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:1607.06968 · Physical review. E

Phase transitions of the random-bond Potts chain with long-range interactions

Jean-Christian Anglès d’Auriac, Ferenc Iglói

Published Dec 19, 2016• 8 citations
arXivHTMLPDF

Abstract

We study phase transitions of the ferromagnetic $q$-state Potts chain with random nearest-neighbor couplings having a variance ${\mathrm{\ensuremath{\Delta}}}^{2}$ and with homogeneous long-range interactions, which decay with distance as a power ${r}^{\ensuremath{-}(1+\ensuremath{\sigma})}, \ensuremath{\sigma}>0$. In the large-$q$ limit the free-energy of random samples of length $L\ensuremath{\le}2048$ is calculated exactly by a combinatorial optimization algorithm. The phase transition stays first order for $\ensuremath{\sigma}<{\ensuremath{\sigma}}_{c}(\mathrm{\ensuremath{\Delta}})\ensuremath{\le}0.5$, while the correlation length becomes divergent at the transition point for ${\ensuremath{\sigma}}_{c}(\mathrm{\ensuremath{\Delta}})<\ensuremath{\sigma}<1$. In the latter regime the average magnetization is continuous for small enough $\mathrm{\ensuremath{\Delta}}$, but for larger $\mathrm{\ensuremath{\Delta}}$---according to the numerical results---it becomes discontinuous at the transition point, thus the phase transition is expected of mixed order.

DOI
10.1103/physreve.94.062126

Related papers

Phase Transition in the 2D Random Potts Model in the Large- q Limit

J C Anglès d'Auriac, Ferenc Iglói

Phase transition in the two-dimensional q-state Potts model with random ferromagnetic couplings is studied in the large-q limit by a combinatorial optimization algorithm and by approximate mappings. We conjecture that the critical behavior of the model is controlled by the isotropic version of the infinite randomness fixed point of the random transverse-field Ising spin chain and the critical exponents are exactly given by beta=(3-sqrt[5])/4, beta(s)=1/2, and nu=1. The specific heat has a logarithmic singularity, but at the transition point there are very strong sample-to-sample fluctuations. Discretized randomness results in discontinuities in the internal energy.

arXiv:cond-mat/0211543•Physical Review Letters•May 13, 2003•32 citations

Disorder-induced rounding of the phase transition in the large- q -state Potts model

Maria Teresa Mercaldo, J C Anglès d'Auriac, Ferenc Iglói

The phase transition in the q -state Potts model with homogeneous ferromagnetic couplings is strongly first order for large q, while it is rounded in the presence of quenched disorder. Here we study this phenomenon on different two-dimensional lattices by using the fact that the partition function of the model is dominated by a single diagram of the high-temperature expansion, which is calculated by an efficient combinatorial optimization algorithm. For a given finite sample with discrete randomness the free energy is a piecewise linear function of the temperature, which is rounded after averaging, however, the discontinuity of the internal energy at the transition point (i.e., the latent heat) stays finite even in the thermodynamic limit. For a continuous disorder, instead, the latent heat vanishes. At the phase transition point the dominant diagram percolates and the total magnetic moment is related to the size of the percolating cluster. Its fractal dimension is found d(f) = ( 5 + square root of 5)/4 and it is independent of the type of the lattice and the form of disorder. We argue that the critical behavior is exclusively determined by disorder and the corresponding fixed point is the isotropic version of the so-called infinite randomness fixed point, which is realized in random quantum spin chains. From this mapping we conjecture the values of the critical exponents as beta=2- d(f), beta(s) =1/2, and nu=1.

arXiv:cond-mat/0312532•Physical Review E•May 24, 2004•25 citations

Ferromagnetic phase transition in random Potts spin chains

Jian-Cheng Lin, Philip L. Taylor, Rafael Rangel

We study the ferromagnetic phase transition in a model consisting of one-dimensional three-state Potts spin chains with random intrachain couplings and comparatively weak ferromagnetic interchain couplings. Mean-field theory is employed to decouple approximately the interchain couplings. The transfer-matrix method is then used to study the resulting effectively one-dimensional random-bond Potts model. The free energy as a function of the ferromagnetic order is calculated numerically and a first-order ferromagnetic-paramagnetic phase transition is found over a wide range of degree of randomness.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Feb 1, 1993•3 citations

