Free energy fluxes and the Kubo–Martin–Schwinger relation

Let the system possess a set of extensive conserved quantities (or charges) Q_i in involution, [Q_i , Q_j ] = 0 for all i, j. Here i lies in some index set; it may be taken to be infinite in integrable models ² , although we will not discuss any convergence issue that may arise in this case. In relaxation processes towards thermodynamic states, only a certain category of conserved quantities are physically relevant (excluding for instance, in quantum systems, projections onto eigenstates of the Hamiltonian): those which are extensive. For our purpose, we assume that each charge has an associated density q_i ( x ) satisfying

$$\begin{equation}{Q}_{i}=\int {\mathrm{d}}^{d}x\enspace {q}_{i}\left(\boldsymbol{x}\right).\end{equation} \tag{ 4 }$$

The charge density q_i ( x ) is a local observable at the point $\boldsymbol{x}\in {\mathbb{R}}^{d}$. In the discrete case we take $\int {\mathrm{d}}^{d}x\to {\sum }_{\boldsymbol{x}\in {\mathbb{Z}}^{d}}$. As a notion of locality, we will assume for simplicity that the observable is supported on a finite region; but the arguments can be extended to include quasi-local observables, whose projections on regions $R\subset {\mathbb{R}}^{d}$ away from x have norms that decay exponentially with the distance of R to x .

Extensive conserved charges can be given a mathematically accurate definition and completed to a Hilbert space [6, 28], and are rigorously shown to occur in the Boltzmann–Gibbs principle and the linearised Euler equations in quantum spin chains in [35]. Here we do not use this precise formulation, and it is not necessary for the set of charges Q_i that we consider to be complete (to form a basis for the Hilbert space of extensive conserved charges).

There always exists a Hamiltonian which generates time translations, and there may exist a total momentum vector; that is,

$$\begin{equation}\begin{gathered}H={Q}_{2}\quad \left(\text{Hamiltonian}\right)\text{,}\\ {P}_{\alpha }={Q}_{{1}_{\alpha }},\quad \alpha =1,2,\dots ,d\quad \left(\text{total}\;\text{momentum}\;\text{vector}\right)\text{.}\end{gathered}\end{equation} \tag{ 5 }$$

In this notation, the numerals 0, 1, 2, ... organise the conserved charges according to the local extent of their densities, and the index α is the spatial direction ³ . In lattice models, due to the discreteness of space there is no microscopic, continuous translation invariance, and hence no microscopically defined momentum operator. We will assume the presence of a conserved momentum mainly for the interpretation of some of the results.

Because the charges are assumed to be in involution, we assume that each conserved quantity Q_k generates a current j _ki ( x ) by its action on each conserved density q_i ( x ). These satisfy the continuity equations

$$\begin{equation}\frac{\mathrm{i}}{\hslash }\left[{Q}_{k},{q}_{i}\left(\boldsymbol{x}\right)\right]+\nabla \cdot {\boldsymbol{j}}_{ki}\left(\boldsymbol{x}\right)=0.\end{equation} \tag{ 6 }$$

Here we use the quantum notation, but a similar equation holds in classical systems; see below. To our knowledge, the 'generalised' currents j _ki ( x ) were first discussed in the context of integrable models in [30], though the concept clearly does not rely on integrability. Again, the same definition holds on discrete space with the discrete derivative $\partial o\left(\boldsymbol{x}\right)/\partial {x}^{\alpha }\to o\left(\boldsymbol{x}\right)-o\left(\boldsymbol{x}-{\boldsymbol{e}}_{\alpha }\right)$, where e _α is a unit vector along the α'th direction, $o$ is any observable, and $o\left(\boldsymbol{x}\right)$ is the translation of $o$ by $\boldsymbol{x}\in {\mathbb{Z}}^{d}$. We note that in quantum spin chains, the existence of local currents associated to local conserved densities is proven rigorously, as elements of the Gelfand–Naimark–Segal Hilbert space, in [35]. Below, we will denote the physical currents, generated by the Hamiltonian, as

$$\begin{equation}{\boldsymbol{j}}_{i}={\boldsymbol{j}}_{2i}\quad \left(\text{physical}\;\text{currents}\right)\text{.}\end{equation} \tag{ 7 }$$

In this notation, the spatial components of the stress tensor are

$$\begin{equation}{\mathcal{T}}_{\alpha }^{\gamma }={j}_{{1}_{\alpha }}^{\gamma }.\end{equation} \tag{ 8 }$$

As the momentum operator P_α generates space translations, $\partial o/\partial {x}^{\alpha }=-\mathrm{i}{\hslash }^{-1}\left[{P}_{\alpha },o\right]$, then (6) shows that we can choose, for the currents with respect to this generator,

$$\begin{equation}{\boldsymbol{j}}_{{1}_{\alpha }i}={\boldsymbol{e}}_{\alpha }{q}_{i}.\end{equation} \tag{ 9 }$$

The states of interest are the MES. Physically, these are states which emerge after relaxation in infinite-volume systems. There are strong indications that extensive conserved charges are sufficient in order to characterise MES's [7, 27–29]. They are expected to take the Gibbs form (1); more precisely, the infinite-volume limit is taken on averages with respect to density matrices (1), in order to obtain averages of local observables in the thermodynamic limit. We denote the resulting infinite-volume averages by ⟨...⟩. In quantum chains with finite-dimensional local space, for instance, the infinite-volume limit is proven to exist and to give a unique state whenever βⁱ Q_i (here and below, summation over repeated indices is implied unless otherwise stated) has finite-range or exponentially decaying interaction [36]. Here we assume a state ⟨...⟩ is given, the thermodynamic limit having been taken.

Maximal entropy states of thermodynamic systems are expected to satisfy a number of crucial properties [33]. The properties of interest for the state ⟨...⟩ involve the extensive conserved charges Q_i , as well as the particular charge

$$\begin{equation}W={\beta }^{i}{Q}_{i}\end{equation} \tag{ 10 }$$

which specifies the state. These concepts hold both for quantum and classical systems.

The first required property is that the state be invariant under evolution with respect to all charges Q_i , and be translation invariant (homogeneous):

$$\begin{equation}\langle \left[{Q}_{i},o\right]\rangle =0,\qquad \nabla \langle o\left(\boldsymbol{x}\right)\rangle =0\end{equation} \tag{ 11 }$$

where again $o$ is any local observable or product thereof. The state is also clustering: connected correlation functions vanish at large distances,

$$\begin{equation}{\langle {o}_{1}\left(\boldsymbol{x}\right){o}_{2}\left(0\right)\rangle }^{\text{c}}\to 0\quad \left(\vert x\vert \to \infty \right),\end{equation} \tag{ 12 }$$

where ${\langle {o}_{1}\left(\boldsymbol{x}\right){o}_{2}\left(0\right)\rangle }^{\text{c}}=\langle {o}_{1}\left(\boldsymbol{x}\right){o}_{2}\left(0\right)\rangle -\langle {o}_{1}\left(\boldsymbol{x}\right)\rangle \langle {o}_{2}\left(0\right)\rangle$. For our purposes we require that the vanishing be at least integrable, i.e. faster than 1/| x |^d . In quantum spin chains, it is exponential [36].

