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Vladimir V. Mangazeev

Publications and source records attributed to Vladimir V. Mangazeev.

At least 19 recordsLinked to original sources
arXiv2026

Quantum Dilogarithms and New Integrable Lattice Models in Three Dimensions

In this paper we introduce a new class of integrable 3D lattice models, possessing continuous families of commuting layer-to-layer transfer matrices. Algebraically, this commutativity is based on a very special construction of local Boltzmann weights in terms of quantum dilogarithms satisfying the inversion and pentagon identities. We give three examples of such quantum dilogarithms, leading to integrable 3D lattice models. The partition function per site in these models can be exactly calculated in the limit of an infinite lattice by using the functional relations, symmetry and factorization properties of the transfer matrix. The results of such calculations for 3D models associated with the Faddeev modular quantum dilogarithm are briefly presented.

math-ph↗
arXiv2024

Corner Transfer Matrix Approach to the Yang-Lee Singularity in the 2D Ising Model in a magnetic field

We study the 2D Ising model in a complex magnetic field in the vicinity of the Yang-Lee edge singularity. By using Baxter's variational corner transfer matrix method combined with analytic techniques, we numerically calculate the scaling function and obtain an accurate estimate of the location of the Yang-Lee singularity. The existing series expansions for susceptibility of the 2D Ising model on a triangular lattice by Chan, Guttmann, Nickel and Perk allowed us to substantially enhance the accuracy of our calculations. Our results are in excellent agreement with the Ising field theory calculations by Fonseca, Zamolodchikov and the recent work by Xu and Zamolodchikov. In particular, we numerically confirm an agreement between the leading singular behavior of the scaling function and the predictions of the ${\mathcal M}_{2/5}$ conformal field theory.

hep-th↗
arXiv2021

Hidden symmetry operators for asymmetric generalised quantum Rabi models

The hidden $\mathbb{Z}_2$ symmetry of the asymmetric quantum Rabi model (AQRM) has recently been revealed via a systematic construction of the underlying symmetry operator. Based on the AQRM result, we propose an ansatz for the general form of the symmetry operators for AQRM-related models. Applying this ansatz we obtain the symmetry operator for three models: the anisotropic AQRM, the asymmetric Rabi-Stark model (ARSM) and the anisotropic ARSM.

quant-ph↗
arXiv2021

Hidden symmetry in the biased Dicke model

The symmetry operators generating the hidden $\mathbb{Z}_2$ symmetry of the asymmetric quantum Rabi model (AQRM) at bias $ε\in \frac{1}{2}\mathbb{Z}$ have recently been constructed by V. V. Mangazeev et al. [J. Phys. A: Math. Theor. 54 12LT01 (2021)]. We start with this result to determine symmetry operators for the $N$-qubit generalisation of the AQRM, also known as the biased Dicke model, at special biases. We also prove for general $N$ that the symmetry operators, which commute with the Hamiltonian of the biased Dicke model, generate a $\mathbb{Z}_2$ symmetry.

quant-ph↗
arXiv2021

The hidden symmetry of the asymmetric quantum Rabi model

The asymmetric quantum Rabi model (AQRM) exhibits level crossings in the eigenspectrum for the values $ε\in\frac{1}{2}\mathbb{Z}$ of the bias parameter $ε$. Such level crossings are expected to be associated with some hidden symmetry of the model. The origin of this hidden symmetry is established by finding the operators which commute with the AQRM hamiltonian at these special values. The construction is given explicitly for the first several cases and can be applied to other related light-matter interaction models for which similar level crossings have been observed in the presence of a bias term.

quant-ph↗
arXiv2019

Boundary matrices for the higher spin six vertex model

In this paper we consider solutions to the reflection equation related to the higher spin stochastic six vertex model. The corresponding higher spin $R$-matrix is associated with the affine quantum algebra $U_q(\widehat{sl(2)})$. The explicit formulas for boundary $K$-matrices for spins $s=1/2,1$ are well known. We derive difference equations for the generating function of matrix elements of the $K$-matrix for any spin $s$ and solve them in terms of hypergeometric functions. As a result we derive the explicit formula for matrix elements of the $K$-matrix for arbitrary spin. In the lower- and upper- triangular cases, the $K$-matrix simplifies and reduces to simple products of $q$-Pochhammer symbols.