Long-range random transverse-field Ising model in three dimensions

István A. Kovács, Róbert Juhász, Ferenc Iglói

We consider the random transverse-field Ising model in $d=3$ dimensions with long-range ferromagnetic interactions which decay as a power $\ensuremath{\alpha}>d$ with the distance. Using a variant of the strong-disorder renormalization group method we study numerically the phase-transition point from the paramagnetic side. We find that the fixed point controlling the transition is of the strong-disorder type, and based on experience with other similar systems, we expect the results to be qualitatively correct, but probably not asymptotically exact. The distribution of the (sample dependent) pseudocritical points is found to scale with $1/lnL, L$ being the linear size of the sample. Similarly, the critical magnetization scales with ${(lnL)}^{\ensuremath{\chi}}/{L}^{d}$ and the excitation energy behaves as ${L}^{\ensuremath{-}\ensuremath{\alpha}}$. Using extreme-value statistics we argue that extrapolating from the ferromagnetic side the magnetization approaches a finite limiting value and thus the transition is of mixed order.

arXiv:1601.04206•Physical review. B./Physical review. B•May 31, 2016•29 citations

First-order phase transition in $1d$ Potts model with long-range interactions

Katarina Uzelac, Zvonko Glumac

The first-order phase transition in the one-dimensional $q$-state Potts model with long-range interactions decaying with distance as $1/r^{1+σ}$ has been studied by Monte Carlo numerical simulations for $0 < σ\le 1$ and integer values of $q > 2$. On the basis of finite-size scaling analysis of interface free energy $ΔF_L$, specific heat and Binder's fourth order cumulant, we obtain the first-order transition which occurs for $σ$ below a threshold value $σ_c(q)$.

arXiv:cond-mat/9807416•Fizika B•Jul 31, 1998•3 citations

First-order transitions in large-q one-dimensional Potts model with power-law decaying interactions

Zvonko Glumac, Katarina Uzelac

We have performed exact analytical calculations of the cluster probability distribution and interface free energy in the finite q-state Potts chains with interactions decaying with distance r as r^(1+sigma). By presenting the partition function in form of the polynomial in q, the advantage of using the Hamiltonian configurations compared with graph or connectivity approach was examined. Scaling analysis of the probability performed for large q (q >= 16) yields the first-order phase transition for 0 < sigma < 1.

