Necessity of eigenstate thermalisation for equilibration towards unique expectation values when starting from generic initial states

Abstract

We investigate dynamical equilibration of expectation values in closed quantum systems for realistic non-equilibrium initial states. Thereby we find that the corresponding long time expectation values depend on the initial expectation values if eigenstate thermalization is violated. An analytical expression for the deviation from the expected ensemble value is derived for small displacements from equilibrium. Additional numerics for magnetization and energy equilibration in an asymmetric anisotropic spin-1/2-ladder demonstrate that the analytical predictions persist beyond the limits of the theory. The results suggest eigenstate thermalization as physically necessary condition for initial state independent equilibration. One approach to this topic is the eigenstate thermalization hypothesis (ETH) [13][14][15]. The ETH implies that expectation values of an observable of interest A with regard to energy eigenstates |n vary slowly as a function of the corresponding eigenenergies E n , which means that all diagonal elements of A in the energy eigenbasis, i.e., A nn = n|A|n , are approximately constant within some energy regime which the system's state is restricted to. For open system situations, i.e., when the global system is divided into system plus bath, there is an alternative formulation of the ETH. If the reduced density matrices of eigenstates of the global system Tr B {|n n|} vary slowly as a function of the global eigenenergies E n in terms of a suitable operator norm, then the ETH is valid. This is equivalent to the statement that the previous definition is fulfilled for all observables acting solely on the systemIf the ETH is fulfilled, then there is initial state independent (ISI) equilibration of expectation values with regard to A [15]. If the ETH is not fulfilled, there are still initial states which yield ISI equilibration. They even form the majority of possible initial states in a statistical sense according to the Haar measure [6,16]. However, in those cases it is usually required that the Hamiltonian, the regarded observable and the initial state are uncorrelated meaning that the mutual orientations of the eigenbases of those operators can be viewed as more or less random and therefore statistically independent [6,17].While this assumption is mathematically reasonable, in the sense of a high relative frequency with respect to the unitary Haar measure, its physical justification is at the heart of the present investigation. Moreover, some investigations rely on rather strong restrictions on the initial state like identical amplitudes on all energy eigenstates in the regarded energy shell [17,18].In this Letter we address the question whether the validity of the ETH is needed for ISI equilibration of expectation values in a realistic setting, i.e., under realistic modeling of the initial state. To this end we first introduce a parametrized class of initial states (1) and derive an analytical expression which quantifies the initial state dependence for small deviations from equilibrium. Furthermore...

Dynamical Typicality Approach to Eigenstate Thermalization

“…In Ref. [11], somewhat similar results have been obtained for some particular initial (mixed) states which arise by certain, very small perturbation of a canonical density operator [44].…”

“…In conclusion, the weak ETH has been established as a necessary and sufficient prerequisite for thermalization in isolated many-body quantum systems in the following sense: The vast majority of all pure states, which exhibit the same initial expectation value for some observable A, closely approach the pertinent microcanonical expectation value of A for practically all sufficiently large times. It is remarkable that also in several other related studies it is the weak rather than the strong ETH which naturally arises [11,24,26,45]. Note that the necessity of the (weak or strong) ETH for thermalization is not something that one might have expected a priori due to some intuitively quite obvious reasons [2,9].…”

“…It is generally taken for granted that the ETH guarantees thermalization for any initial state with a macroscopically well defined system energy. Whether the ETH is also necessary for thermalization is a question of considerable current interest [2,[7][8][9][10][11][12][13]. Here, we will provide examples implying that ETH (in its most common version) should be considered neither as sufficient nor as necessary for thermalization without any further specification of the admitted initial states.…”

“…Since y(0) = 0 (see below (11)), we can expand y(a) as y ′ (0)a + y ′′ (0)a 2 /2 + ... and the denominator in (10) as a geometric series. Substituting all this into (11) and comparing terms with equal powers of a yields equations for y ′ (0), y ′′ (0),... which can be iteratively solved. As a result, Eq.…”

Sudden removal of a static force in a disordered system: Induced dynamics, thermalization, and transport

“…(Note that, due to PBC, this particular choice is arbitrary.) Then, at equilibrium, this situation is described by the density matrix [73][74][75][76]…”