The properties of a simple fluid or magnet with strong one-axis anisotropy can be studied by means of the Ising model. Such a model, but in an confined geometry with identical boundary fields, is studied along various thermodynamic paths using the density-matrix renormalization group technique. It has been found that also in the presence of the capillary condensation the critical exponent β of the bulk system can be found. Introduction In the thermodynamic limit of the two-dimensional system the correlation length and some other quantities diverge at the critical point (T = Tc, H = 0) [1]. One of them is magnetization that vanishes above Tc in zero bulk field (the paramagnetic regime). Phase coexistence occurs for temperatures T T). It results in a jump of the magnetization m = –∂f/∂H at H = 0, whereas the A. Drzewinski, K. Szota 30 limit value mb = m (T = const, H → 0) is finite and named the spontaneous magnetization. In the other words, when the thermodynamic path for fixed but very small H (the dashed arrow at Fig. 1) is closer and closer to the H = 0 axis, then the magnetization resembles more and more the spontaneous magnetization (full dots in Fig. 2). Fig. 1. Numerically determined coexistence curves for finite systems in the (T, H) phase diagram. The thick solid line indicates the bulk coexistence line. The dashed arrow denotes the thermodynamic path for the infinite systems but others for finite systems Fig. 2. Spontaneous magnetization mb as a function of temperature along the thermodynamic paths presented in the Fig. 1 with a constant shift –10 away from the coexistence lines There is an interesting question. What happens when the system is confined in the presence of the boundary fields. Is it possible to apply the same procedure and, exploiting the magnetizations along the (curvilinear) coexistence lines, reconstruct the behaviour of the spontaneous magnetization? Bulk critical exponents in a presence of the capillary condensation 31 1. The model and the method Our model Hamiltonian for the infinitely long strip has the form: − − ∑ ∑ ∑ ∑ l k, l k, k L k, 1 k k l' , k' l k, σ H + σ h + σ h σ σ J = H ,1 1 where the first sum is over all nearest-neighbour pairs and we have measured the surface field h1 and the bulk field H in units of J. In order to find out the free energy f and the longitudinal correlation length ξ|| we have used the density-matrix renormalization group (DMRG) [3]. It was originally proposed by White as a new tool for diagonalization of quantum spin chains [4] and was later adapted to classical two-dimensional equilibrium statistical mechanics by Nishino [5]. The DMRG allows one to study much larger systems (up to L = 650 in the present paper) than is possible with standard exact diagonalization methods (typically L ~50) and provides data with remarkable accuracy. We estimate that the errors in the plots are smaller than the symbol size.
The properties of a simple fluid or magnet with strong one-axis anisotropy can be studied by means of the Ising model. Such a model, but in an confined geometry with identical boundary fields, is studied along various thermodynamic paths using the density-matrix renormalization group technique. It has been found that also in the presence of the capillary condensation the critical exponent β of the bulk system can be found. Introduction In the thermodynamic limit of the two-dimensional system the correlation length and some other quantities diverge at the critical point (T = Tc, H = 0) [1]. One of them is magnetization that vanishes above Tc in zero bulk field (the paramagnetic regime). Phase coexistence occurs for temperatures T T). It results in a jump of the magnetization m = –∂f/∂H at H = 0, whereas the A. Drzewinski, K. Szota 30 limit value mb = m (T = const, H → 0) is finite and named the spontaneous magnetization. In the other words, when the thermodynamic path for fixed but very small H (the dashed arrow at Fig. 1) is closer and closer to the H = 0 axis, then the magnetization resembles more and more the spontaneous magnetization (full dots in Fig. 2). Fig. 1. Numerically determined coexistence curves for finite systems in the (T, H) phase diagram. The thick solid line indicates the bulk coexistence line. The dashed arrow denotes the thermodynamic path for the infinite systems but others for finite systems Fig. 2. Spontaneous magnetization mb as a function of temperature along the thermodynamic paths presented in the Fig. 1 with a constant shift –10 away from the coexistence lines There is an interesting question. What happens when the system is confined in the presence of the boundary fields. Is it possible to apply the same procedure and, exploiting the magnetizations along the (curvilinear) coexistence lines, reconstruct the behaviour of the spontaneous magnetization? Bulk critical exponents in a presence of the capillary condensation 31 1. The model and the method Our model Hamiltonian for the infinitely long strip has the form: − − ∑ ∑ ∑ ∑ l k, l k, k L k, 1 k k l' , k' l k, σ H + σ h + σ h σ σ J = H ,1 1 where the first sum is over all nearest-neighbour pairs and we have measured the surface field h1 and the bulk field H in units of J. In order to find out the free energy f and the longitudinal correlation length ξ|| we have used the density-matrix renormalization group (DMRG) [3]. It was originally proposed by White as a new tool for diagonalization of quantum spin chains [4] and was later adapted to classical two-dimensional equilibrium statistical mechanics by Nishino [5]. The DMRG allows one to study much larger systems (up to L = 650 in the present paper) than is possible with standard exact diagonalization methods (typically L ~50) and provides data with remarkable accuracy. We estimate that the errors in the plots are smaller than the symbol size.
