Locality of Temperature

Kliesch, Martin; Gogolin, Christian; Kastoryano, Michael J.; Riera, Arnau; Eisert, Jens

doi:10.1103/physrevx.4.031019

Cited by 165 publications

(248 citation statements)

References 48 publications

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“…Ground states of interacting gapped local Hamiltonians [13,14] as well as thermal states of arbitrary non-quadratic fermionic systems [15] at sufficiently high temperature have exponential clustering of correlations as defined in Definition 1. Thus the initial state could be prepared within a quench scenario where the Hamiltonian is changed from a gapped interacting model to a quadratic Hamiltonian which governs the non-equilibrium dynamics.…”

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Equilibration via Gaussification in Fermionic Lattice Systems

et al. 2016

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In this work, we present a result on the non-equilibrium dynamics causing equilibration and Gaussification of quadratic non-interacting fermionic Hamiltonians. Specifically, based on two basic assumptions -clustering of correlations in the initial state and the Hamiltonian exhibiting delocalizing transport -we prove that nonGaussian initial states become locally indistinguishable from fermionic Gaussian states after a short and well controlled time. This relaxation dynamics is governed by a power-law independent of the system size. Our argument is general enough to allow for pure and mixed initial states, including thermal and ground states of interacting Hamiltonians on and large classes of lattices as well as certain spin systems. The argument gives rise to rigorously proven instances of a convergence to a generalized Gibbs ensemble. Our results allow to develop an intuition of equilibration that is expected to be more generally valid and relates to current experiments of cold atoms in optical lattices.Despite the great complexity of quantum many-body systems out-of-equilibrium, local expectation values in such systems show the remarkable tendency to equilibrate to stationary values that do not depend on the microscopic details of the initial state, but rather can be described with few parameters using thermal states or generalized Gibbs ensembles [1][2][3]. Such behavior has been successfully studied in many settings theoretically and experimentally, most notably in instances of quantum simulations in optical lattices [2,4,5].By now, it is clear that, despite the unitary nature of quantum mechanical evolution, local expectation values equilibrate due to a dephasing between the eigenstates [3,[6][7][8][9][10][11]. So far it is, however, unclear why this dephasing tends to happen so rapidly. In fact, experiments often observe equilibration after very short times which are independent of the system size [5,12], while even the best theoretical bounds for general initial states of concrete systems diverge exponentially [2,11]. This discrepancy poses the challenge of precisely identifying the equilibration time, which constitutes one of the main open questions in the field [1][2][3].What is more, only little is known about how exactly the equilibrium expectation values emerge. Due to the exponentially many constants of motion present in quantum manybody systems, corresponding to the overlaps with the eigenvectors of the system, there seems to be no obvious reason why equilibrium values often only depend on few macroscopic properties such as temperature or particle number. In short: It is unclear how precisely the memory of the initial conditions is lost during time evolution.To make progress towards a solution of these two problems, it is instructive to study the behavior of non-interacting particles captured by so-called quadratic or free models. In these models the time evolution of so called Gaussian states, which are fully described by their correlation matrix, is particularly simple to describe. While studying...

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Equilibration via Gaussification in Fermionic Lattice Systems

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“…[40][41][42], that the ground state can be approximated by a PEPS of small bond dimension. Even the mild 1 We note Ref.…”

Section: Discussionmentioning

confidence: 99%

“…Reference [41], which builds on Refs. [40,42], shows that ρ T can be approximated with error ε in trace norm by a PEPO of bond dimension (n/ε) O( log(n)) .…”

Section: Peps Approximationmentioning

confidence: 99%

Entanglement area law from specific heat capacity

Brandão

Cramer

2015

Phys. Rev. B

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We study the scaling of entanglement in low-energy states of quantum many-body models on lattices of arbitrary dimensions. We allow for unbounded Hamiltonians such that systems with bosonic degrees of freedom are included. We show that, if at low enough temperatures the specific heat capacity of the model decays exponentially with inverse temperature, the entanglement in every low-energy state satisfies an area law (with a logarithmic correction). This behavior of the heat capacity is typically observed in gapped systems. Assuming merely that the low-temperature specific heat decays polynomially with temperature, we find a subvolume scaling of entanglement. Our results give experimentally verifiable conditions for area laws, show that they are a generic property of low-energy states of matter, and constitute proof of an area law for unbounded Hamiltonians beyond those that are integrable.

