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 Complete spectrum of quantum integrable lattice models associated to Y(gl(n)) by separation of variables

2019 ◽  
Vol 6 (6) ◽  
Author(s):  
Jean Michel Maillet ◽  
Giuliano Niccoli
Keyword(s):  
Transfer Matrix ◽  
Separation Of Variables ◽  
Lattice Models ◽  
Complete Characterization ◽  
Simple Spectrum ◽  
Transfer Matrices ◽  
Integrable Lattice ◽  
Transfer Matrix Spectrum ◽  
Integrable Lattice Models ◽  
Matrix Spectrum

We apply our new approach of quantum Separation of Variables (SoV) to the complete characterization of the transfer matrix spectrum of quantum integrable lattice models associated to \bm{gl_n}𝐠𝐥𝐧-invariant \bm{R}𝐑-matrices in the fundamental representations. We consider lattices with \bm{N}𝐍-sites and general quasi-periodic boundary conditions associated to an arbitrary twist matrix \bm{K}𝐊 having simple spectrum (but not necessarily diagonalizable). In our approach the SoV basis is constructed in an universal manner starting from the direct use of the conserved charges of the models, e.g. from the commuting family of transfer matrices. Using the integrable structure of the models, incarnated in the hierarchy of transfer matrices fusion relations, we prove that our SoV basis indeed separates the spectrum of the corresponding transfer matrices. Moreover, the combined use of the fusion rules, of the known analytic properties of the transfer matrices and of the SoV basis allows us to obtain the complete characterization of the transfer matrix spectrum and to prove its simplicity. Any transfer matrix eigenvalue is completely characterized as a solution of a so-called quantum spectral curve equation that we obtain as a difference functional equation of order \bm{n}𝐧. Namely, any eigenvalue satisfies this equation and any solution of this equation having prescribed properties that we give leads to an eigenvalue. We construct the associated eigenvector, unique up to normalization, of the transfer matrices by computing its decomposition on the SoV basis that is of a factorized form written in terms of the powers of the corresponding eigenvalues. Finally, if the twist matrix \bm{K}𝐊 is diagonalizable with simple spectrum we prove that the transfer matrix is also diagonalizable with simple spectrum. In that case, we give a construction of the Baxter \bm{Q}𝐐-operator and show that it satisfies a \bm{T}𝐓-\bm{Q}𝐐 equation of order \bm{n}𝐧, the quantum spectral curve equation, involving the hierarchy of the fused transfer matrices.

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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