Second, the KMS relation holds,

$$\begin{equation}\langle {o}_{1}{o}_{2}\rangle =\langle {\tau }_{-\mathrm{i}\hslash }{o}_{2}\enspace {o}_{1}\rangle \end{equation} \tag{ 13 }$$

or equivalently

$$\begin{equation}\langle \left[{o}_{1},{o}_{2}\right]\rangle =\langle \left(1-{\tau }_{-\mathrm{i}\hslash }\right){o}_{1}\enspace {o}_{2}\rangle .\end{equation} \tag{ 14 }$$

Here τ_t is the evolution with respect to the generator W,

$$\begin{equation}{\tau }_{t}o={\mathrm{e}}^{\mathrm{i}t\left[W,\cdot \right]/\hslash }o.\end{equation} \tag{ 15 }$$

The KMS relation has an important role in the full characterisation of Gibbs states in the operator algebra formulation of quantum statistical mechanics [32, 33]. It can be taken as a definition of the potentials {βⁱ } via (10), once a basis {Q_i } is chosen.

Third, a 'tangent-manifold' relation holds, as was studied at length in [28], see also [32]. It states that derivatives of averages in the state ⟨...⟩ with respect to βⁱ are given by connected correlation functions with the conserved quantity Q_i ,

$$\begin{equation}-\frac{\partial }{\partial {\beta }^{i}}\langle o\rangle ={\langle o{Q}_{i}\rangle }^{\text{c}},\end{equation} \tag{ 16 }$$

where the derivative exists and is continuous in all βⁱ 's. The right-hand side is ${\langle o{Q}_{i}\rangle }^{\text{c}}=\int {\mathrm{d}}^{d}x\enspace {\langle o{q}_{i}\left(\boldsymbol{x}\right)\rangle }^{\text{c}}$, which can be seen to exist by the clustering property described above. This provides an alternative definition of the βⁱ 's, which has a clearer conceptual link to the exponential form of the GGE density matrix. Relations (13) and (16) are related in that they give rise to consistent potentials {βⁱ }, see appendix B.

The main hypotheses on the state ⟨...⟩ are properties (11)–(13) and (16). Precise formulations of clustering and of the KMS relation require further considerations, see e.g. [28, 33]. For our purposes, we assume that the KMS relation holds for local observables, and we further assume that clustering (12) holds for imaginary-time evolved local observables: with ${o}_{1}\left(\boldsymbol{x}\right)={\tau }_{-\mathrm{i}\hslash s}{\tilde {o}}_{1}\left(\boldsymbol{x}\right)$ where ${\tilde {o}}_{1}$ is a local observable, uniformly for s ∈ [0, 1].

Finally, we make two additional assumptions. (1) The manifold of states is a connected submanifold $\mathcal{M}$ of the space of values of multiplets β^•, and $\mathcal{M}$ contains the 'infinite-temperature state' βⁱ = 0 ∀ i, which is assumed to have the trace property. In general $\mathcal{M}$ may be nontrivial, however in quantum spin chains, considering a finite number of local conserved charges, one can take all ${\beta }^{i}\in \mathbb{R}$, see appendix A for a discussion. (2) The currents j _ij as defined by (6) are defined only up to the addition of terms proportional to the identity observable, and we make the assumption that it is possible to fix these constant terms so that all currents have vanishing averages in the infinite-temperature state—this is the trace state in quantum lattice models. The gauge ambiguity of the currents is thus fixed by

$$\begin{equation}{\langle {\boldsymbol{j}}_{ij}\rangle }_{\left({\beta }^{{\bullet}}=0\right)}=0.\end{equation} \tag{ 17 }$$

Appendix A contains a discussion of rigorous results on these properties for MES. In particular, it is shown that all assumptions and required structures are proven rigorously in quantum spin chains with finite-range interactions.

All hypotheses above are assumed to hold also in the classical realm. In this case, all observables are taken to be phase-space functions instead of operators, and one makes the replacements

$$\begin{equation}\frac{\mathrm{i}}{\hslash }\left[a,b\right]\to \left\{a,b\right\}\quad \text{and,}\;\text{in}\;\left(\text{14}\right)\text{,}\quad \left(1-{\tau }_{-\mathrm{i}\hslash }\right){o}_{2}\to \mathrm{i}\hslash \left\{W,{o}_{2}\right\}\end{equation} \tag{ 18 }$$

where {⋅, ⋅} is the Poisson bracket. The classical KMS relation is equivalent to relation (14) to order ℏ; this relation can be shown to fully characterise the Gibbs states [34]. From the viewpoint of the operator algebra in the quantum (classical) case, observables form a non-commuting (commuting) algebra, and one uses commutators (Poisson bracket), as per equation (18).

For simplicity, in the subsequent calculations we consider only quantum systems, and set ℏ = k_B = 1.

A number of fundamental relations hold for the averages of currents and densities in a MES, see for instance [37]. In fact, many of these relations hold more generally in homogeneous, Q_k -invariant, clustering states.

One of the most important is a symmetry relation involving correlation functions of currents and densities. We consider the integrated density-charge correlator

$$\begin{equation}{\boldsymbol{B}}_{kij}=\int {\mathrm{d}}^{d}x\enspace {\langle {\boldsymbol{j}}_{ki}\left(0\right){q}_{j}\left(\boldsymbol{x}\right)\rangle }^{\text{c}}=-\frac{\partial }{\partial {\beta }^{j}}\langle {\boldsymbol{j}}_{ki}\rangle ,\end{equation} \tag{ 19 }$$

from which we form the set of vector-valued matrices B _k••, where k is the index of the charge generating the currents. These matrices are symmetric in the last two indices,

$$\begin{equation}{\boldsymbol{B}}_{kij}={\boldsymbol{B}}_{kji}.\end{equation} \tag{ 20 }$$

In one dimension of space such a symmetry was shown in a family of classical, stochastic, interacting particle systems in [38, 39]. There, the importance of this Onsager-type relation for the physical consistency of the emerging hyperbolic system of hydrodynamic equations was pointed out. In the present context, equation (20) was shown in [12, 40] for one dimensional systems. The extension of such derivations to higher dimensions is immediate. The most general relations of this type were obtained in [41], where in particular the assumption of space-translation invariance is relaxed. See also [14, 37, 42, 43] for discussions of hydrodynamic matrices. For completeness we provide a simple derivation of (20) in appendix C.

