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Chinese Physics B

Exact ground state properties of the t – J model with open boundary conditions

Pei Sun, Yuanyuan Lei, Xiaotian Xu, Junpeng Cao, Tao Yang, Wen‐Li Yang

Published Jan 14, 2026• 0 citations
Publisher

Abstract

Abstract We develop a new method to study the ground state energy of the one-dimensional supersymmetric t – J model with open boundary conditions. The eigenvalues of the nested transfer matrix are characterized by the zero roots of corresponding polynomials instead of the T – Q relation and Bethe roots. The distribution of zero roots at the ground state is studied. We find that the zero roots form two-string pairs, finite pure real and pure imaginary boundary strings. Based on the distribution of zero roots, we obtain the ground state energy of the system in the thermodynamic limit.

DOI
10.1088/1674-1056/ae37fa

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Abstract We develop a new method to study the ground state energy of the one-dimensional supersymmetric t – J model with open boundary conditions. The eigenvalues of the nested transfer matrix are characterized by the zero roots of corresponding polynomials instead of the T – Q relation and Bethe roots. The distribution of zero roots at the ground state is studied. We find that the zero roots form two-string pairs, finite pure real and pure imaginary boundary strings. Based on the distribution of zero roots, we obtain the ground state energy of the system in the thermodynamic limit.
We propose a novel characterization of the exact solution of the one-dimensional supersymmetric t − J model with generic non-diagonal boundary conditions. Instead of using Bethe roots, we parameterize the eigenvalues of transfer matrix in the spin sector with their zero-roots. By constructing the t − W relation, we obtain the explicit forms of the eigenvalues of the system and the homogeneous Bethe ansatz equations. This scheme is universal and can be applied to other quantum integrable systems with or without U ( 1 ) symmetry.
We investigate the t–W scheme for the anti-ferromagnetic XXX spin chain under both periodic and open boundary conditions. We propose a new parametrization of the eigenvalues of the transfer matrix. Based on it, we obtain the exact solution of the system. By analyzing the distribution of zero roots at the ground state, we obtain the explicit expressions of the eigenfunctions of the transfer matrix and the associated W operator (see Eqs. (10) and (70)) in the thermodynamic limit. We find that the ratio of the quantum determinant with the eigenvalue of W operator for the ground state exhibits exponential decay behavior. Thus this fact ensures that the so-called inversion relation (the t–W relation without the W-term) can be used to study the ground state properties of quantum integrable systems with/without U(1)-symmetry in the thermodynamic limit.
A bstract The thermodynamic limits of the XYZ spin chain with periodic or twisted boundary conditions are studied. By using the technique of characterizing the eigenvalue of the transfer matrix by the T − Q relation and by the zeros of the associated polynomial, we obtain the constraints of the Bethe roots and the zeros for the eigenvalues. With the help of structure of Bethe roots, we obtain the distribution patterns of zeros. Based on them, the physical quantities such as the surface energy and excitation energy are calculated. We find that both of them depend on the parity of sites number due to the topological long-range Neel order on the Mobius manifold in the spin space. We also check our results with those obtaining by the density matrix renormalization group. The method provided in this paper can be applied to study the thermodynamic properties at the thermal equilibrium state with finite temperature.
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In this paper, we study the surface energy of the one-dimensional supersymmetric t-J model with unparallel boundary magnetic fields, which is a typical U(1)-symmetry broken quantum integrable strongly correlated electron system. It is shown that at the ground state, the contribution of inhomogeneous term in the Bethe ansatz solution of eigenvalues of transfer matrix satisfies the finite size scaling law L^β where β<0. Based on it, the physical quantities of the system in the thermodynamic limit are calculated. We obtain the patterns of Bethe roots and the analytical expressions of density of states, ground state energy and surface energy. We also find that there exist the stable boundary bound states if the boundary fields satisfy some constraints.