From Symmetries to Commutant Algebras in Standard Hamiltonians

Abstract

In this work, we revisit several families of standard Hamiltonians that appear in the literature and discuss their symmetries and conserved quantities in the language of commutant algebras. In particular, we start with families of Hamiltonians defined by parts that are local, and study the algebra of operators that separately commute with each part. The families of models we discuss include the spin-1/2 Heisenberg model and its deformations, several types of spinless and spinful free-fermion models, and the Hubbard model. This language enables a decomposition of the Hilbert space into dynamically disconnected sectors that reduce to the conventional quantum number sectors for regular symmetries. In addition, we find examples of non-standard conserved quantities even in some simple cases, which demonstrates the need to enlarge the usual definitions of symmetries and conserved quantities. In the case of free-fermion models, this decomposition is related to the decompositions of Hilbert space via irreducible representations of certain Lie groups proposed in earlier works, while the algebra perspective applies more broadly, in particular also to arbitrary interacting models. Further, the von Neumann Double Commutant Theorem (DCT) enables a systematic construction of local operators with a given symmetry or commutant algebra, potentially eliminating the need for "brute-force" numerical searches carried out in the literature, and we show examples of such applications of the DCT. This paper paves the way for both systematic construction of families of models with exact scars and characterization of such families in terms of non-standard symmetries, pursued in a parallel paper.

arXiv comment18+23 pages, v2: Some sections reorganized, references added

References in corpus (46)

• Fault-tolerant quantum computation by anyons
• Chaos and Quantum Thermalization
• Thermalization and its mechanism for generic isolated quantum systems
• Nonequilibrium dynamics of closed interacting quantum systems
• From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics
• Reference frames, superselection rules, and quantum information
• Quantum Many-Body Scars and Weak Breaking of Ergodicity
• Fractons
• Thermalization and prethermalization in isolated quantum systems: a theoretical overview
• Localization from Hilbert space shattering: from theory to physical realizations
• Operator Quantum Error Correcting Subsystems for Self-Correcting Quantum Memories
• 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
• The Ryu-Takayanagi Formula from Quantum Error Correction
• Stabilizer Formalism for Operator Quantum Error Correction
• Generalized Symmetries in Condensed Matter
• Fracton Phases of Matter
• Non-stationary coherent quantum many-body dynamics through dissipation
• Virtual Quantum Subsystems
• Topological phases with parafermions: theory and blueprints
• Review of Decoherence Free Subspaces, Noiseless Subsystems, and Dynamical Decoupling
• An introduction to the SYK model
• Many Body Scars as a Group Invariant Sector of Hilbert Space
• Determining a local Hamiltonian from a single eigenstate
• Eta-pairing states as true scars in an extended Hubbard Model
• Strong zero modes and eigenstate phase transitions in the XYZ/interacting Majorana chain
• The Bond-Algebraic Approach to Dualities
• The isolated Heisenberg magnet as a quantum time crystal
• Spectra of Operators in Large $N$ Tensor Models
• Symmetry Principles in Quantum Systems Theory
• The Folded Spin-1/2 XXZ Model: I. Diagonalisation, Jamming, and Ground State Properties
• Group theoretic approach to many-body scar states in fermionic lattice models
• Unified approach to Quantum and Classical Dualities
• Matrix product operator symmetries and intertwiners in string-nets with domain walls
• Spectra of Eigenstates in Fermionic Tensor Quantum Mechanics
• Useful relations among the generators in the defining and adjoint representations of SU(N)
• Symmetry criteria for quantum simulability of effective interactions
• Lecture notes on C*-algebras, Hilbert C*-modules, and quantum mechanics
• Spectrum of Majorana Quantum Mechanics with $O(4{)}^3$ Symmetry
• Symmetry protected topological phases beyond groups: The q-deformed Affleck-Kennedy-Lieb-Tasaki model
• Fractal Quantum Phase Transitions: Critical Phenomena Beyond Renormalization
• Majorana Scars as Group Singlets
• Hagedorn Temperature in Large $N$ Majorana Quantum Mechanics
• Engineering Topological Models with a General-Purpose Symmetry-to-Hamiltonian Approach
• More on the Isomorphism $SU(2)\otimes SU(2)\cong SO(4)$
• Reductions in finite-dimensional quantum mechanics: from symmetries to operator algebras and beyond

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• Classification of dynamical Lie algebras for translation-invariant 2-local spin systems in one dimension
• Exhaustive Characterization of Quantum Many-Body Scars using Commutant Algebras
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• Stability of the many-body scars in fermionic spin-1/2 models
• Exotic quantum liquids in Bose-Hubbard models with spatially-modulated symmetries
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• Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries
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• Symmetry Properties of Quantum Dynamical Entropy
• Additional quantum many-body scars of the spin-$1$ $XY$ model with Fock-space cages and commutant algebras