $d$-dimensional hypercube, but it can happen also with other configurations that require fewer vertices. Then our method cannot recover any of these frequency vectors via parallel projections. To resolve this problem, we introduce [*tilted projections*]{}: instead of simple projection onto axes we project frequency vectors onto tilted lines or planes so that there is no collision after the projection. We shall call this the [*tilting method*]{} and provide the details in the next section. After introducing these projection methods, we explore which combination of these methods is likely to be optimal.
TWO DIMENSIONAL SUBLINEAR SPARSE FOURIER ALGORITHM {#3}
==================================================
As means of explanation, we introduce the two-dimensional case in this section and extend this to higher dimensions in Section \[4\]. The basic two-dimensional algorithm using a parallel projection is introduced in Section \[3.1\], the full unwrapping method is introduced in Section \[3.2\] and the tilting method for the worst case is discussed in Section \[3.3\].
![Process of the basic algorithm in 2D[]{data-label="fig1"}](phaseshift_multi_d_normal2d.png){width="99.00000%"}
\[2Dbasic\]
Basic Algorithm Using Parallel Projection {#3.1}
-----------------------------------------
Our basic two-dimensional sublinear algorithm excludes certain worst case scenarios. In most cases, we can recover frequencies in the 2-D plane by projecting them onto each horizontal axis or vertical axis. Figure \[2Dbasic\] is a simple illustration. Here we have three frequency vectors where $\mathbf w_1$ and $\mathbf w_3$ are colliding with each other when they are projected onto the horizontal axis and $\mathbf w_1$ and $\mathbf w_2$ are when they are projected onto the vertical axis. The first step is to project the frequency vectors onto the horizontal axis and recover $\mathbf w_2$ and its corresponding coefficient $a_2$ only, since it is not colliding. After subtracting $\mathbf w_2$ from the data, we project the remaining frequency vectors onto the vertical axis and then find both $\mathbf w_1$ and $\mathbf w_3$.
Now let us consider the generalized two-dimensional basic algorithm. Assume that we have a two-dimensional function $f$ with sparsity $k$ : $$\label{(10)}
f({{\mathbf t}})~=~\sum^{k}_{j=1}a_j e^{2\pi i {{\mathbf w}_j}\cdot{{\mathbf t}}},\quad a_j\in {\mathbb C},\quad {{\mathbf w}_j} \in {\Big{(}\Big[-\frac{N}{2},\frac{N}{2}{\Big )}\cap{\mathbb Z}\Big{)}}^2.$$ For now, let us focus on one frequency vector with index $j'$ which is not collided with any other pairs when they are projected onto the horizontal axis. To clarify put ${{\mathbf t}}=(t_1, 0)$ with ${{\mathbf w}_j}=(w_{j1}, w_{j2})$ into (\[(10)\]), $$\label{(11)}
f^1(t_1)~:=~f(t_1,0)~=~\sum^{k}_{j=1}a_j e^{2\pi i {w_{j1}} {t_1}},$$ which gives the same effect of parallel projection of frequency vectors. Now, we can consider this function as a one-dimensional function $f^1$ so that we can use the original one dimensional sparse Fourier algorithm to find the first component of ${{\mathbf w}_{j'}}$. We get the samples ${\mathbf f}^1_{p,0}$ and ${\mathbf f}^1_{p,\epsilon}$ with and without shift by $\epsilon$. We can find these in the form of sequences in (\[(2)\]), apply the DFT to them, and then recover the first component of the frequency pair and its coefficient as follows, $$\begin{aligned}
w_{j'1} &~=~ \frac{1}{2\pi \epsilon}\rm Arg\Big( \frac{{\mathcal F}({\mathbf f}^1_{p,\epsilon})[h]} {{\mathcal F}({\mathbf f}^1_{p,0})[h]} \Big), \nonumber\\
a_{j'} &~=~ \frac{1}{p}{\mathcal F}({\mathbf f}^1_{p,0})[h].\label{(12)}\end{aligned}$$ At the same time, we need to find the second component. In (\[(11)\]), we replace $0$ by $\epsilon$. Then $$\begin{aligned}
f^2(t_1)~ &:= ~ f(t_1,\epsilon) ~ = ~ \sum^{k}_{j=1}a_j e^{2\pi i ({w_{j1}}t_1+{w_{j2}}\epsilon)}\nonumber,\\
{\mathcal F}({\mathbf f}^2_{p,\epsilon})[h]~ &= ~ pa_{j'}e^{2\pi i w_{j'2}\epsilon}\nonumber,\\
w_{j'2}~ &= ~ \frac{1}{2\pi\epsilon} {\rm Arg}\Big( \frac{{\mathcal F}({\mathbf f}^2_{p,\epsilon})[h]}{ {\mathcal F}({\mathbf f}^1_{p,0})[h]} \Big),\label{(16)}\end{aligned}$$ where ${\mathbf f}^2_{p,\epsilon}$ are samples shifted by $\epsilon$ in the vertical sense with rate $1/p$ from the function $f^2$. (\[(12)\]) holds only when $w_{j'1}$ is the only one congruent to $h$ modulo $p$ among every first component of $k$ frequency pairs and (\[(16)\]) holds only when the previous condition is satisfied and ${\mathbf w}_{j'}=(w_{j'1},w_{j'2})$ does not collide with other frequency pairs from the parallel projection.
Now we have two kinds of collisions. The first one is from taking modulo $p$ after the parallel projection and the second one is from the projection. Thus we need two tests. To determine whether there are both kinds of collisions, we use similar tests as (\[(8)\]). If there are at least two different $w_{j1}$ congruent to $h$ modulo $p$, then the second equality in the following is not satisfied for almost all $\epsilon$, just as (\[(8)\]), $$\label{(17)}
\frac{\vert{\mathcal F}({\mathbf f}^1_{p,\epsilon})[h]\vert}{\vert{\mathcal F}({\mathbf f}^1_{p,0})[h]\vert}~=~ \frac{\vert p \sum_{w_{j1} = h(\bmod p)} a_j e^{2 \pi i \epsilon w_{j1}} \vert}{\vert p \sum_{w_{j1} = h(\bmod p)} a_j \vert} ~=~ 1.$$ Likewise, if there is a collision from the projection, i.e., the first components $w_{j1}$’s of at least two frequency vectors are identical and the corresponding $w_{j2}$’s are different, the following second equality does not hold for almost all $\epsilon$, $$\label{(18)}
\frac{\vert{\mathcal F}({\mathbf f}^2_{p,\epsilon})[h]\vert}{\vert{\mathcal F}({\mathbf f}^1_{p,0})[h]\vert}~=~ \frac{\vert p \sum_{w_{j1} = h (\bmod p)} a_j e^{2 \pi i \epsilon w_{j2}} \vert}{\vert p \sum_{w_{j1} = h (\bmod p)} a_j \vert} ~=~ 1.$$ The two tests above are both satisfied only when there is no collision both from taking modulo $p$ and the projection. We use these for the complete recovery of the objective frequencies. So far we project the frequencies onto the horizontal axis. After we find the non-collided frequencies from the first projection, we subtract a function consisting of found frequencies and their coefficients from the original function $f$ to get a new function. Next we project this new function onto the vertical axis and do a similar process. The difference is to exchange $1$ and $2$ in the super-indices and sub-indices respectively in (\[(11)\]) through (\[(18)\]). Again, find the remaining non-collided frequencies, change the axis again and keep doing this until we recover all of the frequencies.
Full Unwrapping Method {#3.2}
----------------------
We introduce another kind of projection which is one-to-one. The full unwrapping method uses one-to-one projections onto one-dimensional lines instead of the parallel projection onto axes from the previous method. We consider the $k$ pairs of frequencies $(w_{j1}, w_{j2})$, $j=1,2,\cdots,k$ and transform them as follows $$\label{(19)}
(w_{j1}, w_{j2}) ~\rightarrow~ w_{j1}+Nw_{j2}.$$ This transformation in frequency space can be considered as the transformation in phase or time space. That is, from the function in (\[(10)\]) $$\label{(20)}
g(t)~:=~f(t, Nt)~=~\sum^{k}_{j=1}a_j e^{2\pi i (w_{j1}+Nw_{j2})t}.$$ The function $g(t)$ is a one-dimensional function with sparsity $k$ and bandwidth bounded by $N^2$. We can apply the algorithm in Section \[2.1\] on $g$ so that we recover $k$ frequencies of the form on the right side of the arrow in (\[(19)\]). Whether unwrapped or not, the coefficients are the same, so we can find them easily. In the end we need to wrap the unwrapped frequencies to get the original pairs. Remember that unwrapping transformation is one-to-one. Thus we can wrap them without any collisions.
Since the pairs of the frequencies are projected onto the one-dimensional line directly, we call this method the “full unwrapping method". Problem with this method occurs when the dimension $d$ gets large. From the above description, we see that after the one-to-one unwrapping the total bandwidth of the two dimensional signal increases from $N$ in each dimension to $N^2$. If the full unwrapping method is applied to a function in $d$-dimensions, then to guarantee the one-to-one transformation the bandwidth will be $N^{d}$. Theoretically this does not matter. However, since $\epsilon$ is dependent on the bandwidth, in the case where $d$ is large, we need to consider the limit of machine precision for practical implementations. As a result, we need to introduce the partial unwrapping method to prevent the bandwidth from becoming too large. The partial unwrapping method is discussed in Section \[4\].
![Worst case scenario in $2D$ and solving it through tilting[]{data-label="2Dtilting"}](phaseshift_multi_d_worstcasesol2.png){width="90.00000%"}
Tilting Method for the Worst Case {#3.3}
---------------------------------
Up till now, we have assumed that we do not encounter the worst case, i.e., that we do not encounter the case where any frequency pair has collisions from the parallel projection for all coordinate axes. This makes the algorithm break down. The following method is for finding those frequency pairs. Basically, we rotate axes of the frequency plane and thus use a projection onto a one-dimension system which is a tilted line with the tilt chosen so that there are no collisions. If the horizontal and vertical axes are rotated with angle $\theta$ then the frequency pair ${\mathbf w}_j=(w_{j1}, w_{j2})$ can be relabeled with new coordinates as the right side of the following $$\label{(21)}
(w_{j1}, w_{j2}) ~\rightarrow~ (\cos\theta w_{j1}-\sin\theta w_{j2}, \sin\theta w_{j1}+\cos\theta w_{j2}).$$ In phase-sense, this rotation can be written as $$\begin{aligned}
g({\tilde t}_1, {\tilde t}_2)& ~:=~
f(\cos\theta {\tilde t}_1+\sin\theta{\tilde t}_2, -\sin\theta{\tilde t}_1 + \cos \theta{\tilde t}_2)\nonumber \\
&~=~\sum^{k}_{j=1}a_j e^{2\pi i\{ w_{j1}(\cos\theta {\tilde t}_1+\sin\theta{\tilde t}_2)+w_{j2}(-\sin\theta{\tilde t}_1 + \cos \theta{\tilde t}_2)\}},\nonumber\\
&~=~\sum^{k}_{j=1}a_j e^{2\pi i \{(\cos\theta w_{j1}-\sin\theta w_{j2}){\tilde t}_1+(\sin\theta w_{j1}+\cos\theta w_{j2}){\tilde t}_2\}}.\label{(22)}\end{aligned}$$ We can apply the basic algorithm in Section \[3.1\] to the function $g$ to get the frequency pairs in the form of the right side of the arrow in (\[(21)\]).
One problem we face is that the components of the projected frequency pairs should be integers to apply the method, since we assume integer frequencies in the first place. To guarantee injectivity for both projections, $\tan\theta$ should be irrational, however, the projected frequencies become irrational. Thus, we should try rational $\tan\theta$, and to make them integer it is inevitable to increase the bandwidth by multiplying the least common multiple of the denominators of $\sin\theta$ and $\cos\theta$. We assume the following with integers $a$, $b$ and $c$ $$\label{(25)}
\sin\theta ~=~ \frac{a}{c},\quad \cos\theta ~=~ \frac{b}{c},\quad \gcd(a,c)~=~\gcd(b,c)~=~1.$$ Multiplying $c$ to both inputs in the right-hand side of (\[(22)\]) we obtain $$\begin{aligned}
{\hat g}({\tilde t}_1, {\tilde t}_2)&~:=~f(c(\cos\theta {\tilde t}_1+\sin\theta{\tilde t}_2), c(-\sin\theta{\tilde t}_1 + \cos \theta{\tilde t}_2))\nonumber\\
&~=~\sum^{k}_{j=1}a_j e^{2\pi i \{(c\cos\theta w_{j1}-c\sin\theta w_{j2}){\tilde t}_1+(c\sin\theta w_{j1}+c\cos\theta w_{j2}){\tilde t}_2\}}.\label{(27)}\end{aligned}$$ As long as there is no collision for at least one projection, the frequency pairs, $(c\cos\theta w_{j1} - c\sin \theta w_{j2}, c\sin \theta w_{j1} + c\cos \theta w_{j2})$, can be found by applying the basic algorithm in Section \[3.2\] on ${\tilde g}$. Due to machine precision the integer $c$ should not be too large, or the bandwidth gets too large resulting in failure of the algorithm. If four pairs of frequencies are at vertices of a rectangle aligned with coordinate axes before the rotation, then they are not aligned after the rotation with $0<\theta<\pi/2$. Thus we can assure finding whole frequencies whether they are in the worst case or not.
\
[**Input:**]{}[$f, c, k, N, d, \epsilon$, integers $base$, $height$, $hypo$]{}\
[**Output:**]{}[$R$]{} $R \gets \emptyset$ $i \gets 1$ $\cos \gets {base}$, $\sin \gets {height} $ $k^{\ast} \gets k - |R|$ $p \gets {\it{i}th}$ prime number $\geq c k^{\ast}$ $ m \gets (i$ mod 2)+1 $g({\mathbf{t}})=\sum_{({\mathbf w},a_{\mathbf w})\in R} a_{\mathbf w} e^{2 \pi i {\mathbf w}\cdot {\mathbf t}}$ $m' \gets m$ mod $2$, $m'' \gets m+1$ mod $2$, $n' \gets n$ mod $2$, $n'' \gets n+1$ mod $2$ ${f}^{m,n}_{p,\epsilon}[h] = $ $f((\frac{h-1}{p}m'+\epsilon n') \cos+ (\frac{h-1}{p}m''+\epsilon n'') \sin, -(\frac{h-1}{p}m'+\epsilon n') \sin+ (\frac{h-1}{p}m''+\epsilon n'') \cos)$ $- g(\frac{h-1}{p}{\mathbf e}_{m}+\epsilon{\mathbf e}_{n} )$ ${f}^{m,n}_{p,0}[h] = f(\frac{h-1}{p}m' \cos +\frac{h-1}{p}m'' \sin , -\frac{h-1}{p}m' \sin +\frac{h-1}{p}m'' \cos ) - g(\frac{h-1}{p}{\mathbf e}_m)$ $\mathcal{F}({f}^{m,n}_{p, \epsilon})=FFT({f}^{m,n}_{p, \epsilon})$ $\mathcal{F}({f}^{m,n}_{p, 0})=FFT({f}^{m,n}_{p, 0})$ $\mathcal{F}^{sort}({f}^{m,n}_{p, 0})=SORT(\mathcal{F}({f}^{m,n}_{p, 0}))$ $\ell \gets 0$ $\ell \gets \ell +1$ $\tilde{w}_{n}=\frac{1}{2\pi\epsilon}\rm Arg\Big( \frac{\mathcal{F}^{sort}({f}^{m,n}_{p, \epsilon})[h]}{\mathcal{F}^{sort}({f}^{m,n}_{p, 0})[h]} \Big)$ $a=\frac{1}{p}\mathcal{F}^{sort}({f}^{m,n}_{p, 0})[h]$ $R \gets R\cup ({\mathbf{ \tilde{w}}}, a)$ $i \gets i+1$ $\cos \gets \frac{base}{hypo}$, $\sin \gets \frac{height}{hypo} $
The pseudo code of the 2D tilting method is shown in Algorithm \[Algorithm 3\]. The lines 14 and 15 mean that each frequency pair $(w_{j1},w_{j2})$ is rotated by a matrix \[$\cos$ ${-\sin}$; $\sin$ $\cos$\] and scaled to make the rotated components integers. Thus we first find the frequency pairs in the form of ${\mathbf {\tilde{w}}}=(\cos w_{j1}-\sin w_{j2}, \sin w_{j1}+\cos w_{j2})$ and after finding all of them, we rotate them back into the original pairs with the matrix \[$\cos$ $\sin$; ${-\sin}$ $\cos$\] in line 39.
This tilting method is a straight forward way to resolve the worst case problem. We only introduced the tilting method in the two-dimensional case, but the idea of rotating the axes can be extended to the general $d$-dimensional case with some effort. On the other hand, we may notice that the probability of this worst case is very low, especially when the number of dimensions $d$ is large. Its details are shown in Section \[4\]. Thus, as we recover the frequencies as much as possible from the basic algorithm. If we cannot get any frequency pairs for several projection switching among each axis then, assuming that the worst case happens, we apply the tilting method with several angles until all $k$ frequency pairs are found.
PARTIAL UNWRAPPING METHOD FOR HIGH DIMENSIONAL ALGORITHM {#4}
========================================================
In this section we present the [*partial unwrapping*]{} method for a sublinear sparse Fourier algorithm for very high dimensional data. As we have already mentioned, while full unwrapping converts a multi-dimensional problem into a single dimensional problem, it is severely limited in its viability when the dimension is large or when the bandwidth is already high because of the increased bandwidth. Partial unwrapping is introduced here to overcome this problem and other problems. In Section \[4.1\] we give a four dimensional version of the algorithm using the partial unwrapping method as well as a generalize it to $d$ dimension. In Section \[4.2\], the probability of the worst case in $d$ dimension is analyzed.
Partial Unwrapping Method {#4.1}
-------------------------
To see the benefit of partial unwrapping we need to examine the main difficulties we may encounter in developing sublinear sparse Fourier algorithms. For this let us consider a hypothetical case of sparse FFT where we have $k=100$ frequencies in a 20-dimensional Fourier series distributed in $[-10, 10)^{20}$. When we perform the parallel projection method, because the bandwidth is small, there will be a lot of collisions after the projections. It is often impossible to separate any frequency after each projection, and the task could thus not be completed. This, ironically, is a [*curse of small bandwidth*]{} for sparse Fourier algorithm. On the other hand, if we do the full unwrapping we would have increased the bandwidth to $N=20^{20}$, which is impossible to do within reasonable accuracy because $N$ is too large.
However, a partial unwrapping would reap the benefit of both worlds. Let us now break down the 20 dimensions into 5 lower 4-dimensional subspaces, namely we write $$[-10,10)^{20} = \left( [-10,10)^4\right)^{5}.$$ In each subspace we perform the full unwrapping, which yields bandwidth $N=20^4=160,000$ in the subspace. This bandwidth $N$ is large enough compared with $k$, so when projection method is used there is a very good probability that collision will occur only for a small percentage of the frequencies, allowing them to be reconstructed. On the other hand, $N$ is not so large that the phase-shift method will incur significant error.
One of the greatest advantage of partial unwrapping is to turn the curse of dimensionality into the [*blessing*]{} of dimensionality.
Note that in the above example, the 4 dimensions that for any of the subspaces do not have to follow the natural order. By randomizing (if necessary) the order of the dimensions it may achieve the same goal as the tilting method would. Also note that the dimension for each subspace needs not be uniform. For example, we can break down the above 20-dimensional example into four $3$-dimensional subspaces and two $4$-dimensional subspaces, i.e. $$[-10,10)^{20} = \left( [-10,10)^3\right)^{4} \times \left( [-10,10)^4\right)^{2}.$$ This will lead to further flexibility.
### Example of 4-D Case {#4.1.1}
Before introducing the generalized partial unwrapping algorithm for dimension $d$, let us think about the simple case of $4$ dimensions. We assume that $k$ frequency vectors are in $4$-dimensional space ($d=4$). Then, a function $f$ constructed from these frequency vectors is as follows, $$\label{(28)}
f({\mathbf t})=\sum^{k}_{j=1}a_j e^{2\pi i {\mathbf w}_j\cdot{\mathbf t}},\ a_j\in {\mathbb C},\ {\mathbf w}_j \in {\Big{(}\Big[-\frac{N}{2},\frac{N}{2}{\Big )}\cap{\mathbb Z}\Big{)}}^4.$$ Since $4=2\times 2$, the frequency pairs of the two-two dimensional spaces are both unwrapped onto one-dimensional spaces. Here, $4$ dimensions is projected onto $2$ dimensions as follows $$\begin{aligned}
g(t_1,t_2)&~:=~f(t_1, Nt_1, t_2, Nt_2)\nonumber\\
&~=~\sum^{k}_{j=1}a_j e^{2\pi i\{ (w_{j1}+Nw_{j2})t_1+ (w_{j3}+Nw_{j4})t_2\}}\nonumber\\
&~=~ \sum^{k}_{j=1}a_j e^{2\pi i(\tilde{w}_{j1}t_1+ \tilde{w}_{j2}t_2)},\label{(31)}\end{aligned}$$ where $\tilde{w}_{j1}=w_{j1}+Nw_{j2}$ and $\tilde{w}_{j2}=w_{j3}+Nw_{j4}$. Note that this projection is one-to-one so as to guarantee the inverse transformation.
Now we can apply the basic projection method in Section \[3.1\] to this function $g$ re-defined as the $2$-dimensional one. To make this algorithm work, ${\mathbf {\tilde{w}}}_j=(\tilde{w}_{j1},\tilde{w}_{j2})$ should not collide with any other frequency pair after the projection onto either the horizontal or vertical axes. If not, we can consider using the tilting method. After finding all the frequencies in the form of $(\tilde{w}_{j1},\tilde{w}_{j2})$, it can be transformed to $(w_{j1},w_{j2},w_{j3},w_{j4})$.
