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Physical review. B, Condensed matter

Wall effects in critical systems: Scaling in Ising model strips

Helen Au-Yang, Michael Ellis Fisher

Published May 1, 1980• 112 citations
Publisher

Abstract

A scaling theory and a related conjecture presented recently by de Gennes and Fisher to predict the effects of walls inserted in two near-critical binary fluid mixtures, are checked theoretically by exact analytic calculations for $n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$, two-dimensional Ising model strips with a surface field, ${h}_{1}$, imposed on the first layer; exact calculations with a second surface field, ${h}_{n}$, imposed on the nth layer are also reported. It is verified, in particular, that the effects on properties observed close to one wall of a second wall at distance $D$ decay, at the critical point, as $\frac{1}{{D}^{d}}$, where $d$ is the spatial dimensionality. In addition to the scaling limits, the leading corrections are calculated explicitly and presented graphically.

DOI
10.1103/physrevb.21.3956

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The specific heat, energy, and free energy of an infinitely long, square-lattice ferromagnetic Ising strip consisting of nnn parallel layers with free boundary conditions and a surface magnetic field H1=kBTh1{H}_{1}={k}_{B}{\mathrm{Th}}_{1}H1​=kB​Th1​ imposed on the last layer, is analyzed for large nnn in the light of finite-size scaling theory. It is shown rigorously that the free energy (and, similarly, the energy and specific heat) can be written asymptotically in the scaling form $\ensuremath{\Delta}f(n,t,{h}_{1})\ensuremath{\approx}{n}^{\ensuremath{-}2}(\mathrm{ln}n){W}_{1}(nt,{n}^{\frac{1}{2}}{h}_{1})+{n}^{\ensuremath{-}2}{W}_{2}(nt,{n}^{\frac{1}{2}}{h}_{1})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$. The scaling functions Wi(x,y){W}_{i}(x,y)Wi​(x,y) are computed in explicit closed form and shown to verify all the analyticity and asymptotic requirements anticipated by scaling theory. Furthermore, in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ at fixed $t\ensuremath{\ne}0$, the bulk and surface contributions to the thermodynamic properties are found to account for all except a correction of order ${e}^{\frac{\mathrm{pna}}{\ensuremath{\xi}(T)}}$, where aaa is the lattice spacing and $\ensuremath{\xi}(T)$ is the bulk correlation length; the value of the small rational constant ppp is interpreted in terms of interference effects between the two opposite boundaries (or "surfaces").
The properties of a fluid, or Ising magnet, confined in an $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ geometry with opposing surface fields at the walls, is studied by density matrix renormalization techniques. In particular we focus on the effect of gravity on the system, which is modeled by a bulk field whose strength varies linearly with the distance from the walls. It is well known that, in the absence of gravity, phase coexistence is restricted to temperatures below the wetting temperature. We find that gravity restores phase coexistence up to the bulk critical temperature, in agreement with previous mean field results. A detailed study of the scaling to the critical point, as $\stackrel{\ensuremath{\rightarrow}}{L}\ensuremath{\infty}$, is performed. The temperature shift scales as 1/LyT{1/L}^{{y}_{T}}1/LyT​, while the gravitational constant scales as 1/L1+yH{1/L}^{{1+y}_{H}}1/L1+yH​, with yT{y}_{T}yT​ and yH{y}_{H}yH​ the bulk thermal and magnetic exponents, respectively. For weak surface fields and LLL not too large, we also observe a regime where the gravitational constant scales as ${1/L}^{{1+y}_{H}\ensuremath{-}{\ensuremath{\Delta}}_{1}{y}_{T}}$ $({\ensuremath{\Delta}}_{1}$ is the surface gap exponent), with a crossover, for sufficiently large LLL, to a scaling of type 1/L1+yH{1/L}^{{1+y}_{H}}1/L1+yH​.
Part I of this article studied the specific heats of planar alternating layered Ising models with strips of strong coupling J_{1} sandwiched between strips of weak coupling J_{2}, to illustrate qualitatively the effects of connectivity, proximity, and enhancement in analogy to those seen in extensive experiments on superfluid helium by Gasparini and co-workers. It was demonstrated graphically that finite-size scaling descriptions hold in a variety of temperature regions including in the vicinity of the two specific heat maxima. Here we provide exact theoretical analyses and asymptotics of the specific heat that support and confirm the graphical findings. Specifically, at the overall or bulk critical point, the anticipated (and always present) logarithmic singularity is shown to vanish exponentially fast as the width of the stronger strips increases.
A quest for microcanonical functionals describing near-critical behavior has led to an exact local variational functional for square lattice Ising models in zero field with varying bond strengths along the z axis. In the continuum limit near TC{\mathit{T}}_{\mathit{C}}TC​ this yields a second-order differential equation for the transverse Fourier coefficients of the energy density. Applications to a ``grain boundary'' between differing regions and to a wall with a surface field h1{\mathit{h}}_{1}h1​ give bulk and surface exponents and scaling-limit energy-density profiles which exhibit rich crossovers and nonmonotonic variation.
The specific heats of exactly solvable alternating layered planar Ising models with strips of width m_{1} lattice spacings and "strong" couplings J_{1} sandwiched between strips of width m_{2} and "weak" coupling J_{2}, have been studied numerically to investigate the effects of connectivity and proximity. We find that the enhancements of the specific heats of the strong layers and of the overall or "bulk" critical temperature T_{c}(J_{1},J_{2};m_{1},m_{2}) arising from the collective effects reflect the observations of Gasparini and co-workers in experiments on confined superfluid helium. Explicitly, we demonstrate that finite-size scaling holds in the vicinity of the upper limiting critical point T_{1c} (∝J_{1}/k_{B}) and close to the corresponding lower critical limit T_{2c} (∝J_{2}/k_{B}) when m_{1} and m_{2} increase. However, the residual enhancement, defined via appropriate subtractions of leading contributions from the total specific heat, is dominated (away from T_{1c} and T_{2c}) by a decay factor 1/(m_{1}+m_{2}) arising from the seams (or boundaries) separating the strips; close to T_{1c} and T_{2c} the decay is slower by a factor lnm_{1} and lnm_{2}, respectively.