Quench action and Renyi entropies in integrable systems
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 Rényi entropies. We develop a new technique to calculate the diagonal Rényi entropy in the quench action formalism. In the spirit of the replica treatment for the entanglement entropy, the diagonal Rényi entropies are generalised free energies evaluated over a thermodynamic macrostate which depends on the Rényi 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 Rényi 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 Rényi entropy.
Introduction.–An intriguing question for an out-of-equilibrium isolated system evolving from a pure state 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 (2); (3); (4); (5); (6); (7); (8) or integrable (9); (10); (11); (12); (13); (14); (15); (16); (17); (18); (19); (20); (21); (22); (23); (24); (25); (26); (27); (28); (29); (30); (31); (32); (33); (34); (35); (36) 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 (37) the stationary value of the entanglement of a large subsystem embedded in an infinite system (38); (40); (39); (41); (42); (43); (44); (45); (46); (47). This entanglement is quantified by the von Neumann entropy of the reduced density matrix of the subsystem . 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 (46) as carefully tested against numerical simulations in the XXZ spin-chain (46).
An alternative and natural definition of entropy after a quench is the diagonal entropy (48), which is the von Neumann entropy of the diagonal ensemble
Here the states are the eigenstates of the post-quench Hamiltonian and the coefficients are the overlaps with the initial state . 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 (48), as tested also in numerical simulations (49); (40). Contrarily, for quenches to integrable models it has been found numerically and analytically that the diagonal entropy is half of the thermodynamic one (40); (50); (51); (52); (41); (42); (53). A solid physical argument to explain this factor has been provided by Gurarie (50), but so far there was no proof, neither a complete understanding of its generality.
For a generic density matrix , the Rényi entropies
represent a more general class of entropies generalising the von Neumann one. Here is the index of the Rényi entropy and it is an arbitrary positive real number. In the limit , (2) gives back the von Neumann entropy. Although Rényi entropies for generic 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 ) are entanglement monotones (54), 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 , they are the core of the replica approach to the entanglement entropy (55). While the replica method was introduced mainly as a theoretical trick (55), it became a useful tool for the experimental measurement of entanglement: Rényi entanglement entropies (for ) have been measured experimentally with cold atoms, both in equilibrium (56) and after a quantum quench (44).
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 (57); (58)) and we derive analytically the diagonal Rényi entropies (i.e. (2) with ). 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 (59).
The Thermodynamic Bethe ansatz (TBA).–In Bethe ansatz solvable models, the eigenstates are in correspondence with a set of pseudomomenta , usually called rapidities. These rapidities satisfy a set of non-linear coupled equation known as Bethe equations. In the thermodynamic limit (60) the rapidities form a continuum and the eigenstates are characterised by the densities of particles and of holes (holes are unoccupied Bethe modes that in an interacting model are not simply related to ). With bold symbols and we denoted the full set of all stable Bethe quasi-particles, i.e. , where 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 and 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 and is , with the thermodynamic Yang-Yang entropy of the macrostate (61)
where the total density is . For systems in thermal tequilibrium, is the thermal entropy (62).
In the thermodynamic limit, a sum over eigenstates is replaced by a functional integral over the rapidity densities as
where the factor 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 (57); (58). Although this approach can be used to study the time evolution (as done in a few cases (63); (64); (65)), 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 (66); (67); (68); (69); (70); (71); (72); (73); (77); (74); (75); (76). The stationary value of a local observable , i.e. its value in the diagonal ensemble, is
According to the recipe (4), in the thermodynamic limit this becomes
where we defined the thermodynamic limit of the logarithm of the overlaps as (60)
This functional integral can be evaluated in the limit using the saddle point approximation, since both the and are generally estensive. One has to minimise the functional appearing in the exponential in (6), i.e.
which has been termed quench action. The minimisation is carried out solving
under the constraint that the thermodynamic Bethe equations hold. Notice that 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 ) have non-zero overlap (78); (79); (77); (80); (81); (82); (83); (84); (85); (86). This fact can be effectively taken into account either by performing the integral defining (3) only on positive ’s or multiplying by . We will adopt the latter choice.