This symmetry implies that there must exist, for every k, a differentiable function g _k from $\mathcal{M}$ to ${\mathbb{R}}^{d}$ which generates the currents according to [12],

$$\begin{equation}\langle {\boldsymbol{j}}_{ki}\rangle =\frac{\partial {\boldsymbol{g}}_{k}}{\partial {\beta }^{i}}.\end{equation} \tag{ 21 }$$

This parallels the situation for the specific free energy f, where the more apparent symmetry of the static covariance matrix ${\mathsf{C}}_{ij}={\mathsf{C}}_{ji}=\int {\mathrm{d}}^{d}x\enspace {\langle {q}_{i}\left(0\right){q}_{j}\left(\boldsymbol{x}\right)\rangle }^{\text{c}}=-\frac{\partial }{\partial {\beta }^{j}}\langle {q}_{i}\rangle$ implies that there exists ⁴ f such that $\langle {q}_{i}\rangle =\frac{\partial }{\partial {\beta }^{i}}f$. The quantities g _k are referred to as 'free energy fluxes'.

The free energy fluxes appear in the universal definition of the entropy current. That the symmetry (20) implies the existence of a Lax entropy (see e.g. [44]) for the hydrodynamic equations was first noticed in [38, 39], see [14] for a discussion in the present context. The entropy current ${\textbf{\textsf{j}}}_{k}^{s}$ with respect to the flow generated by Q_k is defined as

$$\begin{equation}{\textbf{\textsf{j}}}_{k}^{s}={\beta }^{j}\langle {\boldsymbol{j}}_{kj}\rangle -{\boldsymbol{g}}_{k}.\end{equation} \tag{ 22 }$$

This parallels the definition of the usual entropy density

$$\begin{equation}s={\beta }^{j}\langle {q}_{j}\rangle -f.\end{equation} \tag{ 23 }$$

Again, for the physical entropy current generated by the Hamiltonian we have:

$$\begin{equation}{\textbf{\textsf{j}}}^{s}={\textbf{\textsf{j}}}_{2}^{s}\quad \left(\text{physical}\;\text{entropy}\;\text{current}\right)\text{.}\end{equation} \tag{ 24 }$$

The physical meaning of the entropy currents and density is clearest when considering the hydrodynamic equations that arise at the Euler scale, see subsection 4.3.

Recalling our choice of the currents with respect to space translation, equation (9), using (21) we identify the free energy flux with respect to the momentum as the free energy (if a conserved momentum is present),

$$\begin{equation}{\boldsymbol{g}}_{{1}_{\alpha }}={\boldsymbol{e}}_{\alpha }f\quad \left(\text{recall}\;\text{that}\;{Q}_{{1}_{\alpha }}={P}_{\alpha }\;\text{is}\;\text{the}\;\text{total}\;\text{momentum}\right).\end{equation} \tag{ 25 }$$

In particular, combining with (9), this shows that ${\textbf{\textsf{j}}}_{{1}_{\alpha }}^{s}={\boldsymbol{e}}_{\alpha }s$.

Consider the manifold of states $\mathcal{M}$ built from only the particle number N = Q₀ (if admitted by the system), the total momentum ${P}_{\alpha }={Q}_{{1}_{\alpha }}$ and the energy H = Q₂, and no other conserved charge (that is, βⁱ = 0 for all other indices i). Typical conventional non-integrable systems admit only these as conserved charges. Here we interpret N physically as a particle number, and hence its density is 'ultra local': it does not involve any coupling between points in space. As a consequence, the flow generated by N has trivial currents j _0i = 0, and therefore we take g ₀ = 0. This can be taken as the definition of Q₀ as a conserved quantity with an ultra-local density. We denote β² = β = 1/T (the inverse temperature), ${\beta }^{{1}_{\alpha }}=-\beta {\nu }^{\alpha }$ (ν^α is a parameterisation of the boost, if Galilean or relativistic invariance is present), and β⁰ = −βμ (μ is the chemical potential, if there is particle number conservation). The density matrix for the Gibbs state is of the form

$$\begin{equation}\rho \propto {\text{e}}^{-\beta \left(H-\boldsymbol{\nu }\cdot \boldsymbol{P}-\mu N\right)}.\end{equation} \tag{ 36 }$$

The results (25) and (26) then imply

$$\begin{equation}{\boldsymbol{g}}_{2}=T\boldsymbol{G}+\boldsymbol{\nu }f.\end{equation} \tag{ 37 }$$

Recall that, by (28), the quantity −2GT is the entropy current j ^s in an equilibrium thermal state at temperature T. Again, if for instance the Hamiltonian is parity symmetric, then this current vanishes, G = 0. Evaluating the currents as ⟨ j _i ⟩ = ∂ g ₂/∂βⁱ , we obtain, for the average currents of particles ⟨ j ₀⟩, of the direction-α momentum component $\langle {\mathcal{T}}_{\alpha }^{\gamma }\rangle$, and of the energy ⟨ j ₂⟩, the expressions

$$\begin{equation}\langle {\boldsymbol{j}}_{0}\rangle =\boldsymbol{\nu }\langle {q}_{0}\rangle ,\qquad \langle {\mathcal{T}}_{\alpha }^{\gamma }\rangle ={\nu }^{\gamma }\langle {q}_{{1}_{\alpha }}\rangle -Tf{\delta }_{\alpha }^{\gamma },\qquad \langle {\boldsymbol{j}}_{2}\rangle =\boldsymbol{\nu }\langle {q}_{2}\rangle -T\boldsymbol{\nu }f-{T}^{2}\boldsymbol{G}.\end{equation} \tag{ 38 }$$

Knowledge of the free energy f, or equivalently of the static covariance matrix C _ij , would allow us to write ν , T in terms of the average particle density ⟨q₀⟩, direction-α momentum density component $\langle {q}_{{1}_{\alpha }}\rangle$, and energy density ⟨q₂⟩. Therefore, the equations of state are fully fixed by the free energy f (and the constant vector G characterising the thermal entropy current, which vanishes if the Hamiltonian is parity symmetric). In Galilean or relativistically invariant systems, space-time symmetries can be used to establish the form (38) of the currents. Here however, we find that the currents are fixed in this manner solely as a consequence of the KMS relation, independently of the presence or otherwise of any space-time boost symmetries.

In one-dimensional (d = 1) integrable models, an explicit expression for the free energy fluxes is known within the framework of the TBA [11, 12]. For simplicity, let us assume that the TBA formulation involves only one quasiparticle type, with rapidities ranging over $\mathbb{R}$. This covers many models, for instance in the Lieb–Liniger model of quantum gases [8] and the Toda model of classical particles (see e.g. [46–48] and references therein)—it is simple to extend the discussion to more complicated spectra.