### Generalization {#4.1.2}
We introduce the final version of the multidimensional algorithm in this section. Its pseudo code and detailed explanation are given in Algorithm \[Algorithm2\] and Section \[5.0\], respectively. We start with a $d$-dimensional function $f$, $$\label{(32)}
f({\mathbf t})~=~\sum^{k}_{j=1}a_j e^{2\pi i {\mathbf w}_j\cdot{\mathbf t}},\quad a_j\in {\mathbb C},\quad {\mathbf w}_j \in {\Big{(}\Big[-\frac{N}{2},\frac{N}{2}{\Big )}\cap{\mathbb Z}\Big{)}}^d.$$ Let us assume that $d$ can be divided into $d_1$ and $d_2$ - the case of $d$ being a prime number will be mentioned at the end of this section. The domain of frequencies can be considered as ${([-N/2,N/2)\cap{\mathbb Z})}^d=({([-N/2,N/2)\cap{\mathbb Z})}^{d_1})^{d_2}$ and ${([-N/2,N/2)\cap{\mathbb Z})}^{d_1}$ will be reduced to one dimension, as $d_1$ is in the 4 dimensional case. Each of the $d_1$ elements of a frequency vector, ${\mathbf w}_j = (w_{j1}, w_{j2}, \cdots, w_{jd} )$, is unwrapped as $$\begin{aligned}
& (w_{j(d_1q+1)}, w_{j(d_1q+2)}, w_{j(d_1q+3)}, \cdots, w_{j(d_1q+d_1)})\nonumber\\
&~\rightarrow~ w_{j(d_1q+1)}+Nw_{j(d_1q+2)}+N^2w_{j(d_1q+3)}+\cdots+N^{d_1-1}w_{j(d_1q+d_1)}\nonumber\\
&~=:~\tilde{w}_{j(q+1)}\label{(35)}\end{aligned}$$ with $q=0,1,2,\cdots,d_2-1$, increasing the respective bandwidth from $N$ to $N^{d_1}$ and having injectivity. We rewrite this transformation in terms of the phase. With ${\mathbf t}=(t_1,t_2, \cdots, t_d)$ and put the following into $t_\ell$ $$\label{(36)}
N^{R(\ell-1,d_1)}\tilde{t}_{Q(\ell,d_1)}$$ for all $\ell = 1, 2, \cdots, d$, where $R(\ell-1,d_1)$ and $Q(\ell,d_1)$ are the remainder from dividing $\ell-1$ by $d_1$ and quotient from dividing $\ell$ by $d_1$ respectively, and ${\mathbf {\tilde{t}}}=(\tilde{t}_1, \tilde{t}_2,\cdots, \tilde{t}_{d_2})$ is a phase vector in $d_2$ dimensions after projection. Define a function $g$ on $d_2$ dimension as $$\begin{aligned}
g({\mathbf {\tilde{t}}})&~:=~ f(\cdots, N^{R(\ell-1,d_1)}\tilde{t}_{Q(\ell,d_1)}, \cdots)\nonumber\\
&~=~\sum^{k}_{j=1}a_j e^{2\pi i \sum^{d_2-1}_{q=0} \big( \sum^{d_1-1}_{r =0} w_{j (d_1q+r+1)}N^{r}\big)\tilde{t}_{q+1}},\label{(38)}\end{aligned}$$ where $N^{R(\ell-1,d_1)}\tilde{t}_{Q(\ell,d_1)}$ is the $\ell$th element of the input of $f$. If we project frequency vectors of $g$ onto the $m$th axis then the $n$th element of a frequency vector ${\mathbf {\tilde{w}}}_j$ can be found in the following computation, $$\begin{aligned}
{\mathbf g}^{m,n}_{p, \epsilon} &~=~ \Big( g(0{\mathbf e}_m + \epsilon{\mathbf e}_n), g(\frac{1}{p}{\mathbf e}_m + \epsilon{\mathbf e}_n), \cdots, g(\frac{p-1}{p}{\mathbf e}_m + \epsilon{\mathbf e}_n)\Big) \nonumber\\
\tilde{w}_{jn} &~=~ \frac{1}{2\pi \epsilon}{\rm Arg}\Big( \frac{{\mathcal F}({\mathbf g}^{m,n}_{p,\epsilon})[h]} {{\mathcal F}({\mathbf g}^{m,n}_{p,0})[h]} \Big) \nonumber\\
a_j &~=~ \frac{1}{p}{\mathcal F}({\mathbf g}^{m,n}_{p,0})[h],\label{(41)}\end{aligned}$$ where ${\mathbf e}_m$ is the $m$-th unit vector with length $d_2$, i.e., all elements are zero except the $m$-th one with entry $1$. (\[(41)\]) holds as long as $\tilde{w}_{jn}$ is the only one congruent to $h$ modulo $p$ among all $n$-th elements of the frequency vectors and ${\mathbf {\tilde{w}}}_j$ does not collide with any other frequency vector due to the projection onto the $m$-th axis. The test for checking whether these conditions are satisfied is $$\label{(42)}
\frac{\vert{\mathcal F}({\mathbf g}^{m,n}_{p,\epsilon})[h]\vert}{\vert{\mathcal F}({\mathbf g}^{m,n}_{p,0})[h]\vert} ~=~1$$ for all $1\leq n \leq d_2$. The projections onto the $m$-th axis, where $m=1, \cdots,d_2$, take turns until we recover all frequency vectors and their coefficients. After that we wrap the unwrapped frequency vectors up from $d_2$ to $d$ dimension. Since the unwrapping transformation is one-to-one, this inverse transformation is well-defined.
So far, we assumed that dimension $d$ can be divided into two integers, $d_1$ and $d_2$. For the case that $d$ is a prime number or both $d_1$ and $d_2$ are so large that the unwrapped data has a bandwidth such that $\epsilon$ is below the machine precision, a strategy of divide and conquer can be applied. In that case we can think about applying partial unwrapping method in a way that each unwrapped component has a different size of bandwidth. If $d$ is $3$, for example, then we can unwrap the first two components of the frequency vector onto one dimension and the last one lies in the same dimension. In that case, the unwrapped data is in two dimensions, and the bandwidth of the first component is bounded by $N^2$ and that of second component is bounded by $N$. In this case we can choose a shift $\epsilon< 1/N^2$ where $N^2$ is the largest bandwidth. We can extend this to the general case, so the partial unwrapping method has a variety of choices balancing the bandwidth and machine precision.
Probability of Worst Case Scenario {#4.2}
----------------------------------
In this section, we give an upper bound of the probability of the worst case assuming that we randomly choose a partial unwrapping method. As addressed in the Section \[4.1\], there is flexibility in choosing certain partial unwrapping method. Assuming a certain partial unwrapping method and considering a stronger condition to avoid its failure, we can find the upper bound of the probability of the worst case where there is a collision for each parallel projection.
For simple explanation, consider a two dimensional problem. Choosing the first frequency vector $(w_{11},w_{12})$ on a two dimensional plane, if the second frequency vector, $(w_{21},w_{22})$, is not on the vertical line crossing $(w_{11},0)$ and the horizontal line crossing $(0,w_{12})$, then the projection method works. Then if the third frequency vector is not on four lines, those two lines mentioned before, the vertical line crossing $(w_{21},0)$ and the horizontal line crossing $(0,w_{22})$, then again the projection method works. We keep choosing next frequency vector in this way, excluding the lines containing previous frequencies. Thus, letting such event $A$, the probability that the projection method fails is bounded above by $1-\mathbb{P}(A)$.
Generally, let us assume that we randomly choose a partial unwrapping, without loss of generality, the total dimension is $d=d_1+d_2+\cdots +d_r$ where $r$ is the number of subspaces and $d_1, d_2,\cdots, d_r$ are the dimensions of each subspace. That is, partially unwrapped frequency vectors are in $r<d$ dimension and each bandwidth is $N^{d_1}, N^{d_2},\cdots, N^{d_r}$, respectively, which is integer strictly larger than $1$. Then, the failure probability of projection method is bounded above by $$\begin{aligned}
1-\prod_{j=1}^{k} \mathbb{P}(A_j)& ~\leq~ 1-\prod_{j=1}^{k} \frac{{N^d-(i-1)(N^{d_1}+N^{d_2}+\cdots+N^{d_r})}}{N^d}\nonumber\\
&~=~1- \frac{1}{\tau^{s-1}}\frac{(\tau-1)!}{(\tau-(s-2))!} \quad \left(\tau:=\frac{N^d}{N^{d_1}+N^{d_2}+\cdots+N^{d_r}}\right)\nonumber\\
&~\sim~ 1- \frac{1}{\tau^{s-1}} \sqrt{\frac{\tau-1}{\tau-(s-2)}}\frac{\left( \frac{\tau-1}{e} \right)^{\tau-1}}{\left( \frac{\tau-(s-2)}{e}\right)^{\tau-(s-2)}} \quad(\text{Sterling's formula})\nonumber\\
&~=~1-\frac{1}{e^{s-3}}\frac{(\tau-1)^{\tau-1/2}}{\tau^{s-1}(\tau-s+2))^{\tau-s+5/2}}\end{aligned}$$ where $A_i$ is the event that we choose $i$th frequency not on the lines, crossing formerly chosen frequency vectors and parallel to each coordinate axis. Noting $N^d=N^{d_1}\times{N^{d_2}}\times \cdots \times {N^{d_r}}$, sparsity $k$ is relatively small compared to $N^d$, and $\tau$ is large, we can see that the upper bound above gets closer to $0$ as $d$ or $N$ grows to infinity.
\
[**Input:**]{}[$f, c, k, N, d, d_1, d_2, \epsilon$]{}\
[**Output:**]{}[$R$]{} $R \gets \emptyset$ $i \gets 1$ $k^{\ast} \gets k - |R|$ $p \gets {\it{i}th}$ prime number $\geq c k^{\ast}$ $ m \gets (i$ mod $d_2)+1$ $g({\mathbf{t}})=\sum_{({\mathbf w},a_{\mathbf w})\in R} a_{\mathbf w} e^{2 \pi i {\mathbf w}\cdot {\mathbf t}}$ ${f}^{m,n}_{p,\epsilon}[h] = f(\sum^{d_1}_{\ell=1} N^\ell \frac{h-1}{p}{\mathbf e}_{d_1(m-1)+\ell}+ \epsilon \sum^{d_1}_{\ell=1} N^\ell {\mathbf e}_{d_1(n-1)+\ell} ) - g(\frac{h-1}{p}{\mathbf e}_m+\epsilon{\mathbf e}_n) $ ${f}^{m,n}_{p,0}[h] = f(\sum^{d_1}_{\ell=1} N^\ell \frac{h-1}{p}{\mathbf e}_{d_1(m-1)+\ell} ) - g(\frac{h-1}{p}{\mathbf e}_m)$ $\mathcal{F}({f}^{m,n}_{p, \epsilon})=FFT({f}^{m,n}_{p, \epsilon})$ $\mathcal{F}({f}^{m,n}_{p, 0})=FFT({f}^{m,n}_{p, 0})$ $\mathcal{F}^{sort}({f}^{m,n}_{p, 0})=SORT(\mathcal{F}({f}^{m,n}_{p, 0}))$ $\ell \gets 0$ $\ell \gets \ell +1$ $\tilde{w}_{n}=\frac{1}{2\pi\epsilon}\rm Arg\Big( \frac{\mathcal{F}^{sort}({f}^{m,n}_{p, \epsilon})[h]}{\mathcal{F}^{sort}({f}^{m,n}_{p, 0})[h]} \Big)$ $a=\frac{1}{p}\mathcal{F}^{sort}({f}^{m,n}_{p, 0})[h]$ $R \gets R\cup ({\mathbf {\tilde{w}}}, a)$ $i \gets i+1$ and restore it in $R$
EMPIRICAL RESULT {#5}
================
The partial unwrapping method is implemented in the C language. The pseudo code of this algorithm is shown in Algorithm \[Algorithm2\]. It is explained in detail in Section \[5.0\]. In our experiment, dimension $d$ is set to $100$ and $1000$, $d_1$ is $5$ and $d_2$ is $20$ and $200$, accordingly. Frequency bandwidth $N$ in each dimension is $20$ and sparsity $k$ varies as $1, 2, 2^2, \cdots, 2^{10}$. The value of $\epsilon$ for shifting is set to $1/2N^{d_1}$ and the constant number $c$ determining the prime number $p$ is set to $5$.
We randomly choose $k$ frequency vectors ${\mathbf w}_j \in {\Big{(}\Big[-\frac{N}{2},\frac{N}{2}{\Big )}\cap{\mathbb Z}\Big{)}}^d$ and corresponding coefficients $\ a_j=e^{2\pi i\theta_j}\in {\mathbb C}$ from randomly chosen angles $\theta_j\in [0, 1)$ so that the magnitude of each $a_j$ is $1$. For each $d$ and $k$ we have $100$ trials. We get the result by averaging $\ell^2$ errors, the number of samples used and CPU TICKS out of $100$ trials.
Since it is difficult to implement high dimensional FFT and there is no practical high dimensional sparse Fourier transform it is hard to compare the result of ours with others, as so far no one else was able to do FFT on this large data set. Thus we cannot help but show ours only. From Figure \[fig4\] we can see that the average $\ell^2$ errors are below $2^{-52}$. Those errors are from all differences of frequency vectors and coefficients of the original and recovered values. Since all frequency components are integers and thus the least difference is $1$, we can conclude that our algorithm recover the frequency vectors perfectly. Those errors are only from the coefficients. In Figure \[fig5\] the average sampling complexity is shown. We can see that the logarithm of the number of samples is almost proportional to that of sparsity. Note that the traditional FFT would show the same sampling complexity even though sparsity $k$ varies since it only depends on the bandwidth $N$ and dimension $d$. In Figure \[fig6\] the average CPU TICKS are shown. We can see the the logarithm of CPU TICKS is also almost proportional to that of sparsity. Note that the traditional FFT might show the same CPU TICKS even though sparsity $k$ varies since it also depends on the bandwidth $N$ and dimension $d$ only.
Algorithm {#5.0}
---------
In this section, the explanation of Algorithm \[Algorithm2\] is given. In [@lawlor2013adaptive] several versions of 1D algorithms are shown. Among them, non-adaptive and adaptive algorithms are introduced where the input function $f$ is not modified throughout the whole iteration, and is modified by subtracting the function constructed from the data in registry $R$, respectively. In our multidimensional algorithm, however, the adaptive version is mandatory since excluding the contribution of the currently recovered data is the key of our algorithm to avoid the collision of frequencies through projections, whose simple pictorial description is given in Figure \[2Dbasic\]. In Algorithm \[Algorithm2\], the function $g$ is the one constructed from the data in the registry $R$.
Our algorithm begins with entering inputs, a function $f$, a constant number $c$ determining $p$, a sparsity $k$, a bandwidth $N$ of each dimension, a dimension $d$, factors $d_1$ and $d_2$ of $d$ and a shifting number $\epsilon<1/N$. For each iteration of the algorithm, the number of frequencies to find is updated as $k^{\ast} = k-|R|$. It stops when $|R|$ becomes equal to the sparsity $k$. The prime number $p$ is determined depending on this new $k^{\ast}$ as $p\geq ck^{\ast}$ and is chosen as the next larger prime number. The lines 13 and 14 of Algorithm \[Algorithm2\] represent the partial unwrapping and sampling with and without shifting from the function where the contribution of former data is excluded. After applying the FFT on each sequence, sorting them according to the magnitude of $\mathcal{F}({f}^{m,n}_{p, 0})$, we check the ratio between the FFT’s of the unshifted and shifted sequences to determined whether there is a collision, either from modulo $p$ or a parallel projection. If all tests are passed, then we find each frequency component and corresponding coefficient for the data that passed and store them in $R$. After several iterations, we find all the data and the final wrapping process gives the original frequency vectors in $d$ dimensions.
![Average $\ell^2$ error[]{data-label="fig4"}](result_error.png){width="90.00000%"}
![Average sampling complexity[]{data-label="fig5"}](result_sample.png){width="90.00000%"}
![Average CPU TICKS[]{data-label="fig6"}](result_time.png){width="95.00000%"}
Accuracy {#5.1}
--------
We assume that there is no noise on the data that we want to recover. Figure \[fig4\] shows that we can find frequencies perfectly and the $\ell^2$ error from coefficients are significantly small. This error is what we average out over 100 trials for each $d=100, 1000$ and $k=1,2^1,2^2,\cdots,2^{10}$ when $N$ is fixed to $20$. The horizontal axis represents the logarithm with base $2$ of $k$ and the vertical axis represents the logarithm with base $2$ of the $\ell^2$ error. It is increasing as the sparsity $k$ is increasing since the number of nonzero coefficients increases. The red graph in the Figure \[fig4\] shows the error when the number of dimensions is $100$ and the blue one shows the error when the number of dimensions is $1000$. Thus, we see that the errors are not substantially impacted by the dimensions.
Sampling Complexity {#5.2}
-------------------
Figure \[fig5\] shows the sampling complexity of our algorithm averaged out from 100 tests for each dimension and sparsity. The horizontal axis means the logarithm with base $2$ of $k$ and the vertical axis represents the logarithm with base $2$ of the total number of samples from the randomly constructed function which are used to find all frequencies and coefficients. The red graph in the Figure \[fig5\] shows the sampling complexity when the number of dimensions is $100$ and the blue one shows the one when the number of dimensions is $1000$. Both graphs increase as $k$ increases. When $d$ is large, we see that it requires more samples since there are more frequency components to find. From the graphs, we see that the scaling seems to be proportional to $d$.
Runtime Complexity {#5.3}
------------------
In Figure \[fig6\], we plot the runtime complexity of the main part of our algorithm averaged over 100 tests for each dimension and sparsity. Main part” means that we have excluded the time for constructing a function consisting of frequencies and coefficients and the time associated with getting samples from it. The horizontal axis is the logarithm, base $2$, of $k$ and the vertical axis is the logarithm, base $2$, of CPU TICKS. The red curve shows the runtime when we set the number of dimensions to 100 and the blue one shows the same thing when the number of dimensions to 1000. Both plots increase as $k$ increases. When $d$ is larger, the plots show that it takes more time to run the algorithm. From the graphs we see that the runtime looks proportional to $d$.
Unfortunately, the sampling process of getting the samples from continuous functions dominates the runtime of the whole algorithm instead of the main algorithm. To show the runtime of our main algorithm, however, we showed CPU TICKS without sampling process. Reducing the time for sampling is still a problem. In [@iwen2010combinatorial] the fully discrete Fourier transform is introduced that we expect to use to reduce it. Exploring how to use this will be one part of our future work.
CONCLUSION {#6}
==========
In this paper we show how to extend our deterministic $1D$ sublinear sparse Fourier algorithm to the general $d$ dimensional case. The method projects $d$ dimensional frequency vectors onto lower dimensions. In this process we encounter several obstacles. Thus we introduced tilting method” for the worst case problems and the partial unwrapping method” to reduce the chance of collisions and to increase the frequency bandwidth within the limit of computation. In this way we can overcome the obstacles as well as maintain the advantage of the $1D$ algorithm. In [@lawlor2013adaptive] the sampling complexity is $O(k)$ and the runtime complexity is $O(k{\log}k)$. Extended this estimation from our $1D$ algorithm, we have $O(dk)$ sampling complexity and a runtime complexity of $O(dk{\log}k)$.
Multidimensional sparse Fourier algorithms have not been discussed much so far, so there is a lot of room for future work. The algorithms in this paper are for recovering data from a noiseless environment only. However most of the actual data contains noise. Thus, the next step will be developing an algorithm for noisy multidimensional data. As mentioned in the previous section, reducing sampling time is another problem to consider. Furthermore, algorithms for fully discrete or nonuniform data will be explored. In the end, it is expected that we apply them to real problems like astrophysical data or MRI data.
[**ACKNOWLEDGEMENTS**]{} We would like to thank Mark Iwen for his valuable advice. This research is supported in part by AFOSR grants FA9550-11-1-0281, FA9550-12-1-0343 and FA9550-12-1-0455, NSF grant DMS-1115709, and MSU Foundation grant SPG-RG100059, as well as Hong Kong Research Grant Council grants 16306415 and 16317416.
[10]{}
David Lawlor, Yang Wang, and Andrew Christlieb. Adaptive sub-linear time fourier algorithms. , 5(01):1350003, 2013.
Casey S Greene, Arjun Krishnan, Aaron K Wong, Emanuela Ricciotti, Rene A Zelaya, Daniel S Himmelstein, Ran Zhang, Boris M Hartmann, Elena Zaslavsky, Stuart C Sealfon, et al. Understanding multicellular function and disease with human tissue-specific networks. , 47(6):569–576, 2015.
LSST. Large synoptic survey telescope.
Anna C Gilbert, Sudipto Guha, Piotr Indyk, S Muthukrishnan, and Martin Strauss. Near-optimal sparse fourier representations via sampling. In [*Proceedings of the thiry-fourth annual ACM symposium on Theory of computing*]{}, pages 152–161. ACM, 2002.
Anna C Gilbert, S Muthukrishnan, and Martin Strauss. Improved time bounds for near-optimal sparse fourier representations. In [*Optics & Photonics 2005*]{}, pages 59141A–59141A. International Society for Optics and Photonics, 2005.
Haitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price. Nearly optimal sparse fourier transform. In [*Proceedings of the forty-fourth annual ACM symposium on Theory of computing*]{}, pages 563–578. ACM, 2012.
Haitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price. Simple and practical algorithm for sparse fourier transform. In [*Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms*]{}, pages 1183–1194. SIAM, 2012.
Mark A Iwen. Combinatorial sublinear-time fourier algorithms. , 10(3):303–338, 2010.
Andrew Christlieb, David Lawlor, and Yang Wang. A multiscale sub-linear time fourier algorithm for noisy data. , 40(3):553–574, 2016.
Badih Ghazi, Haitham Hassanieh, Piotr Indyk, Dina Katabi, Eric Price, and Lixin Shi. Sample-optimal average-case sparse fourier transform in two dimensions. In [*Communication, Control, and Computing (Allerton), 2013 51st Annual Allerton Conference on*]{}, pages 1258–1265. IEEE, 2013.
SDSS. Sloan digital sky survey.
|
---
abstract: 'Fifty years after the Apollo program, space exploration has recently been regaining popularity thanks to missions with high media coverage. Future space exploration and space station missions will require specific networks to interconnect Earth with other objects and planets in the solar system. The interconnections of these networks form the core of an Interplanetary Internet (IPN). More specifically, we consider the IPN as the combination of physical infrastructure, network architecture, and technologies to provide communication and navigation services for missions and further applications. Compared to the current implementation of the Internet, nodes composing the core of the IPN are highly heterogeneous (base stations on planets, satellites etc.). Moreover, nodes are in constant motion over intersecting elliptical planes, which results in highly variable delays and even temporary unavailability of parts of the network. As such, an IPN has to overcome the challenges of conventional opportunistic networks, with much higher latency and jitter (from a couple of minutes to several days) and the additional constraint of long-term autonomous operations. In this paper, we highlight the challenges of IPN, demonstrate the elements to deploy within the areas of interest, and propose the technologies to handle deep space communication. We provide recommendations for an evolutionary IPN implementation, coherent with specific milestones of space exploration.'
author:
- 'Ahmad ALHILAL$^{*}$, Tristan BRAUD'
- Pan HUI
bibliography:
- 'references.bib'
title: |
**The Sky is NOT the Limit Anymore:\
Future Architecture of the Interplanetary Internet**
---
|
---
author:
- |
R. Jackiw\
*Department of Physics\
*Massachusetts Institute of Technology\
*Cambridge, MA 02139\
MIT-CTP-3779***
title: 'Homage to Ettore Majorana[^1]'
---
I am privileged to be in this interesting place honoring Ettore Majorana. Of course I have had no personal contact with him — he disappeared before I appeared. However, it is surprising, that I did not encounter his name nor his achievements during my physics education. He is hardly mentioned in the usual textbooks, at least in the American ones. This is a great loss! To me it was a special loss for the following reason. When I was preparing with Hans Bethe our quantum mechanics textbook, I became fascinated by the Thomas-Fermi theory, and I strived to give a complete discussion in our text. But at that time I knew nothing about Majorana’s work in this area, and so could not include it. Again, when we were writing the chapters on Dirac theory, I wondered why only charged fermions are considered. The resolution of my puzzlement lay in the Majorana representation, about which I learned only later, principally through Julian Schwinger’s writings. Schwinger apparently appreciated the Majorana approach and in his discussions of Dirac theory, the charge carrying fermion field is usually presented as the complex superposition of two real fields, in complete analogy to the description of charged boson fields. Another connection between Majorana and Schwinger can be noted. The last topic that Schwinger researched concerned corrections to the Thomas-Fermi model.
