Crossover finite-size scaling at first-order transitions

In a recent paper we developed a method which allows one to control rigorously the finite-size behavior in long cylinders near first-order phase transitions at low temperature. Here we apply this method to asymmetric transitions with two competing phases, and to theq-state Potts model as a typical model of a temperature-driven transition, whereq low-temperature phases compete with one high-temperature phase. We obtain the finite-size scaling of the firstN eigenvalues (whereN is the number of competing phases) of the transfer matrix in a periodic box of volumeL × ... ×L ×t, and, as a corollary, the finite-size scaling of the shape of the order parameter in a hypercubic box (t=L), the infinite cylinder (t=), and the crossover regime from hypercubic to cylindrical scaling. For the two-phase case (N=2 we find that the crossover length _L is given by O(L^w)exp(L^v), where is the inverse temperature, is the surface tension, and w=1/2 if v+1=2 whilew=0 if v+1 >2. For the standard Ising model we also consider free boundary conditions, showing that _L=exp[L^v+O(L^v– 1)] for any dimension v+12. For v+1=2 we finally discuss a class of boundary conditions which interpolate between free (corresponding to the interpolating parameter g=0) and periodic boundary conditions (corresponding to g=1), finding that _L=O(L^w)exp(L ^v) withw=0 forg=0 andw=1/2 for 0<g1.     

Surface-induced finite-size effects for first-order phase transitions

We consider classical lattice models describing first-order phase transitions, and study the finite-size scaling of the magnetization and susceptibility. In order to model the effects of an actual surface in systems such as small magnetic clusters, we consider models with free boundary conditions. For a field-driven transition with two coexisting phases at the infinite-volume transition pointh=h _t , we prove that the low-temperature, finite-volume magnetizationm _free(L, h) per site in a cubic volume of sizeL ^d behaves like $$m_{free} (L,h) = \frac{{m_ + + m_ - }}{2} + \frac{{m_ + - m_ - }}{2}tanh\left[ {\frac{{m_ + - m_ - }}{2}L^d (h - h_\chi (L))} \right] + O\left( {\frac{1}{L}} \right)$$ whereh _x (L) is the position of the maximum of the (finite-volume) susceptibility andm _± are the infinite-volume magnetizations ath=h _t +0 andh=h _t ?0, respectively. We show thath _x (L) is shifted by an amount proportional to 1/L with respect to the infinite-volume transition pointh _t provided the surface free energies of the two phases at the transition point are different. This should be compared with the shift for periodic boundary conditions, which for an asymmetric transition with two coexisting phases is proportional only to 1/L ^2d . One can consider also other definitions of finite-volume transition points, for example, the positionh _U (L) of the maximum of the so-called Binder cumulantU _free(L,h). Whileh _U (L) is again shifted by an amount proportional to 1/L with respect to the infinite-volume transition pointh _t , its shift with respect toh _χ (L) is of the much smaller order 1/L ^2d . We give explicit formulas for the proportionality factors, and show that, in the leading 1/L ^2d term, the relative shift is the same as that for periodic boundary conditions.     

A rigorous theory of finite-size scaling at first-order phase transitions

A large class of classical lattice models describing the coexistence of a finite number of stable states at low temperatures is considered. The dependence of the finite-volume magnetizationM _per(h, L) in cubes of sizeL ^d under periodic boundary conditions on the external fieldh is analyzed. For the case where two phases coexist at the infinite-volume transition pointh _t , we prove that, independent of the details of the model, the finite-volume magnetization per lattice site behaves likeM _per(h _t )+M tanh[ML ^d (h–h_t)] withM _per(h) denoting the infinite-volume magnetization and M=1/2[M _per(h _t +0)–M _per(h _t –0)]. Introducing the finite-size transition pointh _m (L) as the point where the finite-volume susceptibility attains the maximum, we show that, in the case of asymmetric field-driven transitions, its shift ish _t –h _m (L)=O(L ^–2d ), in contrast to claims in the literature. Starting from the obvious observation that the number of stable phases has a local maximum at the transition point, we propose a new way of determining the pointh _t from finite-size data with a shift that is exponentially small inL. Finally, the finite-size effects are discussed also in the case where more than two phases coexist.On leave from: Institut für Theoretische Physik, FU-Berlin, D-1000 Berlin 33, Federal Republic of Germany.     

The finite-size effects of goldstone bosons in Monte Carlo simulations

It is shown how one can, in practice, extract the value of the pion decay constant ⨍, from the finite-size behavior of the magnetic susceptibility. The method is based on a finite-size soft-pion theorem and is successfully tested on the Gell-Mann-Levy linear sigma model.     

Finite-size effects at first-order transitions

Finite-size rounding of a first-order phase transition is studied in “block”- and “cylinder”-shaped ferromagnetic scalar spin systems. Crossover in shape is investigated and the universal form of the rounded susceptibility peak is obtained. Scaling forms on the low-temperature side of the critical point are considered both above and below the borderline dimensionality,d _>=4. A method of phenomenological renormalization, applicable to both odd and even field derivatives, is suggested and used to estimate universal amplitudes for two-dimensional Ising models atT=T_c.     

