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 2 1We 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 2 1

Sep 13, 2013333 citations

Bulk gap stability and spectral flow for fermions with weak long-range interactions

Stefan Teufel

We prove that weak long-range many-body interactions preserve the bulk gap of a free fermion system and that a quasi-local spectral flow transports the free ground state to ground states of the interacting system. Let $Φ_h$ be a free fermionic interaction, possibly including pairing terms, whose infinite-volume Majorana coefficient operator $h$ has no spectrum in $(-g,g)$ , and let $\mathcal V$ be an even self-adjoint many-body interaction. For sufficiently small $|λ|$ , we construct pure ground states $ω_λ$ of the interacting bulk system $Φ_λ=Φ_h+λ\mathcal V$ and quasi-local automorphisms $β_{0,λ}$ such that $ω_λ=ω_0\circβ_{0,λ}$ , where $ω_0$ is the free ground state of $Φ_h$ . Each $ω_λ$ has a GNS Hamiltonian with gap at least $g-C_*|λ| \|\mathcal V\|_*$ , where $\|\cdot\|_*$ is the relevant interaction norm. Our result allows polynomial decay of both the free coefficients and the many-body interaction. For two-site interactions on a uniformly discrete point set $Γ\subset\mathbb{R}^d$ , a sufficient condition is decay as $(1+|x-y|)^{-\varpi}$ with $\varpi>2d$ , for both the hopping and pairing coefficients and the interaction terms. The spectral-flow generator preserves the spatial exponent of the polynomial interaction norm, and its automorphisms satisfy locality bounds at that exponent. If $h$ and $\mathcal V$ have stretched-exponential decay, so does the spectral-flow generator. The proof constructs the spectral flow and establishes the gap through an excitation-number comparison, using polynomial propagation bounds for evolved interactions proved in a companion paper.We prove that weak long-range many-body interactions preserve the bulk gap of a free fermion system and that a quasi-local spectral flow transports the free ground state to ground states of the interacting system. Let $Φ_h$ be a free fermionic interaction, possibly including pairing terms, whose infinite-volume Majorana coefficient operator $h$ has no spectrum in $(-g,g)$, and let $\mathcal V$ be an even self-adjoint many-body interaction. For sufficiently small $|λ|$, we construct pure ground states $ω_λ$ of the interacting bulk system $Φ_λ=Φ_h+λ\mathcal V$ and quasi-local automorphisms $β_{0,λ}$ such that $ω_λ=ω_0\circβ_{0,λ}$, where $ω_0$ is the free ground state of $Φ_h$. Each $ω_λ$ has a GNS Hamiltonian with gap at least $g-C_*|λ| \|\mathcal V\|_*$, where $\|\cdot\|_*$ is the relevant interaction norm. Our result allows polynomial decay of both the free coefficients and the many-body interaction. For two-site interactions on a uniformly discrete point set $Γ\subset\mathbb{R}^d$, a sufficient condition is decay as $(1+|x-y|)^{-\varpi}$ with $\varpi>2d$, for both the hopping and pairing coefficients and the interaction terms. The spectral-flow generator preserves the spatial exponent of the polynomial interaction norm, and its automorphisms satisfy locality bounds at that exponent. If $h$ and $\mathcal V$ have stretched-exponential decay, so does the spectral-flow generator. The proof constructs the spectral flow and establishes the gap through an excitation-number comparison, using polynomial propagation bounds for evolved interactions proved in a companion paper.

math-phCross-listedarXiv:2610.40299Oct 30, 20260 citations

Probabilities of many-body decays of fermions

K. R. Nasriddinov, N.A. Sarikov

Uzbekiston Fizika ZhurnaliJan 1, 19920 citations

Interacting Composite Fermions

Chia‐Chen Chang

Jun 22, 20060 citations

Spectral Analysis Beyond the Spectral Gap: Rapid Mixing of Quantum Gibbs Samplers in One Dimension

