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.     

Multidimensional kinetic theory of first-order phase transitions

A theory is developed to calculate the steady-state nucleation rate in the multidimensional space of variables describing a nucleus. The nucleation rate, a stationary distribution of nuclei, and the direction of the nucleus flux are calculated within this theory. The expression derived for the nucleation rate is invariant with respect to the dimensionality of the space and includes the result obtained in the one-dimensional theory. The stationary distribution function is expressed in terms of the initial physical variables. The nucleation rate is calculated using a new method that requires neither separation of the variables nor taking into account the symmetry of the diffusion matrix $\hat D$ . However, it is demonstrated that the theory is consistent only if the matrix $\hat D$ is symmetric. The symmetry of this matrix is discussed in relation to the constraints imposed on the direction of the nucleus flux. The normalization of the equilibrium distribution functions is discussed, and the relation between the multi-and one-dimensional theories is shown.     

Ginzburg-Landau theory of phase transitions in pseudo-one-dimensional systems

The static and dynamic properties of weakly coupled chains undergoing a phase transition are reviewed. The discussion is based on the functional generalization of the Ginzburg-Landau theory, including systems with real and complex order parameter. Various predictions of the theory, such as static correlations, renormalized phonon frequencies and a central resonance in the dynamic form factor near structural instabilities are discussed and compared with recent experiments on linear conductors that undergo a Peierls transition. New results are obtained for the thermodynamic anomalies near the onset of 3-d ordering and for the dynamic form factor of systems with an incommensurate Peierls distortion.     

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.     

Field theory of fluctuations during phase transitions

The evolution of the most probable amplitudes of the hydrodynamic mode is investigated in the self-consistent scheme. Fluctuations of the amplitude of the conjugate force are also investigated. Fiz. Tverd. Tela (St. Petersburg) 41, 1689–1692 (September 1999)     

On the multidimensional theory of the first-order phase transitions

Multidimensional theory of first-order phase transitions in the vicinity of a one-dimensional saddle point is considered. Transformations of the variables describing new-phase nuclei are suggested; these transformations allow one to completely separate the variables in the Fokker-Planck equation and reduce the problem to a one-dimensional one. The distribution function and the nucleation rate are found for both stationary and non-stationary nucleation stages. As an illustration, the problem of boiling of a volatile liquid is considered in the case where new-phase nuclei are characterized by two parameters.     

Manifestation of critical behavior at first-order phase transitions

We present in this Letter further results which are in good agreement with our earlier observations on the critical behavior at a strong first-order phase transition. More elaborate data analysis has been used here. A quantitative measure of the strength of the transition within the context of Landau theory is also given.     

Description of thermal conductivity in the Ginzburg-Landau theory of phase transitions

A generalization of the Ginzburg-Landau theory of phase transitions is presented which allows one to describe states with variable temperatures. The approach is based on an expression for the entropy in the form of a functional which depends on the temperature gradient and the order parameter. It is shown that the theory is compatible with the zero-th law of thermodynamics (constancy of the temperature in equilibrium). For equilibrium thermodynamically stable states the results of the theory agree with the results of the isothermal approach based on the free energy functional. General limitations on the possible form of the nonlinear dynamic equations are given, in particular for the heat flux vector, and possible particular versions are given. The dynamics of linear fluctuations has been included, including temperature fluctuations. O. Yu. Shmidt Institute of Earth Physics, Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 54–59, June, 1998.     

Singularities in first-order phase transitions

Several theories of phase transitions and their inter-relations have been criticized, focusing on the problem of whether z _c, the value of the fugacity corresponding to the point of condensation, is given by z _s, the smallest real positive singularity of the analytic function defined by the power series using volume-independent cluster integrals, or not. The present situation has been analysed and it is made clear that none of the existing theories can give the answer to this problem. Plausibility arguments for an affirmative or negative answer are discussed.     

Method of virtual media in the theory of first-order phase transitions

The method of deriving a set of equations describing the first-order phase transitions under different conditions has been proposed. A set of equations is obtained for a medium in which a decrease in the pressure leads to the formation of bubbles filled with a dissolved gas.     

