148 results on '"TRANSFER matrix"'
Search Results
2. Exact closed forms for the transmittance of electromagnetic waves in one-dimensional anisotropic periodic media.
- Author
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Torres-Guzmán, J C, Díaz-de-Anda, A, and Arriaga, J
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ELECTROMAGNETIC waves , *TRANSFER matrix , *MATRIX multiplications , *FARADAY effect , *ELECTROMAGNETIC fields , *EIGENVALUES , *ANISOTROPY - Abstract
In this work, we obtain closed expressions for the transfer matrix and the transmittance of electromagnetic waves propagating in finite one-dimensional anisotropic periodic stratified media with an arbitrary number of cells. By invoking the Cayley–Hamilton theorem on the transfer matrix for the electromagnetic field in a periodic stratified media formed by N cells, we obtain a fourth-order recursive relation for the matrix coefficients that defines the so-called Tetranacci polynomials (TPs). In the symmetric case, corresponding to a unit-cell transfer matrix with a characteristic polynomial where the coefficients of the linear and cubic terms are equal, closed expressions for the solutions to the recursive relation, known as symmetric TPs, have recently been derived, allowing us to write the transfer matrix and transmittance in a closed form. We show as sufficient conditions that the 4 × 4 differential propagation matrix of each layer in the binary unit cell, Δ , a) has eigenvalues of the form ± p 1 , ± p 2 , with p 1 ≠ p 2 , and b) its off-diagonal 2 × 2 block matrices possess the same symmetric structure in both layers. Otherwise, the recursive relations are still solvable for any 4 × 4 matrix and provide an algorithm to compute the N th power of the transfer matrix without carrying out explicitly the matrix multiplication of N matrices. We obtain analytical expressions for the dispersion relation and transmittance, in closed form, for two finite periodic systems: the first one consists of two birefringent uniaxial media with their optical axis perpendicular to the z -axis, and the second consists of two isotropic media subject to an external magnetic field oriented along the z -axis and exhibiting the Faraday effect. Our formalism applies also to lossy media, magnetic anisotropy or optical activity. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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3. Two-step relaxation in local many-body Floquet systems.
- Author
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Žnidarič, Marko
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DECAY constants , *TRANSFER matrix , *STATISTICAL physics , *QUBITS , *STATISTICAL correlation , *DECAY rates (Radioactivity) - Abstract
We want to understand how relaxation process from an initial non-generic state proceeds towards a long-time typical state reached under unitary quantum evolution. One would expect that after some initial correlation time relaxation will be a simple exponential decay with constant decay rate. We show that this is not necessarily the case. Studying various Floquet systems with fixed two-qubit gates, and focusing on purity and out-of-time-ordered correlation functions, we find that in many situations relaxation proceeds in two phases of exponential decay having different relaxation rates. Namely, in the thermodynamic limit the relaxation rate exhibits a change at a critical time proportional to system's size. The initial thermodynamically relevant rate can be slower or faster than the asymptotic one, demonstrating that the recently discovered phantom relaxation, in which the decay is slower than predicted by a nonzero transfer matrix gap, is not limited to only random circuits. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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4. Higher order Morita approximation and its validity for random copolymer adsorption onto homogeneous and periodic heterogeneous surfaces.
- Author
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Polotsky, Alexey A and Ivanova, Anna S
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ADSORPTION (Chemistry) , *GENERATING functions , *TRANSFER matrix , *MARKOV processes , *PROBLEM solving , *BLOCK copolymers - Abstract
Adsorption of a single AB random copolymer (RC) chain onto homogeneous and inhomogeneous ab surfaces with a regular periodic pattern is studied theoretically. For the averaging over disorder in the RC sequence, the constrained annealed approximation, known as the Morita approximation is employed. A general scheme for constructing the Morita approximation of an arbitrary order m is proposed; it is based on representation of the RC monomer sequence as a Markov chain of overlapping (m –1)-ads. The problem is solved within the framework of the generating functions approach for the two-dimensional partially directed walk model of the polymer on a square lattice. Temperature dependences of various adsorption characteristics are obtained. The accuracy of the Morita approximation is assessed by comparison with the numerical results for RC with quenched sequences obtained by the transfer matrix approach. It is shown that for RC adsorption on a homogeneous surface for AB -copolymers with adsorbing A and neutral B blocks, it is sufficient to use the Morita approximation of second or third order. If the non-adsorbing B block is repelled from the surface, then a higher order of the approximation (5th–6th) is required. An important indicator of validity of the Morita approximation is the entropy: if the order of the approximation is not high enough, the entropy becomes negative in the low-temperature regime which means that the Morita approximation is wrong in this order. In the case when the surface is periodically inhomogeneous, the Morita approximation works worse and very high orders are required, which can be beyond the computational capabilities. Moreover, the Morita approximations become degenerate: approximations of two or more consecutive orders give the same result. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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5. Integrable boundary conditions for staggered vertex models.
- Author
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Frahm, Holger and Gehrmann, Sascha
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HAMILTONIAN systems , *ALGEBRA , *TRANSFER matrix - Abstract
Yang–Baxter integrable vertex models with a generic Z 2 -staggering can be expressed in terms of composite R -matrices given in terms of the elementary R -matrices. Similarly, integrable open boundary conditions can be constructed through generalized reflection algebras based on these objects and their representations in terms of composite boundary matrices K ± . We show that only two types of staggering yield a local Hamiltonian with integrable open boundary conditions in this approach. The staggering in the underlying model allows for a second hierarchy of commuting integrals of motion (in addition to the one including the Hamiltonian obtained from the usual transfer matrix), starting with the so-called quasi momentum operator. In this paper, we show that this quasi momentum operator can be obtained together with the Hamiltonian for both periodic and open models in a unified way from enlarged Yang–Baxter or reflection algebras in the composite picture. For the special case of the staggered six-vertex model, this allows constructing an integrable spectral flow between the two local cases. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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6. Existence of the transfer matrix for a class of nonlocal potentials in two dimensions.