The excess adsorption Gamma in two-dimensional Ising strips (infinityxL), subject to identical boundary fields at both one-dimensional surfaces decaying in the orthogonal direction j as -h1j(-p), is studied for various values of p and along various thermodynamic paths below the bulk critical point by means of the density-matrix renormalization-group method. The crossover behavior between the complete-wetting and critical-adsorption regimes, occurring in semi-infinite systems, is strongly influenced by confinement effects. Along isotherms T=const the asymptotic power-law dependences on the external bulk field, which characterize these two regimes, are pre-empted by capillary condensation. Along the pseudo-first-order phase-coexistence line of the strips, which varies with temperature, we find a broad crossover regime in which both the thickness of the wetting film and Gamma increase as functions of the reduced temperature tau but do not follow any power law. Above the wetting temperature the order-parameter profiles are not slablike but exhibit wide interfacial variations and pronounced tails.
Two-dimensional Ising strips subject to identical surface fields h(1)=h(2) > or =0 are studied for temperatures above and below the bulk critical temperature T(c) and a range of bulk fields h by means of the density-matrix renormalization-group method. In the case of nonvanishing surface fields, the near-critical behavior of the solvation force f(solv), total adsorption Gamma, inverse longitudinal correlation length xi(parallel)( -1) and specific heat C(H) is strongly influenced by the (pseudo) capillary condensation that occurs below T(c). We obtain scaling functions of f(solv), Gamma, and xi(parallel)(-1). C(H) exhibits a weakly rounded singularity on crossing the pseudocoexistence line. We contrast these results with those for the case of free boundaries where, for temperatures slightly below T(c), f(solv) and C(H) exhibit a sharp extremum away from h=0. Our results have direct repercussions for the properties of near-critical Ising films in three dimensions and we argue that the long-ranged solvation (Casimir) force in confined fluids should be more attractive in the neighborhood of the capillary critical point than exactly at the bulk critical point.
Two-dimensional Ising films $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ in a nonvanishing bulk magnetic field H are studied at the bulk critical temperature Tc for two choices of surface fields (a) H1=HL=0 (ordinary transition), and (b) ${H}_{1}{=H}_{L}=\ensuremath{\infty}$ (normal transition) by the density-matrix renormalization-group method. Universal scaling functions for magnetization profiles, the excess magnetization $\ensuremath{\Gamma},$ the longitudinal correlation length ${\ensuremath{\xi}}_{\ensuremath{\Vert}},$ and for the analog of the solvation force fsolv are found and discussed. When H1=0 the scaling function for fsolv has two symmetric minima at $y=\mathrm{sgn}(H)L|H{|}^{\ensuremath{\nu}/\ensuremath{\Delta}}\ensuremath{\approx}\ifmmode\pm\else\textpm\fi{}1$ with an amplitude at the minimum about 3.8 times the value at H=0, the Casimir amplitude. For the normal transition the scaling function for fsolv has a single minimum near the continuation of the pseudocoexistence (capillary condensation) line, with an amplitude about 100 times the Casimir amplitude.
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, while the gravitational constant scales as 1/L1+yH, with yT and yH the bulk thermal and magnetic exponents, respectively. For weak surface fields and L 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 L, to a scaling of type 1/L1+yH.
First order transitions are rounded in finite cells or in infinite strips of finite width (cylinders). In this work we have considered the rounding of the jump of the magnetization of an Ising system at a fixed temperature T below Tc when the external field h goes from a small positive, to a small negative value. This rounding is well-known in a finite cell (block geometry) but the situation is more complicated, and more interesting, in a cylinder. Indeed it is known from the work of Privman and Fisher that these rounding effects are related to the behaviour of the correlation length along the axis of the cylinder which increases exponentially with the cross-sectional area A of the strip. It then follows that if one studies, for a given magnetization M, the external field h, this field approaches exponentially with A its bulk limit, but the coefficient inside the exponential is different if M is smaller, larger than or equal to the bulk spontaneous magnetization M0. We present numerical data for the two-dimensional case (or equivalently for the one-dimensional quantum model in a transverse field) which are in satisfactory agreement with the theory.
Universality classes encompass the analogous thermodynamic behavior of unlike physical systems, at different spatial dimensions d, in the vicinity of their critical point. Critical exponents define these classes, with the Ising model being the outstanding prototype that elucidates the differences from the mean-field category, believed to be valid above a critical dimension only. Here, in apparent striking contradiction to the Ising universality class, we demonstrate that the critical behavior of a finite Ising system of N spins in d=3 obeys mean-field Landau theory in the vicinity of its critical point, with classical critical exponents. Yet, when expressed in terms of the linear size L of the system, the free energy unveils its proper finite-size scaling form, from which the thermodynamic limit critical temperature Tc and the Ising critical exponents ν, γ and β can be identified. We find that the larger the size L, the smaller the mean-field region, shrinking to zero in the thermodynamic limit. These conclusions are achieved via the use of an alternative approach to collect data from a Monte Carlo simulation of a three-dimensional Ising model that allows for the evaluation of the free energy per spin f=f(T,m;L) and of the coexistence curve, or spontaneous magnetization at zero magnetic field, mcoex=m(T;L) as functions of temperature T and magnetization per spin m=M/N. Our results suggest a revision of the role of mean-field theory in the elucidation of critical phenomena.