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

2002 ◽  
Vol 16 (14n15) ◽  
pp. 1899-1905 ◽  
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.

scholarly journals Hard Squares with Negative Activity and Rhombus Tilings of the Plane

10.37236/1093 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Jakob Jonsson
Keyword(s):  
Partition Function ◽  
Statistical Mechanics ◽  
Simplicial Complex ◽  
Transfer Matrix ◽  
Euler Characteristic ◽  
Independent Sets ◽  
Roots Of Unity ◽  
Hard Squares ◽  
Vertex Set

Let $S_{m,n}$ be the graph on the vertex set ${\Bbb Z}_m \times {\Bbb Z}_n$ in which there is an edge between $(a,b)$ and $(c,d)$ if and only if either $(a,b) = (c,d\pm 1)$ or $(a,b) = (c \pm 1,d)$ modulo $(m,n)$. We present a formula for the Euler characteristic of the simplicial complex $\Sigma_{m,n}$ of independent sets in $S_{m,n}$. In particular, we show that the unreduced Euler characteristic of $\Sigma_{m,n}$ vanishes whenever $m$ and $n$ are coprime, thereby settling a conjecture in statistical mechanics due to Fendley, Schoutens and van Eerten. For general $m$ and $n$, we relate the Euler characteristic of $\Sigma_{m,n}$ to certain periodic rhombus tilings of the plane. Using this correspondence, we settle another conjecture due to Fendley et al., which states that all roots of $\det (xI-T_m)$ are roots of unity, where $T_m$ is a certain transfer matrix associated to $\{\Sigma_{m,n} : n \ge 1\}$. In the language of statistical mechanics, the reduced Euler characteristic of $\Sigma_{m,n}$ coincides with minus the partition function of the corresponding hard square model with activity $-1$.

Quantum Tavis–Cummings Model

2019 ◽  
pp. 474-488
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Bethe Ansatz ◽  
Transfer Matrix ◽  
Strong Interaction ◽  
Building Blocks ◽  
Vertex Model ◽  
Algebraic Bethe Ansatz ◽  
Quantum Matter ◽  
Ansatz Approach

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.

scholarly journals ADVANCED DYNAMIC ALGORITHMS FOR THE DECAY OF METASTABLE PHASES IN DISCRETE SPIN MODELS: BRIDGING DISPARATE TIME SCALES

1999 ◽  
Vol 10 (08) ◽  
pp. 1483-1493 ◽  
Author(s):  
M. A. NOVOTNY
Keyword(s):  
Monte Carlo ◽  
Markov Chains ◽  
Transfer Matrix ◽  
Time Scales ◽  
System Size ◽  
Metastable Phases ◽  
Spin Models ◽  
Dynamic Algorithms ◽  
Absorbing Markov Chains

An overview of advanced dynamical algorithms capable of spanning the widely disparate time scales that govern the decay of metastable phases in discrete spin models is presented. The algorithms discussed include constrained transfer-matrix, Monte Carlo with Absorbing Markov Chains (MCAMC), and projective dynamics (PD) methods. The strengths and weaknesses of each of these algorithms are discussed, with particular emphasis on identifying the parameter regimes (system size, temperature, and field) in which each algorithm works best.

The transfer matrix of the symmetric 16-vertex model

1973 ◽  
Vol 44 (6) ◽  
pp. 437-438 ◽  
Author(s):  
B.U. Felderhof
Keyword(s):  
Transfer Matrix ◽  
Vertex Model

scholarly journals FREQUENCY DEPENDENT EFFECTIVE CONDUCTIVITY OF TWO-DIMENSIONAL METAL–DIELECTRIC COMPOSITES

2005 ◽  
Vol 16 (06) ◽  
pp. 991-998 ◽  
Author(s):  
LOTFI ZEKRI
Keyword(s):  
Lattice Model ◽  
Effective Conductivity ◽  
System Size ◽  
Effective Medium ◽  
Two Dimensional ◽  
Metal Insulator ◽  
Frequency Dependent ◽  
Low Frequencies

We analyze a random resistor–inductor–capacitor (RLC) lattice model of two-dimensional metal–insulator composites. The results are compared with Bruggeman's and Landauer's Effective Medium Approximations where a discrepancy was observed for some frequency zones. Such a discrepancy is mainly caused by the strong conductivity fluctuations. Indeed, a two-branches distribution is observed for low frequencies. We show also by increasing the system size that at pc the so-called Drude peak vanishes; it increases for vanishing losses.

scholarly journals Solution of Eight-vertex Lattice Model Without Elliptic Functions

1974 ◽  
Vol 27 (4) ◽  
pp. 433 ◽  
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.

Six-Vertex Model

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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