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arXiv:0810.4220 · Journal of Physics A Mathematical and Theoretical

A vertex operator approach for correlation functions of Belavin's (ℤ/nℤ)-symmetric model

Yas-Hiro Quano

Published Apr 1, 2009• 4 citations
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Abstract

Belavin's (ℤ/ n ℤ)-symmetric model is considered on the basis of bosonization of vertex operators in the A (1) n −1 model and vertex–face transformation. The corner transfer matrix (CTM) Hamiltonian of the (ℤ/ n ℤ)-symmetric model and tail operators are expressed in terms of bosonized vertex operators in the A (1) n −1 model. Correlation functions of the (ℤ/ n ℤ)-symmetric model can be obtained by using these objects, in principle. In particular, we calculate spontaneous polarization, which reproduces the result we obtained in 1993.

DOI
10.1088/1751-8113/42/16/165211

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Belavin's (ℤ/ n ℤ)-symmetric model is considered on the basis of bosonization of vertex operators in the A (1) n −1 model and vertex–face transformation. The corner transfer matrix (CTM) Hamiltonian of the (ℤ/ n ℤ)-symmetric model and tail operators are expressed in terms of bosonized vertex operators in the A (1) n −1 model. Correlation functions of the (ℤ/ n ℤ)-symmetric model can be obtained by using these objects, in principle. In particular, we calculate spontaneous polarization, which reproduces the result we obtained in 1993.
Belavin's n -symmetric elliptic model with boundary reflection is considered on the basis of the boundary CTM bootstrap. We find non-diagonal K -matrices for n >2 that satisfy the reflection equation (boundary Yang-Baxter equation), and also find non-diagonal Boltzmann weights for the A (1) n -1 -face model even for n ⩾2. We derive difference equations of the quantum Knizhnik-Zamolodchikov type for correlation functions of the boundary model. The boundary spontaneous polarization is obtained by solving the simplest difference equations in the case of the free boundary condition. The resulting quantity is the square of the spontaneous polarization for the bulk n -symmetric model, up to a phase factor.
Eigenstates of the corner transfer matrix (CTM) for the six-vertex model are constructed and their relation to the Bethe ansatz eigenstates of the XXZ Hamiltonian is discussed. In the ferromagnetic regime ( Delta >1), eigenstates with any finite number of overturned arrows are constructed. They are related to the Bethe ansatz states by Fourier transformation over spectral (rapidity) variables. This exhibits the role of the CTM as a lattice boost operator. For the antiferromagnetic regime, the relation between CTM eigenstates and those of the XXZ Hamiltonian is complicated by the filling of the sea and associated Bethe ansatz integral equations. For this case, the authors construct CTM eigenstates by first showing that the Hamiltonian ground state is also an eigenstate of the CTM. Other CTM eigenstates are constructed from excited eigenstates of the Hamiltonian by Fourier transforming over the rapidity of each dressed excitation. They discuss the relationship between CTM eigenstates and a Heisenberg algebra of bosonic oscillators.
A free field representation for the type III vertex operators and the corner transfer matrices of the eight-vertex model is proposed. The construction uses the vertex-face correspondence, which makes it possible to expresses correlation functions of the eight-vertex model in terms of correlation functions of the SOS model with a nonlocal insertion. This new nonlocal insertion admits of a free field representation in terms of Lukyanov's screening operator. The spectrum of the corner transfer matrix and the Baxter-Kelland formula for the average staggered polarization have been reproduced.