Crossover between field theories with short-range and long-range exchange or correlations

Honkonen, Juha; Nalimov, Mikhail Yur'evich

doi:10.1088/0305-4470/22/6/024

Cited by 83 publications

(134 citation statements)

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“…According to the analysis of Fisher et al [2], universality classes are parametrized by s, and the following three distinct regimes were identified: (a) The classical regime; the upper critical dimension is given by d u 2s, so that mean-field-type critical behavior occurs for s # d͞2. (b) The intermediate regime d͞2 , s , 2; here the critical exponents are continuous functions of s. (c) The shortrange regime; for s $ 2, the universal properties are those of the model with short-range interactions, e.g., between nearest neighbors only; thus one observes that, for d 3, van der Waals interactions (decaying as 1͞r 6 ) actually lie quite close to the boundary between regimes (b) and (c).…”

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confidence: 99%

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Boundary between Long-Range and Short-Range Critical Behavior in Systems with Algebraic Interactions

2002

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We investigate phase transitions of two-dimensional Ising models with power-law interactions, using an efficient Monte Carlo algorithm. For slow decay, the transition is of the mean-field type; for fast decay, it belongs to the short-range Ising universality class. We focus on the intermediate range, where the critical exponents depend continuously on the power law. We find that the boundary with short-range critical behavior occurs for interactions depending on distance r as r 215͞4 . This answers a long-standing controversy between mutually conflicting renormalization-group analyses. DOI: 10.1103/PhysRevLett.89.025703 PACS numbers: 64.60.Ak, 05.70.Jk, 64.60.Fr, 75.10.Hk Many of the intermolecular forces that play a central role in large areas of chemistry, physics, and biology have a long-ranged nature. Well-known examples are electrostatic interactions, polarization forces, and van der Waals forces. Remarkably, there are still considerable deficiencies in our knowledge of the critical behavior induced by these interactions, which include the prominent case of Coulombic criticality (see Ref.[1] for a review).But also for the simpler case of purely attractive, long-range interactions, there exists a smoldering controversy, which we aim to resolve in this work. The present understanding of critical behavior in systems with such algebraically decaying interactions is largely based on the renormalization-group (RG) calculations for the O͑n͒ model by Fisher, Ma, and Nickel [2]. Their analysis revealed that different regimes can be identified for the universal critical properties, characterized by the decay power of the interactions. In view of the small number of global parameters determining the universality class, the location of the boundaries between these regimes is of considerable interest. It is, therefore, disturbing to note that there appears to be still no consensus on the theoretical side regarding the precise location of the boundary between shortrange and long-range critical behavior.In this Letter, we address this issue by means of numerical calculations, which allow us, with minimal prior assumptions, to decide between contradictory RG scenarios. Concretely, we demonstrate that the onset of the longrange regime occurs for interactions that decay more slowly than found in Ref.[2]. This result not only bears upon condensed-matter systems with long-range interactions, but also has implications for the renormalizationgroup treatment of competing fixed points in general, including random systems and gauge theories.Our approach specializes to the Ising model, n 1, in d dimensions, described by the reduced Hamiltonian,where the spins s k 61 are labeled by the lattice site k, the sum runs over all spin pairs, and the pair coupling depends on the distance r ij j r i 2 r j j between the spins. According to the analysis of Fisher et al. [2], universality classes are parametrized by s, and the following three distinct regimes were identified: (a) The classical regime; the upper critical dimension is given by...

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confidence: 99%

“…A more serious objection was raised by Gusmão and Theumann [5] (but ignored in Ref. [6]), who argued that the parameter 2 2 s (i.e., 2a in the notation of Ref. [6]) is not a valid expansion parameter.…”

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confidence: 99%

Boundary between Long-Range and Short-Range Critical Behavior in Systems with Algebraic Interactions

2002

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“…While the limit α → 1 presents no difficulties, the opposite limit α → 0 is far more subtle: we will see that, just below the upper critical dimension d c (α), two nontrivial renormalizations are required to render the α = 0 theory finite, in contrast to a single one if α is positive and of O(1), leading to an apparent discontinuity in the critical exponents. A similar situation arises in standard φ 4 -theory with long-ranged interactions [10]. There, however, the discontinuity is entirely spurious and can be removed: letting ε = d c (0) − d denote the distance from the upper critical dimension of the short-range theory, the key is to recognize [10] that there is a region of small α = O (ε), where a careful analysis of the RG flow reveals a smooth crossover between the α = 0 and the α = O (1) theories.…”

Section: Introductionmentioning

confidence: 59%

“…A similar situation arises in standard φ 4 -theory with long-ranged interactions [10]. There, however, the discontinuity is entirely spurious and can be removed: letting ε = d c (0) − d denote the distance from the upper critical dimension of the short-range theory, the key is to recognize [10] that there is a region of small α = O (ε), where a careful analysis of the RG flow reveals a smooth crossover between the α = 0 and the α = O (1) theories. Thus, all universal scaling properties are shown to depend continuously on α and d. Here, in contrast, the limit α → 0 is even more intricate: while the short-range theory is controlled by two fixed points, only a single one remains in the long-range model.…”

Section: Introductionmentioning

confidence: 59%

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Janssen

Schmittmann

1999

Journal of Statistical Physics

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The diffusion of hard-core particles subject to a global bias is described by a nonlinear, anisotropic generalization of the diffusion equation with conserved, local noise. Using renormalization group techniques, we analyze the effect of an additional noise term, with spatially long-ranged correlations, on the long-time, long-wavelength behavior of this model. Above an upper critical dimension dLR, the long-ranged noise is always relevant. In contrast, for d < dLR, we find a "weak noise" regime dominated by short-range noise. As the range of the noise correlations increases, an intricate sequence of stability exchanges between different fixed points of the renormalization group occurs. Both smooth and discontinuous crossovers between the associated universality classes are observed, reflected in the scaling exponents. We discuss the necessary techniques in some detail since they are applicable to a much wider range of problems.

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“…At first sight it would seem that the generic propagator (18) allows for several choices for the bare propagator. Analysis of similar situations in the field theory of static critical phenomena has revealed, however, that the consistent way to proceed is to separate the fractional terms to interaction [11,12] …”

Section: Renormalizationmentioning

confidence: 99%

Fractional Stochastic Field Theory

Honkonen

2018

EPJ Web Conf.

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Abstract. Models describing evolution of physical, chemical, biological, social and financial processes are often formulated as differential equations with the understanding that they are large-scale equations for averages of quantities describing intrinsically random processes. Explicit account of randomness may lead to significant changes in the asymptotic behaviour (anomalous scaling) in such models especially in low spatial dimensions, which in many cases may be captured with the use of the renormalization group. Anomalous scaling and memory effects may also be introduced with the use of fractional derivatives and fractional noise. Construction of renormalized stochastic field theory with fractional derivatives and fractional noise in the underlying stochastic differential equations and master equations and the interplay between fluctuation-induced and built-in anomalous scaling behaviour is reviewed and discussed.

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Crossover between field theories with short-range and long-range exchange or correlations

Cited by 83 publications

References 12 publications

Boundary between Long-Range and Short-Range Critical Behavior in Systems with Algebraic Interactions

Boundary between Long-Range and Short-Range Critical Behavior in Systems with Algebraic Interactions

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Fractional Stochastic Field Theory

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