Effect of Rare Fluctuations on the Thermalization of Isolated Quantum Systems

Effect of Rare Fluctuations on the Thermalization of Isolated Quantum Systems

Giulio Biroli Institut de Physique Théorique, CEA/DSM/IPhT-CNRS/URA 2306 CEA-Saclay, F-91191 Gif-sur-Yvette, France    Corinna Kollath Centre de Physique Théorique, CNRS, École Polytechnique, 91128 Palaiseau Cedex, France    Andreas M. Läuchli Max Planck Institut für Physik komplexer Systeme, D-01187 Dresden, Germany
July 11, 2019

We consider the question of thermalization for isolated quantum systems after a sudden parameter change, a so-called quantum quench. In particular we investigate the pre-requisites for thermalization focusing on the statistical properties of the time-averaged density matrix and of the expectation values of observables in the final eigenstates. We find that eigenstates, which are rare compared to the typical ones sampled by the micro-canonical distribution, are responsible for the absence of thermalization of some infinite integrable models and play an important role for some non-integrable systems of finite size, such as the Bose-Hubbard model. We stress the importance of finite size effects for the thermalization of isolated quantum systems and discuss two alternative scenarios for thermalization, as well as ways to prune down the correct one.

0.5.30.-d, 05.70.Ln, 67.40.Fd

The microscopic description of many particle systems is very involved. In many situations, in particular at equilibrium, one can rely on statistical ensembles that provide a framework to compute time-averaged observables and obtain general results like fluctuation-dissipation relations. The use of statistical ensembles relies on the hypothesis that on long timescales physical systems thermalize. In classical statistical physics a very good understanding of thermalization was reached in the last century Castiglione et al. (2008): under certain chaoticity conditions, an isolated system thermalizes at long times within the micro-canonical ensemble. Furthermore, a large single portion of a (much larger) isolated system thermalizes within the grand-canonical ensemble. Instead for quantum systems, it is fair to state that the comprehension of thermalization and its pre-requisites are still open problems Reimann (2008); Goldstein et al. (2009), except for important results obtained in the semi-classical limit Srednicki (1994); Nonnenmacher (2008) or for the coupling to a thermal bath Weiss (2008) footnote1 (). And this is the case despite a lot of effort especially in the mathematical physics literature starting from the Quantum Ergodic Theorem of von Neumann von Neumann (1929) (see Goldstein et al. (2009) for a very recent account and new results).
The interest in these fundamental questions revived recently due to their direct relevance for experiments in ultracold atomic gases Bloch et al. (2008). The almost perfect decoupling of these gases from their environment enables the investigation of the quantum dynamics of isolated systems. In a fascinating experiment by Kinoshita et al. Kinoshita et al. (2006) it was observed that two counter-oscillating clouds of bosonic atoms confined in a one-dimensional harmonic trapping potential relax to a state different from the thermal one. Up to now the absence of thermalization footnote2 () has been mainly attributed to the presence of infinitely many conserved quantities, i.e. to the integrability of the system (see Calabrese and Cardy (2007) and references therein). For non-integrable isolated models the presence of thermalization after a global quench, i.e. a sudden global parameter change, is still debated Kollath et al. (2007); Manmana et al. (2007); Cramer et al. (2008); Moeckel and Kehrein (2008); Roux (2009); M. Eckstein (2009); Rigol (2009). The origin of thermalization (and its absence) after a global quench was proposed to stem from statistical properties of the time averaged density matrix and the so-called ’eigenstate-thermalization hypothesis’ (ETH) Deutsch (1991); Srednicki (1994); M. Rigol (2008); Rigol (2009). ETH, roughly speaking, says that all eigenstates with the same intensive energies are thermal, meaning that expectation values of all local observables within the eigenstate coincide with the ones in the corresponding Gibbs ensemble (see later for a precise definition).
The aim of our work is to understand to what extent ETH is a necessary and sufficient condition for thermalization. ETH can be interpreted in two different ways: a weak one which we show to be verified even for integrable models and which states that the fraction of the non-thermal states vanishes in the thermodynamic limit, and a strong one which states that non-thermal states completely disappear in the thermodynamic limit. The former interpretation does not imply thermalization. The reason is the possible existence of rare non-thermal states that can have a high overlap with the initial condition for the dynamics. We shall show that this is the origin of non-thermalization of some, and maybe all, integrable models and of some non-integrable systems of finite size, such as the Bose Hubbard Model. Our results reveal the crucial importance of finite size effects in the study of thermalization and allow us to point out two alternative routes for thermalization of quantum systems, as well as ways to prune down the correct one.
We consider the general situation where a system starts evolving at time from a density matrix .

