Quench action and Renyi entropies in integrable systems

TL;DR

This paper introduces a new method to calculate Renyi entropies in integrable quantum systems after a quench, revealing that different Renyi entropies probe different spectral regions and confirming that diagonal entropy is half of the thermodynamic entropy.

Contribution

It develops a novel technique within the quench action formalism to compute diagonal Renyi entropies, generalizing previous results and providing insights into spectral information encoding.

Findings

Diagonal Renyi entropies depend on a macrostate different from the von Neumann entropy.

Different Renyi entropies encode information about different spectral regions.

Diagonal entropy is confirmed to be half of the thermodynamic entropy in integrable systems.

Abstract

Entropy is a fundamental concept in equilibrium statistical mechanics, yet its origin in the non-equilibrium dynamics of isolated quantum systems is not fully understood. A strong consensus is emerging around the idea that the stationary thermodynamic entropy is the von Neumann entanglement entropy of a large subsystem embedded in an infinite system. Also motivated by cold-atom experiments, here we consider the generalisation to Renyi entropies. We develop a new technique to calculate the diagonal Renyi entropy in the quench action formalism. In the spirit of the replica treatment for the entanglement entropy, the diagonal Renyi entropies are generalised free energies evaluated over a thermodynamic macrostate which depends on the Renyi index and, in particular, it is not the same describing the von Neumann entropy. The technical reason for this, maybe surprising, result is that the evaluation of the moments of the diagonal density matrix shifts the saddle point of the quench action. An interesting consequence is that different Renyi entropies encode information about different regions of the spectrum of the post-quench Hamiltonian. Our approach provides a very simple proof of the long-standing issue that, for integrable systems, the diagonal entropy is half of the thermodynamic one and it allows us to generalise this result to the case of arbitrary Renyi entropy.

Equations

$${\rho }_d=\sum _{m=1}^{\infty }{w}_m|m⟩⟨m|\text{with}{w}_m\equiv |⟨{\Psi }_0|m⟩{|}^2.$$

$$S^{(\alpha )}\equiv \frac{1}{1-\alpha }\ln \text{Tr}{\rho }^{\alpha },$$

$$S_{YY}[\boldsymbol{\rho}]\equiv L\sum_{n}\int d\lambda\Big{[}\rho_{n}^{{}_{(t)}}\ln\rho_{n}^{{}_{(t)}}\\ -\rho_{n}\ln\rho_{n}-\rho_{n}^{{}_{(h)}}\ln\rho_{n}^{{}_{(h)}}\Big{]},$$

$$\sum _m\to \int \mathcal{D}\mathbf{\rho }{e}^{S_{YY}[\mathbf{\rho }]},$$

$$⟨\mathcal{O}{⟩}_d=\sum _m|⟨{\Psi }_0|m⟩{|}^2⟨m|\mathcal{O}|m⟩.$$

$$\quad\langle{\mathcal{O}}\rangle_{d}=\int{\mathcal{D}}\boldsymbol{\rho}\exp\Big{[}-2{\cal E}(\boldsymbol{\rho})+S_{YY}(\boldsymbol{\rho})\Big{]}\langle\boldsymbol{\rho}|{\mathcal{O}}|\boldsymbol{\rho}\rangle,$$

$${\cal E}(\boldsymbol{\rho})\equiv-\lim_{\rm th}\Big{[}{\rm Re}(\ln\langle\boldsymbol{\rho}|\Psi_{0}\rangle)\Big{]}.$$

$${\mathcal{S}}_Q(\mathbf{\rho })\equiv -2\mathcal{E}(\mathbf{\rho })+S_{YY}(\mathbf{\rho }),$$

$$\frac{\delta{\mathcal{S}_{Q}}(\boldsymbol{\rho})}{\delta\boldsymbol{\rho}}\bigg{|}_{\boldsymbol{\rho}=\boldsymbol{\rho^{*}}}=0,$$

$$⟨\mathcal{O}{⟩}_d=⟨{\mathbf{\rho }}^{\ast }|\mathcal{O}|{\mathbf{\rho }}^{\ast }⟩,$$

