scholarly journals FUSION OF A-D-E LATTICE MODELS

1994 ◽  
Vol 08 (25n26) ◽  
pp. 3531-3577 ◽  
Author(s):  
YU-KUI ZHOU ◽  
PAUL A. PEARCE
Keyword(s):  
Ising Model ◽  
Potts Model ◽  
Functional Equations ◽  
Lattice Models ◽  
Transfer Matrices ◽  
The Face

Fusion hierarchies of A-D-E face models are constructed. The fused critical D, E and elliptic D models yield new solutions of the Yang-Baxter equations with bond variables on the edges of faces in addition to the spin variables on the corners. It is shown directly that the row transfer matrices of the fused models satisfy special functional equations. Intertwiners between the fused A-D-E models are constructed by fusing the cells that intertwine the elementary face weights. As an example, we calculate explicitly the fused 2×2 face weights of the 3-state Potts model associated with the D4 diagram as well as the fused intertwiner cells for the A5-D4 intertwiner. Remarkably, this 2×2 fusion yields the face weights of both the Ising model and 3-state CSOS models.

Free-fermion, checkerboard and Z -invariant lattice models in statistical mechanics

1986 ◽  
Vol 404 (1826) ◽  
pp. 1-33 ◽  
Keyword(s):  
Ising Model ◽  
Potts Model ◽  
Three Dimensional ◽  
Lattice Models ◽  
Square Lattice ◽  
Free Fermion ◽  
Local Correlations ◽  
Free Fermion Model ◽  
Zamolodchikov Model ◽  
Special Case

It is shown that the two-dimensional free fermion model is equivalent to a checkerboard Ising model, which is a special case of the general ‘ Z -invariant’ Ising model. Expressions are given for the partition function and local correlations in terms of those of the regular square lattice Ising model. Corresponding results are given for the self-dual Potts model, and the application of the methods to the three-dimensional Zamolodchikov model is discussed. The paper ends with a discussion of the critical and disorder surfaces of the checkerboard Potts model.

scholarly journals INTERTWINERS AND A-D-E LATTICE MODELS

1993 ◽  
Vol 07 (20n21) ◽  
pp. 3649-3705 ◽  
Author(s):  
PAUL A. PEARCE ◽  
YU-KUI ZHOU
Keyword(s):  
General Theory ◽  
Graphical Representation ◽  
Lattice Models ◽  
Finite System ◽  
Transfer Matrices ◽  
Adjacency Matrices ◽  
The Face

Intertwiners between A-D-E lattice models are presented and the general theory developed. The intertwiners are discussed at three levels: at the level of the adjacency matrices, at the level of the cell calculus intertwining the face algebras and at the level of the row transfer matrices. A convenient graphical representation of the intertwining cells is introduced. The utility of the intertwining relations in studying the spectra of the A-D-E models is emphasized. In particular, it is shown that the existence of an intertwiner implies that many eigenvalues of the A-D-E row transfer matrices are exactly in common for a finite system and, consequently, that the corresponding central charges and scaling dimensions can be identified.

scholarly journals Parallel family trees for transfer matrices in the Potts model

2015 ◽  
Vol 187 ◽  
pp. 55-71 ◽  
Author(s):  
Cristobal A. Navarro ◽  
Fabrizio Canfora ◽  
Nancy Hitschfeld ◽  
Gonzalo Navarro
Keyword(s):  
Potts Model ◽  
Transfer Matrices ◽  
Family Trees

scholarly journals GENERALIZED YANG-BAXTER EQUATION

1993 ◽  
Vol 08 (24) ◽  
pp. 2299-2309 ◽  
Author(s):  
R. M. KASHAEV ◽  
YU. G. STROGANOV
Keyword(s):  
Single Layer ◽  
Lattice Models ◽  
Baxter Equation ◽  
Explicit Solutions ◽  
Generalized Equations ◽  
Two Dimensional ◽  
Layer Transfer ◽  
Transfer Matrices ◽  
Dimensional Lattice ◽  
Potts Models

A generalization of the Yang-Baxter equation is proposed. It enables us to construct integrable two-dimensional lattice models with commuting two-layer transfer matrices, while single-layer ones are not necessarily commutative. Explicit solutions to the generalized equations are found. They are related with Boltzmann weights of the sl (3) chiral Potts models.