- Author
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Loran, Farhang and Mostafazadeh, Ali
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TRANSFER matrix , *QUANTUM theory , *S-matrix theory , *SCATTERING (Mathematics) , *SYSTEM dynamics , *HAMILTONIAN operator , *INVERSE scattering transform - Abstract
Evanescent waves are waves that decay or grow exponentially in regions of the space void of interaction. In potential scattering defined by the Schrödinger equation, (â' ⇠2 + v) Ď = k 2 Ď for a local potential v, they arise in dimensions greater than one and are generally present regardless of the details of v. The approximation in which one ignores the contributions of the evanescent waves to the scattering process corresponds to replacing v with a certain energy-dependent nonlocal potential V k ˆ . We present a dynamical formulation of the stationary scattering for V ˆ k in two dimensions, where the scattering data are related to the dynamics of a quantum system having a non-self-adjoint, unbounded, and nonstationary Hamiltonian operator. The evolution operator for this system determines a two-dimensional analog of the transfer matrix of stationary scattering in one dimension which contains the information about the scattering properties of the potential. Under rather general conditions on v, we establish the strong convergence of the Dyson series expansion of the evolution operator and prove the existence of the transfer matrix for V ˆ k as a densely-defined operator acting in C 2 ⊗ L 2 (â' k , k). [ABSTRACT FROM AUTHOR]
- Published
- 2022
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7. Exact site-percolation probability on the square lattice.
- Author
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Mertens, Stephan
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PERCOLATION theory , *PERCOLATION , *PROBABILITY theory , *TRANSFER matrix - Abstract
We present an algorithm to compute the exact probability R n (p) for a site percolation cluster to span an n × n square lattice at occupancy p. The algorithm has time and space complexity O (λ n ) with λ ≠2.6. It allows us to compute R n (p) up to n = 24. We use the data to compute estimates for the percolation threshold p c that are several orders of magnitude more precise than estimates based on Monte-Carlo simulations. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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8. Singularity-free treatment of delta-function point scatterers in two dimensions and its conceptual implications.
- Author
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Loran, Farhang and Mostafazadeh, Ali
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COUPLING constants , *TRANSFER matrix , *SCATTERING (Mathematics) - Abstract
In two dimensions, the standard treatment of the scattering problem for a delta-function potential, v (r) = z δ (r) , leads to a logarithmic singularity which is subsequently removed by a renormalization of the coupling constant z. Recently, we have developed a dynamical formulation of stationary scattering (DFSS) which offers a singularity-free treatment of this potential. We elucidate the basic mechanism responsible for the implicit regularization property of DFSS that makes it avoid the logarithmic singularity one encounters in the standard approach to this problem. We provide an alternative interpretation of this singularity showing that it arises, because the standard treatment of the problem takes into account contributions to the scattered wave whose momentum is parallel to the detectors’ screen. The renormalization schemes used for removing this singularity has the effect of subtracting these unphysical contributions, while DFSS has a built-in mechanics that achieves this goal. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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9. Localization of space-inhomogeneous three-state quantum walks.
- Author
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Kiumi, Chusei
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TRANSFER matrix , *MATHEMATICAL analysis , *EIGENVALUES , *LOCALIZATION (Mathematics) - Abstract
Mathematical analysis on the existence of eigenvalues is essential because it is deeply related to localization, which is an exceptionally crucial property of quantum walks (QWs). We construct the method for the eigenvalue problem via the transfer matrix for space-inhomogeneous three-state QWs in one dimension with a self-loop, which is an extension of the technique in a previous study (Kiumi and Saito 2021 Quantum Inf. Process. 20 171). This method reveals the necessary and sufficient condition for the eigenvalue problem of a two-phase three-state QW with one defect whose time evolution varies in the negative part, positive part, and at the origin. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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10. Low-frequency scattering defined by the Helmholtz equation in one dimension.
- Author
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Loran, Farhang and Mostafazadeh, Ali
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ELECTROMAGNETIC wave propagation , *HELMHOLTZ equation , *SCATTERING (Mathematics) , *QUANTUM scattering , *TRANSFER matrix , *ABSORPTION coefficients , *SCHRODINGER equation , *SCATTERING (Physics) - Abstract
The Helmholtz equation in one dimension, which describes the propagation of electromagnetic waves in effectively one-dimensional systems, is equivalent to the time-independent Schrödinger equation. The fact that the potential term entering the latter is energy-dependent obstructs the application of the results on low-energy quantum scattering in the study of the low-frequency waves satisfying the Helmholtz equation. We use a recently developed dynamical formulation of stationary scattering to offer a comprehensive treatment of the low-frequency scattering of these waves for a general finite-range scatterer. In particular, we give explicit formulas for the coefficients of the low-frequency series expansion of the transfer matrix of the system which in turn allow for determining the low-frequency expansions of its reflection, transmission, and absorption coefficients. Our general results reveal a number of interesting physical aspects of low-frequency scattering particularly in relation to permittivity profiles having balanced gain and loss. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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11. Simultaneous block diagonalization of matrices of finite order.
- Author
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Bischer, Ingolf, Döring, Christian, and Trautner, Andreas
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TRANSFER matrix , *MATRICES (Mathematics) , *TIME reversal , *PARTICLE physics , *INVARIANT subspaces , *AUTOMORPHISMS - Abstract
It is well known that a set of non-defect matrices can be simultaneously diagonalized if and only if the matrices commute. In the case of non-commuting matrices, the best that can be achieved is simultaneous block diagonalization. Here we give an efficient algorithm to explicitly compute a transfer matrix which realizes the simultaneous block diagonalization of unitary matrices whose decomposition in irreducible blocks (common invariant subspaces) is known from elsewhere. Our main motivation lies in particle physics, where the resulting transfer matrix must be known explicitly in order to unequivocally determine the action of outer automorphisms such as parity, charge conjugation, or time reversal on the particle spectrum. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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12. Transfer matrix for long-range potentials.
- Author
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Loran, Farhang and Mostafazadeh, Ali
- Subjects
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SCATTERING (Mathematics) , *REFLECTANCE , *TRANSFER matrix - Abstract
We extend the notion of the transfer matrix of potential scattering to a large class of long-range potentials v(x) and derive its basic properties. We outline a dynamical formulation of the time-independent scattering theory for this class of potentials where we identify their transfer matrix with the S-matrix of a certain effective non-unitary two-level quantum system. For sufficiently large values of |x|, we express v(x) as the sum of a short-range potential and an exactly solvable long-range potential. Using this result and the composition property of the transfer matrix, we outline an approximation scheme for solving the scattering problem for v(x). To demonstrate the effectiveness of this scheme, we construct an exactly solvable long-range potential and compare the exact values of its reflection and transmission coefficients with those we obtain using our approximation scheme. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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13. Lattice models, deformed Virasoro algebra and reduction equation.
