We study perturbative Wilsonian renormalization group (RG) for the scalar ϕ 4 theory at finite temperature to one loop order in the Schwinger-Keldysh closed-time-path (CTP) formalism. By explicitly integrating out the UV modes, we show how effective interactions coupling the two branches of the CTP contour arise. This provides a microscopic derivation of the type of effective CTP actions used in the literature. While the full effective theory has no critical points, we instead proceed to study critical properties of the RG flow in a reduced space of complex couplings that characterize certain time-ordered correlators and show the existence of two novel fixed points in d = 4 spacetime dimensions. We relate one to the Wilson-Fisher fixed point known to exist only in 4 − ε dimensions, and the other to the standard Gaussian fixed point.
We study perturbative Wilsonian renormalization group (RG) for the scalar ϕ 4 theory at finite temperature to one loop order in the Schwinger-Keldysh closed-time-path (CTP) formalism. By explicitly integrating out the UV modes, we show how effective interactions coupling the two branches of the CTP contour arise. This provides a microscopic derivation of the type of effective CTP actions used in the literature. While the full effective theory has no critical points, we instead proceed to study critical properties of the RG flow in a reduced space of complex couplings that characterize certain time-ordered correlators and show the existence of two novel fixed points in d = 4 spacetime dimensions. We relate one to the Wilson-Fisher fixed point known to exist only in 4 − ε dimensions, and the other to the standard Gaussian fixed point.
Real-time dynamics of the O(4) scalar theory within the fRG approach
In this paper, the real-time dynamics of the O(4) O ( 4 ) scalar theory is studied within the functional renormalization group formulated on the Schwinger-Keldysh closed time path. The flow equations for the effective action and its n n -point correlation functions are derived in terms of the "classical'' and "quantum’’ fields, and a concise diagrammatic representation is presented. An analytic expression for the flow of the four-point vertex is obtained. Spectral functions with different values of temperature and momentum are obtained. Moreover, we calculate the dynamical critical exponent for the phase transition near the critical temperature in the O(4) O ( 4 ) scalar theory in 3+1 3 + 1 dimensions, and the value is found to be z\simeq 2.023 z ≃ 2.023 .
Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.
We consider a scalar field theory in Minkowski spacetime and define a coarse-grained closed time path (CTP) effective action by integrating quantum fluctuations of wavelengths shorter than a critical value. We derive an exact CTP renormalization group equation for the dependence of the effective action on the coarsegraining scale. We solve this equation using a derivative expansion approach. Explicit calculation is performed for the $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ theory. We discuss the relevance of the CTP average action in the study of nonequilibrium aspects of phase transitions in quantum field theory.
We use a perturbative momentum shell renormalization group (RG) approach to study the properties of a driven quantum system at zero temperature. To illustrate the technique, we consider a bosonic ϕ(4) theory with an arbitrary time dependent interaction parameter λ(t) = λ f(ω0t), where ω0 is the drive frequency and we derive the RG equations for the system using a Keldysh diagrammatic technique. We show that the scaling of ω0 is analogous to that of temperature for a system in thermal equilibrium and its presence provides a cutoff scale for the RG flow. We analyze the resultant RG equations, derive an analytical condition for such a drive to take the system out of the gaussian regime, and show that the onset of the non-gaussian regime occurs concomitantly with the appearance of non-perturbative mode coupling terms in the effective action of the system. We supplement the above-mentioned results by obtaining them from equations of motion of the bosons and discuss their significance for systems near critical points described by time-dependent Landau-Ginzburg theories.
Using the local potential approximation of the exact renormalization group (RG) equation, we show various domains of values of the parameters of the O(1) -symmetric scalar Hamiltonian.In three dimensions, in addition to the usual critical surface S c (attraction domain of the Wilson-Fisher fixed point), we explicitly show the existence of a first-order phase transition domain S f separated from S c by the tricritical surface S t (attraction domain of the Gaussian fixed point).S f and S c are two distinct domains of repulsion for the Gaussian fixed point, but S f is not the basin of attraction of a fixed point.S f is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the ϕ 4 -coupling.This renormalized trajectory also exists in four dimensions making the Gaussian fixed point ultra-violet stable (and the ϕ 4 4 renormalized field theory asymptotically free but with a wrong sign of the perfect action).We also show that a very retarded classical-to-Ising crossover may exist in three dimensions (in fact below four dimensions).This could be an explanation of the unexpected classical critical behaviour observed in some ionic systems.
We study the problem of disorder-free metals near a continuous quantum critical point. We depart from the standard paradigm [Hertz, Phys. Rev. B 14, 1165 (1976) and Millis, Phys. Rev. B 48, 7183 (1993)] and treat both fermions and bosons (i.e., order parameter fields) on equal footing. We construct a Wilsonian effective field theory that integrates out only high-energy boson and fermion modes. Below the upper critical dimension of the theory (d=3 spatial dimensions), we find new fixed points in which the bosons are described by the Wilson-Fisher fixed point and are coupled to a non-Fermi-liquid metal. We describe subtleties with the renormalization group flow of four-Fermi interactions, which can be surmounted in a controlled large-N limit. In this limit, we find that the theory has no superconducting instability.