Functional Renormalization Group

Abstract

The functional renormalization group is a particular implementation of the renormalization group concept which combines functional methods of quantum field theory with the renormalization group idea of Kenneth Wilson. It interpolates smoothly between the known microscopic laws and the complex macroscopic phenomena in physical systems. The renormalization group is formulated directly for a continuum field theory—no lattice regularization is required. The flow from microscopic to macroscopic scales is given by technically demanding flow equations. We derive the Polchinski equation for the scale-dependent Schwinger functional and the Wetterich equation for the scale-dependent effective action. We use the latter to calculate the effective potential, ground state energy and energy gap of quantum mechanical systems. Next we consider scalar fields and calculate the flow of the effective potential and several critical exponents in the local potential approximation. Then we present an exact solution for the scale-dependent effective potential of O(N) models and the critical exponents in the large-N limit. All numerical results were obtained with the matlab/octave programs listed at the end of the chapter.

Appendix: A Momentum Integral

In this appendix we calculate the O(p ²) contribution to the integral

$$ F(p)=\int\mathrm{d}^d q\, \partial_k \bigl \{Z_kR_k(q) \bigr\} G^2_0(q) G_0(p+q) $$

(12.116)

for the optimized regulator function (12.10). The integrand is only non-zero for q ²≤k ² and in this region

$$ \partial_k \bigl\{Z_kR_k(q) \bigr\} G^2_0(q)= \bigl( \bigl(k^2-q^2 \bigr)\partial_k Z_k+2kZ_k \bigr) \varDelta_0^2 . $$

(12.117)

To proceed we need to consider two cases: |p+q|≤k and |p+q|>k separately.

The Case |p+q|<k

This is the region located inside of both spheres in Fig. 12.11 where the Green function G ₀(q)=G ₀(p+q)=Δ ₀ is independent of the integration variable q. Let us decompose this variable as q=q _∥+q _⊥, where q _∥ is parallel and q _⊥ perpendicular to the fixed momentum p. Then the integral has the form

$$ I_1=\operatorname{Vol} (S_{d-2} )\varDelta_0^3 \int\mathrm{d}q_\parallel\int\mathrm{d}\vert q_\perp\vert \vert q_\perp\vert^{d-2} \bigl( \bigl(k^2-q^2 \bigr)\partial_k Z_k+2kZ_k \bigr) , $$

(12.118)

where $q^{2}=q_{\parallel}^{2}+q_{\perp}^{2}$. The volume of the unit sphere S _d−2⊂ℝ^d−1 originates from the integration over the directions of q _⊥. Now we split the integration domain inside of both spheres, the region marked gray in Fig. 12.11, into two spherical caps,

The domain of integration in (12.118) is just the union of these two caps. After a shift q _∥→q _∥−p in the integral over the second cap we are left with the one-dimensional integral

Its second derivative with respect to p at p=0 is

$$ \frac{\mathrm{d}^2 I_1}{\mathrm{d}p^2} \bigg\vert_{p=0}=-k^dV(B_d) \varDelta_0^3 \partial_k Z_k . $$

(12.119)

Fig. 12.11

Sketch of the integration regions in momentum space. Only momenta inside of the ball centered at the origin contribute to the integral. The Green functions are constant in the gray region inside of both spheres. Inside the sickle-shaped light-gray region G ₀(p,q) is momentum dependent. The two spheres intersect at the lower-dimensional sphere defined by $\{q_{\parallel}^{*},q_{\perp}^{*} \}= \{-p/2,k^{2}-p^{2}/4\}$

The Case |p+q|<k

This is the sickle-shaped region inside of the sphere centered at the origin but outside the displaced sphere, marked light-gray in Fig. 12.11. The integral over this region can be written as a difference of two integrals as follows:

The integrand of both integrals is

$$h(q_\parallel,q_\perp)=\frac{ (k^2-q_\parallel^2-q_\perp^2 )\partial_k Z_k +2kZ_k}{Z_k(q_\parallel+p)^2+Z_k q_\perp^2+a_2} . $$

It is convenient to shift the variable q _∥ in the second integral by −p such that

The second derivative of this integral with respect to p at p=0 is given by

$$ \frac{\mathrm{d}^2 I_2}{\mathrm{d}p^2} \bigg\vert_{p=0}=k^dV(B_d) \varDelta_0^4 \bigl(a\partial_k Z_k+k^2 Z_k\partial_k Z_k-2kZ_k^2 \bigr) . $$

(12.120)

Adding the two results (12.119) and (12.120) leads to the simple expression

$$ \frac{\mathrm{d}^2 F}{\mathrm{d}p^2} \bigg\vert_{p=0}=-2k^{d+1}V(B_d) \varDelta_0^4 Z^2_k $$

(12.121)

for the curvature of F(p) in (12.116) at the origin in momentum space.