Numerical solution of the one-dimensional saddle point equation of the Ginzburg–Landau Hamiltonian with random temperature

Abstract

The one-dimensional saddle point equation of the Ginzburg–Landau Hamiltonian with random temperature is studied with a numerical method. The random temperature is correlated with a finite range l. The distribution width of the random temperature is � . The ground state of the saddle point equation is solved. The average, fluctuation and auto-correlation of the order parameter are obtained. It is found that the auto-correlation function behaves like ∼exp −x 2 ξ 2 φ .F or� � 1 /l 2 , where l is dimensionless, the correlation length is given by ξφ ∝ l .F or �< 1 /l 2 ,a st = 0, the correlation length ξφ ∝ l� −α , where α = 0.65. All the saddle point solutions for �< 1 /l 2 can be mapped to that for � = 1 /l 2 by using a coarse-grained approximation.