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arXiv:2607.01038 · Physical review. B./Physical review. B

Exact interlayer triplet-pairing eigenstates in the extended Hubbard model

F. X. Liu, Z. Song

Published Sep 21, 2026• 0 citations
arXivHTMLPDF

Abstract

$η$-pairing symmetry generalizes the pairing mechanisms in superconductivity but is broken in the presence of interlayer interactions. In this work, we extend this approach to triplet pairs. We propose interlayer triplet-pairing operators for the multi-layer extended Hubbard model. We find that a set of exact condensate-pair eigenstates can be constructed, which exhibit off-diagonal long-range order. In contrast to the $η$-pairing mechanism, this originates from restricted spectrum generating algebra and is only available for bilayer and trilayer systems in the presence of interlayer Hubbard interactions. Nevertheless, the system also retains the original on-site $η$-pairing symmetry in the absence of interlayer interactions. Consequently, both singlet and triplet pairs coexist in the eigenstates of the multi-layer Hubbard model. We employ quench dynamics to demonstrate the results through numerical simulations. Our findings open avenues for the study of exact condensate-pair states in strongly correlated systems.

DOI
10.1103/c7kj-ph6v

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ηηη-pairing symmetry generalizes the pairing mechanisms in superconductivity but is broken in the presence of interlayer interactions. In this work, we extend this approach to triplet pairs. We propose interlayer triplet-pairing operators for the multi-layer extended Hubbard model. We find that a set of exact condensate-pair eigenstates can be constructed, which exhibit off-diagonal long-range order. In contrast to the ηηη-pairing mechanism, this originates from restricted spectrum generating algebra and is only available for bilayer and trilayer systems in the presence of interlayer Hubbard interactions. Nevertheless, the system also retains the original on-site ηηη-pairing symmetry in the absence of interlayer interactions. Consequently, both singlet and triplet pairs coexist in the eigenstates of the multi-layer Hubbard model. We employ quench dynamics to demonstrate the results through numerical simulations. Our findings open avenues for the study of exact condensate-pair states in strongly correlated systems.
Yang's η pairing operator is generalized to explore off-diagonal long-range order in the Hubbard bilayer with an arbitrary chemical potential. With this operator and a constraint condition on annihilation and creation operators, we construct explicitly eigenstates which possess simultaneously three kinds of off-diagonal long-range order, i.e., the intralayer one and the interlayer one for on-site pairing, and that for interlayer nearest-neighbor pairing. As in the simple Hubbard model there is also an SO(4) symmetry, with the generators properly defined. A sufficient condition leads to at least one of the above three kinds of off-diagonal long-range order. A constraint relation among different kinds of off-diagonal long-range order is also given. There exists a triplet of collective modes if the U(1) symmetry of a subgroup is spontaneously broken.
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.
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 consider NNN-particle generalizations of $\ensuremath{\eta}$-pairing states in a chain of NNN-component fermions and show that these states are exact (high-energy) eigenstates of an extended SU(N)\mathrm{SU}(N)SU(N) Hubbard model. We compute the singlet correlation function of the states and find that its behavior is qualitatively different for even and odd NNN. When NNN is even, these states exhibit off-diagonal long-range order in NNN-particle reduced density matrix. On the other hand, when NNN is odd, the correlations decay exponentially with distance in the bulk, but end-to-end correlations do not vanish in the thermodynamic limit. Finally, we prove that these states are the unique ground states of suitably tailored Hamiltonians.
We consider the Hubbard model and its extensions on bipartite lattices. We define a dynamical group based on the -pairing operators introduced by Yang, and define coherent pairing states, which are combinations of eigenfunctions of -operators. These states permit exact calculations of numerous physical properties of the system, including energy, various fluctuations and correlation functions, including pairing off-diagonal long-range order to all orders. This approach is complementary to BCS, in that these are superconducting coherent states associated with the exact model, although they are not eigenstates of the Hamiltonian.
We present a systematic framework to construct model Hamiltonians that have unconventional superconducting pairing states as exact energy eigenstates, by incorporating multibody interactions (i.e., interactions among more than two particles). The multibody interactions are introduced in a form of the local density-density coupling in such a way that any pair configuration in real space has a constant interaction energy by canceling the two-body and multibody interactions. Our approach is applicable to both spinless and spinful models in any spatial dimensions and on any bipartite lattices, facilitating an exhaustive extension of Yang's s − wave η − pairing state to various other unconventional pairing symmetries ( p − wave , d − wave , f − wave , etc.). Particularly, the constructed eigenstates have off-site pairs with finite center-of-mass momentum, which leads to superconducting states with either even-parity and spin-triplet or odd-parity and spin-singlet symmetry. We verify that the two-dimensional spinful Hubbard model on a square lattice with the multibody interactions has the spin-triplet d − wave pairing state as an energy eigenstate, which can be regarded as a quantum many-body scar state as evidenced from the numerical analysis of the pair correlation function, entanglement entropy, and level statistics. We also discuss other examples, including spin-triplet f − wave pairing states on a honeycomb lattice and spin-singlet p − wave pairing states in a one-dimensional chain. These findings open up the possibility of realizing nonequilibrium unconventional superconductivity in a long-lived manner protected against thermalization.