$${\mathcal{S}}_Q({\mathbf{\rho }}^{\ast })=0.$$

$$\text{Tr}{\rho }_d^{\alpha }=\int \mathcal{D}\mathbf{\rho }{e}^{-2\alpha \mathcal{E}[\mathbf{\rho }]+\frac{1}{2}{S}_{YY}[\mathbf{\rho }]},$$

$${\mathcal{S}}_Q^{(\alpha )}(\mathbf{\rho })\equiv -2\alpha \mathcal{E}(\mathbf{\rho })+\frac{1}{2}{S}_{YY}(\mathbf{\rho }),$$

$$\frac{\delta{\mathcal{S}_{Q}^{(\alpha)}}(\boldsymbol{\rho})}{\delta\boldsymbol{\rho}}\bigg{|}_{\boldsymbol{\rho}=\boldsymbol{\rho^{*}_{\alpha}}}=0.$$

$$S_{d}^{(\alpha)}=\frac{{\cal S}_{Q}^{(\alpha)}(\boldsymbol{\rho}^{*}_{\alpha})}{1-\alpha}=\frac{1}{1-\alpha}\Big{[}-2\alpha{\cal E}(\boldsymbol{\rho}^{*}_{\alpha})+\frac{1}{2}S_{YY}(\boldsymbol{\rho}^{*}_{\alpha})\Big{]}.$$

$$S_d^{(\alpha )}=\frac{{\mathcal{S}}_Q^{(\alpha )}({\mathbf{\rho }}_{\alpha }^{\ast })-\alpha {\mathcal{S}}_Q^{(1)}({\mathbf{\rho }}_1^{\ast })}{1-\alpha },$$

$$\underset{\alpha \to 1}{\lim }{S}_d^{(\alpha )}=S_d\equiv -\mathrm{Tr}{\rho }_d\ln {\rho }_d.$$

$${\cal S}_{Q}^{(\alpha)}(\boldsymbol{\rho}^{*}_{\alpha})=\frac{d{\cal S}_{Q}^{(\alpha)}(\boldsymbol{\rho}^{*}_{\alpha})}{d\alpha}\bigg{|}_{\alpha=1}(\alpha-1)+O((\alpha-1)^{2}).$$

$$\frac{d{\cal S}_{Q}^{(\alpha)}(\boldsymbol{\rho^{*}_{\alpha}})}{d\alpha}=\frac{\delta{\cal S}_{Q}^{(\alpha)}(\boldsymbol{\rho})}{\delta\boldsymbol{\rho}}\bigg{|}_{\boldsymbol{\rho}=\boldsymbol{\rho^{*}_{\alpha}}}\frac{\partial{\boldsymbol{\rho^{*}_{\alpha}}}}{\partial\alpha}+\frac{\partial{\cal S}_{Q}^{(\alpha)}(\boldsymbol{\rho})}{\partial\alpha}\bigg{|}_{\boldsymbol{\rho}=\boldsymbol{\rho^{*}_{\alpha}}},$$

$${\mathcal{S}}_Q^{(\alpha )}({\mathbf{\rho }}_{\alpha }^{\ast })=-2\mathcal{E}({\mathbf{\rho }}_1^{\ast })(\alpha -1)+O((\alpha -1{)}^2),$$

$$\underset{\alpha \to 1}{\lim }{S}_d^{(\alpha )}=-2\mathcal{E}({\mathbf{\rho }}_1^{\ast })=\frac{1}{2}{S}_{YY}({\mathbf{\rho }}_1^{\ast }),$$

$$S_d^{(\alpha )}=\frac{L}{1-\alpha }\int \frac{dk}{2\pi }\ln [n_k^{\alpha }+(1-n_k{)}^{\alpha }],$$

$$S_{YY}=-L{\int }_{-\infty }^{\infty }\frac{dk}{2\pi }[\tilde{\rho }\ln \tilde{\rho }+(1-\tilde{\rho })\ln (1-\tilde{\rho })],$$

$${\cal E}=-L\Big{[}\int_{0}^{\infty}\frac{dk}{2\pi}\tilde{\rho}(k)\ln g(k)\Big{]},$$

