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Register a new userNon-perturbative renormalization group preserving full-momentum dependence: implementation and quantitative evaluation
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We present in detail the implementation of the Blaizot-Mendez-Wschebor (BMW) approximation scheme of the nonperturbative renormalization group, which allows for the computation of the full momentum dependence of correlation functions. We discuss its signification and its relation with other schemes, in particular the derivative expansion. Quantitative results are presented for the testground of scalar O(N) theories. Besides critical exponents which are zero-momentum quantities, we compute in three dimensions in the whole momentum range the two-point function at criticality and, in the high temperature phase, the universal structure factor. In all cases, we find very good agreement with the best existing results.
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We demonstrate the power of a recently-proposed approximation scheme for the non-perturbative renormalization group that gives access to correlation functions over their full momentum range. We solve numerically the leading-order flow equations obtained within this scheme, and compute the two-point functions of the O(N) theories at criticality, in two and three dimensions. Excellent results are obtained for both universal and non-universal quantities at modest numerical cost.
The non-perturbative renormalization-group approach is extended to lattice models, considering as an example a $phi^4$ theory defined on a $d$-dimensional hypercubic lattice. Within a simple approximation for the effective action, we solve the flow equations and obtain the renormalized dispersion $eps(q)$ over the whole Brillouin zone of the reciprocal lattice. In the long-distance limit, where the lattice does not matter any more, we reproduce the usual flow equations of the continuum model. We show how the numerical solution of the flow equations can be simplified by expanding the dispersion in a finite number of circular harmonics.
We propose a modification of the non-perturbative renormalization-group (NPRG) which applies to lattice models. Contrary to the usual NPRG approach where the initial condition of the RG flow is the mean-field solution, the lattice NPRG uses the (local) limit of decoupled sites as the (initial) reference system. In the long-distance limit, it is equivalent to the usual NPRG formulation and therefore yields identical results for the critical properties. We discuss both a lattice field theory defined on a $d$-dimensional hypercubic lattice and classical spin systems. The simplest approximation, the local potential approximation, is sufficient to obtain the critical temperature and the magnetization of the 3D Ising, XY and Heisenberg models to an accuracy of the order of one percent. We show how the local potential approximation can be improved to include a non-zero anomalous dimension $eta$ and discuss the Berezinskii-Kosterlitz-Thouless transition of the 2D XY model on a square lattice.
In this paper we study the $c$-function of the sine-Gordon model taking explicitly into account the periodicity of the interaction potential. The integration of the $c$-function along trajectories of the non-perturbative renormalization group flow gives access to the central charges of the model in the fixed points. The results at vanishing frequency $beta^2$, where the periodicity does not play a role, are retrieved and the independence on the cutoff regulator for small frequencies is discussed. Our findings show that the central charge obtained integrating the trajectories starting from the repulsive low-frequencies fixed points ($beta^2 <8pi$) to the infrared limit is in good quantitative agreement with the expected $Delta c=1$ result. The behavior of the $c$-function in the other parts of the flow diagram is also discussed. Finally, we point out that also including higher harmonics in the renormalization group treatment at the level of local potential approximation is not sufficient to give reasonable results, even if the periodicity is taken into account. Rather, incorporating the wave-function renormalization (i. e. going beyond local potential approximation) is crucial to get sensible results even when a single frequency is used.
We use a non-perturbative renormalization-group technique to study interacting bosons at zero temperature. Our approach reveals the instability of the Bogoliubov fixed point when $dleq 3$ and yields the exact infrared behavior in all dimensions $d>1$ within a rather simple theoretical framework. It also enables to compute the low-energy properties in terms of the parameters of a microscopic model. In one-dimension and for not too strong interactions, it yields a good picture of the Luttinger-liquid behavior of the superfluid phase.
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