The following time-evolution of any observable can be expressed as

Here are the eigenvectors of the Hamiltonian with corresponding eigenvalues (we use ). In order to be concrete we will focus on the experimentally relevant situation of a quantum quench, which corresponds to a sudden parameter change of the Hamiltonian at time for a system that is in the ground state for . In this case, , where is the overlap between the eigenstate of the Hamiltonian after the quench and the ground state of the Hamiltonian before the quench (). Our results can be generalized straightforwardly to a general . The typical time behavior of consists in damped or overdamped oscillations that converge towards a constant average value at long times. Assuming no degeneracy in eigenenergies, the long-time value of can be computed using the time averaged density matrix,  von Neumann (1929) footnote3 (). Following Ref. M. Rigol (2008) we call ”diagonal ensemble averages” all averages with respect to and we use with . An important property of the diagonal ensemble is that under very general conditions M. Rigol (2008) the energy per particle has vanishing fluctuations:


Here denotes the number of sites and the thermodynamic limit is taken at constant particle density . Property (1) means that the distribution of intensive eigenenergies with weights is peaked for large system sizes.
As already anticipated in the introduction the ’eigenstate thermalization hypothesis’ says for generic non-integrable interacting many body systems that the matrix elements of a few body observable with respect to any eigenstate with eigenenergy equals the microcanonical ensemble average taken at that energy . This was first conjectured based on studies of semiclassical systems Deutsch (1991); Srednicki (1994) and recently shown numerically to hold for a specific non-integrable system of finite size M. Rigol (2008). Were this hypothesis true, an immediate consequence of property (1) would be that averages in the diagonal ensembles coincide with averages in the microcanonical ensemble at the same energy per particle. This was the explanation of thermalization given for generic non-integrable systems and demonstrated for a specific example M. Rigol (2008). In contrast a finite width distribution for specific observables was found numerically for a finite size integrable system and claimed to be at the origin of the absence of thermalization for this model.
Note, however, that for a finite system there are always finite fluctuations of , whether the system is integrable or not. It follows that a precise characterization of ETH has to involve statements about the evolution of the distribution of upon approaching the thermodynamic limit. One can prove that generically the width of the distribution of vanishes in the thermodynamic limit:


where is an intensive local few body Hermitian operator (or observable); the sum is taken over eigenstates with eigenenergies where is the considered energy per particle and is a small number that can be taken to zero after the thermodynamic limit. Our detailed proof presented in EPAPS () is based on the vanishing of the fluctuations in the microcanonical ensemble footnote4 (). Note that Eq. (2) implies that the fraction of states characterized by a value of different from the microcanonical average vanishes in the thermodynamic limit. However, states with different values may and actually do exist, as we shall show in the following in concrete examples. They are just rare compared to the other ones. This is not a minor fact since if the s distribution gives an important weight to these rare states, the diagonal ensemble averages will be different from the micro-canonical one. They keep a memory of the initial state. As a consequence, an interpretation of ETH stating that the fraction of thermal states has to vanish would not guarantee thermalization. Instead the stronger interpretation of ETH, stating that the support of the distribution of the shrinks around the thermal microcanonical value in the thermodynamic limit, does so because states leading to non-thermal averages disappear. In the following we shall show, in concrete examples, that these rare states indeed do exist and prevent thermalization in some integrable infinite systems and in some finite size non-integrable models, such as the Bose Hubbard one.
Our first example is a chain of harmonic with a mass and coupling strength described by

