A convenient way to study phase transitions is by considering finite spin systems of linear size $L$ with boundary conditions that produce a system-size interface. In this paper, we study the statistical properties of such an interface in a disordered Potts ferromagnet in dimension $d=2$ within Migdal--Kadanoff real-space renormalization. We first focus on the interface free-energy and energy to measure the singularities of the average and random contributions, as well as the corresponding histograms, both in the low-temperature phase and at criticality. We then consider the critical behavior of the interfacial adsorption of nonboundary states. Our main conclusion is that all singularities involve the correlation length ${\ensuremath{\xi}}_{\mathit{av}}(T)\ensuremath{\sim}{({T}_{c}\ensuremath{-}T)}^{\ensuremath{-}\ensuremath{\nu}}$ appearing in the average free-energy $\overline{F}\ensuremath{\sim}{[L∕{\ensuremath{\xi}}_{\mathit{av}}(T)]}^{{d}_{s}}$ of the interface of dimension ${d}_{s}=d\ensuremath{-}1$, except for the free-energy width $\ensuremath{\Delta}F\ensuremath{\sim}{[L∕{\ensuremath{\xi}}_{\mathit{var}}(T)]}^{\ensuremath{\theta}}$ that involves the droplet exponent $\ensuremath{\theta}$ and another correlation length ${\ensuremath{\xi}}_{\mathit{var}}(T)$ that diverges more rapidly than ${\ensuremath{\xi}}_{\mathit{av}}(T)$. We compare with the spin-glass transition in $d=3$ , where ${\ensuremath{\xi}}_{\mathit{var}}(T)$ is the ``true'' correlation length and where the interface energy presents unconventional scaling with a chaos critical exponent ${\ensuremath{\zeta}}_{c}>1∕\ensuremath{\nu}$ [M. Nifle and H. J. Hilhorst, Phys. Rev. Lett. 68, 2992 (1992)]. The common feature is that in both cases, the characteristic length scale ${L}_{\mathit{ch}}(T)$ associated with the chaotic nature of the low-temperature phase diverges more slowly than the correlation length.A convenient way to study phase transitions is by considering finite spin systems of linear size $L$ with boundary conditions that produce a system-size interface. In this paper, we study the statistical properties of such an interface in a disordered Potts ferromagnet in dimension $d=2$ within Migdal--Kadanoff real-space renormalization. We first focus on the interface free-energy and energy to measure the singularities of the average and random contributions, as well as the corresponding histograms, both in the low-temperature phase and at criticality. We then consider the critical behavior of the interfacial adsorption of nonboundary states. Our main conclusion is that all singularities involve the correlation length ${\ensuremath{\xi}}_{\mathit{av}}(T)\ensuremath{\sim}{({T}_{c}\ensuremath{-}T)}^{\ensuremath{-}\ensuremath{\nu}}$ appearing in the average free-energy $\overline{F}\ensuremath{\sim}{[L∕{\ensuremath{\xi}}_{\mathit{av}}(T)]}^{{d}_{s}}$ of the interface of dimension ${d}_{s}=d\ensuremath{-}1$, except for the free-energy width $\ensuremath{\Delta}F\ensuremath{\sim}{[L∕{\ensuremath{\xi}}_{\mathit{var}}(T)]}^{\ensuremath{\theta}}$ that involves the droplet exponent $\ensuremath{\theta}$ and another correlation length ${\ensuremath{\xi}}_{\mathit{var}}(T)$ that diverges more rapidly than ${\ensuremath{\xi}}_{\mathit{av}}(T)$. We compare with the spin-glass transition in $d=3$, where ${\ensuremath{\xi}}_{\mathit{var}}(T)$ is the ``true'' correlation length and where the interface energy presents unconventional scaling with a chaos critical exponent ${\ensuremath{\zeta}}_{c}>1∕\ensuremath{\nu}$ [M. Nifle and H. J. Hilhorst, Phys. Rev. Lett. 68, 2992 (1992)]. The common feature is that in both cases, the characteristic length scale ${L}_{\mathit{ch}}(T)$ associated with the chaotic nature of the low-temperature phase diverges more slowly than the correlation length.