- Author
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Lashkevich, Michael, Pugai, Yaroslav, Shiraishi, Jun'ichi, and Tutiya, Yohei
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ALGEBRA , *TRANSFER matrix , *EQUATIONS - Abstract
We study the fused currents of the deformed Virasoro algebra. By constructing a homotopy operator we show that for special values of the parameter of the algebra fused currents pairwise coincide on the cohomologies of the Felder resolution. Within the algebraic approach to lattice models these currents are known to describe neutral excitations of the solid-on-solid (SOS) models in the transfer-matrix picture. It allows us to prove the closeness of the system of excitations for a special nonunitary series of restricted SOS models. Though the results of the algebraic approach to lattice models were consistent with the results of other methods, the lack of such proof had been an essential gap in its construction. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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14. Adiabatic approximation, semiclassical scattering, and unidirectional invisibility.
- Author
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Mostafazadeh, Ali
- Subjects
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MATHEMATICS theorems , *SCATTERING (Mathematics) , *TRANSFER matrix , *HAMILTON'S principle function , *EIGENVECTORS , *WAVE functions - Abstract
The transfer matrix of a possibly complex and energy-dependent scattering potential can be identified with the S-matrix of a two-level time-dependent non-Hermitian Hamiltonian H(τ). We show that the application of the adiabatic approximation to H(τ) corresponds to the semiclassical description of the original scattering problem. In particular, the geometric part of the phase of the evolving eigenvectors of H(τ) gives the pre-exponential factor of the WKB wave functions. We use these observations to give an explicit semiclassical expression for the transfer matrix. This allows for a detailed study of the semiclassical unidirectional reflectionlessness and invisibility. We examine concrete realizations of the latter in the realm of optics. [ABSTRACT FROM AUTHOR]
- Published
- 2014
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15. Phase diagram and strong-coupling fixed point in the disordered O(n) loop model.
- Author
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Shimada, H, Jacobsen, J L, and Kamiya, Y
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PHASE diagrams , *FIXED point theory , *DIMENSIONAL analysis , *RENORMALIZATION group , *TRANSFER matrix , *ISING model - Abstract
We study the phase diagram and critical properties of the two-dimensional disordered O(n) loop model. The renormalization group (RG) flow is extracted from the landscape of the effective central charge c obtained by the transfer matrix method. We find a line of multicritical fixed points (FPs) at strong randomness for n > nc ∼ 0.5. We also find a line of stable random FPs for nc < n < 1, whose c and critical exponents agree well with the 1 − n expansion results. The multicritical FP at n = 1 has c = 0.4612(4), which suggests that it belongs to the universality class of the Nishimori point in the random-bond Ising model. For n > 2, we find another critical line that connects the hard-hexagon FP in the pure model to a finite-randomness zero-temperature FP . [ABSTRACT FROM AUTHOR]
- Published
- 2014
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16. SOV approach for integrable quantum models associated with general representations on spin-1/2 chains of the 8-vertex reflection algebra.
- Author
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Faldella, S and Niccoli, G
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QUANTUM information theory , *SEPARATION of variables , *EIGENVALUES , *MATHEMATICAL analysis , *TRANSFER matrix , *QUANTUM scattering - Abstract
The analysis of the transfer matrices associated with the most general representations of the 8-vertex reflection algebra on spin-1/2 chains is here implemented by introducing a quantum separation of variables (SOV) method, which generalizes to these integrable quantum models the method first introduced by Sklyanin. For representations reproducing in their homogeneous limits the open XYZ spin-1/2 quantum chains with the most general integrable boundary conditions, we explicitly construct representations of the 8-vertex reflection algebras, for which the transfer matrix spectral problem is separated. Then, in these SOV representations we get the complete characterization of the transfer matrix spectrum (eigenvalues and eigenstates) and its non-degeneracy. Moreover, we present the first fundamental step toward the characterization of the dynamics of these models by deriving determinant formulae for the matrix elements of the identity on separated states, which particularly apply to transfer matrix eigenstates. A comparison of our analysis of the 8-vertex reflection algebra with that of (Niccoli G 2012 J. Stat. Mech. P10025, Faldella S et al 2014 J. Stat. Mech. P01011) for the 6-vertex leads to an interesting remark in that there is a profound similarity in both the characterization of the spectral problems and the scalar products, which exists for these two different realizations of the reflection algebra once they are described by the SOV method. As will be shown in a future publication, this remarkable similarity will be the basis of a simultaneous determination of the form factors of local operators of integrable quantum models associated with general reflection algebra representations of both 8-vertex and 6-vertex type. [ABSTRACT FROM AUTHOR]
- Published
- 2014
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17. Transfer matrix analysis of one-dimensional majority cellular automata with thermal noise.
- Author
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Lemoy, Rémi, Mozeika, Alexander, and Seki, Shinnosuke
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THERMAL noise , *CELLULAR automata , *PERTURBATION theory , *MATHEMATICAL physics , *TRANSFER matrix - Abstract
Thermal noise in a cellular automaton (CA) refers to a random perturbation in its function which eventually leads the automaton to an equilibrium state controlled by a temperature parameter. We study the one-dimensional majority-3 CA under this model of noise. Without noise, each cell in the automaton decides its next state by majority voting among itself and its left and right neighbour cells. Transfer matrix analysis shows that the automaton always reaches a state in which every cell is in one of its two states with probability 1/2 and thus cannot remember even one bit of information. Numerical experiments, however, support the possibility of reliable computation for a long but finite time. [ABSTRACT FROM AUTHOR]
- Published
- 2014
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18. Algebraic Bethe ansatz for the six vertex model with upper triangular K-matrices.
- Author
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Pimenta, R. A. and Lima-Santos, A.