$${\tilde{\rho }}_{\alpha }^{\ast }(k)=\frac{1}{1+[g(k){]}^{\alpha }}.$$

$${\mathcal{S}}_{\alpha }({\tilde{\rho }}_{\alpha }^{\ast })=-L{\int }_0^{\infty }\frac{dk}{2\pi }\ln (1-{\tilde{\rho }}_{\alpha }^{\ast }(k)).$$

$${\tilde{\rho }}_{\alpha }^{\ast }(k)=\frac{n_k^{\alpha }}{n_k^{\alpha }+(1-n_k{)}^{\alpha }}.$$

$$\begin{gathered} S_d^{(\alpha )}=-\frac{L}{1-\alpha }{\int }_0^{\infty }\frac{dk}{2\pi }\ln \frac{1-{\tilde{\rho }}_{\alpha }^{\ast }(k)}{(1-{\tilde{\rho }}_1^{\ast }(k){)}^{\alpha }} \\ =\frac{L}{1-\alpha }{\int }_0^{\infty }\frac{dk}{2\pi }\ln [n_k^{\alpha }+(1-n_k{)}^{\alpha }], \end{gathered}$$

$$\ln {\eta }_n(\lambda )=g_n(\lambda )+\sum _{m=1}^{\infty }{a}_{nm}\star \ln (1+{\eta }_m^{-1})(\lambda ),$$

$$2\mathcal{E}[\mathbf{\rho }]=L\sum _{n=1}^{\infty }{\int }_0^{\infty }d\lambda {\rho }_n(\lambda )g_n(\lambda ).$$

Full text

Introduction.–An intriguing question for an out-of-equilibrium isolated system evolving from a pure state $|\Psi_{0}\rangle$ is to understand microscopically the thermodynamic entropy of the emergent statistical ensemble. Indeed, it has been undoubtedly established that for large times the local properties of a quenched system are correctly captured by a statistical ensemble which is either the Gibbs or generalised Gibbs depending on whether the system is generic  v-29 ; eth0 ; deutsch-1991 ; srednicki-1994 ; rdo-08 ; rigol-2012 ; dalessio-2015 or integrable rigol-2007 ; cazalilla-2006 ; barthel-2008 ; cramer-2008 ; fagotti-2008 ; cazalilla-2012a ; calabrese-2012 ; sotiriadis-2012 ; collura-2013 ; fagotti-2013 ; p-13 ; fe-13b ; kcc14 ; sotiriadis-2014 ; fcec-14 ; ilievski-2015a ; alba-2015 ; langen-15 ; essler-2015 ; cardy-2015 ; sotiriadis-2016 ; bastianello-2016 ; vernier-2016 ; pvw-17 ; vidmar-2016 ; gogolin-2015 ; calabrese-2016 ; ef-16 respectively. However, since the entire system is always in a zero-entropy pure state because of the unitary time evolution, it is not obvious whether the entropy of the statistical ensemble has any physical meaning. This apparent paradox has been nowadays clarified and it is widely accepted that the thermodynamic entropy is foot1 the stationary value of the entanglement of a large subsystem $A$ embedded in an infinite system calabrese-2005 ; spr-12 ; dls-13 ; collura-2014 ; kormos-2014 ; bam-15 ; kaufman-2016 ; cc-16 ; alba-2016 ; d-17 . This entanglement is quantified by the von Neumann entropy $S_{A}\equiv-\textrm{Tr}\rho_{A}\ln\rho_{A}$ of the reduced density matrix $\rho_{A}$ of the subsystem $A$. Very recently, the equivalence between stationary entanglement entropy and thermodynamic entropy has been exploited to provide analytic exact predictions for the entire time dependence of the entanglement entropy in integrable models alba-2016 as carefully tested against numerical simulations in the XXZ spin-chain alba-2016 .

