Field Theory of Non-equilibrium Systems Alex Kamenev Frontmatter More information xiv Preface experiments could easily increase the volume by a factor. Finally, the bibliography list is not meant to be exhaustive or complete. In most cases the references are given to the original works, where the presented results were obtained, and to their immediate extensions and the review articles. I sincerely apologize to many authors whose works I was not able to cover. Finally, this is an opportunity to express my deep appreciation to all of my coauthors and colleagues, from whom I learned a great deal about the subjects of this book.

The problem of electron-electron lifetime in a quantum dot is studied beyond perturbation theory by mapping onto the problem of localization in the Fock space. Localized and delocalized regimes are identified, corresponding to quasiparticle spectral peaks of zero and finite width, respectively. In the localized regime, quasiparticle states are single-particle-like. In the delocalized regime, each eigenstate is a superposition of states with very different quasiparticle content. The transition energy is e c Ӎ D͑g͞ ln g͒ 1͞2 , where D is mean level spacing, and g is the dimensionless conductance. Near e c there is a broad critical region not described by the golden rule. [S0031-9007(97)02895-0] PACS numbers: 72.15.Lh, 72.15.Rn, Quasiparticle in a Fermi liquid is not an eigenstate: it decays into two quasiparticles and a hole. In an infinite clean system, by using the golden rule (GR), quasiparticle decay rate is estimated as g͑e͒ ϳ e 2 ͞e F , where e is quasiparticle energy and e F is Fermi energy [1]. However, in a finite system the eigenstate spectrum is discrete. In this case, quasiparticles may be viewed as wave packets constructed of such states, the packet width being determined by the lifetime in an infinite system: de Ӎ g͑e͒. In this paper we attempt to clarify the relation between quasiparticles and many-particle states, and find that at different energies it has different meanings.Conventionally, quasiparticles are well defined provided g͑e͒ ø e. However, to resolve quasiparticles in a mesoscopic system, a more stringent condition is required: g͑e͒ , D, the quasiparticle level spacing. As an example, consider quasiparticle peaks in tunneling conductance of a quantum dot [2,3]. The peaks observed in nonlinear conductance at certain bias are interpreted as the quasiparticle tunneling density of states (DOS). Each peak corresponds to a "quasiparticle state," and its width measures the lifetime of the state. Below we consider an isolated Fermi liquid, ignoring any contributions to the quasiparticle decay due to finite escape rate, phonons, etc. [4].The meaning of quasiparticle lifetime needs clarification: strictly speaking, since a quantum dot is a finite system, any many-particle eigenstate gives rise to an infinitely narrow conductance peak. However, we will see that only a small fraction of those states overlap with one-particle excitations strongly enough to be detected by a finite sensitivity measurement. Under certain conditions, these strong peaks group into clusters of the width ϳg͑e͒ that can be interpreted as quasiparticle peaks.Before discussing possible regimes let us review the GR approach. Recently Sivan et al. [5], adopting the quasiparticle picture to a finite size geometry and relying on the earlier work [6] on electron-electron scattering rate in diffusive conductors, found that g͑e͒ ഠ D͑e͞gD͒ 2 , e , gD ,where D is the mean single-particle level spacing near the Fermi level and g ¿ 1 is the dimensionless conductance, for a finite system defined by g E c ͞D, where E c is the Thouless energy...

We develop an efficient method to calculate probabilities of large deviations from the typical behavior (rare events) in reaction-diffusion systems. The method is based on a semiclassical treatment of underlying "quantum" Hamiltonian, encoding the system's evolution. To this end we formulate corresponding canonical dynamical system and investigate its phase portrait. The method is presented for a number of pedagogical examples.

The purpose of this review is to provide a comprehensive pedagogical introduction into Keldysh technique for interacting out-of-equilibrium fermionic and bosonic systems. The emphasis is placed on a functional integral representation of underlying microscopic models. A large part of the review is devoted to derivation and applications of the non-linear σ-model for disordered metals and superconductors. We discuss such topics as transport properties, mesoscopic effects, counting statistics, interaction corrections, kinetic equation, etc. The chapter devoted to disordered superconductors includes Usadel equation, fluctuation corrections, time-dependent Ginzburg-Landau theory, proximity and Josephson effects, etc. (This review is a substantial extension of arXiv:cond-mat/0412296.)

We show that the proper inclusion of soft reparameterization modes in the Sachdev-Ye-Kitaev model of N randomly interacting Majorana fermions reduces its long-time behavior to that of Liouville quantum mechanics. As a result, all zero temperature correlation functions decay with the universal exponent ∝ τ −3/2 for times larger than the inverse single particle level spacing τ N ln N. In the particular case of the single particle Green function this behavior is manifestation of the zero-bias anomaly, or scaling in energy as 1/2 . We also present exact diagonalization study supporting our conclusions.