The free energy fluxes take the form [12]

$$\begin{equation}{g}_{k}=\int \mathrm{d}\theta \enspace {\partial }_{\theta }{h}_{k}\left(\theta \right)\enspace F\left({\epsilon}\left(\theta \right)\right),\end{equation} \tag{ 39 }$$

where h_k (θ) is the one-particle eigenvalue of Q_k on a state with a quasiparticle of rapidity θ, and F() is the free energy function of the pseudo-energy (θ). For example, in fermionic integrable models F() = −log(1 + e⁻ ) and in classical particle systems F() = −e⁻ , see e.g. [14]. Generally the pseudo-energy solves the equation:

$$\begin{equation}{\epsilon}\left(\theta \right)=\sum\limits _{i}\enspace {\beta }^{i}{h}_{i}\left(\theta \right)+\int \mathrm{d}{\theta }^{\prime }\enspace \varphi \left({\theta }^{\prime },\theta \right)F\left({\epsilon}\left({\theta }^{\prime }\right)\right)\end{equation} \tag{ 40 }$$

with φ(θ', θ) the two-particle differential scattering phase of the model. Further, we assume that

$$\begin{equation}{\int }_{{\epsilon}\left(-\infty \right)}^{{\epsilon}\left(\infty \right)}\mathrm{d}{\epsilon}\enspace F\left({\epsilon}\right)=0.\end{equation} \tag{ 41 }$$

A sufficient condition for this to hold is that lim_θ→±∞ (θ) = ∞, and lim_→∞ F() = 0 (sufficiently fast). The former condition is seen to hold in many physically relevant states (see e.g. [8–10]), and the latter is immediate with the free energy functions above.

The relation (26) then implies

$$\begin{align}\hfill {\beta }^{k}{g}_{k}& =\sum\limits _{k}{\beta }^{k}\int \mathrm{d}\theta \enspace {\partial }_{\theta }{h}_{k}\left(\theta \right)\enspace F\left({\epsilon}\left(\theta \right)\right)\hfill \\ \hfill & =\int \mathrm{d}\theta \enspace \left({\partial }_{\theta }{\epsilon}\left(\theta \right)+\int \mathrm{d}{\theta }^{\prime }\enspace {\partial }_{\theta }\varphi \left({\theta }^{\prime },\theta \right)F\left({\epsilon}\left({\theta }^{\prime }\right)\right)\right)F\left({\epsilon}\left(\theta \right)\right)\hfill \\ \hfill & =\int \mathrm{d}\theta \int \mathrm{d}{\theta }^{\prime }\enspace {\partial }_{\theta }\varphi \left({\theta }^{\prime },\theta \right)F\left({\epsilon}\left({\theta }^{\prime }\right)\right)F\left({\epsilon}\left(\theta \right)\right).\hfill \end{align} \tag{ 42 }$$

Since φ(θ', θ) = ∂_θ' ϕ(θ', θ), where ϕ(θ', θ) = −i log S(θ', θ) is the complex phase of the two-body scattering amplitude S(θ', θ), the requirement that (42) vanish in arbitrary states is equivalent to the condition

$$\begin{equation}{\partial }_{{\theta }^{\prime }}{\partial }_{\theta }\phi \left({\theta }^{\prime },\theta \right)=-{\partial }_{{\theta }^{\prime }}{\partial }_{\theta }\phi \left(\theta ,{\theta }^{\prime }\right).\end{equation} \tag{ 43 }$$

Thus the EKMS relation implies that the scattering phase ϕ(θ', θ) be anti-symmetric, up to 'factorised' terms of the type u(θ') + v(θ). Anti-symmetry of ϕ(θ', θ) is, in quantum systems, the condition of unitarity of the two-body scattering amplitude, and therefore we find that the EKMS relation implies unitarity of the TBA scattering amplitude up to these factorised terms. The conclusion holds also for classical systems. Factorised terms would deserve further studies, although they likely may be argued to vanish by appealing to other physical properties of the scattering amplitude; for instance, one generally expects that the scattering amplitude tends to a constant at large rapidity.

The Euler-scale hydrodynamic equations constitute one of the most important applications of the expressions of currents in MES. In this subsection, we obtain general results on Euler-scale hydrodynamics that follow from the EKMS relation.

Euler-scale hydrodynamic equations are hard to show rigorously, and we do not make any attempt at rigour here. For completeness, we nevertheless present the heuristic arguments, based on local relaxation, underlying the universal form of the Euler-scale hydrodynamic equations with external force coupled to arbitrary charges. Explicit forms of Euler-scale hydrodynamic equations depend on the explicit expressions of currents in a MES, which is also in general a hard problem (but see subsection 4.1). Here we consider the universal form, which depends abstractly on the average currents, without having to evaluate them.

The Euler equations are based on the assumption of local entropy maximisation [37] with respect to the time-evolution generator U. In our discussion above we took U to be the Hamiltonian. Let us here include more generally space-varying external potentials u^k ( x ),

$$\begin{equation}U=\int {\mathrm{d}}^{d}x\enspace {u}^{k}\left(\boldsymbol{x}\right){q}_{k}\left(\boldsymbol{x}\right).\end{equation} \tag{ 44 }$$

For example, we could take U = H + V, where V represents a coupling to external spatially-varying fields V = ∫d^d x u⁰( x )q₀( x ) (in the notation of subsection 4.1); or a spatially-varying Hamiltonian strength U = ∫d^d x u²( x )q₂( x ).

Assuming local entropy maximisation, averages of time-evolved local observables, $o\left(\boldsymbol{x},t\right)={\mathrm{e}}^{\mathrm{i}Ut}o\left(\boldsymbol{x}\right){\mathrm{e}}^{-\mathrm{i}Ut}$, in non-homogeneous initial states ⟨...⟩_ini, are well described by space-time dependent MES ⟨...⟩ _{x ,t} , that is ${\langle o\left(\boldsymbol{x},t\right)\rangle }_{\text{ini}}\approx {\langle o\rangle }_{\boldsymbol{x},t}$. Physically, ⟨...⟩ _{x ,t} is the state in the local fluid cell at x , t, according to the hydrodynamic separation of scales. The dynamics of the space-time dependent state are fixed by the conservation laws, yielding the Euler equations of the system. Below we drop the indices x , t for lightness of notation, and ⟨...⟩ represents the local state in the fluid cell.

Local entropy maximisation is expected to hold approximately only under certain conditions; such that the initial state ⟨...⟩_ini and potentials u^k (x) vary only at long wavelengths, and the time t is large enough for local relaxation to have occurred.

