Universal Crossover Research Articles

Onsager approach to the universal cross-over in variable-range hopping with Coulomb interactions

A model describing low-temperature dc conductivity in a disordered hopping system is formulated. The model considers one-particle excitations as a variable range hopping with a constant density of states near the Fermi level and accounts for Coulomb interaction of charge carriers on the basis of Onsager dissociation probability. The resulting temperature dependence of the conductivity reveals a universal crossover from Mott's T −1/4 to Efros–Shklovskii's T −1/2 laws with decreasing temperature and increasing density of localized states.

Crossover SAFT Equation of State and Thermodynamic Properties of Propan-1-ol

In this work we have developed a new equation of state (EOS) for propan-1-ol on the basis of the crossover modification (CR) of the statistical-associating-fluid-theory (SAFT) EOS recently developed and applied to n-alkanes. The CR SAFT EOS reproduces the nonanalytical scaling laws in the asymptotic critical region and reduces to the analytical-classical SAFT EOS far away from the critical point. Unlike the previous crossover EOS, the new CR SAFT EOS is based on the parametric sine model for the universal crossover function and is able to represent analytically connected van der Waals loops in the metastable fluid region. The CR SAFT EOS contains 10 system-dependent parameters and allows an accurate representation of the thermodynamic properties of propan-1-ol over a wide range thermodynamic states including the asymptotic singular behavior in the nearest vicinity of the critical point. The EOS was tested against experimental isochoric and isobaric specific heats, speed of sound, PVT, and VLE data in and beyond the critical region. In the one-phase region, the CR SAFT equation represents the experimental values of pressure with an average absolute deviation (AAD) of less than 1% in the critical and supercritical regions and the liquid densities with an AAD of about 1%. A corresponding states principle is used for the extension of the new CR SAFT EOS for propan-1-ol to higher n-alkanols.

Mean-field expansion for spin models with medium-range interactions

We study the critical crossover between the Gaussian and the Wilson-Fisher fixed point for general O( N)-invariant spin models with medium-range interactions. We perform a systematic expansion around the mean-field solution, obtaining the universal crossover curves and their leading corrections. In particular we show that, in three dimensions, the leading correction scales as R −3, R being the range of the interactions. We compare our results with the existing numerical ones obtained by Monte Carlo simulations and present a critical discussion of other approaches.

Crossover from Fermi Liquid to Wigner Molecule Behavior in Quantum Dots

The crossover from weak to strong correlations in parabolic quantum dots at zero magnetic field is studied by numerically exact path-integral Monte Carlo simulations for up to eight electrons. By the use of a multilevel blocking algorithm, the simulations are carried out free of the fermion sign problem. We obtain a universal crossover governed only by the density parameter ${r}_{s}$. For ${r}_{s}>{r}_{c}$, the data are consistent with a Wigner molecule description, while, for ${r}_{s}<{r}_{c}$, Fermi liquid behavior is recovered. The crossover value ${r}_{c}\ensuremath{\approx}4$ is surprisingly small.

Revision of a multiparameter equation of state to improve the representation in the critical region: application to water

We have developed a crossover formalism for the thermodynamic surface of pure fluids, which can be applied to any multiparameter equation of state. This procedure has been used to incorporate scaling law behavior into a representation of the thermodynamic properties of water and steam developed by Pruss and Wagner (PW EOS) and adopted recently by the International Association for the Properties of Water and Steam. Our revision to this equation retains most of the functional form and coefficients of the PW EOS, but replaces two of the terms with a crossover representation of scaling law behavior. In order to develop this model, we first developed a new crossover formulation for steam in the critical region, and second, we have incorporated universal crossover functions into the original PW EOS. In the modified form, the PW equation of state reproduces the scaling laws down to dimensionless temperatures τ=10 −7. Far from the critical point the equations practically coincide.

Finite Temperature Transport at the Superconductor-Insulator Transition in Disordered Systems

I argue that the incoherent, zero-frequency limit of the universal crossover function in the temperature-dependent conductivity at the superconductor-insulator transition in disordered systems may be understood as an analytic function of dimensionality of system d, with a simple pole at d=1. Combining the exact result for the crossover function in d=1 with the recursion relations in d=1+\epsilon, the leading term in the Laurent series in the small parameter \epsilon for this quantity is computed for the systems of disordered bosons with short-range and Coulomb interactions. The universal, low-temperature, dc critical conductivity for the dirty boson system with Coulomb interaction in d=2 is estimated to be 0.69 (2e)^2 /h, in relatively good agreement with many experiments on thin films. The next order correction is likely to somewhat increase the result, possibly bringing it closer to the self-dual value.

Screening of the Coulomb interaction in two-dimensional variable-range hopping

A Coulomb gap in the density of states is inferred from variable-range-hopping measurements in two dimensions in a gated GaAs/Al${}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}$ heterostructure. For large hopping lengths the Coulomb interaction is screened by a metallic gate, and a universal crossover to Mott hopping at low temperatures is observed. Excellent agreement with theory is achieved by adjusting a single combination of constants associated with variable-range hopping. This change in constants is explained by correlated hopping. The Coulomb interaction is not screened by the two-dimensional electron layer.