math-ph↗
arXiv2017

Density and current profiles in $U_q(A^{(1)}_2)$ zero range process

The stochastic $R$ matrix for $U_q(A^{(1)}_n)$ introduced recently gives rise to an integrable zero range process of $n$ classes of particles in one dimension. For $n=2$ we investigate how finitely many first class particles fixed as defects influence the grand canonical ensemble of the second class particles. By using the matrix product stationary probabilities involving infinite products of $q$-bosons, exact formulas are derived for the local density and current of the second class particles in the large volume limit.

math-ph↗
arXiv2017

Integrable structure of products of finite complex Ginibre random matrices

We consider the squared singular values of the product of $M$ standard complex Gaussian matrices. Since the squared singular values form a determinantal point process with a particular Meijer G-function kernel, the gap probabilities are given by a Fredholm determinant based on this kernel. It was shown by Strahov \cite{St14} that a hard edge scaling limit of the gap probabilities is described by Hamiltonian differential equations which can be formulated as an isomonodromic deformation system similar to the theory of the Kyoto school. We generalize this result to the case of finite matrices by first finding a representation of the finite kernel in integrable form. As a result we obtain the Hamiltonian structure for a finite size matrices and formulate it in terms of a $(M+1) \times (M+1)$ matrix Schlesinger system. The case $M=1$ reproduces the Tracy and Widom theory which results in the Painlevé V equation for the $(0,s)$ gap probability. Some integrals of motion for $M = 2$ are identified, and a coupled system of differential equations in two unknowns is presented which uniquely determines the corresponding $(0,s)$ gap probability.

math-ph↗
arXiv2016

Construction of $R$-matrices for symmetric tensor representations related to $U_{q}(\widehat{sl_{n}})$

In this paper we construct a new factorized representation of the $R$-matrix related to the affine algebra $U_{q}(\widehat{sl_{n}})$ for symmetric tensor representations with arbitrary weights. Using the 3D approach we obtain explicit formulas for the matrix elements of the $R$-matrix and give a simple proof that a "twisted" $R$-matrix is stochastic. We also discuss symmetries of the $R$-matrix, its degenerations and compare our formulas with other results available in the literature.

math-ph↗
arXiv2016

Stochastic $R$ matrix for $U_q(A^{(1)}_n)$

We show that the quantum $R$ matrix for symmetric tensor representations of $U_q(A^{(1)}_n)$ satisfies the sum rule required for its stochastic interpretation under a suitable gauge. Its matrix elements at a special point of the spectral parameter are found to factorize into the form that naturally extends Povolotsky's local transition rate in the $q$-Hahn process for $n=1$. Based on these results we formulate new discrete and continuous time integrable Markov processes on a one-dimensional chain in terms of $n$ species of particles obeying asymmetric stochastic dynamics. Bethe ansatz eigenvalues of the Markov matrices are also given.

math.QA↗
arXiv2014

On the Yang-Baxter equation for the six-vertex model

In this paper we review the theory of the Yang-Baxter equation related to the 6-vertex model and its higher spin generalizations. We employ a 3D approach to the problem. Starting with the 3D R-matrix, we consider a two-layer projection of the corresponding 3D lattice model. As a result, we obtain a new expression for the higher spin $R$-matrix associated with the affine quantum algebra $U_q(\widehat{sl(2)})$. In the simplest case of the spin $s=1/2$ this $R$-matrix naturally reduces to the $R$-matrix of the 6-vertex model. Taking a special limit in our construction we also obtain new formulas for the $Q$-operators acting in the representation space of arbitrary (half-)integer spin. Remarkably, this construction can be naturally extended to any complex values of spin $s$. We also give all functional equations satisfied by the transfer-matrices and $Q$-operators.

math-ph↗
arXiv2014

Q-operators in the six-vertex model

In this paper we continue the study of $Q$-operators in the six-vertex model and its higher spin generalizations. In [1] we derived a new expression for the higher spin $R$-matrix associated with the affine quantum algebra $U_q(\widehat{sl(2)})$. Taking a special limit in this $R$-matrix we obtained new formulas for the $Q$-operators acting in the tensor product of representation spaces with arbitrary complex spin. Here we use a different strategy and construct $Q$-operators as integral operators with factorized kernels based on the original Baxter's method used in the solution of the eight-vertex model. We compare this approach with the method developed in [1] and find the explicit connection between two constructions. We also discuss a reduction to the case of finite-dimensional representations with (half-) integer spins.

