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The Ladder Operator Approach to Constructing Conserved Operators in Integrable One-Dimensional Lattice Models

M. C. Takizawa

Published Jan 1, 2005• 2 citations
Publisher

Abstract

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.

DOI
10.14264/107294

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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.
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.
Solutions of the classical Yang–Baxter equation provide a systematic method to construct integrable quantum systems in an algebraic manner. A Lie algebra can be associated with any solution of the classical Yang–Baxter equation, from which commuting transfer matrices may be constructed. This procedure is reviewed, specifically for solutions without skew-symmetry. A particular solution with an exotic symmetry is identified, which is not obtained as a limiting expansion of the usual Yang–Baxter equation. This solution facilitates the construction of commuting transfer matrices which will be used to establish the integrability of a multi-species boson tunnelling model. The model generalises the well-known two-site Bose–Hubbard model, to which it reduces in the one-species limit. Due to the lack of an apparent reference state, application of the algebraic Bethe Ansatz to solve the model is prohibitive. Instead, the Bethe Ansatz solution is obtained by the use of operator identities and tensor product decompositions.
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.
The generalization of the Yang-Baxter equations (YBE) in the presence of Z_2 grading along both chain and time directions is presented and an integrable model of t-J type with staggered disposition along a chain of shifts of the spectral parameter is constructed. The Hamiltonian of the model is computed in fermionic formulation. It involves three neighbour site interactions and therefore can be considered as a zig-zag ladder model. The Algebraic Bethe Ansatz technique is applied and the eigenstates, along with eigenvalues of the transfer matrix of the model are found. In the thermodynamic limit, the lowest energy of the model is formed by the quarter filling of the states by fermions instead of usual half filling.
The generalization of the Yang-Baxter equations (YBE) in the presence of Z2 grading along both chain and time directions is presented and an integrable model of t − J type with staggered disposition along a chain of shifts of the spectral parameter is constructed. The Hamiltonian of the model is computed in fermionic formulation. It involves three neighbour site interactions and therefore can be considered as a zig-zag ladder model. The Algebraic Bethe Ansatz technique is applied and the eigenstates, along with eigenvalues of the transfer matrix of the model are found. In the thermodynamic limit, the lowest energy of the model is formed by the quarter filling of the states by fermions instead of usual half filling. 1