arXiv:2003.02506

The elastic backbone phase transition in the Ising model

M. N. Najafi, Jafar Cheraghalizadeh, Hans Jürgen Herrmann

Published Mar 5, 2020• 0 citations

Abstract

The two-dimensional (zero magnetic field) Ising model is known to undergo a second order para-ferromagnetic phase transition, which is accompanied by a correlated percolation transition for the Fortuin-Kasteleyn (FK) clusters. In this paper we uncover that there exists also a second temperature $T_{\text{eb}}

DOI: 10.48550/arxiv.2003.02506

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arXiv:2003.02506

The elastic backbone phase transition in the Ising model

M. N. Najafi, Jafar Cheraghalizadeh, Hans Jürgen Herrmann

Published Mar 5, 2020• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.2003.02506

Related papers

Elastic backbone phase transition in the Ising model

M. N. Najafi, Jafar Cheraghalizadeh, Hans Jürgen Herrmann

The two-dimensional (zero magnetic field) Ising model is known to undergo a second-order paraferromagnetic phase transition, which is accompanied by a correlated percolation transition for the Fortuin-Kasteleyn (FK) clusters. In this paper we uncover that there exists also a second temperature ${T}_{\text{eb}}<{T}_{c}$ at which the elastic backbone of FK clusters undergoes a second-order phase transition to a dense phase. The corresponding universality class, which is characterized by determining various percolation exponents, is shown to be completely different from directed percolation, which leads us to propose a new anisotropic universality class with $\ensuremath{\beta}=0.54\ifmmode\pm\else\textpm\fi{}0.02$, ${\ensuremath{\nu}}_{||}=1.86\ifmmode\pm\else\textpm\fi{}0.01$, ${\ensuremath{\nu}}_{\ensuremath{\perp}}=1.21\ifmmode\pm\else\textpm\fi{}0.04$, and ${d}_{f}=1.53\ifmmode\pm\else\textpm\fi{}0.03$. All tested hyperscaling relations are shown to be valid.The two-dimensional (zero magnetic field) Ising model is known to undergo a second-order paraferromagnetic phase transition, which is accompanied by a correlated percolation transition for the Fortuin-Kasteleyn (FK) clusters. In this paper we uncover that there exists also a second temperature ${T}_{\text{eb}}<{T}_{c}$ at which the elastic backbone of FK clusters undergoes a second-order phase transition to a dense phase. The corresponding universality class, which is characterized by determining various percolation exponents, is shown to be completely different from directed percolation, which leads us to propose a new anisotropic universality class with $\ensuremath{\beta}=0.54\ifmmode\pm\else\textpm\fi{}0.02$, ${\ensuremath{\nu}}_{||}=1.86\ifmmode\pm\else\textpm\fi{}0.01$, ${\ensuremath{\nu}}_{\ensuremath{\perp}}=1.21\ifmmode\pm\else\textpm\fi{}0.04$, and ${d}_{f}=1.53\ifmmode\pm\else\textpm\fi{}0.03$. All tested hyperscaling relations are shown to be valid.

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Santo Fortunato

The properties of the pure-site clusters of spin models, i.e., the clusters which are obtained by joining nearest-neighbor spins of the same sign, are here investigated. In the Ising model in two dimensions it is known that such clusters undergo a percolation transition exactly at the critical point. We show that this result is valid for a wide class of bidimensional systems undergoing a continuous magnetization transition. We provide numerical evidence for discrete as well as for continuous spin models, including $\mathrm{SU}(N)$ lattice gauge theories. The critical percolation exponents do not coincide with the ones of the thermal transition, but they are the same for models belonging to the same universality class.The properties of the pure-site clusters of spin models, i.e., the clusters which are obtained by joining nearest-neighbor spins of the same sign, are here investigated. In the Ising model in two dimensions it is known that such clusters undergo a percolation transition exactly at the critical point. We show that this result is valid for a wide class of bidimensional systems undergoing a continuous magnetization transition. We provide numerical evidence for discrete as well as for continuous spin models, including $\mathrm{SU}(N)$ lattice gauge theories. The critical percolation exponents do not coincide with the ones of the thermal transition, but they are the same for models belonging to the same universality class.

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{T}_{\text{eb}}<{T}_{c}

at which the elastic backbone of FK clusters undergoes a second-order phase transition to a dense phase. The corresponding universality class, which is characterized by determining various percolation exponents, is shown to be completely different from directed percolation, which leads us to propose a new anisotropic universality class with $\ensuremath{\beta}=0.54\ifmmode\pm\else\textpm\fi{}0.02$, ${\ensuremath{\nu}}_{||}=1.86\ifmmode\pm\else\textpm\fi{}0.01$, ${\ensuremath{\nu}}_{\ensuremath{\perp}}=1.21\ifmmode\pm\else\textpm\fi{}0.04$, and ${d}_{f}=1.53\ifmmode\pm\else\textpm\fi{}0.03$. All tested hyperscaling relations are shown to be valid.

({J}_{x})

in the

x

direction and antiferromagnetic interaction

({J}_{y})

in the

y

direction. The phase diagram in the $T\text{\ensuremath{-}}H$ plane is obtained for the particular case

{J}_{x}={J}_{y}

. Special focus is given in the low-temperature region of the phase diagram, where a first-order phase transition is observed using the mean-field approximation, which is in disagreement with the linear chain approximation (LCA). Our results indicate a second-order phase transition for all values of

H∕J∊[0,2]

, with the presence of a reentrant behavior only observed in the BP approximation in accordance with the results of the LCA and exact solution. The null field critical temperature is an increasing function of

r={J}_{y}∕{J}_{x}

, and in the $r\ensuremath{\rightarrow}0$ limit we have found, using BP and EFT, the approximate form ${k}_{B}{T}_{N}∕{J}_{x}\ensuremath{\simeq}A∕\mathrm{ln}(1∕r)$ in accordance with the exact result of Onsager.

p

depending on the NN coupling constant

J

and the temperature

T

, we formally show that the partition function of the Ising model is the generating function of bond-correlated percolation model (BCPM) with a bond probability $p=1\ensuremath{-}\mathrm{exp}(\frac{\ensuremath{-}2J}{\mathrm{kT}})$. The BCPM has the Ising critical temperature and exponents, including $\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}},\ensuremath{\eta},\ensuremath{\beta},\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}}$, and $\ensuremath{\gamma}$ (perhaps also ${\ensuremath{\gamma}}^{\ensuremath{'}}$). From the connection between the Ising model and the BCPM, we also derive and hence give a geometrical meaning of the finite-size scaling and broadening at first-order phase transitions of the Ising model. Our approach may easily be extended to many spin models and give geometrical meaning to other properties of spin models.

\mathrm{SU}(N)

lattice gauge theories. The critical percolation exponents do not coincide with the ones of the thermal transition, but they are the same for models belonging to the same universality class.