Ab-initio Calculations of Critical Amplitudes

• One important contribution of the Renormalization Group (RG) method is to show that there exists a close connection between statistical mechanics near criticality and Euclidean field theory in the large cut-off limit.
• The determination of the renormalized quantities at zero momentum amounts to the determination of a certain number of parameters appearing in the scaling laws. Some of these parameters are universal (the critical exponents) and much effort has been successfully devoted to their calculation. On the other hand, new techniques need to be developed in order to reliably calculate the non-universal parameters.
• The calculation of the critical amplitudes requires a detailed representation of the RG flows. This is a difficult nonlinear problem. A common strategy in problems involving nonlinear flows near a singular point, is to construct a new system of coordinates for which the governing equations become linear.
• We proposed to combine the nonlinear scaling fields associated with the high-temperature (HT) fixed point, with those associated with the unstable fixed point, in order to calculate the susceptibility and other thermodynamic quantities. The general strategy relies on simple linear relations between the HT scaling fields and the thermodynamic quantities, and the estimation of RG invariants formed out of the two sets of scaling fields. This estimation requires convergent expansions in overlapping domains. If, in addition, the initial values of the scaling fields associated with the unstable fixed point can be calculated from the temperature and the parameters appearing in the microscopic Hamiltonian, one can estimate the critical amplitudes.
• This strategy has been developed using Dyson's hierarchical model where all the steps can be approximately implemented with good accuracy. We have shown numerically that for this model (and a simplified version of it), the required overlap apparently occurs, allowing an accurate determination of the critical amplitudes.
• For more details see: Y. Meurice and S. Niermann, From Nonlinear Scaling Fields to Critical Amplitudes, J. Stat. Phys. 108, 213 (2002).

Work in progress, Plans The question of the interpolation between the Gaussian fixed point and the other two fixed point is being investigated by Li Li. We are planning to investigate the method of dimensional regularization as a way to regularize the small denominator problem.