Completing the SO(4) analysis of the Hubbard model: classifying all interaction-independent eigenstates and determining the ordered region R

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The Hubbard model, the minimal lattice model of correlated electrons, has an SO(4) internal symmetry generated by the spin SU(2) and the pseudospin (eta-pairing) SU(2), as established by Yang and Zhang in 1990. Starting from fully spin-polarized free-fermion states and applying the eta-pairing operator and the spin raising operator, they constructed $2^{M-2}(M^{2}+3M+4)$ exact eigenstates on $M$ sites whose eigenvectors do not depend on the interaction strength $W$ , and proved that the minimizing region $R$ of the thermodynamic ground-state energy density $f(W,\rho_a,\rho_b)$ is convex and symmetric, while leaving open whether their family exhausts all $W$ -independent eigenstates and what the actual shape of $R$ is. The exhaustiveness belief was refuted: Ye and Lin (2018) constructed many further interaction-independent eigenstates (sublattice Slater determinants and submaximal-spin two-particle states) lying in SO(4) sectors with $j+j'<M/2$ . Vafek, Regnault and Bernevig (2017) proved the Yang-Zhang family saturates the sector $j+j'=M/2$ in any dimension. Essler, Korepin and Schoutens (1992) completely classified the one-dimensional spectrum via the SO(4)-extended Bethe ansatz. The region $R$ is pinned only by partial rigorous results at attractive interaction. The surviving open problem, corrected accordingly, is twofold: a complete classification of all $W$ -independent eigenstates of the Hubbard model — equivalently, of the common eigenstates of its kinetic and interaction operators — across all SO(4) sectors with $j+j'<M/2$ ; and a determination of the region $R$ , the domain of eta-pairing off-diagonal long-range order (superconductivity) in the density plane, as a function of interaction strength.

Background

The Hubbard model is the minimal lattice model of interacting electrons: spin-1/2 fermions (denoted $a$ and $b$ , i.e. spin up and down) hop between nearest-neighbor sites through a kinetic operator $T'$ , and two electrons on the same site interact with strength $W$ through an on-site interaction operator. Yang and Zhang analyzed it in the particle-hole-symmetric form $H'=T'+V'$ with $V'=2W\sum_{\mathbf r}(n^a_{\mathbf r}-\tfrac12)(n^b_{\mathbf r}-\tfrac12)$ on the cubic lattice, equivalent to the standard Hubbard model with on-site coupling $U=2W$ up to a chemical-potential shift. They proved that $H'$ commutes with two mutually commuting SU(2) algebras: the usual spin algebra, whose raising operator is $\zeta^{+}=\sum_{\mathbf k}a^{+}_{\mathbf k}b_{\mathbf k}$ , which converts a spin-down particle into a spin-up particle, and a pseudospin algebra, whose raising operator is Yang's eta-pairing operator $\eta^{+}=\sum_{\mathbf r}e^{i\boldsymbol\pi\cdot\mathbf r}a^{+}_{\mathbf r}b^{+}_{\mathbf r}$ , which creates an on-site electron pair (a doublon) with a staggered phase, i.e. crystal momentum $\boldsymbol\pi$ . The two raising operators are exchanged by a particle-hole transformation of a single spin species (exchanging particles and holes, i.e. occupied and empty single-particle states, of say the spin-down fermions), which is why $\zeta^{+}$ is called the particle-hole partner of $\eta^{+}$ . The two algebras together generate the internal symmetry group $SO(4)=(SU(2)\times SU(2))/Z_2$ (the quotient by the common two-element center), so simultaneous eigenstates of $H'$ , of the momentum, and of the two angular momenta organize into multiplets $\{j,j'\}$ , with $j$ the pseudospin and $j'$ the spin angular momentum, subject to $j+j'$ integer.

Because the sector with no $a$ -particles ( $N_a=0$ , i.e. all electrons spin-down) contains only noninteracting fermions, each of its $\binom{M}{N_b}$ Slater determinants (a Slater determinant is a free-fermion wavefunction, an antisymmetrized product of single-particle states) is an exact eigenstate; applying $\eta^{+}$ and $\zeta^{+}$ to them generates entire multiplets of exact eigenstates, $2^{M-2}(M^{2}+3M+4)$ states in total on $M$ sites. These states are independent of the interaction strength: their eigenvectors do not depend on $W$ , which is equivalent to being simultaneous eigenstates of both $T'$ and $V'$ . They are called $W$ -independent (or interaction-independent) eigenstates. For suitable multiplets the pseudospin quantum numbers satisfy $j^{2}-j_{z}^{2}=O(M^{2})$ , which Yang and Zhang showed implies off-diagonal long-range order (ODLRO): the largest eigenvalue of the two-particle reduced density matrix becomes of order $M$ (macroscopic), the defining property of a superconductor, with magnetic flux quantization in units $ch/2e$ ; the analogous spin condition instead signals particle-hole (magnetic) ordering.