Transfer Matrix of the Two-dimensional Ising Model

2020 ◽  
pp. 211-234
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Ising Model ◽  
Critical Point ◽  
Transfer Matrix ◽  
Functional Equations ◽  
Baxter Equation ◽  
Matrix Formalism ◽  
Two Dimensional ◽  
R Matrix ◽  
Transfer Matrix Formalism ◽  
Lowest Eigenvalue

This chapter deals with the exact solution of the two-dimensional Ising model as it is achieved through the transfer matrix formalism. It discusses the crucial role played by the commutative properties of the transfer matrices, which lead to a functional equation for their eigenvalues. The exact free energy of the Ising model and its critical point can be identified by means of the lowest eigenvalue. The chapter covers Baxter's approach, the Yang–Baxter equation and its relation to the Boltzmann weights, the R-matrix, and discusses activity away from the critical point, the six-vertex model, as well as functional equations and symmetries.

scholarly journals TRACE MAPS, INVARIANTS, AND SOME OF THEIR APPLICATIONS

1993 ◽  
Vol 07 (06n07) ◽  
pp. 1527-1550 ◽  
Author(s):  
M. BAAKE ◽  
U. GRIMM ◽  
D. JOSEPH
Keyword(s):  
Ising Model ◽  
Algebraic Structure ◽  
Electronic Spectra ◽  
Transfer Matrices ◽  
Two Level Systems ◽  
Commuting Transfer Matrices ◽  
Trace Maps

Trace maps of two-letter substitution rules are investigated with special emphasis on the underlying algebraic structure and on the existence of invariants. We illustrate the results with the generalized Fibonacci chains and show that the well-known Fricke character I(x, y, z)=x2+y2+z2−2xyz−1 is not the only type of invariant that can occur. We discuss several physical applications to electronic spectra including the gap-labeling theorem, to kicked two-level systems, and to the classical 1D Ising model with non-commuting transfer matrices.

LATTICE ALGEBRAS AND THE HIDDEN SYMMETRY OF THE 2D ISING MODEL IN A MAGNETIC FIELD

1991 ◽  
Vol 06 (28) ◽  
pp. 5127-5153 ◽  
Author(s):  
DAN LEVY
Keyword(s):  
Magnetic Field ◽  
Ising Model ◽  
External Magnetic Field ◽  
Lattice Models ◽  
Natural Generalization ◽  
Xxz Model ◽  
Affine Hecke Algebra ◽  
Finite Dimensional ◽  
Hidden Symmetry ◽  
Boundary Terms

Lattice algebras are defined and used to study the symmetries of 2D lattice models. New and interesting examples of such algebras are provided by the affine Hecke algebra, owing to the possibility of constructing braid generators out of its generators. I propose an Ansatz for the braid generators and derive some solutions. A particular finite-dimensional quotient is shown to be a natural generalization of the Temperley-Lieb-Jones algebra. It is used to give a unified picture of known and unknown symmetries of the spin-½ xxz model with boundary terms. The Ising model in an external magnetic field is also a representation of this quotient.

CONFORMAL PROPERTIES OF CRITICAL ISING MODEL CORNER TRANSFER MATRICES

1990 ◽  
Vol 04 (05) ◽  
pp. 907-912
Author(s):  
Brian DAVIES ◽  
Paul A. PEARCE
Keyword(s):  
Ising Model ◽  
Central Charge ◽  
Finite Size ◽  
Summation Formula ◽  
Transfer Matrices ◽  
Corner Transfer Matrices ◽  
Free Boundary Conditions ◽  
Large N ◽  
Fermion Algebra ◽  
Virasoro Characters

The scaling spectra of finite-size Ising model corner transfer matrices (CTMs) are studied at criticality, using the fermion algebra. The low-lying eigenvalues collapse like 1/ log N for large N as predicted by conformal invariance. The shift in the largest eigenvalue is evaluated analytically using a generalized Euler-Maclaurin summation formula giving πc/6 log N with central charge c=1/2. The spectrum generating functions, for both fixed and free boundary conditions, are expressed simply in terms of the c=1/2 Virasoro characters χ∆(q) with modular parameter q= exp (−π/ log N) and conformal dimensions ∆=0, 1/2, 1/16.

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