Nonequilibrium Transport Through Kondo Quantum Dots: A Real-Time Renormalization Group Analysis

My PhD-talk about my work during the PhD at RWTH Aachen University and Utrecht University. The talk was at the 19.11.2014 at Utrecht University in preparation for the official defence at 10.12.2014.

subject:
- Kondo physics
- nonequilibrium method
- real-time renormalizaton group method
- over screened and fully screened Kondo model

Transcript

1. Nonequilibrium Transport Through Kondo Quantum Dots:
 A Real-Time Renormalization Group Analysis Christoph B. M. Hörig PhD Talk – 19th November 2014 Promotor: C. Morais Smith, Copromotor: D. Schuricht
2. Motivation • experiments on quantum dots, single electron transistors • transport properties and current & conductance measurable • concentrate on many-body interaction ➡ focus: Kondo physiscs GaAs Confining Electrodes Gate Source Drain AlGaAs AlGaAs GaAs 2DEG Electron Density
3. Motivation • experiments on quantum dots, single electron transistors • transport properties and current & conductance measurable • concentrate on many-body interaction ➡ focus: Kondo physiscs GaAs Confining Electrodes Gate Source Drain AlGaAs AlGaAs GaAs 2DEG Electron Density
4. Motivation • experiments on quantum dots, single electron transistors • transport properties and current & conductance measurable • concentrate on many-body interaction ➡ focus: Kondo physiscs GaAs Confining Electrodes Gate Source Drain AlGaAs AlGaAs GaAs 2DEG Electron Density source dot drain
5. Motivation • experiments on quantum dots, single electron transistors • transport properties and current & conductance measurable • concentrate on many-body interaction ➡ focus: Kondo physiscs GaAs Confining Electrodes Gate Source Drain AlGaAs AlGaAs GaAs 2DEG Electron Density current source dot drain
6. Motivation • experiments on quantum dots, single electron transistors • transport properties and current & conductance measurable • concentrate on many-body interaction ➡ focus: Kondo physiscs GaAs Confining Electrodes Gate Source Drain AlGaAs AlGaAs GaAs 2DEG Electron Density low-bias hI/hV map of the even region inside the dotted rectangle of Fig. 1c. This clearly displays two distinct regions, which (in anticipa- tion of our results) we associate with the singlet and triplet ground states. The possibility of gate-tuning the singlet–triplet splitting ET 2 ES was demonstrated previously both for lateral quantum dots18 and carbon nanotubes19, and may originate in an asymmetric coup- ling of the molecular levels to the electrodes20. The magnetic states cross sharply at a critical gate voltage Vc g <1:9 V. In the singlet region, a finite-bias conductance anomaly appears when Vb coincides with ET 2 ES ; this is due to a non-equilibrium Kondo effect involving excitations into the spin-degenerate triplet. This effect was recently studied in a carbon-nanotube quantum dot in the singlet state21 (see Supplementary Information). In the triplet region, two kinds of resonance are observed: a finite-bias hI/hV anomaly, which is interpreted as a singlet–triplet non-equilibrium Kondo effect that disperses like ES 2 ET in the Vg –Vb plane, and a sharp, zero-bias hI/hV peak, which is related to a partially screened spin-1 Kondo effect22, as indicated by the narrowness of the conduc- tance peak. To precisely identify these spin states, and justify our analysis in the framework of quantum criticality near the singlet–triplet crossing point, we present a detailed magneto-transport investigation of the even region. Owing to the high g-factor (g < 2) of C60 molecules, it is easy to lift the degeneracy of the triplet state for a C60 quantum dot using the Zeeman effect (see Supplementary Information). Figure 2b, d displays the evolution of the different conductance anomalies in the even region. Figure 2b shows hI/hV as a function of B and Vb for a constant gate voltage Vg chosen in the singlet region. A Zeeman-induced transition from the singlet state j0, 0æ to the lowest-m triplet state j1, 21æ occurs the clear level crossing in the conductance map. The splitting of triplet is also apparent, and the various spectroscopic lines are c sistent with the spin selection rules at both low and high magn field, where j0, 0æ and j1, 21æ are the respective ground states. In Fig. 2d we investigate the gate-induced singlet–triplet cross for constant magnetic field. In the singlet region, the Zeeman-s triplet states are clearly seen as three parallel lines, and the transit lines from the ground state j1, 21æ at higher gate voltage are agreement with the energy levels depicted in Fig. 2c, confirm the singlet to triplet crossing inside the even Coulomb diamo We note the absence of a large enhancement of the zero-bias c ductance at the singlet–triplet crossing in Figs 1d and 2b. Su features were, however, observed in previous experiments of vert semiconductor quantum dots4, where a field-induced orbital eff can be used to make the non-degenerate triplet coincide with singlet state, leading to a large Kondo enhancement of the cond tance that is intimately related to the existence of two screen channels23. In carbon nanotubes5, the Zeeman effect dominates o the orbital effect, so the transition involves the lowest-m triplet st only and Kondo signatures arise from a single channel, as in the c of well-balanced couplings of the two orbital states in the quant dot to the electrodes24. The lack of either type of singlet–triplet Kon effect in our data indicates