Lieb-Robinson Bound at Finite Temperature
The Lieb-Robinson bound shows that the speed of propagating information in a nonrelativistic quantum lattice system is bounded by a finite velocity, which entails the clustering of correlations. In this paper, we extend the Lieb-Robinson bound to quantum systems at finite temperature by calculating the dynamical correlation function at nonzero temperature for systems whose interactions are respectively short-range, exponentially-decaying and also long-range. We introduce a simple way of counting the clusters in a cluster expansion by using the combinatoric generating functions of graphs. As an application, we also applied the obtained bound at finite temperature to the proof of the second law for quantum pure states initially satisfying the eigenstate thermalization hypothesis, and obtain a tighter bound on the deviation parameter. Limitations and possible future applications of the obtained bound are also discussed.
pacs:02.10.Ox, 05.50.+q, 71.10.-w
In relativistic physics, the speed of light defines a light cone for propagating particles, and hence sets the causality of the theory. For nonrelativistic quantum spin systems, the Lieb-Robinson bound (1) defines a similar upper bound on the speed of signals/information propagating on a lattice. For a given lattice and a quantum spin system living on with Hamiltonian , if two observables represented by self-adjoint operators are supported respectively on sublattices , the Lieb-Robinson bound says that for the time evolution of , i.e. , the operator norm of the commutator can be bounded as
where are positive constants and is the distance between and . Here the constant is the Lieb-Robinson velocity, which sets the upper bound of propagation of information (say, perturbations) in this system.
The causality defined by a bounded speed of information propagation (plus unitarity) implies the localizability principle, even in the quantum case (for which see e.g. (2)), that everything can be constructed from local mechanisms. Using the Lieb-Robinson bound, one can show the exponential decay of correlations (3); (4), which manifests the localizability of the lattice system. More explicitly, if we denote by the expectation value of the observable in a state , then
where are positive constants. This inequality (2) holds for gapped ground states in systems with respectively short-range and exponentially decaying interactions.
The exponential decay or clustering of correlations is an example of how the Lieb-Robinson bound helps us in understanding structures and dynamics of nonrelativistic quantum systems. There are many other applications of the Lieb-Robinson bound, for which one is referred to the review articles (5); (6). Among these applications, however, the Lieb-Robinson bound for quantum systems at finite temperature has not been considered particularly. Some moments of reflection will show some possible reasons for this and also some relevant progresses: On the one hand, as in quantum thermodynamics, the time evolution of an observable at finite temperature could be stochastic, which might not be as simple as being unitary for a closed system. In (7) a first step has been made that the Lieb-Robinson bound holds for general Markovian quantum evolutions. On the other hand, from the standpoint of quantum statistical mechanics, since the Lieb-Robinson bound depends only on the Hamiltonian of the system, one could expect that this bound holds for certain equilibrium temperature. Indeed, for two (fermionic) operators at finite temperature in a system with short-ranged interactions, the correlation at finite temperature also decreases exponentially with respect to temperature (8),
where is the thermal correlation function and is a constant velocity. But this inequality (3) is not dynamical and hence unable to manifest the causal propagation of information. Besides, the Lieb-Robinson bound enters the proof in a peculiar way that the time parameter is finally integrated out. An alternative proof of the exponential decay of equal-time correlations at finite temperature is recently provided by Kliesch et al. (9). Using the method of cluster expansion, they prove that the exponential decay of generalized thermal covariance for operators in a system with short-range interactions,
where the generalized covariance with and , and is the Schatten -norm. From (4) one can find a critical temperature for localization in this system.
