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

Journal of Physics A Mathematical and General

Droplet theory of correlations in ordered phases

Ann Bruce

Published Dec 1, 1984• 5 citations
Publisher

Abstract

The explicit droplet theory of low-dimensional phase transitions, developed recently, is extended to yield a description of the pair correlation function for the universality class of q-state Potts models in the ordered phase and in zero external field. It is shown that, in d=1+ epsilon dimensions, the short distance behaviour of the correlation function and in particular, the exponent eta , are controlled by nested nearly spherical droplets. In contrast, it is argued that the spatial dependence of the large distance behaviour, in d=2 or above, is controlled by highly anisotropic droplets, thus illuminating the Widom relation, sigma xi d-1 approximately= constant, linking the dimensionless surface tension sigma and the bulk correlation length xi . In the particular case of d=2 the statistical weight of the relevant droplets is determined as that of two appropriately interacting strings, in the spirit of recent independent, arguments by Abraham and by Fisher: a non-Ornstein-Zernike correlation function prefactor, and the result of sigma xi = 1 / 2 follow in accord with exact results for the q=2 (Ising) case, and with implications for the q to 1 (percolation) problem.

DOI
10.1088/0305-4470/17/17/014

Related papers

Droplet theory in low dimensions: Ising systems in zero field

Ann Bruce, D. J. Wallace

The authors develop a theory of the universal configurational physics underlying critical point phenomena in the Ising universality class. The theory is formally justifiable in d=1+ epsilon dimensions and may be regarded as the natural continuation of the kink-based theory of one dimension, which it incorporates as a limiting case. In d=1+ epsilon the configurational building block is the droplet. The typical droplet is not spherical and the many-droplet assembly is not dilute: the implied problems are handled with renormalisation group methods. It is found that droplet shape fluctuation effects control the correlation length exponent nu , while the nesting of droplets within droplets control the order parameter exponent beta . The exponents nu and beta thus effectively define, respectively, the fractal dimensions of the droplet surface and the droplet volume. The theory is used to determine and illuminate the critical behaviour of further quantities including the free energy, the susceptibility, the droplet number distribution and the distribution of the intra-droplet order.

Journal of Physics A Mathematical and General•Jun 1, 1983•41 citations

Droplet Theory of Low-Dimensional Ising Models

Ann Bruce, D. J. Wallace

A droplet theory of Ising systems is constructed, valid near space dimension $d=1$, and based on a model incorporating only the configurational energy of surface tension. The droplet distribution function is calculated explicitly, with use of renormalization-group methods to control the droplet-shape fluctuation effects essential to its scaling form. Universal quantities are calculated; those (e.g., $\ensuremath{\beta}$) which reflect the droplet concentration have an essential singularity in $\ensuremath{\epsilon}=d\ensuremath{-}1$.

Physical Review Letters•Dec 14, 1981•40 citations

2D Potts-Model Correlation Lengths: Numerical Evidence for ξ o = ξ d at β t

W Janke, S Kappler

We have studied spin-spin correlation functions in the ordered phase of the two-dimensional q -state Potts model with q = 10, 15, and 20 at the first-order transition point β t . Through extensive Monte Carlo simulations we obtain strong numerical evidence that the correlation length in the ordered phase agrees with the exactly known and recently numerically confirmed correlation length in the disordered phase: ξ o (β t ) = ξ d (β t ). As a by-product we find the energy moments in the ordered phase at β t in very good agreement with a recent large q -expansion.

arXiv:hep-lat/9509056•Europhysics Letters (EPL)•Sep 1, 1995•15 citations

Boundary and bulk phase transitions in the two-dimensional Q -state Potts model ( Q > 4 )

Ferenc Iglói, Enrico Carlon

The surface and bulk properties of the two-dimensional $Q>4$ state Potts model in the vicinity of the first-order bulk transition point have been studied by exact calculations and by density-matrix renormalization-group techniques. For the surface transition the complete analytical solution of the problem is presented in the $\stackrel{\ensuremath{\rightarrow}}{Q}\ensuremath{\infty}$ limit, including the critical and tricritical exponents, magnetization profiles, and scaling functions. We have shown in an accurate numerical study that the universality class of the surface transition is independent of the value of $Q>4.$ For the bulk transition we have numerically calculated the latent heat and the magnetization discontinuity and we have shown that the correlation lengths in the ordered and disordered phases are identical at the transition point.

