This tutorial review gives an elementary and self-contained derivation of the standard identities (ψη(x) ∼ Fηe −iφη (x) , etc.) for abelian bosonization in 1 dimension in a system of finite size L, following and simplifying Haldane's constructive approach. As a non-trivial application, we rigorously resolve (following Furusaki) a recent controversy regarding the tunneling density of states, ρ dos (ω), at the site of an impurity in a Tomonaga-Luttinger liquid: we use finite-size refermionization to show exactly that for g = 1 2 its asymptotic low-energy behavior is ρ dos (ω) ∼ ω. This agrees with the results of Fabrizio & Gogolin and of Furusaki, but not with those of Oreg and Finkel'stein (probably because we capture effects not included in their mean-field treatment of the Coulomb gas that they obtained by an exact mapping; their treatment of anti-commutation relations in this mapping is correct, however, contrary to recent suggestions in the literature).-The tutorial is addressed to readers with little or no prior knowledge of bosonization, who are interested in seeing "all the details" explicitly; it is written at the level of beginning graduate students, requiring only knowledge of second quantization, but not of field theory (which is not needed here). At the same time, we hope that experts too might find useful our explicit treatment of certain subtleties that can often be swept under the rug, but are crucial for some applications, such as the calculation of ρ dos (ω)-these include the proper treatment of the so-called Klein factors that act as fermion-number ladder operators (and also ensure the anti-commutation of different species of fermion fields), the retention of terms of order 1/L, and a novel, rigorous formulation of finite-size refermionization of both F e −iΦ(x) and the boson field Φ(x) itself. Changes relative to first version of cond-mat/9805275: We have substantially revised our discussion of the controversy regarding the tunneling density of states ρ dos at the site of an impurity in a Luttinger liquid, with regard to the following points: (1) In a new Appendix K, we confirm explicitly that Oreg and Finkel'stein's treatment of fermionic anti-commutation relations is correct, contrary to recent suggestions (including our own). (2) To try to understand why their result for ρ dos differs from that of Fabrizio & Gogolin, Furusaki and (for g=1/2) ourselves, we make a new suggestion in Sections 1.B and 10.D: this is probably because of effects not captured by their mean-field treatment of their Coulomb gas. (3) In Sections 10.C and 10.D we have replaced the first version of our calculation of ρ dos by a more explicit one (the result is unchanged), in which we refermionize not only the exponential e iΦ but, for the first time, also the field Φ itself (Section 10.C.4); this allows us to calculate various correlation functions involving Φ explicitly in terms of fermion operators (a new Appendix J contains several detailed examples, and a new Figure 4 showing the corresponding Feynman diagrams).

Coulomb blockade phenomena and quantum fluctuations are studied in mesoscopic metallic tunnel junctions with high charging energies. If the resistance of the barriers is large compared to the quantum resistance, transport can be described by sequential tunneling. Here we study the influence of quantum fluctuations. They are important when the resistance is small or the temperature very low. A real-time approach is developed which allows the diagrammatic classification of "inelastic resonant tunneling" processes where different electrons tunnel coherently back and forth between the leads and the metallic island. With the help of a nonperturbative resummation technique we evaluate the spectral density which describes the charge excitations of the system. From it physical quantities of interest like current and average charge can be deduced. Our main conclusions are: An energy renormalization leads to a logarithmic temperature dependence of the renormalized system parameters. A finite lifetime broadening can change the classical picture drastically. It gives rise to a strong flattening of the Coulomb oscillations for low resistances, but in the Coulomb blockade regime inelastic electron cotunneling persists. The temperature where these effects are important are accessible in experiments.

