Communications in Theoretical Physics

Lax Pairs for Integrable Six-Vertex Model with Color Parameters

Ju Guo-Xing

Published Mar 15, 2003• 0 citations

Abstract

Using the colored Yang–Baxter equation, the Lax pairs for the system with color parameters and periodic boundary conditions are derived. The explicit expressions of the Lax pairs for the six-vertex standard matrices of both the Baxter and free-Fermion types are given.

DOI: 10.1088/0253-6102/39/3/285

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Communications in Theoretical Physics

Lax Pairs for Integrable Six-Vertex Model with Color Parameters

Ju Guo-Xing

Published Mar 15, 2003• 0 citations

Publisher

Abstract

DOI: 10.1088/0253-6102/39/3/285

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The Six-Vertex Yang-Baxter Groupoid

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\mathfrak{G}

together with a map

π:\mathfrak{G}\to\operatorname{End}(V\otimes V)

for some vector space

V

such that the Yang-Baxter commutator

[[ π(u),π(w),π(v)]]=0

u,v\in\mathfrak{G}

are such that the groupoid composition

w=u\star v

is defined. An important role is played by an object map

Δ:\mathfrak{G}\to M

for some set

M

such that

Δ(u)=Δ(v')

Δ(w)=Δ(v)

and

Δ(w')=Δ(u')

, where

v\mapsto v'

is the groupoid inverse map. There are two main regimes of the six-vertex model: the free-fermionic point, and everything else. For the free-fermionic point, there exists a parametrized Yang-Baxter equation with a large parameter group

\operatorname{GL}(2)\times\operatorname{GL}(1)

. For non-free-fermionic six-vertex matrices, there are also well-known (group) parametrized Yang-Baxter equations, but these do not account for all possible interactions. Instead we will construct a groupoid parametrized Yang-Baxter equation that accounts for essentially all possible Yang-Baxter equations in the six-vertex model. We will also exhibit a separate groupoid for the five-vertex model. We will show how to construct solvable lattice models based on groupoid parametrized Yang-Baxter equations.