Gauge-invariant critical exponents for the Ginzburg-Landau model

The critical behavior of the Ginzburg-Landau model is described in a manifestly gauge-invariant manner. The gauge-invariant correlation-function exponent is computed to first order in the 4-d and 1/n expansion, and found to agree with the ordinary exponent obtained in the covariant gauge, with the parameter alpha=1-d in the gauge-fixing term ( partial differential (mu)A(mu))(2)/2alpha.     

The critical exponents of the matrix-valued Gross-Neveu model

We study the large-N limit of the matrix-valued Gross-Neveu model in d > 2 dimensions. The method employed is a combination of the approximate recursion formula of Polyakov and Wilson with the solution to the zero-dimensional large-N counting problem of Makeenko and Zarembo. The model is found to have a phase transition at a finite value for the critical temperature and the critical exponents are approximated by ν = 1/(2(d − 2)) and η = d − 2. We test the validity of the approximation by applying it to the usual vector models where it is found to yield exact results to leading order in 1/N.     

Anomalous critical exponents in the anisotropic Ashkin-Teller model

We perform a rigorous computation of the specific heat of the Ashkin-Teller model in the case of small interaction and we explain how the universality-nonuniversality crossover is realized when the isotropic limit is reached. We prove that, even in the region where universality for the specific heat holds, anomalous critical exponents appear: for instance, we predict the existence of a previously unknown anomalous exponent, continuously varying with the strength of the interaction, describing how the difference between the critical temperatures rescales with the anisotropy parameter.     

Quantum effect,critical dynamics and one-particle excitation in 1/n expansion

By quantizing Ma's Hamiltonian, quantum effect on η, the energy spectrum of one-particle excitation and the dynamic scaling law are studied up to O(1/n). The case just at the critical point is considered.     

The n-loop expansion of the Reggeon calculus

We apply the technique known in solid state physics as the n-loop expansion to calculate the critical indices of the φ³ Gribov Reggeon calculus directly in two transverse dimensions. Infrared pathologies of the massless theory require the calculation to be done in the infinite momentum limit of the massive theory. For n = 1 the results are close to those of the ε-expansion in O(ε). For n = 2 the β function has no zero, analogously to the case in solid state physics. Use of a Padé approximant for β yields σ_tot ≈ (1n s)^0.27 at infinity, close to the O(ε²) result.     

High-temperature expansion for the Coulomb lattice

It is shown that the high-temperature expansion for the Coulomb lattice is Tr exp{−βH} = αβ^−3/2[1 − C₁β + C₂β² − C₃β^5/2 + O(β³logβ)]. C₁ and C₂ are the usual Korteweg-de Vries constants of motion. C₃ reflects the local singularity of the Coulomb potential and is absent for smooth periodic fields. C₃ turns out to be independent of the details of the potential.     

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.     

High-temperature series expansion for the relaxation times of the two dimensional Ising model

We derive the high temperature series expansions for the two relaxation times of the single spin-flip kinetic Ising model on the square lattice. The series for the linear relaxation time _l is obtained with 20 non-trivial terms, and the analysis yields 2.183±0.005 as the value of the critical exponent _l , which is equal to the dynamical critical exponentz in the two-dimensional case. For the non-linear relaxation time we obtain 15 non-trivial terms, and the analysis leads to the results _nl = 2.08 ± 0.07. The scaling relation _l – _nl = ( being the exponent of the order parameter) seems to be fulfilled, though the error bars of _nl are still quite substantial. In addition, we obtain the series expansion of the linear relaxation time on the honeycomb lattice with 22 non-trivial terms. The result for the critical exponent is close to the value obtained on the square lattice, which is expected from universality.     

Scale-invariant Lyapunov exponents for classical hamiltonian systems

The Lyapunov exponent for classical hamiltonian systems is made dimensionless by introducing a characteristic time τ_c. This modification yields an energy-independent exponent for systems with scale invariance.     

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.     

Lifshitz-point correlation length exponents from the large-n expansion

The large-n expansion is applied to the calculation of thermal critical exponents describing the critical behavior of spatially anisotropic d-dimensional systems at m  -axial Lifshitz points. We derive the leading non-trivial 1/n$1/n$ correction for the perpendicular correlation-length exponent _νL2${\nu }_{L2}$ and hence several related thermal exponents to order O(1/n)$O(1/n)$. The results are consistent with known large-n expansions for d  -dimensional critical points and isotropic Lifshitz points, as well as with the second-order epsilon expansion about the upper critical dimension d^?=4+m/2$d^{?}=4+m/2$ for generic m∈[0,d]$m\in [0,d]$. Analytical results are given for the special case d=4$d=4$, m=1$m=1$. For uniaxial Lifshitz points in three dimensions, 1/n$1/n$ coefficients are calculated numerically. The estimates of critical exponents at d=3$d=3$, m=1$m=1$ and n=3$n=3$ are discussed.     

The 1/D expansion for low-dimensional classical magnets

The physical characteristics of two-dimensional classical ferro- and antiferro-magnets have been calculated in the whole temperature range by an analytical approach based on the expansion in powers of 1/D, whereD is the number of spin components. This approach works rather well since it yields exact results for antiferromagnetic susceptibility and specific heat atT=0 already in the first order in 1/D and it is consistent with HTSE at high temperatures. For the quantities singular atT=0, such as ferromagnetic susceptibility and correlation length, the 1/D expansion supports their general-D functional form in the low-temperature range obtained by Fukugita and Oyanagi. The critical index calculated in the first order in 1/D proves to be temperature dependent: =20/(D) (=T/T _c ^(MFT) ,T _c ^(MFT) =J ₀/D, J ₀ is the zero Fourier component of the exchange interaction).     

Largest lyapunov exponents for lattices of interacting classical spins

We investigate how generic the onset of chaos in interacting many-body classical systems is in the context of lattices of classical spins with nearest-neighbor anisotropic couplings. Seven large lattices in different spatial dimensions were considered. For each lattice, more than 2000 largest Lyapunov exponents for randomly sampled Hamiltonians were numerically computed. Our results strongly suggest the absence of integrable nearest-neighbor Hamiltonians for the infinite lattices except for the trivial Ising case. In the vicinity of the Ising case, the largest Lyapunov exponents exhibit a power-law growth, while further away they become rather weakly sensitive to the Hamiltonian anisotropy. We also provide an analytical derivation of these results.     

QCD chiral phase transition and critical exponents within the nonextensive Polyakov-Nambu-Jona-Lasinio model

We present a nonextensive version of the Polyakov-Nambu-Jona-Lasinio model that is based on nonextentive statistical mechanics.This new statistics model is characterized by a dimensionless nonextensivity parameter q that accounts for all possible effects violating the assumptions of the Boltzmann-Gibbs(BG) statistics(for q→ 1,it returns to the BG case).Based on the nonextensive Polyakov-Nambu-Jona-Lasinio model,we discussed the influence of nonextensive effects on the curvature of the phase diagram at μ=0 and especially on the location of the critical end point(CEP).A new and interesting phenomenon we found is that with an increase in q,the CEP position initially shifts toward the direction of larger chemical potential and lower temperature.However,when q is larger than a critical value q_c,the CEP position moves in the opposite direction.In other words,as q increases,the CEP position moves in the direction of smaller chemical potential and higher temperature.This U-turn phenomenon may be important for the search of CEP in relativistic heavy-ion collisions,in which the validity of BG statistics is questionable due to strong fluctuations and long-range correlations,and nonextensive effects begin to manifest themselves.In addition,we calculated the influence of the nonextensive effects on the critical exponents and found that they remain almost constant with q.