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arXiv:0904.2262 · Physical Review B

Quantum dot in a two-dimensional topological insulator: The two-channel Kondo fixed point

K. T. Law, Chien-Yeah Seng, Patrick A. Lee, Tai Kai Ng

Published Jan 14, 2010• 41 citations
arXivHTMLPDF

Abstract

A quantum dot coupled to two helical edge states of a two-dimensional topological insulator through electron tunnelings is studied. We show that if the electron interactions on the edge states are repulsive, with Luttinger liquid parameter $K<1$, the system reaches a stable two-channel Kondo fixed point at low temperatures. This is in contrast to the Luttinger liquid leads case in which $K<1/2$ is needed. This two-channel fixed point is described by a boundary sine-Gordon Hamiltonian with a $K$ dependent boundary term. The impurity entropy, the impurity specific heat and the conductance are calculated.

DOI
10.1103/physrevb.81.041305

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A quantum dot coupled to two helical edge states of a two-dimensional topological insulator through electron tunnelings is studied. We show that if the electron interactions on the edge states are repulsive, with Luttinger liquid parameter K<1K<1K<1, the system reaches a stable two-channel Kondo fixed point at low temperatures. This is in contrast to the Luttinger liquid leads case in which K<1/2K<1/2K<1/2 is needed. This two-channel fixed point is described by a boundary sine-Gordon Hamiltonian with a KKK dependent boundary term. The impurity entropy, the impurity specific heat and the conductance are calculated.
We study a quantum dot coupled to two edge states of a quantum spin Hall insulator through electron tunnelings in the presence of a Rashba spin-orbital interaction induced by an external electric field. We show that if the electron interactions on the edge states are repulsive, there are two possible phases, depending on the Luttinger liquid parameter KKK. For 1/2<K<11/2<K<11/2<K<1, the low-temperature physics is controlled by a previously identified two-channel Kondo fixed point. For the edge states with even stronger repulsive interactions, i.e., 1/4<K<1/21/4<K<1/21/4<K<1/2, the system reaches another phase at low temperatures, described by a new two-channel Kondo fixed point. This phase is separated from the original one by a continuous phase transition upon varying the value of KKK through the external electric field. The corresponding critical point is described by a free Dirac fermion backscattered by a local potential. We investigate the low-temperature properties associated with this new fixed point and also discuss the scaling behaviors of the system at the critical point.
We investigate theoretically the quantum phase transition (QPT) between the one-channel Kondo (1CK) and two-channel Kondo (2CK) fixed points in a quantum dot coupled to helical edge states of interacting 2D topological insulators (2DTI) with Luttinger parameter 0 < K < 1 . The model was studied by Law et al (2010 Phys. Rev. B 81 041305(R) ), and was mapped onto an anisotropic two-channel Kondo model via bosonization. For K < 1 , the strong coupling 2CK fixed point was argued to be stable for infinitesimally weak tunnelings between the dot and the 2DTI based on a simple scaling dimensional analysis (Law et al 2010 Phys. Rev. B 81 041305(R) . We re-examine this model beyond the bare scaling dimension analysis via a one-loop renormalization group (RG) approach combined with bosonization and re-fermionization techniques near weak-coupling and strong-coupling (2CK) fixed points. We find for a fixed value of K < 1 that the 2CK fixed point can be unstable towards the 1CK fixed point and the system is expected to undergo a quantum phase transition between 1CK and 2CK fixed points with changing Kondo couplings. Our RG approach is controlled near K = 1. In general, this QPT can also occur upon tuning the Luttinger parameter K to a critical value K c smaller than unity ( 0 < K c < 1 ) for fixed Konodo couplings. The QPT in our model comes as a result of the combined Kondo and the helical Luttinger physics in 2DTI, and it serves as the first example of the 1CK-2CK QPT that is accessible by the controlled RG approach. We extract quantum critical and crossover behaviors from various thermodynamical quantities near the transition. Our results are robust against particle-hole asymmetry for 1 2 < K < 1 .
We revisit the problem of a single quantum impurity on the edge of a two-dimensional time-reversal invariant topological insulator and show that the zero-temperature phase diagram contains a large local moment region for antiferromagnetic Kondo coupling, which was missed by previous poor man's scaling treatments. The combination of an exact solution at the so-called decoupling point and a renormalization group analysis \`a la Anderson-Yuval-Hamann allows us to access the regime of strong electron-electron interactions on the edge and strong Kondo coupling. We apply similar methods to the problem of a regular one-dimensional array of quantum impurities interacting with the edge liquid. When the edge electrons are at half-filling with respect to the impurity lattice, the system remains gapless unless the Luttinger parameter of the edge is less than 1/21/21/2 in which case two-particle backscattering effects drive the system to a gapped phase with long-range Ising antiferromagnetic order. This is in marked contrast with the gapped disordered ground state of the ordinary half-filled one-dimensional Kondo lattice.
We study a quantum impurity coupled to the edge states of a two-dimensional helical topological superconductor, that is, to a pair of counterpropagating Majorana fermion edge channels with opposite spin polarizations. For an impurity described by the Anderson impurity model, we show that the problem maps onto a variant of the interacting resonant two-level model which, in turn, maps onto the ferromagnetic Kondo model. Both magnetic and nonmagnetic impurities are considered. For magnetic impurities, an analysis relying on bosonization and the numerical renormalization group shows that the system flows to a fixed point characterized by a residual ln2\mathrm{ln}2ln2 entropy and anisotropic static and dynamical impurity magnetic susceptibilities. For nonmagnetic impurities, the system flows instead to a fixed point with no residual entropy and we find diamagnetic impurity response at low temperatures. We comment on the Schrieffer-Wolff transformation for problems with nonstandard conduction band continua and on the differences which arise when we describe the impurities by either Anderson or Kondo impurity models.
We study the physics of electron tunneling between multiple quantum dots and the edge of a quantum Hall state. Our results generalize earlier work [Fiete, Bishara, and Nayak, Phys. Rev. Lett. 101, 176801 (2008)] in which it was shown that a single quantum dot tunnel coupled to a non-Abelian quantum Hall state can realize a stable multichannel Kondo fixed point at low energy. In this work, we investigate the physics of multiple dots and find that a rich set of possible low-energy fixed points arises, including those with non-Fermi-liquid properties. Previously unidentified fixed points may also be among the possibilities. We examine both the situation where the dots are spatially separated and where they are in close proximity. We discuss the relation to previous work on two-impurity Kondo models in Fermi liquids and highlight new research directions in multiple quantum impurity problems.
We investigate the effects of the electron-electron interactions on the quantum transport through a charge Kondo circuit. The setup consists of a quantum dot sandwiched between two leads by two nearly transparent single mode quantum point contacts. The size of the interacting area LLL in the Luttinger liquid formed in the vicinities of the narrow constrictions is assumed to be much smaller compared to the size of the quantum dot aaa. We predict that the interplay between the electron-electron interactions in the Luttinger liquid and the fingerprints of the non-Fermi liquid behavior in the vicinity of the two channel Kondo intermediate coupling fixed point allows one to determine the interaction strength through the power-law temperature scaling of the electric conductance.