From the assumption of local entropy maximisation, a heuristic derivation of the Euler equations follows as in [30]. The Heisenberg equation of motion (with the replacement (18) for classical systems) for the charge densities reads:

$$\begin{equation}{\partial }_{t}{q}_{i}\left(\boldsymbol{x}\right)-\mathrm{i}\int {\mathrm{d}}^{d}y\enspace {u}^{k}\left(\boldsymbol{y}\right)\left[{q}_{k}\left(\boldsymbol{y}\right),{q}_{i}\left(\boldsymbol{x}\right)\right]=0.\end{equation} \tag{ 45 }$$

We now assume that u^k ( x ) varies only at long wavelengths, and take a derivative expansion,

$$\begin{equation}{\partial }_{t}{q}_{i}\left(\boldsymbol{x}\right)-\mathrm{i}\int {\mathrm{d}}^{d}y\enspace \left({u}^{k}\left(\boldsymbol{x}\right)+\left(\boldsymbol{y}-\boldsymbol{x}\right)\cdot \nabla {u}^{k}\left(\boldsymbol{x}\right)\right)\left[{q}_{k}\left(\boldsymbol{y}\right),{q}_{i}\left(\boldsymbol{x}\right)\right]=O\left({\nabla }^{2}\right).\end{equation} \tag{ 46 }$$

The Euler-scale hydrodynamic equation is obtained by taking averages, and making the local entropy maximisation assumption at the first-derivative order. By (6) the first term in the parenthesis gives, after integration over y and averaging, the divergence of the current u^k ∇ ⋅ ⟨ j _ki ⟩; recall that ⟨...⟩ is the MES of the local fluid cell. For the second term, as the coefficient ∇u^k already contains a derivative, the integral over y may be assumed to lie entirely within the local, homogeneous fluid cell, ∫d^d y ( y − x ) ⋅ ⟨[q_k ( y ), q_i ( x )]⟩. Using the result (62) of section 5, we then obtain

$$\begin{equation}{\partial }_{t}\langle {q}_{i}\rangle +\nabla \cdot \left({u}^{k}\langle {\boldsymbol{j}}_{ki}\rangle \right)+\nabla {u}^{k}\cdot \langle {\boldsymbol{j}}_{ik}\rangle =0.\end{equation} \tag{ 47 }$$

This is the universal form of the Euler equation for the many-body system. As far as we are aware, it was first written in [30] (where the relation (62) was argued for under slightly stronger assumptions than those made here). Importantly, the total charges Q_i are no longer conserved in spatially-varying external potentials, as the extra term ∇u^k ⋅ ⟨ j _ik ⟩ is not a total derivative; however it is simple to see that the inhomogeneous operator U is conserved. Thus, if the homogeneous model is integrable, the inhomogeneous time-evolution operator breaks integrability. In the context of integrable systems, an extension of this equation to the diffusive order, with certain large-wavelength external potentials, was studied in [26].

In the special case where the u^k are position-independent the equation reduces to

$$\begin{equation}{\partial }_{t}\langle {q}_{i}\rangle +\nabla \cdot \left({u}^{k}\langle {\boldsymbol{j}}_{ki}\rangle \right)=0.\end{equation} \tag{ 48 }$$

In this case all charges are conserved, as the evolution is generated by the conserved charge U = u^k Q_k . We note that u^k j _ki is the ith current with respect to evolution generated by the charge U. At space-time points where the solution is smooth, it is straightforward to show (see e.g. [14, 38, 39]) that this equation implies conservation of entropy,

$$\begin{equation}{\partial }_{t}s+\nabla \cdot \left({u}^{k}{\textbf{\textsf{j}}}_{k}^{s}\right)=0,\end{equation} \tag{ 49 }$$

where we use (22) and (23). The quantity s is a Lax entropy for the hyperbolic system (48) [38, 39, 44]. Thus, ${u}^{k}{\textbf{\textsf{j}}}_{k}^{s}$ is interpreted as the entropy current with respect to the evolution generated by the charge U.

Using (26), we obtain two general results under (47):

• (a)  
  At every point where the solution is smooth, the entropy density satisfies an exact continuity equation, and thus the total entropy is conserved except possibly at non-smooth points.
• (b)  
  Under mild genericity assumptions, the space-dependent MES ⟨...⟩ _x which is a stationary solution of (47), can be seen as the fluid cell approximation (local density approximation) of a thermal state with respect to the space-varying generator U. Its formal density function is ${\rho }_{\boldsymbol{x}}\propto {\mathrm{e}}^{-\bar{\beta }\left({u}^{k}\left(\boldsymbol{x}\right){Q}_{k}-\bar{\mu }{Q}_{0}\right)}$ for some inverse temperature $\bar{\beta }$ and (if there is particle conservation Q₀) chemical potential $\bar{\mu }$, both independent of x .

The first result is natural at the Euler scale; despite total charges not being conserved in external inhomogeneous fields, one expects entropy to still be preserved at this scale by virtue of the lack of any diffusive effects and the local maximisation of entropy ⁵ . This is obtained here in full generality with arbitrary, long-wavelength inhomogeneous external fields. The second result suggests that if a weak irreversibility is added to the Euler equation, the stationary solution approached must be thermal. These solutions are typically inaccessible by the Euler dynamics due to the entropy reduction required to access them, however they are approached if diffusive effects are added [26].

In order to show statement (a), we consider the definitions (22) and (23) of the entropy density and currents. Applying the time derivative to (23) we find

$$\begin{equation}{\partial }_{t}s={\beta }^{i}{\partial }_{t}\langle {q}_{i}\rangle =-{\beta }^{i}\left(\nabla \cdot \left({u}^{k}\langle {\boldsymbol{j}}_{ki}\rangle \right)+\nabla {u}^{k}\cdot \langle {\boldsymbol{j}}_{ik}\rangle \right).\end{equation} \tag{ 50 }$$

On the other hand, the divergence of the entropy current ${u}^{k}{\textbf{\textsf{j}}}_{k}^{s}$ generated by U is

$$\begin{equation}\nabla \cdot \left({\beta }^{i}{u}^{k}\langle {\boldsymbol{j}}_{ki}\rangle -{u}^{k}{\boldsymbol{g}}_{k}\right)={\beta }^{i}\nabla \cdot \left({u}^{k}\langle {\boldsymbol{j}}_{ki}\rangle \right)-\nabla {u}^{k}\cdot {\boldsymbol{g}}_{k}.\end{equation} \tag{ 51 }$$

Using (30) in the form g _k = −βⁱ ⟨ j _ik ⟩, we obtain the required cancellation

$$\begin{equation}{\partial }_{t}s+\nabla \cdot \left({u}^{k}{\textbf{\textsf{j}}}_{k}^{s}\right)=0.\end{equation} \tag{ 52 }$$

In order to show statement (b), we use the chain rule and the identity (34) for the B matrix. The condition for stationarity ∂_t ⟨q_i ⟩ = 0 of (47) then reads

$$\begin{equation}\left({u}^{k}\nabla {\beta }^{j}-{\beta }^{k}\nabla {u}^{j}\right)\cdot {\boldsymbol{B}}_{kij}=0.\end{equation} \tag{ 53 }$$

As this must hold for all i, generically, it must be true for each ordered pair (j, k) separately; and if the B -matrices are generic enough, the parenthesis must vanish when projected onto a generic vector. Thus, we have the solution

$$\begin{equation}{\beta }^{k}=\bar{\beta }{u}^{k},\end{equation} \tag{ 54 }$$

with $\bar{\beta }$ a k-independent thermalisation constant. This is the locally thermal state with respect to the generator U, and indicates that the density matrix of the stationary state is of the form $\rho \propto {\mathrm{e}}^{-\bar{\beta }U}$. Thus $\bar{\beta }$ is identified with the inverse temperature of the stationary state.

