arXiv subjects

Helen Au-Yang

Publications and source records attributed to Helen Au-Yang.

31 records · Page 2Linked to original sources

arXiv2009

New Results for the Correlation Functions of the Ising Model and the Transverse Ising Chain

In this paper we show how an infinite system of coupled Toda-type nonlinear differential equations derived by one of us can be used efficiently to calculate the time-dependent pair-correlations in the Ising chain in a transverse field. The results are seen to match extremely well long large-time asymptotic expansions newly derived here. For our initial conditions we use new long asymptotic expansions for the equal-time pair correlation functions of the transverse Ising chain, extending an old result of T.T. Wu for the 2d Ising model. Using this one can also study the equal-time wavevector-dependent correlation function of the quantum chain, a.k.a. the q-dependent diagonal susceptibility in the 2d Ising model, in great detail with very little computational effort.

cond-mat.stat-mech↗

arXiv2008

Eigenvectors in the Superintegrable Model I: sl_2 Generators

In order to calculate correlation functions of the chiral Potts model, one only needs to study the eigenvectors of the superintegrable model. Here we start this study by looking for eigenvectors of the transfer matrix of the periodic tau_2(t)model which commutes with the chiral Potts transfer matrix. We show that the degeneracy of the eigenspace of tau_2(t) in the Q=0 sector is 2^r, with r=(N-1)L/N when the size of the transfer matrix L is a multiple of N. We introduce chiral Potts model operators, different from the more commonly used generators of quantum group U-tilde_q(sl-hat(2)). From these we can form the generators of a loop algebra L(sl(2)). For this algebra, we then use the roots of the Drinfeld polynomial to give new explicit expressions for the generators representing the loop algebra as the direct sum of r copies of the simple algebra sl(2).

math-ph↗

arXiv2006

Q-Dependent Susceptibility in Z-Invariant Pentagrid Ising Model

We study the q-dependent susceptibility chi(q) of a Z-invariant ferromagnetic Ising model on a Penrose tiling, as first introduced by Korepin using de Bruijn's pentagrid for the rapidity lines. The pair-correlation function for this model can be calculated exactly using the quadratic difference equations from our previous papers. Its Fourier transform chi(q) is studied using a novel way to calculate the joint probability for the pentagrid neighborhoods of the two spins, reducing this calculation to linear programming. Since the lattice is quasiperiodic, we find that chi(q) is aperiodic and has everywhere dense peaks, which are not all visible at very low or high temperatures. More and more peaks become visible as the correlation length increases--that is, as the temperature approaches the critical temperature.

cond-mat.stat-mech↗

arXiv2006

Q-Dependent Susceptibilities in Ferromagnetic Quasiperiodic Z-Invariant Ising Models

We study the q-dependent susceptibility chi(q) of a series of quasiperiodic Ising models on the square lattice. Several different kinds of aperiodic sequences of couplings are studied, including the Fibonacci and silver-mean sequences. Some identities and theorems are generalized and simpler derivations are presented. We find that the q-dependent susceptibilities are periodic, with the commensurate peaks of chi(q) located at the same positions as for the regular Ising models. Hence, incommensurate everywhere-dense peaks can only occur in cases with mixed ferromagnetic-antiferromagnetic interactions or if the underlying lattice is aperiodic. For mixed-interaction models the positions of the peaks depend strongly on the aperiodic sequence chosen.

cond-mat.stat-mech↗

arXiv2006

The N to Infinity Limit of the Chiral Potts Model

We consider the N to infinity limits of the N-state chiral Potts model. We find new weights that satisfy the star-triangle relations with spin variables either taking all the integer values or having values from a continous interval. The models provide chiral generalizations of Zamolodchikov's Fishnet Model. (For the more complete version, see math.QA/9906029, where the misprints in eq. (12) are also corrected.)

hep-th↗

arXiv2006

Yang-Baxter Equations

Preprint of an Encyclopedia article (Encyclopedia of Mathematical Physics, eds. J.-P. Françoise, G.L. Naber and Tsou S.T., Oxford: Elsevier, 2006 (ISBN 978-0-1251-2666-3), volume 5, pages 465-473) extended with an appendix on relations of electric networks and solvable models. We welcome comments, especially on the added appendix. Please, note that the reference list has been restricted to only giving a few historical items and sources explicitly used. We apologize in advance for the many omissions.

