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arXiv:2406.06415 · Physical Review X

Rigorous Lower Bound on Dynamical Exponents in Gapless Frustration-Free Systems

Rintaro Masaoka, Tomohiro Soejima, Haruki Watanabe

Published Dec 2, 2025• 4 citations
arXivHTMLPDF

Abstract

This work rigorously establishes a universal lower bound z ≥ 2 for the dynamical exponent in frustration-free quantum many-body systems whose ground states exhibit power-law decaying correlations. The derivation relies on the Gosset-Huang inequality, providing a unified framework applicable across various lattice structures and spatial dimensions, independent of specific boundary conditions. Remarkably, our result can be applied to prove bounds for dynamics of classical stochastic processes. Specifically, we utilize a well-established mapping from the time evolution of local Markov processes with detailed balance to that of frustration-free quantum Hamiltonians, known as Rokhsar-Kivelson Hamiltonians. This proves z ≥ 2 for such Markov processes, which is an improvement over existing bounds. Beyond these applications, the quantum analysis of the z ≥ 2 bound is further broadened to include systems exhibiting hidden correlations, which may not be evident from purely local operators.

DOI
10.1103/d4c4-5p2r

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•Sep 25, 2015•2 citations
This work rigorously establishes a universal lower bound z ≥ 2 for the dynamical exponent in frustration-free quantum many-body systems whose ground states exhibit power-law decaying correlations. The derivation relies on the Gosset-Huang inequality, providing a unified framework applicable across various lattice structures and spatial dimensions, independent of specific boundary conditions. Remarkably, our result can be applied to prove bounds for dynamics of classical stochastic processes. Specifically, we utilize a well-established mapping from the time evolution of local Markov processes with detailed balance to that of frustration-free quantum Hamiltonians, known as Rokhsar-Kivelson Hamiltonians. This proves z ≥ 2 for such Markov processes, which is an improvement over existing bounds. Beyond these applications, the quantum analysis of the z ≥ 2 bound is further broadened to include systems exhibiting hidden correlations, which may not be evident from purely local operators.
Abstract We study the kinetic Ising model under Glauber dynamics and establish an upper bound on the spectral gap for finite systems. This bound implies the critical exponent inequality z≥2z \ge 2z≥2 z ≥ 2 , thereby rigorously improving the previously known estimate z≥2−ηz \ge 2 - \eta z≥2−η z ≥ 2 - η . Our proof relies on the mapping from stochastic processes to frustration-free quantum systems and leverages the Simon–Lieb and Gosset–Huang inequalities.
We discuss a general scheme for the study of frustration in quantum systems. We introduce a universal measure of frustration for arbitrary quantum systems with Hamiltonians made of sums of local two-body contributions in terms of the overlap between the unfrustrated ground state of the pair Hamiltonians and the frustrated ground state of the total Hamiltonian. We derive an exact lower bound this overlap measure of frutration in degenerate ground states of quantum many-body systems. The bound results in the sum of two contributions: entanglement and classical correlations arising from local measurements. We show that average frustration properties are completely determined by the behavior of the maximally mixed ground state. We identify sufficient conditions for a quantum spin system to saturate the bound, and for models with twofold degeneracy we prove that average and local frustration coincide. If all the (pure) ground states of a given Hamiltonian saturate the inequality, then the system is said to be inequality saturating. These conditions provide a generalization to the quantum domain of the Toulouse criteria for classical frustration-free systems. The models satisfying these conditions can be reasonably identified as geometrically unfrustrated and subject to frustration of purely quantum origin. Our results therefore establish a unified framework for studying the intertwining of geometric and quantum contributions to frustration. The mathematical scheme can be put to test by relating the overlap measure and the inequalities it obeys to global and local observables in terms of the static structure factor and transverse n-point correlation functions.
Frustration-free Hamiltonians provide pivotal models for understanding quantum many-body systems. In this paper, we establish a general framework for frustration-free fermionic systems. First, we derive a necessary and sufficient condition for a free fermion model to be frustration-free. In the case of translation-invariant, noninteracting systems, we show that any band touching between the valence and conduction bands is at least quadratic. Furthermore, by extending the Gosset-Huang inequality to fermionic systems, we demonstrate that even in interacting and non-translation-invariant cases, the finite-size gap of gapless excitations scales as O((log⁡L)2/L2)O((\log L)^2/L^2)O((logL)2/L2), provided the ground-state correlation function exhibits a power-law decay. Our results provide a foundation for studying frustration-free fermionic systems, including flat-band ferromagnetism and ηηη-pairing states.
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.
A broad range of quantum optimization problems can be phrased as the question of whether a specific system has a ground state at zero energy, i.e., whether its Hamiltonian is frustration-free. Frustration-free Hamiltonians, in turn, play a central role for constructing and understanding new phases of matter in quantum many-body physics. Unfortunately, determining whether this is the case is known to be a complexity-theoretically intractable problem. This makes it highly desirable to search for efficient heuristics and algorithms to, at least, partially answer this question. Here we prove a general criterion-a sufficient condition-under which a local Hamiltonian is guaranteed to be frustration-free by lifting Shearer's theorem from classical probability theory to the quantum world. Remarkably, evaluating this condition proceeds via a fully classical analysis of a hardcore lattice gas at negative fugacity on the Hamiltonian's interaction graph, which, as a statistical mechanics problem, is of interest in its own right. We concretely apply this criterion to local Hamiltonians on various regular lattices, while bringing to bear the tools of spin glass physics that permit us to obtain new bounds on the satisfiable to unsatisfiable transition in random quantum satisfiability. We are then led to natural conjectures for when such bounds will be tight, as well as to a novel notion of universality for these computer science problems. Besides providing concrete algorithms leading to detailed and quantitative insights, this work underscores the power of marrying classical statistical mechanics with quantum computation and complexity theory.
A broad range of quantum optimisation problems can be phrased as the question whether a specific system has a ground state at zero energy, i.e. whether its Hamiltonian is frustration free. Frustration-free Hamiltonians, in turn, play a central role for constructing and understanding new phases of matter in quantum many-body physics. Unfortunately, determining whether this is the case is known to be a complexity-theoretically intractable problem. This makes it highly desirable to search for efficient heuristics and algorithms in order to, at least, partially answer this question. Here we prove a general criterion -- a sufficient condition -- under which a local Hamiltonian is guaranteed to be frustration free by lifting Shearer's theorem from classical probability theory to the quantum world. Remarkably, evaluating this condition proceeds via a fully classical analysis of a hard-core lattice gas at negative fugacity on the Hamiltonian's interaction graph which as a statistical mechanics problem is of interest in their own right. We concretely apply this criterion to local Hamiltonians on various regular lattices, while bringing to bear the tools of spin glass theory which permit us to obtain new bounds on the SAT/UNSAT transition in random quantum satisfiability. These also lead us to natural conjectures for when such bounds will be exact (tight), as well as to a novel notion of universality for these computer science problems. Besides providing concrete algorithms leading to detailed and quantitative insights, this underscores the power of marrying classical statistical mechanics with quantum computation and complexity theory.