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LINEAR algebra , *TRIANGULARIZATION (Mathematics) , *MATHEMATICAL formulas , *MATHEMATICAL forms , *GEOMETRICAL constructions , *GENERALIZATION , *TRANSFER matrix - Abstract
We consider a formulation of the algebraic Bethe ansatz for the six vertex model with non-diagonal open boundaries. Specifically, we study the case where both left and right K-matrices have an upper triangular form. We show that the main difficulty entailed by those forms of the K-matrices is the construction of the excited states. However, it is possible to treat this problem with the aid of an auxiliary transfer matrix and by means of a generalized creation operator. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
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19. The hard hexagon partition function for complex fugacity.
- Author
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Assis, M., Jacobsen, J. L., Jensen, I., Maillard, J-M, and McCoy, B. M.
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HEXAGONS , *PARTITION functions , *FUGACITY , *TRANSFER matrix , *EIGENVALUES , *THERMODYNAMICS - Abstract
We study the analyticity of the partition function of the hard hexagon model in the complex fugacity plane by computing zeros and transfer matrix eigenvalues for large finite size systems. We find that the partition function per site computed by Baxter in the thermodynamic limit for positive real values of the fugacity is not sufficient to describe the analyticity in the full complex fugacity plane. We also obtain a new algebraic equation for the low density partition function per site. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
20. An inhomogeneous T-Q equation for the open XXX chain with general boundary terms: completeness and arbitrary spin.
- Author
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Nepomechie, Rafael I.
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INHOMOGENEOUS materials , *NUCLEAR spin , *BOUNDARY value problems , *TRANSFER matrix , *EIGENVALUES , *NUMERICAL solutions to equations - Abstract
An inhomogeneous T-Q equation has recently been proposed by Cao, Yang, Shi and Wang for the open spin-1/2 XXX chain with general (nondiagonal) boundary terms. We argue that a simplified version of this equation describes all the eigenvalues of the transfer matrix of this model. We also propose a generating function for the inhomogeneous T-Q equations of arbitrary spin. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
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21. Identities and exponential bounds for transfer matrices.
- Author
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Guido Molinari, Luca
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TRANSFER matrix , *NUMERICAL analysis , *EIGENVALUES , *BOUNDARY value problems , *VECTOR spaces - Abstract
This paper is about analytic properties of single transfer matrices originating from general block-tridiagonal or banded matrices. Such matrices occur in various applications in physics and numerical analysis. The eigenvalues of the transfer matrix describe localization of eigenstates and are linked to the spectrum of the block tridiagonal matrix by a determinantal identity. If the block tridiagonal matrix is invertible, it is shown that half of the singular values of the transfer matrix have a lower bound exponentially large in the length of the chain, and the other half have an upper bound that is exponentially small. This is a consequence of a theorem by Demko, Moss and Smith on the decay of matrix elements of the inverse of banded matrices. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
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22. Series expansions from the corner transfer matrix renormalization group method: II. Asymmetry and high-density hard squares.
- Author
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Yao-ban Chan
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SERIES expansion (Mathematics) , *TRANSFER matrix , *RENORMALIZATION group , *STATISTICAL mechanics - Abstract
The corner transfer matrix renormalization group method is a method, claimed to be subexponential, for numerically calculating physical quantities of statistical mechanical models. In a previous paper, we extended this method to generate series expansions. Here, we show how to extend both the original and series methods to deal with asymmetry in any model. We discuss the cases of rotational, translational and reflection asymmetry, and give some improvements to the method. This is demonstrated by an application of the method to generate series for the hard square model in the high-density regime, producing 51 terms of the partition function and 48 terms of the order parameter. These series are analysed, producing estimates of the critical point and exponents, and showing the likely presence of a confluent singularity with exponent 17/8 in the order parameter. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
23. Transfer matrix computation of critical polynomials for two-dimensional Potts models.
- Author
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Jacobsen, Jesper Lykke and Scullard, Christian R
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TRANSFER matrix , *POLYNOMIALS , *POTTS model , *TWO-dimensional models , *MANIFOLDS (Mathematics) , *GRAPH theory - Abstract
In our previous work [1] we have shown that critical manifolds of the q-state Potts model can be studied by means of a graph polynomial PB(q, v), henceforth referred to as the critical polynomial. This polynomial may be defined on any periodic two-dimensional lattice. It depends on a finite subgraph B, called the basis, and the manner in which B is tiled to construct the lattice. The real roots v = eK - 1 of PB(q, v) either give the exact critical points for the lattice, or provide approximations that, in principle, can be made arbitrarily accurate by increasing the size of B in an appropriate way. In earlier work, PB(q, v) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give a probabilistic definition of PB(q, v), which facilitates its computation, using the transfer matrix, on much larger B than was previously possible. We present results for the critical polynomial on the (4, 8²), kagome, and (3, 12²) lattices for bases of up to respectively 96, 162, and 243 edges, compared to the limit of 36 edges with contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. The critical temperatures vc obtained for ferromagnetic (u > 0) Potts models are at least as precise as the best available results from Monte Carlo simulations or series expansions. For instance, with q = 3 we obtain vc(4, 8²) = 3.742 489 (4), vc(kagome) = 1.876 459 7 (2), and vc(3, 12²) = 5.033 078 49 (4), the precision being comparable or superior to the best simulation results. More generally, we trace the critical manifolds in the real (q, v) plane and discuss the intricate structure of the phase diagram in the antiferromagnetic (v < 0) region. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
24. Surface critical behaviour of the vertex-interacting self-avoiding walk on the square lattice.
- Author
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Foster, D. P. and Pinettes, C.
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GEOMETRIC surfaces , *SELF-avoiding walks (Mathematics) , *PHASE diagrams , *DENSITY matrices , *EXPONENTS , *RENORMALIZATION group , *TRANSFER matrix - Abstract
The phase diagram and surface critical behaviour of the vertex-interacting selfavoiding walk are examined using transfer matrix methods extended using the density matrix renormalization group method and coupled with finite-size scaling. Particular attention is paid to the critical exponents at the ordinary and special points along the collapse transition line. The question of the bulk exponents (? and ? ) is addressed, and the results found are at variance with previously conjectured exact values. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
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25. The completely packed O(n) loop model on the square lattice.
