Exhaustive Characterization of Quantum Many-Body Scars using Commutant Algebras

Abstract

We study Quantum Many-Body Scars (QMBS) in the language of commutant algebras, which are defined as symmetry algebras of families 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 "non-thermal" subspace, which are attributed to the existence of unconventional non-local conserved quantities in the commutant; hence this 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 (DCT) to formally write down the exhaustive algebra of all 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 $\pi$-bimagnon, and the electronic $\eta$-pairing towers of states; and 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 (ETH), and we discuss its implications to dynamics.

26+22 pages, 1 figure. v2: Improved presentation, references added. v3: Minor clarifications, close to published version

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• Theory of robust quantum many-body scars in long-range interacting systems
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• Quantum Many-Body Scarring in a Non-Abelian Lattice Gauge Theory
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• Stabilizer Scars
• Exceptional stationary state in a dephasing many-body open quantum system
• Detecting Many-Body Scars from Fisher Zeros
• Exact volume-law entangled zero-energy eigenstates in a large class of spin models
• Systematic construction of asymptotic quantum many-body scar states and their relation to supersymmetric quantum mechanics
• Hilbert subspace ergodicity
• Taking the temperature of quantum many-body scars
• Non-Universality from Conserved Superoperators in Unitary Circuits
• Entanglement dynamics from universal low-lying modes
• Geometric fragmentation and anomalous thermalization in cubic dimer model
• Dynamics in the presence of local symmetry-breaking impurities
• ScarFinder: a detector of optimal scar trajectories in quantum many-body dynamics
• Temperature and conditions for thermalization after canonical quenches
• Additional quantum many-body scars of the spin-$1$ $XY$ model with Fock-space cages and commutant algebras
• Basis dependence of eigenstate thermalization
• Transport in a System with a Tower of Quantum Many-Body Scars
• Certain BCS wavefunctions are quantum many-body scars
• Quantum many-body scarring from Kramers-Wannier duality
• Unconventional superconducting correlations in fermionic many-body scars