Quantum ergodicity and localization in conservative systems: the Wigner band random matrix model

We explore the role of the initial state on the onset of thermalization in isolated quantum many-body systems after a quench. The initial state is an eigenstate of an initial HamiltonianĤI and it evolves according to a different final HamiltonianĤF . If the initial state has a chaotic structure with respect toĤF , i.e., if it fills the energy shell ergodically, thermalization is certain to occur. This happens whenĤI is a full random matrix, because its states projected ontoĤF are fully delocalized. The results for the observables then agree with those obtained with thermal states at infinite temperature. However, finite real systems with few-body interactions, as the ones considered here, are deprived of fully extended eigenstates, even when described by a nonintegrable Hamiltonian. We examine how the initial state delocalizes as it gets closer to the middle of the spectrum ofĤF , causing the observables to approach thermal averages, be the models integrable or chaotic. Our numerical studies are based on initial states with energies that cover the entire lower half of the spectrum of one-dimensional Heisenberg spin-1/2 systems.

“…When dealing with realistic systems of few-body interactions, a completely chaotic state is defined as a state that fills the energy shell [31][32][33][34].…”

Section: Energy Shellmentioning

confidence: 99%

Effects of the interplay between initial state and Hamiltonian on the thermalization of isolated quantum many-body systems

Torres-Herrera

Santos

2013

“…We have studied the width of the LDOS, which defines the time scale for the decay of the FA, and showed the appearance of three different regimes, depending on the strength of the perturbation, for the chaotic Hamiltonian. These regimes were also observed in a banded random matrix model defined by Wigner [5,11]. On the other hand, for integrable Hamiltonians the width of the LDOS increases linearly with perturbation.…”

mentioning

confidence: 59%

Sensitivity to perturbations and quantum phase transitions

Wisniacki

Roncaglia

2013

The local density of states or its Fourier transform, usually called fidelity amplitude, are important measures of quantum irreversibility due to imperfect evolution. In this Rapid Communication we study both quantities in a paradigmatic many body system, the Dicke Hamiltonian, where a single-mode bosonic field interacts with an ensemble of N two-level atoms. This model exhibits a quantum phase transition in the thermodynamic limit, while for finite instances the system undergoes a transition from quasi-integrability to quantum chaotic. We show that the width of the local density of states clearly points out the imprints of the transition from integrability to chaos but no trace remains of the quantum phase transition. The connection with the decay of the fidelity amplitude is also established. Sensitivity to perturbations is one of the major impediments to full control of quantum systems. With the advent of quantum information and its technological development, which enable the manipulation of many body systems such as cold atoms in optical lattices [1], a deep understanding of the sources that perturb and deteriorate quantum evolutions is required [2,3]. This would help us to develop strategies to protect and manipulate quantum systems, but also by analyzing the response to perturbations one would be able to extract information from the actual dynamics.In quantum evolutions, the effects of perturbations can be analyzed by measuring how difficult it is to reverse a given dynamics, as was proposed by Peres [4]. To this end several figures of merit have been defined. Among them, the so-called local density of states (LDOS) or strength function, defined by Wigner [5] to describe the statistical behavior of perturbed eigenfunctions, has been extensively studied due to its connections with fundamental problems such as irreversibility, thermalization, or dissipation in quantum systems [6,7]. Moreover, the LDOS provides significant information in quantum quenches, one of the simplest nonequilibrium quantum phenomena [8]. Consider a one parameter dependent Hamiltonian H (λ), with eigenenergies E j (λ) and eigenstates |j (λ) . The LDOS of an eigenstate |i(λ 0 ) , which we call unperturbed, is defined aswhere. It is the distribution of the overlaps squared between the unperturbed and perturbed eigenstates. The LDOS has been studied in several systems with different perturbations [5,[9][10][11][12][13], and it is equivalent to the probability of work for a quantum quench [8]. This quantity is also intimately related to other measures of irreversibility. In fact, the averaged LDOS is equal to the Fourier transform of the fidelity amplitude (FA),where U λ 0 (t) is the evolution operator corresponding to the Hamiltonian H (λ 0 ), and U λ 0 +δλ (t) corresponds the perturbed one that governs the backward evolution. Further, O(t) is connected with other well-known quantity, the Loschmidt where d is the dimension of the Hilbert space. Thus, the width of the LDOS gives the characteristic time scale for the decay of the FA and the...

“…In fact, as shown in Ref. [13], results of the method of averaging with respect to centroids of EFs should be a little narrower than those of the other method.…”

Section: Numerical Study For the Perturbative And Non-perturbatimentioning

confidence: 83%

“…Specifically, when averaging is done with respect to centers of energy shell, for strong interaction central parts of averaged EFs are close to a form predicted for the LDOS by the semicircle law. On the other hand, for the case of averaging with respect to centroids of EFs, central parts of averaged EFs are obviously narrower than the corresponding LDOS when the so-called Wigner parameter is large and an ergodicity parameter is small [13].…”

Section: Introductionmentioning

confidence: 94%

See 1 more Smart Citation

Perturbative and nonperturbative parts of eigenstates and local spectral density of states: The Wigner-band random-matrix model

Wang

2000

A generalization of Brillouin-Wigner perturbation theory is applied numerically to the Wigner Band Random Matrix model. The perturbation theory tells that a perturbed energy eigenstate can be divided into a perturbative part and a non-perturbative part with the perturbative part expressed as a perturbation expansion. Numerically it is found that such a division is important in understanding many properties of both eigenstates and the so-called local spectral density of states (LDOS). For the average shape of eigenstates, its central part is found to be composed of its non-perturbative part and a region of its perturbative part, which is close to the non-perturbative part. A relationship between the average shape of eigenstates and that of LDOS can be explained. Numerical results also show that the transition for the average shape of LDOS from the Breit-Wigner form to the semicircle form is related to a qualitative change in some properties of the perturbation expansion of the perturbative parts of eigenstates. The transition for the half-width of the LDOS from quadratic dependence to linear dependence on the perturbation strength is accompanied by a transition of a similar form for the average size of the non-perturbative parts of eigenstates. For both transitions, the same critical perturbation strength λ b has been found to play important roles. When perturbation strength is larger than λ b , the average shape of LDOS obeys an approximate scaling law.PACS number 05.45.+b