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arXiv:2512.09595 · Physical review. B./Physical review. B

Unified theory of local integrals of motion

Ben Craps, Oleg Evnin, Dmitry Kovrizhin, Gabriele Pascuzzi

cond-mat.dis-nnmath-phquant-phSubmitted Dec 10, 2025• Announced Dec 11, 2025• 0 citations• Revised Aug 25, 2026 (v2)
v1 · Dec 10, 2025v2 · Aug 25, 2026
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Abstract

Conservation laws are of paramount importance in our understanding of classical and quantum dynamics. Here, we present a general framework for constructing exact quantum integrals of motion with the desired locality and quantum numbers, which will be illustrated for the case of many-body-localization (MBL). The latter has been understood theoretically in terms of the existence of an extensive number of local integrals of motion (LIOMs). Using our approach, we show that one can express the task of finding LIOMs as an optimization problem. For some specifications, this problem surprisingly connects to the question of finding classical ground states of spin-glass models. Our work unifies previous results obtained in the MBL context and reveals intriguing connections between many-body localization, spin-glass physics, and constrained optimization problems.

Journal reference
Phys. Rev. B 114 (2026) L140201
DOI
10.1103/xmqc-99hq
Comments
v2: accepted for publication

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Conservation laws are of paramount importance in our understanding of classical and quantum dynamics. Here, we present a general framework for constructing exact quantum integrals of motion with the desired locality and quantum numbers, which will be illustrated for the case of many-body-localization (MBL). The latter has been understood theoretically in terms of the existence of an extensive number of local integrals of motion (LIOMs). Using our approach, we show that one can express the task of finding LIOMs as an optimization problem. For some specifications, this problem surprisingly connects to the question of finding classical ground states of spin-glass models. Our work unifies previous results obtained in the MBL context and reveals intriguing connections between many-body localization, spin-glass physics, and constrained optimization problems.
Many-body localization provides a generic mechanism of ergodicity breaking in quantum systems. In contrast to conventional ergodic systems, many-body-localized (MBL) systems are characterized by extensively many local integrals of motion (LIOM), which underlie the absence of transport and thermalization in these systems. Here we report a physically motivated construction of local integrals of motion in the MBL phase. We show that any local operator (e.g., a local particle number or a spin-flip operator), evolved with the system's Hamiltonian and averaged over time, becomes a LIOM in the MBL phase. Such operators have a clear physical meaning, describing the response of the MBL system to a local perturbation. In particular, when a local operator represents a density of some globally conserved quantity, the corresponding LIOM describes how this conserved quantity propagates through the MBL phase. Being uniquely defined and experimentally measurable, these LIOMs provide a natural tool for characterizing the properties of the MBL phase, in both experiments and numerical simulations. We demonstrate the latter by numerically constructing an extensive set of LIOMs in the MBL phase of a disordered spin-chain model. We show that the resulting LIOMs are quasilocal and use their decay to extract the localization length and establish the location of the transition between the MBL and ergodic phases.
Many interesting properties of many-body localization (MBL) can be understood using a phenomenological model, involving local integrals of motion (LIOMs). Usually, LIOMs can only be computed approximately. Here, the authors explicitly compute a complete set of LIOMs nonperturbatively by maximizing their overlap with physical spin operators. The scheme is immune to accidental resonances and can be applied even at the phase transition point, providing a powerful tool to study microscopic features of MBL as characterized by various localization lengths.
Abstract The emergent integrability in a many-body localized (MBL) system can be well characterized by the existence of the complete set of local integrals of motion (LIOMs). Such exactly conserved and exponentially localized operators are often understood as quasiparticle operators which can be expanded in terms of single-particle operators dressed with different numbers of particle-hole pairs. Here, we consider a one-dimensional XXZ spin- <?CDATA 12\frac{1}{2}21​?> 1 2 Heisenberg chain in the presence of a random field and try to quantify the corrections needed to be considered in the picture of quasiparticles associated with LIOMs due to the presence of particle-hole excitations. To this end, we explicitly present the multibody expansion of LIOM creation operators of the system in the MBL regime. We analytically obtain the coefficients of this expansion and discuss the effect of higher-order corrections associated with different numbers of particle-hole excitations. Our analysis shows that depending on the localization length of the system, there exist a regime in which the contributions that come from higher-order terms can break down the effective one-particle description of the LIOMs and such quasiparticles become essentially many-body-like.
Many-body localization in disordered systems in one spatial dimension is typically understood in terms of the existence of an extensive number of (quasi) local integrals of motion (LIOMs) which are thought to decay exponentially with distance and interact only weakly with one another. By contrast, little is known about the form of the integrals of motion in disorder-free systems which exhibit localization. Here, we explicitly compute the LIOMs for disorder-free localized systems, focusing on the case of a linearly increasing potential. We show that while in the absence of interactions, the LIOMs decay faster than exponentially, the addition of interactions leads to the formation of a slow-decaying plateau at short distances. We study how the localization properties of the LIOMs depend on the linear slope, finding that there is a significant finite-size dependence, and present evidence that adding a weak harmonic potential does not result in typical many-body localization phenomenology. By contrast, the addition of disorder has a qualitatively different effect, dramatically modifying the properties of the LIOMS.
We review the current (as of Fall 2016) status of the studies on the emergent integrability in many‐body localized models. We start by explaining how the phenomenology of fully many‐body localized systems can be recovered if one assumes the existence of a complete set of (quasi)local operators which commute with the Hamiltonian (local integrals of motions, or LIOMs). We describe the evolution of this idea from the initial conjecture, to the perturbative constructions, to the mathematical proof given for a disordered spin chain. We discuss the proposed numerical algorithms for the construction of LIOMs and the status of the debate on the existence and nature of such operators in systems with a many‐body mobility edge, and in dimensions larger than one. image
We construct a complete set of local integrals of motion that characterize the many-body localized (MBL) phase. Our approach relies on the assumption that local perturbations act locally on the eigenstates in the MBL phase, which is supported by numerical simulations of the random-field XXZ spin chain. We describe the structure of the eigenstates in the MBL phase and discuss the implications of local conservation laws for its nonequilibrium quantum dynamics. We argue that the many-body localization can be used to protect coherence in the system by suppressing relaxation between eigenstates with different local integrals of motion.