Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1412.2534 · Journal of Mathematical Physics

Some properties of correlations of quantum lattice systems in thermal equilibrium

Jürg Martin Fröhlich, Daniel Ueltschi

Published May 1, 2015• 41 citations
arXivHTMLPDF

Abstract

Simple proofs of uniqueness of the thermodynamic limit of KMS states and of the decay of equilibrium correlations are presented for a large class of quantum lattice systems at high temperatures. New quantum correlation inequalities for general Heisenberg models are described. Finally, a simplified derivation of a general result on power-law decay of correlations in 2D quantum lattice systems with continuous symmetries is given, extending results of McBryan and Spencer for the 2D classical XY model.

DOI
10.1063/1.4921305

Related papers

Decay of Correlations in 2D Quantum Systems with Continuous Symmetry

Costanza Benassi, Jürg Martin Fröhlich, Daniel Ueltschi

We study a large class of models of two-dimensional quantum lattice systems with continuous symmetries, and we prove a general McBryan–Spencer–Koma–Tasaki theorem concerning algebraic decay of correlations. We present applications of our main result to the Heisenberg, Hubbard, and t - J models, and to certain models of random loops.

arXiv:1612.02478•Annales Henri Poincaré•Apr 8, 2017•10 citations

Correlation Decay in Fermionic Lattice Systems with Power-Law Interactions at Nonzero Temperature

Senaida Hernández-Santana, Christian Gogolin, J. I. Cirac +1 more

We study correlations in fermionic lattice systems with long-range interactions in thermal equilibrium. We prove a bound on the correlation decay between anticommuting operators and generalize a long-range Lieb-Robinson-type bound. Our results show that in these systems of spatial dimension D with, not necessarily translation invariant, two-site interactions decaying algebraically with the distance with an exponent α≥2D, correlations between such operators decay at least algebraically to 0 with an exponent arbitrarily close to α at any nonzero temperature. Our bound is asymptotically tight, which we demonstrate by a high temperature expansion and by numerically analyzing density-density correlations in the one-dimensional quadratic (free, exactly solvable) Kitaev chain with long-range pairing.

arXiv:1702.00371•Physical Review Letters•Sep 13, 2017•25 citations

On classical and quantum lattice spin systems

Costanza Benassi

This thesis focuses on some results about quantum and classical lattice spin systems. We study a wide class of two-dimensional quantum models which enjoy a U(1) symmetry. Using the so called complex rotation method we show that the decay of the relevant correlation functions is at least algebraically fast. We provide some examples of relevant models which belong to our class. We review some results present in the literature concerning the so called GriffthsGinibre inequalities for the classical XY model and propose a generalisation to its quantum counterpart. Correlation inequalities indeed hold for the quantum XY model with spin- 1 2 and for the ground state of the spin-1 system. We propose some applications of these results, namely that the infinite volume limit of some correlation functions exists and that it is possible to compare quenched and annealed averages for a quantum XY model with random couplings. We investigate loop representations for O(n) classical spin systems. We propose a generalised random current representation and show its relationship with the Brydges-Fröhlich-Spencer one. We review some conjectures regarding the expected behaviour of these loop models { namely that macroscopic loops appear and their lengths are distributed according to a Poisson-Dirichlet distribution. We propose some arguments in favour of these conjectures for O(n) loop models, using a mix of exact results and heuristic considerations. In order to do so we de ne a stochastic process which is an effective split-merge process for macroscopic loops and we reformulate some correlation functions for the O(2) spin system in terms of loop properties.

•Sep 14, 2026•1 citation

Mean-field approximations and multipartite thermal correlations

Vlatko Vedral

The relationship between the mean-field approximations in various interacting models of statistical physics and measures of classical and quantum correlations is explored. We present a method that allows us to find an upper bound for the total amount of correlations (and hence entanglement) in a physical system in thermal equilibrium at some temperature in terms of its free energy and internal energy. This method is first illustrated by using two qubits interacting through the Heisenberg coupling, where entanglement and correlations can be computed exactly. It is then applied to the one-dimensional (1D) Ising model in a transverse magnetic field, for which entanglement and correlations cannot be obtained by exact methods. We analyse the behaviour of correlations in various regimes and identify critical regions, comparing them with already known results. Finally, we present a general discussion of the effects of entanglement on the macroscopic, thermodynamical features of solid-state systems. In particular, we exploit the fact that a d -dimensional quantum system in thermal equilibrium can be made to correspond to a ( d +1)-dimensional classical system in equilibrium to substitute all entanglement for classical correlations.

arXiv:quant-ph/0312104•New Journal of Physics•Feb 10, 2004•34 citations

Locality of KMS states for infinite fermionic lattice systems

Lennart Becker

We study how thermal equilibrium states of infinite fermionic lattices respond to perturbations of the Hamiltonian. Our main result concerns extensive perturbations of a chosen KMS state. We show that exponential decay of correlations and exponential local indistinguishability under the extensive perturbation yield stretched-exponential decay of correlations for the perturbed state. Neither the initial nor the perturbed state is required to be unique, allowing stability to be studied in the presence of phase coexistence. Our approach uses Quantum Belief Propagation (QBP), which we establish directly on the underlying $C^*$-algebra. Combining QBP with Lieb--Robinson bounds, we show implications between decay of correlations, the principle that local perturbations perturb locally, and local indistinguishability, closing the cycle for perturbations of the tracial state under the stated decay, support and uniformity assumptions. For extensive perturbations under uniform clustering conditions, we also obtain a differentiable path of local equilibrium expectations and bound their change linearly in the interaction norm of the perturbation.

