Electronic Transport and Quantum Phase Transitions of Quantum Dots in Kondo Regime Chung-Hou Chung 1. Institut für Theorie der Kondensierten Materie Universität Karlsruhe, Karlsruhe, Germany 2. Electrophysics Dept. National Chiao-Tung University, HsinChu, Taiwan, R.O.C. Collaborators: Walter Hofstetter (Frankfurt), Gergely Zarand (Budapest), Peter Woelfle (TKM, Karlsruhe) Acknowledgement: Michael Sindel, Matthias Vojta

Introduction Electronic transport and quantum phase transitions in coupled quantum dots: Model (I): parallel coupled quantum dots, 2-channel Kondo, non-trivial quantum critical point Model (II): side-coupled quantum dots, 1-channel Kondo, Kosterlitz-Thouless quantum transition Conclusions and Outlook Outline

Kondo effect in quantum dot even odd Coulomb blockade Single quantum dot conductance anomalies Goldhaber-Gorden et al. nature (1998) Glazman et al. Physics world 2001 L.Kouwenhoven et al. science 289, 2105 (2000)  d +U dd Kondo effect VgVg V SD

Kondo effect in metals with magnetic impurities At low T, spin-flip scattering off impurities enhances Ground state is spin-singlet Resistance increases as T is lowered electron-impurity scattering via spin exchange coupling logT (Kondo, 1964) (Glazman et al. Physics world 2001)

Kondo effect in quantum dot (J. von Delft)

Kondo effect in quantum dot

Anderson Model local energy level : charging energy : level width : All tunable! Γ= 2πV 2 ρ d U  d ∝ V g New energy scale: T k ≈ Dexp  -  U   ) For T < T k : Impurity spin is screened (Kondo screening) Spin-singlet ground state Local density of states developes Kondo resonance

Spectral density at T=0 Kondo Resonance of a single quantum dot phase shift Fredel sum rule particle-hole symmetry Universal scaling of T/T k L. Kouwenhoven et al. science 2000M. Sindel P-H symmetry  /2

Double quantum dots / Multi-level quantum dot: Singlet-triplet Kondo effect and Quantum phase transitions Interesting topics/questions Non-equilibrium Kondo effect Kondo effect in carbon nanotubes V 12 VVt

Quantum phase transitions c T g g Non-analyticity in ground state properties as a function of some control parameter g True level crossing: Usually a first-order transition Avoided level crossing which becomes sharp in the infinite volume limit: Second-order transition Critical point is a novel state of matter Critical excitations control dynamics in the wide quantum-critical region at non-zero temperatures Quantum critical region exhibits universal power-law behaviors Sachdev, quantum phase transitions, Cambridge Univ. press, 1999

Recent experiments on coupled quantum dots Two quantum dots coupled through an open conducting region which mediates an antiferromagnetic spin-spin coupling For odd number of electrons on both dots, splitting of zero bias Kondo resonance is observed for strong spin exchange coupling. (I). C.M. Macrus et al. Science, 304, 565 (2004)

A quantum dot coupled to magnetic impurities in the leads Antiferromagnetic spin coupling between impurity and dot suppresses Kondo effect Kondo peak restored at finite temperatures and magnetic fields (II). Von der Zant et al. cond-mat/ , (PRL, 2005)

Model system (I): 2-channel parallel coupled quantum dots Model system (II): 1-channel side-coupled quantum dots Coupled quantum dots L1L1 L2L2 R2R2 R1R1 C.H. C and W. Hofstetter, cond-mat/ G. Zarand, C.H. C, P. Simon, M. Vojta, cond-mat/

Numerical Renormalization Group (NRG)  Non-perturbative numerical method by Wilson to treat quantum impurity problem  Anderson impurity model is mapped onto a linear chain of fermions  Logarithmic discretization of the conduction band  Iteratively diagonalize the chain and keep low energy levels K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975) W. Hofstetter, Advances in solid state physics 41, 27 (2001)

Transport properties Transmission coefficient: Current through the quantum dots: Linear conductance :

Model System (I) Two quantum dots (1 and 2) couple to two-channel leads Antiferrimagnetic exchange interaction J, Magnetic field B 2-channel Kondo physics, complete Kondo screening for B = J = 0 L1L1 L2L2 R1R1 R2R2 Izumida and Sakai PRL 87, (2001) Vavilov and Glazman PRL 94, (2005) Simon et al. cond-mat/ triplet states Hofstetter and Schoeller, PRL 88, (2002) singlet state

2-impurity Kondo problem even 1 (L1+R1) even 2 (L2+R2) For V1 = V2 and with p-h symmetry J c = 2.2 T k Non-fermi liquid Jc J T Spin-singletKondo 12 L2L2 L1L1 R1R1 R2R2 Affleck et al. PRB 52, 9528 (1995) Jones and Varma, PRL 58, 843 (1989) Jump of phase shift at Jc J J C,   Quantum phase transition as J is tuned Jones and Varma, PRB 40, 324 (1989) Sakai et al. J. Phys. Soc. Japan 61, 7, 2333 (1992); ibdb. 61, 7, 2348 (1992) Specific heat coefficient  - 2 J-J c

JCJC NRG Flow of the lowest energy Phase shift  0  JJcJc  J<J C J>J C Two stable fixed points (Kondo and spin-singlet phases ) One unstable fixed point (critical fixed point) Jc, controlling the quantum phase transition Jump of phase shift in both channels at Jc Kondo Spin-singlet Kondo Spin-singlet Crossover energy scale T* J-J c