General theory of lee-yang zeros in models with first-order phase transitions

We present a general, rigorous theory of Lee-Yang zeros for models with first-order phase transitions that admit convergent contour expansions. We derive formulas for the positions and the density of the zeros. In particular, we show that, for models without symmetry, the curves on which the zeros lie are generically not circles, and can have topologically nontrivial features, such as bifurcation. Our results are illustrated in three models in a complex field: the low-temperature Ising and Blume-Capel models, and the q-state Potts model for large q.     

Renormalization group theory for temperature-driven first-order phase transitions in scalar models

We study the scaling and universal behavior of temperature-driven first-order phase transitions in scalar models. These transitions are found to exhibit rich phenomena, though they are controlled by a single complex-conjugate pair of imaginary fixed points of ? ³ theory. Scaling theories and renormalization group theories are developed to account for the phenomena, and three universality classes with their own hysteresis exponents are found: a field-like thermal class, a partly thermal class, and a purely thermal class, designated, respectively, as Thermal Classes I, II, and III. The first two classes arise from the opposite limits of the scaling forms proposed and may cross over to each other depending on the temperature sweep rate. They are both described by a massless model and a purely massive model, both of which are equivalent and are derived from ? ³ theory via symmetry. Thermal Class III characterizes the cooling transitions in the absence of applied external fields and is described by purely thermal models, which include cases in which the order parameters possess different symmetries and thus exhibit different universality classes. For the purely thermal models whose free energies contain odd-symmetry terms, Thermal Class III emerges only at the mean-field level and is identical to Thermal Class II. Fluctuations change the model into the other two models. Using the extant three- and two-loop results for the static and dynamic exponents for the Yang–Lee edge singularity, respectively, which falls into the same universality class as ? ³ theory, we estimate the thermal hysteresis exponents of the various classes to the same precision. Comparisons with numerical results and experiments are briefly discussed.     

Self consistent theory of fluctuations near D-dimensional superconducting phase transitions

Nonlinear fluctuation contributions in the vicinity of the transition temperature for D-dimensional superconductors (D = 0…3) are treated in self-consistent Hartree approximation to the Ginzburg-Landau free energy. The results concern specific heat, correlation length, and the effect of a magnetic field on the specific heat (for D = 2).     

General approach for studying first-order phase transitions at low temperatures

By combining different ideas, a general and efficient protocol to deal with discontinuous phase transitions at low temperatures is proposed. For small T's, it is possible to derive a generic analytic expression for appropriate order parameters, whose coefficients are obtained from simple simulations. Once in such regimes simulations by standard algorithms are not reliable; an enhanced tempering method, the parallel tempering-accurate for small and intermediate system sizes with rather low computational cost-is used. Finally, from finite size analysis, one can obtain the thermodynamic limit. The procedure is illustrated for four distinct models, demonstrating its power, e.g., to locate coexistence lines and the phase density at the coexistence.     

Phenomenological theory of first-order phase transitions characterized by a one-component order parameter

The Landau theory of phase transitions is generalized with regard to the change Δρ_inv in the charge distribution probability density that does not break the symmetry of the high-symmetry phase. It is shown that the inclusion of Δρ_inv makes it possible to describe first-order phase transitions in terms of a fourth-order non-equilibrium potential. The suggested theory leads to the conclusion that the ferroelastic transition in TeO₂ is a first-order phase transition.     

Weak first-order superfluid-solid quantum phase transitions

We study superfluid-solid zero-temperature transitions in two-dimensional lattice boson-spin models using worm-algorithm Monte Carlo simulations. We observe that such transitions are typically first order with the exception of special high-symmetry points which require fine-tuning in the Hamiltonian parameter space. We present evidence that the superfluid-checkerboard solid and superfluid-valence-bond solid transitions at half-integer filling factor are extremely weak first-order transitions and in small systems can be confused with continuous or high-symmetry points.