The diagrammatic linear response formalism for the Coulomb drag in two{layer systems is developed. This technique can be used to treat both elastic disorder and intralayer interaction e ects. In the absence of intralayer electron{electron correlations we reproduce earlier results, obtained using the kinetic equation and the memory{function formalism. In addition we calculate weak{localization corrections to the drag coe cient and the Hall drag coe cient in a perpendicular magnetic eld. As an example of the intralayer interaction e ects we consider a situation where one (or both) layers are close to (but above) the superconducting transition temperature. Fluctuation corrections, analogous to the Aslamazov{Larkin corrections, to the drag coecient are calculated. Although the uctuation corrections do not enhance the drag coe cient for normal{superconductor systems, a dramatic enhancement is found for superconductor{superconductor structures.Typeset using REVT E X 1

We develop a field theory formalism for the disordered interacting electron liquid in the dynamical Keldysh formulation. This formalism is an alternative to the previously used replica technique. In addition it naturally allows for the treatment of non-equilibrium effects. Employing the gauge invariance of the theory and carefully choosing the saddle point in the Q-matrix manifold, we separate purely phase effects of the fluctuating potential from the ones that change quasi-particle dynamics. As a result, the cancellation of super-divergent diagrams (double logarithms in d = 2) is automatically build in the formalism. As a byproduct we derive a non-perturbative expression for the single particle density of states. The remaining low-energy σ-model describes the quantum fluctuations of the electron distribution function. Its saddle point equation appears to be the quantum kinetic equation with an appropriate collision integral along with collisionless terms. Altshuler-Aronov corrections to conductivity are shown to arise from the one-loop quantum fluctuation effects.

We evaluate the dynamic structure factor S(q, ω) of interacting one-dimensional spinless fermions with a nonlinear dispersion relation. The combined effect of the nonlinear dispersion and of the interactions leads to new universal features of S(q, ω). The sharp peak S(q, ω) ∝ qδ(ω − uq), characteristic for the Tomonaga-Luttinger model, broadens up; S(q, ω) for a fixed q becomes finite at arbitrarily large ω. The main spectral weight, however, is confined to a narrow frequency interval of the width δω ∼ q 2 /m. At the boundaries of this interval the structure factor exhibits power-law singularities with exponents depending on the interaction strength and on the wave number q.PACS numbers: 71.10.Pm, 72.15.Nj Low-energy properties of fermionic systems are sensitive to interactions between fermions. The effect of interactions is the strongest in one dimension (1D), where single-particle correlation functions exhibit power-law singularities, in a striking departure from the behavior in higher dimensions. Much of our current understanding of 1D fermions is based on the Tomonaga-Luttinger (TL) model [1]. The crucial ingredient of the model is the assumption of a strictly linear fermionic dispersion relation. The TL model, often used in conjunction with a powerful bosonization technique [2], allows one to evaluate various correlation functions, such as momentum-resolved [3,4] and local [5] single-particle densities of states.Unlike the single-particle correlation functions, the two-particle correlation functions of the TL model exhibit behavior rather compatible with that expected for a Fermi liquid with the linear spectrum of quasiparticles. For example, the dynamic structure factor (the densitydensity correlation function)at small q takes the form [3] S TL (q, ω) ∝ qδ(ω − uq). It means that the quanta of density waves propagating with plasma velocity u are true eigenstates of the TL model; these bosonic excitations have an infinite lifetime. Below we show that such a simple behavior is an artefact of the linear spectrum approximation. In reality, the spectrum of fermions always has some nonlinearity,where the upper/lower sign corresponds to the right/left movers (R/L), and k = p ∓ p F are momenta measured from the Fermi points ± p F . (For Galilean-invariant systems the expansion (2) terminates at k 2 .) The finite curvature (1/m = 0) affects drastically the functional form of S(q, ω). In a clear deviation from the results of TL model, power-law singularities now arise not only in the single-particle correlation functions, but in the structure factor as well. We show that the singularities in these two very different objects have a common origin, proliferation of low-energy particle-hole pairs, and evaluate the corresponding exponents.Because of the success of the bosonization technique [2], it is tempting to treat the spectrum nonlinearity as a weak interaction between the TL bosons. Indeed, the nonlinearity gives rise to a three-boson interaction with the coupling constant ∝ 1/m [6]. However, attempts to treat this ...