Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2004.13800 · Physical review. B./Physical review. B

η -pairing states as true scars in an extended Hubbard model

Daniel K. Mark, Olexei I. Motrunich

Published Aug 24, 2020• 133 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.102.075132

Related papers

Two generalizations of η pairing in extended Hubbard models

Hui Zhai

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.

arXiv:cond-mat/0406461•Physical Review B•Jan 26, 2005•14 citations

η -pairing in Hubbard models: From spectrum generating algebras to quantum many-body scars

Sanjay Moudgalya, Nicolas Regnault, Bogdan Andrei Bernevig

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.

arXiv:2004.13727•Physical review. B./Physical review. B•Aug 20, 2020•171 citations

Quantum many-body scars with unconventional superconducting pairing symmetries via multibody interactions

Shohei Imai, Naoto Tsuji

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.

arXiv:2404.02914•Physical Review Research•Jan 16, 2025•10 citations

Generalized η-pairing eigenstates in three-component Hubbard models under a transverse field

F. X. Liu, Z. Song

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.

arXiv:2609.33568•Sep 27, 2026•0 citations

Exact quantum many-body scars in higher-spin kinetically constrained models

Yuan Dong, Shunyao Zhang, Dong-Ling Deng

We discover a variety of exact quantum many-body scars in higher-spin kinetically constrained models through the recently developed DMRG-S algorithm [Zhang et al., Phys. Rev. Lett. 131, 020402 (2023)]. Specifically, for the higher-spin PXP model on arbitrary bipartite lattices of any spatial dimension, we find exact many-body scars that are equidistantly spaced in the energy spectrum and exhibit similar structures to the ground state of the Affleck-Kennedy-Lieb-Tasaki model. For the one-dimensional Fermi-Hubbard model with a tilted potential in a certain parameter regime, whose effective model is equivalent to a kinetically constrained spin model with four degrees of freedom on each site, we find several many-body scars at energy $E=0$ and $E=\ifmmode\pm\else\textpm\fi{}\sqrt{2}$ that can be exactly represented as matrix product states with finite bond dimensions. Our results demonstrate that larger local degrees of freedom in the kinetically constrained models provide a much broader space for the emergence of quantum many-body scars and weak ergodicity breaking.

arXiv:2307.06357•Physical review. B./Physical review. B•Nov 16, 2023•5 citations

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

Kiryl Pakrouski, P. N. Pallegar, Fedor K. Popov +1 more

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.

arXiv:2106.10300•Physical Review Research•Dec 3, 2021•68 citations
The \ensuremathη\ensuremath{\eta}\ensuremathη-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 \ensuremathη\ensuremath{\eta}\ensuremathη-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 \ensuremathη\ensuremath{\eta}\ensuremathη-pairing states survive without the \ensuremathη\ensuremath{\eta}\ensuremathη-pairing symmetry and become true scar states. We also discuss similarities between the \ensuremathη\ensuremath{\eta}\ensuremathη-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 J1\ensuremath−J2{J}_{1}\ensuremath{-}{J}_{2}J1​\ensuremath−J2​ model with Dzyaloshinskii-Moriya interaction, in realistic quantum magnet settings in one and two dimensions.
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.
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.
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.
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 discover a variety of exact quantum many-body scars in higher-spin kinetically constrained models through the recently developed DMRG-S algorithm [Zhang et al., Phys. Rev. Lett. 131, 020402 (2023)]. Specifically, for the higher-spin PXP model on arbitrary bipartite lattices of any spatial dimension, we find exact many-body scars that are equidistantly spaced in the energy spectrum and exhibit similar structures to the ground state of the Affleck-Kennedy-Lieb-Tasaki model. For the one-dimensional Fermi-Hubbard model with a tilted potential in a certain parameter regime, whose effective model is equivalent to a kinetically constrained spin model with four degrees of freedom on each site, we find several many-body scars at energy E=0E=0E=0 and $E=\ifmmode\pm\else\textpm\fi{}\sqrt{2}$ that can be exactly represented as matrix product states with finite bond dimensions. Our results demonstrate that larger local degrees of freedom in the kinetically constrained models provide a much broader space for the emergence of quantum many-body scars and weak ergodicity breaking.
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 $\ensuremath{\eta}$-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 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.