math-phCross-listedarXiv:2609.31858•Sep 29, 2026•0 citations

Spectral gap and decay of correlations in U(1)-symmetric lattice systems in dimensions D<2

Tohru Koma

We consider many-body systems with a global U(1) symmetry on a class of lattices with the (fractal) dimension D<2 and their zero temperature correlations whose observables behave as a vector under the U(1) rotation. For a wide class of the models, we prove that if there exists a spectral gap above the ground state, then the correlation functions have a stretched exponentially decaying upper bound. This is an extension of the McBryan-Spencer method at finite temperatures to zero temperature. The class includes quantum spin and electron models on the lattices, and our method also allows finite or infinite (quasi)degeneracy of the ground state. The resulting bounds rule out the possibility of the corresponding magnetic and electric long-range orders.

arXiv:math-ph/0505022•Journal of Mathematical Physics•Feb 1, 2007•10 citations
Simple proofs of uniqueness of the thermodynamic limit of KMS states and of the decay of equilibrium correlations are presented for a large class of quantum lattice systems at high temperatures. New quantum correlation inequalities for general Heisenberg models are described. Finally, a simplified derivation of a general result on power-law decay of correlations in 2D quantum lattice systems with continuous symmetries is given, extending results of McBryan and Spencer for the 2D classical XY model.
We study correlations in fermionic lattice systems with long-range interactions in thermal equilibrium. We prove a bound on the correlation decay between anticommuting operators and generalize a long-range Lieb-Robinson-type bound. Our results show that in these systems of spatial dimension D with, not necessarily translation invariant, two-site interactions decaying algebraically with the distance with an exponent α≥2D, correlations between such operators decay at least algebraically to 0 with an exponent arbitrarily close to α at any nonzero temperature. Our bound is asymptotically tight, which we demonstrate by a high temperature expansion and by numerically analyzing density-density correlations in the one-dimensional quadratic (free, exactly solvable) Kitaev chain with long-range pairing.
This thesis focuses on some results about quantum and classical lattice spin systems. We study a wide class of two-dimensional quantum models which enjoy a U(1) symmetry. Using the so called complex rotation method we show that the decay of the relevant correlation functions is at least algebraically fast. We provide some examples of relevant models which belong to our class. We review some results present in the literature concerning the so called GriffthsGinibre inequalities for the classical XY model and propose a generalisation to its quantum counterpart. Correlation inequalities indeed hold for the quantum XY model with spin- 1 2 and for the ground state of the spin-1 system. We propose some applications of these results, namely that the infinite volume limit of some correlation functions exists and that it is possible to compare quenched and annealed averages for a quantum XY model with random couplings. We investigate loop representations for O(n) classical spin systems. We propose a generalised random current representation and show its relationship with the Brydges-Fröhlich-Spencer one. We review some conjectures regarding the expected behaviour of these loop models { namely that macroscopic loops appear and their lengths are distributed according to a Poisson-Dirichlet distribution. We propose some arguments in favour of these conjectures for O(n) loop models, using a mix of exact results and heuristic considerations. In order to do so we de ne a stochastic process which is an effective split-merge process for macroscopic loops and we reformulate some correlation functions for the O(2) spin system in terms of loop properties.
The relationship between the mean-field approximations in various interacting models of statistical physics and measures of classical and quantum correlations is explored. We present a method that allows us to find an upper bound for the total amount of correlations (and hence entanglement) in a physical system in thermal equilibrium at some temperature in terms of its free energy and internal energy. This method is first illustrated by using two qubits interacting through the Heisenberg coupling, where entanglement and correlations can be computed exactly. It is then applied to the one-dimensional (1D) Ising model in a transverse magnetic field, for which entanglement and correlations cannot be obtained by exact methods. We analyse the behaviour of correlations in various regimes and identify critical regions, comparing them with already known results. Finally, we present a general discussion of the effects of entanglement on the macroscopic, thermodynamical features of solid-state systems. In particular, we exploit the fact that a d -dimensional quantum system in thermal equilibrium can be made to correspond to a ( d +1)-dimensional classical system in equilibrium to substitute all entanglement for classical correlations.
We study how thermal equilibrium states of infinite fermionic lattices respond to perturbations of the Hamiltonian. Our main result concerns extensive perturbations of a chosen KMS state. We show that exponential decay of correlations and exponential local indistinguishability under the extensive perturbation yield stretched-exponential decay of correlations for the perturbed state. Neither the initial nor the perturbed state is required to be unique, allowing stability to be studied in the presence of phase coexistence. Our approach uses Quantum Belief Propagation (QBP), which we establish directly on the underlying C∗C^*C∗-algebra. Combining QBP with Lieb--Robinson bounds, we show implications between decay of correlations, the principle that local perturbations perturb locally, and local indistinguishability, closing the cycle for perturbations of the tracial state under the stated decay, support and uniformity assumptions. For extensive perturbations under uniform clustering conditions, we also obtain a differentiable path of local equilibrium expectations and bound their change linearly in the interaction norm of the perturbation.
We consider many-body systems with a global U(1) symmetry on a class of lattices with the (fractal) dimension D<2 and their zero temperature correlations whose observables behave as a vector under the U(1) rotation. For a wide class of the models, we prove that if there exists a spectral gap above the ground state, then the correlation functions have a stretched exponentially decaying upper bound. This is an extension of the McBryan-Spencer method at finite temperatures to zero temperature. The class includes quantum spin and electron models on the lattices, and our method also allows finite or infinite (quasi)degeneracy of the ground state. The resulting bounds rule out the possibility of the corresponding magnetic and electric long-range orders.