Although the exponential decays in (3) and (4) hold at a nonzero temperature, they do not manifests the time-evolution of observables. However, the study of dynamical correlation functions at finite temperature is a common practice in statistical mechanics and thermal field theories. Therefore, in this paper, we will combine the iterating integral representations in proving the Lieb-Robinson bound (4) and the cluster expansions (9) to give a Lieb-Robinson bound at finite temperature in the systems with respectively power-law decaying, exponentially decaying and short-range interactions. We prove that the exponential decay or power-law decaying of correlations always persists at high enough temperature and short enough time. For instance, for systems with exponentially decaying interactions we have
where , are positive constants, and is a function of whose explicit form can be found in Sec. III.3, (36). In systems with other interactions the results are similar to (5). In comparison with (1) as well as (3),(4), we see that it gives both the critical temperatures and the Lieb-Robinson velocities. And when the temperature is fixed, the result reduces to the original Lieb-Robinson bound (and times a constant factor).
There are many recent applications of the Lieb-Robinson bound to situations at finite temperature (see also Sec. V). In quantum statistical meachnical settings, a nonzero temperature will affect the localizations in such system, in which case a bound at finite temperature is desirable. As an application we apply the Lieb-Robinson bound at finite temperature to the proof of the second law of thermodynamics for pure quantum states satisfying the eigenstate thermalization hypothesis (ETH) (10). We can tighten the bound on errors that measures the deviation from the second law. Notice that in (10) the setting is in fact that of quantum thermodynamics, but in comparing the bath in canonical ensamble and the bath in pure states, our bound at finite temperature can be applied.
In the next section, we first introduce a cluster expansion of the partition function defined on a graph by exploiting the generating function for graphs, which greatly simplifies the proofs involving cluster expansions as those in (9). In Sec. III, we prove the main results of this paper, the Lieb-Robinson bounds at finite temperature for systems with interactions of various type. In Sec. IV, we describe a simple application to the prove of second law in ETH. And finally in Sec. V, we discuss some possible applications of the Lieb-Robinson bound at finite temperature and conclude the paper.
Ii Cluster expansion on graphs
For a general quantum lattice system, there is a set of the sites (or vertices) and a set of their bonds (or edges). The Hamiltonian is defined on the tensor product of finite dimensional Hilbert spaces at the sites . The interaction is defined by the sum of mutually commuting local Hamiltonians on the bonds
Here every bond corresponds to a pair of sites , so that or is defined on . If we say an meets the as , it means , which further means that the bonds can meet on sites only.
Now that the interactions are defined on the bonds, the cluster expansion of partition function, either in the real time case for time evolutions or in the imaginary time case for statistical mechanics, can be expressed in terms of the sequences of bonds, which amounts to the evaluation of the expansion in terms of (sub)graphs,
where is the set of all possible graphs, is the number of bonds in , and by factorizability of bonds . A graph here is marked by a sequences of bonds . However, not all the graphs are involved in the concrete physical process. There will be many graphs that are simply unphysical. We can put some constraints to pick out the subset that is physical. To this end, if we can bound the interactions by some bond-independent number such as , the problem will be translated into finding the generating function of the graphs , and setting . In view of (7), the constraints can be enforced on the form of the generating function such that
with , the number of graphs with bonds, unspecified.
The generating functions are used to generate different graphical structure from a particular graphical structure. A schematical example is the following: Given a non-connected graphs with restriction rules (say, on the possible linking of bonds), let us define a generating function . Similarly, for a connected graph with the same restriction rules as , define a generating function . Then we can decompose the non-connected into many connected ’s. In terms of generating functions, this is an algebraic equation where denotes some algebraic operations that transform to . (E.g. in the case of decomposing non-connected to connected graphs if the connected subgraphs do not overlap each other.) Let us outline several simple but concrete examples of specifying the and the corresponding generating function for some graphs. For more details, one is referred to (11). For a single bond, if we repeat it times with only two fixed sites, it gives one kind of graphs . So, the generating function for such graphs is . If we include the empty bond, then , and the generating function is . Similarly, for graphs with different bonds, each of which is repeated times, the generating function is simply . We thus see that the graphs with bonds are obtained from the through exponentials.
