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Spectral Gap and Exponential Decay of Correlations

Matthew B. Hastings, Tohru Koma

TL;DR

This work investigates how a uniform spectral gap above the ground state enforces decay of correlations in quantum spin and fermion lattice systems, including those with long-range interactions. By proving exponential clustering and coupling it with a self-similarity framework, the authors derive explicit bounds showing exponential (or power-law) decay for both fermionic and bosonic ground-state correlations under the gap, and provide conditions under which full correlators decay exponentially rather than only connected parts. The results extend earlier D<2 bounds under U(1) symmetry and leverage Hastings’ clustering approach and Lieb-Robinson-type bounds to handle power-law decays, with applications to models like the Heisenberg and Hubbard lattices. A central contribution is a mechanism to upgrade bound decay to full correlator decay via vanishing ground-state matrix elements in the infinite-volume limit, under self-similarity and finite degeneracy assumptions.

Abstract

We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short-range interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance, then the nonvanishing spectral gap implies exponential decay of the corresponding correlation. When two observables commute with each other at large distance, the connected correlation function decays exponentially under the gap assumption. If the observables behave as a vector under the U(1) rotation of a global symmetry of the system, we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself, rather than just the connected correlation function, decays exponentially under the gap assumption on a lattice with a certain self-similarity in (fractal) dimensions D<2. In particular, if the system is translationally invariant in one of the spatial directions, then this self-similarity condition is automatically satisfied. We also treat systems with long-range, power-law decaying interactions.

Spectral Gap and Exponential Decay of Correlations

TL;DR

This work investigates how a uniform spectral gap above the ground state enforces decay of correlations in quantum spin and fermion lattice systems, including those with long-range interactions. By proving exponential clustering and coupling it with a self-similarity framework, the authors derive explicit bounds showing exponential (or power-law) decay for both fermionic and bosonic ground-state correlations under the gap, and provide conditions under which full correlators decay exponentially rather than only connected parts. The results extend earlier D<2 bounds under U(1) symmetry and leverage Hastings’ clustering approach and Lieb-Robinson-type bounds to handle power-law decays, with applications to models like the Heisenberg and Hubbard lattices. A central contribution is a mechanism to upgrade bound decay to full correlator decay via vanishing ground-state matrix elements in the infinite-volume limit, under self-similarity and finite degeneracy assumptions.

Abstract

We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short-range interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance, then the nonvanishing spectral gap implies exponential decay of the corresponding correlation. When two observables commute with each other at large distance, the connected correlation function decays exponentially under the gap assumption. If the observables behave as a vector under the U(1) rotation of a global symmetry of the system, we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself, rather than just the connected correlation function, decays exponentially under the gap assumption on a lattice with a certain self-similarity in (fractal) dimensions D<2. In particular, if the system is translationally invariant in one of the spatial directions, then this self-similarity condition is automatically satisfied. We also treat systems with long-range, power-law decaying interactions.

Paper Structure

This paper contains 5 sections, 11 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Models and main results
  3. Clustering of correlations
  4. Vanishing of the matrix elements in the ground state
  5. Lieb-Robinson bound for group velocity

Key Result

Theorem 2.6

(Clustering of fermionic correlations): Let $A_X,B_Y$ be fermionic observables with a compact support. Assume that there exists a uniform spectral gap $\Delta E>0$ above the ground state sector in the spectrum of the Hamiltonian $H_\Lambda$ in the sense of Definition definition. Let $\omega$ be a gr where $P_0$ is the projection onto the sector of the infinite-volume ground state,$\omega(\cdots P_

Theorems & Definitions (13)

  • Definition 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Theorem 2.8
  • Definition 2.9
  • Theorem 2.10
  • Theorem 2.11
  • Theorem 2.12
  • Theorem 2.13
  • Lemma 3.1
  • ...and 3 more