For equilibrium/ground state systems, a (Wilson) renormalization group transformation produces a series of systems (flow of Hamiltonians/couplings $H_{\Lambda}$ where $\Lambda$ is the cut-off) such that long-wave/asymptotic behaviour of $H_{\Lambda}$ is the same as of $H_{\Lambda'}$ after rescaling by $\Lambda/\Lambda'$. The idea of this definition implies an exact starting point for RG formalisms, with technical details varying between the fields and approximation methods. (For examples, see arXiv:1012.5604 and Wikipedia article).

Now, for non-equilibrium condensed-matter systems there is research direction aiming at generalization of the RG approach to a steady state, e.g., a voltage-biased strongly interacting quantum dot (or Kondo impurity). For examples, see arXiv:0902.1446 and related references.

I would like to understand the conceptual foundations for the non-equlibrium RG.

What is the definition of an RG transformation in a non-equilibirum, steady state ensemble?

I see a problem in the fact that the non-equilibirum density matrix which is used to define the problem is not a function of the Hamiltonian alone, thus it is not clear to me how is the effect of the change in the cut-off is split between the Hamiltonian (running couplings) and the density matrix (renormalization of the boundary/external conditions?)

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    $\begingroup$ Something I know about! I'm on a phone right now, but I'll just leave a reference and expand on an answer later: m.iopscience.iop.org/1751-8121/40/9/002 $\endgroup$
    – genneth
    Commented Nov 4, 2011 at 0:42
  • $\begingroup$ Looks refreshingly interesting, haven't seen this applied to non-equilibrium quantum transport problems. $\endgroup$
    – Slaviks
    Commented Nov 4, 2011 at 11:10
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    $\begingroup$ @genneth: Please, answer before the bounty falls away... $\endgroup$
    – András Bátkai
    Commented Nov 5, 2011 at 8:32

3 Answers 3


This is less ambitious than your question (general non-equilibrium states): Near equilibrium correlation functions are described by hydrodynamic theories with stochastic forces, for example the famous models A-J of Hohenberg and Halperin (Reviews of Modern Physics 49, 435 (1977)). In these models I can use the standard RG technology of integrating out short range modes and get running coupling constants. This is known as the "dynamic RG" or sometimes "mode-coupling" theory. The most important result is the critical scaling of transport coefficients (thermal conductivity, sound attenuation, etc.) near second order phase transitions. There have also been attempts to write down RG equations for the CTP (a.k.a. Schwinger-Keldysh) effective action, see for example Dalvit, Mazzitelli, "Exact CTP renormalization group equation for the coarse grained effective action", Phys. Rev. D54, 6338 (1996), arXiv:hep-th/9605024.


In this with the corresponding Arxiv version, (Berges and Mesterhazy, 2012) present an introduction to the nonequilibrium functional renormalization group for quantum systems specified by a given density matrix at initial time . They derive a generating functional to obtain renormalization equations for real time correlation functions and show that nonequilibrium dynamics such as the evolution of a system from nonequilibrium to thermal equilibrium can be described by a hierarchy of fixed points.


Projection techniques for nonequilibrium renormalization group equations are discussed in http://arxiv.org/abs/cond-mat/9612129

See also http://arxiv.org/abs/nucl-th/9505009


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