We have used arguments of genericity. Of course, if the B matrices satisfy certain relations, or do not span a large enough vector space, then different, special solutions may exist; it would be interesting to investigate these solutions and their physical meaning. A case which is important is that where an ultra-local conserved density exists, such as from particle conservation, N = Q₀. As mentioned, this means g ₀ = 0, hence B _0ij = 0 for all i, j. For β⁰, the generic stationarity requirement becomes, instead of (54),

$$\begin{equation}{\beta }^{0}=\bar{\beta }\left({u}^{0}-\bar{\mu }\right),\end{equation} \tag{ 55 }$$

where an additional constant—the chemical potential of the stationary state—appears. In this case, the stationary density matrix is of the form $\rho \propto {\mathrm{e}}^{-\bar{\beta }\left(U-\bar{\mu }{Q}_{0}\right)}$.

We note that the statements (a) and (b) were shown in [30] in the context of integrable models, by using the specific properties of generalised hydrodynamics. Here they are shown to arise from fully general principles of many-body physics. One expects corresponding statements (increase of entropy, thermal stationary state) for the hydrodynamic equation that includes the diffusive corrections. With ultra-local space-varying external field, such statements were shown in [26] in the context of integrable models. The EKMS relation can likewise be used to obtain these in the more general context of many-component hydrodynamic equations, without the use of integrability. We plan on presenting a more extensive discussion of this and related aspects in a future work.

The result (29) can be shown explicitly in conformal hydrodynamics. Consider a Lorentz-invariant conformal field theory in d > 1 spatial dimensions. The MES's of this model are the boosted thermal states, where we can use the rotational symmetry of the theory to consider a boost only along the x¹-direction (with x⁰ = t), thus involving only two potentials ⁶ βⁱ for i = 1, 2:

$$\begin{equation}\rho \propto {\text{e}}^{-{\beta }^{1}{P}_{1}-{\beta }^{2}H}={\text{e}}^{-{\beta }_{\text{rest}}\left(\mathrm{cosh}\left(\theta \right)H-\mathrm{sinh}\left(\theta \right){P}_{1}\right)}.\end{equation} \tag{ 56 }$$

Here β_rest is the inverse temperature in the rest-frame, and θ is the rapidity of the boost. In terms of the energy–momentum tensor ${\mathcal{T}}^{\mu \nu }$, we have $H={Q}_{2}=\int {\mathrm{d}}^{d}\boldsymbol{x}\enspace {\mathcal{T}}^{00}\left(\boldsymbol{x}\right)$ and ${P}_{1}={Q}_{1}=\int {\mathrm{d}}^{d}\boldsymbol{x}\enspace {\mathcal{T}}^{01}\left(\boldsymbol{x}\right)$. The charge and current averages within the state (56) are constrained as follows by the conformal symmetry:

$$\begin{equation}\langle {\mathcal{T}}^{\mu \nu }\rangle =a{\beta }_{\text{rest}}^{-d-1}\left(\left(d+1\right){u}^{\mu }{u}^{\nu }+{\eta }^{\mu \nu }\right),\qquad {\eta }^{\mu \nu }=\mathrm{diag}\left(-1,1,1,\dots ,1\right)\end{equation} \tag{ 57 }$$

where a is a model-dependent constant and u^μ = (cosh θ, sinh θ, 0, 0, ..., 0)^μ . In particular,

$$\begin{equation}\langle {q}_{1}\rangle =\langle {j}_{2}\rangle =a\left(d+1\right){\beta }_{\text{rest}}^{-\left(d+1\right)}\enspace \mathrm{cosh}\left(\theta \right)\mathrm{sinh}\left(\theta \right),\end{equation} \tag{ 58 }$$

$$\begin{equation}\langle {q}_{2}\rangle =a{\beta }_{\text{rest}}^{-\left(d+1\right)}\left(d\enspace {\mathrm{cosh}}^{2}\left(\theta \right)+{\mathrm{sinh}}^{2}\left(\theta \right)\right),\end{equation} \tag{ 59 }$$

$$\begin{equation}\langle {j}_{1}\rangle =a{\beta }_{\text{rest}}^{-\left(d+1\right)}\left({\mathrm{cosh}}^{2}\left(\theta \right)+d\enspace {\mathrm{sinh}}^{2}\left(\theta \right)\right).\end{equation} \tag{ 60 }$$

We would like to verify the general relation (26) in this model of hydrodynamics. As parity symmetry is generically present, we have G = 0. As we only have two potentials, we concentrate on the free energy fluxes g₁ and g₂. It turns out that these have been evaluated in [45], and take the form

$$\begin{equation}{g}_{1}=f=-{\beta }_{\text{rest}}^{-d}\enspace \mathrm{cosh}\left(\theta \right),\qquad {g}_{2}=-{\beta }_{\text{rest}}^{-d}\enspace \mathrm{sinh}\left(\theta \right).\end{equation} \tag{ 61 }$$

It is a simple matter to verify that the EKMS (26) with G = 0, relation β¹ g₁ + β² g₂ = 0 is indeed satisfied. In particular, it is clear in this example that the EKMS relation implies that the knowledge of f immediate gives g, hence the average currents, without explicitly using relativistic and scale invariance. This is a special case of the situation considered in section 4.1.

The quantum chain is characterised by a UHF algebra $\mathfrak{U}$, which may be seen as the completion with respect to the operator norm of the algebra $\mathfrak{L}={\left(\mathrm{E}\mathrm{n}\mathrm{d}\enspace {\mathbb{C}}^{2}\right)}^{\mathbb{Z}}$ of local observables on $\mathbb{Z}$. Space translations form a representation of $\mathbb{Z}$ on algebra ∗-automorphisms. We assume that the model possesses a certain number of extensive, homogeneous (translation-invariant) conserved charges in involution. It is sufficient to provide the associated conserved densities and currents, so we are given ${q}_{i}\left(x\right),{j}_{ki}\left(x\right)\in \mathfrak{L}$ for k, i in some finite index set I and $x\in \mathbb{Z}$, such that relations (6) are satisfied, and that q_i (x), j_ki (x) are x-translates of q_i (0), j_ki (0). In particular, in (6) the commutator is a local observable, $\left[{Q}_{k},{q}_{i}\left(x\right)\right]\in \mathfrak{L}$, as only a finite number of terms in the extensive charge (4) contribute.

