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arXiv:2302.03028 · Physical review. B./Physical review. B

Numerical methods for detecting symmetries and commutant algebras

Sanjay Moudgalya, Olexei I. Motrunich

Published Jun 27, 2023• 19 citations
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Abstract

Utilizing the understanding of symmetries in the framework of commutant algebras, the authors introduce here general methods to numerically construct symmetry operators and quantum number sectors. One method uses simultaneous block diagonalization of two generic Hamiltonians to construct symmetry sectors, while another method uses the fact that symmetries in the commutant framework are frustration-free ground states of local superoperators, which leads to efficient algorithms. These methods are applied to conventional on-site symmetries, as well as unconventional symmetries such as Hilbert space fragmentation and quantum many-body scars.

DOI
10.1103/physrevb.107.224312

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Utilizing the understanding of symmetries in the framework of commutant algebras, the authors introduce here general methods to numerically construct symmetry operators and quantum number sectors. One method uses simultaneous block diagonalization of two generic Hamiltonians to construct symmetry sectors, while another method uses the fact that symmetries in the commutant framework are frustration-free ground states of local superoperators, which leads to efficient algorithms. These methods are applied to conventional on-site symmetries, as well as unconventional symmetries such as Hilbert space fragmentation and quantum many-body scars.
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Symmetry algebras of quantum many-body systems with locality can be understood using commutant algebras, which are defined as algebras of operators that commute with a given set of local operators. In this work, we show that these symmetry algebras can be expressed as frustration-free ground states of a local superoperator, which we refer to as a “super-Hamiltonian.” We demonstrate this for conventional symmetries such as Z 2 , U ( 1 ) , and S U ( 2 ) , where the symmetry algebras map to various kinds of ferromagnetic ground states, as well as for unconventional ones that lead to weak ergodicity-breaking phenomena of Hilbert-space fragmentation (HSF) and quantum many-body scars. In addition, we show that the low-energy excitations of this super-Hamiltonian can be understood as approximate symmetries, which in turn are related to slowly relaxing hydrodynamic modes in symmetric systems. This connection is made precise by relating the super-Hamiltonian to the superoperator that governs the operator relaxation in noisy symmetric Brownian circuits and this physical interpretation also provides a novel interpretation for Mazur bounds for autocorrelation functions. We find examples of gapped (gapless) super-Hamiltonians indicating the absence (presence) of slow modes, which happens in the presence of discrete (continuous) symmetries. In the gapless cases, we recover hydrodynamic modes such as diffusion, tracer diffusion, and asymptotic scars in the presence of U ( 1 ) symmetry, HSF, and a tower of quantum scars, respectively. In all, this demonstrates the power of the commutant-algebra framework in obtaining a comprehensive understanding of exact symmetries and associated approximate symmetries and hydrodynamic modes, and their dynamical consequences in systems with locality. Published by the American Physical Society 2024
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We study quantum many-body scars (QMBS) in the language of commutant algebras, which are defined as symmetry algebras of of local Hamiltonians. This framework explains the origin of dynamically disconnected subspaces seen in models with exact QMBS, i.e., the large “thermal” subspace and the small “nonthermal” subspace, which are attributed to the existence of unconventional nonlocal conserved quantities in the commutant; hence, it unifies the study of conventional symmetries and weak ergodicity-breaking phenomena into a single framework. Furthermore, this language enables us to use the von Neumann double commutant theorem to formally write down the exhaustive algebra of Hamiltonians with a desired set of QMBS, which demonstrates that QMBS survive under large classes of local perturbations. We illustrate this using several standard examples of QMBS, including the spin- 1 / 2 ferromagnetic, AKLT, spin-1 XY π -bimagnon, and the electronic η -pairing towers of states; in each of these cases, we explicitly write down a set of generators for the full algebra of Hamiltonians with these QMBS. Understanding this hidden structure in QMBS Hamiltonians also allows us to recover results of previous “brute-force” numerical searches for such Hamiltonians. In addition, this language clearly demonstrates the equivalence of several unified formalisms for QMBS proposed in the literature and also illustrates the connection between two apparently distinct classes of QMBS Hamiltonians—those that are captured by the so-called Shiraishi-Mori construction and those that lie beyond. Finally, we show that this framework motivates a precise definition for QMBS that automatically implies that they violate the conventional eigenstate thermalization hypothesis, and we discuss its implications to dynamics. Published by the American Physical Society 2024