Capacitance of a quantum dot from the channel-anisotropic two-channel Kondo model

We investigate the charge fluctuations of a large quantum dot coupled to a two-dimensional electron gas via a quantum point contact following the work of Matveev $[$K. A. Matveev, Phys. Rev. B 51, 1743 (1995); Z. Éksp. Teor. Fiz. 98, 1598 (1990) [Sov. Phys. JETP 72, 892 (1991)]$].$ We limit our discussion to the case where exactly two channels enter the dot and we discuss the role of an anisotropy between the transmission coefficients (for these two channels) at the constriction. Experimentally, a channel anisotropy can be introduced by applying a relatively weak in-plane magnetic field to the system when only one “orbital” channel is open. The magnetic field leads to different transmission amplitudes for spin-up and spin-down electrons. In a strong magnetic field the anisotropic two-channel limit corresponds to two (spin polarized) orbital channels entering the dot. The physics of the charge fluctuations can be captured using a mapping on the channel-anisotropic two-channel Kondo model. For the case of weak reflection at the point contact this has already been briefly stressed by one of us [K. Le Hur, Phys. Rev. B 64, 161302(R) (2001)]. This mapping is also appropriate to discuss the conductance behavior of a two-contact setup in strong magnetic field. Here, we elaborate on this approach and also discuss an alternative solution using a mapping on a channel-isotropic Kondo model. In addition we consider the limit of weak transmission. We show that the Coulomb-staircase behavior of the charge in the dot as a function of the gate voltage, is already smeared out by a small channel anisotropy both in the weak- and strong-transmission limits.