Abstract. We study a generic problem of dissipative quantum mechanics, a small local quantum system with discrete states coupled in an arbitrary way (i.e. not necessarily linear) to several infinitely large particle or heat reservoirs. For both bosonic or fermionic reservoirs we develop a quantum field-theoretical diagrammatic formulation in Liouville space by expanding systematically in the reservoirsystem coupling and integrating out the reservoir degrees of freedom. As a result we obtain a kinetic equation for the reduced density matrix of the quantum system. Based on this formalism, we present a formally exact perturbative renormalization group (RG) method from which the kernel of this kinetic equation can be calculated. It is demonstrated how the nonequilibrium stationary state (induced by several reservoirs kept at different chemical potentials or temperatures), arbitrary observables such as the transport current, and the time evolution into the stationary state can be calculated. Most importantly, we show how RG equations for the relaxation and dephasing rates can be derived and how they cut off generically the RG flow of the vertices. The method is based on a previously derived real-time RG technique [1,2,3,4] but formulated here in Laplace space and generalized to arbitrary reservoir-system couplings. Furthermore, for fermionic reservoirs with flat density of states, we make use of a recently introduced cutoff scheme on the imaginary frequency axis [5] which has several technical advantages. Besides the formal set-up of the RG equations for generic problems of dissipative quantum mechanics, we demonstrate the method by applying it to the nonequilibrium isotropic Kondo model. We present a systematic way to solve the RG equations analytically in the weak-coupling limit and provide an outlook of the applicability to the strong-coupling case.

We study resonant tunneling through a single-level quantum dot in the presence of strong Coulomb repulsion beyond the perturbative regime. The level is either spin-degenerate or can be split by a magnetic field. We, furthermore, discuss the influence of a bosonic environment. Using a real-time diagrammatic formulation we calculate transition rates, the spectral density and the nonlinear I − V characteristic. The spectral density shows a multiplet of Kondo peaks split by the transport voltage and the boson frequencies, and shifted by the magnetic field. This leads to zerobias anomalies in the differential conductance, which agree well with recent experimental results for the electron transport through single-charge traps. Furthermore, we predict that the sign of the zero-bias anomaly depends on the level position relative to the Fermi level of the leads.

We study resonant tunneling through a quantum dot with one degenerate level in the presence of a strong Coulomb repulsion and a bosonic environment. Using a real-time approach we calculate the spectral density and the nonlinear current within a conserving approximation. The spectral density shows a multiplet of Kondo peaks split by the transport voltage and boson frequencies. As a consequence we find a zero-bias anomaly in the differential conductance which can show a local maximum or minimum depending on the level position. The results are compared with recent experiments. 72.15.Qm, 73.20.Dx, 73.40.Gk, 73.50.Fq Transport phenomena through discrete energy levels in quantum dots have been studied by perturbation theory [1,2] and beyond [3][4][5]. In general, resonant tunneling phenomena and Kondo effects in nonequilibrium become important, which have been measured recently by Ralph & Buhrman [6]. In metallic islands, the Coulomb blockade is strongly influenced by inelastic interactions with bosonic degrees of freedom, such as fluctuations of the electrodynamic environment [7] or applied timedependent fields [8]. The study of inelastic interactions in quantum dots with few levels has started only recently, either for the nondegenerate case [9,10] or more general, in the presence of time-dependent fields and Coulomb blockade [2,5]. In earlier work we have studied the influence of bosonic fields in the nonequilibrium Anderson model in the perturbative regime [11] and found resonant side peaks in the Coulomb oscillations.The purpose of the present letter is to investigate the influence of external quantum-mechanical fields on transport phenomena through ultrasmall quantum dots at low temperatures and frequencies (compared to the intrinsic broadening of the resonant state in the dot). This requires a description of the Kondo effect, generalized to nonequilibrium situations and including coupling to bosonic fields. For the nonperturbative treatment of the tunneling we apply a real-time, nonequilibrium manybody approach developed recently [12,13] to a quantum dot with one level and spin degeneracy M . For M ≥ 2 and low lying dot level ǫ we obtain the usual Kondo peaks at the Fermi levels µ α of the reservoirs [4]. However, the emission of bosons causes additional Kondo singularities, for a one mode field at µ α + nω B (n = ±1, ±2, . . .).Furthermore, we will analyze the effect of the singularities in the spectral density on the differential conduc-tance as function of the bias voltage. For a low lying level we obtain the well-known zero bias maximum [4-6], whereas for a level close to the chemical potentials of the reservoirs we find a zero bias minimum. The coupling to bosons gives rise to satellite anomalies, which can be traced back to the corresponding satellite peaks in the spectral density. In a certain range of gate voltages, for M = 2 and in the absence of bosons, we find that the temperature and bias voltage dependence of the conductance coincides with recent measurements of zero-bias minima in point-contacts...