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 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.

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.

scholarly journals THE BROKEN PHASE OF THE TOPOLOGICAL σ MODEL

1995 ◽  
Vol 10 (11) ◽  
pp. 1553-1575 ◽  
Author(s):  
A.V. YUNG
Keyword(s):  
Moduli Space ◽  
Correlation Functions ◽  
Vacuum Energy ◽  
Brst Symmetry ◽  
Vacuum State ◽  
Target Space ◽  
Topological Phase ◽  
Coordinate Dependence ◽  
Vacuum States ◽  
Hole Geometry

The topological σ model with the semi-infinite cigar-like target space (black hole geometry) is considered. It is shown to possess unsuppressed instantons. The noncompactness of the moduli space of these instantons is responsible for an unusual physics. There is a stable vacuum state in which the vacuum energy is 0, and correlation functions are numbers; thus the model is in the topological phase. However, there are other vacuum states in which correlation functions show the coordinate dependence. Estimation of the vacuum energy indicates that it is nonzero. The states are interpreted as the ones with broken BRST symmetry.

scholarly journals Trace spectrum of 1D Transfer Matrices for wave propagation in layered media

2021 ◽  
Author(s):  
Joaquin Garcia-Suarez
Keyword(s):  
Wave Propagation ◽  
Transfer Matrix ◽  
Rational Design ◽  
Single Layer ◽  
Matrix Product ◽  
Matrix Formalism ◽  
Transfer Matrices ◽  
Sensitivity Studies ◽  
Media Applications ◽  
Transfer Matrix Formalism

The Transfer Matrix formalism is ubiquitous when considering wave propagation in various stratified media, applications ranging from Seismology to Quantum Mechanics. The relation between variables at two points in the laminate can be established via a matrix, termed (global) transfer matrix (product of "atomic'' single-layer matrices). As a matter of convenience, we focus on 1D phononic structures, but our derivation can be extended to other fields where the formalism applies. We present exact expressions for entries of the global propagator for N layers. When the layering corresponds to a representative repeated cell in an otherwise infinitely periodic medium, the trace of the cumulative multi-layer matrix is known to control the dispersion relation. We show how this trace has a discrete spectrum made up of distinct 2**(N-1) harmonics (not necessarily orthogonal to each other in any sense) which we characterize exactly both in terms of the periods that they contain and their amplitudes; we also show that the phase shift among harmonics is either zero or pi. This definite appraisal of the spectrum of the trace opens the path for rational design of band gaps, going beyond parametric or sensitivity studies.

scholarly journals MODIFIED TETRAHEDRON EQUATION AND RELATED 3D INTEGRABLE MODELS, II

1996 ◽  
Vol 11 (02) ◽  
pp. 313-327 ◽  
Author(s):  
H.E. BOOS
Keyword(s):  
General Model ◽  
Spin Model ◽  
Parametric Family ◽  
Layer Transfer ◽  
Transfer Matrices ◽  
Tetrahedron Equation ◽  
Cubic Lattice ◽  
Analytical Formulae ◽  
Integrable Spin ◽  
Free Parameters

This work is a continuation of Ref. 13, where the Boltzmann weights for the N-state integrable spin model on the cubic lattice has been obtained only numerically. In this paper we present the analytical formulae for this model in a particular case. Here the Boltzmann weights depend on six free parameters including the elliptic modulus. The obtained solution allows us to construct a two-parametric family of the commuting two-layer transfer matrices. The presented model is expected to be simpler for a further investigation in comparison with a more general model mentioned above.

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