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arXiv:1601.02532 · Nuclear Physics B

Cluster expansion for ground states of local Hamiltonians

Alvise Bastianello, Spyros Sotiriadis

Published Jun 24, 2016• 11 citations
arXivHTMLPDF

Abstract

A central problem in many-body quantum physics is the determination of the ground state of a thermodynamically large physical system. We construct a cluster expansion for ground states of local Hamiltonians, which naturally incorporates physical requirements inherited by locality as conditions on its cluster amplitudes. Applying a diagrammatic technique we derive the relation of these amplitudes to thermodynamic quantities and local observables. Moreover we derive a set of functional equations that determine the cluster amplitudes for a general Hamiltonian, verify the consistency with perturbation theory and discuss non-perturbative approaches. Lastly we verify the persistence of locality features of the cluster expansion under unitary evolution with a local Hamiltonian and provide applications to out-of-equilibrium problems: a simplified proof of equilibration to the GGE and a cumulant expansion for the statistics of work, for an interacting-to-free quantum quench.

DOI
10.1016/j.nuclphysb.2016.06.021

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A central problem in many-body quantum physics is the determination of the ground state of a thermodynamically large physical system. We construct a cluster expansion for ground states of local Hamiltonians, which naturally incorporates physical requirements inherited by locality as conditions on its cluster amplitudes. Applying a diagrammatic technique we derive the relation of these amplitudes to thermodynamic quantities and local observables. Moreover we derive a set of functional equations that determine the cluster amplitudes for a general Hamiltonian, verify the consistency with perturbation theory and discuss non-perturbative approaches. Lastly we verify the persistence of locality features of the cluster expansion under unitary evolution with a local Hamiltonian and provide applications to out-of-equilibrium problems: a simplified proof of equilibration to the GGE and a cumulant expansion for the statistics of work, for an interacting-to-free quantum quench.
Traditional quantum physics solves ground states for a given Hamiltonian, while quantum information science asks for the existence and construction of certain Hamiltonians for given ground states. In practical situations, one would be mainly interested in local Hamiltonians with certain interaction patterns, such as nearest-neighbor interactions on some types of lattices. A necessary condition for a space VVV to be the ground-state space of some local Hamiltonian with a given interaction pattern is that the maximally mixed state supported on VVV is uniquely determined by its reduced density matrices associated with the given pattern, based on the principle of maximum entropy. However, it is unclear whether this condition is in general also sufficient. We examine the situations for the existence of such a local Hamiltonian to have VVV satisfying the necessary condition mentioned above as its ground-state space by linking to faces of the convex body of the local reduced states. We further discuss some methods for constructing the corresponding local Hamiltonians with given interaction patterns, mainly from physical points of view, including constructions related to perturbation methods, local frustration-free Hamiltonians, as well as thermodynamical ensembles.
Local currents are used to describe nonrelativistic many-body quantum mechanics in the thermodynamic limit. The problem of determining a representation of the local currents corresponding to a given Hamiltonian is studied. We formulate the dynamics in such a way that one solves simultaneously for the ground state and the representation of the local currents. This leads to two coupled functional equations relating the generating functional to a functional which describes the ground state. Together these functionals determine a representation of the local currents in which the Hamiltonian is a well-defined operator. The functional equations are equivalent to a set of integro–differential equations for expansion coefficients of the two functionals.
Abstract Ground states ofinteracting QFTs are non-Gaussian states, i.e. their connected n -point correlation functions do not vanish for , in contrast to the free QFT case. We show that, when the ground state of an interacting QFT evolves under a free massive QFT for a long time (a scenario that can be realised by a quantum quench), the connected correlation functions decay and all local physical observables equilibrate to values that are given by a Gaussian density matrix that retains memory only of the two-point initial correlation function. The argument hinges upon the fundamental physical principle of cluster decomposition, which is valid for the ground state of a general QFT. An analogous result was already known to be valid in the case of d = 1 spatial dimensions, where it is a special case of the so-called generalised Gibbs ensemble (GGE) hypothesis, and we now generalise it to higher dimensions. Moreover, in the case of massless free evolution, despite the fact that the evolution may lead not to equilibration but instead to unbounded increase of correlations with time, the GGE gives correctly the leading-order asymptotic behaviour of correlation functions in the thermodynamic and large time limit. The demonstration is performed in the context of a bosonic relativistic QFT, but the arguments apply more generally.
Using the perturbative expansion of the wave function from a single determinantal reference, it is possible to define a genealogical hierarchy of model spaces. The corresponding configuration-interaction matrices are dressed in the frame of intermediate Hamiltonian theory from estimates of the coefficients of the outer space determinants, which may take a coupled cluster (CC) or a perturbative form. The deviations from the additivity of transition energies are taken into account as a modification of coupled-cluster amplitudes, and higher-order exclusion-principle-violating corrections are introduced. The resulting equations relative to the first and second generations are applied to anisotropic Heisenberg Hamiltonians for one-dimensional and square two-dimensional spin lattices. The results show the efficiency of the method, which gives as accurate results with adjacent two-body operators as the traditional CC expansion with local four-body operators. With local four-body operators we obtain results comparable to those obtained with local six-body operators in normal CC treatments. \textcopyright{} 1996 The American Physical Society.
In quantum many-body physics, perturbative expansions organized in diagrammatic series provide a systematic way of computing corrections to observables in the ground state or at thermal equilibrium. Though non-Hermitian systems are generally far from equilibrium, this work establishes that it is still possible to describe their steady-state by a diagrammatic expansion. Applying this framework to exceptional points, it is found that these are generically translated in momentum space due to correlation effects.
We study many-body correlations in the ground state of a general quantum system of bosons or fermions by including an additional Jastrow function in our recently proposed variational coupled-cluster method. Our approach combines the advantages of state-dependent correlations in the coupled-cluster theory and of the strong, short-ranged correlations of the Jastrow function. We apply a generalized linked-cluster expansion for the Jastrow wave function and provide a detailed analysis for practical evaluation of the Hamiltonian expectation value as an energy functional of the Jastrow function and the bare density-distribution functions introduced and calculated in our earlier publications; a simple, first-order energy functional is derived and detailed formulas for the higher-order contributions are provided. Our energy functional does not suffer the divergence as most coupled-cluster calculations often do when applying to Hamiltonians with hardcore potentials. We also discuss possible applications of our technique, including applications to strongly correlated fermion systems.