•Jan 1, 2005•0 citations
We study phase transitions of the ferromagnetic qqq-state Potts chain with random nearest-neighbor couplings having a variance \ensuremathΔ2{\mathrm{\ensuremath{\Delta}}}^{2}\ensuremathΔ2 and with homogeneous long-range interactions, which decay with distance as a power r\ensuremath−(1+\ensuremathσ),\ensuremathσ>0{r}^{\ensuremath{-}(1+\ensuremath{\sigma})}, \ensuremath{\sigma}>0r\ensuremath−(1+\ensuremathσ),\ensuremathσ>0. In the large-qqq limit the free-energy of random samples of length L\ensuremath≤2048L\ensuremath{\le}2048L\ensuremath≤2048 is calculated exactly by a combinatorial optimization algorithm. The phase transition stays first order for \ensuremathσ<\ensuremathσc(\ensuremathΔ)\ensuremath≤0.5\ensuremath{\sigma}<{\ensuremath{\sigma}}_{c}(\mathrm{\ensuremath{\Delta}})\ensuremath{\le}0.5\ensuremathσ<\ensuremathσc​(\ensuremathΔ)\ensuremath≤0.5, while the correlation length becomes divergent at the transition point for \ensuremathσc(\ensuremathΔ)<\ensuremathσ<1{\ensuremath{\sigma}}_{c}(\mathrm{\ensuremath{\Delta}})<\ensuremath{\sigma}<1\ensuremathσc​(\ensuremathΔ)<\ensuremathσ<1. In the latter regime the average magnetization is continuous for small enough \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ, but for larger \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ---according to the numerical results---it becomes discontinuous at the transition point, thus the phase transition is expected of mixed order.
Phase transition in the two-dimensional q-state Potts model with random ferromagnetic couplings is studied in the large-q limit by a combinatorial optimization algorithm and by approximate mappings. We conjecture that the critical behavior of the model is controlled by the isotropic version of the infinite randomness fixed point of the random transverse-field Ising spin chain and the critical exponents are exactly given by beta=(3-sqrt[5])/4, beta(s)=1/2, and nu=1. The specific heat has a logarithmic singularity, but at the transition point there are very strong sample-to-sample fluctuations. Discretized randomness results in discontinuities in the internal energy.
The phase transition in the q -state Potts model with homogeneous ferromagnetic couplings is strongly first order for large q, while it is rounded in the presence of quenched disorder. Here we study this phenomenon on different two-dimensional lattices by using the fact that the partition function of the model is dominated by a single diagram of the high-temperature expansion, which is calculated by an efficient combinatorial optimization algorithm. For a given finite sample with discrete randomness the free energy is a piecewise linear function of the temperature, which is rounded after averaging, however, the discontinuity of the internal energy at the transition point (i.e., the latent heat) stays finite even in the thermodynamic limit. For a continuous disorder, instead, the latent heat vanishes. At the phase transition point the dominant diagram percolates and the total magnetic moment is related to the size of the percolating cluster. Its fractal dimension is found d(f) = ( 5 + square root of 5)/4 and it is independent of the type of the lattice and the form of disorder. We argue that the critical behavior is exclusively determined by disorder and the corresponding fixed point is the isotropic version of the so-called infinite randomness fixed point, which is realized in random quantum spin chains. From this mapping we conjecture the values of the critical exponents as beta=2- d(f), beta(s) =1/2, and nu=1.
We study the ferromagnetic phase transition in a model consisting of one-dimensional three-state Potts spin chains with random intrachain couplings and comparatively weak ferromagnetic interchain couplings. Mean-field theory is employed to decouple approximately the interchain couplings. The transfer-matrix method is then used to study the resulting effectively one-dimensional random-bond Potts model. The free energy as a function of the ferromagnetic order is calculated numerically and a first-order ferromagnetic-paramagnetic phase transition is found over a wide range of degree of randomness.
We consider the random transverse-field Ising model in d=3d=3d=3 dimensions with long-range ferromagnetic interactions which decay as a power $\ensuremath{\alpha}>d$ with the distance. Using a variant of the strong-disorder renormalization group method we study numerically the phase-transition point from the paramagnetic side. We find that the fixed point controlling the transition is of the strong-disorder type, and based on experience with other similar systems, we expect the results to be qualitatively correct, but probably not asymptotically exact. The distribution of the (sample dependent) pseudocritical points is found to scale with 1/lnL,L1/lnL, L1/lnL,L being the linear size of the sample. Similarly, the critical magnetization scales with ${(lnL)}^{\ensuremath{\chi}}/{L}^{d}$ and the excitation energy behaves as ${L}^{\ensuremath{-}\ensuremath{\alpha}}$. Using extreme-value statistics we argue that extrapolating from the ferromagnetic side the magnetization approaches a finite limiting value and thus the transition is of mixed order.
First-order phase transition in 1d1d1d Potts model with long-range interactions
The first-order phase transition in the one-dimensional qqq-state Potts model with long-range interactions decaying with distance as 1/r1+σ1/r^{1+σ}1/r1+σ has been studied by Monte Carlo numerical simulations for 0<σ≤10 < σ\le 10<σ≤1 and integer values of q>2q > 2q>2. On the basis of finite-size scaling analysis of interface free energy ΔFLΔF_LΔFL​, specific heat and Binder's fourth order cumulant, we obtain the first-order transition which occurs for σσσ below a threshold value σc(q)σ_c(q)σc​(q).
We have performed exact analytical calculations of the cluster probability distribution and interface free energy in the finite q-state Potts chains with interactions decaying with distance r as r^(1+sigma). By presenting the partition function in form of the polynomial in q, the advantage of using the Hamiltonian configurations compared with graph or connectivity approach was examined. Scaling analysis of the probability performed for large q (q >= 16) yields the first-order phase transition for 0 < sigma < 1.