arXiv subjects

G. von Gehlen

Publications and source records attributed to G. von Gehlen.

17 recordsLinked to original sources

arXiv2009

Spin operator matrix elements in the superintegrable chiral Potts quantum chain

We derive spin operator matrix elements between general eigenstates of the superintegrable Z_N-symmetric chiral Potts quantum chain of finite length. Our starting point is the extended Onsager algebra recently proposed by R.Baxter. For each pair of spaces (Onsager sectors) of the irreducible representations of the Onsager algebra, we calculate the spin matrix elements between the eigenstates of the Hamiltonian of the quantum chain in factorized form, up to an overall scalar factor. This factor is known for the ground state Onsager sectors. For the matrix elements between the ground states of these sectors we perform the thermodynamic limit and obtain the formula for the order parameters. For the Ising quantum chain in a transverse field (N=2 case) the factorized form for the matrix elements coincides with the corresponding expressions obtained recently by the Separation of Variables Method.

cond-mat.stat-mech↗

arXiv2008

Form-factors in the Baxter-Bazhanov-Stroganov model II: Ising model on the finite lattice

We continue our investigation of the Baxter-Bazhanov-Stroganov or τ^{(2)}-model using the method of separation of variables [nlin/0603028,arXiv:0708.4342]. In this paper we derive for the first time the factorized formula for form-factors of the Ising model on a finite lattice conjectured previously by A.Bugrij and O.Lisovyy in [arXiv:0708.3625,arXiv:0708.3643]. We also find the matrix elements of the spin operator for the finite quantum Ising chain in a transverse field.

nlin.SI↗

arXiv2007

Form-factors in the Baxter-Bazhanov-Stroganov model I: Norms and matrix elements

We continue our investigation of the Z_N-Baxter-Bazhanov-Stroganov model using the method of separation of variables [nlin/0603028]. In this paper we calculate the norms and matrix elements of a local Z_N-spin operator between eigenvectors of the auxiliary problem. For the norm the multiple sums over the intermediate states are performed explicitly. In the case N=2 we solve the Baxter equation and obtain form-factors of the spin operator of the periodic Ising model on a finite lattice.

nlin.SI↗

arXiv2006

Baxter-Bazhanov-Stroganov model: Separation of Variables and Baxter Equation

The Baxter-Bazhanov-Stroganov model (also known as the τ^(2) model) has attracted much interest because it provides a tool for solving the integrable chiral Z_N-Potts model. It can be formulated as a face spin model or via cyclic L-operators. Using the latter formulation and the Sklyanin-Kharchev-Lebedev approach, we give the explicit derivation of the eigenvectors of the component B_n(λ) of the monodromy matrix for the fully inhomogeneous chain of finite length. For the periodic chain we obtain the Baxter T-Q-equations via separation of variables. The functional relations for the transfer matrices of the τ^(2) model guarantee non-trivial solutions to the Baxter equations. For the N=2 case, which is free fermion point of a generalized Ising model, the Baxter equations are solved explicitly.

nlin.SI↗

arXiv2005

Bazhanov-Stroganov model from 3D approach

We apply a 3-dimensional approach to describe a new parametrization of the L-operators for the 2-dimensional Bazhanov-Stroganov (BS) integrable spin model related to the chiral Potts model. This parametrization is based on the solution of the associated classical discrete integrable system. Using a 3-dimensional vertex satisfying a modified tetrahedron equation, we construct an operator which generalizes the BS quantum intertwining matrix S. This operator describes the isospectral deformations of the integrable BS model.

nlin.SI↗

arXiv2003

Theta function parameterization and fusion for 3-D integrable Boltzmann weights

We report progress in constructing Boltzmann weights for integrable 3-dimensional lattice spin models. We show that a large class of vertex solutions to the modified tetrahedron equation can be conveniently parameterized in terms of N-th roots of theta-functions on the Jacobian of a compact algebraic curve. Fay's identity guarantees the Fermat relations and the classical equations of motion for the parameters determining the Boltzmann weights. Our parameterization allows to write a simple formula for fused Boltzmann weights R which describe the partition function of an arbitrary open box and which also obey the modified tetrahedron equation. Imposing periodic boundary conditions we observe that the R satisfy the normal tetrahedron equation. The scheme described contains the Zamolodchikov-Baxter-Bazhanov model and the Chessboard model as special cases.