- Author
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Blöte, Henk W. J., Yougang Wang, and Wenan Guo
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MATHEMATICAL models , *LATTICE field theory , *TRANSFER matrix , *FINITE size scaling (Statistical physics) , *NUMERICAL analysis , *INFINITY (Mathematics) , *STOCHASTIC convergence - Abstract
We present an investigation of the completely packed O(n) loop model on the square lattice by means of the transfer-matrix method and finite-size scaling. We investigate the model for a number of n values covering a wide range. This model is known to be equivalent with the q-state Potts model with q = n2, but here we also investigate the range n < 0, including rather large negative numbers. In the critical range |n| < 2, we find an energy-like scaling dimension X = 4, which is the leading one for n < 1 and the second leading one for 1 < n < 2. The point n = -2 is special, with a conformal anomaly c = -8. For n < -2, the model is no longer critical, as evidenced e.g. by the exponentially fast convergence of the finite-size estimates of the free energy density to the infinite-system value. For |n| > 2, the system is in an ordered phase, where the majority of the loops cover part of the elementary faces of the lattice in one of two checkerboard patterns that are in phase coexistence. Furthermore, we find that the numerical results for the free energy density are in agreement with the expressions obtained from the exact analysis of the equivalent six-vertex model. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
26. Transfer matrix computation of generalized critical polynomials in percolation.
- Author
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Scullard, Christian R. and Jacobsen, Jesper Lykke
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TRANSFER matrix , *POLYNOMIALS , *PERCOLATION theory , *GRAPH theory , *FINITE fields , *APPROXIMATION theory - Abstract
Percolation thresholds have recently been studied by means of a graph polynomial PB(p), henceforth referred to as the critical polynomial, that may be defined on any periodic lattice. The polynomial depends on a finite subgraph B, called the basis, and the way in which the basis is tiled to form the lattice. The unique root of PB(p) in [0, 1] either gives the exact percolation threshold for the lattice, or provides an approximation that becomes more accurate with appropriately increasing size of B. Initially PB(p) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give an alternative probabilistic definition of PB(p), which allows for much more efficient computations, by using the transfer matrix, than was previously possible with contraction-deletion. We present bond percolation polynomials for the (4, 8²), kagome, and (3, 12²) lattices for bases of up to respectively 96, 162 and 243 edges, much larger than the previous limit of 36 edges using contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. For the largest bases, we obtain the thresholds pc(4, 8²) = 0.676 803 329 . . ., pc(kagome) = 0.524 404 998 . . ., pc(3, 122) = 0.740 420 798 . . ., comparable to the best simulation results. We also show that the alternative definition of PB(p) can be applied to study site percolation problems. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
27. Critical properties of joint spin and Fortuin-Kasteleyn observables in the two-dimensional Potts model.
- Author
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Vasseur, Romain and Jacobsen, Jesper Lykke
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- *
TWO-dimensional models , *MATHEMATICAL reformulation , *PARTITIONS (Mathematics) , *MONTE Carlo method , *TRANSFER matrix , *SET theory - Abstract
The two-dimensional Potts model can be studied either in terms of the original Q-component spins or in the geometrical reformulation via Fortuin-Kasteleyn (FK) clusters. While the FK representation makes sense for arbitrary real values of Q by construction, it was only shown very recently that the spin representation can be promoted to the same level of generality. In this paper we show how to define the Potts model in terms of observables that simultaneously keep track of the spin and FK degrees of freedom. This is first done algebraically in terms of a transfer matrix that couples three different representations of a partition algebra. Using this, one can study correlation functions involving any given number of propagating spin clusters with prescribed colours, each of which contains any given number of distinct FK clusters. For 0 ≤ Q ≤ 4, the corresponding critical exponents are all of the Kac form hr,s'with integer indices r, s that we determine exactly both in the bulk and boundary versions of the problem. In particular, we find that the set of points where an FK cluster touches the hull of its surrounding spin cluster has fractal dimension d2,1 = 2 - 2h2,1. If one constrains this set to points where the neighbouring spin cluster extends to infinity, we show that the dimension becomes d1,3 = 2 - 2h1,3. Our results are supported by extensive transfer matrix and Monte Carlo computations. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
28. A new transfer-matrix algorithm for exact enumerations: self-avoiding polygons on the square lattice.
- Author
-
Clisby, Nathan and Jensen, Iwan
- Subjects
- *
TRANSFER matrix , *POLYGONS , *LATTICE theory , *ALGORITHMS , *MATHEMATICAL analysis , *MATHEMATICAL transformations - Abstract
We present a newand more efficient implementation of transfer-matrix methods for exact enumerations of lattice objects. The new method is illustrated by an application to the enumeration of self-avoiding polygons on the square lattice. A detailed comparison with the previous best algorithm shows significant improvement in the running time of the algorithm. The new algorithm is used to extend the enumeration of polygons to length 130 from the previous record of 110. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
29. Series expansions from the corner transfer matrix renormalization group method: the hard-squares model.
- Author
-
Yao-ban Chan
- Subjects
- *
RENORMALIZATION group , *MATHEMATICAL series , *MATHEMATICAL singularities , *NUMERICAL calculations , *TRANSFER matrix , *MATHEMATICAL models , *NUMBER theory - Abstract
The corner transfermatrix renormalization group method is an efficient method for evaluating physical quantities in statistical mechanical models. It originates from Baxter's corner transfer matrix equations and method, and was developed by Nishino and Okunishi in 1996. In this paper, we review and adapt this method, previously used for numerical calculations, to derive series expansions. We use this to calculate 92 terms of the partition function of the hard-squares model. We use the resulting series to provide evidence supporting the claim that the method is subexponential in the number of generated terms, and briefly analyse the singularities of the function. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
30. Statistical mechanical analysis of the linear vector channel in digital communication.
- Author
-
Koujin Takeda, Atsushi Hatabu, and Yoshiyuki Kabashima
- Subjects
- *
STATISTICAL mechanics , *VECTOR spaces , *DIGITAL communications , *WIRELESS communications , *CODE division multiple access , *TRANSFER matrix , *STATISTICAL correlation , *MIMO systems - Abstract
A statistical mechanical framework to analyze linear vector channel models in digital wireless communication is proposed for a large system. The framework is a generalization of that proposed for code-division multiple-access systems in Takeda et al (2006 Europhys. Lett. 76 1193) and enables the analysis of the system in which the elements of the channel transfer matrix are statistically correlated with each other. The significance of the proposed scheme is demonstrated by assessing the performance of an existing model of multi-input multi-output communication systems. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
