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An extensive number of independent local integrals of motion in a nonintegrable quantum many-body system

Oleg Valentinovich Lychkovskiy

Published Dec 11, 2018• 0 citations
OpenAlex

Abstract

Studies of integrable quantum many-body systems have a long history with an impressive record of success. However, quite surprisingly, an unambiguous definition of quantum integrability remains a matter of an ongoing debate. We contribute to this debate by presenting several observation motivated by a particular quantum model of $N$ spins $1/2$. Remarkably, although this system is nonintegrable, it possesses a large number of mutually commuting local binary integrals of motion. We address the question of the number of independent integrals of motion. Strikingly, we find that, according to a common definition of independence of quantum integrals of motion (QIMs), this number equals to the number of spins, $N$. A common wisdom would then suggest that the system is completely integrable, which is not the case. To resolve this conundrum, we dwell upon the notion of independence of QIMs. We argue that the common definition should be amended in order to match the intuitive notion of independence. We suggest a new definition, which, in particular, implies that some of the $N$ QIMs of the considered model are dependent. Still, even according to this strict definition, $N/2$ integrals of motion remain independent. This challenges the intuition on the integrable quantum many-body systems as well as some of the most elaborated definitions of quantum integrability.

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Studies of integrable quantum many-body systems have a long history with an impressive record of success. However, quite surprisingly, an unambiguous definition of quantum integrability remains a matter of an ongoing debate. We contribute to this debate by presenting several observation motivated by a particular quantum model of NNN spins 1/21/21/2. Remarkably, although this system is nonintegrable, it possesses a large number of mutually commuting local binary integrals of motion. We address the question of the number of independent integrals of motion. Strikingly, we find that, according to a common definition of independence of quantum integrals of motion (QIMs), this number equals to the number of spins, NNN. A common wisdom would then suggest that the system is completely integrable, which is not the case. To resolve this conundrum, we dwell upon the notion of independence of QIMs. We argue that the common definition should be amended in order to match the intuitive notion of independence. We suggest a new definition, which, in particular, implies that some of the NNN QIMs of the considered model are dependent. Still, even according to this strict definition, N/2N/2N/2 integrals of motion remain independent. This challenges the intuition on the integrable quantum many-body systems as well as some of the most elaborated definitions of quantum integrability.
We study general quantum integrable Hamiltonians linear in a coupling constant and represented by finite NxN real symmetric matrices. The restriction on the coupling dependence leads to a natural notion of nontrivial integrals of motion and classification of integrable families into Types according to the number of such integrals. A Type M family in our definition is formed by N-M nontrivial mutually commuting operators linear in the coupling. Working from this definition alone, we parameterize Type M operators, i.e. resolve the commutation relations, and obtain an exact solution for their eigenvalues and eigenvectors. We show that our parameterization covers all Type 1, 2, and 3 integrable models and discuss the extent to which it is complete for other types. We also present robust numerical observation on the number of energy level crossings in Type M integrable systems and analyze the taxonomy of types in the 1d Hubbard model.
In this paper we discuss the concepts of quantum integrability and nonintegrability. Based on the concept of a complete set of commuting observables and the Hilbert-space structure of a quantum system, the definitions are given for the quantum-dynamical degrees of freedom and quantum phase space from which the quantum integrability is defined. A criterion for quantum integrability then emerges; the system is integrable if it possesses dynamical symmetry. Breaking of dynamical symmetry is connected with the nonintegrability of systems and thus is the inherent mechanism of chaotic motion. A number of examples are discussed.
We study general quantum integrable Hamiltonians linear in a coupling constant and represented by finite N × N real symmetric matrices. The restriction on the coupling dependence leads to a natural notion of nontrivial integrals of motion and classification of integrable families into types according to the number of such integrals. A type M family in our definition is formed by N – M nontrivial mutually commuting operators linear in the coupling. Working from this definition alone, we parameterize type M operators, i.e. resolve the commutation relations, and obtain an exact solution for their eigenvalues and eigenvectors. We show that our parameterization covers all type 1, 2 and 3 integrable models and discuss the extent to which it is complete for other types. We also present robust numerical observation on the number of energy-level crossings in type M integrable systems and analyze the taxonomy of types in the 1D Hubbard model.
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.
The integrability of a quantum many-body system, which is characterized by the presence or absence of local conserved quantities, drastically impacts the dynamics of isolated systems, including thermalization. Nevertheless, a rigorous and comprehensive method for determining integrability or nonintegrability has remained elusive. In this paper, we address this challenge by introducing rigorously provable tests for integrability and nonintegrability of quantum spin systems with finite-range interactions. Our results significantly simplify existing proofs of nonintegrability, such as those for the S=1/2S=1/2S=1/2 Heisenberg chain with nearest-and next-nearest-neighbor interactions, the S=1S=1S=1 bilinear-biquadratic chain and the S=1/2S=1/2S=1/2 XYZ model in two or higher dimensions. Moreover, our results also yield the first proof of nonintegrability for models such as the S=1/2S=1/2S=1/2 Heisenberg chain with a non-uniform magnetic field, the S=1/2S=1/2S=1/2 XYZ model on the triangular lattice, and the general spin XYZ model. This work also offers a partial resolution to the long-standing conjecture that integrability is governed by the existence of local conserved quantities with small support. Our framework ensures that the nonintegrability of one-dimensional spin systems with translational symmetry can be verified algorithmically, independently of system size.
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