Finite-size effects on the phase transition in a four- and six-fermion interaction model

We consider four- and six-fermion interacting models at finite temperature and density. We construct the corresponding free energies and investigate the appearance of first- and second-order phase transitions. Finite-size effects on the phase structure are investigated using methods of quantum field theory on toroidal topologies.     

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.     

Finite-size,chemical-potential and magnetic effects on the phase transition in a four-fermion interacting model

We study effects coming from finite size, chemical potential and from a magnetic background on a massive version of a four-fermion interacting model. This is performed in four dimensions as an application of recent developments for dealing with field theories defined on toroidal spaces. We study effects of the magnetic field and chemical potential on the size-dependent phase structure of the model, in particular, how the applied magnetic field affects the size-dependent critical temperature. A connection with some aspects of the hadronic phase transition is established.     

Strain effects on phase transitions in transition metal dichalcogenides

We perform density functional theory calculation to investigate the structural and electronic properties of various two-dimensional transition metal dichalcogenides, MX₂ (M=Ti, V, Cr, Zr, Nb, Mo, Hf, Ta, or W, and X=S or Se), and their strain-induced phase transitions. We evaluate the relative stability and the activation barrier between the octahedral-T and the trigonal-H phases of each MX₂. It is found that the equilibrium and phase transition characteristics of MX₂ can be classified by the group to which its metal element M belongs in the periodic table. MX₂ with M in the group 4 (Ti, Zr, or Hf), forms an octahedral-T phase, while that with an M in the group 6 (Cr, Mo, or W) does a trigonal-H phase. On the other hand, MX₂ with M in the group 5 (V, Nb, or Ta), which is in-between the groups 4 and 6, may form either phase with a similar stability. It is also found that their electronic structures are strongly correlated to the structural configurations: mostly metallic in the T phase, while semiconducting in the H phase, although there are some exceptions. We also explore the effects of an applied stress and find for some MX₂ materials that the resultant strain, either tensile or compressive, may induce a structural phase transition by reducing the transition energy barrier, which is, in some cases, accompanied by its metal-insulator transition.     

Absence of first-order phase transitions for antiferromagnetic systems

We consider a spin system with nearest-neighbor antiferromagnetic pair interactions in a two-dimensional lattice. We prove that the free energy of this system is differentiable with respect to the uniform external fieldh, for all temperatures and allh. This implies the absence of a first-order phase transition in this system.     

Finite-size effects on nucleation in a first-order phase transition

We discuss finite-size effects on homogeneous nucleation in first-order phase transitions. We study their implications for cosmological phase transitions and to the hadronization of a quark–gluon plasma generated in high-energy heavy ion collisions. Very general arguments allow us to show that the finite size of the early universe has virtually no relevance in the process of nucleation and in the growth of cosmological bubbles during the primordial quark–hadron and the electroweak phase transitions. In the case of high-energy heavy ion collisions, finite-size effects play an important role in the late-stage growth of hadronic bubbles.     

Kinetic equation approach to phase transitions

An exact mathematical discussion of the linearized Enskog-Vlasov equation is given. A criterion for the occurrence of the linear instability is related to a criterion for the occurrence of the bifurcation of the equilibrium stationary solution to the nonlinear Enskog-Vlasov equation. Mathematical results are interpreted physically in connection with phase transitions.     

Crystal imperfections and structural phase transitions in perovskites

High‐resolution X-ray scattering measurements of the antiferrodistortive phase transitions in the perovskites SrTiO₃, RbCaF₃ and KMnF₃ have recently revealed the existence of two length scales in the critical scattering above T_e . Here we review these observations and discuss how they might be related to the well known existence of two time scales (the "central peak" problem) in the critical dynamic response function above T_e . We reach the tentative conclusion that within a few degrees of T_e both the time and length scales of the fluctuations are strongly influenced by defects and that the intrinsic critical behaviour is swamped by these effects in most crystals.     

Finite-size scaling study of a first-order temperature-driven symmetry-breaking structural phase transition

We present a study of finite-size effects in a model exhibiting a first-order temperature-driven symmetry-breaking structural phase transition in theL × cylindrical geometry in theL limit. Exact studies demonstrate the applicability of our scaling ansatz even in the one-dimensional limit, making this model ideal for studying finite-size effects. The scaling ansatz, similar to the previously developed ansatz for field-driven transitions, demonstrates that latent heat is crucial in driving these transitions. This ansatz is supported by a 2×2 phenomenological transfer matrix based upon the symmetries of the system; this produces an analytic free energy which has the scaling form. Order parameter probability distributions show that the high- and low-temperature phases coexist only in a small finite-size-affected regime near the bulk transition temperature; this regime vanishes exponentially fast asL diverges.     

Finite-size scaling for first-order transitions: Potts model

The finite-size scaling algorithm based on bulk and surface renormalization of de Oliveira is tesed onq-state Potts models in dimensionsD=2 and 3. Our Monte Carlo data clearly distinguish between first- and second-order phase transitions. Continuous-q analytic calculations performed for small lattices show a clear tendency of the magnetic exponentY=D-/v to reach a plateau for increasing values ofq, which is consistent with the first-order transition valueY=D. Monte Carlo data confirm this trend.     