31. What is the probability of connecting two points?
- Subjects
- *
GRAPH connectivity , *PROBABILITY theory , *PERCOLATION theory , *TRANSFER matrix , *ASYMPTOTIC expansions , *EIGENVALUES , *SCALING laws (Statistical physics) - Abstract
The two-terminal reliability, known as the pair connectedness or connectivity function in percolation theory, may actually be expressed as a product of transfer matrices in which the probability of operation of each link and site is exactly taken into account. When link and site probabilities are p and r, it obeys an asymptotic power-law behaviour, for which the scaling factor is the transfer matrix's eigenvalue of largest modulus. The location of the complex zeros of the two-terminal reliability polynomial exhibits structural transitions as 0 [?] r [?] 1. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
32. Corner transfer matrices in statistical mechanics.
- Subjects
- *
STATISTICAL mechanics , *TRANSFER matrix , *MATHEMATICAL models , *CHIRALITY , *DIMENSIONAL analysis - Abstract
Corner transfer matrices are a useful tool in the statistical mechanics of simple two-dimensional models. They can be a very effective way of obtaining series expansions of unsolved models, and of calculating the order parameters of solved ones. Here we review these features and discuss the reason why the method fails to give the order parameter of the chiral Potts model. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
33. Polynomial solutions of qKZ equation and ground state of XXZ spin chain at ? = ?1/2.
- Author
-
A V Razumov, Yu G Stroganov, and P Zinn
- Subjects
- *
NUMERICAL solutions to difference equations , *POLYNOMIALS , *ENERGY levels (Quantum mechanics) , *R-matrices , *DEFORMATION potential , *EIGENVECTORS , *TRANSFER matrix , *ANTIFERROMAGNETISM , *ANISOTROPY - Abstract
Integral formulae for polynomial solutions of the quantum Knizhnik-Zamolodchikov equations associated with the R-matrix of the six-vertex model are considered. It is proved that when the deformation parameter q is equal to \rme^{\pm 2 \pi \rmi/3} and the number of vertical lines of the lattice is odd, the solution under consideration is an eigenvector of the inhomogeneous transfer matrix of the six-vertex model. In the homogeneous limit, it is a ground-state eigenvector of the antiferromagnetic XXZ spin chain with the anisotropy parameter D equal to [?]1/2 and an odd number of sites. The obtained integral representations for the components of this eigenvector allow us to prove some conjectures on its properties formulated earlier. A new statement relating the ground-state components of XXZ spin chains and Temperley-Lieb loop models is formulated and proved. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
34. Family of commuting operators for the totally asymmetric exclusion process.
- Author
-
O Golinelli and K Mallick
- Subjects
- *
HAMILTONIAN operator , *MARKOV operators , *ORDERED algebraic structures , *BETHE-ansatz technique , *TRANSFER matrix , *COMBINATORICS - Abstract
The algebraic structure underlying the totally asymmetric exclusion process is studied by using the Bethe Ansatz technique. From the properties of the algebra generated by the local jump operators, we explicitly construct the hierarchy of operators (called generalized Hamiltonians) that commute with the Markov operator. The transfer matrix, which is the generating function of these operators, is shown to represent a discrete Markov process with long-range jumps. We give a general combinatorial formula for the connectedHamiltonians obtained by taking the logarithm of the transfer matrix. This formula is proved using a symbolic calculation program for the first ten connected operators. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
35. Creation operators and algebraic Bethe ansatz for the elliptic quantum group E?,?(so3).
- Author
-
Nenand Manojlovi and Zoltán Nagy
- Subjects
- *
BETHE-ansatz technique , *QUANTUM groups , *MATHEMATICAL physics , *TRANSFER matrix , *POLYNOMIALS , *EIGENVECTORS , *EIGENVALUES - Abstract
We define the elliptic quantum group E?,?(so3) and the transfer matrix corresponding to its simplest highest weight representation. We use the Bethe ansatz method to construct the creation operators as polynomials of the Lax matrix elements expressed through a recurrence relation. We give common eigenvectors and eigenvalues of the family of commuting transfer matrices. [ABSTRACT FROM AUTHOR]
- Published
- 2007
36. A Q-operator for the quantum transfer matrix.
- Subjects
- *
TRANSFER matrix , *EIGENVALUES , *QUANTUM groups , *MAGNETIC fields , *MATHEMATICAL physics , *PHYSICS research - Abstract
Baxter's concept of a Q-operator is generalized to the quantum transfer matrix of the XXZ spin-chain by employing the representation theory of quantum groups. The spectrum of this Q-operator is discussed and novel functional relations which describe the finite temperature regime of the XXZ spin-chain are derived. For a non-vanishing magnetic field the previously known Bethe ansatz equations can be replaced by a system of quadratic equations which is an important advantage for numerical studies. For vanishing magnetic field and rational coupling values it is argued that the quantum transfer matrix exhibits a loop algebra symmetry closely related to the one of the classical six-vertex transfer matrix at roots of unity. The quantum-classical crossover is also discussed in terms of the eigenvalues of the Q-operator for a few special examples. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
37. Transfer-matrix formulation of the scattering of electromagnetic waves and broadband invisibility in three dimensions.
- Author
-
Farhang Loran and Ali Mostafazadeh
- Subjects
- *
ELECTROMAGNETIC wave scattering , *POLARIZATION of electromagnetic waves , *INVISIBILITY , *LINEAR operators , *TRANSFER matrix , *GREEN'S functions , *MASS media - Abstract
We develop a transfer-matrix formulation of the scattering of electromagnetic waves by a general isotropic medium which makes use of a notion of electromagnetic transfer matrix that does not involve slicing of the scattering medium or discretization of some of the position- or momentum-space variables. This is a linear operator that we can express as a matrix with operator entries and identify with the S-matrix of an effective nonunitary quantum system. We use this observation to establish the composition property of , obtain an exact solution of the scattering problem for a non-magnetic point scatterer that avoids the divergences of the Green’s function approaches, and prove a general invisibility theorem. The latter allows for an explicit characterization of a class of isotropic media displaying perfect broadband invisibility for electromagnetic waves of arbitrary polarization provided that their wavenumber k does not exceed a preassigned critical value , i.e. behaves exactly like vacuum for . Generalizing this phenomenon, we introduce and study -equivalent media that, by definition, have identical scattering features for . [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
