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Higher-Order Calculations for Weakly Random System in 1/n Expansion

K. Ideura, Yutaka Okabe, R. Abe

In the scheme of 1/n expansion the effects of weak randomness on critical exponents are investigated. By extending the lowest-order argument of Abe, the higher-order terms of the O(a3) and of the O(1/n) are studied(a being a measure of the randomness). Though each diagram constituting higher-order terms shows the logarithmic dependence, which has the possibility of modifying critical exponents, they are canceled out when summed up. It is concluded that random impurities are irrelevant to critical exponents.In the scheme of 1/n expansion the effects of weak randomness on critical exponents are investigated. By extending the lowest-order argument of Abe, the higher-order terms of the O(a3) and of the O(1/n) are studied(a being a measure of the randomness). Though each diagram constituting higher-order terms shows the logarithmic dependence, which has the possibility of modifying critical exponents, they are canceled out when summed up. It is concluded that random impurities are irrelevant to critical exponents.

Progress of Theoretical PhysicsApr 1, 19810 citations

Recent theoretical estimates of the critical exponents

V. A. Rabinovich, Yu. E. Sheludyak

Journal of Engineering Physics and ThermophysicsNov 1, 19861 citation

Critical exponents from power spectra

K. B. Lauritsen, Hans C. Fogedby

Journal of Statistical PhysicsJul 1, 199325 citations

Critical exponents for higher order phase transitions: Landau theory and RG flow

Joydeep Chakravarty, Diksha Jain

Abstract In this work, we define and calculate critical exponents associated with higher order thermodynamic phase transitions. Such phase transitions can be classified into two classes: with or without a local order parameter. For phase transitions involving a local order parameter, we write down the Landau theory and calculate critical exponents using the saddle point approximation. Further, we investigate fluctuations about the saddle point and demarcate when such fluctuations dominate over saddle point calculations by introducing the generalized Ginzburg criteria. We use Wilsonian renormalization group (RG) to derive scaling forms for observables near criticality and obtain scaling relations between the critical exponents. Afterwards, we find out fixed points of the RG flow using the one-loop beta function and calculate critical exponents about the fixed points for third and fourth order phase transitions.Abstract In this work, we define and calculate critical exponents associated with higher order thermodynamic phase transitions. Such phase transitions can be classified into two classes: with or without a local order parameter. For phase transitions involving a local order parameter, we write down the Landau theory and calculate critical exponents using the saddle point approximation. Further, we investigate fluctuations about the saddle point and demarcate when such fluctuations dominate over saddle point calculations by introducing the generalized Ginzburg criteria. We use Wilsonian renormalization group (RG) to derive scaling forms for observables near criticality and obtain scaling relations between the critical exponents. Afterwards, we find out fixed points of the RG flow using the one-loop beta function and calculate critical exponents about the fixed points for third and fourth order phase transitions.

arXiv:2102.08398Journal of Statistical Mechanics Theory and ExperimentSep 1, 20218 citations

Critical behaviour of a 3D Ising-like system in the ρ^6 model approximation: Role of the correction for the potential averaging

Ihor V. Pylyuk, Ulyak

The critical behaviour of systems belonging to the three-dimensional Ising universality class is studied theoretically using the collective variables (CV) method.The partition function of a one-component spin system is calculated by the integration over the layers of the CV phase space in the approximation of the non-Gaussian sextic distribution of order-parameter fluctuations (the ρ 6 model).A specific feature of the proposed calculation consists in making allowance for the dependence of the Fourier transform of the interaction potential on the wave vector.The inclusion of the correction for the potential averaging leads to a nonzero critical exponent of the correlation function η and the renormalization of the values of other critical exponents.The contributions from this correction to the recurrence relations for the ρ 6 model, fixed-point coordinates and elements of the renormalization-group linear transformation matrix are singled out.The expression for a small critical exponent η is obtained in a higher non-Gaussian approximation.The critical behaviour of systems belonging to the three-dimensional Ising universality class is studied theoretically using the collective variables (CV) method.The partition function of a one-component spin system is calculated by the integration over the layers of the CV phase space in the approximation of the non-Gaussian sextic distribution of order-parameter fluctuations (the ρ 6 model).A specific feature of the proposed calculation consists in making allowance for the dependence of the Fourier transform of the interaction potential on the wave vector.The inclusion of the correction for the potential averaging leads to a nonzero critical exponent of the correlation function η and the renormalization of the values of other critical exponents.The contributions from this correction to the recurrence relations for the ρ 6 model, fixed-point coordinates and elements of the renormalization-group linear transformation matrix are singled out.The expression for a small critical exponent η is obtained in a higher non-Gaussian approximation.

arXiv:1212.6139Condensed Matter PhysicsDec 1, 20121 citation

Calculation of Critical Exponents from Field Theory

Jean Zinn‐Justin

Springer eBooksJan 1, 19822 citations

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