Schwinger was also interested in the problem of mass generation, a topic which these days is linked to Majorana’s name. I shall use this point of contact between the two scientists to review Schwinger’s mass generation mechanism [@Schwinger:1962tp], to expose its topological underpinings and to present an interesting generalization [@Dvali:2006ps]. Majorana deconstructed the complex Dirac equation into its real components. Here I deconstruct Schwinger’s mass generation into its topological ingredients. I think that Majorana would have liked these results.
Schwinger Model Resumé
=======================
In the Schwinger model, an Abelian vector potential $A_{\mu}$ interacts with a vector current $\mathcal{J}^{\mu}$ constructed from massless Dirac fields $\psi$. The Lagrange density is gauge invariant and reads $$\label{lagrangian}
\mathcal{L} \, = \, -{1\over 4} F^{\mu\nu}F_{\mu\nu}\, + \, i \bar{\psi} \gamma_{\mu} (\partial_{\mu}
\, + \, i e A_{\mu})\psi, ~~~~ F_{\mu\nu} \equiv \partial_{\mu} A_{\nu} \, - \, \partial_{\nu}A_{\mu},$$ $$\label{l1}
\mathcal{L}_I \, = \, - e \mathcal{J}^{\mu} A_{\mu}, ~~~ \mathcal{J}^{\mu} \, = \, \bar{\psi}\gamma^{\mu}\psi.$$ The model is defined on an unphysical 2-dimensional space-time, where the Dirac matrices are $2+2$ and the Dirac spinor $\psi$ possesses tow components. The traditional solution of the model proceeds by functionally integrating the Dirac fields, giving an effective action. $$\label{ieff}
\mathcal{I}_{eff}(A) \, = \, -{1\over 4} \int F^{\mu\nu} F_{\mu\nu} \, - \, i \
ln \ det [\gamma^{\mu}(\partial_{\mu}\, + \, i e A_{\mu})]$$ The functional determinant can be computed because the only non vanishing Feynman diagram is the vacuum polarization graph. (This is a special feature of two dimensions.)\
\[PDFpar\] [Figure 1: Vacuum polarization graph generates the polarization operator $\Pi^{\mu\nu}(p) \, \propto \, (g^{\mu\nu} \, - \, p^{\mu}p^{\nu}/p^2)$.]{}\
This generates the polarization tensor $\Pi^{\, \mu\nu}(p) \, \propto \, (g^{\mu\nu} \, - \, p^{\mu}p^{\nu}/p^2)$. The coefficient of $g^{\mu\nu}$ is evaluation dependent (the diagram is superficially divergent), but it becomes fixed by the gauge invariance requirement that the vector current correlator (whose proper part is $\Pi^{\ \mu\nu}$) be transverse. The effective action $$\label{ieff1}
\mathcal{I}_{eff}(A) \, = \, \int \left [ -\, {1 \over 4} F^{\mu\nu} F_{\mu\nu} \,
+ \, {e^2 \over 2\pi} A_{\mu}(g^{\mu\nu} \, - \, {\partial^{\mu}\partial^{\nu} \over \partial^2} ) A_{\nu}\right ],$$ exhibits the generated mass, $m^{\, 2} \, = \, {e^2 \over \pi}$. Thus Schwinger showed that a gauge invariant theory may nevertheless possess a mass gap — a result known to superconductivity experts, as emphasized by Philip Anderson. Although usually one says that the “photon” acquires a mass, in two dimensions the “photon” field $A_{\mu}$ can be decomposed as $A_{\mu} \, = \, \partial_{\mu}\theta \, + \, \epsilon_{\mu\nu}\partial^{\nu}\eta'$. The gauge part decouples; only the pseudoscalar $\eta'$ remains. So one could just as well say that a pseudoscalar excitation acquires the mass.
It is important to appreciate that the axial vector current $\mathcal{J}_{\alpha}^{\, 5} = \bar{\psi}\, \gamma_\alpha \gamma^5 \psi$, which is conserved with massless fermions with unquantized $\psi$, acquires an anomalous divergence upon quantization. This is immediately seen when the 2-dimensional duality relation between axial and vector currents is used. $$\label{axialvector}
\mathcal{J}_{\alpha}^5\, = \, \epsilon_{\alpha\mu} \mathcal{J}^{\mu}$$ Formula (\[axialvector\]) is a consequence of 2-dimensional geometry: when $\mathcal{J}^{\mu} $ is a vector, $\mathcal{J}_{\alpha}^5$ defined by (\[axialvector\]) is an axial vector. More explicitly, (\[axialvector\]) is seen in a 2-dimensional gamma matrix identity. $$\label{gamma}
\gamma_{\alpha}\gamma^5 \, = \, \epsilon_{\alpha\mu}\gamma^{\mu}$$ Therefore, the correlator ${^5\Pi}_{\alpha}^{\ \nu}$ of $\mathcal{J}_{\alpha}^5$ with $\mathcal{J}^{\nu} $ can be simply obtained from $\Pi^{\mu \nu}$ as $^5\Pi_{\alpha}^{\ \nu} \, = \, \epsilon_{\alpha\mu} \Pi^{\mu\nu}$. Moreover, once a transverse form for $\Pi^{\mu\nu}$ is fixed by gauge invariance, $^5\Pi_{\alpha}^{\ \nu}$ fails to be transverse in the $\alpha$ index; the divergence of the axial vector current is anomalous. $$\label{anomaly}
\partial^{\alpha} \mathcal{J}_{\alpha}^5 \, = \, -{e \over 2\pi} \epsilon^{\mu\nu} F_{\mu\nu}\, = \,
{e \over \pi} F$$ In the second equality we have introduced the (pseudo) scalar $F$, dual in two dimensions to the anti symmetric $F_{\mu\nu} \, \equiv \, \epsilon_{\mu\nu} F$
The anomaly provides an immediate derivation of the mass [@Farhi:ws1982]. We begin with the gauge field equation of motion that follows from (\[lagrangian\]). $$\label{equation}
\partial_{\mu} F^{\mu\nu} \, = \, e \mathcal{J}^{\nu}$$ In terms of the dual field strength $F$ this reads $$\label{equation1}
\epsilon^{\mu\nu} \partial_{\mu} F \, = \, e \mathcal{J}^{\nu}.$$ The $\epsilon$ symbol may be transferred to the right side and $\mathcal{J}^{\nu}$ becomes replaced by its dual $\mathcal{J}_{\alpha}^5$. $$\label{equationdual}
\partial_{\alpha} F \, = \, -\, e \mathcal{J}_{\alpha}^{5}$$ A further divergence gives the d’Alembertian on the left and the anomaly (\[anomaly\]) on the right. $$\label{Fequation}
\partial^2 F \, + \, {e^2 \over \pi} F \, = \, 0$$ This demonstrates that the pseudoscalar $F$ acquires a mass, $m^{\, 2} \, = \, {e^2 \over \pi}$.
Topological Entities in the Schwinger Model
===========================================
The 2-dimensional anomaly is proportional to $- F \, = \, {1\over 2} \epsilon^{\mu\nu}
F_{\mu\nu}$, which is recognized as the 2-dimensional Chern-Pontryagin density $\mathcal{P}_{\, 2}$. $$\label{p2}
\mathcal{P}_2 \, = \, {1 \over 2} \, \epsilon^{\mu\nu} F_{\mu\nu}.$$ Furthermore, the gauge potential $A_{\mu}$ is dual to the Chern-Simons current $\mathcal{C}_{\ 2}^{\alpha}$, $$\label{c2duality }
\mathcal{C}_2^{\alpha} \, \equiv \, \epsilon^{\alpha\mu} A_{\mu},$$ whose divergence forms the Chern-Pontryagin density [@'tHooft:2005cq]. $$\label{cdiv}
\partial_{\alpha} \mathcal{C}_2^{\alpha} \, = \,\epsilon^{\alpha\mu} \partial_{\alpha} A_{\mu} \, =
\, {1\over 2} \epsilon^{\alpha\mu} F_{\alpha\mu} \, = \, \mathcal{P}_2$$
The bosonic portion of the Lagrange density for the Schwinger model may be written in terms of these topological entities.
$$\begin{aligned}
\label{p2lagrangian}
\mathcal{L}_2 \, &= &\, -{1\over 4} F^{\mu\nu}F_{\mu\nu}\, - e \, \mathcal{J}^{\mu} A_{\mu} \, =
{1\over 2} F^2 \, - \, e A_{\mu} \epsilon^{\mu\alpha} \mathcal{J}_{\alpha}^5 \\ \nonumber
&=& \,
{1\over 2} \, \mathcal{P}_2^2 \, + \, e \, \mathcal{C}_2^{\alpha} \mathcal{J}_{\alpha}^5\end{aligned}$$
Moreover, since $\mathcal{C}_2^{\alpha}$ and $A_{\mu}$ are linearly related, it makes no difference which one is the fundamental variable. Thus varying $\mathcal{C}_2^{\alpha}$ in (\[p2lagrangian\]) gives (\[equationdual\]) directly as the equation of motion. $$\label{p2equation}
-\partial_{\alpha} \mathcal{P}_2 \, + \, e\, \mathcal{J}_{\alpha}^5 \, = \, 0$$ A further divergence and the anomaly equation (\[anomaly\]) reproduce (\[Fequation\]), since $\mathcal{P}_{\, 2} \, = \, - \, F$.
It is this last, topological reformulation of the Schwinger model that we shall take to four dimensions. However, we must still address an important point that will arise in the 4-dimensional theory. Observe that the equation of motion (\[equationdual\]) or (\[p2equation\]) entails an integrability condition: Since the (axial) vector $\mathcal{J}_{\alpha}^5$ is set equal to a gradient of (the pseudoscalar) $\mathcal{P}_2$, it must be that the curl of the axial vector vanishes. Equivalently, the dual of the axial vector must be divergence-free; [*viz.*]{} the vector current must be conserved. Of course the same integrability condition is seen in the original vector formulation of the model, with equation of motion (\[equation\]), which entails conservation of the vector current (dual to the axial vector current).
But let us suppose that we have dynamical information only about the topological variables, and do not know whether the current dual to the axial vector current is conserved. (This is the situation that we shall meet in four dimensions.) Then we must reformulate our theory in such a way that the integrability condition is avoided.
This reformulation in two dimensions proceeds by introducing two Stückelberg fields $\textsl{p}$ and $\textsl{q}$ into $\mathcal{L}_2$. $$\label{stuck2}
\mathcal{L}_2' \, = \, {1 \over 2} \mathcal{P}_2^2 \, + \, e (\mathcal{C}_2^{\alpha} \, + \,
\epsilon^{\alpha\beta} \partial_{\beta} \textsl{p}) (\mathcal{J}_{\alpha}^5 \, + \, \epsilon_{\alpha\gamma}
\partial^{\gamma} \textsl{q})$$ Upon varying $\mathcal{C}_2^{\alpha}$, (\[p2equation\]) becomes replaced by $$\label{p2equation1}
-\partial_{\alpha} \mathcal{P}_2 \, + \, e\, (\mathcal{J}_{\alpha}^5 \, + \, \epsilon_{\alpha\gamma} \partial^{\gamma} \textsl{q}) \, = \, 0.$$ Additionally, variation of *p* and *q* give, respectively $$\label{peq}
\partial_{\alpha} \epsilon^{\alpha\beta} \mathcal{J}_{\beta}^5 \, + \, \partial^2 \textsl{q}\, = \, 0,$$ $$\label{qeq}
\partial^{\alpha} \epsilon_{\alpha\beta} \mathcal{C}_2^{\beta} \, + \, \partial^2 \textsl{p}\, = \, 0.$$
The integrability condition on (\[p2equation1\]) demands that the curl of $\mathcal{J}_{\, \alpha}^{\, 5} \, + \, \epsilon_{\alpha\gamma}\partial^{\gamma} \textsl{q}$ vanish, but this is secured by (\[peq\]). This equation determines a non-trivial value for $\textsl{q}$ if the curl of $\mathcal{J}_{\, \alpha}^{\, 5}$ is non-vanishing, while (\[qeq\]) fixes an innocuous value for $\textsl{p}$. Finally we observe that the divergence of (\[p2equation1\]) annihilates the $\textsl{q}$ - dependent term, leaving in the end the previous equation (\[Fequation\]).
We may understand the role of the Stückelberg fields by reverting to the original vector variables. Then the interaction part of $\mathcal{L}_{\, 2}'$ in (\[stuck2\]) reads $$\label{intl2}
\mathcal{L}_{2I}' \, = \, - e (\mathcal{J}^{\mu} \, + \, \partial^{\mu}\textsl{q})
(A_{\mu} \, + \, \partial_{\mu} \textsl{p}),$$ and (\[peq\]), (\[qeq\]) have respective counterparts in $$\label{peq1}
\partial_{\mu} \mathcal{J}^{\mu} \, + \, \partial^2 \textsl{q}\, = \, 0,$$ $$\label{qeq1}
\partial^{\mu} \mathcal{A}_{\mu} \, + \, \partial^2 \textsl{p}\, = \, 0.$$ Eliminating $\textsl{p}$ and $\textsl{q}$ from (\[intl2\]) with the help of (\[peq1\]), (\[qeq1\]) leaves $$\label{l2eff }
\mathcal{L}_{2I}' \, = \, - e \mathcal{J}^{\mu} \left (\delta_{\mu}^{\nu} \, - \, {\partial_{\mu}\partial^{\nu}
\over \partial^2}\right ) A_{\nu}.$$
This shows that the Stückelberg fields ensure that the interaction occurs only between transverse components of $\mathcal{J}^{\mu}$ and $A_{\mu}$. For yet another perspective on the role of the Stückelberg fields, note that $-e \int \mathcal{J}^{\mu} A_{\mu}$ is not gauge invariant ( $A_{\mu} \rightarrow A_{\mu} \, + \, \partial_{\mu}\theta$) when $\mathcal{J}^{\mu}$ is not conserved. However, the combination $A_{\mu} \, + \, \partial_{\mu} \mbox{ \it p}$ is always gauge invariant because $\mbox{\it p}$ can transform as $\mbox{\it p} \, \mbox{-} \, \theta$. Finally observe that eliminating the Stückelberg fields in (\[p2equation\]) with the help of (
\[peq\]) and the anomaly equation (\[anomaly\]) leaves $$\label{P2eq }
\partial_{\mu} \left ( \mathcal{P}_2 \, + \, {e^2/\pi \over \partial^2} \mathcal{P}_2\right ) \, =\, 0$$ This is equivalent to (\[p2equation\]), but carries no integrability condition. Thus we see that the Stückelberg modification overcomes difficulties, which arise when the current dual to the axial vector is not conserved.
For a 4-dimensional generalization of the previous, we adopt the formulation of the 2-dimensional model, presented in Section 2 in terms of the Chern-Pontryagin density and Chern-Simons current, now promoted to four dimensions, $\mathcal{P}_4$ and $\mathcal{C}_4^{\alpha}$ respectively, with the latter coupling to an axial vector current $\mathcal{J}_{\alpha}^5$ whose divergence is anomalous. The topological entities are constructed from gauge potentials, which we take to be Abelian or non-Abelian; in either case $\mathcal{P}_4$ and $\mathcal{C}_4^{\alpha}$ remain gauge singlets. $$\begin{aligned}
\label{p4}
\mathcal{P}_4 &\equiv& {1 \over 2} \epsilon^{\alpha\beta\mu\nu} F_{\alpha\beta}^aF_{\mu\nu}^a \,
=\, ^*F^{\mu\nu~a} F_{\mu\nu}^a \\ \nonumber
F_{\mu\nu}^a \, &\equiv& \, \partial_{\mu} A_{\nu}^a \, - \partial_{\nu} A_{\mu}^a\, + \, f^{abc} A_{\mu}^bA_{\nu}^c,
~~~~^*F^{\alpha\beta} \equiv {1\over 2} \epsilon^{\alpha\beta\mu\nu} F_{\mu\nu}\end{aligned}$$
$$\label{c4}
\mathcal{C}_4^{\alpha} \equiv 2 \epsilon^{\alpha\mu\nu\omega} (A_{\mu}^a\partial_{\nu} A_{\omega}^a
\, + \, {1\over 3} f^{abc} A_{\mu}^aA_{\nu}^bA_{\omega}^c)$$
$$\label{c4div}
\partial_{\alpha} \mathcal{C}_4^{\alpha} \, = \, \mathcal{P}_4$$
Here $f^{abc}$ are the structure constants of the appropriate Lie algebra.
Unlike in the 2-dimansional case, the Chern-Simons current is not linear in the gauge vector potential; nevertheless we remain with the potential as the fundamental dynamical variable (see however below). The variation of the Chern-Simons current reads $$\label{deltac4}
\delta \mathcal{C}_4^{\alpha} \, = \, 4 ^*F^{\alpha\mu~a}\delta A_{\mu}^a \, - \, 2 \epsilon^{\alpha\nu\omega\mu} \partial_{\nu} (A_{\omega}^a\delta A_{\mu}^a).$$ A further difference from the Schwinger model is that there is no reason to suppose that the dual to the 4-dimensional axial vector current is conserved. On the level of 4-dimensional gamma matrices, the duality relation is $$\label{epsilon}
\epsilon^{\mu\nu\omega\alpha}\gamma_{\alpha}\gamma^5\, = \, g^{\mu\nu}\gamma^{\omega}\, -\,
g^{\mu\omega}\gamma^{\nu} \, + \, g^{\nu\omega}\gamma^{\mu}\, - \, \gamma^{\mu}\gamma^{\nu} \gamma^{\omega}.$$ It is improbable that fermion dynamics (here unspecified) would leave conserved the current dual to the axial vector current. But this is not an obstacle to our construction, because we can employ the Stückelberg formalism, as explained in the previous Section, to overcome the difficulty.
Thus the Lagrange density that we adopt is $$\label{L4}
\mathcal{L}_4' \, = \, {1\over 2} \mathcal{P}_4^2 \, + \, \Lambda^2 ( \mathcal{C}_4^{\alpha}\,
+ \, \partial_{\beta} \textsl{p}^{\alpha\beta}) (\mathcal{J}_{\alpha}^5 \, +
\, \partial^{\gamma} \textsl{q}_{\alpha\gamma}).$$ The Stückelberg fields $\textsl{p}^{\alpha\beta}$ and $\textsl{q}_{\alpha\gamma}$ are anti symmetric in their indices; $\Lambda^2$ carries mass-squared dimension; the axial vector current possesses an anomalous divergence. $$\label{anomaly4}
\partial^{\alpha} \mathcal{J}_{\alpha}^5 \, = \, - N~^*F^{\mu\nu~a}F_{\mu\nu}^a\, = \, -N\mathcal{P}_4$$ $N$ is a numerical coupling constant, taken positive.
Variation of the $\mathcal{L}_4'$ action with respect to $A_{\mu}^a$ gives , with the help of (\[deltac4\]), $$\label{varl4 }
\int \left ( -\partial_{\alpha} \mathcal{P}_4\, + \, \Lambda^2 (\mathcal{J}_{\alpha}^5 \, + \, \partial^{\gamma} \textsl{q}_{\alpha\gamma}) \right ) \delta \mathcal{C}_4^{\alpha} \, =$$ $$\nonumber
\int \left [ 4 \left ( -\partial_{\alpha} \mathcal{P}_4\, + \, \Lambda^2 (\mathcal{J}_{\alpha}^5 \, + \, \partial^{\gamma} \textsl{q}_{\alpha\gamma}) \right )\, ^*F^{\alpha\mu~a}\, -
2\epsilon^{\alpha\nu\omega\mu} A_{\nu}^a\partial_{\omega}
\left ( -\partial_{\alpha} \mathcal{P}_4\, + \, \Lambda^2 (\mathcal{J}_{\alpha}^5 \, + \, \partial^{\gamma} \textsl{q}_{\alpha\gamma}) \right ) \right ] \delta A_{\mu}^a,$$ so that the equation of motion demands $$\label{demand}
2 \left ( -\partial_{\alpha} \mathcal{P}_4\, + \, \Lambda^2 (\mathcal{J}_{\alpha}^5 \, + \, \partial^{\gamma} \textsl{q}_{\alpha\gamma}) \right )\, ^*F^{\alpha\mu~a}\, -
\epsilon^{\alpha\mu\nu\omega} A_{\nu}^a\partial_{\omega} \, \Lambda^2 (\mathcal{J}_{\alpha}^5 \, + \, \partial^{\gamma} \textsl{q}_{\alpha\gamma}) \, = \, 0.$$ Variation of the two Stückelberg fields yields the equations $$\label{deltap}
\partial_{\alpha} (\mathcal{J}_{\beta}^5 \, + \, \partial^{\gamma}\textsl{q}_{\beta\gamma}) \, - \,
\alpha \leftarrow\rightarrow \beta \, =\, 0,$$
$$\label{deltaq}
\partial^{\alpha} (\mathcal{C}_4^{\beta} \, + \, \partial_{\gamma}\textsl{p}^{\beta\gamma}) \, - \,
\alpha \leftarrow\rightarrow \beta \, = \, 0.$$
The first of these allows setting to zero the second member of (\[demand\]), while in the first member of that equation we may strip away $^*F^{\alpha\mu~a}$ with the help of the identity $$\label{Fidentity}
^*F^{\alpha\mu}F_{\mu\nu} \, = \, -{1\over 4} \delta_{\nu}^{\alpha}\, \mathcal{P}_4.$$ Consequently (provided $\mathcal{P}_4 \, \neq 0$) we are left with $$\label{gradp4}
-\partial_{\alpha} \mathcal{P}_4\, + \, \Lambda^2 (\mathcal{J}_{\alpha}^5 \, + \, \partial^{\gamma} \textsl{q}_{\alpha\gamma})\, = \, 0.$$ (Even though we varied $A^a_\mu$, which enters non-lineary into $\mathcal{C}^\alpha_4$, the final equation (\[gradp4\]) also results by simply varying the composite $\mathcal{C}^\alpha_4$ in (\[L4\]). This demonstrates the robustness of the derivation.)
