Thermalization of interacting fermions and delocalization in Fock space

Neuenhahn, Clemens; Marquardt, Florian

doi:10.1103/physreve.85.060101

Cited by 87 publications

(115 citation statements)

References 25 publications

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“…Despite the presence of interactions that have a power-law decay with distance, we find that the behavior of eigenstate expectation values of few-body observables, as well as thermalization properties of the systems described by Hamiltonian (1) of the main text, are qualitatively similar to those already seen in models with short-range (nearest and nextnearest-neighbor) interactions [15,41].…”

Section: Thermalizationsupporting

confidence: 62%

Fluctuation-Dissipation Theorem in an Isolated System of Quantum Dipolar Bosons after a Quench

et al. 2013

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We examine the validity of fluctuation-dissipation relations in isolated quantum systems taken out of equilibrium by a sudden quench. We focus on the dynamics of trapped hard-core bosons in one-dimensional lattices with dipolar interactions whose strength is changed during the quench. We find indications that fluctuationdissipation relations hold if the system is nonintegrable after the quench, as well as if it is integrable after the quench if the initial state is an equilibrium state of a nonintegrable Hamiltonian. On the other hand, we find indications that they fail if the system is integrable both before and after quenching. PACS numbers: 05.30.Jp, 03.75.Kk, The fluctuation-dissipation theorem (FDT) [1-3] is a fundamental relation in statistical mechanics which states that typical deviations from the equilibrium state caused by an external perturbation (within the linear response regime) dissipate in time in the same way as random fluctuations. The theorem applies to both classical and quantum systems as long as they are in thermal equilibrium. Fluctuation-dissipation relations are not, in general, satisfied for out-of-equilibrium systems. In particular, if a system is isolated, it is not clear whether once taken far from equilibrium fluctuation-dissipation relations apply at any later time. Studies of integrable models such as a Luttinger liquid [4] and the transverse field Ising chain [5] have shown that the use of fluctuation-dissipation relations to define temperature leads to values of the temperature that depend on the momentum mode and/or the frequency being considered. More recently, Essler et al. [6] have shown that for a subsystem of an isolated infinite system, the basic form of the FDT holds, and that the same ensemble that describes the static properties also describes the dynamics.The question of the applicability of the FDT to isolated quantum systems is particularly relevant to experiments with cold atomic gases [7,8], whose dynamics is considered to be, to a good approximation, unitary [9]. In that context, the description of observables after relaxation (whenever relaxation to a time-independent value occurs) has been intensively explored in the recent literature [10]. This is because, for isolated quantum systems out of equilibrium, it is not apparent that thermalization can take place. For example, if the system is prepared in an initial pure state |φ ini that is not an eigenstate of the HamiltonianĤ (Ĥ|ψ α = E α |ψ α ) (as in Ref. [9]), then the infinite-time average of the evolution of the observableÔ can be written as Ô (t) = ∑ α |c α | 2 O αα ≡ O diag , where c α = ψ α |φ ini , O αα = ψ α |Ô|ψ α , and we have assumed that the spectrum is nondegenerate. The outcome of the infinite-time average can be thought of as the prediction of a "diagonal" ensemble [11]. O diag depends on the initial state through the c α 's (there is an exponentially large number of them), while the thermal predictions depend only on the total energy φ ini |Ĥ|φ ini ; i.e., they need not agree.The lack of thermal...

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Section: Thermalizationsupporting

confidence: 62%

Fluctuation-Dissipation Theorem in an Isolated System of Quantum Dipolar Bosons after a Quench

et al. 2013

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“…This is often called the "diagonal ensemble" value of the long-time limit [77]. The eigenstate thermalization hypothesis (ETH), which is expected to hold for non-integrable systems, postulates that the diagonal matrix elements n|O|n are smooth functions of eigenenergy, for large enough systems [77][78][79][80][81]. The question of thermalization can then reduce to the question of how narrow in energy the overlap distribution is.…”

