Transfer matrix spectrum for the finite-width Ising model with adjustable boundary conditions: Exact solution

1989 ◽  
Vol 56 (5-6) ◽  
pp. 563-587 ◽  
Author(s):  
D. B. Abraham ◽  
L. F. Ko ◽  
N. M. Švrakić
Keyword(s):  
Exact Solution ◽  
Ising Model ◽  
Boundary Conditions ◽  
Transfer Matrix ◽  
Finite Width ◽  
Transfer Matrix Spectrum ◽  
Matrix Spectrum

scholarly journals Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra II

2018 ◽  
Vol 5 (3) ◽  
Author(s):  
Jean Michel Maillet ◽  
Giuliano Niccoli ◽  
Baptiste Pezelier
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Transfer Matrix ◽  
Separation Of Variables ◽  
General Boundary ◽  
Triangular Form ◽  
Transfer Matrix Spectrum ◽  
Lax Operator ◽  
Matrix Spectrum ◽  
Reflection Algebra

This article is a direct continuation of where we begun the study of the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazhanov-Stroganov Lax operator. There we addressed this problem for the case where one of the KK-matrices describing the boundary conditions is triangular. In the present article we consider the most general integrable boundary conditions, namely the most general boundary KK-matrices satisfying the reflection equation. The spectral analysis is developed by implementing the method of Separation of Variables (SoV). We first design a suitable gauge transformation that enable us to put into correspondence the spectral problem for the most general boundary conditions with another one having one boundary KK-matrix in a triangular form. In these settings the SoV resolution can be obtained along an extension of the method described in . The transfer matrix spectrum is then completely characterized in terms of the set of solutions to a discrete system of polynomial equations in a given class of functions and equivalently as the set of solutions to an analogue of Baxter’s T-Q functional equation. We further describe scalar product properties of the separate states including eigenstates of the transfer matrix.

scholarly journals Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra I

2017 ◽  
Vol 2 (1) ◽  
Author(s):  
Jean Michel Maillet ◽  
Giuliano Niccoli ◽  
Baptiste Pezelier
Keyword(s):  
Spectral Analysis ◽  
Boundary Conditions ◽  
Transfer Matrix ◽  
Separation Of Variables ◽  
Open Boundary ◽  
Open Boundary Conditions ◽  
Transfer Matrix Spectrum ◽  
Lax Operator ◽  
Matrix Spectrum ◽  
Reflection Algebra

We study the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazanov-Stroganov Lax operator. The results apply as well to the spectral analysis of the lattice sine-Gordon model with integrable open boundary conditions. This spectral analysis is developed by implementing the method of separation of variables (SoV). The transfer matrix spectrum (both eigenvalues and eigenstates) is completely characterized in terms of the set of solutions to a discrete system of polynomial equations in a given class of functions. Moreover, we prove an equivalent characterization as the set of solutions to a Baxter’s like T-Q functional equation and rewrite the transfer matrix eigenstates in an algebraic Bethe ansatz form. In order to explain our method in a simple case, the present paper is restricted to representations containing one constraint on the boundary parameters and on the parameters of the Bazanov-Stroganov Lax operator. In a next article, some more technical tools (like Baxter’s gauge transformations) will be introduced to extend our approach to general integrable boundary conditions.

scholarly journals Exact solution of the critical Ising model with special toroidal boundary conditions

2017 ◽  
Vol 96 (6) ◽  
Author(s):  
Armen Poghosyan ◽  
Nickolay Izmailian ◽  
Ralph Kenna
Keyword(s):  
Exact Solution ◽  
Ising Model ◽  
Boundary Conditions

scholarly journals FLUCTUATING INTERFACES, SURFACE TENSION, AND CAPILLARY WAVES: AN INTRODUCTION

1992 ◽  
Vol 03 (05) ◽  
pp. 857-877 ◽  
Author(s):  
VLADIMIR PRIVMAN
Keyword(s):  
Surface Tension ◽  
Transfer Matrix ◽  
Lattice Models ◽  
Wave Model ◽  
Capillary Waves ◽  
Roughening Transition ◽  
Transfer Matrix Spectrum ◽  
Model Transfer ◽  
Matrix Spectrum ◽  
Long Cylinder

We present an introduction to modern theories of interfacial fluctuations and the associated interfacial parameters: surface tension and surface stiffness, as well as their interpretation within the capillary wave model. Transfer matrix spectrum properties due to fluctuation of an interface in a long-cylinder geometry are reviewed. The roughening transition and properties of rigid interfaces below the roughening temperature in 3d lattice models are surveyed with emphasis on differences in fluctuations and transfer matrix spectral properties of rigid vs. rough interfaces.

scholarly journals On quantum separation of variables beyond fundamental representations

2021 ◽  
Vol 10 (2) ◽  
Author(s):  
Jean Michel Maillet ◽  
Giuliano Niccoli
Keyword(s):  
Functional Equation ◽  
Transfer Matrix ◽  
Separation Of Variables ◽  
Operator Family ◽  
Transfer Matrices ◽  
Conserved Charges ◽  
Transfer Matrix Spectrum ◽  
Lax Operator ◽  
The One ◽  
Matrix Spectrum

We describe the extension, beyond fundamental representations of the Yang-Baxter algebra, of our new construction of separation of variables bases for quantum integrable lattice models. The key idea underlying our approach is to use the commuting conserved charges of the quantum integrable models to generate bases in which their spectral problem is separated, i.e. in which the wave functions are factorized in terms of specific solutions of a functional equation. For the so-called “non-fundamental” models we construct two different types of SoV bases. The first is given from the fundamental quantum Lax operator having isomorphic auxiliary and quantum spaces and that can be obtained by fusion of the original quantum Lax operator. The construction essentially follows the one we used previously for fundamental models and allows us to derive the simplicity and diagonalizability of the transfer matrix spectrum. Then, starting from the original quantum Lax operator and using the full tower of the fused transfer matrices, we introduce a second type of SoV bases for which the proof of the separation of the transfer matrix spectrum is naturally derived. We show that, under some special choice, this second type of SoV bases coincides with the one associated to Sklyanin’s approach. Moreover, we derive the finite difference type (quantum spectral curve) functional equation and the set of its solutions defining the complete transfer matrix spectrum. This is explicitly implemented for the integrable quantum models associated to the higher spin representations of the general quasi-periodic Y(gl_{2})Y(gl2) Yang-Baxter algebra. Our SoV approach also leads to the construction of a QQ-operator in terms of the fused transfer matrices. Finally, we show that the QQ-operator family can be equivalently used as the family of commuting conserved charges enabling to construct our SoV bases.

Transfer-Matrix Spectrum for Systems with Interfaces

1989 ◽  
Vol 62 (6) ◽  
pp. 633-636 ◽  
Author(s):  
V. Privman ◽  
N. M. Švrakić
Keyword(s):  
Transfer Matrix ◽  
Transfer Matrix Spectrum ◽  
Matrix Spectrum
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