that the predominant coupling is betw a single screening channel and one of the two spin states of the sing molecule quantum dot, leading to a Kosterlitz–Thouless quant phase transition at the singlet–triplet crossing, as predicted by theory2,3. Although the peculiar magnetic response associated w this transition is not directly accessible in our scheme, we demo strate that very specific characteristics of the Kosterlitz–Thou transition can be observed in transport. The basic factor in 100 nm 1 µm V b V g a b c d Odd Even Singlet Triplet E T – E S E S – E T Quantum critical point |A〉 lc l –30 –1.0 –0.5 |B〉 C 60 1 nm NATURE PHYSICS DOI: 10.1038/NPHYS1880 5.5 Vsd (mV) add (meV) 4N0 ∼ 180 +4 +8 ¬4 ¬8 S Gate B θ D 2 ¬2 0 8 a b c Fig. 3 f sys- sizes d be- Petta 1995; opar- dots wen- et al., ateral m dot reser- he de- and a FIG. 2. Lateral quantum dot device defined by metal surface electrodes. ͑a͒ Schematic view. Negative voltages applied to metal gate electrodes ͑dark gray͒ lead to depleted regions pins in few-electron quantum dots [R. Hanson (2007)] [N. Roch(2008)] [T. S. Jespersen (2011)] current source dot drain
7. Motivation • experiments on quantum dots, single electron transistors • transport properties and current & conductance measurable • concentrate on many-body interaction ➡ focus: Kondo physiscs GaAs Confining Electrodes Gate Source Drain AlGaAs AlGaAs GaAs 2DEG Electron Density low-bias hI/hV map of the even region inside the dotted rectangle of Fig. 1c. This clearly displays two distinct regions, which (in anticipa- tion of our results) we associate with the singlet and triplet ground states. The possibility of gate-tuning the singlet–triplet splitting ET 2 ES was demonstrated previously both for lateral quantum dots18 and carbon nanotubes19, and may originate in an asymmetric coup- ling of the molecular levels to the electrodes20. The magnetic states cross sharply at a critical gate voltage Vc g <1:9 V. In the singlet region, a finite-bias conductance anomaly appears when Vb coincides with ET 2 ES ; this is due to a non-equilibrium Kondo effect involving excitations into the spin-degenerate triplet. This effect was recently studied in a carbon-nanotube quantum dot in the singlet state21 (see Supplementary Information). In the triplet region, two kinds of resonance are observed: a finite-bias hI/hV anomaly, which is interpreted as a singlet–triplet non-equilibrium Kondo effect that disperses like ES 2 ET in the Vg –Vb plane, and a sharp, zero-bias hI/hV peak, which is related to a partially screened spin-1 Kondo effect22, as indicated by the narrowness of the conduc- tance peak. To precisely identify these spin states, and justify our analysis in the framework of quantum criticality near the singlet–triplet crossing point, we present a detailed magneto-transport investigation of the even region. Owing to the high g-factor (g < 2) of C60 molecules, it is easy to lift the degeneracy of the triplet state for a C60 quantum dot using the Zeeman effect (see Supplementary Information). Figure 2b, d displays the evolution of the different conductance anomalies in the even region. Figure 2b shows hI/hV as a function of B and Vb for a constant gate voltage Vg chosen in the singlet region. A Zeeman-induced transition from the singlet state j0, 0æ to the lowest-m triplet state j1, 21æ occurs the clear level crossing in the conductance map. The splitting of triplet is also apparent, and the various spectroscopic lines are c sistent with the spin selection rules at both low and high magn field, where j0, 0æ and j1, 21æ are the respective ground states. In Fig. 2d we investigate the gate-induced singlet–triplet cross for constant magnetic field. In the singlet region, the Zeeman-s triplet states are clearly seen as three parallel lines, and the transit lines from the ground state j1, 21æ at higher gate voltage are agreement with the energy levels depicted in Fig. 2c, confirm the singlet to triplet crossing inside the even Coulomb diamo We note the absence of a large enhancement of the zero-bias c ductance at the singlet–triplet crossing in Figs 1d and 2b. Su features were, however, observed in previous experiments of vert semiconductor quantum dots4, where a field-induced orbital eff can be used to make the non-degenerate triplet coincide with singlet state, leading to a large Kondo enhancement of the cond tance that is intimately related to the existence of two screen channels23. In carbon nanotubes5, the Zeeman effect dominates o the orbital effect, so the transition involves the lowest-m triplet st only and Kondo signatures arise from a single channel, as in the c of well-balanced couplings of the two orbital states in the quant dot to the electrodes24. The lack of either type of singlet–triplet Kon effect in our data indicates that the predominant coupling is betw a single screening channel and one of the two spin states of the sing molecule quantum dot, leading to a Kosterlitz–Thouless quant phase transition at the singlet–triplet crossing, as predicted by theory2,3. Although the peculiar magnetic response associated w this transition is not directly accessible in our scheme, we demo strate that very specific characteristics of the Kosterlitz–Thou transition can be observed in transport. The basic factor in 100 nm 1 µm V b V g a b c d Odd Even Singlet Triplet E T – E S E S – E T Quantum critical point |A〉 lc l –30 –1.0 –0.5 |B〉 C 60 1 nm NATURE PHYSICS DOI: 10.1038/NPHYS1880 5.5 Vsd (mV) add (meV) 4N0 ∼ 180 +4 +8 ¬4 ¬8 S Gate B θ D 2 ¬2 0 8 a b c Fig. 3 f sys- sizes d be- Petta 1995; opar- dots wen- et al., ateral m dot reser- he de- and a FIG. 2. Lateral quantum dot device defined by metal surface electrodes. ͑a͒ Schematic view. Negative voltages applied to metal gate electrodes ͑dark gray͒ lead to depleted regions pins in few-electron quantum dots [R. Hanson (2007)] [N. Roch(2008)] [T. S. Jespersen (2011)] current How can we approach many-body transport from theory? source dot drain