An alternative and natural definition of entropy after a quench is the diagonal entropy $S_{d}$ polkovnikov-2011 , which is the von Neumann entropy $S_{d}=-\textrm{Tr}\rho_{d}\ln\rho_{d}$ of the diagonal ensemble

[TABLE]

Here the states $|m\rangle$ are the eigenstates of the post-quench Hamiltonian $H$ and the coefficients $w_{m}$ are the overlaps with the initial state $|\Psi_{0}\rangle$. The great advantage of the diagonal entropy is that can be computed from the initial state, without solving the complicated quench dynamics. A lot of effort has been devoted to understand the relation between the diagonal entropy and the stationary subsystem entanglement entropy (which, as already stressed, is the entropy of the stationary ensemble). It has been suggested that for generic systems relaxing to a Gibbs ensemble the diagonal entropy is the same as the thermodynamic one polkovnikov-2011 , as tested also in numerical simulations santos-2011 ; spr-12 . Contrarily, for quenches to integrable models it has been found numerically and analytically that the diagonal entropy is half of the thermodynamic one spr-12 ; gurarie-2013 ; fagotti-2013b ; dora-2014 ; collura-2014 ; kormos-2014 ; piroli-2016 . A solid physical argument to explain this factor $1/2$ has been provided by Gurarie gurarie-2013 , but so far there was no proof, neither a complete understanding of its generality.

For a generic density matrix $\rho$, the Rényi entropies

[TABLE]

represent a more general class of entropies generalising the von Neumann one. Here $\alpha$ is the index of the Rényi entropy and it is an arbitrary positive real number. In the limit $\alpha\to 1$, (2) gives back the von Neumann entropy. Although Rényi entropies for generic $\alpha$ do not have some important properties of the von Neumann one (the most physical one being the strong subadditivity), they are attracting a lot of attention, especially in connection with entanglement. Indeed, Rényi entanglement entropies (i.e. (2) with $\rho=\rho_{A}$) are entanglement monotones v-00 , i.e. quantities that do not increase under local operations and classical communication which is the technical requirement for any good measure of entanglement. Furthermore, for integer $\alpha$, they are the core of the replica approach to the entanglement entropy cc-04 . While the replica method was introduced mainly as a theoretical trick cc-04 , it became a useful tool for the experimental measurement of entanglement: Rényi entanglement entropies (for $\alpha=2$) have been measured experimentally with cold atoms, both in equilibrium islam-2015 and after a quantum quench kaufman-2016 .

The main goal of this paper is to initiate the investigation of Rényi entropies in interacting integrable models also, but not only, to provide predictions testable in experiments. To this aim, we adapt the Thermodynamic Bethe Ansatz (TBA) techniques for quantum quenches (overlap TBA or Quench Action method caux-2013 ; caux-16 ) and we derive analytically the diagonal Rényi entropies (i.e. (2) with $\rho=\rho_{d}$). In order to make this paper as readable as possible, we report only the general framework for calculating the diagonal entropies and we discuss in details the main physical consequences of our findings. We postpone a detailed analytical and numerical technical analysis of the TBA solutions for specific integrable models to forthcoming publications ac-prep .

The Thermodynamic Bethe ansatz (TBA).–In Bethe ansatz solvable models, the eigenstates are in correspondence with a set of pseudomomenta $\lambda$, usually called rapidities. These rapidities satisfy a set of non-linear coupled equation known as Bethe equations. In the thermodynamic limit foot2 the rapidities form a continuum and the eigenstates are characterised by the densities of particles $\boldsymbol{\rho}(\lambda)$ and of holes $\boldsymbol{\rho}_{h}(\lambda)$ (holes are unoccupied Bethe modes that in an interacting model are not simply related to $\boldsymbol{\rho}$). With bold symbols $\boldsymbol{\rho}$ and $\boldsymbol{\rho}^{(h)}$ we denoted the full set of all stable Bethe quasi-particles, i.e. $\boldsymbol{\rho}\equiv\{\rho_{n}\}_{n=1}^{\cal N}$, where ${\cal N}$ depends on the considered model (e.g. there is only one species for the repulsive Lieb-Liniger model, but infinite for the gapped XXZ chain). In a given eigenstate, the densities $\boldsymbol{\rho}$ and $\boldsymbol{\rho}^{(h)}$ are related by the continuum limit of the Bethe equation. A set of densities identifies a thermodynamic macro-state that corresponds to an exponentially large number of microscopic eigenstates. The total number of microstates with the same $\boldsymbol{\rho}$ and $\boldsymbol{\rho}^{(h)}$ is $e^{S_{YY}}$, with $S_{YY}$ the thermodynamic Yang-Yang entropy of the macrostate yy-69

[TABLE]

where the total density is $\rho_{n}^{{}_{(t)}}\equiv\rho_{n}+\rho_{h}^{(h)}$. For systems in thermal tequilibrium, $S_{YY}$ is the thermal entropy taka-book .