math-ph↗

arXiv2006

Some Recent Results on Pair Correlation Functions and Susceptibilities in Exactly Solvable Models

Using detailed exact results on pair-correlation functions of Z-invariant Ising models, we can write and run algorithms of polynomial complexity to obtain wavevector-dependent susceptibilities for a variety of Ising systems. Reviewing recent work we compare various periodic and quasiperiodic models, where the couplings and/or the lattice may be aperiodic, and where the Ising couplings may be either ferromagnetic, or antiferromagnetic, or of mixed sign. We present some of our results on the square-lattice fully-frustrated Ising model. Finally, we make a few remarks on our recent works on the pentagrid Ising model and on overlapping unit cells in three dimensions and how these works can be utilized once more detailed results for pair correlations in, e.g., the eight-vertex model or the chiral Potts model or even three-dimensional Yang-Baxter integrable models become available.

math-ph↗

arXiv2006

Overlapping Unit Cells in 3d Quasicrystal Structure

A 3-dimensional quasiperiodic lattice, with overlapping unit cells and periodic in one direction, is constructed using grid and projection methods pioneered by de Bruijn. Each unit cell consists of 26 points, of which 22 are the vertices of a convex polytope P, and 4 are interior points also shared with other neighboring unit cells. Using Kronecker's theorem the frequencies of all possible types of overlapping are found.

cond-mat.other↗

arXiv2006

Recent Developments on Ising and Chiral Potts Model

After briefly reviewing selected Ising and chiral Potts model results, we discuss a number of properties of cyclic hypergeometric functions which appear naturally in the description of the integrable chiral Potts model and its three-dimensional generalizations.

math-ph↗

arXiv2006

Quasicrystals: Projections of 5-d Lattice into 2 and 3 Dimensions

We show that generalized Penrose tilings can be obtained by the projection of a cut plane of a 5-dimensional lattice into two dimensions, while 3-d quasiperiodic lattices with overlapping unit cells are its projections into 3d. The frequencies of all possible vertex types in the generalized Penrose tilings, and the frequencies of all possible types of overlapping 3-d unit cells are also given here. The generalized Penrose tilings are found to be nonconvertable to kite and dart patterns, nor can they be described by the overlapping decagons of Gummelt.

math-ph↗

arXiv1999

The Large N Limits of the Chiral Potts Model

In this paper we study the large-N limits of the integrable N-state chiral Potts model. Three chiral solutions of the star-triangle equations are derived, with states taken from all integers, or from a finite or infinite real interval. These solutions are expected to be chiral-field lattice deformations of parafermionic conformal field theories. A new two-sided hypergeometric identity is derived as a corollary.

math.QA↗

arXiv1996

The Many Faces of the Chiral Potts Model

In this talk, we give a brief overview of several aspects of the theory of the chiral Potts model, including higher-genus solutions of the star-triangle and tetrahedron equations, cyclic representations of affine quantum groups, basic hypergeometric functions at root of unity, and possible applications.

q-alg↗

arXiv1994

The Chiral Potts Models Revisited

In honor of Onsager's ninetieth birthday, we like to review some exact results obtained so far in the chiral Potts models and to translate these results into language more transparent to physicists, so that experts in Monte Carlo calculations, high and low temperature expansions, and various other methods, can use them. We shall pay special attention to the interfacial tension $ε_r$ between the $k$ state and the $k-r$ state. By examining the ground states, it is seen that the integrable line ends at a superwetting point, on which the relation $ε_r=rε_1$ is satisfied, so that it is energetically neutral to have one interface or more. We present also some partial results on the meaning of the integrable line for low temperatures where it lives in the non-wet regime. We make Baxter's exact results more explicit for the symmetric case. By performing a Bethe Ansatz calculation with open boundary conditions we confirm a dilogarithm identity for the low-temperature expansion which may be new. We propose a new model for numerical studies. This model has only two variables and exhibits commensurate and incommensurate phase transitions and wetting transitions near zero temperature. It appears to be not integrable, except at one point, and at each temperature there is a point, where it is almost identical with the integrable chiral Potts model.

cond-mat↗