We assume periodic boundary conditions and the usual commutation relations between the operators and given by . Using a suitable standard transformation one can rewrite the Hamiltonian as with the new creation and annihilation operators and and . As a consequence the eigenstates of the Hamiltonian at are characterized by occupation numbers for the and operators. Following Calabrese and Cardy Calabrese and Cardy (2007), we consider now a quantum quench where the system is in the ground state at a certain initial value of that we switch instantaneously to the final value , i.e. . We focus on the coupling between next-nearest neighbour R-oscillators which reads with footnote5 (). The diagonal matrix element for a state is . In the large system size limit the number of eigenstates with and respectively between and and and has the form of a large deviation function, i.e. it is proportional to , (cf. EPAPS ()). Physically is just related to the entropy of the system with intensive energy and an average coupling between next-nearest neighbour equal to . Thus the distribution of is strongly peaked around the maximum of and has a width of the order , but its tails extend to non thermal values of . Therefore, this is indeed a case where the width of the distribution of the matrix elements vanishes but the support does not due to the existence of rare states. Additionally, all the weights can be computed exactly EPAPS (). Their typical value is exponentially small in the size of the system. Thus they can bias significantly the micro-canonical ensemble distribution footnote6 () by counterbalancing the difference in cardinality between rare and typical states, which is also exponential in the system size. This is indeed what happens as it can be explicitly checked by computing the average value of in the diagonal ensemble: . We find that the distributions of in the micro-canonical and diagonal ensembles become infinitely peaked but around two different values, in agreement with Calabrese and Cardy (2007), thus explaining the absence of thermalization in this model (see EPAPS () for details).
The other example we discuss is the one-dimensional Bose-Hubbard model with one particle per site:

where and are the bosonic creation and annihilation operators, and the number operators on site . For most values of and , this model has been shown to be non-integrable Kolovsky and Buchleitner (2004). Only in special points, e.g. () and (), this model is integrable. The first case we consider is a quench from the superfluid state to . For this quench a non-thermal steady state has been found for long-times Kollath et al. (2007); Roux (2009). The correlations in this non-thermal state (system sizes up to , solid horizontal line) do agree well with their diagonal ensemble average (, dashed horizontal line), but not with the microcanonical distribution (shaded region). In this non-integrable situation it is more difficult to disentangle the role of rare states and finite size effects in the formation of a non-thermal state as we show in the following.

Figure 1: (Color online) Full diagonalization results of versus energy in the even parity, momentum sector for the final Hamiltonian characterized by (upper panels) and correlation of versus for a quench from (lower panel). Additionally the average energy (dashed-dotted line) after the quench and the average value of obtained from the t-DMRG time-evolution (solid line) and the diagonal ensemble for (dashed line) are shown. The shaded region corresponds to the microcanonical average footnote7 (). The t-DMRG calculations are performed as detailed in Ref. Kollath et al. (2007). Upper right panel: Distribution of values for the kinetic energy the values . The average value is removed from the distribution and the histograms are shifted vertically for visibility.

Figure 2: (Color online) Left panel: full diagonalization results of versus energy in the even parity, momentum sector for the final Hamiltonian characterized by . Right panel: distribution of values for the kinetic energy for values between . The linear trend of the observable in the considered window is removed from the distribution and the histograms are shifted vertically for visibility.

