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arXiv:cond-mat/9704138

Exact solutions of 1-D Hubbard model with open boundary conditions and the conformal dimensions under boundary magnetic fields

Tetsuo Deguchi, Rui‐Hong Yue

Published Apr 16, 1997• 7 citations

Abstract

The Bethe ansatz equations of the 1-D Hubbard model under open boundary conditions are systematically derived by diagonalizing the inhomogeneous transfer matrix of the XXX model with open boundaries. Through the finite-size correction, we obtain the energy spectrum of the open chain and discuss the effects of boundary magnetic fields applied only at the edges of the chain. Several physical implications of the finite-size spectrum are discussed from the viewpoint of the boundary conformal field theories (BCFT); the conformal dimension of the spin excitation is quite sensitive to the boundary magnetic fields, and this sensitivity can be explained in terms of the $π/2$-phase shift of the BCFT. For several limiting cases such as for the half-filling case, the conformal dimensions are explicitly calculated.

DOI: 10.48550/arxiv.cond-mat/9704138

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The Bethe ansatz equations of the 1-D Hubbard model under open boundary conditions are systematically derived by diagonalizing the inhomogeneous transfer matrix of the XXX model with open boundaries. Through the finite-size correction, we obtain the energy spectrum of the open chain and discuss the effects of boundary magnetic fields applied only at the edges of the chain. Several physical implications of the finite-size spectrum are discussed from the viewpoint of the boundary conformal field theories (BCFT); the conformal dimension of the spin excitation is quite sensitive to the boundary magnetic fields, and this sensitivity can be explained in terms of the

π/2

-phase shift of the BCFT. For several limiting cases such as for the half-filling case, the conformal dimensions are explicitly calculated.

We use the Bethe ansatz solution for the one-dimensional Hubbard model with open boundary conditions and applied boundary fields to study the spectrum of bound states at the boundary. Depending on the strength of the boundary potentials, one finds that the true ground state contains a single charge or, for boundary potentials comparable with the Hubbard interaction, a pair of electrons in a bound state. If these are left unoccupied one finds holon and spinon bound states. We compute the finite size corrections to the low-lying energies in this system and use the predictions of boundary conformal field theory to study the exponents related to the orthogonality catastrophe.

We use the Bethe Ansatz solution for the one dimensional Hubbard model with open boundary conditions and applied boundary fields to study the spectrum of bound states at the boundary. Depending on the strength of the boundary potentials one finds that the true ground state contains a single charge or, for boundary potentials comparable to the Hubbard interaction, a pair of electrons in a bound state. If these are left unoccupied one finds holon and spinon bound states. We compute the finite size corrections to the low lying energies in this system and use the predictions of boundary conformal field theory to study the exponents related to the orthogonality catastrophe.

Using the Bethe ansatz we study the spectrum of boundary bound states of the SU ( N ) generalization of the one-dimensional Hubbard model with open boundaries and applied boundary potentials. The number of electrons bound by the boundary field is found from the surface contribution to the energy. Finite size corrections to the low-lying energies in this system and use of the predictions of boundary conformal field theory allows us to study the exponents related to the orthogonality catastrophe and absorption edges.

We study the spectrum of the integrable open XXX Heisenberg spin chain subject to non-diagonal boundary magnetic fields. The spectral problem for this model can be formulated in terms of functional equations obtained by separation of variables or, equivalently, from the fusion of transfer matrices. For generic boundary conditions the eigenvalues cannot be obtained from the solution of finitely many algebraic Bethe equations. Based on careful finite size studies of the analytic properties of the underlying hierarchy of transfer matrices we devise two approaches to analyze the functional equations. First we introduce a truncation method leading to Bethe-type equations determining the energy spectrum of the spin chain. In a second approach, the hierarchy of functional equations is mapped to an infinite system of nonlinear integral equations of TBA type. The two schemes have complementary ranges of applicability and facilitate an efficient numerical analysis for a wide range of boundary parameters. Some data are presented on the finite-size corrections to the energy of the state which evolves into the antiferromagnetic ground state in the limit of parallel boundary fields.