8. Outline 1) story of Kondo physics in a nutshell 2) real-time renormalization group for nonequilibrium • basic framework • some aspects of RG flow schemes 3) my nonequilibrium results on two Kondo setups: • overscreened Kondo model • screened Kondo model
9. Outline 1) story of Kondo physics in a nutshell 2) real-time renormalization group for nonequilibrium • basic framework • some aspects of RG flow schemes 3) my nonequilibrium results on two Kondo setups: • overscreened Kondo model • screened Kondo model 3 TECHNICAL SLIDES
10. Story of Kondo Physics measurement of resistance minimum Anderson impurity model Kondo model Kondo Problem Poor Man’s scaling solution full solution by numerical renormalization group quantum dots nonequilibrium - 1934: - 1961: - 1961: - 1964: - 1970: - 1975: - 1988: - present:
11. • de Haas, de Boer, and van den Berg measured gold wires in Leiden • expected monotonic decrease • dominated by electron-phonon interaction (Bloch, Grüneisen) 1934: Resistance Minimum THE ELECTRICAL RESISTANCE OF GOLD, COPPER AND LEAD 1 1 ] g It is of course very interesting to investigatethe influence of the purity of the metal on this minimum (the residual resistance of our material was rather high). M e i s s n e r 1) has investigated resist- ances of very pure gold (gold of M y 1 i u s). The accuracy of his measurements, however, is too small to ascertain a minimum. Nevertheless, the earliest series of measurements points to an in- crease of the resistance at decreasing temperatures. / 31.5 285 . ~ ' 275 . ~ OAUl ' ~ 6 7 8 9 11 ~oK Fig. 2. Resistance of Au between 4°K. and 12°K. §3. Copper. We have measured (table 3) three resistances Cu2, Cu3 and Cu4. Besides we have given in tables 4 and 5 the [Physica 1, 1115 (1934)] R(T) = R0 + cn Tn, n > 0
12. • de Haas, de Boer, and van den Berg measured gold wires in Leiden • expected monotonic decrease • dominated by electron-phonon interaction (Bloch, Grüneisen) • found minimum at 4 K 1934: Resistance Minimum THE ELECTRICAL RESISTANCE OF GOLD, COPPER AND LEAD 1 1 ] g It is of course very interesting to investigatethe influence of the purity of the metal on this minimum (the residual resistance of our material was rather high). M e i s s n e r 1) has investigated resist- ances of very pure gold (gold of M y 1 i u s). The accuracy of his measurements, however, is too small to ascertain a minimum. Nevertheless, the earliest series of measurements points to an in- crease of the resistance at decreasing temperatures. / 31.5 285 . ~ ' 275 . ~ OAUl ' ~ 6 7 8 9 11 ~oK Fig. 2. Resistance of Au between 4°K. and 12°K. §3. Copper. We have measured (table 3) three resistances Cu2, Cu3 and Cu4. Besides we have given in tables 4 and 5 the I I 16 W.j. DE HAAS, ]. DE BOER AND G. ]. VAN DEN BERG 26~ 26~ 26.'7 2S{~ 255 ,o"% 26 TABLE 1 (An1) TABLE 2 (An2) T°K 1104 R/Rooc T°K. 104 R / Ro°c H8 J, % 1.63 26.66 2.39 26.52 3.12 26.46 3.77 26.44 4.23 26.445 4.81 26.51 6.08 26.74 6.54 26.85 7.28 27.14 8.83 28,26 9.38 28.90 9.95 29.59 I0.00 29,67 I 1.84 33.20 I 1.95 33.48 12.10 33.86 15,17 45.64 16.05 50.76 17.03 57.30 18.08 65.73 20.44 90.48 He J~ fi S 1.75 2.86 3.49 3.76 4.22 4.24 6.28 6.70 8.48 9.41 9.76 10.12 10,24 10.68 11.11 11.15 12.66 14.25 16.02 18.15 20.45 26.85 26.61 26.58 26.57 26.57 26.58 27.02 27.15 28.20 29.14 29.62 30.05 30.22 30.96 31.73 31.83 35.42 41.40 50.67 66.42 90.48 / --~Z 2 Fig. 1. Resistance of Au between I°K. and 5°K. e Au, s A u I 3 4 5"K A minimum? What is going on here?! “The resistance curve of the gold wires measured (not very pure) has a minimum.” [Physica 1, 1115 (1934)] R(T) = R0 + cn Tn, n > 0
13. • formally derived from Anderson impurity model (d: ) via Schrieffer-Wolff transformation • projection onto fixed charge ( ) with spin S and effective spin-spin interactions J0 > 0 (antiferromagnetic) 1961: Kondo Model impurity electrons in host metal [P. W. Anderson, Phys. Rev. 124, 41 (1961)] H = X ↵k ✏k c† ↵k c↵k + h0 Sz + J0 2⌫0 X ↵↵0kk0 0 ~ S · ~ 0 c† ↵k c↵0k0 0 |0i , |#i , |"i , |2i |0i , |#i , |"i , |2i
14. • formally derived from Anderson impurity model (d: ) via Schrieffer-Wolff transformation • projection onto fixed charge ( ) with spin S and effective spin-spin interactions J0 > 0 (antiferromagnetic) 1961: Kondo Model impurity electrons in host metal [P. W. Anderson, Phys. Rev. 124, 41 (1961)] H = X ↵k ✏k c† ↵k c↵k + h0 Sz + J0 2⌫0 X ↵↵0kk0 0 ~ S · ~ 0 c† ↵k c↵0k0 0 |0i , |#i , |"i , |2i |0i , |#i , |"i , |2i
15. • formally derived from Anderson impurity model (d: ) via Schrieffer-Wolff transformation • projection onto fixed charge ( ) with spin S and effective spin-spin interactions J0 > 0 (antiferromagnetic) 1961: Kondo Model J0 impurity electrons in host metal [P. W. Anderson, Phys. Rev. 124, 41 (1961)] H = X ↵k ✏k c† ↵k c↵k + h0 Sz + J0 2⌫0 X ↵↵0kk0 0 ~ S · ~ 0 c† ↵k c↵0k0 0 |0i , |#i , |"i , |2i |0i , |#i , |"i , |2i
