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.

A NEW INTEGRABLE MODEL OF STRONGLY CORRELATED ELECTRONS WITH HUBBARD-LIKE INTERACTION

2001 ◽  
Vol 15 (06n07) ◽  
pp. 213-218
Author(s):  
XIANG-YU GE
Keyword(s):  
Bethe Ansatz ◽  
Integrable Model ◽  
Strongly Correlated Electrons ◽  
The Other ◽  
Competitive Interactions ◽  
Correlated Electrons ◽  
Strongly Correlated ◽  
Transfer Matrices ◽  
Ansatz Method ◽  
Commuting Transfer Matrices

A new completely integrable model of strongly correlated electrons is proposed which describes two competitive interactions: one is the correlated one-particle hopping, the other is the Hubbard-like interaction. The integrability follows from the fact that the Hamiltonian is derivable from a one-parameter family of commuting transfer matrices. The Bethe ansatz equations are derived by algebraic Bethe ansatz method.

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 Correlations and commuting transfer matrices in integrable unitary circuits

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
Pieter W. Claeys ◽  
Jonah Herzog-Arbeitman ◽  
Austen Lamacraft
Keyword(s):  
Correlation Functions ◽  
Exceptional Point ◽  
Baxter Equation ◽  
Matrix Formalism ◽  
Transfer Matrices ◽  
Local Conservation ◽  
Vacuum States ◽  
Integrable Spin ◽  
Commuting Transfer Matrices ◽  
Bethe Equations

We consider a unitary circuit where the underlying gates are chosen to be \check{R}Ř-matrices satisfying the Yang-Baxter equation and correlation functions can be expressed through a transfer matrix formalism. These transfer matrices are no longer Hermitian and differ from the ones guaranteeing local conservation laws, but remain mutually commuting at different values of the spectral parameter defining the circuit. Exact eigenstates can still be constructed as a Bethe ansatz, but while these transfer matrices are diagonalizable in the inhomogeneous case, the homogeneous limit corresponds to an exceptional point where multiple eigenstates coalesce and Jordan blocks appear. Remarkably, the complete set of (generalized) eigenstates is only obtained when taking into account a combinatorial number of nontrivial vacuum states. In all cases, the Bethe equations reduce to those of the integrable spin-1 chain and exhibit a global SU(2) symmetry, significantly reducing the total number of eigenstates required in the calculation of correlation functions. A similar construction is shown to hold for the calculation of out-of-time-order correlations.

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.

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