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arXiv:2006.01730 · Nuclear Physics B

Integrability of spin- 12 fermions with charge pairing and Hubbard interaction

Marcio Jose Martins

Published Jul 7, 2020• 0 citations

Abstract

In this paper we study the exact solution of a one-dimensional model of spin-12 electrons composed by a nearest-neighbor triplet pairing term and the on-site Hubbard interaction. We argue that this model admits a Bethe ansatz solution through a mapping to a Hubbard chain with imaginary kinetic hopping terms. The Bethe equations are similar to that found by Lieb and Wu [3] but with additional twist phases which are dependent on the ring size. We have studied the spectrum of the model with repulsive interaction by exact diagonalization and through the Bethe equations for large lattice sizes. One feature of the model is that it is possible to define the charge gap for even and odd lattice sites and both converge to the same value in the infinite size limit. We analyze the finite-size corrections to the low-lying spin excitations and argue that they are equivalent to that of the spin-12 isotropic Heisenberg model with a boundary twist depending on the lattice parity. We present the classical statistical mechanics model whose transfer matrix commutes with the model Hamiltonian. To this end we have used the construction employed by Shastry [6], [7] for the Hubbard model. In our case, however, the building block is a free-fermion eight-vertex model with a particular null weight.

DOI: 10.1016/j.nuclphysb.2020.115109

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In this paper we study the exact solution of a one-dimensional model of spin-12 electrons composed by a nearest-neighbor triplet pairing term and the on-site Hubbard interaction. We argue that this model admits a Bethe ansatz solution through a mapping to a Hubbard chain with imaginary kinetic hopping terms. The Bethe equations are similar to that found by Lieb and Wu [3] but with additional twist phases which are dependent on the ring size. We have studied the spectrum of the model with repulsive interaction by exact diagonalization and through the Bethe equations for large lattice sizes. One feature of the model is that it is possible to define the charge gap for even and odd lattice sites and both converge to the same value in the infinite size limit. We analyze the finite-size corrections to the low-lying spin excitations and argue that they are equivalent to that of the spin-12 isotropic Heisenberg model with a boundary twist depending on the lattice parity. We present the classical statistical mechanics model whose transfer matrix commutes with the model Hamiltonian. To this end we have used the construction employed by Shastry [6], [7] for the Hubbard model. In our case, however, the building block is a free-fermion eight-vertex model with a particular null weight.

In this work, we study two models of strongly correlated fermions: Hubbard and FalicovKimball. We present a proof of the existence of real and ordered solutions to the nested Bethe Ansatz equations for the one dimensional Hubbard model on a finite lattice, with periodic boundary conditions. The existence of a continuous set of solutions extending from any U > 0 to U = ∞ is also shown, where U is the strength of the interaction between fermions at the same site. We use this continuity property, combined with the proof that the norm of the wavefunction obtained with the generalized Bethe Ansatz is not zero, to prove that the solution gives us the ground state of the finite system, as assumed by Lieb and Wu. For the absolute ground state at half-filling, we show that the solution converges to a distribution in the thermodynamic limit. This limit distribution satisfies the integral equations that led to the Lieb-Wu solution of the 1D Hubbard model. We also obtain a lower bound for the ground state energy of the Falicov-Kimball model. This bound is given by a bulk term, plus a term proportional to the boundary. A numerical value for the coefficient of the boundary term is determined. The explicit derivation is important in the proof of the conjecture of segregation of the two kinds of fermions in the Falicov-Kimball model, for sufficiently large interactions.

We calculate spin and charge excitation spectra of the one-dimensional (1D) quarter-filled Hubbard model with nearest-neighbor

t

and next-nearest-neighbor ${t}^{\ensuremath{'}}$ hopping integrals using the dynamical density-matrix renormalization group technique. We consider a case where

t

(>0)

is much smaller than ${t}^{\ensuremath{'}}$

(>0)

. First, we examine the accuracy of our method based on comparison between our result and exact noninteracting spectrum. Next, we investigate the spectra with the on-site Coulomb interaction. We find that the spin and charge excitation spectra are essentially the same as those of the 1D quarter-filled Hubbard (and $t\text{\ensuremath{-}}J$) model for the two 1D chains along the hopping integral ${t}^{\ensuremath{'}}$. However, the hopping integral

t

$(<{t}^{\ensuremath{'}})$ plays a crucial role in the short-range correlations and low-energy excitaions; ferromagnetic correlation between electrons on neighboring sites is enhanced and pairing correlation between the electrons is induced. Consequently, a spin-triplet superconducting state may be derived.

The thermodynamic Bethe-ansatz equations for the one-dimensional Hubbard model are derived for an attractive on-site interaction of the electrons, extending in this way Takahashi's result for the repulsive Hubbard chain. The results are discussed in several special limits, e.g., interaction much larger than the hopping, U\ensuremath{\rightarrow}0 case, and the high- and low-temperature limits. The dominating states at low temperatures consist of spin-paired electrons (resembling Cooper pairs) and bound states of such pairs. The excitation spectrum and its properties are discussed in the context of superconductivity and resonant valence bonds.

The recently introduced concept of exact mean-field theories is applied to the extended simplified Hubbard model, which is a one-band extended Hubbard model (including nearest-neighbor interaction) where only the \ensuremath{\downarrow} electrons can hop. An exact mean-field Hamiltonian is derived for the model on a Bethe lattice in the limit of a large coordination number (Z\ensuremath{\rightarrow}\ensuremath{\infty}). The mean-field Hamiltonian is used to calculate the Green functions and the free energy at half-filling. This exact solution is analyzed analytically at small and large values of the on-site (U) and nearest-neighbor (V) interaction strengths. One finds that a phase transition occurs at sufficiently low temperatures for all U,V>0. The low-temperature phase is a charge-density wave for V>1/2U and a spin-density wave for V1/2U. Special attention is paid to (i) the critical temperature as a function of U and V, (ii) the order parameter as a function of temperature, and (iii) the density of states. The density of states shows that the low-temperature phase is an insulator for all U,V provided that V/U\ensuremath{\ne}1/2, while the high-temperature phase is a paramagnetic metal for all V if U2, and a paramagnetic insulator if U>2. Many detailed results are given.

By defining a graded global R-operator

\mathbb{R}_{ab}^{(2D,2S)}

that couples free-fermion structures and incorporates anisotropic Hubbard interactions while satisfying the Yang--Baxter equation, we construct a strictly solvable two-dimensional lattice model. We then build the layer-to-layer transfer matrix through a bidirectional-monodromy construction and prove the model's integrability via the associated global RTT relations. Using the nested algebraic Bethe ansatz, we obtain the exact eigenvalues of the transfer matrix and derive the corresponding first- and second-level Bethe equations. Finally, by taking the logarithmic derivative of the transfer matrix at the regular point, we recover explicitly a local Hamiltonian that features anisotropic hopping, an on-site Hubbard interaction, and orbital-coupling contributions.