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arXiv:cond-mat/0011368 · Physical Review Letters

Ladder Operator for the One-Dimensional Hubbard Model

Jon Links, Huan-Qiang Zhou, Ross H. McKenzie, M. D. Gould

Published May 28, 2001• 35 citations
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Abstract

The one-dimensional Hubbard model is integrable in the sense that it has an infinite family of conserved currents. We explicitly construct a ladder operator which can be used to iteratively generate all of the conserved current operators. This construction is different from that used for Lorentz invariant systems such as the Heisenberg model. The Hubbard model is not Lorentz invariant, due to the separation of spin and charge excitations. The ladder operator is obtained by a very general formalism which is applicable to any model that can be derived from a solution of the Yang-Baxter equation.

DOI
10.1103/physrevlett.86.5096

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The one-dimensional Hubbard model is integrable in the sense that it has an infinite family of conserved currents. We explicitly construct a ladder operator which can be used to iteratively generate all of the conserved current operators. This construction is different from that used for Lorentz invariant systems such as the Heisenberg model. The Hubbard model is not Lorentz invariant, due to the separation of spin and charge excitations. The ladder operator is obtained by a very general formalism which is applicable to any model that can be derived from a solution of the Yang-Baxter equation.
The theory of constructing integrable systems using solutions to the Yang-Baxter equation (YBE) is well established. The conserved operators can be obtained by a series expansion for the family of commuting transfer matrices. Another approach is to use the ladder operator, which permits a recursive method through repeated commutators, to obtain the conserved operators. The ladder operator for models where the solution of the YBE has the difference property is well known. Recently it has been shown that a ladder operator exists for the Hubbard model, which does not have the difference property, and the present work extends this to a general theory for the construction of the ladder operator for any integrable system obtained through the YBE, with the XYh model as an example. The existence of ladder operators for integrable models constructed from other variants of the YBE, such as closed chains with quantum algebra symmetry and the Heisenberg model with single site impurity, is also considered.
We propose a new type of the Yang-Baxter equation (YBE) and the decorated Yang-Baxter equation (DYBE). Those relations for the fermionic R-operator were introduced recently as a tool to treat the integrability of the fermion models. Using the YBE and the DYBE for the XX fermion model, we construct the fermionic R-operator for the one-dimensional (1D) Hubbard model. It gives another proof of the integrability of the 1D Hubbard model. Furthermore a new approach to the SO(4) symmetry of the 1D Hubbard model is discussed.
A detailed study of an S = ½ spin ladder model is given. The ladder consists of plaquettes formed by nearest-neighbor rungs with all possible SU (2)-invariant interactions. For properly chosen coupling constants, the model is shown to be integrable in the sense that the quantum Yang-Baxter equation holds and one has an infinite number of conserved quantities. The R -matrix and L -operator associated with the model Hamiltonian are given in a limiting case. It is shown that after a simple transformation, the model can be solved via a Bethe ansatz. The phase diagram of the ground state is exactly derived using the Bethe ansatz equation.