Corner transfer matrices for the Ising model

1988 ◽  
Vol 154 (1) ◽  
pp. 1-20 ◽  
Author(s):  
B. Davies
Keyword(s):  
Ising Model ◽  
Transfer Matrices ◽  
Corner Transfer Matrices

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.

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.

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