arXiv:cond-mat/9805083•Physical review. B, Condensed matter•Feb 1, 1999•28 citations

Critical droplets and phase transitions in two dimensions

Santo Fortunato

In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point ${T}_{c}$ of the thermal transition and the percolation exponents belong to a special universality class. By introducing a bond probability ${p}_{B}<1,$ the corresponding site-bond clusters keep on percolating at ${T}_{c}$ and the exponents do not change, until ${p}_{B}{=p}_{\mathrm{CK}}=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}2J/kT):$ for this special expression of the bond weight the critical percolation exponents switch to the two-dimensional Ising universality class. We show here that the result is valid for a wide class of bidimensional models with a continuous magnetization transition: there is a critical bond probability ${p}_{c}$ such that, for any ${p}_{B}>~{p}_{c},$ the onset of percolation of the site-bond clusters coincides with the critical point of the thermal transition. The percolation exponents are the same for ${p}_{c}<{p}_{B}<~1$ but, for ${p}_{B}{=p}_{c},$ they suddenly change to the thermal exponents, so that the corresponding clusters are critical droplets of the phase transition. Our result is based on Monte Carlo simulations of various systems near criticality.

arXiv:cond-mat/0205211•Physical review. B, Condensed matter•Jan 15, 2003•16 citations

Finite-size scaling for the correlation function of the spherical model with long-range interactions

J.G. Brankov, Daniel M. Danchev

Finite-size scaling for the effective correlation length and the pair correlation function of the mean spherical model is studied in the case of general geometry Ld−d′×∞d′, periodic boundary conditions, and long-range interactions decaying like R−d−σ at large distances R, with arbitrary real parameters 0<σ≤2, σ≤d≤2σ, d′≥0. The analytical technique used is based on integral transformations with kernels of Mittag–Leffler type. It makes it possible to easily generalize a number of results available only for short-range interactions or in special cases of the Ld−d′×∞d′ geometry. The effective correlation length is identified at arbitrary temperatures from the finite-size large-distance asymptotic behavior of the pair correlation function. It is explicitly shown that the finite-size scaling functions are not singular in the ε-expansion when ε→0+ neither at d=σ+ε, nor at d=2σ−ε. Moreover, the applicability of the ε-expansion is tested in the specific finite-size case of d′=σ±ε. The definition of the scaled field variable and the finite-size scaling in the regime of the first-order phase transition are also considered.