The integrability condition on this equation is satisfied by virtue of (\[deltap\]). Taking another divergence of (\[gradp4\]) annihilates the Stückelberg field because of its anti symmetry, while (\[anomaly4\]) provides the divergence for $\mathcal{J}_{\alpha}^5$. Thus we are left with $$\label{p4final}
\partial^2 \mathcal{P}_4 \, + \, N\, \Lambda^2 \, \mathcal{P}_4 \, = \, 0.$$ This shows that the pseudoscalar $\mathcal{P}_4$ has acquired the mass, $m^{\, 2} = N\Lambda^2$.
By taking the divergence of (\[deltap\]), we find from (\[anomaly4\]) $$\label{j5eq}
\mathcal{J}_{\beta}^5 \, + \, \partial^{\gamma} \textsl{q}_{\beta\gamma} \, = \, -{N \over \partial^2} \partial_{\beta}\, \mathcal{P}_4.$$ Inserting this in (\[gradp4\]) yields $$\label{p4final1}
\partial_{\alpha} \left ( \mathcal{P}_4 \, + \, {N\, \Lambda^2 \over \partial^2} \, \mathcal{P}_4
\right ) \, = \, 0,$$ which is equivalent to (\[p4final\]) , but does not entail integrability conditions.
Conclusion
==========
While the 4-dimensional transposition of the 2-dimensional Schwinger model succeeds in generating a mass for a pseudoscalar, just as in the 2-dimensional case, there are various shortcomings. To these we now call attention.
The principal defect is the absence of dynamics that should produce the anomaly for the axial vector current. In the Schwinger model, the same dynamics and the same degrees of freedom that generate the mass are also responsible for the anomaly (\[anomaly\]). In the 4-dimensional theory we must posit the anomaly (\[anomaly4\]) separately from the mass generating dynamics. Moreover, our final result is that $\mathcal{P}_4$ propagates as a free massive field. Additional dynamics must be specified to describe interactions.
A related question concerns the role in physical theory for our Lagrangian (\[L4\]). Since it involves dimension eight ($\mathcal{P}_4$) and dimension six ( $\mathcal{C}_4^{\alpha} \mathcal{J}_{\alpha}^5$) operators, it should be viewed as an effective Lagrangian. In this connection, observe that the Born-Infeld action and the radiatively induced Euler-Heisenberg action both contain the Abelian $(^*F^{\mu\nu}F_{\mu\nu})^2$ quantity in a weak-field expansion \[also accompanied by an $(F^{\mu\nu}F_{\mu\nu})^2$ term\].
The kinetic portion of the Lagrangian in the Weyl ($A_{\ 0} \, = \, 0$ ) gauge involves $\dot{A}_i \dot{A}_j B^iB^j$ where $B^i$ is the magnetic field. Canonical analysis and quantization with such a kinetic term faces difficulties because the “metric” on $A_i$ space, [*viz.*]{} $B^iB^j$, is singular. But this poses no problem if our Lagrangian is used for phenomenological purposes, with the semi-classical addition of quantum effects through the chiral anomaly.
The $U(1)$ character of our anomalous current and the presence in our theory of the Chern-Pontryagin quantity suggest that here we are dealing with the problems of the unwanted axial $U(1)$ symmetry and the mass of the $\eta'$ meson. Conventionally these issues are resolved by instantons [@'tHooft:1986nc]. Here we offer a phenomenological description. We relate the axial vector current to the $\eta\ '$ field, $$\label{eta}
\mathcal{J}_{\alpha}^5 \, = \, Z \partial_{\alpha} \eta' / \Lambda$$ ($Z$ is a normalization) and add an $\eta'$ kinetic term to (\[L4\]). $$\label{leta}
\mathcal{L}_{\eta'} \, = \, {1 \over 2} \mathcal{P}_4^2 \, + \, Z \Lambda \mathcal{C}_4^{\alpha}
\partial_{\alpha}\eta' \, + \, {1 \over 2} \partial_{\alpha}\eta'\partial^{\alpha} \eta'$$ \[We dispense with the Stückelberg fields because the dual of the current in (5.1) is conserved.\] Observe that the $\eta'$ field enjoys a constant shift symmetry, as befits the quadratic portion of a Goldstone field Lagrangian. The equations that follow from varying $A_{\ \mu}^{\ a}$ and $\eta'$ respectively, are $$\label{Aeq}
\partial_{\alpha} (\mathcal{P}_4 \, - \, Z\Lambda \eta') \, ^*F^{\alpha\mu~a} \, = \, 0$$ $$\label{etaeq}
\partial^2 \eta' \, + \, Z \Lambda \mathcal{P}_4\, = \, 0$$ Together the two imply $$\label{p4mass}
\partial^2 \mathcal{P}_4 \, + \, Z^2 \Lambda^2 \mathcal{P}_4\, = \, 0.$$ As before, a mass is generated.
This may also be seen by rewriting the Lagrangian in (\[leta\]), apart from a total derivative, as $$\begin{aligned}
\label{leta1}
\mathcal{L}_{\eta'} \, &= &\, {1 \over 2} \mathcal{P}_4^2 \, - \, Z \Lambda \mathcal{P}_4\eta' \, + \, {1 \over 2} \partial_{\alpha}\eta'\partial^{\alpha} \eta'\\ \nonumber
&=& {1 \over 2} (\mathcal{P}_4\, - \, Z\Lambda \eta')^2 \, + \, {1\over 2} \partial_{\alpha}\eta'\partial^{\alpha}\eta' \, - \, {1 \over 2} Z^2\Lambda^2 \eta'^2.\end{aligned}$$ With $Z\Lambda \eta'$ absorbed by $\mathcal{P}_4$, we see that $\eta'$ decouples, but carries a mass [@Witten:etel].
In the case of 4-dimensional QCD with massless quark flavor(s), equation (\[p4mass\]) can be obtained without any assumptions about the dependence of the effective Lagrangian on the $\eta'$ meson. We only need to assume that the effective Lagrangian contains the first $\mathcal{P}_{\ 4}^{\ 2}$ term in (\[leta1\]). The analog of the second term is automatically generated from the anomaly diagram (Fig. 2) that correlates $\mathcal{C}_4^{\alpha}$ and $\mathcal{J}_{\alpha}^5$.\
\[PDFpar\]
\
and $\mathcal{J}_{\beta}^5$
\
The diagram generates the following operator $$\label{contact}
\Lambda^2 \mathcal{C}_4^{\alpha} {\partial_{\alpha} \partial^{\beta}\over \partial^2} \mathcal{J}_{\beta}^5,$$ where $\Lambda^2$ arises as a momentum cut off. This expression is also what one obtains from (\[L4\]) after eliminating the Stückelberg fields $\textsl{p}^{\, \alpha\beta}$ and $\textsl{q}_{\alpha\gamma}$ through their equations of motion (4.10), (4.11). Thus massless quark dynamics due to the anomaly substitute the effect of the Stückelberg fields. Variation with respect to $A_{\mu}^a$ yields the analog of equation (\[gradp4\]). $$\label{gradp4analog}
-\partial_{\alpha} \mathcal{P}_4 \, + \, \Lambda^2 {\partial_{\alpha} \over \partial^2} \partial^{\beta}
\mathcal{J}_{\beta}^5 \, = \, 0$$ Using the anomalous divergence relation (\[anomaly4\]), we arrive to the equation (\[p4final1\]), which is equivalent to (\[p4mass\]). Because $\mathcal{P}_4$ acquires a mass, its expectation value in the QCD vacuum must vanish. This explains why QCD solves both $U(1)$ and the strong CP problems in the zero quark mass limit [@dvali2005].
Similar effects should be present in all even dimensions, but the singularity structure and the required dimensional parameter (analog of the 2- and 4-dimensional $e$ and $\Lambda$) will change.
In conclusion we observe that although both the 2- and 4-dimensional models are formulated in terms of topological entities ($\mathcal{P}, \mathcal{C}^{\, \alpha} $), they are not topological theories. Examining (3.6), (4.6) we see that the Chern-Simons/axial vector interaction term ($\mathcal{C}^{\, \alpha} \mathcal{J}_{\alpha}^5$) is a geometric scalar density and can be integrated over a manifold in a diffeomrphism invariant way, without introducing a metric tensor. However, for the kinetic term ($\mathcal{P}^{\ 2}$) to be a scalar density it must be divided by $\sqrt{g}$. (In this discussion we ignore the Stückelberg terms.) Without this metric factor the theory is not invariant against all diffeomorphisms, but only against the “volume” preserving ones with unit Jacobian.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported by the U.S. Department of Energy under cooperative research agreement No. DE-FG02-05ER41360.
[99]{}
J. Schwinger, [*Phys. Rev.*]{} [**125**]{}, 397 (1962); [*Phys. Rev.*]{} [**128**]{}, 2425 (1962). G. Dvali, R. Jackiw and S.-Y. Pi, [*Phys. Lett.*]{} [**96**]{}, 081602 (2006).
The anomaly-based approach to the Schwinger model is explained in the Introduction to “Dynamical Gauge Symmetry Breaking", E. Farhi and R. Jackiw, eds. (World Scientific, Singapore, 1982).
The relevant topological structures are discussed by [*e.g.*]{} R. Jackiw in “Fifty Years of Yang-Mills Theory", G. ’t Hooft, ed. (World Scientific, Singapore, 2005).
G. ’t Hooft, Phys. Rept. [**142**]{}, 357 (1986). See also E. Witten, [*Nucl. Phys.*]{} [**B156**]{}, 269 (1979); G. Veneziano, [*Nucl. Phys.*]{} [**B159**]{}, 213 (1979) and [*Phys. Lett.*]{} [**B95**]{}, 90 (1980); A. Aurillia, Y. Takahashi and P.K. Townsend, [*Phys. Lett.*]{} [**B95**]{}, 265 (1980); G. Rosenzweig, J. Schechter and C. Trahern, [*Phys. Rev.*]{} [**D21**]{}, 3388 (1980).
G. Dvali, \[arxiv: hep-th/0507215\].
[^1]: Majorana Memorial, Catalina, Italy, October 2006
|
---
author:
- 'F. Haberl'
- 'P. Eger'
- 'W. Pietsch'
- 'R.H.D. Corbet'
- 'M. Sasaki'
bibliography:
- 'general.bib'
- 'myrefereed.bib'
- 'mcs.bib'
- 'hmxb.bib'
- 'ism.bib'
- 'ins.bib'
- 'cv.bib'
date: 'Received 8 February 2008 / Accepted 6 March 2008'
title: |
XMM-Newton observations of the Small Magellanic Cloud:\
XMMUJ004814.1-731003, a 25.55 s Be/X-ray binary pulsar [^1]
---
Introduction
============
The region around the emission nebula N19 in the southwestern part of the SMC shows complex X-ray emission from several supernova remnants and high mass X-ray binaries (HMXBs). Several X-ray pulsars which are most likely associated with Be/X-ray binaries, the major subclass of HMXBs, were detected with ASCA and RXTE, but could not be accurately located and therefore, lack a clear optical identification. Southeast of N19, ASCA detected a 9.1 s pulsar [@2000PASJ...52L..63U] where ROSAT PSPC images show two close hard X-ray sources, both possibly contributing to the ASCA source . Both these objects are most likely Be/X-ray binaries and it is not clear which is the 9.1 s pulsar [@2005MNRAS.356..502C and references therein]. Another nearby ASCA source () was classified as candidate Be/X-ray binary by @2003PASJ...55..161Y from its X-ray spectral properties. In the error circle of the [H${\alpha}$]{} emission line star is located, suggesting an association with a Be star. However, this requires confirmation by an improved X-ray position. ROSAT did not detect any source at the ASCA position but a faint source was seen during an early [XMM-Newton]{} observation in August 2002 [@2005MNRAS.362..879S]. RXTE discovered pulsations with periods of 16.6 s and 25.5 s during an observation in September 2000, viewing the area near the southwestern edge of the SMC [@2002ApJ...567L.129L]. The pulse period from the latter source was later argued to be the harmonic of a 51 s spin period [@2005ApJS..161...96L]. Both pulsars could never be better localized and optically identified.
In the course of our AO5 [XMM-Newton]{} program to investigate candidates for HMXBs in the SMC, we observed the region around N19. The [XMM-Newton]{} source detected by @2005MNRAS.362..879S was seen again and we analysed the EPIC data to investigate the nature of this object, designated . Here we report on results from a temporal and spectral analysis of the X-ray data of and identify as optical counterpart a Be star. This adds to the numerous Be/X-ray binary pulsars known in the SMC .
Data analysis and results
=========================
We observed the field around the emission nebula N19 with [XMM-Newton]{} on 2006 October 5. The EPIC-MOS and EPIC-PN cameras were operated in imaging mode (see Table \[tab-obs\]). For the X-ray analysis we used the [XMM-Newton]{} Science Analysis System (SAS) version 7.1.0 supported by tools from the FTOOL package together with XSPEC version 11.3.2p for spectral modelling.
A source, not seen by ROSAT was detected near the center of the field of view and inside the error circle of the ASCA source [@2003PASJ...55..161Y]. After astrometric boresight correction [see also @2008arXiv0801.4679H] we determined the position of the source to R.A. = 00 48 14.10 and Dec. = $-$73 10 04.0 (J2000.0) using the SAS standard maximum likelihood technique for source detection and assign the name (hereafter ). The 1$\sigma$ position error is 1.2, dominated by the remaining systematic uncertainty of 1.1. Due to background flaring activity during part of the observation we applied a background screening before our source detection analysis, but used the full exposure times for the extraction of light curves and spectra of individual sources. The resulting net exposures are listed in Table \[tab-obs\].
\[sect-obs\]
[cccc]{} & &\
& & &\
& &\
0404680101 & 00 47 36.0 & -73 08 24.0 & 1249\
& & &\
& &\
& & &\
PN FF thin & 00:44:47 & 06:51:09 & 19.04 / 6.84\
M1 FF medium & 00:22:05 & 06:50:49 & 22.76 / 7.66\
M2 FF medium & 08:22:05 & 06:50:54 & 22.77 / 7.67\
$^{(1)}$ FF: full frame CCD readout mode with 73 ms frame time for PN and 2.6 s for MOS; thin and medium optical blocking filters. $^{(2)}$ Left: Full exposure times as used for spectral and temporal analysis. Right: Exposure after removing intervals of high background (used for source detection analysis, see text). \[tab-obs\]
The precise X-ray position inferred from the EPIC data allowed us to identify the optical counterpart of . The source position is incompatible with the emission line star (46 away from the X-ray position) which was suggested as optical counterpart for , but another star with optical properties of a B star is found at the [XMM-Newton]{} X-ray position (Fig. \[xmmp-fc\]). In Table \[tab-ids\] optical brightness and colours taken from the UBVR CCD Survey of the Magellanic Clouds [@2002ApJS..141...81M], the Magellanic Clouds Photometric Survey [MCPS, @2002AJ....123..855Z] and the OGLE BVI photometry catalogue [@1998AcA....48..147U] are given.
To investigate longterm brightness variations of the optical counterpart we retrieved light curves in the I-band from the OGLE photometry database [star 171264; @2005AcA....55...43S; @1997AcA....47..319U] and the B- and R-band from the MACHO survey (star 212.15849.52). The light curves in all bands show a gradual fading over the total observing period with additional variations by $\sim$0.2 magnitudes superimposed (Fig. \[xmmp-opt\]). We applied an FFT analysis to the MACHO R- and B-band data with a magnitude error $< 0.05$ mag to determine whether these changes are periodic. We investigated the period range between 1 day and 1000 days which revealed peaks at short and long periods (Fig. \[xmmp-redfft\]). Peaks at 1.49 days, 1.81 days, 3.02 days and 5.93 days indicate similar periods as seen from other SMC Be/X-ray binaries [@2004AJ....127.3388S], but are probably too short for orbital periods. Broad peaks at long periods of a few hundred days ($\sim$220 days, $\sim$340 days and $\sim$400 days, $\sim$440 days and $\sim$ 680 days) indicate variations which are not strictly periodic.
[llcccccc]{} & & & & & & &\
& UBVR & 00 48 14.10 –73 10 04.0 & 15.25 & $+$0.13 & $-$0.64 & $+$0.12 & –\
& MCPS & 00 48 14.18 –73 10 03.9 & 15.30 & $+$0.26 & $-$0.50 & – & $-$0.21\
& OGLE & 00 48 14.13 –73 10 03.5 & 15.71 & $+$0.01 & – & – & $+$0.09\
\[tab-ids\]
The broad band X-ray light curve of indicates some intensity variations of the flux on time scales of 1$-$2 hours. But due to the low count rate (on average $\sim$0.03 [cts s$^{-1}$]{} in EPIC-PN) the statistics is too low for definite conclusions. We searched for X-ray pulsations in the EPIC data of , after correcting the photon arrival times to the solar system barycenter. The Fourier power spectra obtained from the EPIC-PN data in different energy bands revealed a peak at 25.5 s which is most significant when selecting energies above 1 keV. For the band 1.0$-$10.0 keV a power of $\sim$40 is obtained. To verify the period, also the MOS data were investigated. Peaks at the same frequency were present for the individual MOS light curves, but not significant on their own. Finally, combining the events from the two MOS cameras, also resulted in the highest peak at the same frequency. Combining all EPIC data yields a maximum power of $\sim$60 (Fig. \[xmmp-pspec\]). Although no significant power around 51 s is seen in the power spectrum, we investigated this period range using the Rayleigh $Z^2$ method with fundamental and one harmonic frequency involved. The resulting maximum $Z^2$ power is 69.5 with $Z^2_1$=2.9 for the fundamental (at 51.1 s) and $Z^2_2$=66.6 for the first harmonic (at 25.55 s). We conclude that there is no significant modulation with a period of 51 s. To derive an accurate value and error for the detected pulse period we used the Bayesian method [@1996ApJ...473.1059G] as described in @2000ApJ...540L..25Z. The pulse period is determined to 25.550$\pm$0.002 s (1$\sigma$ error).
We folded the X-ray light curves on the best period in the standard EPIC energy bands (0.2$-$0.5 keV, 0.5$-$1.0 keV, 1.0$-$2.0 keV, 2.0$-$4.5 keV, 4.5$-$10.0 keV and the broad band 0.2$-$10.0 keV) and show the resulting pulse profiles in Fig. \[xmmp-pulse\]. The pulse profile of is dominated by a broad main pulse which varies in shape with energy. At energies below 1 keV the high absorption (see below) strongly reduces the count rate and a modulation is only marginally seen. The pulsed fraction (derived from modelling the pulse profile with two sine waves) in the total energy band is (60$\pm$20)%. There is some indication for a higher pulsed fraction at energies below 1 keV and above 4.5 keV, but the number of counts in the individual bands is insufficient for a statistically justified statement.
For the analysis of the X-ray spectra we extracted pulse-phase averaged EPIC spectra for PN (single + double pixel events, PATTERN 0$-$4) and MOS (PATTERN 0$-$12) disregarding bad CCD pixels and columns (FLAG 0). The three EPIC spectra were simultaneously fit with an absorbed powerlaw model allowing for a constant normalization factor between the spectra. We used two absorption components, accounting for the Galactic foreground absorption [with a fixed hydrogen column density of 6[$\times 10^{20}$ cm$^{-2}$]{} and elemental abundances from @2000ApJ...542..914W] and the SMC absorption [with column density as free parameter in the fit and with metal abundances reduced to 0.2 as typical for the SMC; @1992ApJ...384..508R]. The best-fit (reduced $\chi^2$ = 0.82 for 27 degrees of freedom) powerlaw model yields a high absorption with [N$_{\rm H}$]{} = (5.2$\pm$2.0) [$\times 10^{22}$ cm$^{-2}$]{}, a photon index of 1.33$\pm$0.27, an observed flux of 3.5[$\times 10^{-13}$ erg cm$^{-2}$ s$^{-1}$]{} and a source luminosity of 2.1[$\times 10^{35}$ erg s$^{-1}$]{} [0.2$-$10.0 keV, assumed distance to the SMC of 60 kpc @2005MNRAS.357..304H]. Flux and luminosity refer to the values derived from the EPIC-PN spectrum, MOS1 and MOS2 yield somewhat lower values by 6% and 3%, respectively. Errors for spectral parameters denote 90% confidence limits. The EPIC spectra with the best-fit model are shown in Fig \[xmmp-spec\].
Discussion
==========
We report the discovery of pulsations in the X-ray flux of with a period of 25.550 s and propose as optical counterpart a V$\sim$15.5 mag star. The X-ray spectrum of is well represented by a simple powerlaw, typical for Be/X-ray binary pulsars. However, the high absorption and the relative faintness of the source (the flux of 3.5[$\times 10^{-13}$ erg cm$^{-2}$ s$^{-1}$]{} corresponds to a source luminosity of 2.1[$\times 10^{35}$ erg s$^{-1}$]{} at SMC distance) prevent a more detailed spectral analysis.
Following @2005MNRAS.356..502C, we estimate the spectral class of the optical counterpart from its B$-$V colour index to B3 or later, depending on the observed B$-$V index which is in the range 0.01 to 0.26 (Table \[tab-ids\]). This assumes an extinction correction to the SMC of E(B$-$V)=0.08 and an additional correction of (B$-$V)=$-$0.13 to account for the presence of a circumstellar disc [@2005MNRAS.356..502C]. However, in the Galaxy no Be/X-ray binaries with spectral type later than B2 are found which may indicate that this additional correction for is insufficient. This is supported by the large absorption seen in the X-ray spectrum of the pulsar. The significantly different values measured for B$-$V in the different photometric surveys also suggest, that the extinction varies with time (also a B$-$R colour index derived from the approximate B and R light curves in Fig. \[xmmp-opt\] slowly increased after the end of 1998). At least this variable extinction must originate from matter local to the binary system and is most likely due to changes in the disc of the Be star.
From the relation between spin and orbital period [for a recent version of the ‘Corbet’ diagram of SMC pulsars see @2005AJ....130.2220S], we expect an orbital period between $\sim$25 days and $\sim$150 days for . The optical light curves indicate various periodicities, which are either shorter or longer than this period range. Short periods between 3 and 11 days were also seen in other SMC Be/X-ray binaries and might be caused by a changing view of the Be disk region that is brightened by the neutron star @2004AJ....127.3388S. The star shows optical brightness variations on long time scales of a few hundred days which most likely are also associated with the Be star phenomenon, similar to e.g. XTEJ0103-728 [=SXP6.85; @2008MNRAS.384..821M], but with smaller amplitude.
The optical and X-ray properties identify as Be/X-ray binary pulsar in the SMC. The [XMM-Newton]{} source position is located within the error circle of , but the presence of another Be star (), which might be associated with , leaves it unclear if the ASCA source is identical with . Only a retrospective detection of the pulse period in the ASCA data would allow an unambiguous identification. The pulsar might be identical to the 25.55s pulsar seen by RXTE in September 2000 [@2002ApJ...567L.129L]. The RXTE observation was pointed at R.A. = 00 50 44.64 and Dec. = $-$73 16 04.8 which is 12 away from . Therefore was well within the RXTE PCA field of view of 1 FWHM. However, in more recent literature [@2005ApJS..161...96L; @2005MNRAS.356..502C; @2008arXiv0802.2118G] the RXTE pulsar is listed with a period of 51 s. Therefore, we can not exclude the possibility that the [XMM-Newton]{} and RXTE sources are two pulsars with different period.
