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arXiv:2212.11914 · Physical Review Research

Majorana scars as group singlets

Zimo Sun, Fedor K. Popov, Igor R. Klebanov, Kiryl Pakrouski

Published Dec 6, 2023• 14 citations
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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 [K. Pakrouski et al., Phys. Rev. Lett. 125, 230602 (2020)] 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 $\text{SO}(N)\ifmmode\times\else\texttimes\fi{}\text{SO}(M)$ group, and the scars are the $\text{SO}(N)$ singlets. For any even $M$ there are two families of scars. One of them, which we call the $\ensuremath{\eta}$ states, is symmetric under the group $\mathrm{O}(N)$. The other, the $\ensuremath{\zeta}$ states, has the $\text{SO}(N)$ invariance. For $M=4$, where our construction reduces to spin-$1/2$ fermions on a lattice with local interactions, the former family are the $N+1\phantom{\rule{4pt}{0ex}}\ensuremath{\eta}$-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 formulas for the scar states and use them to calculate the bipartite entanglement entropy analytically. For large $N$, it grows logarithmically with the subsystem 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 nonlocal 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 $N$.

DOI
10.1103/physrevresearch.5.043208

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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 [K. Pakrouski et al., Phys. Rev. Lett. 125, 230602 (2020)] 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 MMM Majorana fermions per site. The Hilbert space for NNN sites may be decomposed under the action of the SO(N)\ifmmode×\else\texttimes\fiSO(M)\text{SO}(N)\ifmmode\times\else\texttimes\fi{}\text{SO}(M)SO(N)\ifmmode×\else\texttimes\fiSO(M) group, and the scars are the SO(N)\text{SO}(N)SO(N) singlets. For any even MMM there are two families of scars. One of them, which we call the \ensuremathη\ensuremath{\eta}\ensuremathη states, is symmetric under the group O(N)\mathrm{O}(N)O(N). The other, the \ensuremathζ\ensuremath{\zeta}\ensuremathζ states, has the SO(N)\text{SO}(N)SO(N) invariance. For M=4M=4M=4, where our construction reduces to spin-1/21/21/2 fermions on a lattice with local interactions, the former family are the N+1\ensuremathηN+1\phantom{\rule{4pt}{0ex}}\ensuremath{\eta}N+1\ensuremathη-pairing states, while the latter are the N+1N+1N+1 states of maximum spin. We generalize this construction to M>4M>4M>4. For M=6M=6M=6 we exhibit explicit formulas for the scar states and use them to calculate the bipartite entanglement entropy analytically. For large NNN, it grows logarithmically with the subsystem 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>6M>6M>6 we find that with local Hamiltonians the scars typically have certain degeneracies. The scar spectrum can be made ergodic by adding a nonlocal 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 NNN.
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Weak ergodicity breaking in interacting quantum systems may occur due to the existence of a subspace dynamically decoupled from the rest of the Hilbert space. In two-orbital spinful lattice systems, we construct such subspaces that are in addition distinguished by strongest inter-orbital and spin-singlet or spin-triplet, long-range superconducting pairing correlations. All unconventional pairing types we consider are local in space and unitary. Alternatively to orbitals, the additional degree of freedom could originate from the presence of two layers or through any other mechanism. Required Hamiltonians are rather non-exotic and include chemical potential, Hubbard, and spin-orbit interactions typically used for two-orbital superconducting materials. Each subspace is spanned by a family of group-invariant quantum many-body scars combining both 2e and 4e pairing/clustering contributions. One of the basis states has the form of a BCS wavefunction and can always be made the ground state by adding a mean-field pairing potential. Analytical results in this work are lattice-, dimension- and (mostly) system size-independent. We confirm them by exact numerical diagonalization in small systems.
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Construction of asymptotic quantum many-body scar states in the SU(NNN) Hubbard model
We construct asymptotic quantum many-body scars (AQMBS) in one-dimensional SU(NNN) Hubbard chains (N≥3N\geq 3N≥3) by embedding the scar subspace into an auxiliary Hilbert subspace HP\mathcal{H}_PHP​ and identifying a parent Hamiltonian within it, together with a corresponding extension of the restricted spectrum-generating algebra to the multi-ladder case. Unlike previous applications of the parent-Hamiltonian scheme, we show that the parent Hamiltonian becomes the SU(NNN) ferromagnetic Heisenberg model rather than the spin-1/2 case, so that its gapless magnons realize explicit AQMBS of the original model. Working in the doublon-holon subspace, we derive this mapping, obtain the one-magnon dispersion for periodic and open boundaries, and prove (i) orthogonality to the scar states, (ii) vanishing energy variance in the thermodynamic limit, and (iii) subvolume entanglement entropy with rigorous MPS/MPO bounds. Our results broaden the parent-Hamiltonian family for AQMBS beyond spin-1/2 and provide analytic, low-entanglement excitations in SU(NNN)-symmetric systems.
A quantum many-body scar system usually contains a special nonthermal subspace (approximately) decoupled from the rest of the Hilbert space. In this work, we propose a general structure called deformed symmetric spaces for the decoupled subspaces hosting quantum many-body scars, which are irreducible sectors of simple Lie groups transformed by matrix-product operators (or projected entangled pair operators), of which the entanglement entropies are proved to obey sub-volume-law scaling and thus violate the eigenstate thermalization hypothesis. A deformed symmetric space, in general, is required to have at least a U(1) sub-Lie-group symmetry to allow coherent periodic dynamics from certain low-entangled initial states. We enumerate several possible deforming transformations based on the subgroup symmetry requirement and recover many exact scar eigenstates in the existing scar models, of which the Hamiltonians can be reconstructed using the standard quantum inverse method. In particular, a two-dimensional scar model is proposed, which hosts a periodic dynamical trajectory on which all states are topologically ordered.