16. • formally derived from Anderson impurity model (d: ) via Schrieffer-Wolff transformation • projection onto fixed charge ( ) with spin S and effective spin-spin interactions J0 > 0 (antiferromagnetic) 1961: Kondo Model J0 impurity electrons in host metal [P. W. Anderson, Phys. Rev. 124, 41 (1961)] H = X ↵k ✏k c† ↵k c↵k + h0 Sz + J0 2⌫0 X ↵↵0kk0 0 ~ S · ~ 0 c† ↵k c↵0k0 0 |0i , |#i , |"i , |2i |0i , |#i , |"i , |2i
17. • formally derived from Anderson impurity model (d: ) via Schrieffer-Wolff transformation • projection onto fixed charge ( ) with spin S and effective spin-spin interactions J0 > 0 (antiferromagnetic) 1961: Kondo Model impurity electrons in host metal [P. W. Anderson, Phys. Rev. 124, 41 (1961)] H = X ↵k ✏k c† ↵k c↵k + h0 Sz + J0 2⌫0 X ↵↵0kk0 0 ~ S · ~ 0 c† ↵k c↵0k0 0 |0i , |#i , |"i , |2i |0i , |#i , |"i , |2i
18. R ( T ) = R0 + aT2 + bT5 cimp ln max {T, E µ} D 1964: Kondo Problem • only spin flip contributes; proportional to Fermi function • logarithm explains resistance minimum for the first time • divergency in resistance → that is the Kondo Problem [J. Kondo, Prog. Theor. Phys. 32, 37 (1964)]
19. R ( T ) = R0 + aT2 + bT5 cimp ln max {T, E µ} D 1964: Kondo Problem • only spin flip contributes; proportional to Fermi function • logarithm explains resistance minimum for the first time • divergency in resistance → that is the Kondo Problem [J. Kondo, Prog. Theor. Phys. 32, 37 (1964)]
20. R ( T ) = R0 + aT2 + bT5 cimp ln max {T, E µ} D 1964: Kondo Problem • only spin flip contributes; proportional to Fermi function • logarithm explains resistance minimum for the first time • divergency in resistance → that is the Kondo Problem [J. Kondo, Prog. Theor. Phys. 32, 37 (1964)] ➡ different energy scales equally important! ➡ considered by renormalization group (RG, NRG) [K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975)] strong coupling J0 ≫1 weak coupling J0 ≪1 T 0
21. R ( T ) = R0 + aT2 + bT5 cimp ln max {T, E µ} D 1964: Kondo Problem • only spin flip contributes; proportional to Fermi function • logarithm explains resistance minimum for the first time • divergency in resistance → that is the Kondo Problem [J. Kondo, Prog. Theor. Phys. 32, 37 (1964)] ➡ different energy scales equally important! ➡ considered by renormalization group (RG, NRG) [K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975)] strong coupling J0 ≫1 weak coupling J0 ≪1 T 0 Kondo temperature TK
22. R ( T ) = R0 + aT2 + bT5 cimp ln max {T, E µ} D 1964: Kondo Problem • only spin flip contributes; proportional to Fermi function • logarithm explains resistance minimum for the first time • divergency in resistance → that is the Kondo Problem [J. Kondo, Prog. Theor. Phys. 32, 37 (1964)] ➡ different energy scales equally important! ➡ considered by renormalization group (RG, NRG) [K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975)] strong coupling J0 ≫1 weak coupling J0 ≪1 T 0 Kondo singlet 1 p 2 |"S #el i |#S "el i Kondo temperature TK
23. 1988: Quantum Dots • proposal: Kondo physics also present in quantum dots [ L. I. Glazman and M. E. Raikh, JETP Letters 47, 452 (1988)] [T. K. Ng and P. A. Lee, Phys. Rev. Lett. 61, 1768 (1988)] 28 introduction to the kondo model temperature resistivity superconductor conductor im purity temperature conductance quantum dot (odd) quantum dot (even) figure 2 . 7 : Analogy between the increase of the resistivity and the conductance in R E P O R T S resistance impurities scatter: Kondo singlet 1 p 2 |"S #el i |#S "el i
24. 1988: Quantum Dots • proposal: Kondo physics also present in quantum dots [ L. I. Glazman and M. E. Raikh, JETP Letters 47, 452 (1988)] [T. K. Ng and P. A. Lee, Phys. Rev. Lett. 61, 1768 (1988)] 28 introduction to the kondo model temperature resistivity superconductor conductor im purity temperature conductance quantum dot (odd) quantum dot (even) figure 2 . 7 : Analogy between the increase of the resistivity and the conductance in R E P O R T S resistance impurities scatter: Kondo singlet 1 p 2 |"S #el i |#S "el i
25. 1988: Quantum Dots • proposal: Kondo physics also present in quantum dots impurities bridge: [ L. I. Glazman and M. E. Raikh, JETP Letters 47, 452 (1988)] [T. K. Ng and P. A. Lee, Phys. Rev. Lett. 61, 1768 (1988)] 28 introduction to the kondo model temperature resistivity superconductor conductor im purity temperature conductance quantum dot (odd) quantum dot (even) figure 2 . 7 : Analogy between the increase of the resistivity and the conductance in R E P O R T S resistance impurities scatter: 28 introduction to the kondo model temperature resistivity superconductor conductor im purity temperature conductance quantum dot (odd) quantum dot (even) figure 2 . 7 : Analogy between the increase of the resistivity and the conductance in R E P O R T S Kondo singlet 1 p 2 |"S #el i |#S "el i source dot drain
26. 1988: Quantum Dots • proposal: Kondo physics also present in quantum dots impurities bridge: [ L. I. Glazman and M. E. Raikh, JETP Letters 47, 452 (1988)] [T. K. Ng and P. A. Lee, Phys. Rev. Lett. 61, 1768 (1988)] 28 introduction to the kondo model temperature resistivity superconductor conductor im purity temperature conductance quantum dot (odd) quantum dot (even) figure 2 . 7 : Analogy between the increase of the resistivity and the conductance in R E P O R T S resistance impurities scatter: 28 introduction to the kondo model temperature resistivity superconductor conductor im purity temperature conductance quantum dot (odd) quantum dot (even) figure 2 . 7 : Analogy between the increase of the resistivity and the conductance in R E P O R T S Kondo singlet 1 p 2 |"S #el i |#S "el i source dot drain