38. Free fermions in disguise.
- Author
-
Paul Fendley
- Subjects
- *
FERMIONS , *CRITICAL exponents , *TRANSFER matrix , *ISING model , *ALGEBRA , *SUPERSYMMETRY - Abstract
I solve a quantum chain whose Hamiltonian is comprised solely of local four-fermi operators by constructing free-fermion raising and lowering operators. The free-fermion operators are both non-local and highly non-linear in the local fermions. This construction yields the complete spectrum of the Hamiltonian and an associated classical transfer matrix. The spatially uniform system is gapless with dynamical critical exponent z = 3/2, while staggering the couplings gives a more conventional free-fermion model with an Ising transition. The Hamiltonian is equivalent to that of a spin-1/2 chain with next-nearest-neighbour interactions, and has a supersymmetry generated by a sum of fermion trilinears. The supercharges are part of a large non-abelian symmetry algebra that results in exponentially large degeneracies. The model is integrable for either open or periodic boundary conditions but the free-fermion construction only works for the former, while for the latter the extended symmetry is broken and the degeneracies split. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
39. A Hubbard model with integrable impurity.
- Author
-
Yahya Öz and Andreas Klümper
- Subjects
- *
HUBBARD model , *TRANSFER matrix , *INTEGRAL equations , *DEGREES of freedom , *EQUATIONS - Abstract
We construct an integrable Hubbard model with impurity site containing spin and charge degrees of freedom. The Bethe ansatz equations for the Hamiltonian are derived and two alternative sets of equations for the thermodynamical properties. For this study, the thermodynamical Bethe ansatz (TBA) and the quantum transfer matrix (QTM) approach are used. The latter approach allows for a consistent treatment by use of a finite set of non-linear integral equations. In both cases, TBA and QTM, the contribution of the impurity to the thermodynamical potential is given by integral expressions. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
40. Complete spectrum of quantum integrable lattice models associated to by separation of variables.
- Author
-
J M Maillet and G Niccoli
- Subjects
- *
SEPARATION of variables , *TRANSFER matrix , *FUNCTIONAL equations , *EIGENVECTORS , *HILBERT space , *FINITE differences - Abstract
In this paper we apply our new separation of variables approach to completely characterize the transfer matrix spectrum for quantum integrable lattice models associated to fundamental evaluation representations of with quasi-periodic boundary conditions. We consider here the case of generic deformations associated to a parameter q which is not a root of unity. The separation of variables (SoV) basis for the transfer matrix spectral problem is generated by using the action of the transfer matrix itself on a generic co-vector of the Hilbert space, following the general procedure described in our paper (Maillet and Niccoli 2018 J. Math. Phys. 59 091417). Such a SoV construction allows to prove that for general values of the parameters defining the model the transfer matrix is diagonalizable and with simple spectrum for any twist matrix which is also diagonalizable with simple spectrum. Then, using together the knowledge of such a SoV basis and of the fusion relations satisfied by the hierarchy of transfer matrices, we derive a complete characterization of the transfer matrix eigenvalues and eigenvectors as solutions of a system of polynomial equations of order n + 1. Moreover, we show that such a SoV discrete spectrum characterization is equivalently reformulated in terms of a finite difference functional equation, the quantum spectral curve equation, under a proper choice of the set of its solutions. A construction of the associated Q-operator induced by our SoV approach is also presented. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
41. Extended T-systems, Q matrices and T-Q relations for models at roots of unity.
- Author
-
Holger Frahm, Alexi Morin-Duchesne, and Paul A Pearce
- Subjects
- *
TRANSFER matrix , *FUNCTIONAL equations , *MATRICES (Mathematics) - Abstract
The mutually commuting fused single and double-row transfer matrices of the critical six-vertex model are considered at roots of unity with crossing parameter a rational fraction of . The transfer matrices of the dense loop model analogs, namely the logarithmic minimal models , are similarly considered. For these models, we find explicit closure relations for the T-system functional equations and obtain extended sets of bilinear T-system identities. We also define extended Q matrices as linear combinations of the fused transfer matrices and obtain extended matrix T-Q relations. These results hold for diagonal twisted boundary conditions on the cylinder as well as invariant/Kac vacuum and off-diagonal/Robin vacuum boundary conditions on the strip. Using our extended T-system and extended T-Q relations for eigenvalues, we deduce the usual scalar Baxter T-Q relation and the Bazhanov–Lukyanov–Zamolodchikov decomposition of the fused transfer matrices and , at fusion level , in terms of the product or Q(u)2. It follows that the zeros of and are comprised of the Bethe roots and complete strings. We also clarify the formal observations of Pronko and Yang–Nepomechie–Zhang and establish, under favourable conditions, the existence of an infinite fusion limit in the auxiliary space of the fused transfer matrices. Despite this connection, the infinite-dimensional oscillator representations are not needed at roots of unity due to finite closure of the functional equations. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
42. Exact solution of the B 1 model.
- Author
-
Panpan Xue, Guang-Liang Li, Junpeng Cao, Kun Hao, Wen-Li Yang, and Kangjie Shi
- Subjects
- *
TRANSFER matrix , *HEISENBERG model , *LIE algebras - Abstract
The B1 vertex model is studied by the off-diagonal Bethe ansatz method. New closed operator product identities of the fused transfer matrices are obtained by using the fusion technique. Based on them and the asymptotic behaviors as well as the values of the fused transfer matrices at certain points, the exact solutions of the B1 model with periodic and with integrable off-diagonal open boundary conditions are obtained. We find that the degree of the polynomial of the inhomogeneous T − Q relation is lower. The B1 model is equivalent to the integrable spin-1 Heisenberg chain, thus the method and the results in this paper can be generalized to the high spin Heisenberg model and the integrable models associated with Bn Lie algebra. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
43. New K-matrices with quantum group symmetry.
- Author
-
Rafael I Nepomechie and Rodrigo A Pimenta
- Subjects
- *
QUANTUM groups , *DYNKIN diagrams , *DUALITY (Logic) , *YANG-Baxter equation , *TRANSFER matrix - Abstract
We propose new families of solutions of the boundary Yang–Baxter equation. The open spin-chain transfer matrices constructed with these K-matrices have quantum group symmetry corresponding to removing one node from the Dynkin diagram, namely, , where . These transfer matrices also have a duality symmetry. These symmetries help to account for the degeneracies in the spectrum of the transfer matrix. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
44. Exact solution of the two-dimensional scattering problem for a class of δ-function potentials supported on subsets of a line.