There are some graphs that are not easy to be counted. We need to neglect some properties of these graphs, so that they become simpler graphs, and then add these properties to finally recover the original graphs. For instance, one can simplify a graph to its skeleton tree, then the recovering process can be viewed as adding new graphs to the tree with some specific rules. These new graphs can also be enumerated with generating functions. Therefore, the generating function of the original graphs can be obtained by multiplying the generating function of the simplified graphs and the added graphs. Take the graphs with bonds and given above as an example. We can choose the simplified graph as the graph of bonds without repetition, and then add a bound to it in the following way: (i) The generating function of bonds without repetition is . (ii) When we add a repeated bond, we have choice, and hence every added bond contributes a factor . The generating function of all the added graphs is then . The graphs generated in this way will have some redundencies. With positive , the generating function is therefore larger than the original one . But this is useful in bounding the cluster expansion, as we shall show in the following.
We can give a short proof of the locality of temperature with the help of generating functions of graphs (because the key step in (9) is the cluster expansion as (13)). The outline of the proof is the following, which is also the procedure we shall follow in the next section. In calculating the covariance, the expansion of partition function is limited to the graphs that have at least bonds, where is the distance of two local operator and . The graphs can be divided into several blocks of connected graphs, some of which connect to the operator and others not. Those blocks that do not connect to can be viewed as all possible graphs substracting the blocks connecting to , i.e. . Every block can be simplified to a connected graph without repetition, which is called an animal in (9). The number of animals in the lattice with bonds are bounded by with being the valence of each site and the is just the natural constant. So we can start from animals whose number of bonds bigger that , and add the repeating bonds to recover the blocks, and and glue the blocks to recover the original graphs.
Iii Lieb-Robinson bound at finite temperature
In this section, we prove a Lieb-Robinson bound at finite temperature in two-dimensional quantum lattice systems using the cluster expansions. We show how to bound the interactions by some bond-independent number, and then utilize the generating function of graphs to approach a Lieb-Robinson bound at finite temperature.
The key quantity in these proof is the dynamical correlation function at finite temperature , so we first study the cluster expansions of these functions in Sec.III.1. In subsequent subsections, we prove the bounds for different types of interaction. The exponential clusterings of correlations then manifest themselves in the intermediate steps of this section.
iii.1 Dynamical correlation functions at finite temperature
Given a (two-dimensional) quantum lattice systems at finite temperature , we can observe some local operator through its thermal expectation . (Notice that from the standpoint of quantum thermodynamics, one needs not to specify the Gibbs canonical state . A non-canonical state in the pertinent Hilbert space will do. Here we stick to the quantum statistical mecahnical approach in analogy to (9).)
When one adds a local perturbation to the system Hamiltonian , the expectation we observe will be changed. It is expected that if the perturbation and observable are separated by a large distance, the observable can not feel the perturbation at once. At high enough temperature and short enough time, the change in the expectation value can be bounded as
We can see that in order to find the effect of the perturbation, we need to analyze the dynamical thermal correlation function where is the time evolution of the local operator .
Here we consider two local operator and , supported on respectively, in a general quantum lattice system at finite temperature. (The locality of the operators can be relaxed in such a way that and could be non-local operators as products of local operators supported on regions and . Cf. Appendix B of (12). This is becuase the final bound only depends on the supports .) The correlation function at finite temperature can be written in the form of norm
where is the Schatten -norm of the operator .
We can give a bound on Eq. (10). In order to do so, we need to count graphs appearing in the cluster expansions of of the time evolution operator and the partition function . Notice that although their forms look similar, their cluster expansions are very different. The graphs in the nontrivial expansion of the time evolution operator is similar to the “paths”, whereas the graphs in the nontrivial expansion of the partition function are “clusters”. To see this, let us start the calculations using the method of (4),
where the are the local interaction Hamiltonians living on the bond that do not commute with . Here in the second line we have used the Hölder’s inequality since . Now iterate the process of Eq. (11), we obtain
where the are sequences of bonds that meet sequentially, i.e. , and . From Eq. (12), we can see that each bond must overlap the nearby bonds so that the graphs of these possible sequences are just like “paths”.