27. 1988: Quantum Dots • indeed: Kondo physics is also present in quantum dots • triggered (again) interest of community ➡ more generalized setups possible ➡ nonequilibrium situations possible Fig. 1 (left). (A) Atomic force microscope image of the device. An AB ring is defined in a 2DEG by dry etching of the dark regions (depth is ϳ75 nm). The 2DEG with electron density n S ϭ 2.6 ϫ 1015 mϪ2 is situated 100 nm below the surface of an AlGaAs/GaAs heterostructure. In both arms of the ring (lithographic width, 0.5 ␮m; inner perimeter, 6.6 ␮m), a quantum dot can be defined by applying negative voltages to gate electrodes. The gates at the entry and exit of the ring are not used. A quantum dot of size ϳ200 nm by 200 nm, containing ϳ100 electrons, is formed in the lower arm using gate voltages V gl and V gr (the central plunger gate was not working). The average energy spacing between single-particle states is ϳ100 ␮eV. The conductance of the upper arm, set by V gu , is kept at zero, except for AB R E P O R T S [D. Goldhaber-Gordon et al., Nature 391, 156 (1998)] temperature decreases Fig. 1 (left). (A) Atomic force microscope image o defined in a 2DEG by dry etching of the dark regio 2DEG with electron density n S ϭ 2.6 ϫ 1015 mϪ2 the surface of an AlGaAs/GaAs heterostructure. (lithographic width, 0.5 ␮m; inner perimeter, 6.6 ␮ [W. G. van der Wiel et al., Science 289 (2000)] el temperature conductance quantum dot (odd) quantum dot (even) ease of the resistivity and the conductance in E P O R T S
28. • So, what’s different for nonequilibrium? • equilibrium: Nonequilibrium • nonequilibrium:
29. Motivation Dynamics in Low-Dimensional Co Quantum Systems • So, what’s different for nonequilibrium? • equilibrium: Nonequilibrium • nonequilibrium:
30. Motivation Dynamics in Low-Dimensional Co Quantum Systems • So, what’s different for nonequilibrium? • equilibrium: Nonequilibrium • nonequilibrium: V = 0
31. Motivation Dynamics in Low-Dimensional Co Quantum Systems V • So, what’s different for nonequilibrium? • equilibrium: Nonequilibrium • nonequilibrium: Motivation ynamics in Low-Dimensional Correlated Quantum Systems V = 0
32. Motivation Dynamics in Low-Dimensional Co Quantum Systems • So, what’s different for nonequilibrium? • equilibrium: Nonequilibrium • nonequilibrium: Motivation ynamics in Low-Dimensional Correlated Quantum Systems V V = 0
33. [P. Nozières, Journal of Low Temperature Physics 17, 31 (1974)] equilibrium methods: • Fermi liquid theory (very low energies) • numerical renormalization group (numerically exact) • Bethe Ansatz present: Available Methods [K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975)] [N. Andrei, K. Furuya, and J. Lowenstein, Rev. Mod. Phys. 55, 331 (1983)] Motivation Dynamics in Low-Dimensional Correlated Quantum Systems 1 / 20
34. [P. Nozières, Journal of Low Temperature Physics 17, 31 (1974)] equilibrium methods: • Fermi liquid theory (very low energies) • numerical renormalization group (numerically exact) • Bethe Ansatz nonequilibrium methods: • no complete solution (possible?) • established methods difficult to extend (NRG, DMRG, …) present: Available Methods [K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975)] [N. Andrei, K. Furuya, and J. Lowenstein, Rev. Mod. Phys. 55, 331 (1983)] Motivation Dynamics in Low-Dimensional Correlated Quantum Systems 1 / 20 vation mensional Correlated m Systems 1 / 20
35. [P. Nozières, Journal of Low Temperature Physics 17, 31 (1974)] equilibrium methods: • Fermi liquid theory (very low energies) • numerical renormalization group (numerically exact) • Bethe Ansatz nonequilibrium methods: • no complete solution (possible?) • established methods difficult to extend (NRG, DMRG, …) • crafted method specific for nonequilibrium: ➡ real-time renormalization group (RTRG) method [H. Schoeller, Eur. Phys. J. Special Topics 168, 179 (2009)] present: Available Methods [K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975)] [N. Andrei, K. Furuya, and J. Lowenstein, Rev. Mod. Phys. 55, 331 (1983)] Motivation Dynamics in Low-Dimensional Correlated Quantum Systems 1 / 20 vation mensional Correlated m Systems 1 / 20
36. Method Used in Thesis ⇢0 e iH(t t0) eiH(t t0) t t0 t = • real-time renormalization group method in frequency space • formally solve von Neumann equation ˙ ⇢(t) = i [H, ⇢(t)] = iL⇢(t), L = [H, ·] [H. Schoeller, Eur. Phys. J. Special Topics 168, 179 (2009)] ⇢(t) = e i H(t t0)⇢(t0)ei H(t t0) = • Liouvillian superoperator → »hide« Keldysh 3 TECHNICAL SLIDES
37. Method Used in Thesis ⇢0 e iH(t t0) eiH(t t0) t t0 t = • real-time renormalization group method in frequency space • formally solve von Neumann equation ˙ ⇢(t) = i [H, ⇢(t)] = iL⇢(t), L = [H, ·] [H. Schoeller, Eur. Phys. J. Special Topics 168, 179 (2009)] ⇢(t) = e i H(t t0)⇢(t0)ei H(t t0) = • Liouvillian superoperator → »hide« Keldysh 3 TECHNICAL SLIDES FOR THE EXPERTS
38. Method Used in Thesis ⇢0 e iH(t t0) eiH(t t0) t t0 t = ⇢0 e iH(t t0) eiH(t t0) t t0 ⇢0 t t0 e iL(t t0) = • real-time renormalization group method in frequency space • formally solve von Neumann equation ˙ ⇢(t) = i [H, ⇢(t)] = iL⇢(t), L = [H, ·] [H. Schoeller, Eur. Phys. J. Special Topics 168, 179 (2009)] ⇢(t) = e i L(t t0)⇢(t0) = • Liouvillian superoperator → »hide« Keldysh 3 TECHNICAL SLIDES FOR THE EXPERTS
