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arXiv:hep-th/0303261

Direct Calculation of the Critical Effective Potential

Hendrik Ballhausen

Published Mar 31, 2003• 0 citations
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Abstract

The critical effective potential is the nonperturbative part of the effective action at a phase transition. It equals the scale invariant effective average potential and can be calculated from the renormalization group flow of the effective average action. In some cases this requires only the solution of an ordinary differential equation without actually simulating the renormalization group flow. Here the Ising model is examined beyond leading order and with full field dependent effective potential.

DOI
10.48550/arxiv.hep-th/0303261

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The critical effective potential is the nonperturbative part of the effective action at a phase transition. It equals the scale invariant effective average potential and can be calculated from the renormalization group flow of the effective average action. In some cases this requires only the solution of an ordinary differential equation without actually simulating the renormalization group flow. Here the Ising model is examined beyond leading order and with full field dependent effective potential.
The desirability of evaluating the effective potential in field theories near a phase transition has been recognized in a number of different areas. We show that recent Monte Carlo simulations for the probability distribution for the order parameter in an equilibrium Ising system, when combined with low-order renormalization group results for an ordinary ${\ensuremath{\varphi}}^{4}$ system, can be used to extract the effective potential. All scaling features are included in the process.
Effective critical exponents for the λϕ^4 scalar field theory are calculated as a function of the renormalization group block size k_o^{-1} and inverse critical temperature β_c. Exact renormalization group equations are presented up to first order in the derivative expansion and numerical solutions are obtained with and without polynomial expansion of the blocked potential. For a finite temperature system in d dimensions, it is shown that \barβ_c = β_c k_o determines whether the d-dimensional (\barβ_c << 1) or (d+1)-dimensional (\barβ_c >> 1) fixed point governs the phase transition. The validity of a polynomial expansion of the blocked potential near criticality is also addressed.