Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1701.09058 · Journal of Physics A Mathematical and Theoretical

One-dimensional q -state Potts model with multi-site interactions

Loïc Turban

Published Apr 3, 2017• 2 citations
arXivHTMLPDF

Abstract

Abstract A one-dimensional (1D) q -state Potts model with N sites, m -site interaction K in a field H is studied for arbitrary values of m . Exact results for the partition function and the two-point correlation function are obtained at H = 0. The system in a field is shown to be self-dual. Using a change of Potts variables, it is mapped onto a standard 2D Potts model, with first-neighbour interactions K and H , on a cylinder with helical boundary conditions (BC). The 2D system has a length N / m and a transverse size m . Thus the Potts chain with multi-site interactions is expected to develop a 2D critical singularity along the self-duality line, ( e q K − 1 ) ( e q H − 1 ) = q , when N / m → ∞ and m → ∞ .

DOI
10.1088/1751-8121/aa6ad1

Related papers

Exact results for quantum chains with multisite interactions

Loïc Turban

Generalised spin-1/2 and Potts quantum chains with m- (n-) site interactions for the z(x) component of the site variables are shown to be self-dual. Through duality, with m arbitrary, n=2 versions of the models are transformed into m independent Ising or Potts chains in a transverse field, allowing one to get the exact values of the critical exponents for any m.

Journal of Physics A Mathematical and General•Aug 21, 1985•11 citations

One-dimensional Ising model with multispin interactions

Loïc Turban

Abstract We study the spin-1/2 Ising chain with multispin interactions K involving the product of m successive spins, for general values of m . Using a change of spin variables the zero-field partition function of a finite chain is obtained for free and periodic boundary conditions and we calculate the two-spin correlation function. When placed in an external field H the system is shown to be self-dual. Using another change of spin variables the one-dimensional Ising model with multispin interactions in a field is mapped onto a zero-field rectangular Ising model with first-neighbour interactions K and H . The 2D system, with size m × N / m , has the topology of a cylinder with helical BC. In the thermodynamic limit N / m → ∞ , m → ∞ , a 2D critical singularity develops on the self-duality line, sinh 2 K sinh 2 H = 1 .

arXiv:1605.05199•Journal of Physics A Mathematical and Theoretical•Aug 8, 2016•50 citations

Self-dual D-dimensional quantum Potts model with multi-spin interactions

Loïc Turban

Introduces a family of D-dimensional q-state quantum Potts models with multi-spin interactions, which are self-dual for any D. These models are introduced using their (d=D+1)-dimensional classical formulations and the transfer matrix technique. The self-duality is proved on the quantum Hamiltonian. In the limit q to infinity the ground state energy is obtained exactly and the phase transition is first-order. A 1/q expansion allows one to get an approximate expression for the line in the (q,D)-plane separating regions where the transition changes from first- to second-order.

Journal of Physics A Mathematical and General•May 1, 1982•0 citations

Double potts chain and exact results for some two-dimensional spin models

M. A. Yurishchev

An exact analytical solution for the q -state Potts model on a 2×∞ ladder with arbitrary two-, three-, and four-site interactions in a unit cell in presented is a closed form. This solution is used to show that the finite-size internal energy equation [6] yields an accurate value of the critical temperature for the triangular Potts lattice with three-site interactions in alternate triangular faces. It is argued that the above equation is exact at least for self-dual models on isotopic strips.

arXiv:cond-mat/0111401•Journal of Experimental and Theoretical Physics•Nov 1, 2001•1 citation

Renormalization-group study of the Hamiltonian version of the Potts model. II. Self-dual renormalization-group treatment

Ferenc Iglói, Jenő Sólyom

The one-dimensional Hamiltonian version of the two-dimensional Potts model is studied by the application of a self-dual renormalization-group transformation. Blocks containing up to 11 sites are considered and mapped onto new, equivalent Potts spins by keeping the lowest-lying levels only. The higher-lying states are also taken into account in a perturbational way for smaller cells. Unfortunately, this renormalization-group treatment cannot reproduce the first-order transition for large values of the number of states, and the critical exponents for the three- and four-state Potts model do not seem to converge to the conjectured results as larger and larger blocks are taken.