The XMM-Newton project is supported by the Bundesministerium für Wirtschaft und Technologie/Deutsches Zentrum für Luft- und Raumfahrt (BMWI/DLR, FKZ 50 OX 0001) and the Max-Planck Society. The “Second Epoch Survey” of the southern sky was produced by the Anglo-Australian Observatory (AAO) using the UK Schmidt Telescope. Plates from this survey have been digitized and compressed by the ST ScI. Produced under Contract No. NAS 5-26555 with the National Aeronautics and Space Administration. This paper utilizes public domain data obtained by the MACHO Project, jointly funded by the US Department of Energy through the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48, by the National Science Foundation through the Center for Particle Astrophysics of the University of California under cooperative agreement AST-8809616, and by the Mount Stromlo and Siding Spring Observatory, part of the Australian National University.
[^1]: Based on observations with XMM-Newton, an ESA Science Mission with instruments and contributions directly funded by ESA Member states and the USA (NASA)
|
---
abstract: 'The quality of machine translation has increased remarkably over the past years, to the degree that it was found to be indistinguishable from professional human translation in a number of empirical investigations. We reassess Hassan et al.’s 2018 investigation into Chinese to English news translation, showing that the finding of human–machine parity was owed to weaknesses in the evaluation design—which is currently considered best practice in the field. We show that the professional human translations contained significantly fewer errors, and that perceived quality in human evaluation depends on the choice of raters, the availability of linguistic context, and the creation of reference translations. Our results call for revisiting current best practices to assess strong machine translation systems in general and human–machine parity in particular, for which we offer a set of recommendations based on our empirical findings.'
author:
- |
Samuel Läubli laeubli@cl.uzh.ch\
Institute of Computational Linguistics, University of Zurich Sheila Castilho sheila.castilho@adaptcentre.ie\
ADAPT Centre, Dublin City University Graham Neubig gneubig@cs.cmu.edu\
Language Technologies Institute, Carnegie Mellon University Rico Sennrich sennrich@cl.uzh.ch\
Institute of Computational Linguistics, University of Zurich Qinlan Shen qinlans@cs.cmu.edu\
Language Technologies Institute, Carnegie Mellon University Antonio Toral a.toral.ruiz@rug.nl\
Center for Language and Cognition, University of Groningen
bibliography:
- 'references.bib'
title: |
A Set of Recommendations for Assessing\
Human–Machine Parity in Language Translation
---
[UTF8]{}[gbsn]{}
Introduction {#sec:Introduction}
============
Machine translation (MT) has made astounding progress in recent years thanks to improvements in neural modelling [@Sutskever2014; @Bahdanau2014; @NIPS2017_7181], and the resulting increase in translation quality is creating new challenges for MT evaluation. Human evaluation remains the gold standard, but there are many design decisions that potentially affect the validity of such a human evaluation.
This paper is a response to two recent human evaluation studies in which some neural machine translation systems reportedly performed at (or above) the level of human translators for news translation from Chinese to English [@hassan2018achieving] and English to Czech [@Popel2018; @WMT2018].
Both evaluations were based on current best practices in the field: they used a source-based direct assessment with non-expert annotators, using data sets and the evaluation protocol of the Conference on Machine Translation (WMT). While the results are intriguing, especially because they are based on best practices in MT evaluation, @WMT2018 [p. 293] warn against taking their results as evidence for human–machine parity, and caution that “for well-resourced language pairs, an update of WMT evaluation style will be needed to keep up with the progress in machine translation.” We concur that these findings have demonstrated the need to critically re-evaluate the design of human MT evaluation.
Our paper investigates three aspects of human MT evaluation, with a special focus on assessing human–machine parity: the choice of raters, the use of linguistic context, and the creation of reference translations. We focus on the data shared by @hassan2018achieving, and empirically test to what extent changes in the evaluation design affect the outcome of the human evaluation.[^1] We find that for all three aspects, human translations are judged more favourably, and significantly better than MT, when we make changes that we believe strengthen the evaluation design. Based on our empirical findings, we formulate a set of recommendations for human MT evaluation in general, and assessing human–machine parity in particular. All of our data are made publicly available for external validation and further analysis.[^2]
Background {#sec:Background}
==========
We first review current methods to assess the quality of machine translation system outputs, and highlight potential issues in using these methods to compare such outputs to translations produced by professional human translators.
Human Evaluation of Machine Translation {#sec:BackgroundEvaluation}
---------------------------------------
The evaluation of MT quality has been the subject of controversial discussions in research and the language services industry for decades due to its high economic importance. While automatic evaluation methods are particularly important in system development, there is consensus that a reliable evaluation should—despite high costs—be carried out by humans.
Various methods have been proposed for the human evaluation of MT quality [c.f. @CASTILHO_2018_TQA]. What they have in common is that the MT output to be rated is paired with a translation hint: the source text or a reference translation. The MT output is then either adapted or scored with reference to the translation hint by human post-editors or raters, respectively.
As part of the large-scale evaluation campaign at WMT, two primary evaluation methods have been used in recent years: relative ranking and direct assessment [@bojar-etal_Cracker:2016]. In the case of relative ranking, raters are presented with outputs from two or more systems, which they are asked to evaluate relative to each other (e.g., to determine system A is better than system B). Ties (e.g., system A is as good or as bad as system B) are typically allowed. Compared to absolute scores on Likert scales, data obtained through relative ranking show better inter- and intra-annotator agreement [@CallisonBurch2007]. However, they do not allow conclusions to be drawn about the order of magnitude of the differences, so that it is not possible to determine *how much* better system A was than system B.
This is one of the reasons why direct assessment has prevailed as an evaluation method more recently. In contrast to relative ranking, the raters are presented with one MT output at a time, to which they assign a score between 0 and 100. To increase homogeneity, each rater’s ratings are standardised [@Graham2013]. Reference translations serve as the basis in the context of WMT, and evaluations are carried out by monolingual raters. To avoid reference bias, the evaluation can be based on source texts instead, which presupposes bilingual raters, but leads to more reliable results overall [@Bentivogli2018].
Assessing Human–Machine Parity {#sec:BackgroundParity}
------------------------------
@hassan2018achieving base their claim of achieving human–machine parity on a source-based direct assessment as described in the previous section, where they found no significant difference in ratings between the output of their MT system and a professional human translation. Similarly, @WMT2018 report that the best-performing English to Czech system submitted to WMT 2018 [@Popel2018] significantly outperforms the human reference translation. However, the authors caution against interpreting their results as evidence of human–machine parity, highlighting potential limitations of the evaluation.
In this study, we address three aspects that we consider to be particularly relevant for human evaluation of MT, with a special focus on testing human–machine parity: the choice of raters, the use of linguistic context, and the construction of reference translations.
#### Choice of Raters
The human evaluation of MT output in research scenarios is typically conducted by crowd workers in order to minimise costs. @CallisonBurch2009 shows that aggregated assessments of bilingual crowd workers are “very similar” to those of MT developers, and @Graham2017, based on experiments with data from WMT 2012, similarly conclude that with proper quality control, MT systems can be evaluated by crowd workers. @hassan2018achieving also use bilingual crowd workers, but the studies supporting the use of crowdsourcing for MT evaluation were performed with older MT systems, and their findings may not carry over to the evaluation of contemporary higher-quality neural machine translation (NMT) systems. In addition, the MT developers to which crowd workers were compared are usually not professional translators. We hypothesise that expert translators will provide more nuanced ratings than non-experts, and that their ratings will show a higher difference between MT outputs and human translations.
#### Linguistic Context
MT has been evaluated almost exclusively at the sentence level, owing to the fact that most MT systems do not yet take context across sentence boundaries into account. However, when machine translations are compared to those of professional translators, the omission of linguistic context—[e.g. ]{}\*, by random ordering of the sentences to be evaluated—does not do justice to humans who, in contrast to most MT systems, can and do take inter-sentential context into account [@VoigtJurafsky2012; @Wang2017]. We hypothesise that an evaluation of sentences in isolation, as applied by @hassan2018achieving, precludes raters from detecting translation errors that become apparent only when inter-sentential context is available, and that they will judge MT quality less favourably when evaluating full documents.
#### Reference Translations
The human reference translations with which machine translations are compared within the scope of a human–machine parity assessment play an important role. @hassan2018achieving used all source texts of the WMT 2017 Chinese–English test set for their experiments, of which only half were originally written in Chinese; the other half were translated from English into Chinese. Since translated texts are usually simpler than their original counterparts [@Laviosa1998b], they should be easier to translate for MT systems. Moreover, different human translations of the same source text sometimes show considerable differences in quality, and a comparison with an MT system only makes sense if the human reference translations are of high quality. @hassan2018achieving, for example, had the WMT source texts re-translated as they were not convinced of the quality of the human translations in the test set. At WMT 2018, the organisers themselves noted that “the manual evaluation included several reports of ill-formed reference translations” [@WMT2018 p. 292]. We hypothesise that the quality of the human translations has a significant effect on findings of human–machine parity, which would indicate that it is necessary to ensure that human translations used to assess parity claims need to be carefully vetted for their quality.
We empirically test and discuss the impact of these factors on human evaluation of MT in Sections \[sec:Raters\]–\[sec:ReferenceTranslations\]. Based on our findings, we then distil a set of recommendations for human evaluation of strong MT systems, with a focus on assessing human–machine parity ([Section \[sec:Recommendations\]]{}).
Translations {#sec:Translations}
------------
We use English translations of the Chinese source texts in the WMT 2017 English–Chinese test set [@Bojar2017] for all experiments presented in this article:
[H$_A$ ]{}
: The professional human translations in the dataset of @hassan2018achieving.
[H$_B$ ]{}
: Professional human translations that we ordered from a different translation vendor, which included a post-hoc native English check. We produced these only for the documents that were originally Chinese, as discussed in more detail in Section \[sec:Directionality\].
[MT$_1$ ]{}
: The machine translations produced by [@hassan2018achieving’s [-@hassan2018achieving]]{} best system (<span style="font-variant:small-caps;">Combo-6</span>),for which the authors found parity with [H$_A$ ]{}\*.
[MT$_2$ ]{}
: The machine translations produced by Google’s production system (Google Translate) in October 2017, as contained in [@hassan2018achieving’s [-@hassan2018achieving]]{} dataset.
Statistical significance is denoted by \* ($p\le.05$), \*\* ($p\le.01$), and \*\*\* ($p\le.001$) throughout this article, unless otherwise stated.
Choice of Raters {#sec:Raters}
================
Both professional and amateur evaluators can be involved in human evaluation of MT quality. However, from published work in the field [@Doherty_2017], it is fair to say that there is a tendency to “rely on students and amateur evaluators, sometimes with an undefined (or self-rated) proficiency in the languages involved, an unknown expertise with the text type" [@CASTILHO_2018_TQA p. 23].
Previous work on evaluation of MT output by professional translators against crowd workers by @Castilho2017crowdsourcing showed that for all language pairs (involving 11 languages) evaluated, crowd workers tend to be more accepting of the MT output by giving higher fluency and adequacy scores and performing very little post-editing. The authors argued that non-expert translators lack knowledge of translation and so might not notice subtle differences that make one translation more suitable than another, and therefore, when confronted with a translation that is hard to post-edit, tend to accept the MT rather than try to improve it.
Evaluation Protocol
-------------------
We test for difference in ratings of MT outputs and human translations between experts and non-experts. We consider professional translators as experts, and both crowd workers and MT researchers as non-experts.[^3]
We conduct a relative ranking experiment using one professional human ([H$_A$ ]{}\*) and two machine translations ([MT$_1$ ]{}and [MT$_2$ ]{}\*), considering the native Chinese part of the WMT 2017 Chinese–English test set (see [Section \[sec:Directionality\]]{} for details). The 299 sentences used in the experiments stem from 41 documents, randomly selected from all the documents in the test set originally written in Chinese, and are shown in their original order. Raters are shown one sentence at a time, and see the original Chinese source alongside the three translations. The previous and next source sentences are also shown, in order to provide the annotator with local inter-sentential context.
Five raters—two experts and three non-experts—participated in the assessment. The experts were professional Chinese to English translators: one native in Chinese with a fluent level of English, the other native in English with a fluent level of Chinese. The non-experts were NLP researchers native in Chinese, working in an English-speaking country.
The ratings are elicited with Appraise [@mtm12_appraise]. We derive an overall score for each translation ([H$_A$ ]{}\*, [MT$_1$ ]{}\*, and [MT$_2$ ]{}\*) based on the rankings. We use the TrueSkill method adapted to MT evaluation [@sakaguchi-post-vandurme:2014:W14-33] following its usage at WMT15,[^4] [i.e. ]{}\*, we run 1,000 iterations of the rankings recorded with Appraise followed by clustering (significance level $\alpha=0.05$).
Results
-------
Table \[t:evaluators\] shows the TrueSkill scores for each translation resulting from the evaluations by expert and non-expert translators. We find that translation expertise affects the judgement of [MT$_1$ ]{}and [H$_A$ ]{}\*, where the rating gap is wider for the expert raters.[^5] This indicates that non-experts disregard translation nuances in the evaluation, which leads to a more tolerant judgement of MT systems and a lower inter-annotator agreement ($\kappa=0.13$ for non-experts versus $\kappa=0.254$ for experts).
It is worth noticing that, regardless of their expertise, the performance of human raters may vary over time. For example, performance may improve or decrease due to learning effects or fatigue, respectively [@GONZALEZ201119]. It is likely that such longitudinal effects are present in our data. They should be accounted for in future work, [e.g. ]{}\*, by using trial number as an additional predictor [@toral_penovel_18].
Linguistic Context {#sec:Context}
==================
Another concern is the unit of evaluation. Historically, machine translation has primarily operated on the level of sentences, and so has machine translation evaluation. However, it has been remarked that human raters do not necessarily understand the intended meaning of a sentence shown out-of-context [@Wu2016], which limits their ability to spot some mistranslations. Also, a sentence-level evaluation will be blind to errors related to textual cohesion and coherence.
While sentence-level evaluation may be good enough when evaluating MT systems of relatively low quality, we hypothesise that with additional context, raters will be able to make more nuanced quality assessments, and will also reward translations that show more textual cohesion and coherence. We believe that this aspect should be considered in evaluation, especially when making claims about human–machine parity, since human translators can and do take inter-sentential context into account [@VoigtJurafsky2012; @Wang2017].
Evaluation Protocol {#sec:ContextEvaluationProtocol}
-------------------
We test if the availability of document-level context affects human–machine parity claims in terms of adequacy and fluency. In a pairwise ranking experiment, we show raters (i) isolated sentences and (ii) entire documents, asking them to choose the better (with ties allowed) from two translation outputs: one produced by a professional translator, the other by a machine translation system. We do not show reference translations as one of the two options is itself a human translation.
We use source sentences and documents from the WMT 2017 Chinese–English test set (see [Section \[sec:Translations\]]{}): documents are full news articles, and sentences are randomly drawn from these news articles, regardless of their position. We only consider articles from the test set that are native Chinese (see [Section \[sec:Directionality\]]{}). In order to compare our results to those of @hassan2018achieving, we use both their professional human ([H$_A$ ]{}\*) and machine translations ([MT$_1$ ]{}\*).
Each rater evaluates both sentences and documents, but never the same text in both conditions so as to avoid repetition priming [@FrancisSaenz2007]. The order of experimental items as well as the placement of choices ([H$_A$ ]{}\*, [MT$_1$ ]{}\*; left, right) are randomised.
We use spam items for quality control [@Kittur2008]: In a small fraction of items, we render one of the two options nonsensical by randomly shuffling the order of all translated words, except for 10at the beginning and end. If a rater marks a spam item as better than or equal to an actual translation, this is a strong indication that they did not read both options carefully.
We recruit professional translators (see [Section \[sec:Raters\]]{}) from [proz.com](proz.com), a well-known online market place for professional freelance translation, considering Chinese to English translators and native English revisers for the adequacy and fluency conditions, respectively. In each condition, four raters evaluate 50 documents (plus 5 spam items) and 104 sentences (plus 16 spam items). We use two non-overlapping sets of documents and two non-overlapping sets of sentences, and each is evaluated by two raters.
Results {#sec:ContextResults}
-------
Results are shown in [Table \[tab:mt-vs-a\]]{}. We note that sentence ratings from two raters are excluded from our analysis because of unintentional textual overlap with documents, meaning we cannot fully rule out that sentence-level decisions were informed by access to the full documents they originated from. Moreover, we exclude document ratings from one rater in the fluency condition because of poor performance on spam items, and recruit an additional rater to re-rate these documents.
We analyse our data using two-tailed Sign Tests, the null hypothesis being that raters do not prefer [MT$_1$ ]{}over [H$_A$ ]{}or vice versa, implying human–machine parity. Following WMT evaluation campaigns that used pairwise ranking [[e.g. ]{}\*, @WMT2013], the number of successes $x$ is the number of ratings in favour of [H$_A$ ]{}\*, and the number of trials $n$ is the number of all ratings except for ties. Adding half of the ties to $x$ and the total number of ties to $n$ [@EmersonSimon1979] does not impact the significance levels reported in this section.
Adequacy raters show no statistically significant preference for [MT$_1$ ]{}or [H$_A$ ]{}when evaluating isolated sentences ($x=86, n=189, p=.244$). This is in accordance with @hassan2018achieving, who found the same in a source-based direct assessment experiment with crowd workers. With the availability of document-level context, however, preference for [MT$_1$ ]{}drops from 49.5 to 37.0and is significantly lower than preference for human translation ($x=104, n=178, p<.05$). This evidences that document-level context cues allow raters to get a signal on adequacy.
Fluency raters prefer [H$_A$ ]{}over [MT$_1$ ]{}both on the level of sentences ($x=106, n=172, p<.01$) and documents ($x=99, n=143, p<.001$). This is somewhat surprising given that increased fluency was found to be one of the main strengths of NMT [@WMT2016], as we further discuss in [Section \[sec:Quality\]]{}. The availability of document-level context decreases fluency raters’ preference for [MT$_1$ ]{}\*, which falls from 31.7 to 22.0, without increasing their preference for [H$_A$ ]{}([Table \[tab:mt-vs-a\]]{}).
Discussion
----------
Our findings emphasise the importance of linguistic context in human evaluation of MT. In terms of adequacy, raters assessing documents as a whole show a significant preference for human translation, but when assessing single sentences in random order, they show no significant preference for human translation.
Document-level evaluation exposes errors to raters which are hard or impossible to spot in a sentence-level evaluation, such as coherent translation of named entities. The example in [Table \[tab:context-example\]]{} shows the first two sentences of a Chinese news article as translated by a professional human translator ([H$_A$ ]{}\*) and [@hassan2018achieving’s [-@hassan2018achieving]]{} NMT system ([MT$_1$ ]{}\*). When looking at both sentences (document-level evaluation), it can be seen that [MT$_1$ ]{}uses two different translations to refer to a cultural festival, “2016盂兰文化节", whereas the human translation uses only one. When assessing the second sentence out of context (sentence-level evaluation), it is hard to penalise [MT$_1$ ]{}for producing “2016 Python Cultural Festival”, particularly for fluency raters without access to the corresponding source text. For further examples, see [Section \[sec:Quality\]]{} and [Table \[tab:quality\_qualitative\]]{}.
Reference Translations {#sec:ReferenceTranslations}
======================
Yet another relevant element in human evaluation is the reference translation used. This is the focus of this section, where we cover two aspects of reference translations that can have an impact on evaluation: quality and directionality.
Quality {#sec:Quality}
-------
Because the translations are created by humans, a number of factors could lead to compromises in quality:
Errors in Understanding:
: If the translator is a non-native speaker of the source language, they may make mistakes in interpreting the original message. This is particularly true if the translator does not normally work in the domain of the text, [e.g. ]{}\*, when a translator who normally works on translating electronic product manuals is asked to translate news.
Errors in Fluency:
: If the translator is a non-native speaker of the target language, they might not be able to generate completely fluent text. This similarly applies to domain-specific terminology.
Limited Resources:
: Unlike computers, human translators have limits in time, attention, and motivation, and will generally do a better job when they have sufficient time to check their work, or are particularly motivated to do a good job, such as when doing a good job is necessary to maintain their reputation as a translator.
Effects of Post-editing:
: In recent years, a large number of human translation jobs are performed by post-editing MT output, which can result in MT artefacts remaining even after manual post-editing [@daems2017translation; @toral-2019-post; @castilho-etal-2019-influences].
In this section, we examine the effect of the quality of underlying translations on the conclusions that can be drawn with regards to human–machine parity. We first do an analysis on (i) how the source of the human translation affects claims of human–machine parity, and (ii) whether significant differences exist between two varieties of human translation. We follow the same protocol as in [Section \[sec:ContextEvaluationProtocol\]]{}, having 4 professional translators per condition, evaluate the translations for adequacy and fluency on both the sentence and document level.[^6]
The results are shown in Table \[tab:quality\_main\]. From this, we can see that the human translation [H$_B$ ]{}\*, which was aggressively edited to ensure target fluency, resulted in lower adequacy ([Table \[tab:a-vs-b\]]{}). With more fluent and less accurate translations, raters do not prefer human over machine translation in terms of adequacy ([Table \[tab:mt-vs-b\]]{}), but have a stronger preference for human translation in terms of fluency (compare Tables \[tab:mt-vs-b\] and \[tab:mt-vs-a\]). In a direct comparison of the two human translations ([Table \[tab:a-vs-b\]]{}), we also find that [H$_A$ ]{}is considered significantly more adequate than [H$_B$ ]{}\*, while there is no significant difference in fluency.
To achieve a finer-grained understanding of what errors the evaluated translations exhibit, we perform a categorisation of 150 randomly sampled sentences based on the classification used by @hassan2018achieving.[^7] We expand the classification with a Context category, which we use to mark errors that are only apparent in larger context ([e.g. ]{}\*, regarding poor register choice, or coreference errors), and which do not clearly fit into one of the other categories. @hassan2018achieving perform this classification only for the machine-translated outputs, and thus the natural question of whether the mistakes that humans and computers make are qualitatively different is left unanswered. Our analysis was performed by one of the co-authors who is a bi-lingual native Chinese/English speaker. Sentences were shown in the context of the document, to make it easier to determine whether the translations were correct based on the context. The analysis was performed on one machine translation ([MT$_1$ ]{}\*) and two human translation outputs ([H$_A$ ]{}\*, [H$_B$ ]{}\*), using the same 150 sentences, but blinding their origin by randomising the order in which the documents were presented. We show the results of this analysis in Table \[tab:quality\_error\_classification\].
