The transfer matrix of the symmetric 16-vertex model

This chapter extends the algebraic Bethe ansatz to the quantum Tavis–Cummings model, an N atom generalization of the Jaynes–Cummings model to describe the strong interaction between light and quantum matter. In the case of the quantum Tavis–Cum- mings model there is no underlying vertex model to suggest the constituent building blocks of the algebraic Bethe ansatz approach, e.g.like the L-matrix or ultimately the transfer matrix. The algebraic Bethe ansatz is then first applied to the Tavis–Cummings Hamiltonian with an added Stark term using a conjecture for the transfer matrix. The original Tavis–Cummings model and its algebraic Bethe ansatz are obtained in the limit of vanishing Stark term, which requires considerable care.

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THE 8V CSOS MODEL AND THE sl2 LOOP ALGEBRA SYMMETRY OF THE SIX-VERTEX MODEL AT ROOTS OF UNITY

International Journal of Modern Physics B ◽

10.1142/s0217979202011615 ◽

2002 ◽

Vol 16 (14n15) ◽

pp. 1899-1905 ◽

Cited By ~ 8

Author(s):

TETSUO DEGUCHI

Keyword(s):

Transfer Matrix ◽

Lattice Model ◽

Algebraic Method ◽

System Size ◽

Spin Operator ◽

Total Spin ◽

Coupling Constants ◽

Roots Of Unity ◽

Vertex Model ◽

Elliptic Quantum

We review an algebraic method for constructing degenerate eigenvectors of the transfer matrix of the eight-vertex Cyclic Solid-on-Solid lattice model (8V CSOS model), where the degeneracy increases exponentially with respect to the system size. We consider the elliptic quantum group Eτ,η(sl2) at the discrete coupling constants: 2N η = m1 + im2τ, where N, m1 and m2 are integers. Then we show that degenerate eigenvectors of the transfer matrix of the six-vertex model at roots of unity in the sector SZ ≡ 0 ( mod N) are derived from those of the 8V CSOS model, through the trigonometric limit. They are associated with the complete N strings. From the result we see that the dimension of a given degenerate eigenspace in the sector SZ ≡ 0 ( mod N) of the six-vertex model at Nth roots of unity is given by [Formula: see text], where [Formula: see text] is the maximal value of the total spin operator SZ in the degenerate eigenspace.

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Solution of Eight-vertex Lattice Model Without Elliptic Functions

Australian Journal of Physics ◽

10.1071/ph740433 ◽

1974 ◽

Vol 27 (4) ◽

pp. 433 ◽

Cited By ~ 1

Author(s):

Kailash Kumar

Keyword(s):

Eigenvalue Problem ◽

Transfer Matrix ◽

Elliptic Functions ◽

Point Of View ◽

Vertex Model ◽

Type Problem ◽

Matrix Eigenvalue Problem ◽

Lattice Statistical Mechanics ◽

Algebraic Operations ◽

Functional Matrix

Baxter's method of solving the eight-vertex model in lattice statistical mechanics is examined from an elementary point of view. It is shown that the algebraic operations in the method can be carried out without recourse to elliptic functions. These include: construction of certain subspaces invariant uti.der the action of the transfer matrix; reduction of the transfer matrix eigenvalue problem to an equivalent ice-type problem and construction of certain matrices which commute with the transfer matrix and satisfy a functional matrix equation.

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Six-Vertex Model

Models of Quantum Matter ◽

10.1093/oso/9780199678839.003.0011 ◽

2019 ◽

pp. 454-473

Author(s):

Hans-Peter Eckle

Keyword(s):

Bethe Ansatz ◽

Transfer Matrix ◽

Spin Chain ◽

Quantum Spin ◽

Square Lattice ◽

Vertex Model ◽

Quantum Spin Chain ◽

Special Value ◽

Quantum Operators ◽

The One

This chapter considers the special case of the six-vertex model on a square lattice using a trigonometric parameterization of the vertex weights. It demonstrates how, by exploiting the Yang-Baxter relations, the six-vertex model is diagonalized and the Bethe ansatz equations are derived. The Hamiltonian of the Heisenberg quantum spin chain is obtained from the transfer matrix for a special value of the spectral parameter together with an infinite set of further conserved quantum operators. By the diagonalization of the transfer matrix the exact solution of the one-dimensional quantum spin chain Hamiltonian has automatically also been obtained, which is given by the same Bethe ansatz equations.

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