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confidence: 94%

“…But the expectation that the physical corner is well approximated by PEPS, dubbed the "PEPS conjecture", is still very much in place: This expectation is usually taken for granted and constitutes the basis of an entire research field. Indeed, for higher temperatures, a variant of this conjecture is actually provably true [25,26].Having said all this, a new obstacle emerges for higherdimensional systems; one that is often seen as a key obstacle, a make-or-break issue when it comes to numerically simulating strongly correlated models with PEPS: Even though PEPS are expected to provide good approximations, they are believed to be not efficiently contractible to compute expectation values of local observables. This is backed up by a proof in worst-case complexity, stating that the contraction of two-dimensional PEPS is #P-complete [27].…”

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Approximating local observables on projected entangled pair states

2017

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Tensor network states are for good reasons believed to capture ground states of gapped local Hamiltonians arising in the condensed matter context, states which are in turn expected to satisfy an entanglement area law. However, the computational hardness of contracting projected entangled pair states in two and higher dimensional systems is often seen as a significant obstacle when devising higher-dimensional variants of the density-matrix renormalisation group method. In this work, we show that for those projected entangled pair states that are expected to provide good approximations of such ground states of local Hamiltonians, one can compute local expectation values in quasi-polynomial time. We therefore provide a complexity-theoretic justification of why state-of-the-art numerical tools work so well in practice. We comment on how the transfer operators of such projected entangled pair states have a gap and discuss notions of local topological quantum order. We finally turn to the computation of local expectation values on quantum computers, providing a meaningful application for a small-scale quantum computer.Recent years have seen an explosion of interest in capturing quantum many-body systems in terms of tensor network states [1][2][3][4]. Such approaches provide powerful numerical tools for simulating strongly correlated quantum systems [2,[5][6][7][8], even for fermionic systems [9][10][11][12][13], overcoming the notorious sign problem that marres Monte Carlo approaches. The success of such approaches is essentially rooted in the entanglement structure that ground states of gapped local Hamiltonian models exhibit: They are expected to satisfy an entanglement area law [14], originating from the locality of interactions. The insight that ground states are very a-typical quantum states is often captured in the phrase that states having this entanglement pattern constitute what is called the "physical corner" of Hilbert space [15]. Indeed, the intuition that tensor network states should approximate this physical corner remarkably well is substantiated by a significant body of numerical work. In this discussion, projected entangled pair states (PEPS) [5][6][7][8]16], the higherdimensional analogues of matrix product states (MPS), take the leading role. Such PEPS not only provide numerical tools, but are also workhorses for understanding phases of matter or notions of topological order [17][18][19].This development can actually be seen as a natural continuation of the established density-matrix renormalisation group (DMRG) method that allows to simulate 1D quantum systems essentially to machine precision [4,20]. For such 1D systems, a deep understanding on the functioning of tensor networks has already been reached, even to full rigor. Area laws for entanglement entropies have been proven to hold for gapped models [21], implying MPS approximations [22]. A polynomial-time classical algorithm computing an MPS approximation of ground states of gapped Hamiltonians [23] can be seen as a DMRG method with a convergen...

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Locality of Temperature

Cited by 165 publications

References 48 publications

Equilibration via Gaussification in Fermionic Lattice Systems

Equilibration via Gaussification in Fermionic Lattice Systems

Entanglement area law from specific heat capacity

Approximating local observables on projected entangled pair states

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