In the thermodynamic limit, a sum over eigenstates is replaced by a functional integral over the rapidity densities as

[TABLE]

where the factor $e^{S_{YY}}$ counts the exponentially large number of microscopic eigenstates leading to the same densities.

The quench action.–The out of equilibrium dynamics of integrable systems can be cast in the TBA language with the quench action method caux-2013 ; caux-16 . Although this approach can be used to study the time evolution (as done in a few cases dc-14 ; dlc-15 ; pc-17 ), it mainly provides a calculable and manageable representation of the stationary state (or equivalently of the diagonal ensemble), as nowadays explicitly worked out for many integrable models BeSE14 ; BePC16 ; pozsgay-13 ; wdbf-14 ; PMWK14 ; ac-16qa ; ppv-17 ; mbpc-17 ; dwbc-14 ; pce-16 ; Bucc16 ; npg-17 . The stationary value of a local observable ${\cal O}$, i.e. its value in the diagonal ensemble, is

[TABLE]

According to the recipe (4), in the thermodynamic limit this becomes

[TABLE]

where we defined the thermodynamic limit of the logarithm of the overlaps as foot2

[TABLE]

This functional integral can be evaluated in the limit $L\to\infty$ using the saddle point approximation, since both the $S_{YY}(\boldsymbol{\rho})$ and ${\cal E}(\boldsymbol{\rho})$ are generally estensive. One has to minimise the functional appearing in the exponential in (6), i.e.

[TABLE]

which has been termed quench action. The minimisation is carried out solving

[TABLE]

under the constraint that the thermodynamic Bethe equations hold. Notice that $S_{YY}$ appearing in (6) and (8) counts only those microstates with a non zero-overlap with the initial state (i.e. those contributing to the sum (5) before taking the thermodynamic limit). This is only a small fraction of the total ones: in all quenches solved so far only parity invariant Bethe states (i.e. those containing only pairs of rapidities with opposite sign, i.e., such that $\{\lambda_{j}\}_{j=1}^{M}=\{-\lambda_{j}\}_{j=1}^{M}$) have non-zero overlap grd-10 ; cd-12 ; dwbc-14 ; pozsgay-14 ; bdwc-14 ; pc-14 ; fz-16 ; dkm-16 ; hst-17 ; msca-16 . This fact can be effectively taken into account either by performing the integral defining $S_{YY}$ (3) only on positive $\lambda$’s or multiplying $S_{YY}$ by $1/2$. We will adopt the latter choice.

Finally for the expectation values of observables (6), one obtains

[TABLE]

under the reasonable assumption that the considered operator does not shift the saddle point (as obvious for local ones). As a particular case, the identity operator in (5) provides the normalisation of the initial state $\langle\Psi_{0}|\Psi_{0}\rangle=1$, implying that at the saddle point the quench action vanishes, i.e.

[TABLE]

Overlap TBA for the diagonal entropies.– So far we have collected all the needed ingredients to study the diagonal entropy with the quench action. Taking the thermodynamic limit (4) of the definition (2) with $\rho=\rho_{d}$ in (1), we simply have

[TABLE]

where we introduced the factor $1/2$ in front of $S_{YY}$ to explicitly take into account that only parity-invariant eigenstates have non-zero overlap with most initial states, as discussed above. The most important aspect of (12) is that the Rényi index $\alpha$ appears in the exponential term and so it shifts the saddle points from the one in (9).

The modified quench action which should be considered in this case explicitly depends on the Rényi index as

[TABLE]

and the quench action saddle-point equation is

[TABLE]

The Rényi diagonal entropies are then the saddle point expectation of this quench action

[TABLE]

${\cal S}_{Q}^{(\alpha)}(\boldsymbol{\rho}^{*}_{\alpha})$ can be thought of as a generalised free energy, with ${\cal E}$ playing the role of the Hamiltonian. Using the property that ${\cal S}_{Q}^{(1)}(\boldsymbol{\rho}^{*}_{1})=0$ as in (11), it is convenient to rewrite (15) as

[TABLE]

which is a form that closely resembles the replica approach to the entanglement entropy cc-04 .

