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Jacques H. H. Perk

Publications and source records attributed to Jacques H. H. Perk.

At least 37 records · Page 2Linked to original sources
arXiv2011

Spontaneous Magnetization of the Integrable Chiral Potts Model

We show how $Z$-invariance in the chiral Potts model provides a strategy to calculate the pair correlation in the general integrable chiral Potts model using only the superintegrable eigenvectors. When the distance between the two spins in the correlation function becomes infinite it becomes the square of the order parameter. In this way, we show that the spontaneous magnetization can be expressed in terms of the inner products of the eigenvectors of the $N$ asymptotically degenerate maximum eigenvalues. Using our previous results on these eigenvectors, we are able to obtain the order parameter as a sum almost identical to the one given by Baxter. This gives the known spontaneous magnetization of the chiral Potts model by an entirely different approach.

math-ph↗
arXiv2010

Quantum Loop Subalgebra and Eigenvectors of the Superintegrable Chiral Potts Transfer Matrices

It has been shown in earlier works that for Q=0 and L a multiple of N, the ground state sector eigenspace of the superintegrable tau_2(t_q) model is highly degenerate and is generated by a quantum loop algebra L(sl_2). Furthermore, this loop algebra can be decomposed into r=(N-1)L/N simple sl_2 algebras. For Q not equal 0, we shall show here that the corresponding eigenspace of tau_2(t_q) is still highly degenerate, but splits into two spaces, each containing 2^{r-1} independent eigenvectors. The generators for the sl_2 subalgebras, and also for the quantum loop subalgebra, are given generalizing those in the Q=0 case. However, the Serre relations for the generators of the loop subalgebra are only proven for some states, tested on small systems and conjectured otherwise. Assuming their validity we construct the eigenvectors of the Q not equal 0 ground state sectors for the transfer matrix of the superintegrable chiral Potts model.

math-ph↗
arXiv2009

Eigenvectors in the Superintegrable Model II: Ground State Sector

In 1993, Baxter gave $2^{m_Q}$ eigenvalues of the transfer matrix of the $N$-state superintegrable chiral Potts model with spin-translation quantum number $Q$, where $m_Q=\lfloor(NL-L-Q)/N\rfloor$. In our previous paper we studied the Q=0 ground state sector, when the size $L$ of the transfer matrix is chosen to be a multiple of $N$. It was shown that the corresponding $τ_2$ matrix has a degenerate eigenspace generated by the generators of $r=m_0$ simple $sl_2$ algebras. These results enable us to express the transfer matrix in the subspace in terms of these generators $E_m^{\pm}$ and $H_m$ for $m=1,...,r$. Moreover, the corresponding $2^r$ eigenvectors of the transfer matrix are expressed in terms of rotated eigenvectors of $H_m$.

math-ph↗
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↗
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↗