Communications in Theoretical Physics

A Free-Fermion Type Solution of Quantum Dynamical Yang–Baxter Equation

Ju Guo-Xing, Luo XuDong, Wang ShiKun, Ke Wu

Published Dec 15, 1999• 2 citations

Abstract

The symmetries of the quantum dynamical Yang-Baxter (QDYB) equation without spectral parameters for gl(2) are discussed. The classification of the six-vertex type solutions to the QDYB equation without spectral parameters is given and a free-fermion type solution is obtained.

DOI: 10.1088/0253-6102/32/4/557

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Communications in Theoretical Physics

A Free-Fermion Type Solution of Quantum Dynamical Yang–Baxter Equation

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Published Dec 15, 1999• 2 citations

Publisher

Abstract

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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.