The set of coupled equations (14) can be solved, at least numerically, by standard methods. This analysis will be presented elsewhere ac-prep , while here we only discuss some general properties of the diagonal Rényi entropies following from (14) and (15).

Proof that the diagonal entropy is half of the Yang-Yang entropy.– We now consider the limit

[TABLE]

The normalisation (11), implying ${\cal S}_{Q}^{(1)}(\boldsymbol{\rho}^{*}_{1})=0$, ensures the limit to be finite. We then need to expand at the first order in $(\alpha-1)$ the saddle point quench action as

[TABLE]

We have

[TABLE]

in which the first term in the rhs is zero because of the saddle-point condition (14) while the second term is simply $-2{\cal E}(\boldsymbol{\rho^{*}_{\alpha}})$ (cf. (13)). Hence we have

[TABLE]

i.e.

[TABLE]

where in the last equality we used the normalisation conditions (11). This proves that the diagonal entropy is half of the thermodynamic one. The sole assumption to derive this general result is that only parity invariant states have non-zero overlaps. While this is a technical assumption, it is exactly on the same line of thoughts of the physical interpretation of Gurarie gurarie-2013 . If for some quenches (still unknown in integrable models) all states would have non-zero overlaps, then the above proof would lead to the equality between the diagonal and the thermodynamic entropy. If one imagine of generalising the quench action approach to non-integrable model (as speculated in caux-16, ) then the above argument would immediately lead to the equality of the two entropies, as numerically observed santos-2011 .

Indeed, our formalism can be used in general to show that Rényi diagonal entropies are half of the thermodynamic Rényi entropies for arbitrary $\alpha$. Anyhow, the derivation is slightly more technical and so we report it in the appendix.

Free fermionic systems.– For translational invariant systems of free fermions (or models mappable to), the diagonal Rényi entropies after a quench can be written in terms of the momentum occupation numbers $n_{k}$ as

[TABLE]

which simply tells that each mode is populated with probability $n_{k}$ and empty with probability $1-n_{k}$. Again, assuming parity invariance, the integral is only on positive $k$. In TBA language, the rapidities $\lambda$ are the real momenta $k$; there is only one root density $\rho(k)=n_{k}/(2\pi)$ and the hole density is $\rho^{(h)}(k)=(1-n_{k})/(2\pi)$.

Thus, Eq. (22) implies that, for free fermions, the Rényi entropies for arbitrary $\alpha$ can be written in terms of the root density at $\alpha=1$ without the need of generalised TBA equations. It is instructive to recover (22) from the generalised TBA saddle point (15) and show that, in this particular case, it is possible to write $\rho_{\alpha}$ in terms of $n_{k},i.e.$ $\rho_{\alpha=1}$. In the following we will work with $\tilde{\rho}(k)=2\pi\rho(k)$ to get rid of factors $2\pi$’s and often omit the $k$ dependence.

The Yang-Yang entropy for free fermions is

[TABLE]

because $\rho^{(t)}=1/(2\pi)$. (For lattice free fermions, the integral runs between $-\pi$ and $\pi$, but this does not influence any of the following results.) Exploiting the parity invariance, the driving term ${\cal E}$ can be always written as

[TABLE]

for some calculable $g(k)$ depending on the initial state (e.g. for the quench from zero to infinite interaction in the Lieb-Liner gas $g(k)=4n^{2}/k^{2}$ grd-10 with $n$ being the density). The solution of the generalised quench action saddle point equation (14) is straightforwardly worked out

[TABLE]

The mode occupation is $\tilde{\rho}^{*}_{1}(k)$, i.e. $n_{k}=1/(1+g(k))$.

In terms of $\tilde{\rho}^{*}_{\alpha}(k)$ the generalised saddle-point free energy ${\cal S}_{\alpha}(\tilde{\rho}^{*}_{\alpha})$ is, after simple algebra,

[TABLE]

One can use the relation between $\tilde{\rho}^{*}_{\alpha}$ and $g(k)$ to write $\tilde{\rho}^{*}_{\alpha}$ as function of $n_{k}$ as

[TABLE]

The Renyi entropy is straightforwardly obtained from (16)

[TABLE]

which is the same as (22). Notice that to show this equality in a simple and general way we had to add to the saddle-point generalised action the term $\alpha S(\tilde{\rho}^{*}_{1})$ which is zero, but written in terms of an integral of a non-zero integrand.