From these results, we can glean a few interesting insights. First, we find significantly larger numbers of errors of the categories of Incorrect Word and Named Entity in [MT$_1$ ]{}\*, indicating that the MT system is less effective at choosing correct translations for individual words than the human translators. An example of this can be found in Table \[tab:incorrect\_word\], where we see that the MT system refers to a singular “point of view" and translates “线路” (channel, route, path) into the semantically similar but inadequate “lines”. Interestingly, [MT$_1$ ]{}has significantly more Word Order errors, one example of this being shown in Table \[tab:reordering\], with the relative placements of “at the end of last year” (去年年底) and “stop production” (停产). This result is particularly notable given previous reports that NMT systems have led to great increases in reordering accuracy compared to previous statistical MT systems [@neubig15wat; @bentivogli16neuralvsphrasebased], demonstrating that the problem of generating correctly ordered output is far from solved even in very strong NMT systems. Moreover, [H$_B$ ]{}had significantly more Missing Word (Semantics) errors than both [H$_A$ ]{}($p<.001$) and [MT$_1$ ]{}($p<.001$), an indication that the proofreading process resulted in drops of content in favour of fluency. An example of this is shown in Table \[tab:missing\_words\], where [H$_B$ ]{}dropped the information that the meetings between Suning and Apple were *recently* (近期) held. Finally, while there was not a significant difference, likely due to the small number of examples overall, it is noticeable that [MT$_1$ ]{}had a higher percentage of Collocation and Context errors, which indicate that the system has more trouble translating words that are dependent on longer-range context. Similarly, some Named Entity errors are also attributable to translation inconsistencies due to lack of longer-range context. Table \[tab:context\] shows an example where we see that the MT system was unable to maintain a consistently gendered or correct pronoun for the female Olympic shooter Zhang Binbin (张彬彬).
Apart from showing qualitative differences between the three translations, the analysis also supports the finding of the pairwise ranking study: [H$_A$ ]{}is both preferred over [MT$_1$ ]{}in the pairwise ranking study, and exhibits fewer translation errors in our error classification. [H$_B$ ]{}has a substantially higher number of missing words than the other two translations, which agrees with the lower perceived adequacy in the pairwise ranking.
However, the analysis not only supports the findings of the pairwise ranking study, but also adds nuance to it. Even though [H$_B$ ]{}has the highest number of deletions, and does worse than the other two translations in a pairwise adequacy ranking, it is similar to [H$_A$ ]{}\*, and better than [MT$_1$ ]{}\*, in terms of most other error categories.
Directionality {#sec:Directionality}
--------------
Translation quality is also affected by the nature of the source text. In this respect, we note that from the 2,001 sentences in the WMT 2017 Chinese–English test set, half were originally written in Chinese; the remaining half were originally written in English and then manually translated into Chinese. This Chinese reference file (half original, half translated) was then manually translated into English by @hassan2018achieving to make up the reference for assessing human–machine parity. Therefore, 50of the reference comprises direct English translations from the original Chinese, while 50are English translations from the human-translated file from English into Chinese, [i.e. ]{}\*, backtranslations of the original English.
According to @Laviosa1998, translated texts differ from their originals in that they are simpler, more explicit, and more normalised. For example, the synonyms used in an original text may be replaced by a single translation. These differences are referred to as translationese, and have been shown to affect translation quality in the field of machine translation [@kurokawa2009automatic; @daems2017translationese; @toral-2019-post; @castilho-etal-2019-influences].
We test whether translationese has an effect on assessing parity between translations produced by humans and machines, using relative rankings of translations in the WMT 2017 Chinese–English test set by five raters (see [Section \[sec:Raters\]]{}). Our hypothesis is that the difference between human and machine translation quality is smaller when source texts are translated English (translationese) rather than original Chinese, because a translationese source text should be simpler and thus easier to translate for an MT system. We confirm Laviosa’s observation that “translationese” Chinese (that started as English) exhibits less lexical variety than “natively” Chinese text and demonstrate that translationese source texts are generally easier for MT systems to score well on.
Table \[tab:orig\_lang\] shows the TrueSkill scores for translations ([H$_A$ ]{}\*, [MT$_1$ ]{}\*, and [MT$_2$ ]{}\*) of the entire test set (Both) versus only the sentences originally written in Chinese or English therein. The human translation [H$_A$ ]{}outperforms the machine translation [MT$_1$ ]{}significantly when the original language is Chinese, while the difference between the two is not significant when the original language is English ([i.e. ]{}\*, translationese input).
We also compare the two subsets of the test set, original and translationese, using type-token ratio (TTR). Our hypothesis is that the TTR will be smaller for the translationese subset, thus its simpler nature getting reflected in a less varied use of language. While both subsets contain a similar number of sentences (1,001 and 1,000), the Chinese subset contains more tokens (26,468) than its English counterpart (22,279). We thus take a subset of the Chinese (840 sentences) containing a similar amount of words to the English data (22,271 words). We then calculate the TTR for these two subsets using bootstrap resampling. The TTR for Chinese ($M=0.1927$, $SD=0.0026$, 95confidence interval $[0.1925,0.1928]$) is 13higher than that for English ($M=0.1710$, $SD=0.0025$, 95confidence interval $[0.1708,0.1711]$).
Our results show that using translationese (Chinese translated from English) rather than original source texts results in higher scores for MT systems in human evaluation, and that the lexical variety of translationese is smaller than that of original text.
Recommendations {#sec:Recommendations}
===============
Our experiments in Sections \[sec:Raters\]–\[sec:ReferenceTranslations\] show that machine translation quality has not yet reached the level of professional human translation, and that human evaluation methods which are currently considered best practice fail to reveal errors in the output of strong NMT systems. In this section, we recommend a set of evaluation design changes that we believe are needed for assessing human–machine parity, and will strengthen the human evaluation of MT in general.
#### (R1) Choose professional translators as raters.
In our blind experiment ([Section \[sec:Raters\]]{}), non-experts assess parity between human and machine translation where professional translators do not, indicating that the former neglect more subtle differences between different translation outputs.
#### (R2) Evaluate documents, not sentences.
When evaluating sentences in random order, professional translators judge machine translation more favourably as they cannot identify errors related to textual coherence and cohesion, such as different translations of the same product name. Our experiments show that using whole documents ([i.e. ]{}\*, full news articles) as unit of evaluation increases the rating gap between human and machine translation ([Section \[sec:Context\]]{}).
#### (R3) Evaluate fluency in addition to adequacy.
Raters who judge target language fluency without access to the source texts show a stronger preference for human translation than raters with access to the source texts (Sections \[sec:Context\] and \[sec:Quality\]). In all of our experiments, raters prefer human translation in terms of fluency while, just as in [@hassan2018achieving’s [-@hassan2018achieving]]{} evaluation, they find no significant difference between human and machine translation in sentence-level adequacy (Tables \[tab:mt-vs-a\] and \[tab:mt-vs-b\]). Our error analysis in [Table \[tab:quality\_qualitative\]]{} also indicates that MT still lags behind human translation in fluency, specifically in grammaticality.
#### (R4) Do not heavily edit reference translations for fluency.
In professional translation workflows, texts are typically revised with a focus on target language fluency after an initial translation step. As shown in our experiment in [Section \[sec:Quality\]]{}, aggressive revision can make translations more fluent but less accurate, to the degree that they become indistinguishable from MT in terms of accuracy ([Table \[tab:mt-vs-b\]]{}).
#### (R5) Use original source texts.
Raters show a significant preference for human over machine translations of texts that were originally written in the source language, but not for source texts that are translations themselves ([Section \[sec:Directionality\]]{}). Our results are further evidence that translated texts tend to be simpler than original texts, and in turn easier to translate with MT.
Our work empirically strengthens and extends the recommendations on human MT evaluation in previous work [@Laeubli2018; @Toral2018], some of which have meanwhile been adopted by the large-scale evaluation campaign at WMT 2019 [@WMT2019]: the new evaluation protocol uses original source texts only (R5) and gives raters access to document-level context (R2). The findings of WMT 2019 provide further evidence in support of our recommendations. In particular, human English to Czech translation was found to be significantly better than MT [@WMT2019 p. 28]; the comparison includes the same MT system (`CUNI-Transformer-T2T-2018`) which outperformed human translation according to the previous protocol [@WMT2018 p. 291]. Results also show a larger difference between human translation and MT in document-level evaluation.[^8]
We note that in contrast to WMT, the judgements in our experiments are provided by a small number of human raters: five in the experiments of Sections \[sec:Raters\] and \[sec:Directionality\], four per condition (adequacy and fluency) in [Section \[sec:Context\]]{}, and one in the fine-grained error analysis presented in [Section \[sec:Quality\]]{}. Moreover, the results presented in this article are based on one text domain (news) and one language direction (Chinese to English), and while a large-scale evaluation with another language pair supports our findings (see above), further experiments with more languages, domains, and raters will be required to increase their external validity.
Conclusion {#sec:Conclusion}
==========
We compared professional human Chinese to English translations to the output of a strong MT system. In a human evaluation following best practices, @hassan2018achieving found no significant difference between the two, concluding that their NMT system had reached parity with professional human translation. Our blind qualitative analysis, however, showed that the machine translation output contained significantly more incorrect words, omissions, mistranslated names, and word order errors.
Our experiments show that recent findings of human–machine parity in language translation are owed to weaknesses in the design of human evaluation campaigns. We empirically tested alternatives to what is currently considered best practice in the field, and found that the choice of raters, the availability of linguistic context, and the creation of reference translations have a strong impact on perceived translation quality. As for the choice of raters, professional translators showed a significant preference for human translation, while non-expert raters did not. In terms of linguistic context, raters found human translation significantly more accurate than machine translation when evaluating full documents, but not when evaluating single sentences out of context. They also found human translation significantly more fluent than machine translation, both when evaluating full documents and single sentences. Moreover, we showed that aggressive editing of human reference translations for target language fluency can decrease adequacy to the point that they become indistinguishable from machine translation, and that raters found human translations significantly better than machine translations of original source texts, but not of source texts that were translations themselves.
Our results strongly suggest that in order to reveal errors in the output of strong MT systems, the design of MT quality assessments with human raters should be revisited. To that end, we have offered a set of recommendations, supported by empirical data, which we believe are needed for assessing human–machine parity, and will strengthen the human evaluation of MT in general. Our recommendations have the aim of increasing the validity of MT evaluation, but we are aware of the high cost of having MT evaluation done by professional translators, and on the level of full documents. We welcome future research into alternative evaluation protocols that can demonstrate their validity at a lower cost.
[^1]: Our results synthesise and extend those reported by @Laeubli2018 and @Toral2018.
[^2]: <https://github.com/ZurichNLP/mt-parity-assessment-data>
[^3]: This terminology is not consistent with other literature, where MT researchers have been referred to as experts and crowd workers as non-experts [[e.g. ]{}\*, @CallisonBurch2009].
[^4]: <https://github.com/mjpost/wmt15>
[^5]: As mentioned before, relative ranking mostly tells whether a translation is better than another but not by how much. The TrueSkill score is able to measure that difference, but may be difficult to interpret.
[^6]: Translators were recruited from [proz.com](proz.com).
[^7]: [@hassan2018achieving’s [-@hassan2018achieving]]{} classification is in turn based on, but significantly different than that proposed by @vilar2006error.
[^8]: Specifically, the absolute difference between HUMAN and `CUNI-Transformer-T2T-2018` in terms of average standardized human scores is 11–22% for segment-level evaluation, 24% for segment-level evaluation with document-level context, and 39% for document-level evaluation [@WMT2019 p. 28].
|
---
abstract: 'In the present paper the problem of the radiative association of atoms of carbon C with electrons $e^{-}$ at an interval of kinetic temperatures $T_{c}$ of $100 < T_{c} < 3000$ K is considered. The calculation of the rate of these associations has been made by using the principle of detailed balance. It is shown that the rate has correct behavior (it increase with the temperature) it is behavior is look-like that for the $H^{-}$ formation rate coefficient, which also increase with the temperature.'
address:
- |
Centro de Investigación en Física, Universidad de Sonora\
Rosales y Blvd. Transversal, Col. Centro, Edif. 3-I 83000 Hermosillo, Sonora, MEXICO\
Apartado Postal 5-088 Tel.: (52-662) 259-21-56. Fax: (52-662) 212-66-49. E-mail: jcampos@cajeme.cifus.uson.mx
- |
Physical Tecnical Institute de Ioffe\
26 polytekhnicheskaya, St Petersburg 194021, Russian Federation\
Fax: (812) 297-10-17, Phone: (812) 297-10-17
author:
- 'J. Campos, A. Lipovka, J. Saucedo'
- 'V. Zalkind'
title: Formación del ion negativo de carbón por asociación radiativa
---
En el presente trabajo es considerado el problema de la asociación radiativa de átomos de carbón C con electrones $e^{-}$ en un intervalo de temperaturas cinéticas $T_{c}$ de $100 < T_{c} < 3000$ K. El cálculo de la razón de dicha asociación se ha hecho, empleando el principio del balance detallado. Se muestra que la razón se comporta de manera correcta conforme la temperatura se incrementa. Sucede algo por el estilo, respecto a la formación del ion negativo del hidrógeno $H^{-}$, donde la razón también crece con la temperatura.
[2]{}
Introducción
============
En los últimos años ha aparecido en la literatura, mucho interés sobre la formación de moléculas de $CH$ en la cosmología. Este interés, es debido a que átomos de carbón, nitrógeno y oxígeno estan formando un pico en la distribución de núcleos que se forman en núcleos prímigenios. Estas considerables abundancias de moléculas basadas en estos átomos pesados, pueden ser los únicos instrumentos para medir condiciones físicas que aparecen en el universo temprano y que nos permiten elegir entre diferentes modelos de nucleosíntesis primordial no estándar (los cuales se discuten mucho ahora). Estas mismas abundancias, también se les puede usar para establecer límites superiores a las abundancias predichas por el modelo estándar de nucleosíntesis primordial.
El ion negativo de carbón $C^{-}$, el cual tiene una energía de amarre de $D_{0}=1.25$ $eV$, es más estable que el ion negativo de hidrógeno $H^{-}$ con $D_{0}=0.75$ $eV$, y esta jugando un papel importante en química del carbón en combustibles y también en química del gas en astrofísica (cascaras de estrellas, de SuperNovas (SN), cinética molecular de nubes moleculares en la galaxia, asi como extragalácticos).
La molécula $CH$ juega un papel principal, como es citado en nuestro artículo anterior \[1\], debido a que por un lado se forma más rápido durante la época pregaláctica y por otro lado el carbón $C$ es una especie muy sensible a los modelos de Big Bang y a inhomogeneidades primigenias. Como fue mencionado en \[1\], los canales principales de formación de $CH$ primordial son: $C+H_{2}\rightarrow CH+H
$, pero, como en el caso del hidrógeno molecular $H_{2}$, el cual se forma por medio de la cadena: $H+e^{-}\rightarrow H^{-}+\gamma $ y $H^{-}+H\rightarrow H_{2}+e^{-}$ formando iones negativos de $H$, en la formación molecular de $CH$ debería jugarse un canal con ayuda de un ion negativo de carbón $C^{-}$. Esta especie se forma por medio de la reacción $C+e^{-}\rightarrow C^{-}+\gamma $. La razón de esta reacción fue calculada en \[2\]. Pero, en tal artículo los autores usaron una aproximación de secciones eficaces para energías bajas, la cual cuenta sólo con un término que corresponde a las velocidades pequeñas de las especies. Tal hecho, los llevo a una razón la cual tiene un comportamiento incorrecto en el caso de las temperaturas de interés (desde $100K$ a $1000K$), las cuales se asocian al caso de formación molecular a través de la época oscura. Por eso es muy importante calcular dicha taza.
En consecuencia del principio del balance detallado, la sección eficaz del proceso puede ser recalculado con la ayuda de la sección eficaz correspondiente a un proceso inverso \[3,4\]. Tal proceso inverso es el desprendimiento radiativo $C^{-}+\gamma \rightarrow C+e^{-}$.
En este artículo estamos calculando la razón de asociación radiativa de $C$ con $e^{-}$, formando un ion negativo de carbón $C^{-}$ usando el principio del balance detallado.
Ecuaciónes para el Cálculo de la Razón
======================================
El proceso de desprendimiento fue considerado en detalle en \[5\]. En este artículo fue calculada la sección eficaz del proceso de desprendimiento de $C^{-}$, y el cálculo coincide muy bien con el experimento.
Consideramos un balance detallado entre reacciones de ambos lados, directa e inversa. El número de cantidad de reacciones de la asociación radiativa en un $cm^{3}/s$ en un intervalo de velocidades desde $v$ hasta $v+dv$ esta dado por
$$Z_{a}=N_{C}N_{e}f\left( v\right) vdv\sigma _{A},\text{ }$$
donde la función de distribución de Maxwell integrada por los ángulos $\theta $ y $\phi $ es
$$f\left( v\right) =4\pi v^{2}\left( \frac{m}{2\pi kT}\right) ^{3/2}\exp
\left( \frac{-mv^{2}}{2kT}\right) ,\text{ }$$
donde $v$ es la velocidad relativa entre dos especies, $N_{C}$ y $N_{e}$ son abundancias de $C$ y $e^{-}$. Por otro lado, el número de reacciones reversas (desprendimiento radiativo) en un $cm^{3}/s$ en el intervalo de frecuencias desde $\nu $ hasta $\nu +d\nu $ esta dado por
$$Z_{d}=N_{nC^{-}}\frac{U_{\nu }}{h\nu }d\nu c\sigma _{d}\left[ 1-\exp \left(
\frac{-h\nu }{kT}\right) \right] ,$$
donde $N_{nC^{-}}$ es la abundancia de $C^{-}$ y $U_{\nu }$ es la densidad de energía de cuerpo negro, la cual es dada por
$$U_{\nu }=\frac{8\pi h\nu ^{3}}{c^{3}}\frac{1}{\left[ \exp \left( \frac{h\nu
}{kT}\right) -1\right] },$$
donde $c$ es la velocidad de la luz.
La densidad de $C^{-}$ esta dada por la relación
$$N_{nC^{-}}=\frac{g_{m}}{Z_{C^{-}}}N_{C^{-}}\exp \left( \frac{-E_{m}+D_{0}}{kT}\right) ,$$
donde $D_{0}$ es la energía de desprendimiento, $g_{m}$ es el peso estadístico, $Z_{C^{-}}$ es la suma estadística y $N_{C^{-}}$ es la abundancia total de $C^{-}$.
Igualando las ecuaciones (1) y (3) y usando la ecuación (5), obtenemos la relación para secciones eficaces para el desprendimiento radiativo
$$\begin{aligned}
\sigma _{a}=\sigma _{d}\frac{g_{m}}{Z_{C^{-}}}\frac{U_{\nu }}{f\left(
v\right) }\left( \frac{d\nu }{dv}\right) \frac{c}{h\nu v}\frac{N_{C^{-}}}{N_{C}N_{e}} \times \mbox{} \nonumber \\
\left[ 1-\exp \left( \frac{-h\nu }{kT}\right) \right] \exp \left(
\frac{-E_{m}+D_{0}}{kT}\right) .\end{aligned}$$
En consecuencia con la ecuación de Saha, las abundancias son de la forma
$$\frac{N_{C^{-}}}{N_{C}N_{e}}=\left( \frac{h^{2}}{2\pi mkT}\right) ^{3/2}\frac{Z_{C^{-}}}{Z_{C}Z_{e}}\exp \left( \frac{D_{0}}{kT}\right) .$$
Introduciendo (7) en (6) y tomando en cuenta la ley de conservación de la energía
$$h\nu =\frac{mv^{2}}{2}-E_{n}.$$
Entonces, tenemos la relación de secciones eficaces
$$\sigma _{a}=\frac{2g_{n}}{Z_{C}Z_{e}}\left( \frac{h\nu }{mcv}\right)
^{2}\sigma _{d},$$
donde $Z_{C}$ es la suma estadística.
Finalmente la razón de la asociación radiativa $C+e\rightarrow
C^{-}+\gamma $, esta dada por la integral
$$R_{a}\left( T_{c}\right) =\int_{0}^{\infty }\sigma _{a}f\left( v\right) vdv.$$
donde $f\left( v\right) $ es la función de distribución dada por la ec. (2) y $v$ es la velocidad del electrón.
Resultados de Cálculo
=====================
Las relaciones (9) y (10) con los datos para la sección eficaz del proceso inverso ($C^{-}+\gamma \rightarrow C+e^{-}$) \[5\] nos permite resolver la tarea y calcular la sección eficaz y la razón para la asociación radiativa $C+e^{-}\rightarrow C^{-}+\gamma $. A diferencia del cálculo realizado en \[2\], nosotros realizamos el cálculo tomando en cuenta el principio del balance detallado y así obtener tal razón con el comportamiento más correcto en la región de temperaturas de interés, comparandolo con resultados de \[2\].
En la figura 1 mostramos el resultado de nuestro cálculo, junto con el obtenido en \[2\]. Dentro de la figura, los cuadritos negros representan a nuestro resultado, el cual fue obtenido usando el principio del balance detallado. Los triangulos blancos representan el resultado ofrecido en \[2\]. La comparación de ambos resultados muestra un buen acuerdo de los cálculos en regiones de baja temperatura cinética $T_{c}$. Sin embargo, conforme dicha temperatura se incrementa, empiezan a ser visibles las discrepancias de ambas predicciones teóricas para la razón. Se ve claramente que mientras la temperatura cinética crece, la razón ofrecida en \[2\] se cae fuertemente. Tal comportamiento (como han mencionado los autores) aparece debido a una simplificación en sus cálculos, es decir, cuando ellos cancelaron la parte de sección eficaz que corresponde a las energías altas. Dicha simplificación realmente no afectó la razón en el régimen de temperaturas pequeñas, pero en el caso de temperaturas altas y medias, la discrepancia se aumenta mucho. Tales discrepancias ilustran la descripción correcta predicha en el límite de altas temperaturas por nuestro cálculo, las cuales podemos comparar con la bien conocida razón para formación de $H^{-}$ ($H+e^{-}\rightarrow H^{-}+\gamma $), la cual es muy parecida a nuestro resultado.
Conclusión
==========
En este trabajo se derivó una expresión analítica que relaciona a las secciones eficaces de asociación y de desprendimiento (ver ec. 9) a partir del principio del balance detallado. Posteriormente, calculamos la razón de asociación radiativa $R_{a}$ de átomos de carbón $C$ con electrones $e^{-}$, a través de la reacción $C+e^{-}\rightarrow
C^{-}+\gamma $ y usando tal principio. Nuestro procedimiento de obtención de la razón de asociación resultó ser bastante correcto, en un amplio intervalo de temperaturas, tanto bajas como altas (desde $100^{\circ }K$ hasta $3000^{\circ }K$), si comparamos con la misma razón de formación de $H^{-}$.