Scaling and crossover functions for the conductance in the directed network model of edge states

We consider the directed network (DN) of edge states on the surface of a cylinder of length L and circumference C. By mapping it to a ferromagnetic superspin chain, and using a scaling analysis, we show its equivalence to a one-dimensional supersymmetric nonlinear sigma model in the scaling limit, for any value of the ratio L/C, except for short systems where L is less than of order C^{1/2}. For the sigma model, the universal crossover functions for the conductance and its variance have been determined previously. We also show that the DN model can be mapped directly onto the random matrix (Fokker-Planck) approach to disordered quasi-one-dimensional wires, which implies that the entire distribution of the conductance is the same as in the latter system, for any value of L/C in the same scaling limit. The results of Chalker and Dohmen are explained quantitatively.

Some recent advances in the theory of homogeneous isotropic turbulence

We review some advances in the theory of homogeneous, isotropic turbulence. Our emphasis is on the new insights that have been gained from recent numerical studies of the three-dimensional Navier Stokes equation and simpler shell models for turbulence. In particular, we examine the status of multiscaling corrections to Kolmogorov scaling, extended self similarity, generalized extended self similarity, and non-Gaussian probability distributions for velocity differences and related quantities. We recount our recent proposal of a wave-vector-space version of generalized extended self similarity and show how it allows us to explore an intriguing and apparently universal crossover from inertial- to dissipation-range asymptotics.

Universal Crossover between Efros-Shklovskii and Mott Variable-Range-Hopping Regimes.

A universal scaling function, describing the crossover between the Mott and the Efros-Shklovskii hopping regimes, is derived using the percolation picture of transport in strongly localized systems. This function agrees very well with experimental data. Quantitative comparison with experiment allows for the possible determination of the role played by quasiparticles in transport.

The Coulomb gap and the transition to Mott hopping

Resistivity data are presented which confirm a Coulomb gap in the single particle density of states in two dimensional variable range hopping. For large hopping lengths the Coulomb interaction is screened by a metallic gate. A universal crossover to Mott hopping is observed at low temperatures. Excellent agreement with theory is achieved with a single parameter fit.

Universal, finite-temperature, crossover functions of the quantum transition in the Ising chain in a transverse field

We consider finite-temperature properties of the Ising chain in a transverse field in the vicinity of its zero-temperature, second-order quantum phase transition. New universal crossover functions for static and dynamic correlators of the “spin” operator are obtained. The static results follow from an early lattice computation of McCoy, and a method of analytic continuation in the space of coupling constants. The dynamic results are in the “renormalized classical” region and follow from a proposed mapping of the quantum dynamics to the Glauber dynamics of a classical Ising chain.

Kinematic theory for scale-invariant patterns in acicular martensitas

We present a kinematic theory which explains the emergence of scale-invariant patterns in acicular martensites which occur, for example, in FeC and FeNi alloys. Scale-invariant structures emerge naturally as a consequence of competition between the average tip-velocity, ν, and the rate of nucleation, I, of the martensite grains. Martensite growth is analyzed in terms of fixed points of a well-defined renormalization group. It is shown that the stationary probability distribution of the martensite grain size P( l), is governed by two fixed points - an unstable (noncritical) fixed point at ν/ I = 0, characterized by a Gamma distribution and a stable (critical) fixed point at ν/ I = ∞, characterized by a Lévy distribution, P( l) ∼ l − α . A universal crossover function describes the SPD at intermediate values of ν/ I. The analysis may also be used to compute the tip-velocity from optical micrograph pictures of martensite grains.

Dimensional crossover in the large-N limit

We consider dimensional crossover for an $O(N)$ Landau-Ginzburg-Wilson model on a $d$-dimensional film geometry of thickness $L$ in the large $N$-limit. We calculate the full universal crossover scaling forms for the free energy and the equation of state. We compare the results obtained using ``environmentally friendly'' renormalization with those found using a direct, non-renormalization group approach. A set of effective critical exponents are calculated and scaling laws for these exponents are shown to hold exactly, thereby yielding non-trivial relations between the various thermodynamic scaling functions.

Transitions in critical size for metal (100) homoepitaxy

Abstract For submonolayer nucleation and growth of islands during metal (100) homoepitaxy, we precisely characterize transitions, with increasing temperature, in the critical size, i , above which islands are effectively stable. For strong adatom bonding, we demonstrate universal crossover behavior from i = 1 directly to a well-defined “ i = 3” regime where tetramers are stable, but not, e.g., pentamers. A criterion is also presented for the ultimate breakdown of “ i = 3” due to double-bond scission. Various transition temperatures and key energies are thereby determined for Ag, Cu, Fe, Ni and Pd.

Screening of the Coulomb gap

Data from resistivity measurements are presented which confirm a Coulomb gap in the density of states in two-dimensional variable-range hopping. For large hopping lengths the Coulomb interaction is screened by a metallic gate. A universal crossover to Mott hopping at low temperatures is observed. Internal consistency of the data and excellent agreement with theory are achieved with a single parameter fit.