- Author
-
Farhang Loran and Ali Mostafazadeh
- Subjects
- *
SCATTERING (Mathematics) , *MATHEMATICAL functions , *TRANSFER matrix - Abstract
We use the transfer matrix formulation of scattering theory in two-dimensions (2D) to treat the scattering problem for a potential of the form where , and b are constants, is the Dirac δ function, and g is a real- or complex-valued function. We map this problem to that of and give its exact (nonapproximate) and analytic (closed-form) solution for the following choices of : (i) a linear combination of δ functions, in which case is a finite linear array of 2D δ functions; (ii) a linear combination of with real; (iii) a general periodic function that has the form of a complex Fourier series. In particular we solve the scattering problem for a potential consisting of an infinite linear periodic array of 2D δ functions. We also prove a general theorem that gives a sufficient condition for different choices of to produce the same scattering amplitude within specific ranges of values of the wavelength λ. For example, we show that for arbitrary real and complex parameters, a and , the potentials and have the same scattering amplitude for . [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
45. On transfer matrices, Bethe ansatz and scale invariance.
- Author
-
Alessandro Torrielli
- Subjects
- *
TRANSFER matrix , *MANY-body perturbation calculations , *SCALE invariance (Statistical physics) , *EIGENVALUES , *S-matrix theory , *CRITICAL theory - Abstract
We explicitly calculate the transfer-matrix eigenvalues in the massless sector using the exact integrable S-matrix, for up to 5 particles. This enables us to conjecture the general pattern. We use the conjectured form of the eigenvalues to write down a set of massless Bethe ansatz equations. The same procedure applies to the relativistic as well as to the non-relativistic situation. In the relativistic case, the right and left modes decouple. We speculate that the relativistic massless Bethe ansatz we obtain in that case might capture the integrable structure of an underlying 2D critical theory. We finally take advantage of some remarkable simplifications to make progress in the massive case as well. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
46. Integrals of motion from quantum toroidal algebras.
- Author
-
B Feigin, M Jimbo, and E Mukhin
- Subjects
- *
TOROIDAL harmonics , *TRANSFER matrix , *QUANTUM states - Abstract
We identify the Taylor coefficients of the transfer matrices corresponding to quantum toroidal algebras with the elliptic local and non-local integrals of motion introduced by Kojima, Shiraishi, Watanabe, and one of the authors. That allows us to prove the Litvinov conjectures on the Intermediate Long Wave model. We also discuss the duality of XXZ models in quantum toroidal setting and the implications for the quantum KdV model. In particular, we conjecture that the spectrum of non-local integrals of motion of Bazhanov, Lukyanov, and Zamolodchikov is described by Gaudin Bethe ansatz equations associated to affine . [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
47. On integrable boundaries in the 2 dimensional O(N) σ-models.
- Author
-
Inês Aniceto, Zoltán Bajnok, Tamás Gombor, Minkyoo Kim, and László Palla
- Subjects
- *
INTEGRABLE functions , *DIMENSIONAL analysis , *MATHEMATICAL models , *YANG-Baxter equation , *TRANSFER matrix - Abstract
We make an attempt to map the integrable boundary conditions for 2 dimensional non-linear σ-models. We do it at various levels: classically, by demanding the existence of infinitely many conserved local charges and also by constructing the double row transfer matrix from the Lax connection, which leads to the spectral curve formulation of the problem; at the quantum level, we describe the solutions of the boundary Yang–Baxter equation and derive the Bethe–Yang equations. We then show how to connect the thermodynamic limit of the boundary Bethe–Yang equations to the spectral curve. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
48. The design of efficient dynamic programming and transfer matrix enumeration algorithms.
- Author
-
Andrew R Conway
- Subjects
- *
TRANSFER matrix , *COMPUTER algorithms , *DYNAMIC programming - Abstract
Many algorithms have been developed for enumerating various combinatorial objects in time exponentially less than the number of objects. Two common classes of algorithms are dynamic programming and the transfer matrix method. This paper covers the design and implementation of such algorithms. A host of general techniques for improving efficiency are described. Three quite different example problems are used for detailed examples: 1324 pattern avoiding permutations, three-dimensional polycubes (using a novel approach), and two-dimensional directed animals. Other examples from the literature are used when appropriate to describe applicability of various techniques, but the paper does not attempt to survey all applications. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
49. Scaling in the vicinity of the four-state Potts fixed point.
- Author
-
H W J Blöte, Wenan Guo, and M P Nightingale
- Subjects
- *
FIXED point theory , *TRANSFER matrix , *LOGARITHMIC functions - Abstract
We study a self-dual generalization of the Baxter–Wu model, employing results obtained by transfer matrix calculations of the magnetic scaling dimension and the free energy. While the pure critical Baxter–Wu model displays the critical behavior of the four-state Potts fixed point in two dimensions, in the sense that logarithmic corrections are absent, the introduction of different couplings in the up- and down triangles moves the model away from this fixed point, so that logarithmic corrections appear. Real couplings move the model into the first-order range, away from the behavior displayed by the nearest-neighbor, four-state Potts model. We also use complex couplings, which bring the model in the opposite direction characterized by the same type of logarithmic corrections as present in the four-state Potts model. Our finite-size analysis confirms in detail the existing renormalization theory describing the immediate vicinity of the four-state Potts fixed point. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
50. Quantum walled Brauer algebra: commuting families, Baxterization, and representations.
- Author
-
A M Semikhatov and I Yu Tipunin
- Subjects
- *
COMMUTING operators (Quantum mechanics) , *BLOWING up (Algebraic geometry) , *TRANSFER matrix - Abstract
For the quantum walled Brauer algebra, we construct its Specht modules and (for generic parameters of the algebra) seminormal modules. The latter construction yields the spectrum of a commuting family of Jucys–Murphy elements. We also propose a Baxterization prescription; it involves representing the quantum walled Brauer algebra in terms of morphisms in a braided monoidal category and introducing parameters into these morphisms, which allows constructing a ‘universal transfer matrix’ that generates commuting elements of the algebra. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
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