For concreteness, one may take the Heisenberg spin chain, whose Hamiltonian density is expressed in terms of the Pauli matrices ${ \overrightarrow {\sigma }}_{x}$ acting on sites $x\in \mathbb{Z}$,

$$\begin{equation}{q}_{2}\left(x\right)={ \overrightarrow {\sigma }}_{x}\cdot { \overrightarrow {\sigma }}_{x+1}.\end{equation} \tag{ 83 }$$

This model possesses a large number of extensive conserved charges thanks to its integrability. In particular, the total spin in every direction is conserved. We may choose one direction, and set the density with label 0 to

$$\begin{equation}{q}_{0}\left(x\right)={\sigma }_{x}^{\text{z}}.\end{equation} \tag{ 84 }$$

Another conserved charge is the one with density

$$\begin{equation}{q}_{4}\left(x\right)={ \overrightarrow {\sigma }}_{x}\cdot \left({ \overrightarrow {\sigma }}_{x+1}{\times}{ \overrightarrow {\sigma }}_{x+2}\right).\end{equation} \tag{ 85 }$$

It turns out that this is (proportional to) the energy current j₂(x), although this will not play any role in our discussion. By transfer matrix methods, it is simple to find more and more local conserved densities, but for concreteness it is sufficient to concentrate on these, with I = {0, 2, 4}. Direct calculations also give the explicit forms of j_ki for k, i ∈ {0, 2, 4}.

As we do not have a momentum operator in quantum spin chains, equations (8) and (9) do not make sense and are omitted; this does not affect the main results.

We consider a set of parameters ${\beta }^{i}\in \mathbb{R}$ for i ∈ I and construct the formal extensive charge W = ∑_i∈I βⁱ Q_i as in (10), where the sum over i is finite. Again, the generator i[W, ⋅] is well defined on $\mathfrak{L}$, with $\mathrm{i}\left[W,\cdot \right]:\mathfrak{L}\to \mathfrak{L}$. As $\mathfrak{L}$ is dense in $\mathfrak{U}$, the closure of this generator generates a unique one-parameter strongly continuous group of ∗-automorphisms $\tau :t\in \mathbb{R}\to {\tau }_{t}$. Here finiteness of the interaction range, that is the fact that $\mathrm{i}\left[W,\cdot \right]:\mathfrak{L}\to \mathfrak{L}$, is used, although weaker conditions are possible. Then, we consider a set of states ⟨...⟩, positive linear functionals $\mathfrak{U}\to \mathbb{C}$, parameterised (implicitly) by ${\beta }^{i}\in \mathbb{R}$ for i ∈ I. These are defined by the KMS relation (13) for every analytic element $o\in \mathfrak{U}$ (entire analytic elements of $\mathfrak{U}$ form a dense subspace). The state is unique for every choice of finite βⁱ , and it is the trace state if βⁱ = 0 ∀ i. See [33, chapter 6.2]. In fact, in a KMS state, the two-point function $\langle {\tau }_{t}{o}_{2}{o}_{1}\rangle$ is analytic in t ∈ [0, −i) for any ${o}_{1},{o}_{2}\in \mathfrak{U}$, and its boundary value satisfies the KMS relation (13) [49].

Further, by [36], the state is invariant under space translations, equation (11), and hence, thanks to (6), it is stationary under every generator [Q_i , ⋅]. Also, as a result of [36] again, the state is exponentially clustering, equation (12). More precisely, one may use the statement from [28, theorem 6.1], restated as follows: let $K\subset {\mathbb{R}}^{\vert I\vert }$ be a compact subset and p > 1. There exists ν, a > 0 such that, for every multiplet β^• ∈ K and every $o,{o}^{\prime }\in \mathfrak{L}$,

$$\begin{equation}\vert {\langle o\left(x\right){o}^{\prime }\left(0\right)\rangle }^{\text{c}}\vert {\leqslant}\nu \vert o{\vert }^{a}\vert {o}^{\prime }{\vert }^{a}{\left(\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\left(o\left(x\right),{o}^{\prime }\left(0\right)\right)\right)}^{-p}\end{equation} \tag{ 86 }$$

where $\vert o\vert$ is the size of the support of $o\left(x\right)$ (it is independent of x), and $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\left(o\left(x\right),o\left(0\right)\right)$ is the distance between the supports of $o\left(x\right)$ and ${o}^{\prime }\left(0\right)$ (it grows like |x| as x → ±∞); see [28] for the details. This is the statement of algebraic clustering for every power p, instead of exponential clustering, but this is sufficient.

As parts of the assumptions of subsection 2.1, clustering is required not only for local observables, but also for imaginary-time evolved ones, with $o\left(x\right)$ replaced by ${\tau }_{-\mathrm{i}s}o\left(x\right)$ for s ∈ [0, 1]. We are not currently aware of a study of clustering of such imaginary-time evolved local observables in thermodynamic states, except for [35]. There, it is shown that, for complex time t with |t| small enough, t-evolved local observables are analytic, and clustering for ${\langle {\tau }_{t}o\left(x\right){o}^{\prime }\left(0\right)\rangle }^{\text{c}}$ holds uniformly. Thus, (86) holds with $o\left(x\right)$ replaced by ${\tau }_{-\mathrm{i}s}o\left(x\right)$ for s ∈ [0, 1], for all β^• in some neighbourhood of 0. The neighbourhood of 0 depends on the particular model under study. This will therefore establish the EKMS relation for β^• in a neighbourhood of 0. But by analyticity of the state in β^• (see below), the relation stays valid for all ${\beta }^{{\bullet}}\in {\mathbb{R}}^{\vert I\vert }$; hence this is sufficient.

We believe that it is possible to establish directly clustering of ${\langle {\tau }_{-\mathrm{i}s}o\left(x\right){o}^{\prime }\left(0\right)\rangle }^{\text{c}}$ for all β^• by using the results of [50]; however this would require a more careful analysis of the infinite-volume limit.

Finally, theorem 6.1 in [28] establishes part of the tangent-manifold relation (16), and can be extended to the full relation ⁷ . The theorem is concerned with the case with βⁱ = 0 for all i ≠ 2, that is, the case W = ∑_i∈I βⁱ Q_i = β² Q₂ ≡ βQ₂; it shows in this case that the derivative $\mathrm{d}\langle o\rangle /\mathrm{d}\beta =-{\langle o{Q}_{2}\rangle }^{\text{c}}$ exists and is analytic in β, for every $o\in \mathfrak{L}$. It is a simple matter to extend the proof to any finite set I, by generalising the steps [28, (equations (70)–(77) in [28])] as follows.

The proof is based on expressing the KMS state ⟨...⟩ as an infinite volume limit, and on theorem 2 in [50]. In equation (66) in [28], the finite volume version of the system, on which the limit is taken, was defined with 'open boundaries'. First, we need to reformulate this to 'periodic boundary conditions', in order to ensure that Q_i , in the finite volume version, are in involution. The choice of boundary condition does not affect this limit, as shown for instance by corollary 2 in [50], and in particular all conditions of theorem 2 in [50] still hold.

