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arXiv:hep-th/0611222

Renormalization group study of damping in nonequilibrium field theory

Juan Zanella, Esteban Calzetta

Published Nov 21, 2006• 8 citations
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Abstract

In this paper we shall study whether dissipation in a $λϕ^{4}$ may be described, in the long wavelength, low frequency limit, with a simple Ohmic term $κ\dotϕ$, as it is usually done, for example, in studies of defect formation in nonequilibrium phase transitions. We shall obtain an effective theory for the long wavelength modes through the coarse graining of shorter wavelengths. We shall implement this coarse graining by iterating a Wilsonian renormalization group transformation, where infinitesimal momentum shells are coarse-grained one at a time, on the influence action describing the dissipative dynamics of the long wavelength modes. To the best of our knowledge, this is the first application of the nonequilibrium renormalization group to the calculation of a damping coefficient in quantum field theory.

DOI
10.48550/arxiv.hep-th/0611222

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In this paper we shall study whether dissipation in a λϕ4λϕ^{4}λϕ4 may be described, in the long wavelength, low frequency limit, with a simple Ohmic term κϕ˙κ\dotϕκϕ˙​, as it is usually done, for example, in studies of defect formation in nonequilibrium phase transitions. We shall obtain an effective theory for the long wavelength modes through the coarse graining of shorter wavelengths. We shall implement this coarse graining by iterating a Wilsonian renormalization group transformation, where infinitesimal momentum shells are coarse-grained one at a time, on the influence action describing the dissipative dynamics of the long wavelength modes. To the best of our knowledge, this is the first application of the nonequilibrium renormalization group to the calculation of a damping coefficient in quantum field theory.
Inspired by holographic Wilsonian renormalization, we consider coarse graining a quantum system divided between short-distance and long-distance degrees of freedom (d.o.f.), coupled via the Hamiltonian. Observations using purely long-distance observables are described by the reduced density matrix that arises from tracing out the short-distance d.o.f. The dynamics of this density matrix is non-Hamiltonian and nonlocal in time, on the order of some short time scale. We describe this dynamics in a model system with a simple hierarchy of energy gaps $\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{UV}}>\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{IR}}$, in which the coupling between high- and low-energy d.o.f. is treated to second order in perturbation theory. We then describe the equations of motion under suitable time averaging, reflecting the limited time resolution of actual experiments, and find an expansion of the master equation in powers of $\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{IR}}/\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{UV}}$, after the fashion of effective field theory. The failure of the system to be Hamiltonian or even Markovian appears at higher orders in this ratio. We compute the evolution of the density matrix in three specific examples: coupled spins, linearly coupled simple harmonic oscillators, and an interacting scalar quantum field theory. Finally, we argue that the logarithm of the Feynman-Vernon influence functional is the correct analog of the Wilsonian effective action for this problem.
We apply Renormalization Group techniques to the Real Time formulation of thermal field theory. Due to the separation between the T = 0 and the T = 0 parts of the propagator in this formalism, one can derive exact evolution equations for the Green functions describing the effect of integrating out thermal fluctuations of increasing wavelengths, the initial conditions being the renormalized Green functions of the T = 0 theory. As a first application, we study the phase transition for the real scalar theory, computing the order of the transition, the critical temperature, and critical exponents, in different approximations to the evolution equations for the scalar potential. CERN-TH/96-23 The dynamics of a second order, or weakly first order, phase transition is governed by long wavelength fluctuations of the order parameter. These fluctuations, whose length scale is much larger than the inverse temperature of the system, are essentially classical, since the
The perturbative approach to the description of long-wavelength excitations at high temperature breaks down near the critical point of a second order phase transition. We study the dynamics of these excitations in a relativistic scalar field theory at and near the critical point via a renormalization group approach at high temperature and an $\ensuremath{\epsilon}$ expansion in $d=5\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions. The long-wavelength physics is determined by a nontrivial fixed point of the renormalization group. At the critical point we find that the dispersion relation and width of quasiparticles of momentum p are ${\ensuremath{\omega}}_{p}\ensuremath{\sim}{p}^{z}$ and ${\ensuremath{\Gamma}}_{p}\ensuremath{\sim}(z\ensuremath{-}1){\ensuremath{\omega}}_{p},$ respectively, and the group velocity of quasiparticles ${v}_{g}\ensuremath{\sim}{p}^{z\ensuremath{-}1}$ vanishes in the long-wavelength limit at the critical point. Away from the critical point for $T\ensuremath{\gtrsim}{T}_{c}$ we find ${\ensuremath{\omega}}_{p}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{-}z}[1+(p\ensuremath{\xi}{)}^{2z}{]}^{1/2}$ and ${\ensuremath{\Gamma}}_{p}\ensuremath{\sim}(z\ensuremath{-}1){\ensuremath{\omega}}_{p}(p\ensuremath{\xi}{)}^{2z}/[1+(p\ensuremath{\xi}{)}^{2z}]$ with $\ensuremath{\xi}$ the finite temperature correlation length $\ensuremath{\xi}\ensuremath{\propto}|T\ensuremath{-}{T}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}.$ The new dynamical exponent z results from anisotropic renormalization in the spatial and time directions. For a theory with O(N)O(N)O(N) symmetry we find $z=1+\ensuremath{\epsilon}{(N+2)/(N+8)}^{2}+\mathcal{O}({\ensuremath{\epsilon}}^{2}).$ This dynamical critical exponent describes a new universality class for dynamical critical phenomena in quantum field theory. Critical slowing down, i.e., a vanishing width in the long-wavelength limit, and the validity of the quasiparticle picture emerge naturally from this analysis.