Monte Carlo study of very weakly first-order transitions in the three-dimensional Ashkin-Teller model

We propose numerical simulations of the Ashkin-Teller model as a foil for theoretical techniques for studying very weakly first-order phase transitions in three dimensions. The Ashkin-Teller model is a simple two-spin model whose parameters can be adjusted so that it has an arbitrarily weakly first-order phase transition. In this limit, there are quantities characterizing the first-order transition which are universal: we measure the relative discontinuity of the specific heat, the correlation length, and the susceptibility across the transition by Monte Carlo simulation.     

Continuum Monte Carlo simulation of phase transitions in rod-like molecules at surfaces

Summary Stiff rod-like chain molecules with harmonic bond length potentials and trigonometric bond angle potentials are used to model Langmuir monolayers at high densities. One end of the rod-like molecules is strongly bound to a flat two-dimensional substrate which represents the air-water interface. A ground-state analysis is performed which suggests phase transitions between phases with and without collective uniform tilt. Large-scale off-lattice Monte Carlo simulations over a wide temperature range show in addition to the tilting transition the presence of a strongly constrained melting transition at high temperatures. The latter transition appears to be related to two-dimensional melting of the head group lattice. These findings show that the model contains both, two- and three-dimensional ergodicity breaking solidification transitions. We discuss our findings with respect to experiment. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Analytic continuation at first-order phase transitions

We study the analytic structure of thermodynamic functions at first-order phase transitions in systems with short-range interactions and in particular in the two-dimensional Ising model. We analyze the nature of the approximation of the d=2 system by anN × strip. Investigation of the structure of the eigenvalues of the transfer matrix in the vicinity of H=0 in the complexH plane allows us to define a new function which provides rapidly convergent approximations to the stable free energyf and its derivatives for allH 0. This new function is used for numerical calculation of the coefficients C_n in the power series expansions of the magnetizationm in the form m(H)=1 + C_n(H-H ₀ )ⁿ for various H₀ 0. The resulting series are studied by conventional methods. We confirm recent series analysis results on the existence of the droplet model type essential singularity at H=0. Evidence is found for a spinodal at H=H_sp(Ti < 0.     

Monte Carlo study of low/high spin transitions

We study the finite temperature property of a model on two dimensional square lattices with two Ising spins at each lattice site by Monte Carlo simulations. When those Ising spins at a lattice site are parallel the site is said to be in the high-spin state (HS), while when they are antiparallel the site is said to be in the low-spin state (LS). Throughout the study, the energy of HS is presumed to be higher than that of LS. Two Ising spins at each site are added to form a total spin, which interacts with its nearest neighbour total spins via spin-spin couplings. The spin-phonon coupling also is introduced via harmonic springs between nearest neighbour sites with spring constants and equilibrium distances depending on the spin states of the sites involved. In this system, we investigate the feature of transitions between LS and HS (to be called low/high spin transition (LHST)) by varying the temperature. As for the ferromagnetic interaction between total spins, the second order phase transition: pure HSmixed state of HS and LS is possible to occur in a pure spin system, as is expected from mean field calculations. The role of lattice distortions by the change of lattice spacings is shown to be essential for LHST: pure LS(pure)HS. In the model investigated, there appears an indication of the strong first order phase transition which reveals a conspicuous hysteresis.     

Monte Carlo studies of liquid water

Computer simulation studies are reported for the Rowlinson and Ben-Naim and Stillinger models of water-water interactions. Particular attention is given to the effects of altering the size of the system and to accounting for some long-range interactions by including the Onsager reaction field. It is shown that both models give a good qualitative account of the structure of liquid water but that neither is able to describe the high dielectric constant. A particularly sensitive property, the dipole-dipole correlation function, demonstrates the problems encountered in truncating the water interactions. Good agreement between the Rowlinson potential and a modified Hartree-Fock calculation suggests that the Rowlinson model is more accurate than the Ben-Naim and Stillinger form.     

Viscosity and thermal conductivity effects at first-order phase transitions in heavy-ion collisions

Effects of viscosity and thermal conductivity on the dynamics of first-order phase transitions are studied. The nuclear gas-liquid and hadron-quark transitions in heavy-ion collisions are considered. We demonstrate that at nonzero thermal conductivity, κ ≠ 0, onset of spinodal instabilities occurs on an isothermal spinodal line, whereas for κ = 0 instabilities take place at lower temperatures, on an adiabatic spinodal.     

Surface induced disordering at first-order bulk transitions

Semi-infinite systems may undergo surface induced disordering transitions. These transitions exhibit both critical surface behaviour and interface delocalization phenomena. As a consequence, various surface exponents can be defined although there are no bulk exponents. It is shown that the corresponding power laws can be derived from a scaling form for the surface free energy where two independent surface exponentsΔ ₁ and α _s enter. In addition, global phase diagrams with finite symmetry breaking fields are also briefly discussed.