[2]{}
[99]{} A. Lipovka, J. Saucedo, J. Campos, RMF 48 (2002) 325-334.
R.K. Janev, H.V. Regemorte, Astron. & Astrophys., 37, 1-6 (1974).
J.K. Martin, Physical Review, 97 (1955) 1446.
F. Coester, Letter to the Editor, (1951).
Yu.V. Moskvin, Opt. Spektrosk., 17, 270 (1964).
|
---
abstract: 'For the Dunkl operator $\Lambda_\alpha$ $(\alpha > -1/2)$ on the space of entire functions on the complex space $\C$, the critical rate of growth for the integral means $M_p(f,r)$ of their hypercyclic functions $f$ is obtained. The rate of growth of the corresponding frequently hypercyclic functions is also analyzed.'
author:
- 'L. Bernal-González and A. Bonilla'
title: Rate of growth of hypercyclic and frequently hypercyclic functions for the Dunkl operator
---
[^1] [^2]
Introduction. {#S-intro}
=============
In this paper, we consider the vector space $H(\C )$ of all entire functions $\C \to \C$, endowed with the topology of local uniform convergence. Under this topology, $H(\C )$ becomes an F-space, that is, a completely metrizable topological vector space. Recall that a (continuous and linear) operator $T$ on a (Hausdorff) topological vector space $X$ is said to be [*hypercyclic*]{} if there exists a vector $x\in X$, also called [*hypercyclic*]{}, whose orbit $\{T^nx:n\in
\mathbb{N}\}$ is dense in $X$. We refer the reader to the books [@BaMa09] and [@GrPe11] for rather complete accounts and further information on hypercyclic operators.
.15cm
In 1952, MacLane [@Mac52] proved that the differentiation operator $D:H(\mathbb{C})\to H(\mathbb{C})$, given by $$Df(z)=f'(z),$$ is hypercyclic. Moreover, he showed that $D$-hypercyclic entire functions can be of exponential type 1. In 1984, Duyos-Ruiz [@Duy84] they cannot be of exponential type less than 1. An optimal result on the possible rates of growth for the differentiation operator was subsequently obtained by Grosse-Erdmann [@Gro90] and, independently, by Shkarin [@Shk93]. Specifically, they obtained that there is no $D$-hypercyclic entire function $f$ for which there exists a constant $C > 0$ satisfying $|f(z)| \le C \, {e^r \over \sqrt{r}}$ for $|z| = r$ large enough but, given a function $\varphi : \R_+ \to \R_+$ with $\varphi (r) \to \infty$ as $r \to \infty$, there exists a $D$-hypercyclic entire function such that $|f(z)| \le \varphi (r) \, {e^r \over \sqrt{r}}$ for $|z| = r$ large enough.
.15cm
The notion of frequent hypercyclicity that was recently introduced by Bayart and Grivaux [@BaGr04], [@BaGr06]. We recall that the lower density of a subset $A$ of $\mathbb{N}$ is defined as $$\underline{\mbox{dens}}\,(A) = \liminf_{N\to\infty}\frac{\#\{n\in A : n\leq N\}}{N},$$ where $\#$ denotes the cardinality of a set.
.15cm
A vector $x \in X$ is called [*frequently hypercyclic*]{} for $T$ if, for every non-empty open subset $U$ of $X$, $$\underline{\mbox{dens}}\,\{n\in \mathbb{N} : T^n x \in U\} > 0.$$ The operator $T$ is called [*frequently hypercyclic*]{} if it possesses a frequently hypercyclic vector.
.15cm
The problem of determining possible rates of growth of frequently hypercyclic entire functions for differentiation operator was studied in [@BlaBoGro], [@BoBo13] and [@drasinsaksman2012]. More generally, the problem of isolating the possible rates of growth of hypercyclic or frequently hypercyclic entire functions for convolution operators on $H(\Bbb C)$ (operators that conmute with the differentiation operator) has been considered in [@BeBo02], [@BoGE06] and [@ChSh91]. Finally, the rates of growth of hypercyclic entire functions for weighted backward shifts $B_w$ on $H(\C )$ –considered as a special sequence space– have been established in [@Gro00]. More classes of hypercyclic non-convolution operators on $H(\C )$ –different from composition operators– have been analyzed by a number of authors, see e.g. [@aronmarkose2004; @fernandezhallack2005; @Kim1; @Kim2; @leonromero2014; @petersson2005; @petersson2005b; @petersson2006].
.15cm
One remarkable example is the [*Dunkl operator*]{} $$\Lambda _{\alpha} : H(\C ) \to H(\C ) \quad (\alpha > - \frac{1}{2}),$$ which is a differential-difference operator given by $$\label{Dunkl}
\Lambda _{\alpha}f(z) = \frac{d}{dz}f(z) + \frac{2\alpha +1}{z}\Big(\frac{f(z) - f(-z)}{2}\Big ).$$ Note that for $\alpha = - \frac{1}{2}$ we get $\Lambda _{\alpha} = D$. The operator $\Lambda _{\alpha}$ was introduced in 1989 by Dunkl [@dunkl1989]. It is connected to the theory of sampling signals, and there are in the literature a lot of papers dealing with the Dunkl operator, see for instance [@ciaurrivarona2008] and the references given in it.
.15cm
Here we will concentrate on the dynamical aspects of $\Lambda_\alpha$. The hypercyclicity of this operator was established by Betancor, Sifi and Trimeche in [@BeSiTri] (see also [@KimNa]and [@ChMeMiTri]). The purpose of this note is to study the rate of growth of hypercyclic and frequently hypercyclic entire functions for the Dunkl operator. In fact, the critical rate of growth for the $p$-integral means $M_p(f,r)$ of their hypercyclic functions $f$ is obtained. Moreover, we get permissible and non-permissible rates of growth for the means of the corresponding frequently hypercyclic functions.
Preliminaries and notation. {#Section-preliminaries}
===========================
As usual, throughout this paper constants $C>0$ can take different values at different occurrences. We write $a_n \sim b_n$ for positive sequences $(a_n)$ and $(b_n$) if $a_n/b_n$ and $b_n/a_n$ are bounded.
.15cm
For an entire function $f$ and $1 \leq p < \infty$ we consider the $p$-integral means $$M_p(f,r)=\Big(\frac{1}{2\pi}\int_0^{2\pi} |f(re^{it})|^p
dt\Big)^{1/p} \quad (r>0)$$ and $$M_\infty(f,r)=\sup_{|z|=r} |f(z)| \quad (r > 0).$$
These means have been considered by Blasco, Bonet, Bonilla and Grosse-Erdmann (see [@BlaBoGro; @BoBo13]) in order to establish the possible rates of growth of hypercyclic or frequently hypercyclic entire functions for the operator $D$. Some of the techniques from [@BlaBoGro; @BoBo13; @Gro90] will be used and generalized in this paper in order to analyze the corresponding problem for $\Lambda_{\alpha}$.
.15cm
With this aim, some properties of the Dunkl operators are needed. Let us define $$\label{dn}
d_{n}(\alpha) = {2^n \, \big([\frac{n}{2}]!\big) \over \Gamma(\alpha+1)} \, \Gamma \left([\frac{n+1}{2}] +\alpha +1 \right) \hbox{ \ for } n\ge 0
\hbox{ \,and\, } \alpha > - \frac{1}{2},$$ and $d_{n}(\alpha ) = 0$ when $n<0$, where $[x]$ denotes the integer part of $x$. Notice that $d_{n}(\alpha )$ tends rapidly to $+\infty$ as $n \to \infty$. Then, for every $k\in \Bbb N$, we have (see e.g. [@BeSiTri pag. 106]) that $$\label{zeta}
\Lambda ^{k} _{\alpha}(z^{n})= {d_n(\alpha) \over d_{n-k}(\alpha)} \, z^{n-k} \hbox{ \ for all }z\in \Bbb C .$$ Moreover, given $f(z)= \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}z^{n} \in H(\C )$, the following holds: $$\Lambda_{\alpha}^{n}f(0) = \frac{f^{(n)}(0)}{n!} \, d_{n}(\alpha).$$ By using Stirling’s asymptotic formula (see e.g. [@ahlfors1979]) $\Gamma (x) = (2\pi )^{1/2} \, x^{x - {1 \over 2}} \, e^{-x} \, \Psi (x)$ for $x > 0$ (where $\Psi (x) \to 1$ as $x \to +\infty$) and the facts $\Gamma (n+1) = n!$, $[{n \over 2}] + [{n+1 \over 2}] = n$ $(n \ge 0)$, one can easily obtain the following lemma, which will be used in the forthcoming sections.
\[Lemma-Stirling\] For each $\alpha > -1/2$, we have the equivalence $$d_n(\alpha ) \sim {(n + \alpha + 1)^{n + \alpha + 1} \over e^{n + \alpha + 1}}.$$
An important tool in the setting of linear dynamics is the Universality Criterion. There are several versions of it, see for instance [@Gro99] or [@GrPe11]. The one given in Theorem \[Univ-Crit\] is sufficient for our goals. Prior to this, it is easy to extend the notion of hypercyclicity. A sequence of continuous linear mappings $T_n: X \to Y$ $(n \ge 1)$ between two topological vector spaces $X,\, Y$ is said to be [*universal*]{} whenever there is a vector $x_0 \in X$ –called universal for $(T_n)$– such that the set $\{T_n x_0: \, n \in \N\}$ is dense in $Y$. Of course, an operator $T:X \to X$ is hypercyclic if and only if the sequence $(T^n)$ of its iterates is universal.
\[Univ-Crit\] Assume that $X$ and $Y$ are topological vector spaces, such that $X$ is a Baire space and $Y$ is separable and metrizable. Let $T_n : X \to Y$ $(n \ge 1)$ be a sequence of continuous linear mappings. Suppose also that there subsets $X_0 \subset X$ and $Y_0 \subset Y$ that are respectively dense in $X$ and $Y$ satisfying that, for any pair $(x_0,y_0) \in X_0 \times Y_0$, there are sequences $\{n_1 < n_2 < \cdots < n_k < \cdots \} \subset \N$ and $(x_k) \subset X$ such that $$x_k \to 0, \,\, T_{n_k}x_0 \to 0 \hbox{ \ and \ } T_{n_k}x_k \to y_0 \hbox{ \ \,as\, \ } n \to \infty .$$ Then $(T_n)$ is universal. In fact, the set of universal vectors for $(T_n)$ is residual [(]{}hence dense[)]{} in $X$.
Corresponding notion and criterium for frequent universal sequences will be given in Section \[Section-freqhc-Dunkl\]. In that section the well-known Hausdorff–Young inequality (see for instance [@Ka76] or [@rudinARC]) will be used several times. We state a version of it in the next theorem for the sake of completeness. Recall that if $p \in [1,\infty ]$ then the conjugate exponent of $p$ is the unique $q \in [1,\infty ]$ satisfying ${1 \over p} + {1 \over q} = 1$.
\[Thm-HausdorffYoung\] Consider the spaces $L^p([0,2\pi ])$ of Lebesgue integrable functions $[0,2\pi ] \to \C$ of order $p$. For each $F \in L^p ([0,2\pi ])$, let $\hat{F} (n) = {1 \over 2\pi} \int_0^{2 \pi} F(t) \, e^{-int} \,dt$ $(n \ge 0)$ be the sequence of its Fourier coefficients. If $1 < p \le 2$ and $q$ is the conjugate exponent of $p$ then $$\Big(\sum_{n=0}^\infty |\hat{F} (n)|^q \Big)^{1/q} \le \Big({1 \over 2\pi} \int_0^{2\pi} |F(t)|^p \,dt \Big)^{1/p}.$$
Hypercyclic entire functions for the Dunkl operator. {#S-Rate}
====================================================
For hypercyclicity with respect to the Dunkl operator, the rate of growth $e^r/r^{\alpha + 1 }$ turns out to be critical. We denote $\R^+ := (0,+\infty )$ and $\N_0 := \N \cup \{0\}$.
\[T-ratehc\] Let $1 \leq p \leq \infty$. Then the following holds:
- For any function $\varphi:\mathbb{R}_+\to\mathbb{R}_+$ with $\varphi(r)\to\infty$ as $r\to\infty$ there is a $\Lambda _{\alpha}$-hypercyclic entire function $f$ with $$M_p(f,r)\leq \varphi(r)\frac{e^r}{r^{\alpha + 1}}\quad\mbox{for $r>0$
sufficiently large}.$$
- There is no $\Lambda _{\alpha}$-hypercyclic entire function $f$ satisfying, for some $C > 0$, that $$M_p(f,r)\leq C \frac{e^r}{r^{\alpha + 1}}\quad\mbox{for all $r>0$.}$$
Thanks to the elementary inequality $$M_p(f,r)\leq M_q (f,r) \hbox{ \,for all } r > 0 \,\,\, (1 \le p \le q)$$ it is enough to prove part (a) for the case $p = \infty$ and part (b) for the case $p=1$.
.15cm
Let us prove (a). We can assume without loss of generality that $\varphi : \Bbb R^+ \rightarrow \Bbb R^+$ with $\varphi(r) \rightarrow \infty$ as $r\rightarrow \infty$ is monotonic and continuous on $[0,+\infty )$ with $\varphi(0) > 0$. Consider the following space $$\widetilde {X} = \left\{ f\in H(\Bbb C): \, \sup_{r > 0} \frac{M_\infty (f,r) \, r^{\alpha + 1}}{\varphi(r) \, e^r} < \infty \right\}.$$ It is not difficult to see that $\widetilde X$ is a Banach space under the norm $\|f\| := \sup_{r > 0} \frac{M_\infty (f,r) \, r^{\alpha + 1}}{\varphi(r) \, e^r}$ that is continuously embedded in $H(\mathbb{C})$.
.15cm
In addition, we define the set $X$ as the closure of the polynomials in $\widetilde X$, and the set $X_0 = Y_{0}\subset H(\mathbb{C})$ as the collection of polynomials in $z$. Note that, trivially, $X_0$ is dense in $X$ and $Y_0$ is dense in $Y := H(\mathbb{C})$. Now, for $n \in \N := \{1,2,3, \dots \}$, we consider the mappings $$T_n : X \to Y, \, T_n = \Lambda _{\alpha}^n|_X,$$ which are continuous, and $$S_n:Y_0\to X, \,\, S_n=S^n \; \; \mbox{ with \,}
S(z^{k}) = \frac{d_{k}(\alpha)}{d_{k+1}(\alpha)}z^{k+1},$$ extended linearly to $Y_0$.
.15cm
Now, fix a pair of polynomials $P,Q$, that is, $(P,Q) \in X_0 \times Y_0$. Then $P(z) = \sum_{k=0}^p a_k z^k$ and $Q(z) = \sum_{k=0}^q b_k z^k$ for certain $a_0, \dots ,a_p,b_0, \dots ,b_q \in \C$. Take as $(n_k)$ the full sequence $\{1,2,3, \dots\}$ of natural numbers and $f_n (z) := S^n Q$ (with $S^n = S \circ S \circ \cdots \circ S$, $n$ times) for all $n \in \N$. It follows from the properties of $\Lambda _{\alpha}$ given in Section \[Section-preliminaries\] that $T_n f_n = Q$ (so, trivially, $T_n f_n \to Q$ as $n \to \infty$) and $T_n P \to 0$ in $H(\C )$ as $n \to \infty$ (because $\Lambda^n_\alpha (z^k) = 0$ if $n \ge k$).
.15cm
Thus the conditions of the Universality Criterion (Theorem \[Univ-Crit\]) are satisfied if we can show that $\{f_n\}_{n \ge 1}$ converges to $0$ in $X$. By linearity, one can assume that $Q(z) = z^k$, a monomial ($k \in \mathbb{N}_0$), in which case $$\lim_{n \rightarrow \infty} S_n Q(z) = \lim_{n \rightarrow \infty}
\frac{d_{k}(\alpha)}{d_{k+n}(\alpha)} z^{k+n}.$$ Therefore, all we need to show is that $$\lim_{n\rightarrow \infty}\frac{z^{n}}{d_{n}(\alpha)}$$ converges to $0$ in $X$.
.15cm
To this end, fix $\varepsilon >0$ as well as an $n \in \N$. We choose $R>0$ such that $\varphi(r)\geq 1/\varepsilon$ for $r\geq R$. Then we have that $$\sup_{r\leq R} \frac{r^{\alpha + 1}}{\varphi(r) e^r} \frac{r^{n}}{d_{n}(\alpha)} \leq \frac{{R^{\alpha +1}}}{\inf_{r>0} \varphi(r)}
\frac{R^n}{d_{n}(\alpha)} \to 0 \quad \hbox{as \, } n \to \infty .$$ Moreover a simple calculation involving the derivative of $r^{n + \alpha + 1} e^{-r}$ shows that $$\sup_{r\geq
R}\frac{1}{\varphi(r)} \frac{r^{\alpha + 1}}{e^r}
\frac{r^{n}}{d_n (\alpha)}\leq \varepsilon \frac{(n+\alpha + 1)^{n + \alpha + 1}}{e^{n+\alpha +1} \, d_n(\alpha)},$$ and Lemma \[Lemma-Stirling\] implies that the last term is bounded by $C \varepsilon$ for any $n \in \mathbb{N}$. This shows that $\lim_{n\rightarrow \infty} \frac{z^{n}}{d_{n}(\alpha)}$ converges to $0$ in $X$.
.15cm
In order to prove part (b), the use of the Cauchy estimates leads us to $$|\Lambda_{\alpha }^{n}f(0)|= \left|\frac{f^{(n)}(0)}{n!} \right|d_{n}(\alpha ) \leq \frac{d_{n}(\alpha)}{r^{n}} M_1(f,r).$$ Assume, by way of contradiction, that there is $C > 0$ such that $M_1(f,r)\leq C \frac{e^r}{r^{\alpha + 1}}$ for all $r > 0$. Hence we find that $$|\Lambda_{\alpha }^{n}f(0)| \leq C\frac{d_{n}(\alpha )}{r^{n+\alpha +1}} e^r \le C\frac{d_{n}(\alpha )}{(n+\alpha +1)^{n+\alpha +1}}
\, e^{n+\alpha +1} \hbox{ \ for all } n \geq 1.$$ Now Lemma 1 implies that the sequence $\{\Lambda_{\alpha }^{n}f(0)\}_{n \ge 1}$ is bounded, so that $f$ cannot be hypercyclic for $\Lambda_{\alpha }$. This is the desired contradiction.
An operator $T:H(\C ) \to H(\C )$ is called a [*weighted backward shift*]{} if there is a sequence $\{a_n\}_{n \ge 1} \subset \C \setminus \{0\}$ such that $$(T\, f)(z) = \sum_{n=0}^\infty a_{n+1} c_{n+1} z^n \quad (z \in \C )$$ provided that $f(z) = \sum_{n=0}^\infty c_n z^n$. We denote this operator by $T = B_{(a_n)}$. It is not difficult to see that $B_{(a_n)}$ is well defined and continuous if and only if $\sup_{n \in \N}
|a_n|^{1/n} < \infty$. In [@Gro00a] (see also [@bernal1996]) is was proved that, under the assumption $\sup_{n \in \N}
|a_n|^{1/n} < \infty$, the operator $B_{(a_n)}$ is hypercyclic if and only if $\limsup_{n \to \infty} \Big| \prod_{k=1}^n a_k \Big|^{1/n} = \infty$. This result contains MacLane’s theorem as a special case because the differentiation operator $D$ is the weighted backward shift $B_{(a_n)}$ with $a_n = n$ $(n \in \N )$. Now, observe that, for $\alpha > -1/2$, the Dunkl operator is $B_{(a_n)}$ with $a_n = {d_n(\alpha ) \over d_{n-1} (\alpha )}$. In this case, from Lemma 1 one easily derives that $\sup_{n \in \N} |a_n|^{1 \over n} < \infty$ and $\limsup_{n \to \infty} \Big| \prod_{k=1}^n a_k \Big|^{1/n} = \limsup_{n \to \infty} d_n(\alpha )^{1 \over n} = C \, \lim_{n \to \infty} (n + \alpha + 1)^{n + \alpha + 1 \over n} = \infty$. Then the Dunkl operator is hypercyclic (note that we get an alternative proof of this result given in [@BeSiTri]; as a matter of fact, a much more general result is shown in [@BeSiTri Theorem 4.1]). Moreover, Grosse-Erdmann proved in [@Gro00a Theorems 1–2] that, under the assumption that $(|a_n|)$ is nondecreasing, the critical rate of growth of $M_\infty (f, \cdot )$ allowed for a $B_{(a_n)}$-hypercyclic entire function $f$ is $\mu (r) = \max_{n \ge 0} |r^n/\prod_{k=1}^n a_k|$. If $B_{(a_n)} = \Lambda_\alpha$ then $(|a_n|)$ is nondecreasing and $\mu (r) = \max_{n \ge 0} |r^n/d_n(\alpha )|$, so the cited theorems of [@Gro00a] are at our disposal. Nevertheless, we have opted for a more direct proof, which in turn yields critical rates for all $M_p(f, \cdot \,)$ $(1 \le p \le \infty )$.
Frequently hypercyclic entire functions for the Dunkl operator. {#Section-freqhc-Dunkl}
===============================================================
First of all, it is easy to extend the notion of frequent hypercyclicity to a sequence of mappings, see [@BoGro].
\[D-fruniv\] Let $X$ and $Y$ be topological spaces, and let $T_n:X \to Y$ $(n \in \N )$ be a sequence of continuous linear mappings. Then an element $x \in X$ is called [*frequently universal*]{} for the sequence $(T_n)$ if, for every non-empty open subset $U$ of $Y$, $$\underline{\mbox{dens}}\{n\in \mathbb{N} : T_nx\in U\}>0.$$ The sequence $(T_n)$ is called [*frequently universal*]{} if it possesses a frequently universal element.
Of course, an operator $T:X \to X$ on a topological vector space $X$ is frequently hypercyclic if and only if the sequences of its iterates $(T^n)$ is frequently universal. In [@BoGro], a [*Frequent Universality Criterion*]{} was obtained that generalizes the Frequent Hypercyclicity Criterion of Bayart and Grivaux [@BaGr06]. We state it here (see Theorem \[T-FUC\] below) only for Fréchet spaces. Recall that a collection of series $\sum_{k=1}^\infty x_{k,j}$ $(j \in I)$ in a Fréchet space $X$ is said to be [*unconditionally convergent, uniformly in $j\in I$,*]{} if for every continuous seminorm $p$ on $X$ and every $\varepsilon > 0$ there is some $N \geq 1$ such that for every finite set $F\subset \mathbb{N}$ with $F \cap \{1,2,\ldots,N\} = \varnothing$ and every $j \in I$ we have that $p(\sum_{k\in F} x_{k,j})<\varepsilon$.
\[T-FUC\] Let $X$ be a Fréchet space, $Y$ a separable Fréchet space, and $T_n : X\to Y$ $(n \in \mathbb{N})$ a sequence of operators. Suppose that there are a dense subset $Y_0$ of $Y$ and mappings $S_n:Y_0\to X$ $(n \in \mathbb{N})$ such that, for all $y \in Y_0$,
- $\sum_{n=1}^k T_{k}S_{k-n}y$ converges unconditionally in $Y$, uniformly in $k\in \mathbb{N}$,
- $\sum_{n=1}^\infty T_kS_{k+n}y$ converges unconditionally in $Y$, uniformly in $k\in \mathbb{N}$,
- $\sum_{n=1}^\infty S_{n}y$ converges unconditionally in $X$,
- $T_nS_ny \to y$ as $n \to \infty$.
