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arXiv:2502.04559

Generalized $η$-pairing approach to interacting non-Hermitian systems in arbitrary dimensions

Kai Lieta

Published Feb 6, 2025• 0 citations
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Abstract

Developing a general and rigorous analytical approach to non-Hermitian many-body systems is a challenging task. Here, we generalize the eta-pairing theory to very general non-Hermitian Hubbard models and find many novel phenomena without Hermitian analogs. For instance, the Hermitian conjugate of an eta-pairing eigenoperator may not be an eigenoperator, eta-pairing eigenoperators can be spatially modulated, and the $SU(2)$ pseudospin symmetry may not be possible even if $H$ commutes with the eta-pairing operators. Remarkably, these novel non-Hermitian phenomena are closely related to each other by several theorems we establish and can lead to, for example, new types of eta-pairing operators (e.g., the notion of non-Hermitian angular-momentum operators) and the Fock space localization of many-body eta-pairing eigenstates. Some issues on the $SO(4)$ and particle-hole symmetries are clarified. Our general eta-pairing theory also reveals a previously unnoticed unification of these symmetries of the Hubbard model. These general results can be illustrated with concrete examples such as the generalized Hatano-Nelson-Hubbard models and a general two-sublattice model. In particular, the general two-sublattice model can reveal the eta-pairing structure [e.g., the $SO(4)$ symmetry] in systems with Hermitian hoppings, including the original eta-pairing theory for square lattice, the extension to triangular lattice, and some topological systems. Our results establish a new and rigorous theoretical framework for studying interacting non-Hermitian systems in arbitrary spatial dimensions, even without bulk translation symmetry.

