Abstract We consider near-critical two-dimensional statistical systems at phase coexistence on the half plane with boundary conditions leading to the formation of a droplet separating coexisting phases. General low-energy properties of two-dimensional field theories are used in order to find exact analytic results for one- and two-point correlation functions of both the energy density and order parameter fields. The subleading finite-size corrections are also computed and interpreted within an exact probabilistic picture in which interfacial fluctuations are characterized by the probability density of a Brownian excursion. The explicit analysis of the closed-form expression for order parameter correlations reveals the long-ranged character of interfacial correlations and their confinement within the interfacial region. The analysis of correlations is then carried out in momentum space through the notion of interface structure factor, which we extend to the case of systems bounded by a flat wall. The presence of the wall and its associated entropic repulsion leads to a specific term in the interface structure factor which we identify.
Abstract We consider near-critical two-dimensional statistical systems with boundary conditions inducing phase separation on the strip. By exploiting low-energy properties of two-dimensional field theories, we compute arbitrary n-point correlation of the order parameter field. Finite-size corrections and mixed correlations involving the stress tensor trace are also discussed. As an explicit illustration of the technique, we provide a closed-form expression for a three-point correlation function and illustrate the explicit form of the long-ranged interfacial fluctuations as well as their confinement within the interfacial region.
The correlation length controlling the long-range decay of the density-density correlation function in the direction parallel to the interface is computed exactly, for T\ensuremath{\le}
Tc, in the two-dimensional Ising M\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} strip periodic in the long direction with a (+/-) boundary condition in the other direction which creates a planar strongly fluctuating interface. A crossover from long-range capillary-wave-like behavior to short-range bulk correlations above
Tc, a new scaling function, and corrections to scaling at
Tc are found.
We study the long-distance behavior of the
O(N) model in the presence of random fields and random anisotropies correlated as $\ensuremath{\sim}1∕{x}^{d\ensuremath{-}\ensuremath{\sigma}}$ for large separation
x using the functional renormalization group. We compute the fixed points and analyze their regions of stability within a double $\ensuremath{\epsilon}=d\ensuremath{-}4$ and $\ensuremath{\sigma}$ expansion. We find that the long-range disorder correlator remains analytic but generates short-range disorder whose correlator develops the usual cusp. This allows us to obtain the phase diagrams in $(d,\ensuremath{\sigma},N)$ parameter space and compute the critical exponents to first order in $\ensuremath{\epsilon}$ and $\ensuremath{\sigma}$. We show that the standard renormalization group methods with a finite number of couplings used in previous studies of systems with long-range correlated random fields fail to capture all critical properties. We argue that our results may be relevant to the behavior of $^{3}\mathrm{He}\text{\ensuremath{-}}\mathrm{A}$ in aerogel.
We show the way perturbation procedures originated by Brout and Hemmer and further developed by Lebowitz and co-workers can be used to study the critical behavior of lattice systems with a potential
v(r) having a weak, long-ranged tail that approaches ${\ensuremath{\gamma}}^{d}\ensuremath{\phi}(\ensuremath{\gamma}\mathrm{r})$ as $\ensuremath{\gamma}$ approaches zero, where
d is dimensionality. We consider both the Ising and the spherical models, and note that the results for the latter model are also those that follow from the Ornstein-Zernike assumption that the direct correlation function behaves like $\ensuremath{-}\frac{v(\mathrm{r})}{\mathrm{kT}}$, when $v(\mathrm{r})\ensuremath{\ll}\mathrm{kT}$. We recover by a graph technique the previously known result that in one dimension, quantities of interest such as the inverse correlation length $\ensuremath{\kappa}$ cannot be expanded in $\ensuremath{\gamma}$ at the mean-field critical temperature
T0 and density ${\ensuremath{\rho}}_{0}$ that characterize the critical point in the $\ensuremath{\gamma}\ensuremath{\rightarrow}0$ limit. Instead, at (${T}_{0}, {\ensuremath{\rho}}_{0}$), $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}^{\frac{4}{3}}$ as $\ensuremath{\gamma}\ensuremath{\rightarrow}0$. This is true for both the Ising and the spherical models. In the two-dimensional spherical model, we find $\ensuremath{\kappa}$ to vary as ${\ensuremath{\gamma}}^{2}{[\mathrm{ln}(\frac{1}{\ensuremath{\gamma}})]}^{\frac{1}{2}}$ when $\ensuremath{\gamma}\ensuremath{\rightarrow}0$, while in three dimensions $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}^{\frac{5}{2}}$. In the Ising case for
d=1, we characterize topologically the infinite sum of graphs that contribute to the $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}^{\frac{4}{3}}$ term in the expansion of the pair correlation function. (In the spherical model, a chain graph gives the entire contribution to the pair correlation function for all
d). For
d=3 we do not attempt to deal with the correlation length in the Ising case, but instead consider the shift in the critical temperature as $\ensuremath{\gamma}\ensuremath{\rightarrow}0$. We find that the spherical-model result that gives a shift of order ${\ensuremath{\gamma}}^{3}$ is exact to that order in $\ensuremath{\gamma}$ for the Ising model. We also find that the lowest-order correction to the spherical-model result is of the form ($\mathrm{const}{\ensuremath{\gamma}}^{6}\mathrm{ln}(\mathrm{const}\ensuremath{\gamma})$, in agreement with recent work by Thouless; but in general we expect to find terms of order ${\ensuremath{\gamma}}^{4}$ and ${\ensuremath{\gamma}}^{5}$ from the spherical-model result dominating this correction.