39. Method Used in Thesis ⇢0 e iH(t t0) eiH(t t0) t t0 t = ⇢0 e iH(t t0) eiH(t t0) t t0 ⇢0 t t0 e iL(t t0) = • real-time renormalization group method in frequency space • formally solve von Neumann equation ˙ ⇢(t) = i [H, ⇢(t)] = iL⇢(t), L = [H, ·] [H. Schoeller, Eur. Phys. J. Special Topics 168, 179 (2009)] • work in Laplace space E ⇢(E) = Z 1 t0 dt ei E(t t0)⇢(t) L(E) = Z 1 t0 dt ei E(t t0)L(t) ⇢(t) = e i L(t t0)⇢(t0) = • Liouvillian superoperator → »hide« Keldysh 3 TECHNICAL SLIDES FOR THE EXPERTS
40. Method Used in Thesis • real-time renormalization group method in frequency space • integrate out the reservoirs & resum self-energy insertions • evaluate trace with Wick’s theorem [H. Schoeller, Eur. Phys. J. Special Topics 168, 179 (2009)] ⇢S(E) = Trres ⇢(E) ) ⇢S(E) = i E LS ⌃(E) ⇢S(t0) 3 TECHNICAL SLIDES
41. Method Used in Thesis • real-time renormalization group method in frequency space • integrate out the reservoirs & resum self-energy insertions • evaluate trace with Wick’s theorem • perturbative in interaction J , use diagrammatic language [H. Schoeller, Eur. Phys. J. Special Topics 168, 179 (2009)] ⇢S(E) = Trres ⇢(E) ) ⇢S(E) = i E LS ⌃(E) ⇢S(t0) S j where j runs over all eigenvalues lj(E) of the effective Liouvillian Leff S (E) = Âj lj(E)Pj(E). The corresponding projectors Pj(E) are obtained by Pj(E) = vj(E)↵ ⌦ ¯ vj(E) , ( 3 . 19 ) where vj(E)↵ and ⌦ ¯ vj(E) are the right and left eigenoperators3 of Leff S (E) respectively. For the evaluations of the contractions in S(E), the frequency dependence of the resolvent ( 3 . 18 ) has to be approximated. Because this is done in sightly different ways in the L- and the E-flow scheme, we will discuss the approximation separately in each section later. Using the diagrammatic rules, the perturbative diagrams of the Liouvillian and the vertex of the Kondo model are given by Leff S (E) = L(0) S + 1 2 + + O(G4), ( 3 . 20 ) G12(E) = 12 + ⇣ 1 2 (1 $ 2)⌘ + 1 2 2 1 + ⇣ + (1 $ 2)⌘ + O(G4), ( 3 . 21 ) 3 TECHNICAL SLIDES
42. Method Used in Thesis • real-time renormalization group method in frequency space • integrate out the reservoirs & resum self-energy insertions • evaluate trace with Wick’s theorem • perturbative in interaction J , use diagrammatic language • calculate with RTRG approach [H. Schoeller, Eur. Phys. J. Special Topics 168, 179 (2009)] ⇢S(E) = Trres ⇢(E) ) ⇢S(E) = i E LS ⌃(E) ⇢S(t0) ⌃(E) S j where j runs over all eigenvalues lj(E) of the effective Liouvillian Leff S (E) = Âj lj(E)Pj(E). The corresponding projectors Pj(E) are obtained by Pj(E) = vj(E)↵ ⌦ ¯ vj(E) , ( 3 . 19 ) where vj(E)↵ and ⌦ ¯ vj(E) are the right and left eigenoperators3 of Leff S (E) respectively. For the evaluations of the contractions in S(E), the frequency dependence of the resolvent ( 3 . 18 ) has to be approximated. Because this is done in sightly different ways in the L- and the E-flow scheme, we will discuss the approximation separately in each section later. Using the diagrammatic rules, the perturbative diagrams of the Liouvillian and the vertex of the Kondo model are given by Leff S (E) = L(0) S + 1 2 + + O(G4), ( 3 . 20 ) G12(E) = 12 + ⇣ 1 2 (1 $ 2)⌘ + 1 2 2 1 + ⇣ + (1 $ 2)⌘ + O(G4), ( 3 . 21 ) 3 TECHNICAL SLIDES
43. Method Used in Thesis • real-time renormalization group method in frequency space • integrate out the reservoirs & resum self-energy insertions • evaluate trace with Wick’s theorem • perturbative in interaction J , use diagrammatic language • calculate with RTRG approach • solve quantum kinetic equation [H. Schoeller, Eur. Phys. J. Special Topics 168, 179 (2009)] ⇢S(E) = Trres ⇢(E) ) ⇢S(E) = i E LS ⌃(E) ⇢S(t0) ⌃(E) d dt ⇢S(t) + i LS⇢S(t) = i Z t t0 dt0 ⌃(t t0) ⇢S(t0) S j where j runs over all eigenvalues lj(E) of the effective Liouvillian Leff S (E) = Âj lj(E)Pj(E). The corresponding projectors Pj(E) are obtained by Pj(E) = vj(E)↵ ⌦ ¯ vj(E) , ( 3 . 19 ) where vj(E)↵ and ⌦ ¯ vj(E) are the right and left eigenoperators3 of Leff S (E) respectively. For the evaluations of the contractions in S(E), the frequency dependence of the resolvent ( 3 . 18 ) has to be approximated. Because this is done in sightly different ways in the L- and the E-flow scheme, we will discuss the approximation separately in each section later. Using the diagrammatic rules, the perturbative diagrams of the Liouvillian and the vertex of the Kondo model are given by Leff S (E) = L(0) S + 1 2 + + O(G4), ( 3 . 20 ) G12(E) = 12 + ⇣ 1 2 (1 $ 2)⌘ + 1 2 2 1 + ⇣ + (1 $ 2)⌘ + O(G4), ( 3 . 21 ) 3 TECHNICAL SLIDES
44. Method Used in Thesis • real-time renormalization group method in frequency space • renormalization group procedure: • derive flow equations to include different energy scales [H. Schoeller, Eur. Phys. J. Special Topics 168, 179 (2009)] 3 TECHNICAL SLIDES
45. Method Used in Thesis • real-time renormalization group method in frequency space • renormalization group procedure: • derive flow equations to include different energy scales • -flow scheme: • cutoff in contraction for high energy divergencies • does not allow treatment of strong coupling regime [H. Schoeller, Eur. Phys. J. Special Topics 168, 179 (2009)] 3 TECHNICAL SLIDES ⇤c = max {T, V, h} TK
46. Method Used in Thesis • real-time renormalization group method in frequency space • renormalization group procedure: • derive flow equations to include different energy scales • -flow scheme: • cutoff in contraction for high energy divergencies • does not allow treatment of strong coupling regime • E-flow scheme: • use Laplace variable E as flow parameter • flow to strong coupling possible [H. Schoeller, Eur. Phys. J. Special Topics 168, 179 (2009)] 3 TECHNICAL SLIDES ⇤c = max {T, V, h} TK [Pletyukhov and Schoeller, Phys. Rev. Lett. 108 (2012)]
