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AIP conference proceedings

Some Applications of the Lambert W-function to Critical Phenomena: a View on the Critical Behavior in the Upper Critical Dimension

Ekaterina S. Pisanova, N. S. Tonchev, Hristo Kisov, Angelos Angelopoulos, Takis Fildisis

Published Jan 1, 2010• 2 citations
Publisher

Abstract

Usually, the critical phenomena properties at their upper critical dimensions are characterized by leading logarithmic terms in the corresponding equations and the Lambert W‐function can be used to obtain exact results. A class of models with long‐range interactions and developed classical and quantum fluctuations is considered at the upper quantum critical dimension The exact results, obtained here in the Lambert W‐function terms, and the asymptotic ones for the inverse susceptibility of a lattice model of structural phase transitions in an anharmonic crystal are compared on the basis of their graphic presentations.

DOI
10.1063/1.3322442

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Usually, the critical phenomena properties at their upper critical dimensions are characterized by leading logarithmic terms in the corresponding equations and the Lambert W‐function can be used to obtain exact results. A class of models with long‐range interactions and developed classical and quantum fluctuations is considered at the upper quantum critical dimension The exact results, obtained here in the Lambert W‐function terms, and the asymptotic ones for the inverse susceptibility of a lattice model of structural phase transitions in an anharmonic crystal are compared on the basis of their graphic presentations.
The pure quantum limit of an exactly solvable lattice model describing structural phase transitions in an anharmonic crystal with long-range interaction is considered. At the upper quantum critical dimension the free energy density at T = 0 in the neighbourhood of the quantum critical point is exactly calculated in terms of the Lambert W-function. For the three real physical dimensions, the exact results, obtained here, and the asymptotic ones are compared. It is pointed out that the Lambert W-function turns out to be a very effective tool for an exact computation of non-universal characteristics in the upper critical dimensions, especially in a broader neighbourhood of the critical region.
The entropy and the specific heat of a quantum spherical model with a long-range interaction (decreasing at large distances r as r –d–σ where d is the space dimensionality and 0 < σ ≤ 2) are studied in the quantum critical region at the upper quantum critical dimension d = 3σ/2. The problem of obtaining the temperature-dependent corrections to the ground state free energy involves the solution of a transcendental equation, the exact solution of which is expressed in terms of the Lambert W-function. The free energy, the entropy and the specific heat in the quantum critical region are derived in terms of the Lambert W-function. For systems in real physical dimensions (chains, thin layers, i.e. films and three-dimensional systems) the exact results for the entropy and the specific heat obtained in terms of the Lambert W-function and the leading asymptotic ones are compared on the basis of the calculated relative errors. It can be concluded that for a class of quantum models at the upper critical dimension the Lambert W-function is a very effective tool for an exact computation of low-temperature critical properties, especially in the finite-temperature quantum critical region.
Abstract Phase transitions and critical phenomena are among the most intriguing phenomena in nature and the renormalization-group theory for them is one of the greatest achievements of theoretical physics. However, the predictions of the theory above an upper critical dimension d c seriously disagree with reality. In addition to its fundamental significance, the problem is also of practical importance because both complex classical systems with long-range spatial or temporal interactions and quantum phase transitions with long-range interactions can substantially lower d c . The extant scenarios built on a dangerous irrelevant variable (DIV) to resolve the problem introduce two sets of critical exponents and even two sets of scaling laws whose origin is obscure. Here, we consider the DIV from a different perspective and clearly unveil the origin of the two sets of exponents and hence the intrinsic inconsistency in those scenarios. We then develop an effective-dimension theory in which critical fluctuations and system volume are fixed at an effective dimension by the DIV. This enables us to account for all the extant results consistently. A novel asymptotic finite-size scaling behavior for a correlation function together with a new anomalous dimension and its associated scaling law is also predicted. We also apply the theory to quantum phase transitions.
An exactly solvable lattice model describing structural phase transitions in an anharmonic crystal with long-range interaction (decreasing at large distances r as r−d−σ, where d is the space dimensionality and 0 < σ ≤ 2) is considered near to its zero-temperature critical point. The low-temperature behavior of the bulk specific heat capacity at the lower classical critical dimension (d = σ) is studied in different regions of the (T, λ)-phase diagram, where T is the temperature and λ is a parameter which switches on quantum fluctuations. From the results obtained one can see that when T → 0+ the specific heat capacity tends to zero in a different way in the regions: (a) renormalized classical region - as T raised to the second power and (b) quantum disordered region - exponentially.
We present the results of extensive Monte Carlo simulations of Ising models with algebraically decaying ferromagnetic interactions in the regime where classical critical behavior is expected for these systems. We corroborate the values for the exponents predicted by renormalization theory for systems in one, two, and three dimensions and accurately observe the predicted logarithmic corrections at the upper critical dimension. We give both theoretical and numerical evidence that above the upper critical dimension the decay of the critical spin-spin correlation function in finite systems consists of two different regimes. For one-dimensional systems our estimates for the critical couplings are more than two orders of magnitude more accurate than existing estimates. In two and three dimensions we are unaware of any other results for the critical couplings.