scholarly journals Separation of variables bases for integrable $gl_{\mathcal{M}|\mathcal{N}}$ and Hubbard models

2020 ◽  
Vol 9 (4) ◽  
Author(s):  
Jean Michel Maillet ◽  
Giuliano Niccoli ◽  
Louis Vignoli
Keyword(s):  
Boundary Conditions ◽  
Hubbard Model ◽  
Spectral Problem ◽  
Transfer Matrix ◽  
Separation Of Variables ◽  
Integrable Models ◽  
Simple Spectrum ◽  
Transfer Matrices ◽  
Matrix Eigenvalue ◽  
Twisted Boundary Conditions

We construct quantum Separation of Variables (SoV) bases for both the fundamental inhomogeneous % gl_{\mathcal{M}|\mathcal{N}}glℳ|𝒩 supersymmetric integrable models and for the inhomogeneous Hubbard model both defined with quasi-periodic twisted boundary conditions given by twist matrices having simple spectrum. The SoV bases are obtained by using the integrable structure of these quantum models, i.e. the associated commuting transfer matrices, following the general scheme introduced in [1]; namely, they are given by set of states generated by the multiple actions of the transfer matrices on a generic co-vector. The existence of such SoV bases implies that the corresponding transfer matrices have non-degenerate spectrum and that they are diagonalizable with simple spectrum if the twist matrices defining the quasi-periodic boundary conditions have that property. Moreover, in these SoV bases the resolution of the transfer matrix eigenvalue problem leads to the resolution of the full spectral problem, i.e. both eigenvalues and eigenvectors. Indeed, to any eigenvalue is associated the unique (up to a trivial overall normalization) eigenvector whose wave-function in the SoV bases is factorized into products of the corresponding transfer matrix eigenvalue computed on the spectrum of the separated variables. As an application, we characterize completely the transfer matrix spectrum in our SoV framework for the fundamental gl_{1|2}gl1|2 supersymmetric integrable model associated to a special class of twist matrices. From these results we also prove the completeness of the Bethe Ansatz for that case. The complete solution of the spectral problem for fundamental inhomogeneous gl_{\mathcal{M}|\mathcal{N}}glℳ|𝒩 supersymmetric integrable models and for the inhomogeneous Hubbard model under the general twisted boundary conditions will be addressed in a future publication.

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 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.

Flexural Vibration Response of Beams With General Boundary Conditions: A Transfer Matrix Approach

1991 ◽  
Author(s):  
T. Önsay
Keyword(s):  
Boundary Conditions ◽  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Vibration Response ◽  
General Boundary ◽  
Phase Variable ◽  
General Boundary Conditions ◽  
Transfer Matrices ◽  
Analytical Results

Abstract The wave-mode representation is utilized to obtain a more efficient form to the conventional transfer matrix method for bending vibrations of beams. The proposed improvement is based on a phase-variable canonical state representation of the equation governing the time-harmonic flexural vibrations of a beam. Transfer matrices are obtained for external forces, step-change of beam properties, intermediate supports and for boundaries. The transfer matrices are utilized to obtain the vibration response of a point-excited single-span beam with general boundary conditions. The general characteristic equation and the transfer mobility of a single-span beam are determined. The application of the analytical results are demonstrated on physical structures with different boundary conditions. A hybrid model is developed to incorporate measured impedance of nonideal boundaries into the transfer matrix method. The analytical results are found to be in excellent agreement with experimental measurements.

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.

scholarly journals On scalar products and form factors by Separation of Variables: the antiperiodic XXZ model

2021 ◽  
Author(s):  
Hao Pei ◽  
Veronique Terras
Keyword(s):  
Boundary Conditions ◽  
Transfer Matrix ◽  
Separation Of Variables ◽  
Form Factors ◽  
Periodic Case ◽  
Matrix Elements ◽  
Heisenberg Chain ◽  
Xxz Model ◽  
Matrix Eigenvalues ◽  
Scalar Products

Abstract We consider the XXZ spin-1/2 Heisenberg chain with antiperiodic boundary conditions. The inhomogeneous version of this model can be solved by Separation of Variables (SoV), and the eigenstates can be constructed in terms of Q-functions, solution of a Baxter TQ-equation, which have double periodicity compared to the periodic case. We compute in this framework the scalar products of a particular class of separate states which notably includes the eigenstates of the transfer matrix. We also compute the form factors of local spin operators, i.e. their matrix elements between two eigenstates of the transfer matrix. We show that these quantities admit determinant representations with rows and columns labelled by the roots of the Q-functions of the corresponding separate states, as in the periodic case, although the form of the determinant are here slightly different. We also propose alternative types of determinant representations written directly in terms of the transfer matrix eigenvalues.