math-ph↗
arXiv2013

An integrable 3D lattice model with positive Boltzmann weights

In this paper we construct a three-dimensional (3D) solvable lattice model with non-negative Boltzmann weights. The spin variables in the model are assigned to edges of the 3D cubic lattice and run over an infinite number of discrete states. The Boltzmann weights satisfy the tetrahedron equation, which is a 3D generalisation of the Yang-Baxter equation. The weights depend on a free parameter 0<q<1 and three continuous field variables. The layer-to-layer transfer matrices of the model form a two-parameter commutative family. This is the first example of a solvable 3D lattice model with non-negative Boltzmann weights.

math-ph↗
arXiv2013

An Elliptic Parameterisation of the Zamolodchikov Model

The Zamolodchikov model describes an exact relativistic factorized scattering theory of straight strings in (2+1)-dimensional space-time. It also defines an integrable 3D lattice model of statistical mechanics and quantum field theory. The three-string S-matrix satisfies the tetrahedron equation which is a 3D analog of the Yang-Baxter equation. Each S-matrix depends on three dihedral angles formed by three intersecting planes, whereas the tetrahedron equation contains five independent spectral parameters, associated with angles of an Euclidean tetrahedron. The vertex weights are given by rather complicated expressions involving square roots of trigonometric function of the spectral parameters, which is quite unusual from the point of view of 2D solvable lattice models. In this paper we consider a particular four-parameter specialization of the tetrahedron equation when one of its vertices goes to infinity and the tetrahedron itself degenerates into an infinite prism. We show that in this limit all the vertex weights in the tetrahedron equation can be represented as meromorphic functions on an elliptic curve. Moreover we show that a special reduction of the tetrahedron equation in this case leads precisely to an example of the tetrahedral Zamolodchikov algebra, previously constructed by Korepanov. This algebra plays important role for a "layered" construction of the Shastry's R-matrix and the 2D S-matrix appearing in the problem of the ADS/CFT correspondence for N=4 SUSY Yang-Mills theory in four dimensions. Possible applications of our results in this field are briefly discussed.

math-ph↗
arXiv2010

Scaling and universality in the 2D Ising model with a magnetic field

The scaling function of the 2D Ising model in a magnetic field on the square and triangular lattices is obtained numerically via Baxter's variational corner transfer matrix approach. The use of the Aharony-Fisher non-linear scaling variables allowed us to perform calculations sufficiently away from the critical point to obtain very high precision data, which convincingly confirm all predictions of the scaling and universality hypotheses. The results are in excellent agreement with the field theory calculations of Fonseca and Zamolodchikov as well as with many previously known exact and numerical results for the 2D Ising model. This includes excellent agreement with the classic analytic results for the magnetic susceptibility by Barouch, McCoy, Tracy and Wu, recently enhanced by Orrick, Nickel, Guttmann and Perk.

cond-mat.stat-mech↗
arXiv2010

Form factor expansions in the 2D Ising model and Painlevé VI

We derive a Toda-type recurrence relation, in both high and low temperature regimes, for the $λ$ - extended diagonal correlation functions $C(N,N;λ)$ of the two-dimensional Ising model, using an earlier connection between diagonal form factor expansions and tau-functions within Painlevé VI (PVI) theory, originally discovered by Jimbo and Miwa. This greatly simplifies the calculation of the diagonal correlation functions, particularly their $λ$-extended counterparts. We also conjecture a closed form expression for the simplest off-diagonal case $C^{\pm}(0,1;λ)$ where a connection to PVI theory is not known. Combined with the results for diagonal correlations these give all the initial conditions required for the $ł$-extended version of quadratic difference equations for the correlation functions discovered by McCoy, Perk and Wu. The results obtained here should provide a further potential algorithmic improvement in the $ł$-extended case, and facilitate other developments.

math-ph↗
arXiv2009

Eight-vertex model and Painlevé VI equation. II. Eigenvector results

We study a special anisotropic XYZ-model on a periodic chain of an odd length and conjecture exact expressions for certain components of the ground state eigenvectors. The results are written in terms of tau-functions associated with Picard's elliptic solutions of the Painlevé VI equation. Connections with other problems related to the eight-vertex model are briefly discussed.

math-ph↗