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arXiv:1311.0432 · Nuclear Physics B

Exact solution of the one-dimensional Hubbard model with arbitrary boundary magnetic fields

Yuanyuan Li, Junpeng Cao, Wen‐Li Yang, Kangjie Shi, Yupeng Wang

Published Dec 11, 2013• 36 citations
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Abstract

The one-dimensional Hubbard model with arbitrary boundary magnetic fields is solved exactly via the Bethe ansatz methods. With the coordinate Bethe ansatz in the charge sector, the second eigenvalue problem associated with the spin sector is constructed. It is shown that the second eigenvalue problem can be transformed into that of the inhomogeneous XXX spin chain with arbitrary boundary fields which can be solved via the off-diagonal Bethe ansatz method.

DOI
10.1016/j.nuclphysb.2013.12.004

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The one-dimensional Hubbard model with arbitrary boundary magnetic fields is solved exactly via the Bethe ansatz methods. With the coordinate Bethe ansatz in the charge sector, the second eigenvalue problem associated with the spin sector is constructed. It is shown that the second eigenvalue problem can be transformed into that of the inhomogeneous XXX spin chain with arbitrary boundary fields which can be solved via the off-diagonal Bethe ansatz method.
We propose a novel characterization of the exact solution of the one-dimensional Hubbard model with unparallel boundary magnetic fields. The non-diagonal boundary terms break the U(1) symmetry of the system and the number of electrons with fixed spin is not conserved. Thus it is difficult to study the physical quantities in the thermodynamic limit based on the previously obtained inhomogeneous T−Q relation and Bethe ansatz equations. In order to solve this problem, we propose the t−W scheme. The basic idea is that the eigenvalues of the transfer matrix can be expressed by its zero-roots instead of the usual Bethe roots. This method allows us to study thermodynamic properties of the system such as the ground state, surface energy, charge and spin excitations. The t−W scheme is universal and can be applied to the systems either with or without U(1) symmetry.
The exact solution of the one-dimensional super-symmetric t-J model under generic integrable boundary conditions is obtained via the Bethe ansatz methods. With the coordinate Bethe ansatz, the corresponding R-matrix and K-matrices are derived for the second eigenvalue problem associated with spin degrees of freedom. It is found that the second eigenvalue problem can be transformed to that of the transfer matrix of the inhomogeneous XXX spin chain, which allows us to obtain the spectrum of the Hamiltonian and the associated Bethe ansatz equations by the off-diagonal Bethe ansatz method.
The Bethe ansatz equations of the 1-D SU(3) Hubbard model are systematically derived by diagonalizing the inhomogeneous transfer matrix of the XXX model. We first derive the scattering matrix of the SU(3) Hubbard model through the coordinate Bethe ansatz method. Then, with the help quantum inverse scattering method we solve the nested transfer matrix and give the eigenvalues, the eigenvectors and the Bethe ansatz equations. Finally, we obtain the exactly analytic solution for the ground state. PACS:05.20.-y; 05.50.+q; 04.20.Jb; 03.65.Fd
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/2π/2π/2-phase shift of the BCFT. For several limiting cases such as for the half-filling case, the conformal dimensions are explicitly calculated.
By means of the Bethe-ansatz method, the one-dimensional Hubbard model with repulsive on-site coupling in an external magnetic field is solved exactly. It is found that the energy of the ground state depends on the magnetic flux \ensuremath{\varphi} through the ring formed by a periodic lattice chain, and the lowest-energy state corresponds to \ensuremath{\varphi}=n(hc/2e), with n an integer and -e the electron charge. The calculations illustrate that the strongly correlated Hubbard ring with periodic boundary conditions in its ground state is a superconducting ring. It can also be shown that the system possesses an internal SU(2)\ifmmode\times\else\texttimes\fi{}SU(2) symmetry.