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arXiv:2106.10300 · Physical Review Research

Group theoretic approach to many-body scar states in fermionic lattice models

Kiryl Pakrouski, P. N. Pallegar, Fedor K. Popov, Igor R. Klebanov

Published Dec 3, 2021• 68 citations
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Abstract

It has been shown [K. Pakrouski et al., Phys. Rev. Lett. 125, 230602 (2020)] that three families of highly symmetric states are many-body scars for any spin-$\frac{1}{2}$ fermionic Hamiltonian of the form ${H}_{0}+OT$, where $T$ is a generator of an appropriate Lie group. One of these families consists of the well-known $\ensuremath{\eta}$-pairing states. In addition to having the usual properties of scars, these families of states are insensitive to electromagnetic noise and have advantages for storing and processing quantum information. In this paper we show that a number of well-known coupling terms, such as the Hubbard and the Heisenberg interactions, and the Hamiltonians containing them, are of the required form and support these states as scars without fine tuning. The explicit ${H}_{0}+OT$ decomposition for a number of most commonly used models, including topological ones, is provided. To facilitate possible experimental implementations, we discuss the conditions for the low-energy subspace of these models to be comprised solely of scars. Further, we write all the generators $T$ that can be used as building blocks for designing new models with scars, most interestingly including the spin-orbit coupled hopping and superconducting pairing terms. We expand this framework to the non-Hermitian open systems and demonstrate that for them the scar subspace continues to undergo coherent time evolution and exhibit the ``revivals.'' A full numerical study of an extended two-dimensional $tJU$ model explicitly illustrates the novel properties of the invariant scars and supports our findings.

DOI
10.1103/physrevresearch.3.043156

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We present a class of Hamiltonians H for which a sector of the Hilbert space invariant under a Lie group G, which is not a symmetry of H, possesses the essential properties of many-body scar states. These include the absence of thermalization and the "revivals" of special initial states in time evolution. A particular class of examples concerns interacting spin-1/2 fermions on a lattice consisting of N sites (it includes deformations of the Fermi-Hubbard model as special cases), and we show that it contains two families of N+1 scar states. One of these families, which was found in recent literature, comprises the well-known η-pairing states. We find another family of scar states that is U(N) invariant. Both families and most of the group-invariant scar states produced by our construction, in general, give rise to the off-diagonal long range order, which survives at high temperatures and is insensitive to the details of the dynamics. Such states could be used for reliable quantum information processing because the information is stored nonlocally and thus cannot be easily erased by local perturbations. In contrast, other scar states we find are product states, which could be easily prepared experimentally. The dimension of scar subspace is directly controlled by the choice of group G and can be made exponentially large.

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It has been shown [K. Pakrouski et al., Phys. Rev. Lett. 125, 230602 (2020)] that three families of highly symmetric states are many-body scars for any spin-12\frac{1}{2}21​ fermionic Hamiltonian of the form H0+OT{H}_{0}+OTH0​+OT, where TTT is a generator of an appropriate Lie group. One of these families consists of the well-known \ensuremathη\ensuremath{\eta}\ensuremathη-pairing states. In addition to having the usual properties of scars, these families of states are insensitive to electromagnetic noise and have advantages for storing and processing quantum information. In this paper we show that a number of well-known coupling terms, such as the Hubbard and the Heisenberg interactions, and the Hamiltonians containing them, are of the required form and support these states as scars without fine tuning. The explicit H0+OT{H}_{0}+OTH0​+OT decomposition for a number of most commonly used models, including topological ones, is provided. To facilitate possible experimental implementations, we discuss the conditions for the low-energy subspace of these models to be comprised solely of scars. Further, we write all the generators TTT that can be used as building blocks for designing new models with scars, most interestingly including the spin-orbit coupled hopping and superconducting pairing terms. We expand this framework to the non-Hermitian open systems and demonstrate that for them the scar subspace continues to undergo coherent time evolution and exhibit the ``revivals.'' A full numerical study of an extended two-dimensional tJUtJUtJU model explicitly illustrates the novel properties of the invariant scars and supports our findings.
We present a class of Hamiltonians H for which a sector of the Hilbert space invariant under a Lie group G, which is not a symmetry of H, possesses the essential properties of many-body scar states. These include the absence of thermalization and the "revivals" of special initial states in time evolution. A particular class of examples concerns interacting spin-1/2 fermions on a lattice consisting of N sites (it includes deformations of the Fermi-Hubbard model as special cases), and we show that it contains two families of N+1 scar states. One of these families, which was found in recent literature, comprises the well-known η-pairing states. We find another family of scar states that is U(N) invariant. Both families and most of the group-invariant scar states produced by our construction, in general, give rise to the off-diagonal long range order, which survives at high temperatures and is insensitive to the details of the dynamics. Such states could be used for reliable quantum information processing because the information is stored nonlocally and thus cannot be easily erased by local perturbations. In contrast, other scar states we find are product states, which could be easily prepared experimentally. The dimension of scar subspace is directly controlled by the choice of group G and can be made exponentially large.
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 $\text{SO}(N)\ifmmode\times\else\texttimes\fi{}\text{SO}(M)$ 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 $\ensuremath{\eta}$ states, is symmetric under the group O(N)\mathrm{O}(N)O(N). The other, the $\ensuremath{\zeta}$ 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 $N+1\phantom{\rule{4pt}{0ex}}\ensuremath{\eta}$-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.
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.
We construct many-body scar states in multi-flavour fermionic lattice models that possess strong magnetic or superconducting correlations of a given type specified by a unitary matrix AAA. One of the states maximizes the one-point correlations over the full Hilbert space and has the form of the BCS wavefunction. It may always be made the ground state by adding the correlations as a "pairing potential" to any Hamiltonian supporting group-invariant scars. In our single-flavour, spin-full fermions example we consider a superconducting AAA. The BCS scar ground state is a linear combination of the well-known ηηη-pairing states. In the multi-orbital fermions example the BCS-like ground state maximizes unconventional magnetic correlations. The broad class of eligible Hamiltonians includes many conventional condensed matter interactions. The part of the Hamiltonian that governs the exact dynamics of the scar subspace coincides with the BCS mean-field Hamiltonian. We therefore show that its eigenstates are many-body scars that are decoupled from the rest of the Hilbert space and thereby protected from thermalization. Our results point out a connection between the fields of superconductivity and weak ergodicity breaking (many-body scars) and will hopefully encourage further investigations. They also provide the first feasible protocol to initialize a fermionic system to a scar state in (a quantum simulator) experiment.
We study the stability of the many-body scars in spin-1/2 fermionic systems under the most typical perturbations in relevant materials. We find that some families of scars are completely insensitive to certain perturbations. In some other cases they are stable to the first order in perturbation theory. Our analytical results apply to a large class of Hamiltonians that are known [arXiv:2106.10300] to support exact many-body scars. For the numerical calculations we choose the deformed t−J−Ut-J-Ut−J−U model that includes both Heisenberg and Hubbard interactions. We propose two new stability measures that are based on physical observables rather than the fidelity to the exact initial wavefunction. They enable the experimental detection of scars and are more reliable from the theoretical and numerical perspectives. One of these measures may potentially find applications in other systems where the exact many-body scars are equally spaced in energy. In small systems and at small perturbations, a regime particularly relevant for quantum simulators, we identify and describe an additional stability exhibited by the many-body scars. For larger perturbation strengths we observe a distinct mode of ergodicity breaking that is consistent with many-body localization.
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.