The nontrivial expansion of the partition function is very different. Consider on a bond ,
where are also the sequences of bonds , and . Now the only constraint is that must contain at least one connected subsequence that connects and , and all the connected subsequences connecting to must also connect to . Such a constraint is not convenient for practical calculations, so we need to divide into several connected subsequences by adding more constraints and lift them at the final stage. Let us put all of the subsequences that does not connect to into a single sequence , and write other distinct subsequences (complementary to ) as . Since is connected, it might contain repeated bonds. If we substract the repeated bonds, such a graph without repeated bonds is called an animal. We denote the animals from sequence by . Now we can see that the expansion is in terms of those , the “clusters” of animals. We can further rewrite Eq. (13) in the form of inclusion-exclusion principle,
where the normalization is omitted on the right hand side.
iii.2 Systems with short-range interactions
For systems with short-range interactions, the local Hamiltonian is nontrivial only on the nearest neighbors of a site. It is easier to first bound the term (14),
With Eq. (16), we can now use the generating functions of graphs to give the bounds. Let us denote by all the graphs that can be added to animals so as to recover those connected graphs. Its generating function is . Therefore, we can change the sum over to the sum over animals,
where is the set of all possible animals connecting to (not restricted to ). By denoting the distance between and by , we should require . The number of possible animals is generally bounded by its size , where is the number of nearest neighbors or the valence of a site. It equal to for D-dimensional cubic lattice. Hence, we can substitute (17) into (16) and use the conditions on the number of animals to obtain
where . Now that , at a high enough temperature, for example , we then have the familiar form
One can easily obtain a universal critical inverse temperature from Eq. (18)
This critical temperature provides a universal upper bound on physical critical temperatures like the Curie temperature.
Let us turn to the calculations about the “paths” in the expansion of the time evolution operator. Because we have , the bound of term , where comes from Eq. (18), is just the generating function of the “paths” connecting to . Different from the “cluster” of animals, a “path” itself is a connected graph. So there is no need to dividing the graphs into blocks and substracting the repeated bonds. Now given bonds, when we add the -th bond to the graph, we have choices since we have to make sure that the -th bond meets the -th bond. Hence the are maximally possible “paths”, and we can take the bound . Then we have, as in Eq. (18), by bounding by ,
Again this critical time can be used to provide a universal lower bound on the Lieb-Robinson velocity in many-body systems.
where and is the lower bound of the number satisfies inequality . Notice that the first term in (12) corresponds to the case and have been absorbed in the exponent in the second term of (12). The lightcone structure now depends on the temperature (see Fig. 1).
iii.3 Systems with exponentially decaying interactions
For quantum lattice systems with exponentially decaying interactions, the interaction Hamiltonian decreases exponentially with the growth of distance,
with some positive constants, , and . We also require the lattice to satisfy
with a positive constant . This equality can be used to derive another useful inequality
(Cf. the assumption 2.1. in (4).)
Because the system might contain an infinite number of sites, and similarly an infinite number of adjacent bonds for each site. The method of generating function of graph can not be directly applied to this situation. In order to find a bound, we need to make use of Eq. (26), which enables us to contract two linked bonds into a single one. Particularly, let us define an equivalence relation between graphs by their connective structures of bonds (say, ordering), that is, two graphs are in the same generic connective graph if their connective structures are the same. Given a generic connective graph starting with and ending on , we can count the graphs in it by choosing all possible sites between and ergodically. Then the problem of counting all possible graphs (which could be infinite) is therefore translated into the problem of finding all possible generic connective graphs, which can be treated using the method introduced in Sec. II. Pointedly stated: (i) We divide generic connective graphs into connected parts. (ii) For the connected generic graphs, we substract the repeated bonds, and for the connected ones without repeated bonds, we further substract the possible loops. (iii) Now the generic graphs are simplified to generic trees. We are going to analyze these tree graphs, add the loops, reactivate repetitions, glue the blocks, and go through all possible sites ergodically. Finally we can recover all kinds of possible graphs thanks to the ergodic choices of intermediate sites.