Bethe ansatz solution for crossover scaling functions of the asymmetric XXZ chain and the Kardar-Parisi-Zhang-type growth model.

A perturbative method is developed to calculate the finite size corrections of the low-lying energies of the asymmetric XXZ Hamiltonian near the stochastic line. The crossover from isotropic to anisotropic, Kardar-Parisi-Zhang (KPZ) scaling of the mass gaps is determined in terms of universal crossover scaling functions. At the stochastic line, the asymmetric XXZ Hamiltonian describes the time evolution of the single-step or body-centered solid-on-solid growth model in one dimension. The mass gaps of the growth model are found as a function of the growth rate and the substrate slope. Higher-order corrections to the growth model mass gaps are also calculated to obtain the first terms of the KPZ to Edwards-Wilkinson crossover scaling function in the large argument expansion in the zero slope sector.

Random-field-crossover scaling in Mn0.35Zn0.65F2.

The weakly anisotropic and highly diluted antiferromagnet ${\mathrm{Mn}}_{0.35}$${\mathrm{Zn}}_{0.65}$${\mathrm{F}}_{2}$ has been studied by dc magnetization measurements under a uniform magnetic field and by ac susceptibility at zero field. Earlier experiments in ${\mathrm{Mn}}_{\mathit{x}}$${\mathrm{Zn}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{F}}_{2}$ for xg0.4 have shown that under an external field applied parallel to the easy direction [001] the critical phase boundary ${\mathit{T}}_{\mathit{c}}$(H) is governed by a random-exchange Ising model (REIM) to random-field Ising model (RFIM) crossover scaling. This crossover scaling stipulates that ${\mathit{T}}_{\mathit{N}}$-${\mathit{T}}_{\mathit{c}}$(H)\ensuremath{\sim}${\mathit{H}}^{2/\mathrm{\ensuremath{\varphi}}}$, where \ensuremath{\varphi}\ensuremath{\simeq}1.4 is the universal REIM-RFIM crossover exponent. We present magnetization results which show that for x=0.35, ${\mathit{T}}_{\mathit{c}}$(H) has a different curvature following the above scaling law but with \ensuremath{\varphi}\ensuremath{\simeq}3.4 (similar to the de Almeida--Thouless line in spin glasses). Zero-field ac susceptibility (${\mathrm{\ensuremath{\chi}}}_{\mathrm{ac}}$) measurements performed on this sample reveal a freezing of spatial fluctuations at low temperatures. The magnetic features of ${\mathrm{Mn}}_{0.35}$${\mathrm{Zn}}_{0.65}$${\mathrm{F}}_{2}$ are distinct in some aspects from the spin-glass behavior found in the isostructural but more anisotropic compound ${\mathrm{Fe}}_{\mathit{x}}$${\mathrm{Zn}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{F}}_{2}$ close to the percolation threshold.

Exact conformal-field-theory results on the multi-channel Kondo effect: Asymptotic three-dimensional space- and time-dependent multi-point and many-particle Green's functions

We have recently developed a conformal field theory approach for quantum impurity problems, which include the multi-channel Kondo effect, allowing us to obtain exact asymptotic Green's functions, amongst other results. The single-fermion Green's function was presented in a separate publication. Here we present the detailed derivation of and explicit expressions for the four-point single-particle and two-point two-particle Green's functions, including the spin-density, charge-density and pair field Green's function. These represent universal crossover functions, from non-interacting (Fermi-liquid) behavior when all operators are far from the impurity, to singular (non-Fermi-liquid) behavior, characterized by Wess-Zumino-Witten conformal field theories, when operators move close to the impurity. These crossover functions show explicitly how “deconfinement” of the spin degrees of freedom near the impurity site occurs. The results are obtained for the general case of an SU( N) impurity spin and k channels, using the Fusion Rule of Kac-Moody conformal field theory, modular transformations, the Verlinde formula, and Braid and Fusion matrices. We also show how to compute exact higher multi-point functions with our approach. From the four-point single-particle function we extract an operator product coefficient, needed for the computation of the resistivity. We also present the result for the Wilson ratio for an SU( N) impurity spin.

The Plateau and Symmetrical Structure of Lyapunov Exponents in 2-Dimensional Homogeneous Dissipative Map11The project supported by National Basic Research Project “Nonlinear Science”

The universal crossover behavior of Lyapunov exponents in transition from conservative limit to dissipative limit of dynamical system is studied. We discover numerically and prove analytically that for homogeneous dissipative two-dimensional maps, along the equal dissipation line in parameter space, two Lyapunov exponents and of periodic orbits possess a plateau structure, and around this exponent plateau value, there is a strict symmetrical relation between and no matter whether the orbit is periodic, quasiperiodic, or chaotic. The method calculating stable window and Lyapunov exponent plateau widths is given. For Hénon map and 2-dimensional circle map, the analytical and numerical results of plateau structure of Lyapunov exponents for period-1, 2 and 3 orbits are presented.