The same derivation above also holds for free bosons with very minor modifications.

A numerical example for an interacting model.– It is far beyond the aim of this paper to present detailed calculations of the Rényi diagonal entropies for interacting integrable models which require a large amount of technicalities. However, it is instructive to show at least one example of the comparison between the quench action approach and the numerical computation for finite systems. We focus on the quench in the spin-$1/2$ XXZ chain starting from the Néel state which has been studied intensively in the literature wdbf-14 ; ac-16qa ; pvc-16 . The exact diagonalisation data for the diagonal entropies are reported in Fig. 1 showing $S_{d}^{(\alpha)}$ for $\alpha=2$ and $\alpha=3$. Finite-size effects are visible for both $\alpha=2$ and $\alpha=3$. The star symbols denote the diagonal entropy densities $s_{\infty}^{(\alpha)}$ in the thermodynamic limit, which are obtained using (15). The dash-dotted lines are linear fits to $S_{d}^{(\alpha)}=s^{(\alpha)}_{\infty}+a_{\alpha}/L$ with $s^{(\alpha)}_{\infty}$ fixed to the TBA prediction. The data are clearly compatible with the quench action prediction for all values of $\Delta$ although large oscillations in $L$ affect the numerical data.

Conclusions.– In this paper we developed a Bethe ansatz framework to calculate the diagonal Rényi entropies after a quantum quench in integrable models. The diagonal Rényi entropies are generalised free energies evaluated over a thermodynamic macrostate which depends on the Rényi index $\alpha$ and it is not the same describing the von Neumann entropy and the local observables. We checked that this prescription reproduces known results for free models and we tested numerically its validity for interacting integrable models. An intriguing consequence is that the steady state contains information about different regions of the spectrum of the $XXZ$ chain, which can be accessed by varying $\alpha$. This is similar, in spirit, to the observation garrison-2015 that a single eigenstate of a generic (non-integrable) Hamiltonian at finite energy density contains information about the full spectrum of the Hamiltonian. A byproduct of our approach is a very simple proof of the long standing issue that the diagonal entropy is half of the thermodynamic one in the steady state. This is a direct consequence of the parity invariance of the Bethe eigenstates with non-zero overlap with the initial state.

This Letter opens several new questions. First, it would be important to understand whether the Rényi entropies can be re-written in terms of the saddle point root densities describing local observables. This does not look neither simple nor reasonable from our approach, but, physically, the thermodynamic Rényi entropy is the same as the entanglement one for a large subsystem embedded in an infinite system. Thus it is, in principle, expected to be describable by the same saddle-point of the von Neumann entropy. Having the Rényi entropies written in terms of the standard root densities would allow to reconstruct the entire time dependence of the entanglement Rényi entropies exploiting the quasi-particle picture calabrese-2005 , similarly to what done for the von Neumann entropy alba-2016 .

Another issue is the role played by the parity invariance of the relevant post-quench eigenstates. Up to now this has been seen only as a technical remark (see however delfino-14, ), but we showed that it affects the ratio between diagonal and thermodynamic entropies. Is it possible to design a solvable quench where non-parity invariant states have non vanishing overlaps so that the ratio between the entropies equals one? (the arguments in delfino-14, suggests this to be unlikely). Alternatively, are there quenches in which only a small subclass of parity invariant eigenstates have non-zero overlap making the ratio between the two entropies even smaller than $1/2$?

Acknowledgments.– We are grateful to Bruno Bertini for suggesting the proof reported in the appendix. PC acknowledges support from the ERC under the Starting Grant No 279391 EDEQS and VA the European Union’s Horizon 2020 under the Marie Sklodowoska-Curie grant agreement No 702612 OEMBS.

Note added.– After the completion of this work, one of us and coworkers found a set of initial states for free-like theories in which the eigenstates with non-zero overlap are different from parity invariant states btc-17 . The analysis of the corresponding entropies is not trivial and still under examination.

