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Journal of Physics A Mathematical and General

Integrability of the one-dimensional Bariev model

Masahiro Shiroishi, Miki Wadati

Published Feb 21, 1997• 16 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/30/4/014

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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.
The integrability of the one-dimensional (1D) fermion chain model is investigated in the framework of the Quantum Inverse Scattering Method (QISM). We introduce a new R-operator for the fermion chain model, which is expressed in terms of the fermion operators. The R-operator satisfies a new type of the Yang-Baxter relation with fermionic L-operator. We derive the fermionic Sutherland equation from the relation, which is equivalent to the fermionic Lax equation. It also provides a mathematical foundation of the boost operator approach for the fermion model. In fact, we obtain some higher conserved quantities of the fermion model using the boost operator.
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.
The 1D Hubbard model is shown to be an exactly integrable system. A "covering" model of 2D statistical mechanics which I proposed recently was shown to provide a one-parameter family of transfer matrices, commuting with the Hamiltonian of the Hubbard model. I show in this work that any two transfer matrices of a family commute mutually. At the root of the commutation relation is the ubiquitous Yang-Baxter factorization condition. The form of the RRR operator is displayed explicitly.
The SO (4) invariance of the transfer matrix for the one-dimensional Hubbard model is clarified from the viewpoint of the quantum inverse scattering method. We demonstrate the SO (4) symmetry by means of the fermionic L -operator and the fermionic R -matrix, which satisfy the graded Yang-Baxter relation. The transformation law of the fermionic L -operator under the SO (4) rotation is identified with a kind of gauge transformation, which determines the corresponding transformation of the fermionic creation and annihilation operators under the SO (4) rotation. The transfer matrix is confirmed to be invariant under the SO (4) rotation, which ensures the SO (4) invariance of the conserved currents including the Hamiltonian. Furthermore, we show that the representation of the higher conserved currents in terms of the Clifford algebra gives manifestly SO (4) invariant forms.
Abstract The quantum inverse scattering method is a scheme for solving integrable models in 1 + 1 dimensions, building on an R-matrix that satisfies the Yang–Baxter equation (YBE) and in terms of which one constructs a commuting family of transfer matrices. In the standard formulation, this R-matrix acts on a tensor product of vector spaces. Here, we relax this tensorial property and develop a framework for describing and analysing integrable models based on planar algebras, allowing non-separable R-operators satisfying generalised YBEs. We also re-evaluate the notion of integrals of motion and characterise when an (algebraic) transfer operator is polynomial in a single integral of motion. We refer to such models as polynomially integrable. In an eight-vertex model, we demonstrate that the corresponding transfer operator is polynomial in the natural Hamiltonian. In the Temperley–Lieb loop model with loop fugacity β ∈ C , we likewise find that, for all but finitely many β-values, the transfer operator is polynomial in the usual Hamiltonian element of the Temperley–Lieb algebra T L n ( β ) , at least for n ⩽ 17 . Moreover, we find that this model admits a second canonical Hamiltonian, and that this Hamiltonian also acts as a polynomial integrability generator for small n and all but finitely many β-values.