Journal of Statistical Mechanics Theory and Experiment

The domain wall partition function for the Izergin–Korepin nineteen-vertex model at a root of unity

Alexandr Garbali

Published Mar 1, 2016• 4 citations

Abstract

Abstract We study the domain wall partition function Z N for the (Izergin–Korepin) integrable nineteen-vertex model on a square lattice of size N . Z N is a symmetric function of two sets of parameters: horizontal and vertical rapidities. For generic values of the parameter q we derive the recurrence relation for the domain wall partition function relating Z N +1 to , where P N is the proportionality factor in the recurrence, which is a polynomial symmetric in two sets of variables and . After setting the recurrence relation simplifies and we solve it in terms of a Jacobi–Trudi-like determinant of polynomials generated by P N .

DOI: 10.1088/1742-5468/2016/03/033112

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Journal of Statistical Mechanics Theory and Experiment

The domain wall partition function for the Izergin–Korepin nineteen-vertex model at a root of unity

Alexandr Garbali

Published Mar 1, 2016• 4 citations

Publisher

Abstract

DOI: 10.1088/1742-5468/2016/03/033112

Related papers

The domain wall partition function for the Izergin-Korepin 19-vertex model at a root of unity

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We study the domain wall partition function

Z_N

for the

U_q(A_2^{(2)})

(Izergin-Korepin) integrable

19

-vertex model on a square lattice of size

N

Z_N

is a symmetric function of two sets of parameters: horizontal

ζ_1,..,ζ_N

and vertical

z_1,..,z_N

rapidities. For generic values of the parameter

q

we derive the recurrence relation for the domain wall partition function relating

Z_{N+1}

P_N Z_N

, where

P_N

is the proportionality factor in the recurrence, which is a polynomial symmetric in two sets of variables

ζ_1,..,ζ_N

and

z_1,..,z_N

. After setting

q^3=-1

the recurrence relation simplifies and we solve it in terms of a Jacobi-Trudi-like determinant of polynomials generated by

P_N

U_q(sl_2)

R

-matrix and a special class of the triangular

K

-matrix. By using its graphical representation, we make the Izergin-Korepin analysis with the help of the Yang-Baxter relation and the reflection equation to give a characterization of the partition function. The explict form of the symmetric function representing the partition function is presented by showing that it satisfies all the required properties.