Hongrui Chen, Zhiyan Ding

We prove rapid mixing of a quantum Gibbs sampler for general one-dimensional non-commuting local spin systems at every fixed positive temperature. In particular, for a system of $n$ sites, we show that the Kubo--Martin--Schwinger (KMS) detailed-balanced Gibbs sampler of~\cite{CKG2023} converges from any initial state to trace-norm error $ε$ in time $O(\log^4(n)+\log(1/ε))$ . This improves the previous $O(n+\log(1/ε))$ bound and establishes rapid mixing beyond high-temperature and weak-interaction regimes. Our proof explains why the slowest decay rate, captured by the spectral gap, can substantially underestimate the speed of equilibration. The key insight is that a state far from equilibrium must have a substantial component outside the span of low-weight Pauli operators, where stronger dissipation estimates apply. This yields two stages of relaxation: a double-exponential convergence from a bad initial state, followed by exponential convergence near equilibrium. We turn this mechanism into a framework for proving rapid mixing through spectral analysis beyond the gap. Our decay estimate is equivalent to exponential convergence in sandwiched Rényi-2 divergence. Together with the spectral gap, it also implies a modified logarithmic Sobolev inequality with rate $Ω(\log^{-4}(n))$ .We prove rapid mixing of a quantum Gibbs sampler for general one-dimensional non-commuting local spin systems at every fixed positive temperature. In particular, for a system of $n$ sites, we show that the Kubo--Martin--Schwinger (KMS) detailed-balanced Gibbs sampler of~\cite{CKG2023} converges from any initial state to trace-norm error $ε$ in time $O(\log^4(n)+\log(1/ε))$. This improves the previous $O(n+\log(1/ε))$ bound and establishes rapid mixing beyond high-temperature and weak-interaction regimes. Our proof explains why the slowest decay rate, captured by the spectral gap, can substantially underestimate the speed of equilibration. The key insight is that a state far from equilibrium must have a substantial component outside the span of low-weight Pauli operators, where stronger dissipation estimates apply. This yields two stages of relaxation: a double-exponential convergence from a bad initial state, followed by exponential convergence near equilibrium. We turn this mechanism into a framework for proving rapid mixing through spectral analysis beyond the gap. Our decay estimate is equivalent to exponential convergence in sandwiched Rényi-2 divergence. Together with the spectral gap, it also implies a modified logarithmic Sobolev inequality with rate $Ω(\log^{-4}(n))$.

quant-pharXiv:2610.05848Oct 6, 20260 citations

Φ_h

be a free fermionic interaction, possibly including pairing terms, whose infinite-volume Majorana coefficient operator

h

has no spectrum in

(-g,g)

, and let

\mathcal V

be an even self-adjoint many-body interaction. For sufficiently small

|λ|

, we construct pure ground states

ω_λ

of the interacting bulk system

Φ_λ=Φ_h+λ\mathcal V

and quasi-local automorphisms

β_{0,λ}

such that

ω_λ=ω_0\circβ_{0,λ}

, where

ω_0

is the free ground state of

Φ_h

. Each

ω_λ

has a GNS Hamiltonian with gap at least

g-C_*|λ| \|\mathcal V\|_*

, where

\|\cdot\|_*

is the relevant interaction norm. Our result allows polynomial decay of both the free coefficients and the many-body interaction. For two-site interactions on a uniformly discrete point set

Γ\subset\mathbb{R}^d

, a sufficient condition is decay as

(1+|x-y|)^{-\varpi}

with

\varpi>2d

, for both the hopping and pairing coefficients and the interaction terms. The spectral-flow generator preserves the spatial exponent of the polynomial interaction norm, and its automorphisms satisfy locality bounds at that exponent. If

h

and

\mathcal V

have stretched-exponential decay, so does the spectral-flow generator. The proof constructs the spectral flow and establishes the gap through an excitation-number comparison, using polynomial propagation bounds for evolved interactions proved in a companion paper.

We prove rapid mixing of a quantum Gibbs sampler for general one-dimensional non-commuting local spin systems at every fixed positive temperature. In particular, for a system of

n

sites, we show that the Kubo--Martin--Schwinger (KMS) detailed-balanced Gibbs sampler of~\cite{CKG2023} converges from any initial state to trace-norm error

ε

in time

O(\log^4(n)+\log(1/ε))

. This improves the previous

O(n+\log(1/ε))

bound and establishes rapid mixing beyond high-temperature and weak-interaction regimes. Our proof explains why the slowest decay rate, captured by the spectral gap, can substantially underestimate the speed of equilibration. The key insight is that a state far from equilibrium must have a substantial component outside the span of low-weight Pauli operators, where stronger dissipation estimates apply. This yields two stages of relaxation: a double-exponential convergence from a bad initial state, followed by exponential convergence near equilibrium. We turn this mechanism into a framework for proving rapid mixing through spectral analysis beyond the gap. Our decay estimate is equivalent to exponential convergence in sandwiched Rényi-2 divergence. Together with the spectral gap, it also implies a modified logarithmic Sobolev inequality with rate

Ω(\log^{-4}(n))