47. screened impurity is removed at very low temperatures Fermi liquid description • degree of freedom: size of spin S & number of reservoirs N Possible Kondo models S=½ 2S = N N=1
48. screened impurity is removed at very low temperatures Fermi liquid description • degree of freedom: size of spin S & number of reservoirs N Possible Kondo models S=½ 2S = N N=1 one reservoir
49. screened impurity is removed at very low temperatures Fermi liquid description • degree of freedom: size of spin S & number of reservoirs N Possible Kondo models S=½ 2S = N N=1
50. screened impurity is removed at very low temperatures Fermi liquid description • degree of freedom: size of spin S & number of reservoirs N Possible Kondo models S=½ 2S = N N=1
51. screened impurity is removed at very low temperatures Fermi liquid description • degree of freedom: size of spin S & number of reservoirs N Possible Kondo models S=½ 2S = N N=1
52. S=1 • degree of freedom: size of spin S & number of reservoirs N Possible Kondo models underscreened switch from antiferromagnetic (J0 > 0) to ferromagnetic (J0 < 0) 2S > N N=1
53. S=1 • degree of freedom: size of spin S & number of reservoirs N Possible Kondo models underscreened switch from antiferromagnetic (J0 > 0) to ferromagnetic (J0 < 0) 2S > N N=1
54. Seff =½ • degree of freedom: size of spin S & number of reservoirs N Possible Kondo models underscreened switch from antiferromagnetic (J0 > 0) to ferromagnetic (J0 < 0) 2S > N N=1
55. Seff =½ • degree of freedom: size of spin S & number of reservoirs N Possible Kondo models underscreened switch from antiferromagnetic (J0 > 0) to ferromagnetic (J0 < 0) NOT IN THIS TALK – BUT IN THESIS 2S > N N=1
56. • degree of freedom: size of spin S & number of reservoirs N Possible Kondo models overscreened too many reservoir spins: non-Fermi liquid non-trivial power laws S=½ 2S < N N=2
57. • degree of freedom: size of spin S & number of reservoirs N Possible Kondo models overscreened too many reservoir spins: non-Fermi liquid non-trivial power laws S=½ 2S < N N=2
58. My Nonequilibrium Results 1) Overscreened Kondo Model • non-Fermi liquid fixed point • power-laws • nonequilibrium • cotunneling features • same voltages • different voltages 2) Screened Kondo Model • flow to strong coupling • different Kondo scales • nonequilibrium • conductance • susceptibility • comparison to other equilibrium methods
59. Overscreened Kondo Model power-law coefficients & coupling between conduction channels
60. • isotropic spin-½ Kondo dot • fixed charge with S =½ on dot • K independent reservoirs at different chemical potentials • in a magnetic field h0 Overscreened Kondo Model S=1/2 +V 1 / 2 +V 2 / 2 +V K / 2 -V 1 / 2 -V 2 / 2 -V K / 2 J J J J J … … ! µi L µi R = Vi H = X i↵k ✏k c† i↵k ci↵k + h0 Sz + J0 2⌫0 X i↵↵0kk0 0 ~ S · ~ 0 c† i↵k ci↵0k0 0 channel / reservoir index
61. • real-time renormalization group method in frequency space • high energy cutoff Scaling Equations ⇤ ⇤ d d⇤ J = (J) = 2J2(1 KJ) strong coupling weak coupling J = 0 J = ∞
62. • real-time renormalization group method in frequency space • high energy cutoff • fixed point ➡ stops flow to strong coupling at • expansion around 0.6 0.8 1 0 2 4 6 8 10 J(Λ) / J* Λ/ TK K= 5 Jc = J ⇤c = max {h, V } Scaling Equations ⇤ = 0 ⇤ ⇤ d d⇤ J = (J) = 2J2(1 KJ) strong coupling weak coupling J⇤ = 1 K J = 0 J = ∞ J*
63. • real-time renormalization group method in frequency space • high energy cutoff • fixed point ➡ stops flow to strong coupling at • expansion around 0.6 0.8 1 0 2 4 6 8 10 J(Λ) / J* Λ/ TK K= 5 Jc = J ⇤c = max {h, V } Scaling Equations ⇤ = 0 ⇤ ⇤ d d⇤ J = (J) = 2J2(1 KJ) strong coupling weak coupling J⇤ = 1 K J = 0 J = ∞ J*
64. • real-time renormalization group method in frequency space • high energy cutoff • fixed point ➡ stops flow to strong coupling at • expansion around 0.6 0.8 1 0 2 4 6 8 10 J(Λ) / J* Λ/ TK K= 5 Jc = J ⇤c = max {h, V } Scaling Equations ⇤ = 0 ⇤ ⇤ d d⇤ J = (J) = 2J2(1 KJ) strong coupling weak coupling TK = ⇤ ✓ eJ J⇤ J ◆K/2 e 1/2J J(⇤) ⇤c ⌧TK = J⇤ " 1 ✓ ⇤ TK ◆ # = 2/K • Kondo temperature (scaling invariant) and power-law J⇤ = 1 K J = 0 J = ∞ J*
65. • identical bias voltages Vi =V: • logarithmic enhancement + broadening in RTRG • elastic cotunneling for V < h • inelastic cotunneling for V > h Differential Conductance V V h h 0 0.2 0.4 0.6 0 1 2 3 G / G∆ V / h K= 20 PT RTRG -Vi /2 +Vi /2 Gi = dIi dVi , G = 3⇡2 16 2 current Ii
66. • identical bias voltages Vi =V: • logarithmic enhancement + broadening in RTRG • elastic cotunneling for V < h • inelastic cotunneling for V > h Differential Conductance V V h h 0 0.2 0.4 0.6 0 1 2 3 G / G∆ V / h K= 20 PT RTRG -Vi /2 +Vi /2 Gi = dIi dVi , G = 3⇡2 16 2 current Ii
67. • identical bias voltages Vi =V: • logarithmic enhancement + broadening in RTRG • elastic cotunneling for V < h • inelastic cotunneling for V > h Differential Conductance V V h h 0 0.2 0.4 0.6 0 1 2 3 G / G∆ V / h K= 20 PT RTRG -Vi /2 +Vi /2 Gi = dIi dVi , G = 3⇡2 16 2 current Ii
68. V V/4 V/3 V/2 • different bias voltages for different reservoirs: • total number of reservoirs K = 20 • 4 different voltages (each 5 reservoirs with identical bias voltage) • inelastic cotunneling: first black, than green, than blue, than pink V, V/2, V/3, V/4 h S=1/2 +V / 2 +V / 4 +V / 8 -V / 2 +V / 6 -V / 6 -V / 4 -V / 8 J J J J J J J J Differential Conductance V= h V = 2h V = 3h V = 4h
