Quantum critical Hall exponents

We investigate a finite size “double scaling” hypothesis using data from an experiment on a quantum Hall system with short range disorder ,  and . For Hall bars of width w at temperature T   the scaling form is w^−μT^−κ$w^{-\mu }{T}^{-\kappa }$, where the critical exponent μ≈0.23$\mu \approx 0.23$ we extract from the data is comparable to the multi-fractal exponent _α0−2${\alpha }_0-2$ obtained from the Chalker–Coddington (CC) model [4]. We also use the data to find the approximate location (in the resistivity plane) of seven quantum critical points, all of which closely agree with the predictions derived long ago from the modular symmetry of a toroidal σ-model with m matter fields [5]. The value _ν8=2.60513…${\nu }_8=2.60513\dots$ of the localisation exponent obtained from the m=8$m=8$ model is in excellent agreement with the best available numerical value _νnum=2.607±0.004${\nu }_{\mathrm{num}}=2.607\pm 0.004$ derived from the CC-model [6]. Existing experimental data appear to favour the m=9$m=9$ model, suggesting that the quantum Hall system is not in the same universality class as the CC-model. We discuss the reason this may not be the case, and propose experimental tests to distinguish between the two possibilities.     

Scaling relationship between effective critical exponents throughout the crossover region in thin Ising films

The Monte Carlo (MC) approach is used to check the validity of the scaling relationship for the effective critical exponents in thin Ising films. We investigate this relationship not just in the critical region but throughout the crossover to the expected two-dimensional behavior. Our results indicate that this scaling relationship is very well-fulfilled throughout the entire crossover temperature region, as predicted by a previous renormalization group analysis. The two-dimensional universality class of Ising films is confirmed by means of data collapsing plots for plates with increasing L, up to L=100. The evolution of the maximum value of the effective critical exponents with film thickness is discussed. Received 22 April 1999     

Computation of critical exponents for two-dimensional ising model on a cellular automaton

Static critical exponents for the two-dimensional Ising model are computed on a cellular automaton. The analysis of the data within the framework of the finite-size scaling theory reproduces their well-established values.     

Rigorous proof of the high-temperature Josephson inequality for critical exponents

I give a rigorous proof of the high-temperature Josephson inequalitydv 2–, following the original ideas of Josephson. The proof is applicable to a class of models including the ferromagnetic Ising model and the ⁴ lattice field theory.Research supported in part by NSF Grant PHY 78-23952.     

Absence of critical exponents in ferroelectrics: experiments of Hilczer and theory of Levanyuk and Sigov

The pioneering breakthrough of Bozena Hilczer and her coworkers showed that the so-called critical phenomena in many ferroelectric crystals are entirely produced by defects. She studied gadolinium aluminum sulphate hexahydrate (GASH), lithium ammonium sulphate, lithium hydrazinium sulphate, PZT, barium titanate, and especially TGS, via neutron irradiation, X-ray irradiation, and e-beam electron irradiation, which showed that the divergences in specific heat or dielectric constant near the Curie temperature T_c are not caused by ‘critical’ fluctuations but by static defects, which can be annealed out and subsequently reproduced by irradiation. This work is rarely cited (modern physicists often feel that literature searches are optional), leading to frequent rediscovery and generally spurious claims of true ‘critical’ phenomena near T_c.     

Evaluation of critical exponents from raman intensities

Theory and experiment are compared for Raman intensities near continuous structural phase transitions. Situations in which the order parameter couples linearly to light or quadratically are considered. Both cases are easily analyzed for the soft modes in ferroelastic LnP₅O₁₄ (Ln = La, Pr, Nd, Tb). The trace polarizability tensor yields β = 0.49 ± 0.02; off-diagonal terms give γ’ = 1.16 ± 0.15 and γ = 1.07 ± 0.10. Mean field results are also obtained for barium sodium niobate near T (incommensurate) = 582 K and for tris-sarcosine calcium chloride near T _c = 128 K.     

Ising-type critical exponents of the fully frustrated spin-1/2 Heisenberg FM/AF square bilayer at a critical magnetic field

The fully frustrated spin-1/2 Heisenberg FM/AF square bilayer in a magnetic field with the ferromagnetic inter-dimer interaction and the antiferromagnetic intra-dimer interaction is explored by the use of localized many-magnon approach, which allows to connect the original purely quantum Heisenberg spin model on a square bilayer with the effective ferromagnetic Ising model on a simple square lattice. Magnetization and specific heat are investigated exactly at a field-driven phase transition from the singlet-dimer phase towards the fully saturated ferromagnetic phase, which changes from a discontinuous phase transition to a continuous one at a certain critical temperature. The mapping correspondence between the spin-1/2 Heisenberg FM/AF square bilayer and the ferromagnetic Ising square lattice suggests for this special critical point of the spin-1/2 Heisenberg FM/AF square bilayer critical exponents from the standard two-dimensional Ising universality class.     

More inequalities for critical exponents

A variety of rigorous inequalities for critical exponents is proved. Most notable is the low-temperature Josephson inequalitydv +2 2–. Others are 1 1 +v, 1 1 , 1,d 1 + 1/ (for d),dv, ₃ + (for d), ₄ , and _{2m 2m+2} (form 2). The hypotheses vary; all inequalities are true for the spin-1/2 Ising model with nearest-neighbor ferromagnetic pair interactions.NSF Predoctoral Fellow (1976–1979). Research supported in part by NSF Grant PHY 78-23952.     

