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arXiv:2504.19786 · Physical review. B./Physical review. B

η -pairing states in the Hubbard model with nonuniform Hubbard interaction

D. K. He, Z. Song

Published Aug 5, 2025• 3 citations
arXivHTMLPDF

Abstract

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 $U$. In this work, we present a generalized Hubbard model containing a variety of pseudospin terms that break the ${\mathrm{SO}}_{4}$ 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.

DOI
10.1103/h3yg-lpf8

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The existence of \ensuremathη\ensuremath{\eta}\ensuremathη-pairing eigenstates in the fermionic Hubbard model is fundamentally rooted in the \ensuremathη\ensuremath{\eta}\ensuremathη-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 \ensuremathη\ensuremath{\eta}\ensuremathη-pairing symmetry. This allows us to construct a variety of correlated systems possessing \ensuremathη\ensuremath{\eta}\ensuremathη-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-\ensuremathη\ensuremath{\eta}\ensuremathη-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 \ensuremathη\ensuremath{\eta}\ensuremathη-pairing wave packet with the aid of a time-dependent Hamiltonian.
We investigate Hubbard models with bond-charge interactions on general graphs. For a Hamiltonian HHH of such a model, we provide the condition on its parameters under which the ηηη-pairing method can be employed to construct its exact eigenstates. We arrive at this condition by finding that the requirement for the ηηη-pairing state (η†)N∣0⟩(η^\dagger)^N |0\rangle(η†)N∣0⟩ to be an eigenstate of HHH is identical to the requirement for it to be an eigenstate of a Hubbard-type Hamiltonian HmH_mHm​. When the condition for (η†)N∣0⟩(η^\dagger)^N |0\rangle(η†)N∣0⟩ to be an eigenstate of the Hubbard-type Hamiltonian HmH_mHm​ is satisfied, we demonstrate that there are additional states, distinct from (η†)N∣0⟩(η^\dagger)^N |0\rangle(η†)N∣0⟩, which are also exact eigenstates of HmH_mHm​. Our results enhance the understanding of Hubbard models on general graphs, both with and without bond-charge interactions.
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 authors study the $\ensuremath{\eta}$-pairing phenomenon in correlated fermion models with spin-orbit coupling. Spin-orbit coupled systems are usually thought to possess no continuous symmetries, except the conventional U(1) particle-number symmetry. The $\ensuremath{\eta}$-pairing reveals an exact SU(2) pseudospin symmetry in a broad class of spin-orbit coupled Hubbard models. Furthermore, the authors construct an exactly solvable interacting Dirac-semimetal model that exhibits an exact pseudospin symmetry with fine-tuned parameters. This model hosts $\ensuremath{\eta}$-pairing and charge density wave ground states, which meet at an exactly solvable multicritical line.
We construct exact eigenstates of multicomponent Hubbard models in arbitrary dimensions by generalizing the η -pairing mechanism. Our models include the SU ( N ) Hubbard model as a special case. Unlike the conventional two-component case, the generalized η -pairing mechanism permits the construction of eigenstates that feature off-diagonal long-range order and magnetic long-range order. These states form fragmented fermionic condensates due to a simultaneous condensation of multicomponent η pairs. While the η -pairing states in the SU(2) Hubbard model are based on the η -pairing symmetry, the exact eigenstates in the N -component system with N ≥ 3 arise not from symmetry of the Hamiltonian but from a spectrum generating algebra defined in a restricted Hilbert space. We exploit this fact to show that the generalized η -pairing eigenstates do not satisfy the eigenstate thermalization hypothesis and serve as quantum many-body scar states. This result indicates a weak breakdown of ergodicity in the N -component Hubbard models for N ≥ 3 . Furthermore, we show that these exact eigenstates constitute integrable subsectors in which the Hubbard Hamiltonian effectively reduces to a noninteracting model. This partial integrability causes various multicomponent Hubbard models to weakly break ergodicity. We propose a method of harnessing dissipation to distill the integrable part of the dynamics and elucidate a mechanism of nonthermalization caused by dissipation. This work establishes the coexistence of off-diagonal long-range order and SU ( N ) magnetism in excited eigenstates of the multicomponent Hubbard models, which presents a possibility of novel out-of-equilibrium pairing states of multicomponent fermions. These models unveil a unique feature of quantum thermalization of multicomponent fermions, which can experimentally be tested with cold-atom quantum simulators. Published by the American Physical Society 2024
We investigate three-component Hubbard models on a bipartite lattice in the presence of a trans?verse field. Unlike the two-component Hubbard model, where the η-pairing symmetry survives the transverse field, the three-component model possesses neither η-pairing symmetry nor conservation of the particle number of each component. We introduce a generalized η-pairing operator, formed as a hybridization of three types of two-component pairing operators, and employ the restricted spectrum-generating algebra to construct a family of exact eigenstates. These eigenstates exhibit off-diagonal long-range order. Furthermore, we construct an exact tensor-product state whose co?herent periodic dynamics provide an exact realization of quantum many-body scar dynamics while simultaneously exhibiting long-range magnetic order. We also propose three protocols for preparing this state via quench dynamics, and verify their effectiveness through numerical simulations.
Abstract We investigate Hubbard models with bond–charge interactions on general graphs. For a Hamiltonian H of such a model, we provide the condition on its parameters under which the η -pairing method can be employed to construct its exact eigenstates. We arrive at this condition by finding that the requirement for the η -pairing state ( η † ) N ∣ 0 〉 to be an eigenstate of H is identical to the requirement for it to be an eigenstate of a Hubbard-type Hamiltonian H m without bond–charge interactions. When the condition for ( η † ) N ∣ 0 〉 to be an eigenstate of the Hubbard-type Hamiltonian H m is satisfied, we demonstrate that there are additional states, distinct from ( η † ) N ∣ 0 〉 , which are also exact eigenstates of H m . Our results enhance the understanding of Hubbard models on general graphs, both with and without bond–charge interactions.