Physical review. B, Condensed matter•Sep 1, 1983•14 citations

Monte Carlo study of the critical behavior of random bond Potts models

T. Olson, A. P. Young

We present results of Monte Carlo simulations of random bond Potts models in two dimensions, for different numbers of Potts states q. We introduce a simple scheme which yields continuous self-dual distributions of the interactions. As expected, we find multifractal behavior of the correlation functions at the critical point and obtain estimates of the exponent ${\ensuremath{\eta}}_{n}$ for several moments n of the correlation functions, including typical $(\stackrel{\ensuremath{\rightarrow}}{n}0)$, average $(n=1)$, and others. In addition, for $q=8$, we find that there is only a single correlation length exponent $\ensuremath{\nu}$ describing the correlation length away from criticality. This is numerically very close to the pure Ising value $\ensuremath{\nu}=1$.

arXiv:cond-mat/9903068•Physical review. B, Condensed matter•Aug 1, 1999•71 citations
Abstract A one-dimensional (1D) q -state Potts model with N sites, m -site interaction K in a field H is studied for arbitrary values of m . Exact results for the partition function and the two-point correlation function are obtained at H = 0. The system in a field is shown to be self-dual. Using a change of Potts variables, it is mapped onto a standard 2D Potts model, with first-neighbour interactions K and H , on a cylinder with helical boundary conditions (BC). The 2D system has a length N / m and a transverse size m . Thus the Potts chain with multi-site interactions is expected to develop a 2D critical singularity along the self-duality line, ( e q K − 1 ) ( e q H − 1 ) = q , when N / m → ∞ and m → ∞ .
Abstract We study the spin-1/2 Ising chain with multispin interactions K involving the product of m successive spins, for general values of m . Using a change of spin variables the zero-field partition function of a finite chain is obtained for free and periodic boundary conditions and we calculate the two-spin correlation function. When placed in an external field H the system is shown to be self-dual. Using another change of spin variables the one-dimensional Ising model with multispin interactions in a field is mapped onto a zero-field rectangular Ising model with first-neighbour interactions K and H . The 2D system, with size m × N / m , has the topology of a cylinder with helical BC. In the thermodynamic limit N / m → ∞ , m → ∞ , a 2D critical singularity develops on the self-duality line, sinh 2 K sinh 2 H = 1 .
Introduces a family of D-dimensional q-state quantum Potts models with multi-spin interactions, which are self-dual for any D. These models are introduced using their (d=D+1)-dimensional classical formulations and the transfer matrix technique. The self-duality is proved on the quantum Hamiltonian. In the limit q to infinity the ground state energy is obtained exactly and the phase transition is first-order. A 1/q expansion allows one to get an approximate expression for the line in the (q,D)-plane separating regions where the transition changes from first- to second-order.
The one-dimensional Hamiltonian version of the two-dimensional Potts model is studied by the application of a self-dual renormalization-group transformation. Blocks containing up to 11 sites are considered and mapped onto new, equivalent Potts spins by keeping the lowest-lying levels only. The higher-lying states are also taken into account in a perturbational way for smaller cells. Unfortunately, this renormalization-group treatment cannot reproduce the first-order transition for large values of the number of states, and the critical exponents for the three- and four-state Potts model do not seem to converge to the conjectured results as larger and larger blocks are taken.
We present results of Monte Carlo simulations of random bond Potts models in two dimensions, for different numbers of Potts states q. We introduce a simple scheme which yields continuous self-dual distributions of the interactions. As expected, we find multifractal behavior of the correlation functions at the critical point and obtain estimates of the exponent ${\ensuremath{\eta}}_{n}$ for several moments n of the correlation functions, including typical $(\stackrel{\ensuremath{\rightarrow}}{n}0)$, average (n=1)(n=1)(n=1), and others. In addition, for q=8q=8q=8, we find that there is only a single correlation length exponent $\ensuremath{\nu}$ describing the correlation length away from criticality. This is numerically very close to the pure Ising value $\ensuremath{\nu}=1$.