arXiv:0711.2878Physical Review BApr 7, 200815 citations

Dynamics of clusters in the two-dimensional Potts model

Y. Gündüç, M. Aydin

arXiv:hep-lat/9605015Physica A Statistical Mechanics and its ApplicationsOct 1, 19963 citations

Interfacial tension and correlation lengths for the three-state Potts model in two dimensions

David Jasnow, K. K. Mon, Martin V. Ferer

Estimates for the critical interfacial tension amplitudes of the three-state Potts model in two dimensions via direct Monte Carlo sampling are presented for the first time. Both the square and triangular lattices are analyzed with use of finite-size scaling considerations. Also presented for the first time are estimates for the correlation-length amplitudes as derived from high-temperature series. These numerical results allow investigation of the ratio U=\ensuremath{\tau}${\ensuremath{\xi}}^{d\mathrm{\ensuremath{-}}1}$/ ${k}_{B}$ T as T\ensuremath{\rightarrow} ${T}_{c}$ . The results confirm the universality of U and are in agreement with exact results for hard hexagons. The finite-size scaling amplitude for the interfacial free energy at criticality is also shown to be universal within the available precision.Estimates for the critical interfacial tension amplitudes of the three-state Potts model in two dimensions via direct Monte Carlo sampling are presented for the first time. Both the square and triangular lattices are analyzed with use of finite-size scaling considerations. Also presented for the first time are estimates for the correlation-length amplitudes as derived from high-temperature series. These numerical results allow investigation of the ratio U=\ensuremath{\tau}${\ensuremath{\xi}}^{d\mathrm{\ensuremath{-}}1}$/${k}_{B}$T as T\ensuremath{\rightarrow}${T}_{c}$. The results confirm the universality of U and are in agreement with exact results for hard hexagons. The finite-size scaling amplitude for the interfacial free energy at criticality is also shown to be universal within the available precision.

Physical review. B, Condensed matterMar 1, 19869 citations

A convenient way to study phase transitions is by considering finite spin systems of linear size

L

with boundary conditions that produce a system-size interface. In this paper, we study the statistical properties of such an interface in a disordered Potts ferromagnet in dimension

d=2

within Migdal--Kadanoff real-space renormalization. We first focus on the interface free-energy and energy to measure the singularities of the average and random contributions, as well as the corresponding histograms, both in the low-temperature phase and at criticality. We then consider the critical behavior of the interfacial adsorption of nonboundary states. Our main conclusion is that all singularities involve the correlation length ${\ensuremath{\xi}}_{\mathit{av}}(T)\ensuremath{\sim}{({T}_{c}\ensuremath{-}T)}^{\ensuremath{-}\ensuremath{\nu}}$ appearing in the average free-energy $\overline{F}\ensuremath{\sim}{[L∕{\ensuremath{\xi}}_{\mathit{av}}(T)]}^{{d}_{s}}$ of the interface of dimension ${d}_{s}=d\ensuremath{-}1$, except for the free-energy width $\ensuremath{\Delta}F\ensuremath{\sim}{[L∕{\ensuremath{\xi}}_{\mathit{var}}(T)]}^{\ensuremath{\theta}}$ that involves the droplet exponent $\ensuremath{\theta}$ and another correlation length ${\ensuremath{\xi}}_{\mathit{var}}(T)$ that diverges more rapidly than ${\ensuremath{\xi}}_{\mathit{av}}(T)$. We compare with the spin-glass transition in

d=3

, where ${\ensuremath{\xi}}_{\mathit{var}}(T)$ is the ``true'' correlation length and where the interface energy presents unconventional scaling with a chaos critical exponent ${\ensuremath{\zeta}}_{c}>1∕\ensuremath{\nu}$ [M. Nifle and H. J. Hilhorst, Phys. Rev. Lett. 68, 2992 (1992)]. The common feature is that in both cases, the characteristic length scale

{L}_{\mathit{ch}}(T)

associated with the chaotic nature of the low-temperature phase diverges more slowly than the correlation length.

{k}_{B}

T as T\ensuremath{\rightarrow}

{T}_{c}

. The results confirm the universality of U and are in agreement with exact results for hard hexagons. The finite-size scaling amplitude for the interfacial free energy at criticality is also shown to be universal within the available precision.