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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Thermodynamic renormalization group

Hiroshi Matsuoka

Published Mar 1, 1996• 0 citations
Publisher

Abstract

For a system with a first or second order phase transition at ${\mathit{T}}^{\mathrm{*}}$, we construct a type of renormalization group transformation that maps the reduced temperature t=(T-${\mathit{T}}^{\mathrm{*}}$)/${\mathit{T}}^{\mathrm{*}}$ and the external field h for a system of volume V to t\ensuremath{'} and h\ensuremath{'} for another system of volume V\ensuremath{'}. Our construction is purely thermodynamic without reference to effective Hamiltonians. Using the transformation, we derive the finite size scaling form for the singular part of the free energy, which leads to the scaling laws in the thermodynamic limit. We thus provide a unified thermodynamic framework for phase transitions. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreve.53.2076

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Scaling behavior for first-order phase transitions can be derived alternatively but consistently from renormalization-group, phenomenological, or finite-size considerations. A general analysis of densities at a renormalization-group fixed point demonstrates that if the coexistence of ppp distinct phases is possible, then ppp distinct eigenvalue exponents must equal the spatial dimensionality. This basic eigenvalue (or scaling) exponent condition can also be derived phenomenologically by various arguments not depending on detailed renormalization-group considerations. A scaling description of first-order phase transitions is presented and extended to finite systems with linear dimensions LLL, leading to a rounding proportional to ${L}^{\ensuremath{-}d}$, response-function maxima varying as ${L}^{d}\ensuremath{\propto}N$, and boundary-condition-dependent shifts which may be as large as $\ensuremath{\sim}{L}^{\ensuremath{-}1}$.
A renormalization-group calculation has been carried out in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions to lowest order in $\ensuremath{\epsilon}$, using the continuum generalization of the Potts model. The equation of state and the various thermodynamic functions have been calculated explicitly by integrating up the recursion relations in both the ordered and disordered phases. The transition is found to be first order for p>2p>2p>2. For 2<p<1032<p<\frac{10}{3}2<p<310​ the transition is first order in spite of the fact that there is an accessible stable fixed point for all p<103p<\frac{10}{3}p<310​. At the first-order transition, instead of the usual expansion in temperature, the various thermodynamic functions are studied as functions of ($p\ensuremath{-}2$). Finally it is shown that the free energy of the Potts model has an essential singularity at the first-order transition temperature Tc{T}_{c}Tc​, due to instantons, which is of the same form as that of the Ising model at H=0H=0H=0.
The description of first-order phase transitions using the linear renormalization group is illustrated with the two-dimensional Ising model. Different linear renormalization groups should be used to describe the system in the neighborhood of criticality and away from it. In the latter case, the renormalization-group trajectories are lines of constant spin expectation value, which map a system below critical temperature in a small magnetic field to a system of free spins at a finite magnetic field. A finite-lattice approximation is also presented. The equation of state and several thermodynamic quantities are obtained; a smooth matching of the behavior near and away from criticality occurs naturally.
Calculations on finite (or partially finite) systems, such as Monte-Carlo simulations or the diagonalization of a finite transfer matrix, are often extrapolated to the thermodynamic limit by the use of scaling assumptions involving the finite sizes L of the system. It is shown here that these finite size scaling laws follow from the fact that : i) in the renormalization group flow equations the length L is not renormalized ; ii) there is no singularity at the fixed point. This second property fails in four (and above four) dimensions and finite size scaling does not hold there. An illustration of these points is presented, with detailed calculations of the large N limit of the N- vector model. As a consequence, the ε-expansion of finite size scaling properties is singular.