J < J c, transport properties reach unitary limit: T(  = 0) 2, G(T = 0) 2G 0 where G 0 = 2e 2 /h. J > J c spins of two dots form singlet ground state, T(  = 0) 0, G(T = 0) 0; and Kondo peak splits up. Quantum phase transition between Kondo (small J) and spin singlet (large J) phase. Quantum phase transition of Model System (I)

NRG ResultExperiment by von der Zant et al. Restoring of Kondo resonance Singlet-triplet crossover at finite temperatures T At T= 0, Kondo peak splits up due to large J. Low energy spectral density increases as temperature increases Kondo resonance reappears when T is of order of J Kondo peak decreases again when T is increased further. T=0.003 T=0.004

Singlet-triplet crossover at finite magnetic fields At T = B = 0, Kondo peak splits up due to large J. T = 0 singlet-triplet crossover at finite magnetic fields. Splitting of Kondo peaks gets smaller as B increases. B J, Kondo resonance restored, T(  = 0) 1 reaches unitary limit of a single-channel S = ½ Kondo effect. B > J, Kondo peak splits again. B J, T(  ) shows 4 peaks in pairs around  = (B J). Effective S=1/2 Kondo effect T k = Jc= Glazman et al. PRB 64, (2001) Hofstetter and Zarand PRB 69, (2002)

Singlet-triplet crossover at finite field and temperature J=0.007, Jc=0.005, Tk=0.0025, T= ,in step of 400 B J close to Jc, smooth crossover Antiferromagnetic J>0 Ferromagnetic J<0 J >> Jc, sharper crossover B in Step of J=-0.005, Tk= EXP: P-h asymmetry NRG: P-h symmetry splitting of Kondo peak due to Zeemann splitting of up and down spins splitting is linearly proportional to B

Two coupled quantum dots, only dot 1 couples to single-channel leads Antiferrimagnetic exchange interaction J 1-channel Kondo physics, dot 2 is Kondo screened for any J > 0. Kosterlitz-Thouless transition, Jc = 0 Model System (II) Vojta, Bulla, and Hofstetter, PRB 65, , (2002) Cornaglia and Grempel, PRB 71, , (2005) 12 VJ even

Anderson's poor man scaling and T k H Anderson Reducing bandwidth by integrating out high energy modes Obtaining equivalent model with effective couplings Scaling equation  < Tk, J diverges, Kondo screening JJ J J  J Anderson 1964

2 stage Kondo effect 1 st stage Kondo screening J k : Kondo coupling D TkTk dip in DOS of dot 1 2nd stage Kondo screening JkJk 4V 2 /U J: AF coupling btw dot 1 and 2 cc 1/ 

Kosterlitz-Thouless quantum transition NRG:Spectral density of Model (II) 8 0 J Kondospin-singlet No 3 rd unstable fixed point corresponding to the critical point Crossover energy scale T* exponentially depends on |J-Jc| U=1  d =-0.5  =0.1 T k =0.006  Log (T*) 1/J

Dip in DOS of dot 1: Perturbation theory self-energy vertex sum over leading logarithmic corrections  n < T k 1 2  when Dip in DOS of dot 1 d1d1 J = 0 J > 0 but weak

Dip in DOS: perturbation theory Excellence agreement between Perturbation theory (PT) and NRG for T* <<  << T k U=1,  d=-0.5,   J=0.0005, Tk=0.006, T*=8.2x PT breaks down for  T* Deviation at larger  > O(T k ) due to interaction U

More general model of 1-channel 2-stage Kondo effect Two-impurity, S=1, underscreened Kondo 1 2 I J k1 J k2 1 2 J k1 J ( J k2 = 0 ) Vojta, Bulla, and Hofstetter, PRB 65, , (2002) Ic ~ J k1 J k2 D I < Ic: T  imp = 1/4 residual spin-1/2 I > Ic: T  imp = 0 spin-singlet

Optical conductivity Linear AC conductivity Sindel, Hofstetter, von Delft, Kindermann, PRL 94, (2005) ‘ ‘ 1 Dot 2 J U=1  d =-0.5  =0.1 T k =0.006 

Comparison between two models 1 2 JkJk J even 1 (L1+R1) even 2 (L2+R2) L2L2 L1L1 R1R1 R2R2 2 impurity, S=1, Two-channel Kondo 2 impurity, S=1, One-channel Kondo 1 2 J J k1 J k2 complete Kondo screening underscreened Kondo quantum critical point K-T transition 8 J Kondospin-singlet   x Jc J Kondospin-singlet 8 T* J-J c Model (I)Model (II)

Conclusions Coupled quantum dots in Kondo regime exhibit quantum phase transition Model system (II): Our results have applications in spintronics and quantum information Quantum phase transition between Kondo and spin-singlet phases Singlet-triplet crossover at finite field and temperatures, qualitatively agree with experiments Kosterlitz-Thouless quantum transition, Provide analytical and numerical understanding of the transition L2L2 L1L1 R1R1 R2R2 2-channel Kondo physics 1-channel Kondo physics, two-stage Kondo effect Model system (I):

Outlook Non-equilibrium transport in various coupled quantum dots Quantum critical and crossover in transport properties near QCP Quantum phase transition out of equilibrium V c T g g Quantum phase transition in quantum dots with dissipation