Typical Relaxation of Isolated Many-Body Systems Which Do Not Thermalize

Abstract

We consider isolated many-body quantum systems which do not thermalize, i.e., expectation values approach an (approximately) steady longtime limit which disagrees with the microcanonical prediction of equilibrium statistical mechanics. A general analytical theory is worked out for the typical temporal relaxation behavior in such cases. The main prerequisites are initial conditions which appreciably populate many energy levels and do not give rise to significant spatial inhomogeneities on macroscopic scales. The theory explains very well the experimental and numerical findings in a trapped-ion quantum simulator exhibiting many-body localization, in ultracold atomic gases, and in integrable hard-core boson and XXZ models.The long-standing task to explain macroscopic equilibration phenomena in terms of the underlying microscopic quantum dynamics is presently regaining considerable attention [1][2][3]. Since open systems are beyond the realm of standard quantum mechanics, the common starting point is an isolated many-body system, possibly incorporating the environment of the subsystem of actual interest. The question whether and how such a system or subsystem approaches some thermal or nonthermal equilibrium state after a sufficiently long time has been at the focus of numerous analytical [4][5][6], numerical [7][8][9][10][11][12], and experimental [13][14][15][16][17][18] studies. Despite the reversible and everlasting motion of the microscopic degrees of freedom, it could be shown in Refs. [19,20] under increasingly weak assumptions about the system Hamiltonian, the initial condition, and the considered observable that expectation values must remain extremely close to a constant value for the vast majority of all sufficiently late times (the exceptional times include initial transients and quantum revivals).The natural next question is whether the system thermalizes, that is, whether the longtime behavior is well approximated by the pertinent microcanonical expectation value from equilibrium statistical mechanics. A first prominent criterion for thermalization is the so-called eigenstate thermalization hypothesis (ETH), postulating that every energy eigenstate yields expectation values close to the corresponding microcanonical values [5][6][7][8].In other words, a violation of ETH is commonly considered an indicator of nonthermalization [1,[6][7][8][9]. A related but different such indicator is the existence of additional conserved quantities (besides the system Hamiltonian) which can be written as sums of local operators, and which play a particularly prominent role for so-called integrable systems [1]. Numerically, it has been found that such systems usually violate the ETH and do not thermalize [7][8][9]. Instead, the longtime behavior is well captured by a so-called generalized Gibbs ensemble (GGE), which is obtained by the standard working recipe to maximize the von Neumann entropy under the constraints that the expectation values of the conserved quantities must be correctly reproduced [7]. Yet another commo...

Transportless equilibration in isolated many-body quantum systems

“…Technically speaking, the crucial idea is to skillfully 'rearrange' the systems's very dense energy eigenvalues and to 'redistribute' the possibly quite heterogeneous populations of the corresponding eigenstates, yielding an effective description in terms of an auxiliary Hamiltonian with approximately equally populated eigenstates. The main result is a unification and substantial amendment of the earlier findings in [14][15][16][17], formally summarized by the compact final equation (74). The decisive quantity, which governs the temporal relaxation via the last term in equation (74), will furthermore be identified in section 7 with the Fourier transform of the system's initial energy distribution, and in case the system is in a pure state, also with the so-called survival probability of the initial state.…”

“…At this point, the assumption (a) from (35) is needed. Namely, due to this assumption and the formal equivalence of (48) with (10), the heuristic considerations from section 3 or the more rigorous treatment in [14,15] can be adopted to arrive at the counterpart of (20), namely Setting t=0 in (51), the above property (i) implies that…”

“…While it makes the derivation short and physically instructive, a more rigorous justification of (15) seems very difficult. On the other hand, the very same formula (20) can also be rigorously obtained by means of a technically very different, more arduous and less instructive approach, see [14,15], using averages over unitary transformations, under which the locality properties of a given Hamiltonian are in general not preserved (see also sections 4 and 5).…”

“…In this section, we collect the main a priori reasons announced above (15), which may invalidate the approximation (15) and hence our main result (20).…”

“…In section 3, the previous rigorous approach to transportless equilibration from [14,15] is revisited in terms of an alternative, non-rigorous but physically much simpler line of reasoning, while in sections 4 and 5 its main preconditions are worked out in considerable more detail than before. A representative comparison of this theory with experimental observations is provided by section 6.…”