In a tree two neighboring bonds can always be contracted and finally one is left with a longest single bond. For example, given a tree connecting to with bonds, we can always find a generic tree . We denote the set of (generic) trees connecting to by . The ergodic choice of intermediate sites of can give all those trees corresponding to . With the help of Eq. (24) and Eq. (26), we can choose the intermediate sites ergodically through the lattice and obtain a bound on the interaction Hamiltonians living on ,
Next, in order to recover the loops and repetitions, we can fix one site in and chosse another site of ergodically. Such a new graph may contain a loop, have a repeated bond in the tree or become a larger tree. Applying Eq. (24) and Eq. (25), we obtain a new factor in the cluster expansion (of the partition function) from the above combinatoric considerations of adding a single bond,
where comes from the indistinguishability of two sites of a bond and the sum over means that the intermediate sites are chosen ergodically. Recalling the example given in Sec. II, we see that the generating function of this combinatoric process is . We can in fact recover all possible loops and repetitions with this procedure. Notice that now the number of all possible (generic) trees with bonds is without the valence, since the choice of sites runs ergodically over the lattice. But the procedure is similar to counting the animals.
Combining these with Eq. (27) and summing over all possible connected (generic) trees with different numbers of bonds, we can get a bound on all possible connected graph in this situation. Consider first the term in the inclusion-exclusion form
where and . Here in the first bracket on the right hand side of (29) there is no redundency so that the generating function of a tree is simply (cf. Sec. II), and the last two terms of (29) contribute to the in (30) since they represent the same process of adding bonds in cases with and without existing bonds respectively. In (31) the factor is the bound of the number of possible plane trees with bonds. In (32) the term in the bracket comes from the sum of the over . With this, it is easy to give a bound to Eq. (14),
Here is the number of sites in the support of . It gives a critical inverse temperature , similar to the in Sec. (III.2). We can also use again the relation to simplify the bound (33). For instance, at a high enough temperature such that , we have
The calculations about the ”paths” are relatively easier. We need only contract the bonds using (26) without the complications of generic connective graphs. The result is
where is the number of sites in the support of .
iii.4 Systems with long-range interactions
For the systems with long-range interactions, is nontrivial for all of the bonds. But it decreases as power-law of distance
Iv Application: the second law of thermodynamics in ETH
We will give an example to show the usefulness of the Lieb-Robinson bounds at finite temperature. A very interesting idea that the second law of thermodynamics in ETH holds within a small error , i.e. , was proposed in (10). The proof, which exploits the original Lieb-Robinson bound, shows that if the bath is infinitely large the error will be infinitely small. With the results of Lieb-Robinson bound at finite temperature, we can give a much tighter bound to this error. Although the original settings of (10) is in quantum thermodynamics where one does not need a bound at finite temperature, it is applicable when comparing the quantum thermodynamic results to the case where the bath is the canonical ensamble. Moreover,we can clearly see how the temperature, the propagation time, the interactions of bath and the size of bath influence the error bound.
One of the most important step in the proof is that the local observation of a system at a short time can not tells the difference between the bath in pure quantum states and the canonical ensemble bath : Denote the observable by and the unitary evolution operator by , then
The bath B here is in fact a bath with finite size , i.e. .
Eq. (38) can be bounded as follows. Let us first do the following manipulations,
The represents a bath with infinitely large size. Following the process of the cluster expansion (12) of the time evolution operator, we have
Since at the initial time the system and the bath are decoupled, it is obvious that . The rest is also strictly equal to zero according to the Corollary 2 of (10).