Finally for the expectation values of observables (6), one obtains
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 , implying that at the saddle point the quench action vanishes, i.e.
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 in (1), we simply have
where we introduced the factor in front of 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 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
and the quench action saddle-point equation is
The Rényi diagonal entropies are then the saddle point expectation of this quench action
which is a form that closely resembles the replica approach to the entanglement entropy (55).
The set of coupled equations (14) can be solved, at least numerically, by standard methods. This analysis will be presented elsewhere (59), 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
The normalisation (11), implying , ensures the limit to be finite. We then need to expand at the first order in the saddle point quench action as
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 (50). 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 (58)) then the above argument would immediately lead to the equality of the two entropies, as numerically observed (49).
Indeed, our formalism can be used in general to show that Rényi diagonal entropies are half of the thermodynamic Rényi entropies for arbitrary . 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 as
which simply tells that each mode is populated with probability and empty with probability . Again, assuming parity invariance, the integral is only on positive . In TBA language, the rapidities are the real momenta ; there is only one root density and the hole density is .
Thus, Eq. (22) implies that, for free fermions, the Rényi entropies for arbitrary can be written in terms of the root density at 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 in terms of . In the following we will work with to get rid of factors ’s and often omit the dependence.
The Yang-Yang entropy for free fermions is
because . (For lattice free fermions, the integral runs between and , but this does not influence any of the following results.) Exploiting the parity invariance, the driving term can be always written as
for some calculable depending on the initial state (e.g. for the quench from zero to infinite interaction in the Lieb-Liner gas (78) with being the density). The solution of the generalised quench action saddle point equation (14) is straightforwardly worked out
The mode occupation is , i.e. .
In terms of the generalised saddle-point free energy is, after simple algebra,
One can use the relation between and to write as function of as
The Renyi entropy is straightforwardly obtained from (16)
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 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- XXZ chain starting from the Néel state which has been studied intensively in the literature (69); (71); (87). The exact diagonalisation data for the diagonal entropies are reported in Fig. 1 showing for and . Finite-size effects are visible for both and . The star symbols denote the diagonal entropy densities in the thermodynamic limit, which are obtained using (15). The dash-dotted lines are linear fits to with fixed to the TBA prediction. The data are clearly compatible with the quench action prediction for all values of although large oscillations in 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 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 chain, which can be accessed by varying . This is similar, in spirit, to the observation (88) 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 (38), similarly to what done for the von Neumann entropy (46).
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 (89)), 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 (89) 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 ?
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 (90). 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 , generalizing the result for 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 and for each string of length . These equations are written in a more compact form in terms of the ratios , assuming the standard form (62); (58)
where the symbol denotes the convolution and we introduced the driving term from
In Eq. (29) the functions encode the interaction in the thermodynamic limit and are simply read from the Bethe equations of the specific model (see e.g. (62)). Consequently the saddle point equation for general in Eq. (14) are simply
The generalized Gibbs ensemble describing local observables after a quench to an integrable model has the form
where the operators ’s form a complete set of local and quasi-local integrals of motion and are fixed by the condition that the charges assume the same value in the initial state and in the GGE, i.e. .
The expectation values of local operator in the GGE are obtained by generalized TBA (91). Analogously to (6) for the diagonal ensemble, we write the expectation value of a local operator as a path integral
where we used the fact that Bethe states are eigenstates of the conserved charges with eigenvalues , i.e.
The resulting saddle point equations are easily read from the above path integral (91)
where we introduced the functions as
These functions 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 above), these functions are simply , i.e. .
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 which implies Eqs. (29) and (35) to be equal. This equality leads to the identification
for where is defined. This identification implies
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 which are odd in do not contribute to the GGE because the considered initial state is even and the corresponding ’s are vanishing. This consideration ensures the validity of (38).
We are finally ready to obtain thermodynamic Rényi entropies. These are given by
Thus, the saddle point equation are the same as in (35) with the replacement for all . 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
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