Journal of Physics A Mathematical and General

The graded Yang - Baxter relation for the one-dimensional Bariev chain of fermions

Huan-Qiang Zhou

Published Sep 7, 1996• 12 citations

Abstract

The graded version of quantum Yang - Baxter relation for the one-dimensional Bariev chain of fermions is formulated. From this relation the commutativity of the transfer matrices for different values of the spectral parameter follows. Our result is consistent with the applicability of the traditional Bethe ansatz method. Further, the first non-trivial conserved current next to the Hamiltonian is constructed.

DOI: 10.1088/0305-4470/29/17/022

Related papers

Integrability of the one-dimensional Bariev model

Masahiro Shiroishi, Miki Wadati

We investigate the exact integrability of the one-dimensional (1D) Bariev model in the framework of the quantum inverse scattering method (QISM). Using the Jordan - Wigner transformation, the 1D Bariev model can be regarded as a coupled spin model. We construct the higher conserved currents which commute with the Hamiltonian. The explicit form of the conserved currents helps us to infer the L-operator of the 1D Bariev model. From the L-operator, we construct a transfer matrix which is a generating function of the conserved currents. We also find the corresponding R-matrix which satisfies the Yang - Baxter relation. Thus the exact integrability of the 1D Bariev model is established. The R-matrix does not have the `difference property' for the spectral parameter, as in the case of the 1D Hubbard model. We also provide the Lax representation and the fermionic formulation of the Yang - Baxter relation.

Journal of Physics A Mathematical and GeneralFeb 21, 199716 citations

The Yang–Baxter integrability of the critical Ising chain

Akash Sinha, T. Clark Justin, Pramod Padmanabhan +1 more

Abstract We demonstrate that the one-dimensional critical transverse field Ising model (TFIM) is Yang–Baxter integrable. This is achieved by constructing commuting transfer matrices derived from an R-matrix that satisfies the Yang–Baxter equation with additive spectral parameters. The R-matrix is nonlocal because it is expressed in terms of Majorana fermions. It is also non-regular. Nevertheless, we show that the quantum inverse scattering method can still be suitably adapted. Using the boost operator method, we recursively obtain the conserved quantities in the infinite volume. Among the conserved charges, we identify the Kramers–Wannier duality and other non-invertible symmetries for the periodic TFIM.

arXiv:2506.03668Journal of Statistical Mechanics Theory and ExperimentOct 1, 20253 citations

Yang-Baxter integrable Lindblad equations

Aleksandra A. Ziolkowska, Fabian H. L. Essler

We consider Lindblad equations for one dimensional fermionic models and quantum spin chains. By employing a (graded) super-operator formalism we identify a number of Lindblad equations than can be mapped onto non-Hermitian interacting Yang-Baxter integrable models. Employing Bethe Ansatz techniques we show that the late-time dynamics of some of these models is diffusive.

arXiv:1911.04883SciPost PhysicsMar 19, 202061 citations

A first course in the Yang–Baxter equation

D Karanth, D Richmond, Jeffrey Robert Schmidt

A crash course on the Yang–Baxter equation and its applications in the statistical mechanics of lattice vertex models is presented. Using the simple example of a one-dimensional lattice gas, the basic terminology and standard mathematical procedures of statistical mechanics are illustrated. The coproduct notation is introduced via a discussion of the two-dimensional dimer model, and the transfer matrix formulation is elaborated further. The algebraic Bethe Ansatz (ABA) is introduced in the context of the ice model, leading to the Yang–Baxter equations. The motivation for introducing the ABA is developed, emphasizing the similarity to the quantum oscillator problem, in features such as the generation of “excited states”, and the determination of the eigenvalue spectrum. Finally, these methods from the theory of lattice models are applied to the case of a quantum many-body problem, the Heisenberg chain. From the condition that the monodromy matrices satisfy the Yang–Baxter equation, a complete set of mutually commuting operators is derived, and the Heisenberg chain problem is completely solved.PACS Nos.: 02.20.Uw, 24.10.Cn, 87.10.Hk, 05.20.–y

Canadian Journal of PhysicsOct 1, 20080 citations

Yang-Baxter Relations for Spin Models and Fermion Models

Eugenio Olmedilla, Miki Wadati, Yasuhiro Akutsu

A systematic method to derive the Yang-Baxter relations for the fermion models which are equivalent to the solvable spin models is presented. The method is applied to prove the Yang-Baxter relation and the commutability of the transfer matrices for the one-dimensional Hubbard model.

Journal of the Physical Society of JapanJul 15, 198788 citations

Six-Vertex Model

Hans-Peter Eckle

Abstract This chapter considers the special case of the six-vertex model on a square lattice using a trigonometric parameterization of the vertex weights. It demonstrates how, by exploiting the Yang-Baxter relations, the six-vertex model is diagonalized and the Bethe ansatz equations are derived. The Hamiltonian of the Heisenberg quantum spin chain is obtained from the transfer matrix for a special value of the spectral parameter together with an infinite set of further conserved quantum operators. By the diagonalization of the transfer matrix the exact solution of the one-dimensional quantum spin chain Hamiltonian has automatically also been obtained, which is given by the same Bethe ansatz equations.

Jul 25, 201912 citations