Then the sequence $(T_n)$ is frequently universal.
We note that the sums in (i) can be understood as infinite series by adding zero terms.
.15cm
As an auxiliary result we shall need the following estimate. In the following, we shall adopt the convention ${1 \over 2p} = 0$ for $p = \infty$.
\[L-Bar\] Let $1 \le q \leq 2$ and $p$ be the conjugate exponent of $q$. Then there is some $C > 0$ such that, for all $r>0$, we have $$\sum_{n=0}^{\infty} \frac{r^{qn}}{(d_n (\alpha ))^q} \le
C \left( \frac{e^r}{r^{\alpha + {1 \over 2} + {1 \over 2p}}} \right)^q.$$
Consider the entire functions $$E_{\alpha}(z;\theta,\beta)=
\sum_{n=0}^\infty\frac{z^n}{(n+\theta)^{\beta}\Gamma(\alpha n+1)}$$ for $\alpha, \theta>0, \beta \in \mathbb{R}$. In the special case of $\beta = 0$ these are the Mittag-Leffler functions, see for instance [@Er55 Section 18.1]. Barnes [@Bar06 pp. 289–292] studied the functions $E_{\alpha}(z;\theta,\beta)$ and, by using Stirling’s formula, he derived asymptotical expansions from which one deduces that, for $0 < \alpha \le 2$, $$\label{eq1}
E_{\alpha}(r;\theta,\beta) = \alpha^{\beta-1}r^{-\beta/\alpha}
e^{r^{1/\alpha}}\big(1 +O(r^{-1/\alpha})\big) \hbox{ \ as \ } r \to +\infty .$$
Now, by the definition of $d_n (\alpha )$, we get $$\sum_{n=0}^\infty \frac{r^{q n}}{(d_{n}(\alpha ))^q} \leq \sum_{n=0}^\infty \frac{(\Gamma(\alpha +1))^q r^{2qn}}{(2^{2n} \, n! \,
\Gamma (n + \alpha + 1))^q}
+ \sum_{n=0}^\infty \frac{(\Gamma(\alpha +1))^q r^{(2n + 1)q}}{(2^{2n+2} \, n! \, \Gamma (n + \alpha + 2))^q}$$ $$=(\Gamma (\alpha + 1))^q \Big(\sum_{n=0}^\infty \frac{r^{2qn}}{2^{2qn} \, n!^q \,(\Gamma(n + \alpha +1))^q}
+ \frac{r^{q}}{2^{2q}} \, \sum_{n=0}^\infty \frac{ r^{2nq}}{(2^{2qn} \, n!^q \, (\Gamma(n + \alpha +2))^q}\Big)$$
Thanks to , we obtain for $r > 0$ that $$\begin{split}
\sum_{n=0}^\infty \frac{ r^{2qn}}{2^{2qn} \, n!^q \, (\Gamma(n +\alpha + 1))^q} &\le C \, \sum_{n=0}^\infty \frac{((qr)^{2q})^{n}}{(n+1)^{q(\alpha +1)-1/2}\Gamma(2qn+1)} \\
&\le C \, \big((qr)^{2q}\big)^{-\frac{q(\alpha +1) - {1 \over 2}}{2q}} e^{\big((qr)^{2q}\big)^{1 \over 2q}} \\
&\le C \, \left( \frac{e^r}{r^{\alpha + {1 \over 2} + {1 \over 2p}}} \right)^q,
\end{split}$$ where the constant $C$ is not necessarily the same in each occurrence.
.15cm
Analogously, we have for all $r > 0$ that $$\frac{r^q}{2^{2q}} \sum_{n=0}^\infty \frac{ r^{2nq}}{(2^{2qn} \, n!^q \, (\Gamma (n + \alpha + 2))^q}
\le C \, r^q \, \left( \frac{e^r}{r^{\alpha + 1 + {1 \over 2} + {1 \over 2p}}} \right)^q
\le C \, \left( \frac{e^r}{r^{\alpha + {1 \over 2} + {1 \over 2p}}} \right)^q .$$ Finally, it is enough to add up both inequalities to get the desired result.
Our first main result in this section gives growth rates for which $\Lambda _{\alpha}$-frequently hypercyclic functions exist.
\[T-RateDFHC\] Let $1\leq p\leq \infty$, and put $a = \alpha + \frac{1}{2} + \frac{1}{2\max\{2,p\}}$. Then, for any function $\varphi:\mathbb{R}_+ \to \mathbb{R}_+$ with $\varphi(r) \to \infty$ as $r\to\infty$, there is an entire function $f$ with $$M_p(f,r)\leq \varphi(r){\frac{e^r}{r^{a}}}\quad \mbox{for $r>0$
sufficiently large}$$ that is frequently hypercyclic for the Dunkl operator.
Since $$M_p(f,r)\leq M_{2}(f,r)\quad \mbox{for $1\leq p< 2$,}$$ we need only prove the result for $p \geq 2$.
.15cm
Thus let $2 \leq p \leq \infty$. We shall make use of the Frequent Universality Criterion (Theorem \[T-FUC\]). Assuming without loss of generality that $\inf_{r>0} \varphi(r)>0$, we consider the vector space $$X := \Big\{ f \in H(\mathbb{C}) : \, \sup_{r>0} \frac{M_p(f,r) \,
r^{\alpha +\frac{1}{2}+ {1 \over 2p}}}{\varphi (r) \, e^r} < \infty \Big\}.$$ If we endow $X$ with the norm $\|f\| :=\sup_{r>0} \frac{M_p(f,r)
r^{\alpha +\frac{1}{2}+ {1 \over 2p}}}{\varphi (r) \, e^r}$ then it is not difficult to see that $(X,\|\cdot\|)$ is a Banach space that is continuously embedded in $H(\mathbb{C})$.
.15cm
Let $Y_0 \subset H(\mathbb{C})$ be the set of polynomials, and we consider the mappings $$T_n = \Lambda _{\alpha}^n|_X : X \to H(\mathbb{C}),$$ which are continuous, and $$S_n:Y_0\to X, \,\,S_n=S^n \; \; \mbox{ with }
S(z^{k}) = \frac{d_{k}(\alpha)}{d_{k+1}(\alpha)}z^{k+1}.$$ Then we have for any polynomial $f$ and any $k \in \Bbb N$ that $$\sum_{n=1}^k T_{k}S_{k-n}f = \sum_{n=1}^k \Lambda _{\alpha}^n f,$$ that converges unconditionally convergent in $H(\Bbb C)$, uniformly for $k\in \Bbb N$, because $\sum_{n=1}^\infty \Lambda _{\alpha}^n f$ is a finite series. Moreover, we have according to the properties given in Section \[Section-preliminaries\] that $$T_n S_nf = f \quad \mbox{for any $n \in \mathbb{N}$},$$ and $$\sum_{n=1}^\infty T_k S_{k+n}f = \sum_{n=1}^\infty S_n f.$$ Thus the conditions (i)–(iv) in Theorem \[T-FUC\] are satisfied if we can show that $\sum_{n=1}^\infty S_n f$ converges unconditionally in $X$, for any polynomial $f(z) =z ^k$ $(k \in \mathbb{N}_0)$. In which case $$\sum_{n=1}^\infty S_n f(z) = \sum_{n=1}^\infty
\frac{d_{k}(\alpha)}{d_{k+n}(\alpha)} \, z^{k+n}.$$
Therefore, it is sufficient to show that $$\sum_{n=1}^\infty \frac{z^{n}}{d_{n}(\alpha)}$$ converges unconditionally in $X$. To this end, let $\varepsilon >0$ and $N \in \mathbb{N}$. By Theorem \[Thm-HausdorffYoung\] we obtain for any finite set $F \subset
\mathbb{N}$ that $$M_p\Big(\sum_{n\in F} \frac{z^n}{d_{n}(\alpha)},r\Big) \le \Big( \sum_{n\in
F}\frac{r^{qn}}{(d_{n}(\alpha))^{q}}\Big)^{1/q},$$ where $q$ is the conjugate exponent of $p$. Hence, if $F \cap \{0,1,\ldots,N\} = \varnothing$, then $$\Big\|\sum_{n\in F} \frac{z^n}{d_{n}(\alpha)}\Big\| \le \Big(\sup_{r>0}
\frac{r^{q((\alpha +\frac{1}{2})+ {1 \over 2p})}}{\varphi(r)^{q}e^{qr}}\sum_{n
> N}\frac{r^{qn}}{(d_{n}(\alpha))^{q}}\Big)^{1/q}.$$
We choose $R > 0$ such that $\varphi(r)^q\geq 1/\varepsilon$ for $r\geq R$. Then we have that $$\sup_{r\leq R} \frac{r^{q((\alpha +\frac{1}{2}) + {1 \over 2p})}}{\varphi(r)^{q} \, e^{qr}} \, \sum_{n
> N}\frac{r^{qn}}{(d_{n}(\alpha))^{q}} \leq\frac{{R^{q((\alpha +\frac{1}{2}) + {1 \over 2p})}}}{\inf_{r > 0} \varphi(r)^{q}}
\sum_{n>N}\frac{R^{qn}}{(d_{n}(\alpha))^{q}} \longrightarrow 0$$ as $N \to \infty$. Moreover, Lemma \[L-Bar\] implies that $$\sup_{r\geq
R}\frac{1}{\varphi(r)^{q}}\frac{r^{q((\alpha +\frac{1}{2})+ {1 \over 2p})}}{e^{qr}}\sum_{n
> N}\frac{r^{qn}}{(d_{n}(\alpha))^{q}}\leq C\varepsilon \quad\mbox{for any $N \in \mathbb{N}$,}$$ where $C$ is a constant only depending on $q$.
.15cm
This shows that $$\Big\|\sum_{n\in F} \frac{z^{n}}{d_{n}(\alpha)}\Big\|^q \leq
(1+C)\, \varepsilon,$$ if $\min F> N$ and $N$ is sufficiently large, so that $\sum_{n=1}^\infty \frac{z^{n}}{d_{n}(\alpha)}$ converges unconditionally in $X$, as required.
The following result, that concludes the paper, gives lower estimates on the possible growth rates.
\[T-RateDFHC2\] Let $1\leq p\leq \infty$, and put $a = \alpha +\frac{1}{2} + \frac{1}{2\min\{2,p\}}$. Assume that $\psi:\mathbb{R}_+\to\mathbb{R}_+$ is a function with $\psi(r)\to 0$ as $r \to\infty$. Then there is no $\Lambda _{\alpha}$-frequently hypercyclic entire function $f$ that satisfies $$\label{eq2}
M_p(f,r)\leq \psi(r){\frac{e^r}{r^{a}}}\quad \mbox{for $r>0$ sufficiently large}.$$
First, for $p=1$ the result follows immediately from Theorem \[T-ratehc\](b) (notice that one may even take $\psi(r)\equiv C$ here). Moreover, since $$M_2(f,r)\leq M_p(f,r)\quad \mbox{for $2< p\leq\infty$,}$$ it suffices to prove the result for $p\leq 2$.
.15cm
Thus let $1 < p\leq 2$ and $q$ is the conjugate exponent of $p$. We obviously may assume that $\psi$ is decreasing. Suppose that $f$ satisfy . With the help of Theorem \[Thm-HausdorffYoung\] we get that $$\Big( \sum_{n=0}^\infty \Big(\frac{|f^{(n)}(0)|}{n!}r^{n}\Big)^q
\Big)^{1/q} \leq M_p(f,r)\leq \psi(r)\frac{e^r}{r^{\alpha +\frac{1}{2} + {1 \over 2p}}}$$ for $r > 0$ sufficiently large. Thus we have that, for large $r$, $$\label{eq3}
\sum_{n=0}^\infty \Big( \frac{|f^{(n)}(0)|}{n!}d_{n}(\alpha )\Big)^q
\cdot \frac{r^{qn+ {q \over 2p} + (\alpha +\frac{1}{2})q}e^{-qr}}{\psi(r)^q (d_{n}(\alpha ))^q} \leq 1.$$
Using Stirling’s formula we see that the function $$g(r) := \frac{r^{qn + {q \over 2p} +(\alpha +\frac{1}{2})q}e^{-qr}}{(d_{n}(\alpha))^q}$$ has its maximum at $a_n := n + {1 \over 2p} + \alpha +\frac{1}{2}$ of order $1/\sqrt{n}$ and an inflection point at $b_n := a_n + \sqrt{\frac{a_{n}}{q}}$. On $I_n:=[a_n,b_n]$, $g$ therefore dominates the linear function $h$ that satisfies $h(a_n) = g(a_n), \,h(b_n)=0$.
.15cm
Now let $m \in \N$. If $m$ is sufficiently large and $m < n \leq 2m$ then $I_n \subset [m,3m]$. Hence we have for these $n$ that $$\begin{split}
\int_m^{3m}\frac{r^{q \, n + {q \over 2p} + (\alpha +\frac{1}{2})q} \, e^{-qr}}{\psi(r)^q(d_{n}(\alpha))^q} \, dr &\geq
\int_{I_n} \frac{h(r)}{\psi(r)^q} \, dr \\
&\geq C \frac{1}{\psi(m)^q}\frac{1}{\sqrt{n}}\sqrt{\frac n q + \frac
{{1 \over 2p}+ \alpha +\frac{1}{2}}{q}} \\
&\geq C\frac{1}{\psi(m)^q}.
\end{split}$$ Now, integrating over $[m,3m]$ we obtain that for $m$ sufficiently large $$\frac{1}{m}\sum_{n=m+1}^{2m} \Big( \frac{|f^{(n)}(0)|}{n!}d_{n}(\alpha)\Big)^q \leq C \, \psi(m)^q.$$ Hence $$\frac{1}{m}\sum_{n=0}^{m} \Big( \frac{|f^{(n)}(0)|}{n!}d_{n}(\alpha)\Big)^q \longrightarrow 0 \hbox{ \ as \ } m \to \infty .$$ Therefore we have $$\begin{aligned}
\underline{\mbox{dens}}\{ n\in \mathbb{N} : \, |\Lambda_{\alpha}^{n}f(0)|>1\} &= \liminf_{m\to\infty} \frac {1}{ m} \# \{n\leq m : |\Lambda_{\alpha}^n f(0)|>1\}\\
&\leq \liminf_{m\to\infty} \frac{1}{m}\sum_{n=0}^{m}
\Big( \frac{|f^{(n)}(0)|}{n!}d_{n}(\alpha)\Big)^q = 0.\end{aligned}$$ If we take $U := (\delta_0)^{-1} (\{z \in \C : \, |z| > 1\})$ (where $\delta_0$ represents the $0$-evaluation functional $g \in H(\C ) \mapsto g(0) \in \C$, that is continuous) and $T := \Lambda_\alpha$, then $U$ is a nonempty open subset of $H(\C )$ and the last display shows that $\underline{\mbox{dens}} \{ n \in \mathbb{N} : \, T_n f \in U\} = 0$, which prevents $f$ to be frequently hypercyclic for the Dunkl operator. This proves the theorem.
.20cm
[**Acknowledgements.**]{} The first author is partially supported by the Plan Andaluz de Investigación de la Junta de Andalucía FQM-127 Grant P08-FQM-03543 and by MEC Grant MTM2012-34847-C02-01. The second author is partially supported by MEC and FEDER, project no. MTM2014-52376-P.
[15]{} , *Complex Analysis*, 3rd edition, [McGraw-Hill]{}, [London]{}, [1979]{}.
, [On universal functions]{}, [*J. Korean Math. Soc.*]{} [**41**]{} (2004), 65–76.
E. W. Barnes, [The asymptotic expansion of integral functions defined by Taylor’s series]{}, *Phil. Trans. Royal Soc. London* A [**206**]{} (1906), 249–297.
F. Bayart and S. Grivaux, [Hypercyclicité: le rôle du spectre ponctuel unimodulaire]{}, *C. R. Math. Acad. Sci. Paris* [**338**]{} (2004), 703–708.
F. Bayart and S. Grivaux, [Frequently hypercyclic operators]{}, *Trans. Amer. Math. Soc.* [**358**]{} (2006), 5083–5117.
, *Dynamics of Linear Operators*, [Cambridge Tracts in Mathematics]{}, [Cambridge University Press]{}, 2009.
, [Universal functions for Taylor shifts]{}, *Complex Variables* [**31**]{} (1996), 121–129.
L. Bernal-González and A. Bonilla, Exponential type of hypercyclic entire functions, *Arch. Math. (Basel)* [**78**]{} (2002), 283–290.
J.J. Betancor, M. Sifi and K. Trimeche, [Hypercyclic and chaotic convolution operators associated with the Dunkl operator on $\Bbb C$]{}, *Acta Math. Hungar.* [**106**]{} (1-2) (2005), 101–116.
O. Blasco, A. Bonilla and K.-G. Grosse-Erdmann, [Rate of growth of frequently hypercyclic functions]{}, *Proc. Edinburgh Math. Soc.* [**53**]{}, (2010), 39–59.
, [Chaos of the differentiation operator on weighted Banach spaces of entire functions]{}, *Complex Analysis Oper. Theory* [**7**]{} (2013), 33–42.
A. Bonilla and K.-G. Grosse-Erdmann, Frequently hypercyclic operators and vectors, *Ergodic Theory Dynam. Systems* [**27**]{} (2007), 383-404.
A. Bonilla and K.-G. Grosse-Erdmann, On a theorem of Godefroy and Shapiro, *Integral Equations Operator Theory* [**56**]{} (2006), 151–162.
K.C. Chan and J.H. Shapiro, The cyclic behavior of translation operators on Hilbert spaces of entire functions, *Indiana Univ. Math. J.* [**40**]{} (1991), 1421–1449.
F. Chouchene, H. Mejjaoli, M. Mili, and K. Trimeche, Hypercyclic and chaotic convolution associated with the Jacobi-Dunkl operator. *Mediterr. J. Math.* [**11**]{} (2014), 577–600. O. Ciaurri and J.L. Varona, Transformadas de Dunkl y teoremas de muestreo, *Bol. Soc. Esp. Mat. Apl.* [**42**]{} (2008), 103–115.
D. Drasin and E. Saksman, Optimal growth of frequently hypercyclic entire functions, *J. Funct. Anal.* [**263**]{} (2012), 3674–3688.
C.F. Dunkl, Differential-difference operators associated with reflections groups, *Trans. Amer. Math. Soc.* [**311**]{} (1989), 167–183.
S.M. Duyos-Ruiz, Universal functions and the structure of the space of entire functions, *Soviet Math. Dokl.* [**30**]{} (1984), 713–716.
A. Erdélyi, *Higher transcendental functions*, Volume III, McGraw-Hill Book Company, New York, 1955.
, [Remarks on a result about hypercyclic non-convolution operators]{}, *J. Math. Anal. Appl.* [**309**]{} (2005), 52–55.
K.-G. Grosse-Erdmann, On the universal functions of G. R. MacLane, *Complex Variables Theory Appl.* [15]{} (1990), 193–196.
K.-G. Grosse-Erdmann, [Universal families and hypercyclic operators]{}, *Bull. Amer. Math. Soc. (N.S.)* [**36**]{} (1999), 345–381.
K.-G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, *Studia Math.* [**139**]{} (2000), 47-68.
K.-G. Grosse-Erdmann, Rate of growth of hypercyclic entire functions, *Indag. Math. (N.S.)* [**11**]{} (2000), 561–571.
K.-G. Grosse-Erdmann and A. Peris, *Linear Chaos*, Springer, London, 2011.
Y. Katznelson, *An introduction to harmonic analysis*, second edition, Dover Publications, New York, 1976.
V.E. Kim, Hypercyclicity and chaotic character of generalized convolution operators generated by Gel’fond-Leont’ev Operators, *Math. Notes* [**85**]{} (6)(2009), 807–813.
, [Commutation relations and hypercyclic operators]{}, [*Arch. Math.*]{} [**99**]{} (2012), 247–253.
V.E. Kim and V.V. Napalkov, Generalization of the Dunkl operator on the space of entire functions, *Doklady Math.* [**77**]{} (3) (2008), 365–367.
, [Fixed points and orbits of non-convolution operators]{}, [*Fixed Point Theory Appl.*]{} [**2014**]{}, 2014:221.
G.R. MacLane, Sequences of derivatives and normal families, *J. Analyse Math.* [**2**]{} (1952/53), 72–87.
, [Supercyclic and hypercyclic non-convolution operators on Hilbert-Schmidt entire functions]{}, [*Math. Proc. R. Irish Acad.*]{} [**105 A**]{} (2005), 87–101.
, [Supercyclic and hypercyclic non-convolution operators on Hilbert-Schmidt entire functions]{}, [*Math. Proc. Royal Irish Acad.*]{} [**105-A**]{} (2) (2005), 87–101.
, [Supercyclic and hypercyclic non-convolution operators]{}, [*J. Operator Theory*]{} [**55**]{} (2006), 135–151.
, *Real and complex analysis*, third edition, [McGraw-Hill Book Co.]{}, [New York]{}, [1987]{}.
S.A. Shkarin, On the growth of $D$-universal functions, *Moscow Univ. Math. Bull.* [**48**]{} (1993), 49–51.
[$$\begin{array}{lr}
\mbox{Luis Bernal-Gonz\'alez } & \mbox{ Antonio Bonilla } \\
\mbox{Departamento de An\'alisis Matem\'atico } & \mbox{ Departamento de An\'alisis Matem\'atico } \\
\mbox{Universidad de Sevilla } & \mbox{ Universidad de La Laguna } \\
\mbox{Facultad de Matem\'aticas, Apdo.~1160 } & \mbox{ C/Astrof\'{\i}sico Francisco S\'anchez, s/n } \\
\mbox{Avda.~Reina Mercedes, 41080 Sevilla, Spain} & \mbox{ 38271 La Laguna, Tenerife, Spain } \\ \mbox{E-mail: {\tt lbernal@us.es} } & \mbox{ E-mail: {\tt abonilla@ull.es} }
\end{array}$$]{}
[^1]: 2010 [*Mathematics Subject Classification.*]{} Primary 47A16; Secondary 30D15, 47B37, 47B38.
[^2]: [*Key words and phrases.*]{} Frequently hypercyclic operator, frequently hypercyclic vector, Frequent Hypercyclicity Criterion, rate of growth, entire function, Dunkl operator.
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.