DOI
10.48550/arxiv.2502.04559

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Generalized ηηη-pairing approach to interacting non-Hermitian systems in arbitrary dimensions
Developing a general and rigorous analytical approach to non-Hermitian many-body systems is a challenging task. Here, we generalize the eta-pairing theory to very general non-Hermitian Hubbard models and find many novel phenomena without Hermitian analogs. For instance, the Hermitian conjugate of an eta-pairing eigenoperator may not be an eigenoperator, eta-pairing eigenoperators can be spatially modulated, and the SU(2)SU(2)SU(2) pseudospin symmetry may not be possible even if HHH commutes with the eta-pairing operators. Remarkably, these novel non-Hermitian phenomena are closely related to each other by several theorems we establish and can lead to, for example, new types of eta-pairing operators (e.g., the notion of non-Hermitian angular-momentum operators) and the Fock space localization of many-body eta-pairing eigenstates. Some issues on the SO(4)SO(4)SO(4) and particle-hole symmetries are clarified. Our general eta-pairing theory also reveals a previously unnoticed unification of these symmetries of the Hubbard model. These general results can be illustrated with concrete examples such as the generalized Hatano-Nelson-Hubbard models and a general two-sublattice model. In particular, the general two-sublattice model can reveal the eta-pairing structure [e.g., the SO(4)SO(4)SO(4) symmetry] in systems with Hermitian hoppings, including the original eta-pairing theory for square lattice, the extension to triangular lattice, and some topological systems. Our results establish a new and rigorous theoretical framework for studying interacting non-Hermitian systems in arbitrary spatial dimensions, even without bulk translation symmetry.
The Hubbard model in bipartite lattice has been rigorously proved to have $\ensuremath{\eta}$-paired states as eigenstates by C. N. Yang [Phys. Rev. Lett. 63, 2144 (1989)]. In this paper, this result is first generalized to a triangular lattice and then even to an arbitrary lattice structure. We also consider its generalization to higher-spin Hubbard models. The conditions under which these models have $\ensuremath{\eta}$-paired states as eigenstates were obtained.
The η pseudo PT symmetry theory, denoted by the symbol η, explores the conditions under which non-Hermitian Hamiltonians can possess real spectra despite the violation of PT symmetry, that is the adjoint of H, denoted H^{†} is expressed as H^{†}=PTHPT. This theory introduces a new symmetry operator, η=PTη, which acts on the Hilbert space. The η pseudo PT symmetry condition requires the Hamiltonian to commute with the η operator, leading to real eigenvalues. We discuss some general implications of our results for the coupled non hermitian harmonic oscillator.
The existence of $\ensuremath{\eta}$-pairing eigenstates in the fermionic Hubbard model is fundamentally rooted in the $\ensuremath{\eta}$-pairing symmetry, which may hold for systems with nonuniform Hubbard interaction UUU. In this work, we present a generalized Hubbard model containing a variety of pseudospin terms that break the SO4{\mathrm{SO}}_{4}SO4​ symmetry but retain the $\ensuremath{\eta}$-pairing symmetry. This allows us to construct a variety of correlated systems possessing $\ensuremath{\eta}$-pairing eigenstates. We exemplify our findings by considering a modified Hubbard model associated with alternative magnetic fields and on-site repulsion. We find that the same quasi-$\ensuremath{\eta}$-pairing eigenstate exhibits two distinct dynamic behaviors in the two models. Numerical results of the time evolution driven by several typical Hamiltonians accord with the analytic predictions and provide a method for the control of an $\ensuremath{\eta}$-pairing wave packet with the aid of a time-dependent Hamiltonian.
The $\ensuremath{\eta}$-pairing states are a set of exactly known eigenstates of the Hubbard model on hypercubic lattices, first discovered by Yang [C. N. Yang, Phys. Rev. Lett. 63, 2144 (1989)]. These states are not many-body scar states in the Hubbard model because they occupy unique symmetry sectors defined by the so-called $\ensuremath{\eta}$-pairing SU(2) symmetry. We study an extended Hubbard model with bond-charge interactions, popularized by Hirsch [J. E. Hirsch, Physica C 158, 326 (1989)], where the $\ensuremath{\eta}$-pairing states survive without the $\ensuremath{\eta}$-pairing symmetry and become true scar states. We also discuss similarities between the $\ensuremath{\eta}$-pairing states and exact scar towers in the spin-1 XY model found by Schecter and Iadecola [M. Schecter and T. Iadecola, Phys. Rev. Lett. 123, 147201 (2019)], and systematically arrive at all nearest-neighbor terms that preserve such scar towers in one dimension. We also generalize these terms to arbitrary bipartite lattices. Our study of the spin-1 XY model also leads us to several scarred models, including a spin-1/2 ${J}_{1}\ensuremath{-}{J}_{2}$ model with Dzyaloshinskii-Moriya interaction, in realistic quantum magnet settings in one and two dimensions.
We develop a generalized Aufbau principle for non-Hermitian systems that allows for building up the configurations of indistinguishable particles. The Aufbau rule of non-Hermitian systems is unexpectedly shown to be identical to that developed in Hermitian systems when the real parts of the complex energy levels are considered. We derive full many-body energy spectra of the fermionic and bosonic Hatano-Nelson models as examples by filling single-particle energy levels in the momentum space. For open boundary conditions, we show that many-body non-Hermitian skin effects persist in all many-body eigenstates for both fermions and bosons. Furthermore, we find surprisingly that the ground state of bosons is an anomalous Bose-Einstein condensation with all of the particles simultaneously localizing in both the real and momentum space beyond the Heisenberg uncertainty principle. For periodic boundary conditions, we show that hard-core bosons cannot be mapped to fermions. This work establishes a general framework for understanding the many-body physics of non-Hermitian systems, revealing rich unique non-Hermitian many-body physics.
We revisit the $\ensuremath{\eta}$-pairing states in Hubbard models and explore their connections to quantum many-body scars to discover a universal scars mechanism. $\ensuremath{\eta}$-pairing occurs due to an algebraic structure known as a spectrum generating algebra (SGA), giving rise to equally spaced towers of eigenstates in the spectrum. We generalize the original $\ensuremath{\eta}$-pairing construction and show that several Hubbard-like models on arbitrary graphs exhibit SGAs, including ones with disorder and spin-orbit coupling. We further define a restricted spectrum generating algebra (RSGA) and give examples of perturbations to the Hubbard-like models that preserve an equally spaced tower of the original model as eigenstates. The states of the surviving tower exhibit a subthermal entanglement entropy, and we analytically obtain parameter regimes for which they lie in the bulk of the spectrum, showing that they are exact quantum many-body scars. The RSGA framework also explains the equally spaced towers of eigenstates in several well-known models of quantum scars, including the Affleck-Kennedy-Lieb-Tasaki model.