In order to treat the second term in (40), let us view the bath as a part of the infinitely large bath . The Hamiltonian of the the infinitely large bath can be written as , where is the part of the infinitely large bath beside the B. When the boundary substracted, and become isolated, and furthermore the Hamiltonian can be taken as perturbed by . Hence, according to the Theorem 1 of (9), the problem can be translated into calculating the following quantity
where with . It is easy to give a bound to it with the help of Eq. (23) and Eq. (36). The expression is complicated and will not be shown here. The bound depends, as we have mentioned, on the temperature, the propagation time, the interactions of bath and the size of bath. Now the important point here is that when the temperature is much higher than critical temperature, evolution time can be much shorter than Lieb-Robinson critical time. This allows us to improve the bound on the deviation from the second law of thermodynamics given in (10).
We have the following results. In systems with respectively short-range interactions and exponentially decaying interactions, the bound is roughly
while in the system with power-law decaying interactions, the bound is roughly
where the is from the factor . Recall that the assumptions of (10) corresponds to the cases with short-range interactions or exponentially decaying interactions here. There the bound is given by (cf. the Observation 1 in the supplementary material of (10)), which is much larger than our result (42). Thus, wee can prove the second law of thermodynamics in ETH with a smaller error.
V Conclusion and outlook
In this paper we have proved the Lieb-Robinson-like bound at finite temperature for systems with respectively short-range, exponentially decaying and also long-range interactions. We exploited the nontrivial technique of graph generating functions to count the number of “paths” and “clusters” in the cluster expansion, which greatly simplifies the proofs.
The temperature dependence of the Lieb-Robinson bound at finite temperature should have a wide range of applications, especially in comparing cases with different temperatures. The application to the proof of the second law in ETH is the first example of application, showing the powerfulness of this result. There are many other situations of direct relevance:
The topological order has been proved to be stable under local perturbations with the help of the original Lieb-Robinson bound (13) or cluster expansions (14). But the topological order do not exists at finite temperature for many two-dimensional lattice models (15). From the Lieb-Robinson bound at finite temperature given in this paper, a moment of reflection shows that there is a possible case where at a finite temperature the Lieb-Robinson velocity is still be relatively large, which implies the possibility long-range entanglement at finite temperature. This is corroborated by the result that the autocorrelation time of topological order at finite temperature, which is characterized by gaugelike symmetries (12); (16), can be very large at a finite temperature below the spectral gap. Given its importance in topological quantum memories at finite temperature, it is desirable if a more rigorous study of the dynamical thermal stability could be given.
The original Lieb-Robinson bound has recently been applied in holographic models, e.g. to bound the butterfly velocity (17) and to bound the diffusivity (18). We think an important missing point of these applications is that, due to the well-known Tolman-Ehrenfest effect (19), in the gravitational dual bulk, say, a black hole spacetime with nonzero Hawking-Unruh temperature, even the equilibrium temperature cannot be the same constant, and not to mention those nonequilibrium effects listed above. So the temperature should have an impact of the localization properties in these cases. Our results have shown part of such impact.
There is, however, a caveat that the obtained bound cannot be applied in quantum thermodynamics where the temperature dependence is not needed. The original Lieb-Robinson bound can be directly applied because one starts with purely quantum setups and derive the emergent thermodynamic results. For recent applications, see e.g. (10); (20). Another drawback of the obtained bound is that the temperature effect and the time evolution are treated separately. In general, it is hoped that these two effects will affect each other, e.g. a Lieb-Robinson velocity depending on temperature. As is shown in (21), the Lieb-Robinson velocity in a dissipative quantum system is a function of the dissipation rate and will decrease due to the local dissipation. So the question of how to relate the bound obtained above, which hold in the framework of quantum statistical mechanics, to the Lieb-Robinson bounds in stochastic dynamics deserves further investigations.