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.     

Supercooling across first-order phase transitions in vortex matter

Hysteresis in cycling through first-order phase transitions in vortex matter, akin to the well-studied phenomenon of supercooling of water, has been discussed in literature. Hysteresis can be seen while varying either temperature T or magnetic field H (and thus the density of vortices). Our recent work on phase transitions with two control variables shows that the observable region of metastability of the supercooled phase would depend on the path followed in H-T space, and will be larger when T is lowered at constant H compared to the case when H is lowered at constant T. We discuss the effect of isothermal field variations on metastable supercooled states produced by field-cooling. This path dependence is not a priori applicable to metastability caused by reduced diffusivity or hindered kinetics.     

One-dimensional chiral models with first-order phase transitions

A family of one-dimensional classical chiral spin models with groupG=U(N) orSU(N) is introduced, having complex nearest-neighbor interaction. TheseG×G invariant systems have self-adjoint positive transfer matrices and satisfy reflection positivity. In the case ofG=U(N), forN=1, 2, 3, sequences of first-order phase transitions are shown to occur.     

Rounding effects of quenched randomness on first-order phase transitions

Frozen-in disorder in an otherwise homogeneous system, is modeled by interaction terms with random coefficients, given by independent random variables with a translation-invariant distribution. For such systems, it is proven that ind=2 dimensions there can be no first-order phase transition associated with discontinuities in the thermal average of a quantity coupled to the randomized parameter. Discontinuities which would amount to a continuous symmetry breaking, in systems which are (stochastically) invariant under the action of a continuous subgroup ofO(N), are suppressed by the randomness in dimensionsd4. Specific implications are found in the Random-Field Ising Model, for which we conclude that ind=2 dimensions at all (,h) the Gibbs state is unique for almost all field configurations, and in the Random-Bond Potts Model where the general phenomenon is manifested in the vanishing of the latent heat at the transition point. The results are explained by the argument of Imry and Ma [1]. The proofs involve the analysis of fluctuations of free energy differences, which are shown (using martingale techniques) to be Gaussian on the suitable scale.Dedicated to R. L. Dobrushin, on the occasion of his 60^th birthdayAlso in the Physics DepartmentResearch partially supported by NSF grants PHY-8896163 and PHY-8912067Work done in part at Rutgers University, Department of Mathematics     

Picturing quantum phase transitions

We discuss the quantum phase transitions (QPT) in N-spin chains from the point of view of collective observables. We show that the measurement space representation is a convenient tool for the analysis of phase transitions, allowing the determination of an appropriate set of macroscopic order parameters (for a given Hamiltonian). Quantum correlations in the vicinity of the critical points are analyzed both in the ground states and low temperature thermal states.     

Coarsening in fluid phase transitions

We review the understanding of the kinetics of fluid phase separation in various space dimensions. Morphological differences, percolating or disconnected domains, based on overall composition in a binary liquid or on density in a vapor–liquid system, are discussed. Depending upon the morphology, various possible mechanisms for domain growth are pointed out and discussions of corresponding theoretical predictions are provided. On the computational front, useful models and simulation methodologies are presented. Theoretically predicted growth laws have been tested via molecular dynamics simulations of vapor–liquid transitions. In the case of a disconnected structure, the mechanism has been confirmed directly.     

Acoustic dispersion near structural phase transitions

We show on various examples, that elastic constant measurements provide a valuable tool for studying the dynamics of solids near phase transitions. Many experimental methods used in the investigation of phase transitions (NMR, neutron scattering, etc.) are able to give information on the dynamics, but in different regions of q- and ω-space, thereby probing different dynamical processes.

A comparison of the dynamics obtained from the elastic measurements with other techniques (NMR) demonstrates this very clearly for the cases of KSCN and C_60.     

Low-temperature neutron diffraction study of the crystal and magnetic phase transitions in DyCrO4

The crystal and magnetic structures of DyCrO₄ were studied using neutron powder diffraction. Complete diffraction data at 3.6, 17, 27, and 40 K show that a crystal structural phase transition from tetragonal I4₁/amd to orthorhombic Imma symmetry is found to take place between 27 and 40 K. This transition does not involve a significant change in the unit cell volume. Strong ferromagnetic reflections are observed at 3.6 and 17 K, and can be fit well using the magnetic model of space group Im'ma', with the moments of both Dy³⁺ and Cr⁵⁺ ions aligning along the y-axis. Detailed temperature dependent magnetic intensities of 101/011 and 211/121 peaks reveal a Curie temperature of T_c=22.35(15) K.     

Driving rate effects in avalanche-mediated first-order phase transitions

We study the driving-rate and temperature dependence of the power-law exponents that characterize the avalanche distribution in first-order phase transitions. Measurements of acoustic emission in structural transitions in Cu-Zn-Al and Cu-Al-Ni are presented. We show how the observed behavior emerges within a general framework of competing time scales of avalanche relaxation, driving rate, and thermal fluctuations. We confirm our findings by numerical simulations of a prototype model.