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.

arXiv comment19+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

References in corpus (116)

• The density-matrix renormalization group in the age of matrix product states
• Generalized Global Symmetries
• Quantum many-body scars
• An RVB phase in the triangular lattice quantum dimer model
• Glassy dynamics of kinetically constrained models
• Lieb-Schultz-Mattis in Higher Dimensions
• Quantum Many-Body Scars and Weak Breaking of Ergodicity
• Fractons
• Localization from Hilbert space shattering: from theory to physical realizations
• Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws
• Local constraints can globally shatter Hilbert space: a new route to quantum information protection
• Quantum Many-Body Scars and Hilbert Space Fragmentation: A Review of Exact Results
• Quantum scarred eigenstates in a Rydberg atom chain: entanglement, breakdown of thermalization, and stability to perturbations
• Symmetries and conserved quantities in Lindblad master equations
• Ergodicity-breaking arising from Hilbert space fragmentation in dipole-conserving Hamiltonians
• Towards the fast scrambling conjecture
• Commensurability, excitation gap and topology in quantum many-particle systems on a periodic lattice
• Keldysh technique and non-linear sigma-model: basic principles and applications
• A note on symmetry reductions of the Lindblad equation: transport in constrained open spin chains
• Entanglement of Exact Excited States of AKLT Models: Exact Results, Many-Body Scars and the Violation of Strong ETH
• Generalized Symmetries in Condensed Matter
• Fracton Phases of Matter
• Non-stationary coherent quantum many-body dynamics through dissipation
• Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation
• Exact Quantum Many-Body Scar States in the Rydberg-Blockaded Atom Chain
• A symmetry principle for Topological Quantum Order
• Weak Ergodicity Breaking and Quantum Many-Body Scars in Spin-1 XY Magnets
• Exact Excited States of Non-Integrable Models
• Fracton hydrodynamics
• Quantum Many-Body Scars: A Quasiparticle Perspective
• Disorder-Free Localization
• Categorical symmetry and non-invertible anomaly in symmetry-breaking and topological phase transitions
• Topological Defects on the Lattice I: The Ising model
• Exotic $U(1)$ Symmetries, Duality, and Fractons in 3+1-Dimensional Quantum Field Theory
• Subsystem symmetry protected topological order
• Anomalous Diffusion in Dipole- and Higher-Moment Conserving Systems
• Statistical localization: from strong fragmentation to strong edge modes
• Symmetry and Topological Order
• Generalized Jordan-Wigner Transformations
• Eta-Pairing in Hubbard Models: From Spectrum Generating Algebras to Quantum Many-Body Scars
• Subdiffusion and heat transport in a tilted 2D Fermi-Hubbard system
• Exact Bethe ansatz spectrum of a tight-binding chain with dephasing noise
• Exact ground states and topological order in interacting Kitaev/Majorana chains
• Spectral statistics and many-body quantum chaos with conserved charge
• Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
• Many Body Scars as a Group Invariant Sector of Hilbert Space
• Eta-pairing states as true scars in an extended Hubbard Model
• Kinetically constrained freezing transition in a dipole-conserving system
• From quantum mechanics to classical statistical physics: generalized Rokhsar-Kivelson Hamiltonians and the "Stochastic Matrix Form" decomposition
• Schwinger-Keldysh formalism I: BRST symmetries and superspace
• Generalized Elitzur's Theorem and Dimensional Reduction
• Quasilocal conservation laws in XXZ spin-1/2 chains: open, periodic and twisted boundary conditions
• Unitary equilibrations: probability distribution of the Loschmidt echo
• Anomalous Subdiffusion from Subsystem Symmetries
• The Bond-Algebraic Approach to Dualities
• From classical to quantum dynamics at Rokhsar-Kivelson points
• Locality, Quantum Fluctuations, and Scrambling
• Strong-weak coupling self-duality in the two-dimensional quantum phase transition of $p+ip$ superconducting arrays
• From tunnels to towers: quantum scars from Lie Algebras and q-deformed Lie Algebras
• Quantum scars as embeddings of weakly "broken" Lie algebra representations
• Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners
• The spectral gap for some spin chains with discrete symmetry breaking
• Enlarged symmetry algebras of spin chains, loop models, and S-matrices
• Bond Algebras and Exact Solvability of Hamiltonians: Spin S=1/2 Multilayer Systems and Other Curiosities
• Quantum chaos in the Brownian SYK model with large finite $N$: OTOCs and tripartite information
• Topological Holography: Towards a Unification of Landau and Beyond-Landau Physics
• Exotic ${\mathbb{Z}}_N$ Symmetries, Duality, and Fractons in 3+1-Dimensional Quantum Field Theory
• Symmetry as a shadow of topological order and a derivation of topological holographic principle
• Thermodyamic bounds on Drude weights in terms of almost-conserved quantities
• Spectral statistics in constrained many-body quantum chaotic systems
• Unified approach to Quantum and Classical Dualities
• Group theoretic approach to many-body scar states in fermionic lattice models
• Operator Dynamics in Brownian Quantum Circuit
• Slow Thermalization of Exact Quantum Many-Body Scar States Under Perturbations
• Lectures on the Bethe Ansatz
• Stochastic dissipative quantum spin chains (I) : Quantum fluctuating discrete hydrodynamics
• Yang-Baxter integrable Lindblad equations
• Quasilocal conservation laws from semicyclic irreducible representations of $U_q({\mathfrak{sl}}_2)$ in $XXZ$ spin-$1/2$ chains
• Equilibrium Fluctuations in Maximally Noisy Extended Quantum Systems
• Subsystem Non-Invertible Symmetry Operators and Defects
• Local gap threshold for frustration-free spin systems
• Random Matrix Spectral Form Factor in Kicked Interacting Fermionic Chains
• Slow Relaxation in a Model with Many Conservation Laws : Deposition and Evaporation of Trimers on a Line
• Conservation Laws and Integrability of a One-dimensional Model of Diffusing Dimers
• Entanglement distribution in the Quantum Symmetric Simple Exclusion Process
• Quantum Phase Diagram of the t-Jz Chain Model
• Hilbert Space Fragmentation in Open Quantum Systems
• Unifying Emergent Hydrodynamics and Lindbladian Low Energy Spectra across Symmetries, Constraints, and Long-Range Interactions
• Quantum integrability in systems with finite number of levels
• From Symmetries to Commutant Algebras in Standard Hamiltonians
• Dynamics of Fluctuations in Quantum Simple Exclusion Processes
• Unfrustrated Qudit Chains and their Ground States
• Gapless line for the anisotropic Heisenberg spin-1/2 chain in a magnetic field and the quantum axial next-nearest-neighbor Ising Chain
• How much entanglement is needed to reduce the energy variance?
• Solving frustration-free spin systems
• Note on entropy dynamics in the Brownian SYK model
• Ergodicity breaking provably robust to arbitrary perturbations
• Matrix product symmetries and breakdown of thermalization from hard rod deformations
• Asymptotic Quantum Many-Body Scars
• Height-conserving quantum dimer models
• Information Scrambling and Entanglement Dynamics of Complex Brownian Sachdev-Ye-Kitaev Models
• Hydrodynamics in long-range interacting systems with center-of-mass conservation
• Revisiting the Mazur bound and the Suzuki equality
• Numerical Methods for Detecting Symmetries and Commutant Algebras
• Emergent Symmetry in Brownian SYK Models and Charge Dependent Scrambling
• Exact results for a $Z_3$ clock-type model and some close relatives
• Charge and spin dynamics in the one-dimensional $t-J_z$ and $t-J$ models
• Entanglement entropy growth in stochastic conformal field theory and the KPZ class
• Divide and conquer method for proving gaps of frustration free Hamiltonians
• Hierarchical hydrodynamics in long-range multipole-conserving systems
• Topologically stable ergodicity breaking from emergent higher-form symmetries in generalized quantum loop models
• The Irreducible String and an Infinity of Additional Constants of Motion in a Deposition-Evaporation Model on a Line
• Charge transport, information scrambling and quantum operator-coherence in a many-body system with U(1) symmetry
• Bounding Entanglement Entropy Using Zeros of Local Correlation Matrices
• Disorder-Free Localization as a Purely Classical Effect
• Coherence generation, symmetry algebras and Hilbert space fragmentation

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