Two of the paper's own stated limitations define the present problem. First, whether the constructed family exhausts all $W$ -independent eigenstates was left as an explicitly unproven belief. Second, writing $E_0(W,N_a,N_b)$ for the lowest eigenvalue at fixed particle numbers and $\rho_a=N_a/M$ , $\rho_b=N_b/M$ for the densities, the thermodynamic-limit ground-state energy density $f(W,\rho_a,\rho_b)=\lim_{M\to\infty}M^{-1}E_0$ was proved convex in the densities and to obey exact symmetry relations: invariance under the spin interchange $\rho_a\leftrightarrow\rho_b$ , under the simultaneous particle-hole reflection $(\rho_a,\rho_b)\mapsto(1-\rho_a,1-\rho_b)$ , and under $W\mapsto -W$ combined with particle-hole conjugation of a single spin species, $f(W,\rho_a,\rho_b)=f(-W,\rho_a,1-\rho_b)$ . These constrain the minimizing region $R=\{(\rho_a,\rho_b): f(W,\rho_a,\rho_b)=f(W,\tfrac12,\tfrac12)\}$ — the set of densities at which the lowest energy per site equals its half-filled value, which by convexity and the reflection symmetry is the global minimum, and $R$ is expected to be the domain where the ground state carries eta-pairing (superconducting) order — to be convex, symmetric, and to contain $(\tfrac12,\tfrac12)$ ; but the actual shape and extent of $R$ were only sketched schematically. Establishing this identification, in particular the phase boundary, is part of the determination. Later work refuted the exhaustiveness belief, settled the sector $j+j'=M/2$ of the classification in all dimensions, completed the full one-dimensional spectral classification via the Bethe ansatz (the exact solution method for one-dimensional integrable lattice models), and established eta-pairing ground states rigorously only in parameter subsets, leaving both questions open in two and higher dimensions, where the model is not integrable.

Problem Statement

Consider the Hubbard model in the particle-hole-symmetric form of Yang and Zhang, $H'=T'+V'$ , on hypercubic lattices with an even number of sites per direction and $M$ sites in total, where $T'=-2\varepsilon\sum_{\mathbf k}\Lambda(\mathbf k)(a^{+}_{\mathbf k}a_{\mathbf k}+b^{+}_{\mathbf k}b_{\mathbf k})$ with $\Lambda(\mathbf k)=\cos k_x+\cos k_y$ in two dimensions and $\Lambda(\mathbf k)=\cos k_x+\cos k_y+\cos k_z$ in three ( $\varepsilon$ the hopping amplitude) is the nearest-neighbor hopping for the two spin species, and $V'=2W\sum_{\mathbf r}(n^{a}_{\mathbf r}-\tfrac12)(n^{b}_{\mathbf r}-\tfrac12)$ is the on-site interaction (equivalent to the standard Hubbard model with $U=2W$ ). Call an eigenstate of $H'$ W-independent if its eigenvector does not depend on the interaction strength $W$ , equivalently if it is a simultaneous eigenstate of $T'$ and $V'$ . The problem is to complete the SO(4)-based exact characterization of this model in the general, non-integrable case of two and higher dimensions:

(1) Classify all W-independent eigenstates. What is known: the Yang-Zhang family — multiplets generated by the eta-pairing raising operator $\eta^{+}$ and the spin raising operator $\zeta^{+}$ (the particle-hole conjugate of $\eta^{+}$ ) from fully spin-polarized free-fermion base states, $2^{M-2}(M^{2}+3M+4)$ states in total — exhausts the SO(4) sector $j+j'=M/2$ in every dimension, where $j$ and $j'$ are the pseudospin and spin angular momenta labelling the multiplet under the two SU(2) algebras whose raising operators are $\eta^{+}$ and $\zeta^{+}$ ; and the Ye-Lin families of sublattice Slater determinants (free-fermion determinants with the two spin species confined to the two complementary checkerboard sublattices, so that no site is doubly occupied) and submaximal-spin two-particle states supply interaction-independent eigenstates in sectors with $j+j'<M/2$ . What is open: determine every SO(4) sector $\{j,j'\}$ with $j+j'<M/2$ that contains W-independent eigenstates, construct all such eigenstates explicitly — that is, either prove that the known families jointly exhaust the W-independent spectrum, or exhibit and classify the further families that complete it. (The original Yang-Zhang conjecture that their family alone is exhaustive is false, refuted by the constructions cited in refs 9 and 10, and is not part of the target.)

(2) Determine the ordered region R. For the thermodynamic-limit lowest energy per site $f(W,\rho_a,\rho_b)=\lim_{M\to\infty}M^{-1}E_{0}(W,N_a,N_b)$ , determine the shape and extent of the minimizing region $R=\{(\rho_a,\rho_b): f(W,\rho_a,\rho_b)=f(W,\tfrac12,\tfrac12)\}$ as a function of the interaction strength $W$ , in particular the phase boundary at which the lowest states acquire the maximal-pseudospin structure, $j^{2}-j_{z}^{2}=O(M^{2})$ , that implies particle-particle off-diagonal long-range order (eta-pairing superconductivity, with flux quantization $ch/2e$ ). The determination must be consistent with the proved convexity and symmetry constraints and with the rigorous special cases at attractive interaction.

An explicit, proved classification of the W-independent spectrum is sought — families with wavefunctions, SO(4) and momentum quantum numbers, energies as functions of $W$ , and a proof of exhaustiveness or a proof that specifically identified additional families complete it — along with a determination of $R$ by rigorous inclusions, exclusions and boundary characterization. The one-dimensional case, where the SO(4)-extended Bethe ansatz already classifies the full spectrum, serves as a consistency boundary case rather than as the target.

References

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