Appendix A Rényi diagonal entropies are half of thermodynamic ones

In this appendix we prove that Rényi diagonal entropies are half of thermodynamic one for arbitrary index $\alpha$, generalizing the result for $\alpha=1$ presented in the main text. However, this proof turn out to be more cumbersome and technical than the one for the von Neumann entropy, mainly because the thermodynamic Rényi entropies must be derived from the GGE since there is no analogous of the Yang-Yang entropy. In particular, the proof requires the manipulation of the explicit TBA equations that we avoided in the main text.

In TBA language, the saddle point equation (9) is a set of coupled integral equations for the root and hole densities $\rho_{n}(\lambda)$ and $\rho^{(h)}_{n}(\lambda)$ for each string of length $n$. These equations are written in a more compact form in terms of the ratios $\eta_{n}(\lambda)\equiv\rho^{(h)}_{n}(\lambda)/\rho_{n}(\lambda)$, assuming the standard form taka-book ; caux-16

[TABLE]

where the symbol $\star$ denotes the convolution and we introduced the driving term $g_{n}(\lambda)$ from

[TABLE]

In Eq. (29) the functions $a_{nm}(\lambda)$ encode the interaction in the thermodynamic limit and are simply read from the Bethe equations of the specific model (see e.g. taka-book, ). Consequently the saddle point equation for general $\alpha$ in Eq. (14) are simply

[TABLE]

The generalized Gibbs ensemble describing local observables after a quench to an integrable model has the form

[TABLE]

where the operators $\hat{I}_{k}$’s form a complete set of local and quasi-local integrals of motion and $\beta_{k}$ are fixed by the condition that the charges assume the same value in the initial state and in the GGE, i.e. $\langle\Psi_{0}|\hat{I}_{k}|\Psi_{0}\rangle={\rm Tr}[\rho_{\rm GGE}\hat{I}_{k}]$.

The expectation values of local operator in the GGE are obtained by generalized TBA mc-12 . Analogously to (6) for the diagonal ensemble, we write the expectation value of a local operator $O$ as a path integral

[TABLE]

where we used the fact that Bethe states $|\boldsymbol{\rho}\rangle$ are eigenstates of the conserved charges with eigenvalues $I_{k}(\boldsymbol{\rho})$, i.e.

[TABLE]

The resulting saddle point equations are easily read from the above path integral mc-12

[TABLE]

where we introduced the functions $f_{kn}(\lambda)$ as

[TABLE]

These functions $f_{kn}(\lambda)$ are the representation of the local and quasi-local integrals of motion in terms of the rapidities and are generically known but can be cumbersome. For example in the case of the repulsive Lieb-Liniger model where there are no strings (i.e. there is not the sum over $n$ above), these functions are simply $f_{k}(\lambda)=\lambda^{k}$, i.e. $I_{k}(\boldsymbol{\rho})=L\int d\lambda\lambda^{k}\rho(\lambda)$.

At this point, since the expectation values of all the local observables in the diagonal ensemble and in the GGE must be equal, these must have the same saddle point root distribution $\boldsymbol{\rho}$ which implies Eqs. (29) and (35) to be equal. This equality leads to the identification

[TABLE]

for $\lambda\geq 0$ where $g_{n}(\lambda)$ is defined. This identification implies

[TABLE]

where we used the fact that the integral in (30) is on the positive axis, while the one in (36) is on the entire real line. Notice that those functions $f_{kn}(\lambda)$ which are odd in $\lambda$ do not contribute to the GGE because the considered initial state is even and the corresponding $\beta_{k}$’s are vanishing. This consideration ensures the validity of (38).

We are finally ready to obtain thermodynamic Rényi entropies. These are given by

[TABLE]

Thus, the saddle point equation are the same as in (35) with the replacement $\beta_{k}\to\alpha\beta_{k}$ for all $k$. Because of the identification (37), these generalised TBA for the GGE equations are the same as those for the diagonal ensemble (31). This equivalence could have been expected, but it is far from trivial because Rényi entropies are global and not local quantities (and indeed we will soon find a factor 2 between the two). At this point, the GGE Rényi entropy is read off from the saddle-point of the above path integral, in analogy to the diagonal one (15), as

[TABLE]

which using (38) is exactly twice the diagonal Rényi entropy (15).