Since its introduction, the Renormalization Group has been the theoretical tool of choice for understanding and describing collective behavior characterized by an emergent scale invariance as in the vicinity of a continuous phase transition. It provides a powerful conceptual framework but, exact results being scarce, the search for generic and efficient approximation schemes has been very active from the very beginning. One recent line of research starts from an exact formulation of the Renormalization Group, in the form of a functional Renormalization Group (FRG) for scale-dependent generating functionals of correlation functions, and introduces potentially {it nonperturbative} approximations through ansatzes for the scale-dependent generating functional under study. The question that we address is to what extent such generic approximation schemes are able to describe specific problems in which the long-distance behavior involves strongly nonuniform configurations with, {it e.g.}, localized excitations.An example of such an approximation scheme within the FRG is the so-called derivative expansion of the effective average action (coarse-grained Gibbs free energy in the language of magnetic systems) which amounts to truncating the functional form of the latter in powers of the external momenta or equivalently in gradients of the fields. The derivative-expansion approximation focuses on the long wavelength properties and, in terms of coarse-grained configurations of the system, works as an expansion about uniform configurations. One may therefore wonder if such a scheme is able to capture the physics associated with nonuniform configurations containing, {it e.g.}, domain walls, spin waves, or localized defects such as the kinks and anti-kinks found in the instanton analysis of the 1-dimensional Ising model.In this work we assess the ability of the derivative expansion of the FRG to quantitatively describe the long-distance physics of a model across the whole range of space dimensions {it without a priori knowledge} of the relevant real-space coarse-grained configurations. To do so we first investigate how low orders of the derivative expansion describe the approach to the lower critical dimension d→dlc of the Ising-like scalar φ4 theory which is expected to be controlled by the proliferation of localized excitations (instantons). We show that the convergence of the fixed-point effective potential describing the critical point of the model is nonuniform in the field when the dimension d approaches dlcdlc, with the emergence of a boundary layer around the minimum of the potential. At the lowest nontrivial approximation level (known as LPA’) this allows us to make analytical predictions for the value of the lower critical dimension dlcdlc and for the behavior of the critical temperature as d→dlc, which are both found in fair agreement with the known exact results. We also consider for the same Ising-like scalar φ4 model the ordered phase in the limit where d→dlc. We focus on the way the effective potential in the FRG becomes convex as the proper spatial fluctuations, which now involve nonuniform configurations with a domain wall separating two phases, are included in the calculation.
We apply the recently developed real-time renormalization-group (RTRG) method to different quantum dissipative systems. In contrast to other renormalization-group techniques the RTRG approach is formulated on the Keldysh contour and therefore allows for the description of nonequilibrium situations. The method is based on an exact kinetic equation which determines the time-dependent reduced density matrix. The kinetic equation involves an integral kernel which describes the phenomenon of dissipation. We calculate this quantity using the RTRG method: After applying Wick's theorem the arising diagrams are accounted for successively with respect to their time scale. During that process the theory is left invariant by renormalizing the operators describing the theory. As a starting point we discuss the polaron problem. We calculate the renormalized ground-state energy and the effective mass. We compare our results of the RTRG with those obtained from the flow equation formalism, which is based on infinitesimal transformations. In the central part of this thesis we consider the spin-boson model which is applicable to various physical and chemical systems. It models a particle in a double-well potential which is coupled to a bosonic environment. We determine static properties such as the static susceptibility. We also present results for the time evolution of the reduced density matrix starting from an arbitrary nonequilibrium state. The asymptotic behaviour of the reduced density matrix can be characterized by an oscillation frequency and two decay constants, all of which are determined. Finally, we calculate equilibrium correlation functions. We find consistency of our results with quantum Monte Carlo simulations and with the Shiba-relation. Additionally, our results show the correct scaling behaviour. Comparisons are also drawn to results of the noninteracting blip approximation and to results obtained for the anisotropic Kondo model. Our approach is very accurate for not too large couplings to the environment. Furthermore, the stationary tunnel current through a double quantum dot coupled to a phonon bath is calculated and compared to experimental data. Eventually, the running couplings of the two-lead Kondo model are discussed.
We study the effects of dissipation on a disordered quantum phase transition with O(N) order-parameter symmetry by applying a strong-disorder renormalization group to the Landau-Ginzburg-Wilson field theory of the problem. We find that Ohmic dissipation results in a nonperturbative infinite-randomness critical point with unconventional activated dynamical scaling while super-Ohmic damping leads to conventional behavior. We discuss applications to the superconductor-metal transition in nanowires and to the Hertz theory of the itinerant antiferromagnetic transition.