First-order phase transitions, the Maxwell construction, and the momentum-space renormalization group

Parola, Alberto; Pini, Davide; Reatto, L.

doi:10.1103/physreve.48.3321

Cited by 45 publications

(95 citation statements)

References 22 publications

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“…Instead, as already discussed in Ref. [6], the sharp cut-off HRT equation predicts the divergence of the compressibility when coexistence is approached.…”

Section: The Coexistence Boundarymentioning

confidence: 83%

“…We now provide an analytical interpretation of the origin of the discontinuity following the procedure devised in [6], for the sharp cut-off formalism. We first write our equation (16) in a scale invariant form.…”

Section: The Coexistence Boundarymentioning

confidence: 99%

“…the transition between a finite solution for |x| > x 0 and the asymptotic form (22) inside the binodal. Following [6], it is useful to zoom-in the region close to x 0 by rescaling the x variable as:…”

Section: The Coexistence Boundarymentioning

confidence: 99%

“…The additive term aQ 2 takes into account the fluctuation corrections to the position of the phase boundary, which are known to play an important role in determining the asymptotic solution on the binodal [6]. Then Eq.…”

Section: The Coexistence Boundarymentioning

confidence: 99%

“…Moreover, the proper treatment of long-wavelength fluctuations forces the free energy to display the correct convexity also in the two-phase region, thereby incorporating Maxwell construction: A major achievement of HRT. Although quite successful, the cur-rent sharp cut-off formulation of HRT predicts a diverging compressibility along the binodal (in any dimension smaller than four) [6] while, for a scalar order parameter, a finite value of the inverse compressibility at coexistence is expected. In this paper we show that this failure of the HRT approach in the sharp cut-off formulation can be overcome by studying the smooth cut-off implementation, already introduced several years ago [7].…”

Section: Introductionmentioning

confidence: 99%

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Smooth cutoff formulation of hierarchical reference theory for a scalarϕ4field theory

et al. 2007

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The φ 4 scalar field theory in three dimensions, prototype for the study of phase transitions, is investigated by means of the Hierarchical Reference Theory (HRT) in its smooth cut-off formulation. The critical behavior is described by scaling laws and critical exponents which compare favorably with the known values of the Ising universality class. The inverse susceptibility vanishes identically inside the coexistence curve, providing a first principle implementation of the Maxwell construction, and shows the expected discontinuity across the phase boundary, at variance with the usual sharp cut-off implementation of HRT. The correct description of first and second order phase transitions within a microscopic, non-perturbative approach is thus achieved in the smooth cut-off HRT.

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“…Instead, as already discussed in Ref. [6], the sharp cut-off HRT equation predicts the divergence of the compressibility when coexistence is approached.…”

Section: The Coexistence Boundarymentioning

confidence: 83%

Section: The Coexistence Boundarymentioning

confidence: 99%

Section: The Coexistence Boundarymentioning

confidence: 99%

Section: The Coexistence Boundarymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Smooth cutoff formulation of hierarchical reference theory for a scalarϕ4field theory

et al. 2007

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Infinite Compressibility States in the Hierarchical Reference Theory of Fluids. I. Analytical Considerations

Reiner

2005

J Stat Phys

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In its customary formulation for one-component fluids, the Hierarchical Reference Theory yields a quasilinear partial differential equation (pde) for an auxiliary quantity f that can be solved even arbitrarily close to the critical point, reproduces non-trivial scaling laws at the critical singularity, and directly locates the binodal without the need for a Maxwell construction. In the present contribution we present a systematic exploration of the possible types of behavior of the pde for thermodynamic states of diverging isothermal compressibility κT as the renormalization group theoretical momentum cutoff approaches zero. By purely analytical means we identify three classes of asymptotic solutions compatible with infinite κT , characterized by uniform or slowly varying bounds on the curvature of f , by monotonicity of the build-up of diverging κT , and by stiffness of the pde in part of its domain, respectively. These scenarios are analzyed and discussed with respect to their numerical properties. A seeming contradiction between two of these alternatives and an asymptotic solution derived earlier [Parola et al., Phys. Rev. E 48, 3321 (1993)] is easily resolved.

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Momentum scale expansion of sharp cutoff flow equations

Morris

1996

Nuclear Physics B

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We show how the exact renormalization group for the effective action with a sharp momentum cutoff, may be organised by expanding one-particle irreducible parts in terms of homogeneous functions of momenta of integer degree (Taylor expansions not being possible). A systematic series of approximations -the O(p M ) approximations -result from discarding from these parts, all terms of higher than the M th degree. These approximations preserve a field reparametrization invariance, ensuring that the field's anomalous dimension is unambiguously determined. The lowest order approximation coincides with the local potential approximation to the Wegner-Houghton equations. We discuss the practical difficulties with extending the approximation beyond O(p 0 ). hep-th/9508017August, 1995. via the one-particle irreducible parts, and the best that one can achieve is an expansion in momentum scale[13] p ∼ √ p µ p µ , the M th approximation resulting from dropping all terms beyond O(p M ). This raises the spectre that the lowest order approximation -O(p 0 ), in which the momentum scale dependence is completely discarded, nevertheless still involvesGreen functions with non-trivial dependence on the angles between their momenta. It turns out that this dependence cancels however, and it is in this way that we recover the local potential approximation [4][5][6][7][8][9][10][11][12].Our second main motivation is unfortunately more negative. We wish to provide the evidence for the claim in ref. [14]: that the momentum scale expansion of the sharp cutoff flow equations is not a practical technique, i.e. beyond O(p 0 ), because only certain truncations (of the field dependence) are calculable in closed form, and this is not good enough[8]. It is not good enough because, in contrast to a momentum scale expansionwhich might be expected to work 5 since it corresponds to an expansion in (an appropriately defined) 'localness' of the effective Lagrangian, truncations of the field dependence have limited accuracy and reliability[8]. We want to emphasise here that our claim that there are practical difficulties in extending the momentum scale expansion beyond O(p 0 ), merely amounts to an admission of our own lack of ability in extending it in a substantive way. It would certainly be worthwhile to improve on these attempts (presented in sect.6), because a simple two-loop example -and general arguments (c.f. ref.[13] and appendix B), suggest that the sharp cutoff momentum scale expansion should yield the fastest convergence (i.e.faster than the smooth cutoff momentum expansions [14,15]).Despite the increasing interest in utilising the exact renormalization group to develop non-perturbative approximations [19], there have so far been relatively few works concerned with going beyond the local potential approximation [11][12][13][14][15][16][17][18]. Part of the point of sect.6 is also pedagogical: we provide simple concrete illustrations of the discussions in earlier sections, but also reemphasise [11,[14][15][16]20] in a simple context the importan...

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First-order phase transitions, the Maxwell construction, and the momentum-space renormalization group

Cited by 45 publications

References 22 publications

Smooth cutoff formulation of hierarchical reference theory for a scalarϕ4field theory

Smooth cutoff formulation of hierarchical reference theory for a scalarϕ4field theory

Infinite Compressibility States in the Hierarchical Reference Theory of Fluids. I. Analytical Considerations

Momentum scale expansion of sharp cutoff flow equations

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