Majorana Scars as Group Singlets

Abstract

In some quantum many-body systems, the Hilbert space breaks up into a large ergodic sector and a much smaller scar subspace. It has been suggested [arXiv:2007.00845] that the two sectors may be distinguished by their transformation properties under a large group whose rank grows with the system size (it is not a symmetry of the Hamiltonian). The quantum many-body scars are invariant under this group, while all other states are not. Here we apply this idea to lattice systems containing $M$ Majorana fermions per site. The Hilbert space for $N$ sites may be decomposed under the action of the O$(N)\times$O$(M)$ group, and the scars are the SO$(N)$ singlets. For any even $M$ there are two families of scars. One of them, which we call the $\eta$ states, is symmetric under the group O$(N)$. The other, the $\zeta$ states, has the SO$(N)$ invariance. For $M=4$, where our construction reduces to spin-$1/2$ fermions on a lattice with local interactions, the former family are the $N+1$ $\eta$-pairing states, while the latter are the $N+1$ states of maximum spin. We generalize this construction to $M>4$. For $M=6$ we exhibit explicit formulae for the scar states and use them to calculate the bipartite entanglement entropy analytically. For large $N$, it grows logarithmically with the sub-system size. We present a general argument that any group-invariant scars should have the entanglement entropy that is parametrically smaller than that of typical states. The energies of the scars we find are not equidistant in general but can be made so by choosing Hamiltonian parameters. For $M>6$ we find that with local Hamiltonians the scars typically have certain degeneracies. The scar spectrum can be made ergodic by adding a non-local interaction term. We derive the dimension of each scar family and show the scars could have a large contribution to the density of states for small $N$.

18 pages, 1 table, 10 figures; v2: changed parameters to improve presentation, added several new results including a bound on entropy for group-invariant scars; v3: version to appear in PRR

References in corpus (27)

• The distribution of the ratio of consecutive level spacings in random matrix ensembles
• Quantum Many-Body Scars and Weak Breaking of Ergodicity
• Quantum Many-Body Scars and Hilbert Space Fragmentation: A Review of Exact Results
• Emergent SU(2) dynamics and perfect quantum many-body scars
• Ultracold fermions and the SU(N) Hubbard model
• Quantum Many-Body Scars: A Quasiparticle Perspective
• Quantum Many-Body Scar States with Emergent Kinetic Constraints and Finite-Entanglement Revivals
• Eta-Pairing in Hubbard Models: From Spectrum Generating Algebras to Quantum Many-Body Scars
• Eta-pairing states as true scars in an extended Hubbard Model
• Quantum scars from zero modes in an Abelian lattice gauge theory on ladders
• Systematic construction of scarred many-body dynamics in 1D lattice models
• Discrete time-crystalline order enabled by quantum many-body scars: entanglement steering via periodic driving
• Group theoretic approach to many-body scar states in fermionic lattice models
• Rainbow Scars: From Area to Volume Law
• Gravitational orbits, double-twist mirage, and many-body scars
• Exact Quantum Scars in the Chiral Non-Linear Luttinger Liquid
• From Symmetries to Commutant Algebras in Standard Hamiltonians
• Deformed Symmetry Structures and Quantum Many-body Scar Subspaces
• Multiple quantum scar states and emergent slow-thermalization in the flat-band system
• Multi-magnon quantum many-body scars from tensor operators
• Entanglement and geometry from subalgebras of the Virasoro algebra
• Holographic Quantum Scars
• Exact eigenstates of extended SU($N$) Hubbard models: Generalizations of $\eta$-pairing states with $N$-particle off-diagonal long-range order
• A Quantum Breakdown Model: from Many-body Localization to Chaos with Scars
• Permutation symmetry in large N Matrix Quantum Mechanics and Partition Algebras
• Bounding Entanglement Entropy Using Zeros of Local Correlation Matrices
• Exact many-body scars based on pairs or multimers in a chain of spinless fermions

Cited by in corpus (9)

• From Symmetries to Commutant Algebras in Standard Hamiltonians
• Stability of the many-body scars in fermionic spin-1/2 models
• Exact eigenstates of multicomponent Hubbard models: SU($N$) magnetic $\eta$ pairing, weak ergodicity breaking, and partial integrability
• Quantum Many-Body Scarring in a Non-Abelian Lattice Gauge Theory
• Quantum many-body scars with unconventional superconducting pairing symmetries via multibody interactions
• Quasi-Nambu-Goldstone modes in many-body scar models
• Stabilizer Scars
• Unconventional superconducting correlations in fermionic many-body scars
• Certain BCS wavefunctions are quantum many-body scars