NTNU, April 2013 with collaborators: Salman A. Silotri (NCTU), Chung-Hou Chung (NCTU, NCTS) Sung Po Chao Helical edge states transport through a quantum dot (QSHI)

The insulating state (Taken from Charlie Kane’s lecture given at Princeton summer school, Aug )

Integer Quantum Hall effect and Edge state (Adapted from Charlie Kane’s seminar given at U Penn, Jan )

Possible Topological state without magnetic field? Earliest proposal by D. Haldane: Based on this model Kane & Mele proposed: (Adapted from Charlie Kane’s seminar given at U Penn, Jan )

Quantum Spin Hall effect in HgTe/CdTe quantum wells Another proposal in quantum wells system: (Taken from Charlie Kane’s lecture given at Princeton summer school, Aug )

Experiments on HgTe/CdTe quantum wells (Taken from Charlie Kane’s lecture given at Princeton summer school, Aug )

Edge nonequilibrium transport through Quantum Dot Model Hamiltonian

Edge nonequilibrium transport through Quantum Dot Dot Hamiltonian Treat the couplings between quantum dot and leads (edge states of QSHI) as the perturbative term: Consider U=0 case (large quantum dot) first in the following discussion. Temperature is assumed to be zero as we are interested in the quantum phase transitions in this system.

Approaches in tackling the problem 1. Edge state Hamiltonian is exactly solvable by bosonization. Dot state is solvable for U=0 case. 2. Perform time dependent Keldysh perturbation on the coupling term and compute steady state physical quantities (ex: current). For U=0 quantum dot the results are exact by summing all perturbation terms through self energy. For small but nonzero U approximation is made through equation of motion method to study charge fluctuations on the quantum dot.

Crash course on Bosonization The interaction term: Particle hole excitation nearby Fermi surface are bosonic: The kinetic term:  Conventional notation for bosonic fields:

Bosonization and Nonequilibrium perturbation Rewrite in terms of bosonic operators and perform time dependent gauge transform to move the chemical potentials to : Do Keldysh perturbation on Green’s function and with Get the current:

Conductivity result for T=0, U=0 case ed=0 case ed=-0.1D case Common feature: Resonance peak occurs when chemical potential of the lead aligns with dot level. Width and height of the peak decreases with the increasing interaction strength of the leads. Differences: Off-resonance requires smaller interaction strength to reach metal-insulator transition at zero voltage.

Conductivity result for T=0, U=0 ; in resonance case ed=0 case nearby critical interaction strength : For dot level in resonance with equilibrium Fermi level the scaling analysis gives:  ( ) Inset shows power law behavior:

Conductivity result for T=0, U=0 ; off resonance case For dot level away from the equilibrium Fermi level we can integrate out dot electron to get: The tunneling current is obtained as (Y.W. Lee et al, PRB 86, (2012))  critical Luttinger 1 Two particles process

Noise for T=0, U=0 case Noise To lowest nonzero order in perturbation: Here and. ( w =0, =-0.05 ) (V=0.2, =-0.05 )

Dephasing through Quantum Dot Imaginary part of the self energy of dot Green function:  -- Measurable by Aharonov-Bohm ring exp or by tunneling density of state. Singular around for

Finite but small U case: Eq. of motion approach For finite but small dot interaction U we first decouple the dot with the leads: Define retarded dot Green’s function From we get with. Take time derivative again on and combines with above result gives Coupling the dot with the leads gives approximately: with

Nonequilibrium dot density of state for T=0, small U case The nonequilibrium dot density of state, which can be probed by STM for and case: Peaks corresponds to resonance structure at two charge fluctuation levels and. Insets show singular/nonsingular behavior for different interaction parameter K at.. Same physics as Lifetime. 

Conductivity result for T=0, small U case case: Again the results are roughly consistent with scaling analysis: 

Very different results for different setup Connecting a quantum dot with edge state and a Fermi liquid lead: (C. Y. Seng and T. K. Ng, EPL (2011)) Critical Value K~1

Summary and Outlook Future direction along this line: Finite temperature results Large U case: Two channel Kondo in large U (ex: KT Law et al., PRB 81,041305; C.H. Chung et al. PRB 82,115325) Summary: Quantum criticality tuned by interactions in the leads and dot level shown in transport through quantum dot. Singular behavior around for noise spectrum and for lifetime (depahsing) in metallic phase. Same singular behavior also observed in nonequilibrium dot density of state. Acknowledgement: Yu-Li Lee(NCUE) and Yu-Wen Lee (Tunghai U )