scholarly journals Complete spectrum of quantum integrable lattice models associated to $\boldsymbol {\mathcal{U}_{q} (\widehat{gl_{n}})}$ by separation of variables

2019 ◽  
Vol 52 (31) ◽  
pp. 315203 ◽  
Author(s):  
J M Maillet ◽  
G Niccoli
Keyword(s):  
Separation Of Variables ◽  
Lattice Models ◽  
Complete Spectrum ◽  
Integrable Lattice ◽  
Integrable Lattice Models

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.

YANG-BAXTER ALGEBRAS, INTEGRABLE THEORIES AND QUANTUM GROUPS

1989 ◽  
Vol 04 (10) ◽  
pp. 2371-2463 ◽  
Author(s):  
H.J. De VEGA
Keyword(s):  
Quantum Groups ◽  
Scaling Limit ◽  
Finite Size ◽  
Lattice Models ◽  
Physical Models ◽  
Quantum Field Theories ◽  
Integrable Lattice ◽  
Ba Construction ◽  
Integrable Lattice Models ◽  
Conformal Models

The Yang-Baxter-Zamolodchikov-Faddeev (YBZF) algebras and their many applications are the subject of this reivew. I start by the solvable lattice statistical models constructed from YBZF algebras. All two-dimensional integrable vertex models follow in this way and are solvable via Bethe Ansatz (BA) and their generalizations. The six-vertex model solution and its q(2q−1) vertex generalization including its nested BA construction are exposed. YBZF algebras and their associated physical models are classified in terms of simple Lie algebras. It is shown how these lattice models yield both solvable massive quantum field theories (QFT) and conformal models in appropriated scaling (continuous) limits within the lattice light-cone approach. The method of finite-size calculations from the BA is described as well as its applications to derive the conformal properties of integrable lattice models. It is conjectured that all integrable QFT and conformal models follow in a scaling limit from these YBZF algebras. A discussion on braid and quantum groups concludes this review.

INTEGRABLE LATTICE MODELS, GRAPHS AND MODULAR INVARIANT CONFORMAL FIELD THEORIES

1992 ◽  
Vol 07 (03) ◽  
pp. 407-500 ◽  
Author(s):  
P. DI FRANCESCO
Keyword(s):  
Lattice Models ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Compact Lie Groups ◽  
Modular Invariant ◽  
Conformally Invariant ◽  
Conformal Field ◽  
Integrable Lattice ◽  
Modular Invariants ◽  
Integrable Lattice Models

We review the construction of integrable height models attached to graphs, in connection with compact Lie groups. The continuum limit of these models yields conformally invariant field theories. A direct relation between graphs and (Kac–Moody or coset) modular invariants is proposed.

GENERALIZED SKLYANIN ALGEBRA AND INTEGRABLE LATTICE MODELS

1994 ◽  
Vol 09 (13) ◽  
pp. 2245-2281 ◽  
Author(s):  
YAS-HIRO QUANO
Keyword(s):  
Inverse Problem ◽  
Lattice Model ◽  
Lattice Models ◽  
Baxter Equation ◽  
Initial Condition ◽  
Infinite Dimensional ◽  
Integrable Lattice ◽  
Crossing Symmetry ◽  
Sklyanin Algebra ◽  
Integrable Lattice Models

We study three properties of the ℤn⊗ℤn-symmetric lattice model; i.e. the initial condition, the unitarity and the crossing symmetry. The scalar factors appearing in the unitarity and the crossing symmetry are explicitly obtained. The [Formula: see text]-Sklyanin algebra is introduced in the natural framework of the inverse problem for this model. We build both finite- and infinite-dimensional representations of the [Formula: see text]-Sklyanin algebra, and construct an [Formula: see text] generalization of the broken ℤN model. Furthermore, the Yang-Baxter equation for this new model is proved.

scholarly journals Approximate conservation laws in perturbed integrable lattice models

2015 ◽  
Vol 92 (19) ◽  
Author(s):  
Marcin Mierzejewski ◽  
Tomaž Prosen ◽  
Peter Prelovšek
Keyword(s):  
Conservation Laws ◽  
Lattice Models ◽  
Integrable Lattice ◽  
Integrable Lattice Models

scholarly journals Conjectures on hidden Onsager algebra symmetries in interacting quantum lattice models

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Yuan Miao
Keyword(s):  
Lattice Models ◽  
Transfer Matrices ◽  
Xxz Model ◽  
Fusion Procedure ◽  
Root Of Unity ◽  
Integrable Lattice ◽  
Onsager Algebra ◽  
Cyclic Transfer ◽  
Integrable Lattice Models ◽  
Matrix Fusion

We conjecture the existence of hidden Onsager algebra symmetries in two interacting quantum integrable lattice models, i.e. spin-1/2 XXZ model and spin-1 Zamolodchikov-Fateev model at arbitrary root of unity values of the anisotropy. The conjectures relate the Onsager generators to the conserved charges obtained from semi-cyclic transfer matrices. The conjectures are motivated by two examples which are spin-1/2 XX model and spin-1 U(1)-invariant clock model. A novel construction of the semi-cyclic transfer matrices of spin-1 Zamolodchikov-Fateev model at arbitrary root of unity values of the anisotropy is carried out via the transfer matrix fusion procedure.

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