Acknowledgements.We thank Zohar Nussinov for helpful comments. This work is supported in part by the National Science Foundation of China.
- E. H. Lieb, D. W. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28, 251 (1972).
- D. Beckman, D. Gottesman, M. A. Nielsen, J. Preskill, Causal and localizable quantum operations, Phys. Rev. A 64, 052309 (2001).
- B. Nachtergaele, R. Sims, Lieb-Robinson bounds and the exponential clustering theorem, Commun. Math. Phys. 265, 119 (2006); B. Nachtergaele, Y. Ogata, R. Sims, Propagation of correlation in quantum lattice systems, J. Stat. Phys. 124, 1 (2006).
- M. B. Hastings, T. Koma, Spectral gap and the exponential decay of correlations, Commun. Math. Phys. 265, 781 (2006).
- B. Nachtergaele, R. Sims, Lieb-Robinson bounds in quantum many-body physics, arXiv:1004.2086; M. B. Hastings, Locality in quantum systems, arXiv:1008.5137.
- M. Kliesch, C. Gogolin, J. Eisert, Lieb-Robinson bounds and the simulation of time-evolution of local observables in lattice systems, in V. Bach, L. Delle Site (eds.) Many-Electron Approaches in Physics, Chemistry and Mathematics, Springer (2014).
- D. Poulin, Lieb-Robinson bound and locality for general Markovian quantum dynamics, Phys. Rev. Lett. 104, 190401 (2010).
- M. B. Hastings, Decay of correlations in Fermi systems at nonzero temperature, Phys. Rev. Lett. 93, 126402 (2004).
- M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, J. Eisert, Locality of temperature, Phys. Rev. X 4, 031019 (2014).
- E. Iyoda, K. Kaneko, T. Sagawa, Fluctuation theorem for many-body pure quantum states, Phys. Rev. Lett. 119, 100601 (2017).
- F. Harary, E.M. Palmer , Graphical Enumeration, Academic Press (1973).
- Z. Nussinov, G. Ortiz, Autocorrelations and thermal fragility of anyonic loops in topologically quantum ordered systems, Phys. Rev. B 77, 064302 (2008).
- S. Bravyi, M. B. Hastings, S. Michalakis, Topological quantum order: Stability under local perturbations, J. Math. Phys. 51, 093512 (2010).
- I. Klich, On the stability of topological phases on a lattice, Ann. Phys. 325, 2120 (2010).
- M. B. Hastings, Topological order at nonzero temperature, Phys. Rev. Lett. 107, 210501 (2011); S. Roberts, B. Yoshida, A. Kubica, S. D. Bartlett, Symmetry-protected topological order at nonzero temperature, Phys. Rev. B 96, 022306 (2017).
- Z. Nussinov, G. Ortiz, Sufficient symmetry conditions for topological quantum order, PNAS 106, 16944 (2009); A symmetry principle for topological quantum order, Ann. Phys. 324, 977 (2009).
- D. A. Roberts, B. Swingle, Lieb-Robinson bound and the butterfly effect in quantum field theories, Phys. Rev. Lett. 117, 091602 (2016).
- T. Hartman, S. A. Hartnoll, R. Mahajan, Upper bound on diffusivity, Phys. Rev. Lett. 119, 141601 (2017).
- R. C. Tolman, On the weight of heat and thermal equilibrium in general relativity, Phys. Rev. 35, 904 (1930); R. C. Tolman, P. Ehrenfest, Temperature equilibrium in a static gravitational field, Phys. Rev. 36, 1791 (1930).
- N. Shiraishi, H. Tajima, Efficiency versus speed in quantum heat engines: Rigorous constraint from Lieb-Robinson bound, Phys. Rev. E 96, 022138 (2017).
- B. Descamps, Asymptotically decreasing Lieb-Robinson velocity for a class of dissipative quantum dynamics, J. Math. Phys. 54, 092202 (2013).