scholarly journals On scalar products in higher rank quantum separation of variables

2020 ◽  
Vol 9 (6) ◽  
Author(s):  
Jean Michel Maillet ◽  
Giuliano Niccoli ◽  
Louis Vignoli
Keyword(s):  
Transfer Matrix ◽  
Separation Of Variables ◽  
Lattice Models ◽  
Simple Spectrum ◽  
Conserved Charges ◽  
Integrable Lattice ◽  
Rank One ◽  
Higher Rank ◽  
Left And Right ◽  
Scalar Products

Using the framework of the quantum separation of variables (SoV) for higher rank quantum integrable lattice models , we introduce some foundations to go beyond the obtained complete transfer matrix spectrum description, and open the way to the computation of matrix elements of local operators. This first amounts to obtain simple expressions for scalar products of the so-called separate states, that are transfer matrix eigenstates or some simple generalization of them. In the higher rank case, left and right SoV bases are expected to be pseudo-orthogonal, that is for a given SoV co-vector \langle\underline{\mathbf{h}}\rangle⟨𝐡̲⟩, there could be more than one non-vanishing overlap \langle{\underline{\mathbf{h}}}|{\underline{\mathbf{k}}}\rangle⟨𝐡̲|𝐤̲⟩ with the vectors |{\underline{\mathbf{k}}}\rangle|𝐤̲⟩ of the chosen right SoV basis. For simplicity, we describe our method to get these pseudo-orthogonality overlaps in the fundamental representations of the \mathcal{Y}(gl_3)𝒴(gl3) lattice model with NN sites, a case of rank 2. The non-zero couplings between the co-vector and vector SoV bases are exactly characterized. While the corresponding SoV-measure stays reasonably simple and of possible practical use, we address the problem of constructing left and right SoV bases which do satisfy standard orthogonality (by standard we mean \langle{\underline{\mathbf{h}}}|{\underline{\mathbf{k}}}\rangle \propto \delta_{\underline{\mathbf{h}}, \underline{\mathbf{k}}}⟨𝐡̲|𝐤̲⟩∝δ𝐡̲,𝐤̲). In our approach, the SoV bases are constructed by using families of conserved charges. This gives us a large freedom in the SoV bases construction, and allows us to look for the choice of a family of conserved charges which leads to orthogonal co-vector/vector SoV bases. We first define such a choice in the case of twist matrices having simple spectrum and zero determinant. Then, we generalize the associated family of conserved charges and orthogonal SoV bases to generic simple spectrum and invertible twist matrices. Under this choice of conserved charges, and of the associated orthogonal SoV bases, the scalar products of separate states simplify considerably and take a form similar to the \mathcal{Y}(gl_2)𝒴(gl2) rank one case.

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 INTEGRABLE MULTIPARAMETRIC SU(N) CHAIN

1996 ◽  
Vol 11 (12) ◽  
pp. 987-993 ◽  
Author(s):  
ANGELA FOERSTER ◽  
ITZHAK RODITI ◽  
LIGIA M.C.S. RODRIGUES
Keyword(s):  
Boundary Conditions ◽  
Algebraic Structure ◽  
Central Element ◽  
Integrable Models ◽  
R Matrix ◽  
Twisted Boundary Conditions

We analyze integrable models associated to a multiparametric SU (N)R-matrix. We show that the Hamiltonians describe SU (N) chains with twisted boundary conditions and that the underlying algebraic structure is the multiparametric deformation of SU (N) enlarged by the introduction of a central element.

On a technique for deriving explicit transfer matrices of orthotropic layers

2015 ◽  
Vol 37 (4) ◽  
pp. 303-315 ◽  
Author(s):  
Pham Chi Vinh ◽  
Nguyen Thi Khanh Linh ◽  
Vu Thi Ngoc Anh
Keyword(s):  
Wave Propagation ◽  
Explicit Form ◽  
Half Space ◽  
Transfer Matrix ◽  
Rayleigh Waves ◽  
Secular Equation ◽  
Transfer Matrices ◽  
Orthotropic Layer ◽  
Wave Propagation Problem ◽  
Arbitrary Thickness

This paper presents  a technique by which the transfer matrix in explicit form of an orthotropic layer can be easily obtained. This transfer matrix is applicable for both the wave propagation problem and the reflection/transmission problem. The obtained transfer matrix is then employed to derive the explicit secular equation of Rayleigh waves propagating in an orthotropic half-space coated by an orthotropic layer of arbitrary thickness.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

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