Symmetries as Ground States of Local Superoperators: Hydrodynamic Implications

Abstract

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 $SU(2)$, 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 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, Hilbert space fragmentation, 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.

19+16 pages, 2 figures v2: Added discussion on interpretation of super-Hamiltonian ground states in terms of spontaneous symmetry breaking. v3: Added connection to approximate block-diagonalization, modified title, added references

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Cited by in corpus (13)

• Strong-to-Weak Spontaneous Symmetry Breaking in Mixed Quantum States
• Hydrodynamics as the effective field theory of strong-to-weak spontaneous symmetry breaking
• Highly entangled stationary states from strong symmetries
• Spontaneous symmetry breaking in open quantum systems: strong, weak, and strong-to-weak
• Strong-to-weak Symmetry Breaking and Entanglement Transitions
• Scheme to Detect the Strong-to-weak Symmetry Breaking via Randomized Measurements
• Systematic construction of asymptotic quantum many-body scar states and their relation to supersymmetric quantum mechanics
• Entanglement dynamics from universal low-lying modes
• Dynamics in the presence of local symmetry-breaking impurities
• Non-Universality from Conserved Superoperators in Unitary Circuits
• Integrals of motion as slow modes in dissipative many-body operator dynamics
• Additional quantum many-body scars of the spin-$1$ $XY$ model with Fock-space cages and commutant algebras
• Construction of asymptotic quantum many-body scar states in the SU($N$) Hubbard model