First, let us start to consider the validity of eq. (2). In Fig. 1 (upper right panel) we show the correlations versus energy . At low energies an (overlapping) bandstructure is seen. The center of the bands are separated by the interaction energy and have a width proportional to . Within these low energy bands s decay almost linearly. For intermediate energies a mixing of these energy bands starts to show up (cf. Fig. 1 upper right panel ) which is weak for small systems and becomes stronger for larger system sizes (cf. already ). In most fixed energy intervals the values of the correlations are spread considerably. In the upper-right panel the predicted narrowing of the half width of the distribution with increasing system size is clearly visible. In contrast the support does not seem to shrink which might point towards the existence of rare states. This is further supported in the lower panel of Fig. 1, where the weight of the initial state on the final eigenstates is strongly correlated with the values of the . The weights are much larger for larger values of , which correspond to the lower energy band edges Roux (2009) and are larger than the microcanonical average (shaded region in Fig. 1). A general decay of the weights towards lower values of the correlations is evident. This shows that the states which are important for the diagonal ensemble average do not have the microcanonical expectation value, i.e. rare states matter. Let us note that for the shown system sizes the average energy after the quench (marked by a vertical line) lies still within the lowest few energy bands. However, we estimated that for the largest sizes considered by t-DMRG (), for which there is still no thermalization at accessible time-scales, the eigenstates with considerable weight will be spread over tens of energy bands and the level statistics close to GOE Calabrese and Cardy (2007). We have also studied cases which should be easier from the point of view of thermalization since they are not close to an integrable point, in particular we discuss (Fig. 2). In this case the distribution of is much more peaked than for and its width decreases when increasing system sizes. Additionally, the support of the distribution seems to decrease pointing towards thermalization. Certainly larger system sizes are needed to make any firm statement.
As a conclusion, we find that the absence of thermalization for finite size systems can be attributed to two sources: (a) the distribution of the weights versus energy and the distribution of in a restricted energy interval may be very broad for finite size systems and (b) states characterized by a value of different from the micro-canonical value may have a considerable weight . All these phenomena clearly are at play for the finite size Bose Hubbard model investigated above. Eq. (2) and property (1) assure that the first origin of non-thermalization will be cured for large enough systems— the distributions will eventually become infinitely peaked—but not necessarily the second one. Indeed we showed that in some integrable models the origin of non-thermalization stems from the existence of non-thermal eigenstates which are less numerous compared to the thermal ones, but still exist and possibly bias a lot the diagonal expectation values. What happens for non-integrable systems and what is the correct requirement on the s in order to have thermalization in the thermodynamic limit is an open question. Our results reveal two possible options to obtain thermalization: (1) the support of the distribution of around the thermal value shrinks to zero in the thermodynamic limit, i.e. rare non-thermal states disappear altogether (as at seemingly), (2) rare states exist but the s do not bias too much the micro-canonical distribution toward them. Since the only apriori distinction between rare and typical states is that the latter are overwhelming more numerous, a plausible (but not necessary) assumption, leading to thermalization is that the s sample rather uniformly states with the same energy. Note that the existence of rare states for very large non-integrable models is not completely unreasonable as suggested by mathematical physics results obtained in the semi-classical limit Faure et al. (2003); Nonnenmacher (2008). Both scenarios are testable in numerical experiments. One has to study how the support of the distribution of evolves with the size of the system to understand whether (1) is realized, see EPAPS () for a first attempt. In order to study (2), one can use the (von Neumann) Kullback-Leibler (KL) entropy Cover and Thomas (2006) of the Gibbs distribution with respect to the diagonal ensemble footnote (). A ’rather uniform sampling’ would correspond to a zero intensive in the thermodynamic limit.
We conclude stressing that thermalization after a quantum quench appears to be a property that emerge for large enough system sizes. Understanding the physics behind this ”finite size thermalization length” and its dependence on the distance from integrability is a very interesting problem worth investigating in the future, specially because some cold atomic systems may well be below this thermalization threshold.
We would like to thank P. Calabrese, D. Huse, S. Kehrein, M. Olshanii, P. Reimann, M. Rigol, and G. Roux for fruitful discussions. This work was partly supported by the ’Triangle de la Physique’, DARPA-OLE and the ANR (’FAMOUS’).


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See pages 1 of epaps_arxiv.pdf

See pages 2 of epaps_arxiv.pdf

See pages 3 of epaps_arxiv.pdf

See pages 4 of epaps_arxiv.pdf

See pages 5 of epaps_arxiv.pdf

See pages 6 of epaps_arxiv.pdf

See pages 7 of epaps_arxiv.pdf

See pages 8 of epaps_arxiv.pdf

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