Wilson's momentum shell renormalization group from Fourier Monte Carlo simulations

Previous attempts to accurately compute critical exponents from Wilson’s momentum shell renormalization prescription suffered from the difficulties posed by the presence of an infinite number of irrelevant couplings. Taking the example of the 1d long-ranged Ising model, we calculate the momentum shell renormalization flow in the plane spanned by the coupling constants (u0, r0) for different values of the momentum shell thickness parameter b by simulation using our recently developed Fourier Monte Carlo algorithm. We report strong anomalies in the b-dependence of the fixed point couplings and the resulting exponents yτ and ω in the vicinity of a shell parameter b∗ < 1 characterizing a thin but finite momentum shell. Evaluation of the exponents for this value of b yields a dramatic improvement of their numerical accuracy, indicating a strong damping of the influence of irrelevant couplings for b = b∗. 1. Wilson’s Momentum Shell Renormalization Group: A Reminder Recall that – in a nutshell – a general momentum shell renormalization group (MSRG) transformation is a mapping R(·, b) : K → K induced by [1, 2] 1. coarse graining, 2. rescaling of lengths/momenta, and 3. renormalization of field(s), on the infinite-dimensional space K = (K0,K1,K2, . . . ) of coupling constants of a given model. In the following we will frequently deal with the parameter b > 1 measuring the “thickness” of the “momentum shell” Λ/b < |k| < Λ at fixed cutoff Λ, which is therefore indicated explicitly in out notation. Nontrivial critical behavior is connected to fixed points (FPs) R(K∗, b) ≡ K∗ of R(·, b) corresponding to a correlation length ξ[K∗] = ∞. Linearizing around such an FP, one obtains a matrix equation δK′ i = ∑ jMi j[K]δK j + O(δK2) where δK = K − K∗, and the eigenvalues λi of Mi j[K] = ∂Ri(K,b) ∂K j ∣∣∣∣ K=K∗ , are related to universal critical exponents yi by λi ≡ byi . These are termed relevant, irrelevant and marginal if yi > 0, yi < 0 or yi = 0, respectively. In what follows, we will focus on a subject which is usually not regarded to be of any importance, namely the effect of choosing a particular value for the parameter b. In fact, all universal quantities as well as K∗ should be independent of such a choice in a (hypothetical) exact calculation. In analytical expansion calculations, choosing b ∼ 1 + δ for infinitesimal δ > 0 is usually computationally convenient. However, what is usually swept under the rug is that in a real world computation, where truncations and approximations are unavoidable, exact invariance with respect to the choice of b may be broken (cf. an early analysis of this situation in Ref. [3, 4])! We thus raise the following question: “What is the qualitative as well as the quantitative influence of a concrete choice of b on the results obtained from the MSRG?” In an analytic approach, there is little hope to answer this question. To make progress, we thus have to resort to simulations. 2. RG Flows From Fourier Monte Carlo Simulations In recent papers [5, 6] we presented a new simulation method that in principle allows to non-perturbatively calculate renormalization group (RG) flows and its fixed points in the space K for Wilson’s MSRG scheme. These as well as the present simulations are based on our recently developed Fourier Monte Carlo (FMC) algorithm. This method, which allows to work exclusively in reciprocal space and thus avoid any reference to real space, is tailor-made for studying coarse graining and criticality for φ4 type models. It is based on the following observations and ideas (see [7, 8, 9, 10] for details): In FMC, the real and imaginary parts of the discrete Fourier amplitudes φ̃(k) = N−1/2 ∑ x φ(x)e−ikx of lattice fields φ(x) act as the basic MC variables. MC moves are constructed by picking a particular wave vector k0 at random and shifting the associated Fourier amplitude according to η̃(k)→ η̃(k) + ξδk,k0 + ξδ−k,k0 (1) As the harmonic part is always diagonalized by Fourier transform, it is easy to keep track of the change of this contribution to any φ4 type of Hamiltonian under this move. The central observation [11] leading to an efficient and manageable MC algorithm for φ4 type models is that the remaining anharmonic part ∼ ∑x φ4(x) = ∑x(φ2(x))2 can also be diagonalized – not in terms of the original Fourier amplitudes, but in terms of the Fourier amplitudes φ̃2(k) = N−1/2 ∑ p φ̃(p)φ̃(k − p) of the squared field φ2(x)! Additional recording and smart bookkeeping of these quadratic modes and their changes under shifts Preprint submitted to Computer Physics Communications June 18, 2010 of type (1) of the basic modes η̃(k) during the course of the simulation thus leads to a manageable algorithm that succeeds in avoiding any reference to the original direct lattice. While this strategy may be quite inefficient for studying problems in lattice systems for which all Fourier modes of fields have to be fully included, it can out–power other approaches in situations where one concentrates on a small subset of Fourier modes, while all others can be put to zero. Such cases include coarse graining problems [11] like the MSRG, in which one sums over modes belonging to comparatively thin shells of momenta inside the Brillouin zone, or problems based on a given effective Hamiltonian defined with respect to a small wave vector cutoff Λ π/a, where a is a measure of the microscopic lattice spacing. In particular, the FMC is tailor-made to study the critical behavior of lattice models of the φ4 type. Other situations in which FMC will be advantageous include problems with a complicated anharmonic coupling structure ([7, 12]) or models involving long-range lattice interactions, as these are conveniently diagonalized by Fourier transform [13]. A representative of the latter class is the long-range φ4 model defined by the dimensionless lattice Hamilonian