Section: O(t) = ψ(T)|o|ψ(t) = ψmentioning

confidence: 99%

Overlap distributions for quantum quenches in the anisotropic Heisenberg chain

et al. 2016

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The dynamics after a quantum quench is determined by the weights of the initial state in the eigenspectrum of the final Hamiltonian, i.e., by the distribution of overlaps in the energy spectrum. We present an analysis of such overlap distributions for quenches of the anisotropy parameter in the one-dimensional anisotropic spin-1/2 Heisenberg model (XXZ chain). We provide an overview of the form of the overlap distribution for quenches from various initial anisotropies to various final ones, using numerical exact diagonalization. We show that if the system is prepared in the antiferromagnetic Néel state (infinite anisotropy) and released into a non-interacting setup (zero anisotropy, XX point) only a small fraction of the final eigenstates gives contributions to the post-quench dynamics, and that these eigenstates have identical overlap magnitudes. We derive expressions for the overlaps, and present the selection rules that determine the final eigenstates having nonzero overlap. We use these results to derive concise expressions for time-dependent quantities (Loschmidt echo, longitudinal and transverse correlators) after the quench. We use perturbative analyses to understand the overlap distribution for quenches from infinite to small nonzero anisotropies, and for quenches from large to zero anisotropy.

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“…Fluctuations are indeed expected to be larger in the integrable regime than in the chaotic one. In the chaotic regime (and away from the edges of the spectrum) all eigenstates of the Hamiltonian that are close in energy have (i) a similar structure, as reflected by the inverse participation ratio and information entropy in different bases [10], and (ii) thermal expectation values of few-body observables [4,5,16,17]. However, this is not the case close to integrability where most quantities fluctuate wildly between eigenstates close in energy [4,5,10,17], and this is affecting our results here.…”

mentioning

confidence: 99%

Weak and strong typicality in quantum systems

2012

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We study the properties of mixed states obtained from eigenstates of many-body lattice Hamiltonians after tracing out part of the lattice. Two scenarios emerge for generic systems: (i) the diagonal entropy becomes equivalent to the thermodynamic entropy when a few sites are traced out (weak typicality); and (ii) the von Neumann (entanglement) entropy becomes equivalent to the thermodynamic entropy when a large fraction of the lattice is traced out (strong typicality). Remarkably, the results for few-body observables obtained with the reduced, diagonal, and canonical density matrices are very similar to each other, no matter which fraction of the lattice is traced out. Hence, for all physical quantities studied here, the results in the diagonal ensemble match the thermal predictions.PACS numbers: 05.70. Ln, 05.45.Mt, 02.30.Ik Despite advances, the emergence of thermodynamics from quantum mechanics is still a subject under debate. How to derive, from first principles, proper ensembles leading to the basic thermodynamic relations is not yet entirely clear. According to the concept of canonical typicality [1], the reduced density matrix of a subsystem of most pure states of manyparticle systems is canonical. The proof of this statement requires a partition of the original system into a small "system" and a large "environment". However, suppose, for example, that our universe is in a pure state and that we trace out only a finite number of degrees of freedom. Can we describe the rest of the universe using statistical mechanics? How much one needs to trace out, how well the notion of canonical typicality works in finite systems, and which quantities will be more or less affected by the fraction of the original system traced out are questions that have received little attention. They are the more pressing given the progress made in experiments with ultracold gases [2].We can discuss these issues in the context of entropy. If one takes an eigenstate of a generic many-body Hamiltonian and traces out the "environment" (E), the grand canonical (GC) ensemble is the appropriate ensemble to describe the system (S) that is left, where the total energy and number of particles fluctuate. One immediately realizes that the GC-entropy, S GC , must be different from the von Neumann entropy, S vN , if only a small fraction of the original system is traced out. In this case, S vN is extensive in the size of the environment, instead of in the size of the system as S GC . This follows from the fact that, for a composite system S + E in a pure stateρ = |Ψ Ψ|, the reduced von Neumann entropy is defined aswhere the Boltzmann constant is set to unity and the reduced density matrixρ S = Tr Eρ and similarlyρ E = Tr Sρ . S vN has been widely used to measure the entanglement in bipartite systems [3]. If the pure state is separable,ρ =ρ S ⊗ρ E , then S vN = 0, while maximum entanglement leads towhere D is the smallest dimension of the two subsystems. The source of the disparity between S vN and S GC is the information present in the off-diago...

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Thermalization of interacting fermions and delocalization in Fock space

Cited by 87 publications

References 25 publications

Fluctuation-Dissipation Theorem in an Isolated System of Quantum Dipolar Bosons after a Quench

Fluctuation-Dissipation Theorem in an Isolated System of Quantum Dipolar Bosons after a Quench

Overlap distributions for quantum quenches in the anisotropic Heisenberg chain

Weak and strong typicality in quantum systems

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