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Exact Spectral Gap and Dynamical Critical Exponent for the Long-Range Ferromagnetic Heisenberg Chain

Srinivas Kothapalli

Published Jul 28, 2026• 0 citations
Publisher

Abstract

Standard direct methods for proving spectral gaps of frustration-free quantum spin systems require finite interaction range. Lemm and Nill~LemmNill2026 initiate the analysis of long-range frustration-free models, prove $z 1$ under summability hypotheses, and highlight quote ``the problem of determining the optimal values of $z$'' quote as a function of the interaction decay, as well as the broader open problem of developing direct gap methods for long-range frustration-free systems. We resolve a precise special case for a non-abelian, genuinely quantum model. For every $N 2$ and $>0$ define the long-range ferromagnetic Heisenberg Hamiltonian on $N$ qubits equation* H_N^() = _1 j3$ (pure-power $z=2$). All assumptions are listed explicitly. The identity $(H_N^())=_2(M_N^())$ is machine-checked against full many-body diagonalisation for all $N 8$ and a grid of $$ values (scripts/verify\_spectrum.py). Supplementary materials: LaTeX source, verification scripts, and unit tests for the long-range ferromagnetic Heisenberg spectral gap results.

DOI
10.5281/zenodo.21645856

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Standard direct methods for proving spectral gaps of frustration-free quantum spin systems require finite interaction range. Lemm and Nill~LemmNill2026 initiate the analysis of long-range frustration-free models, prove z1z 1z1 under summability hypotheses, and highlight quote ``the problem of determining the optimal values of zzz'' quote as a function of the interaction decay, as well as the broader open problem of developing direct gap methods for long-range frustration-free systems. We resolve a precise special case for a non-abelian, genuinely quantum model. For every N2N 2N2 and >0>0>0 define the long-range ferromagnetic Heisenberg Hamiltonian on NNN qubits equation* H_N^() = _1 j3(pure−power(pure-power(pure−powerz=2).Allassumptionsarelistedexplicitly.Theidentity). All assumptions are listed explicitly. The identity ).Allassumptionsarelistedexplicitly.Theidentity(H_N^())=_2(M_N^())ismachine−checkedagainstfullmany−bodydiagonalisationforallis machine-checked against full many-body diagonalisation for allismachine−checkedagainstfullmany−bodydiagonalisationforallN 8$ and a grid of $$ values (scripts/verify_spectrum.py). Supplementary materials: LaTeX source, verification scripts, and unit tests for the long-range ferromagnetic Heisenberg spectral gap results.
The thermodynamic properties of a classical spin-SSS Heisenberg ferromagnetic chain with long-range interactions in the presence of an external magnetic field are investigated by means of the spectral density method within the framework of the two-time Green's functions in classical statistical mechanics. For exchange interactions decaying as ${r}^{\ensuremath{-}\ensuremath{\alpha}},$ it is shown that long-range order exists at finite temperature for $1<\ensuremath{\alpha}<2$ and that no transition to a ferromagnetic phase at any finite temperature occurs for $\ensuremath{\alpha}>~2,$ consistently with a recent extension of the Mermin-Wagner theorem to spin models with long-range interactions. Besides, by means of a suitable modification of the lowest-order moment equations for the spectral density, the critical temperature and the main critical exponents for $1<\ensuremath{\alpha}<2$ and the low-temperature behavior of the paramagnetic susceptibility for $\ensuremath{\alpha}>~2$ are also obtained. Our results appear in a reasonable agreement with the available exact ones and Monte Carlo calculations.
Abstract Providing system-size independent lower bounds on the spectral gap of local Hamiltonian is in general a hard problem. For the case of finite-range, frustration free Hamiltonians on a spin lattice of arbitrary dimension, we show that a property of the ground state space is sufficient to obtain such a bound. We furthermore show that such a condition is necessary and equivalent to a constant spectral gap. Thanks to this equivalence, we can prove that for gapless models in any dimension, the spectral gap on regions of diameter n is at most o ( log ⁡ ( n ) 2 + ε n ) for any positive ϵ.
Estimating spectral gaps of quantum many-body Hamiltonians is a highly challenging computational task, even under assumptions of locality and translation-invariance. Yet, the quest for rigorous gap certificates is motivated by their broad applicability, ranging from many-body physics to quantum computing and classical sampling techniques. Here we present a general method for obtaining lower bounds on the spectral gap of frustration-free quantum Hamiltonians in the thermodynamic limit. We formulate the gap certification problem as a hierarchy of optimization problems (semidefinite programs) in which the certificate – a proof of a lower bound on the gap – is improved with increasing levels. Our approach encompasses existing finite-size methods, such as Knabe's bound and its subsequent improvements, as those appear as particular possible solutions in our optimization, which is thus guaranteed to either match or surpass them. We demonstrate the power of the method on one-dimensional spin-chain models where we observe an improvement by several orders of magnitude over existing finite size criteria in both the accuracy of the lower bound on the gap, as well as the range of parameters in which a gap is detected.
In quantum many-body systems, the existence of a spectral gap above the ground state has far-reaching consequences. In this paper, we discuss “finite-size” criteria for having a spectral gap in frustration-free spin systems and their applications. We extend a criterion that was originally developed for periodic systems by Knabe and Gosset-Mozgunov to systems with a boundary. Our finite-size criterion says that if the spectral gaps at linear system size n exceed an explicit threshold of order n−3/2, then the whole system is gapped. The criterion takes into account both “bulk gaps” and “edge gaps” of the finite system in a precise way. The n−3/2 scaling is robust: it holds in 1D and 2D systems on arbitrary lattices and with arbitrary finite-range interactions. One application of our results is to give a rigorous foundation to the folklore that 2D frustration-free models cannot host chiral edge modes (whose finite-size spectral gap would scale like n−1).
We improve Knabe’s spectral gap bound for frustration-free translation-invariant local Hamiltonians in 1D. The bound is based on a relationship between global and local gaps. The global gap is the spectral gap of a size-m chain with periodic boundary conditions, while the local gap is that of a subchain of size n < m with open boundary conditions. Knabe proved that if the local gap is larger than the threshold value 1/(n − 1) for some n > 2, then the global gap is lower bounded by a positive constant in the thermodynamic limit m → ∞. Here we improve the threshold to 6n(n+1), which is better (smaller) for all n > 3 and which is asymptotically optimal. As a corollary we establish a surprising fact about 1D translation-invariant frustration-free systems that are gapless in the thermodynamic limit: for any such system the spectral gap of a size-n chain with open boundary conditions is upper bounded as O(n−2). This contrasts with gapless frustrated systems where the gap can be Θ(n−1). It also limits the extent to which the area law is violated in these frustration-free systems, since it implies that the half-chain entanglement entropy is O(1/ϵ) as a function of spectral gap ϵ. We extend our results to frustration-free systems on a 2D square lattice.
Antiferromagnetic Hamiltonians with short range, non-frustrating interactions are well known to exhibit long range magnetic order in dimensions d ≥2 but exhibit only quasi-long-range order, with power-law decay of correlations, in d = 1 (for half-integer spin). On the other hand, non-frustrating long range interactions can induce long range order in d = 1. We study Hamiltonians in which the long range interactions have an adjustable amplitude λ, as well as an adjustable power law 1/| x | α , using a combination of quantum Monte Carlo and analytic methods: spin-wave, large N non-linear σ model, and renormalization group methods. We map out the phase diagram in the λ–α plane and study the nature of the critical line separating the phases with long range and quasi-long-range order. We find that this corresponds to a novel line of critical points with continuously varying critical exponents and a dynamical exponent, z <1.