Investigating the nonequilibrium aspects of long-range Potts model: Refinement of critical temperatures and raw exponents

In this work, we analyze the q-state Potts model with long-range interactions through nonequilibrium scaling relations commonly used when studying short-range systems. We determine the critical temperature via an optimization method for short-time Monte Carlo simulations. The study takes into consideration two different boundary conditions and three different values of range parameters of the couplings. We also present estimates of some critical exponents, named as raw exponents for systems with long-range interactions, which confirm the non-universal character of the model. Finally, we provide some preliminary results addressing the relations between the raw exponents and the exponents obtained for systems with short-range interactions. The results assert that the methods employed in this work are suitable to study the considered model and can easily be adapted to other systems with long-range interactions.     

Scaling laws for all Liapunov exponents: Models and measurements

We introduce simple diamond models of random symplectic matrices in order to study the scaling laws of all Liapunov exponents. These universal properties appear in physical problems that are modeled by transfer matrices: dynamical systems, random potentials, random fields, etc. Numerical experiments for the general case are in agreement with the results derived from the models.     

Potts ferromagnet: Transformations and critical exponents in planar hierarchical lattices

We prove that the duality transformation for a Potts ferromagnet on two-rooted planar hierarchical lattices (HL) preserves the thermal eigenvalue. This leads to a relation between the correlation length critical exponents of a HL and its corresponding dual lattice. Using hyperscaling, we show that their specific heat critical exponents coincide. For a smaller class of HL—namely of diamond and tress types—we prove that another transformation also preserves and .     

Scaling relations in the lyapunov exponents of one dimensional maps

We establish numerically the validity of Huberman-Rudnick scaling relation for Lyapunov exponents during the period doubling route to chaos in one dimensional maps. We extend our studies to the context of a combination map, where the scaling index is found to be different.     

Microcanonical Finite-Size Scaling

In the microcanonical ensemble, suitably defined observables show nonanalyticities and power-law behavior even for finite systems. For these observables, a microcanonical finite-size scaling theory is established and combined with the experimentally observed power-law behavior. Scaling laws are obtained which relate exponents of the finite system and critical exponents of the infinite system to the system-size dependence of the affiliated microcanonical observables.     

Lyapunov exponents of stochastic dynamical systems

It is shown that stochastic equations can have stable solutions. In particular, there exists stochastic dynamics for which the motion is both ergodic and stable, so that all trajectories merge with time. We discuss this in the context of Monte Carlo-type dynamics, and study the convergence of nearby trajectories as the number of degrees of freedom goes to infinity and as a critical point is approached. A connection with critical slowdown is suggested.     

Determination of critical exponents of inhomogeneous Gd films

The role of inhomogeneity on the critical behavior is studied for non-epitaxial Gd films. For this purpose, the film inhomogeneity was varied experimentally by annealing otherwise identical samples at different temperatures T_AN=200, 400, and 500 °C. Vibrating sample magnetometry (VSM) was used for magnetization M vs. T measurements at different external fields H. A method based upon the linear superposition of different sample parts having different Curie temperatures T_C was used to extract the critical exponents and the intrinsic distribution of Curie temperatures. We found that this method allows extracting reliable values of the critical exponents for all annealing temperatures, which enabled us to study the effects of disorder onto the universality class of Gd films.     

Measures of critical exponents in the four-dimensional site percolation

Using finite-size scaling methods we measure the thermal and magnetic exponents of the site percolation in four dimensions, obtaining a value for the anomalous dimension very different from the results found in the literature. We also obtain the leading corrections-to-scaling exponent and, with great accuracy, the critical density.     

Critical exponents for the self-avoiding random walk in three dimensions

We compute by direct Monte Carlo simulation the main critical exponents, , ₄, andv and the effective coordination number for the self-avoiding random walk in three dimensions on a cubic lattice. We find both hyperscaling relationsdv=2– anddv– 2 ₄+=0 satisfied ind = 3.     

Scaling and universality of critical fluctuations in granular gases

The total energy fluctuations of a low-density granular gas in the homogeneous cooling state near the threshold of the clustering instability are studied by means of molecular dynamics simulations. The relative dispersion of the fluctuations is shown to exhibit a power-law divergent behavior. Moreover, the probability distribution of the fluctuations presents data collapse as the system approaches the instability, for different values of the inelasticity. The function describing the collapse turns out to be the symmetric of the one found in several molecular equilibrium and nonequilibrium systems.     

临界堆中子增殖系数测量统计涨落数值模拟

建立了临界堆中子增殖系数(参数)计算模型,采用闪烁体加光电倍增管组成的电流型探测器对参数进行测量。采用构造理想函数加统计涨落偏差的方式,利用数值模拟方法,研究了统计涨落对参数测量结果的影响。结果表明：测量信号统计涨落会影响参数测量精度;考虑统计涨落影响,为使参数满足要求,需要使探测器入射粒子数达到一定数量;针对不同测量时间,入射粒子起始强度需要满足不同要求;当起始强度不能满足要求时,可以采用增加测量时间;指数信号带随机涨落时,较大的求值区间能有效减弱统计涨落的影响。