Let ${\mathfrak{L}}_{\left[-N,N\right]}={\left(\mathrm{E}\mathrm{n}\mathrm{d}\enspace {\mathbb{C}}^{2}\right)}^{\left[-N,N\right]\cap \mathbb{Z}}$ be the space of local observables supported on the finite set of sites $\left[-N,N\right]\cap \mathbb{Z}$. For all N large enough, there is n > 0 such that ${q}_{i}\left(x\right),{j}_{ki}\left(x\right)\in {\mathfrak{L}}_{\left[-N,N\right]}$ for all x ∈ [−N + n, N − n] and all i, k ∈ I. Let us define new observables, still denoted q_i (x), j_ki (x), that belong to ${\mathfrak{L}}_{\left[-N,N\right]}$ for x ∈ [−N, N], as translates of q_i (0), j_ki (0), under translation automorphisms that are periodic on $\left[-N,N\right]\cap \mathbb{Z}$. The new q_i (x), j_ki (x) may differ from the old ones only 'near the boundaries', for x ∈ [−N, −N + n − 1] ∪ [N − n + 1, N]. Then, the limit

$$\begin{equation}\langle o\rangle =\underset{N\to \infty }{\mathrm{lim}}\frac{{\mathrm{Tr}}_{{\mathfrak{L}}_{\left[-N,N\right]}}\left({\mathrm{e}}^{-{\sum }_{i\in I}{\beta }^{i}{Q}_{i}^{\left(N\right)}}o\right)}{{\mathrm{Tr}}_{{\mathfrak{L}}_{\left[-N,N\right]}}\left({\mathrm{e}}^{-{\sum }_{i\in I}{\beta }^{i}{Q}_{i}^{\left(N\right)}}\right)},\qquad {Q}_{i}^{\left(N\right)}=\sum\limits _{x\in \left[-N,N\right]}{q}_{i}\left(x\right)\end{equation} \tag{ 87 }$$

exists for every $o\in \mathfrak{L}$ and gives, after completion to $\mathfrak{U}$, the KMS state ⟨...⟩ with the properties discussed above.

As, by construction, (6) holds for Q_i replaced by ${Q}_{i}^{\left(N\right)}$, we have $\left[{Q}_{i}^{\left(N\right)},{Q}_{j}^{\left(N\right)}\right]=0$. Then, the steps (equations (70)–(77) in [28]) follow, with the derivative d/dβ replaced by ∂/∂βⁱ for any i ∈ I, and the trace state replaced by the state where βⁱ = 0. The arguments presented there show that $\partial \langle o\rangle /\partial {\beta }^{i}$ is analytic in ${\beta }^{j}\in \mathbb{R}$ for every j ∈ I and fixed β^k≠j (the argument is that the finite-volume version is analytic, and analyticity subsists in the infinite-volume limit thanks to the uniform clustering of theorem 2 in [50]); hence, in particular, it is continuous. Therefore, (16) holds, and the derivative is continuous.

Thus, all assumptions of subsection 2.1, at the basis of the general results, are satisfied in translation-invariant quantum spin chains with finite-range interactions. The Heisenberg chain, with the choice I = {0, 2, 4} as above, gives an explicit and nontrivial example.

As the main results are concerned with the free energy fluxes g_k , it is important to guarantee that these exist and satisfy the required relations. As explained in subsection 2.2, this is a result of standard arguments, and we simply repeat these arguments here in the context of quantum spin chains for completeness and full mathematical rigour.

First, the symmetry B _kij = B _kji follows from translation invariance and clustering, and it is easy to check that the clustering statement (86) is sufficient in order for the derivation in appendix C to hold rigorously. Then, the equality

$$\begin{equation}\partial \langle {j}_{ki}\rangle /\partial {\beta }^{j}=\partial \langle {j}_{kj}\rangle /\partial {\beta }^{i},\end{equation} \tag{ 88 }$$

which follows from this symmetry, along with continuity of the derivatives, established above, show the existence of a second-differentiable function g_k such that (21) holds. As ⟨j_ki ⟩ are analytic in β^j for every j, so are g_k . Similar arguments hold for the existence of a free energy f, using symmetry of the C _ij matrix. Hence, the discussion in subsection 2.2 is mathematically rigorous (again, omitting (25) as we do not assume the presence of a conserved momentum in quantum spin chains).

Assuming that the main result (26) holds, it is also a simple matter to verify that equations (27)–(35), except (31), are therefore rigorously established.

In section 5 we establish the general identity (62), which plays an important role in the derivation of the main result, the EKMS relation (26). It is a easy to see that all steps are fully rigorous. In one dimension, it is not necessary to define the transverse marginals (64), simplifying the steps slightly. In particular, in (65) and (67) the series are convergent thanks to locality of the conserved densities and clustering. We note for instance that the quantity ∫dx [q_i (0), q_j (x)] is local: supported on a finite number of sites around 0. Equation (68) and the steps leading to (70) are rigorous within $\int \mathrm{d}x\enspace {\langle \left[{q}_{i}\left(0\right),{q}_{j}\left(x\right)\right]\cdot \rangle }^{\text{c}}$. Finally locality of ∫dx [q_i (0), q_j (x)] allows us to use the tangent-manifold relation and obtain (75).

In section 6 the main result is obtained. Clustering (86), in particular analyticity and clustering of imaginary-time evolved observables for β^• near enough to 0, is established rigorously; so are the first-moment relation (62), the tangent-manifold relation (16), and the trace property of the infinite-temperature state. This guarantees that every step is rigorous.

Thus, we have established (26), and hence equations (27)–(35) (except (31)), in translation-invariant quantum spin chains with finite-range interactions, when a finite number of extensive conserved charges Q_i , i ∈ I with local (i.e. finite-range) densities are considered. The arguments presented hold for β^• near enough to 0, where clustering at large distances of imaginary-time evolved observables has been proven in the literature, but by analytic continuation all relations still hold for ${\beta }^{{\bullet}}\in {\mathbb{R}}^{\vert I\vert }$. Again, the Heisenberg chain, with the choice I = {0, 2, 4} as in subsection A.1, gives an explicit and nontrivial example. It is simple to show that the Heisenberg chain has parity symmetry, hence in this case G = 0.

We gather the main result in the following theorem.

Theorem A.1. Let ⟨...⟩ be (τ, 1) − KMS state in a quantum spin chain on infinite volume, parameterised by ${\beta }^{i}\in \mathbb{R}$ for i ∈ I, where I is a finite set. Take τ to be generated by W = ∑_i∈I βⁱ Q_i , where Q_i are extensive conserved quantities in involution, with local (supported on finite numbers of sites) densities q_i (x) and associated traceless local generalised currents j_ki (x). Then there are differentiable real functions ${g}_{k}:{\mathbb{R}}^{\vert I\vert }\to \mathbb{R}$ of βⁱ 's such that ⟨j_ki ⟩ = ∂g_k /∂βⁱ , and there exists $G\in \mathbb{R}$ such that (26) holds, ∑_k∈I β^k g_k = G for all ${\beta }^{{\bullet}}\in {\mathbb{R}}^{\vert I\vert }$. In particular, relations (30), (34) and (35) hold for all ${\beta }^{{\bullet}}\in {\mathbb{R}}^{\vert I\vert }$.