Renormalization group for the φ4 theory with long-range interaction and the critical exponent η of the Ising model

Abstract

We calculate the critical exponent η of the D-dimensional Ising model from a simple truncation of the functional renormalization group flow equations for a scalar field theory with long-range interaction. Our approach relies on the smallness of the inverse range of the interaction and on the assumption that the Ginzburg momentum defining the width of the scaling regime in momentum space is larger than the scale where the renormalized interaction crosses over from long range to short range; the numerical value of η can then be estimated by stopping the renormalization group flow at this scale. In three dimensions our result η=0.03651 is in good agreement with recent conformal bootstrap and Monte Carlo calculations. We extend our calculations to fractional dimensions D and obtain the resulting critical exponent η(D) between two and four dimensions. For dimensions 2≤D≤3 our result for η is consistent with previous calculations.

The nonperturbative functional renormalization group and its applications

“…Let us also mention the following studies: O(N) models with long-range interactions [184][185][186]; fixed points with imaginary couplings [187,188]; non-polynomial perturbations to fixed points [189][190][191][192][193]; O(N) models in finite geometries and critical Casimir forces [172,194]; nonlinear sigma models [195][196][197]; the Potts model [198].…”

Strain fields and critical phenomena in manganites II: spin-lattice-energy Hamiltonians

“…multi-critical universality classes [41], crossover phenomena [42], and stripe phases [43,44]. Further, the effect of LR generalizations of Φ 4 interactions near the PM-FM transition was studied by Goll and Kopietz [45], who observed a finite Fisher exponent in d = 3. Lattice models with LR spin interactions have also been studied by various authors [42,[46][47][48] and shown to exhibit non-universal behavior, e.g.…”

Criticality of the Higgs mass for the long-range quantum XY chain: Amplitude ratio between the Higgs and paramagnetic gaps

“…The spin systems with the long-range interactions have been investigated both theoretically [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] and experimentally [17,18,19,20,21,22]. A main concern is to clarify how the criticality between the ordered and disordered phases is affected by the power of the algebraic decay [7,8,9].…”

“…Actually, relying on the correspondence between the σ-and Dmediated considerations, the authors of Ref. [13] analyzed the criticality for the D = 3 (short-range) Ising model quantitatively. Experimentally, systems with variable-range interactions have been synthesized [18,19,20].…”