Journal of Mathematical Physics•Sep 1, 1991•21 citations
The explicit droplet theory of low-dimensional phase transitions, developed recently, is extended to yield a description of the pair correlation function for the universality class of q-state Potts models in the ordered phase and in zero external field. It is shown that, in d=1+ epsilon dimensions, the short distance behaviour of the correlation function and in particular, the exponent eta , are controlled by nested nearly spherical droplets. In contrast, it is argued that the spatial dependence of the large distance behaviour, in d=2 or above, is controlled by highly anisotropic droplets, thus illuminating the Widom relation, sigma xi d-1 approximately= constant, linking the dimensionless surface tension sigma and the bulk correlation length xi . In the particular case of d=2 the statistical weight of the relevant droplets is determined as that of two appropriately interacting strings, in the spirit of recent independent, arguments by Abraham and by Fisher: a non-Ornstein-Zernike correlation function prefactor, and the result of sigma xi = 1 / 2 follow in accord with exact results for the q=2 (Ising) case, and with implications for the q to 1 (percolation) problem.
The authors develop a theory of the universal configurational physics underlying critical point phenomena in the Ising universality class. The theory is formally justifiable in d=1+ epsilon dimensions and may be regarded as the natural continuation of the kink-based theory of one dimension, which it incorporates as a limiting case. In d=1+ epsilon the configurational building block is the droplet. The typical droplet is not spherical and the many-droplet assembly is not dilute: the implied problems are handled with renormalisation group methods. It is found that droplet shape fluctuation effects control the correlation length exponent nu , while the nesting of droplets within droplets control the order parameter exponent beta . The exponents nu and beta thus effectively define, respectively, the fractal dimensions of the droplet surface and the droplet volume. The theory is used to determine and illuminate the critical behaviour of further quantities including the free energy, the susceptibility, the droplet number distribution and the distribution of the intra-droplet order.
A droplet theory of Ising systems is constructed, valid near space dimension d=1d=1d=1, and based on a model incorporating only the configurational energy of surface tension. The droplet distribution function is calculated explicitly, with use of renormalization-group methods to control the droplet-shape fluctuation effects essential to its scaling form. Universal quantities are calculated; those (e.g., $\ensuremath{\beta}$) which reflect the droplet concentration have an essential singularity in $\ensuremath{\epsilon}=d\ensuremath{-}1$.
We have studied spin-spin correlation functions in the ordered phase of the two-dimensional q -state Potts model with q = 10, 15, and 20 at the first-order transition point β t . Through extensive Monte Carlo simulations we obtain strong numerical evidence that the correlation length in the ordered phase agrees with the exactly known and recently numerically confirmed correlation length in the disordered phase: ξ o (β t ) = ξ d (β t ). As a by-product we find the energy moments in the ordered phase at β t in very good agreement with a recent large q -expansion.
The surface and bulk properties of the two-dimensional Q>4Q>4Q>4 state Potts model in the vicinity of the first-order bulk transition point have been studied by exact calculations and by density-matrix renormalization-group techniques. For the surface transition the complete analytical solution of the problem is presented in the $\stackrel{\ensuremath{\rightarrow}}{Q}\ensuremath{\infty}$ limit, including the critical and tricritical exponents, magnetization profiles, and scaling functions. We have shown in an accurate numerical study that the universality class of the surface transition is independent of the value of Q>4.Q>4.Q>4. For the bulk transition we have numerically calculated the latent heat and the magnetization discontinuity and we have shown that the correlation lengths in the ordered and disordered phases are identical at the transition point.
In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point Tc{T}_{c}Tc​ of the thermal transition and the percolation exponents belong to a special universality class. By introducing a bond probability pB<1,{p}_{B}<1,pB​<1, the corresponding site-bond clusters keep on percolating at Tc{T}_{c}Tc​ and the exponents do not change, until ${p}_{B}{=p}_{\mathrm{CK}}=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}2J/kT):$ for this special expression of the bond weight the critical percolation exponents switch to the two-dimensional Ising universality class. We show here that the result is valid for a wide class of bidimensional models with a continuous magnetization transition: there is a critical bond probability pc{p}_{c}pc​ such that, for any pB> pc,{p}_{B}>~{p}_{c},pB​> pc​, the onset of percolation of the site-bond clusters coincides with the critical point of the thermal transition. The percolation exponents are the same for pc<pB< 1{p}_{c}<{p}_{B}<~1pc​<pB​< 1 but, for pB=pc,{p}_{B}{=p}_{c},pB​=pc​, they suddenly change to the thermal exponents, so that the corresponding clusters are critical droplets of the phase transition. Our result is based on Monte Carlo simulations of various systems near criticality.
Finite-size scaling for the effective correlation length and the pair correlation function of the mean spherical model is studied in the case of general geometry Ld−d′×∞d′, periodic boundary conditions, and long-range interactions decaying like R−d−σ at large distances R, with arbitrary real parameters 0<σ≤2, σ≤d≤2σ, d′≥0. The analytical technique used is based on integral transformations with kernels of Mittag–Leffler type. It makes it possible to easily generalize a number of results available only for short-range interactions or in special cases of the Ld−d′×∞d′ geometry. The effective correlation length is identified at arbitrary temperatures from the finite-size large-distance asymptotic behavior of the pair correlation function. It is explicitly shown that the finite-size scaling functions are not singular in the ε-expansion when ε→0+ neither at d=σ+ε, nor at d=2σ−ε. Moreover, the applicability of the ε-expansion is tested in the specific finite-size case of d′=σ±ε. The definition of the scaled field variable and the finite-size scaling in the regime of the first-order phase transition are also considered.