From integrability to conductance, impurity systems
Introduction
Conductance (conductivity) measurements belong to the easiest and most direct experiments which can be carried out. They attract a lot of attention, due to the fact that in general they can be performed without perturbing very much the behaviour of the system, e.g., a rigid-lattice bulk metal, such that the uncertainty of experimental artifacts is reduced to a minimum. There exist various well-known theoretical descriptions, such as semi-classical transport theories (Landauer [1] and Boltzmann–Drude [2]), dynamical linear-response theory [3], [4] and also Green function linear-response theory [5]. To carry out the latter, in particular at finite temperature, is still poorly understood in generality [6], even in 1+1 space–time dimensions [7], [8]. For a review on Landauer–Büttinger theory and the Kubo formula see, e.g., [4], [9]. Since recent experimental progress allows conductance measurements also in 1+1 space–time dimensions [10], one can on the theoretical side fully exploit the special features of low dimensionality.
It is in particular very suggestive to exploit the full scope of non-perturbative techniques which have been developed in the context of integrable quantum field theories in 1+1 space–time dimensions, such as the thermodynamic Bethe ansatz (TBA) [11], [12] and the form factor bootstrap approach [13], [14]. Generalizing the Landauer transport picture a proposal for the conductance through a quantum wire with a defect (impurity) has been made in [15], [16] which we only modify to accommodate parity breaking, known to occur in integrable lattice models, see, e.g., [17]. This means in particular we allow the transmission amplitudes to be different for a particle of type i with charge qi passing with rapidity θ through a defect of type α from the left Tiα(θ) and right . The density distribution function ρir(θ,T,μi), being a function the temperature T, and the potential at the left μil and right μir constriction of the wire, can be determined by means of the TBA. We have already restricted (1.1) to the Abelian (diagonal) situation. It is clear that the effect resulting from the defect is most interesting when |Tiα(θ)|≠1, which requires the occurrence of simultaneous transmission and reflection (see (2.6), (2.17)). In this paper we will therefore be mainly interested in that situation. One may adapt (1.1) also to the case of pure reflection, which physically describes the influence of the constriction to the conducting process. From the previous statement it is clear that such boundary theories are only interesting in this physical context when they are non-Abelian.
The other prominent way of determining the conductance is a result from linear response theory, which yields an expression for the conductance in form of the Fourier transform of the current–current two-point correlation function. This Kubo formula has been adapted to the situation with a boundary [18]. As we mentioned, this will only capture effects coming from the constriction of the wire, we propose here a generalization to the analogous situation as described in (1.1), i.e., when a defect is present Here the defect operator Zα enters in-between the two currents J within the temperature and mass m dependent correlation function. The Matsubara frequency is denoted by ω.
The main purpose of this manuscript is to compare the two alternative descriptions (1.1) and (1.2) for massive bulk theories with a defect which allows for simultaneous reflection and transmission. There exist various investigations, e.g., [15], [16], [19], [20] for conformal (massless) theories with defect, which exploit the original folding idea of Wong and Affleck [19]. The idea is that a conformal field theory with a purely transmitting or reflecting defect can be mapped into a boundary theory, i.e., a theory living in half space, which has the advantage that the full restriction of modular invariance can be exploited in the construction of boundary states as pioneered by Cardy [21]. Since this folding idea relies on the vanishing of either the reflection or transmission, our considerations do in general not reduce to that set up, even in the conformal limit. As was already pointed out in [19], and as can be seen directly from (1.1) and (1.2), in that case the conductance is less interesting because it is either zero or perfect for Abelian theories.
In Section 2 we outline the procedure of how the defect scattering matrices may be determined, since they are needed as input in both approaches. We demonstrate that the Yang–Baxter system singles out the free Fermion as a very special model, which we treat thereafter extensively. In Section 3 we newly formulate the defect TBA equations and use them to determine the density distribution functions. We evaluate numerically the Landauer formula (1.1) for various defects and provide some analytical approximations in certain regimes. In Section 4 we propose a Kubo formula (1.2) for a configuration in which an impurity is present and compute the current–current two-point correlation functions occurring in there by means of a form factor expansion. We find very good agreement between (1.1) and (1.2) for the complex free Fermion theory with various types of defects. Our final conclusions and an outlook into open problems is provided in Section 5.
Section snippets
Determining the defect scattering matrices
An essential input required in both non-perturbative methods which are exploited to compute the conductance (1.1) and (1.2), that is the TBA and the form factor bootstrap approach, respectively, is the knowledge of the exact (defect) scattering matrix. It is one of the most intriguing facts of two-dimensional quantum field theories that these matrices can be determined exactly to all orders in perturbation theory. In the following section we will recall how much (little) of this approach can be
Conductance through an impurity
The most intuitive way to compute the conductance is via Landauer transport theory [1]. Let us consider a set up as depicted in Fig. 3, that is we place a defect in the middle of a rigid bulk wire, where the two halves might be at different temperatures.
The direct current I through such a quantum wire can be computed simply by determining the difference between the static charge distributions at the right and left constriction of the wire, i.e., I=Qr−Ql. This is based on the assumption [15],
Conductance from the Kubo formula
Having computed the DC conductance by means of a TBA analysis, we want to proceed now by introducing an alternative method for the acquisition of the same quantity, that is the evaluation of the celebrated Kubo formula3 [3] The key quantity needed for the explicit computation of (4.1) is the occurrence of the current–current correlation function 〈J(r)J(0)〉T,m. In
Conclusions
We have exploited the special features of (1+1)-dimensional integrable quantum field theories in order to compute the DC conductance in an impurity system. For this purpose several non-perturbative techniques have been used. As the main tools we employed the thermodynamic Bethe ansatz in a Landauer transport theory computation and the form factor expansion in the Kubo formula.
The comparison between the Landauer formula (1.1) and the Kubo formula (1.2) yields in particular an identical plateau
Acknowledgements
We are grateful to the Deutsche Forschungsgemeinschaft (Sfb288), for financial support. We thank F. Göhmann for discussions.
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