nlin.SI↗

arXiv2003

The modified tetrahedron equation and its solutions

A large class of 3-dimensional integrable lattice spin models is constructed. The starting point is an invertible canonical mapping operator in the space of a triple Weyl algebra. This operator is derived postulating a current branching principle together with a Baxter Z-invariance. The tetrahedron equation for this operator follows without further calculations. If the Weyl parameter is taken to be a root of unity, the mapping operator decomposes into a matrix conjugation and a C-number functional mapping. The operator of the matrix conjugation satisfies a modified tetrahedron equation (MTE) in which the "rapidities" are solutions of a classical integrable Hirota-type equation. The matrix elements of this operator can be represented in terms of the Bazhanov-Baxter Fermat curve cyclic functions, or alternatively in terms of Gauss functions. The paper summarizes several recent publications on the subject.

nlin.SI↗

arXiv2002

Explicit Free Parameterization of the Modified Tetrahedron Equation

The Modified Tetrahedron Equation (MTE) with affine Weyl quantum variables at N-th root of unity is solved by a rational mapping operator which is obtained from the solution of a linear problem. We show that the solutions can be parameterized in terms of eight free parameters and sixteen discrete phase choices, thus providing a broad starting point for the construction of 3-dimensional integrable lattice models. The Fermat curve points parameterizing the representation of the mapping operator in terms of cyclic functions are expressed in terms of the independent parameters. An explicit formula for the density factor of the MTE is derived. For the example N=2 we write the MTE in full detail. We also discuss a solution of the MTE in terms of bosonic continuum functions.

nlin.SI↗

arXiv2002

Onsager's algebra and partially orthogonal polynomials

The energy eigenvalues of the superintegrable chiral Potts model are determined by the zeros of special polynomials which define finite representations of Onsager's algebra. The polynomials determining the low-sector eigenvalues have been given by Baxter in 1988. In the Z_3-case they satisfy 4-term recursion relations and so cannot form orthogonal sequences. However, we show that they are closely related to Jacobi polynomials and satisfy a special "partial orthogonality" with respect to a Jacobi weight function.

hep-th↗

arXiv2001

Correlation Functions and Corner Transfer Matrix of the Chiral Potts Model

We use the Density Matrix Renormalization Group technique to compute correlation functions of the Z_3-chiral Potts quantum chain in the massive regimes. Chains of up to 70 sites are used and a clear oscillatory behavior is found. We also check the relation between the density matrix spectrum and the Corner Transfer Matrix spectrum pointed out by Nishino and Okunishi, using Baxter's low-temperature formulae for the chiral Potts Corner Transfer Matrix eigenvalues. We find very good agreement.

cond-mat↗

arXiv2001

The superintegrable chiral Potts quantum chain and generalized Chebyshev polynomials

Finite-dimensional representations of Onsager's algebra are characterized by the zeros of truncation polynomials. The Z_N-chiral Potts quantum chain hamiltonians (of which the Ising chain hamiltonian is the N=2 case) are the main known interesting representations of Onsager's algebra and the corresponding polynomials have been found by Baxter and Albertini, McCoy and Perk in 1987-89 considering the Yang-Baxter-integrable 2-dimensional chiral Potts model. We study the mathematical nature of these polynomials. We find that for N>2 and fixed charge Q these don't form classical orthogonal sets because their pure recursion relations have at least N+1-terms. However, several basic properties are very similar to those required for orthogonal polynomials. The N+1-term recursions are of the simplest type: like for the Chebyshev polynomials the coefficients are independent of the degree. We find a remarkable partial orthogonality, for N=3,5 with respect to Jacobi-, and for N=4,6 with respect to Chebyshev weight functions. The separation properties of the zeros known from orthogonal polynomials are violated only by the extreme zero at one end of the interval.

hep-th↗

arXiv1998

Finite-size energy levels of the superintegrable chiral Potts model

In the solution of the superintegrable chiral Potts model special polynomials related to the representation theory of the Onsager algebra play a central role. We derive approximate analytic formulae for the zeros of particular polynomials which determine sets of low-lying energy eigenvalues of the chiral Potts quantum chain. These formulae allow the analytic calculation of the leading finite-size corrections to the energy eigenvalues without resorting to a numerical determination of the zeros.

hep-th↗

arXiv1996

Integrable Z_n-Chiral Potts Model: The Missing Rapidity-Momentum Relation

The McCoy-Roan integral representation for gaps of the integrable Z_n- symmetric Chiral Potts quantum chain is used to calculate the boundary of the incommensurable phase for various n. In the limit n -> \infty an analytic formula for this phase boundary is obtained. The McCoy-Roan formula gives the gaps in terms of a rapidity. For the lowest gap we conjecture the relation of this rapidity to the physical momentum in the high-temperature limit using symmetry properties and comparing the McCoy-Roan formula to high-temperature expansions and finite-size data.

hep-th↗

arXiv1994

Excitation Spectrum and Correlation Functions of the Z_3-Chiral Potts Quantum Spin Chain

We study the excitation spectrum and the correlation functions of the Z_3- chiral Potts model in the massive high-temperature phase using perturbation expansions and numerical diagonalization. We are mainly interested in results for general chiral angles but we consider also the superintegrable case. For the parameter values considered, we find that the band structure of the low- lying part of the excitation spectrum has the form expected from a quasiparticle picture with two fundamental particles. Studying the N-dependence of the spectrum, we confirm the stability of the second fundamental particle in a limited range of the momentum, even when its energy becomes so high that it lies very high up among the multiparticle scattering states. This is not a phenomenon restricted to the superintegrable line. Calculating a non-translationally invariant correlation function, we give evidence that it is oscillating. Within our numerical accuracy we find a relation between the oscillation length and the dip position of the momentum dispersion of the lightest particle which seems to be quite independent of the chiral angles.

hep-th↗

arXiv1994

Non-hermitian tricriticality in the Blume-Capel model with imaginary field

Using finite-size-scaling methods, we study the quantum chain version of the spin-$1$-Blume-Capel model coupled to an imaginary field. The aim is to realize higher order non-unitary conformal field theories in a simple Ising-type spin model. We find that the first ground-state level crossing in the high-temperature phase leads to a second-order phase transition of the Yang-Lee universality class (central charge $c=-22/5$). The Yang-Lee transition region ends at a line of a new type of tricriticality, where the {\em three} lowest energy levels become degenerate. The analysis of the spectrum at two points on this line gives good evidence that this line belongs to the universality class of the ${\cal M}_{2,7}$-conformal theory with $c=-68/7$.

hep-th↗

arXiv1992

Multi-particle structure in the Z_n-chiral Potts models

We calculate the lowest translationally invariant levels of the Z_3- and Z_4-symmetrical chiral Potts quantum chains, using numerical diagonalization of the hamiltonian for N <= 12 and N <= 10 sites, respectively, and extrapolating N to infinity. In the high-temperature massive phase we find that the pattern of the low-lying zero momentum levels can be explained assuming the existence of n-1 particles carrying Z_n-charges Q = 1, ... , n-1 (mass m_Q), and their scattering states. In the superintegrable case the masses of the n-1 particles become proportional to their respective charges: m_Q = Q m_1. Exponential convergence in N is observed for the single particle gaps, while power convergence is seen for the scattering levels. We also verify that qualitatively the same pattern appears for the self-dual and integrable cases. For general Z_n we show that the energy-momentum relations of the particles show a parity non-conservation asymmetry which for very high temperatures is exclusive due to the presence of a macroscopic momentum P_m=(1-2Q/n)/ϕ, where ϕis the chiral angle and Q is the Z_n-charge of the respective particle.

hep-th↗

arXiv1992

Phase diagram and two-particle structure of the $Z_3$-chiral Potts model

We calculate the low-lying part of the spectrum of the $Z_3$-symmetrical chiral Potts quantum chain in its self-dual and integrable versions, using numerical diagonalisation of the hamiltonian for $N \leq 12$ sites and extrapolation $N \ra \infty$. From the sequences of levels crossing we show that the massive phases have oscillatory correlation functions. We calculate the wave vector scaling exponent. In the high-temperature massive phase the pattern of the low-lying levels can be explained assuming the existence of two particles, with $Z_3$-charge $Q\!=\!1$ and $Q\!=\!2$, and their scattering states. In the superintegrable case the $Q\!=\!2$-particle has twice the mass of the $Q\!=\!1$-particle. Exponential convergence in $N$ is observed for the single particle gaps, while power convergence is seen for the scattering levels. In the high temperature limit of the self-dual model the parity violation in the particle dispersion relation is equivalent to the presence of a macroscopic momentum $P_m = \pm \vph/3$, where $\vph$ is the chiral angle.

hep-th↗