69. V V/4 V/3 V/2 V V/4 V/3 V/2 • different bias voltages for different reservoirs: • total number of reservoirs K = 20 • 4 different voltages (each 5 reservoirs with identical bias voltage) • inelastic cotunneling: first black, than green, than blue, than pink V, V/2, V/3, V/4 h S=1/2 +V / 2 +V / 4 +V / 8 -V / 2 +V / 6 -V / 6 -V / 4 -V / 8 J J J J J J J J Differential Conductance V= h V = 2h V = 3h V = 4h
70. own resonance: V = h other resonances: V = 2 h V = 3 h V = 4 h • inelastic cotunneling most prominent • additional features from other reservoirs • coupling due to dot magnetization • example: @M @V V, V/2, V/3, V/4 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 6 7 8 Gi / G∆ V / h h0 = 100 TK K= 20 G1... 5 1×10-3 2×10-3 3×10-3 0 1 2 3 4 5 6 7 8 ∂M / ∂V TK V / h h0 = 100 TK K= 20 0 -0.5 0 0 5 10 15 20 M V / h V1…5 =V V6…10 =V/2 V11…15 =V/3 V16…20 =V/4 V V/4 V/3 V/2 Differential Conductance • different bias voltages for different reservoirs: V= h V = 2h V = 3h V = 4h
71. own resonance: V = h other resonances: V = 2 h V = 3 h V = 4 h • inelastic cotunneling most prominent • additional features from other reservoirs • coupling due to dot magnetization • example: @M @V V, V/2, V/3, V/4 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 6 7 8 Gi / G∆ V / h h0 = 100 TK K= 20 G1... 5 1×10-3 2×10-3 3×10-3 0 1 2 3 4 5 6 7 8 ∂M / ∂V TK V / h h0 = 100 TK K= 20 0 -0.5 0 0 5 10 15 20 M V / h V1…5 =V V6…10 =V/2 V11…15 =V/3 V16…20 =V/4 Differential Conductance • different bias voltages for different reservoirs: V= h V = 2h V = 3h V = 4h
72. own resonance: V = 2 h other resonances: V = h V = 3 h V = 4 h • inelastic cotunneling most prominent • additional features from other reservoirs • coupling due to dot magnetization • example: @M @V V, V/2, V/3, V/4 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 6 7 8 Gi / G∆ V / h h0 = 100 TK K= 20 G1... 5 G6... 10 1×10-3 2×10-3 3×10-3 0 1 2 3 4 5 6 7 8 ∂M / ∂V TK V / h h0 = 100 TK K= 20 0 -0.5 0 0 5 10 15 20 M V / h V1…5 =V V6…10 =V/2 V11…15 =V/3 V16…20 =V/4 Differential Conductance • different bias voltages for different reservoirs: V= h V = 2h V = 3h V = 4h
73. • inelastic cotunneling most prominent • additional features from other reservoirs • coupling due to dot magnetization • example: @M @V V, V/2, V/3, V/4 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 6 7 8 Gi / G∆ V / h h0 = 100 TK K= 20 G1... 5 G6... 10 G11... 15 1×10-3 2×10-3 3×10-3 0 1 2 3 4 5 6 7 8 ∂M / ∂V TK V / h h0 = 100 TK K= 20 0 -0.5 0 0 5 10 15 20 M V / h V1…5 =V V6…10 =V/2 V11…15 =V/3 V16…20 =V/4 own resonance: V = 3 h other resonances: V = h V = 2 h V = 4 h Differential Conductance • different bias voltages for different reservoirs: V= h V = 2h V = 3h V = 4h
74. • inelastic cotunneling most prominent • additional features from other reservoirs • coupling due to dot magnetization • example: @M @V V, V/2, V/3, V/4 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 6 7 8 Gi / G∆ V / h h0 = 100 TK K= 20 G1... 5 G6... 10 G11... 15 G16... 20 1×10-3 2×10-3 3×10-3 0 1 2 3 4 5 6 7 8 ∂M / ∂V TK V / h h0 = 100 TK K= 20 0 -0.5 0 0 5 10 15 20 M V / h V1…5 =V V6…10 =V/2 V11…15 =V/3 V16…20 =V/4 own resonance: V = 4 h other resonances: V = h V = 2 h V = 3 h Differential Conductance • different bias voltages for different reservoirs: V= h V = 2h V = 3h V = 4h
75. My Nonequilibrium Results 1) Overscreened Kondo Model • non-Fermi liquid fixed point • power-laws • nonequilibrium • cotunneling features • same voltages • different voltages 2) Screened Kondo Model • flow to strong coupling • different Kondo scales • nonequilibrium • conductance • susceptibility • comparison to other equilibrium methods
76. My Nonequilibrium Results 1) Overscreened Kondo Model • non-Fermi liquid fixed point • power-laws • nonequilibrium • cotunneling features • same voltages • different voltages 2) Screened Kondo Model • flow to strong coupling • different Kondo scales • nonequilibrium • conductance • susceptibility • comparison to other equilibrium methods ✔
77. Screened Kondo Models strong coupling & static spin susceptibility & comparison to (numerical) exact limits
78. • screened models: • no coupling between reservoirs • differential conductance • static spin susceptibility & dynamical correlation function S=1/2 -V/2 V/2 J0 J0 -V/2 -V/2 V/2 V/2 S=1 (a) (b) J0 J0 J0 J0 H = X i↵k ✏k c† i↵k ci↵k + J0 2⌫0 X i↵↵0 kk0 0 ~ S · ~ 0 c† i↵k ci↵0k0 0 ˆ IL ⌘ d dt ˆ NL G = d dV D ˆ IL E N=1 N=2 S(⌦) = Re C+(⌦) (⌦) = iC (⌦) C±(⌦) = Z 0 1 dt e i⌦th[Sz(0)H, Sz(t)H]± ist 2S = N Screened Kondo Models
79. • developed by Pletyukhov and Schoeller [Phys. Rev. Lett. 108 (2012)] • instead of high energy cutoff use Laplace variable E • initial condition flow: fixed by unitary conductance only: RG: E-Flow Scheme ⇤ G(T = V = 0) ⇤=0 = 2e2N h ⌘ G0
80. • developed by Pletyukhov and Schoeller [Phys. Rev. Lett. 108 (2012)] • instead of high energy cutoff use Laplace variable E • initial condition flow: fixed by unitary conductance only: • single model and framework for weak and strong coupling • usually considered: limit models (BA, Fermi liquid theory) • see how far we get with »only« regular Kondo model • expect not to capture all features ➡ especially difficult: quantities at zero energy RG: E-Flow Scheme ⇤ G(T = V = 0) ⇤=0 = 2e2N h ⌘ G0
81. Different Kondo Scales • scaling invariant: • various definitions of Kondo scales in the literature • cross-over quantity → hard to derive accurately ˜ J = ZJ, Z = 1/(1 + d d⇤ ) spin relaxation rate Z d ˜ J d⇤ = 1 ⇤ + 2 ˜ J2 (1 N ˜ J ) TK = (⇤ + ) N ˜ J 1 N ˜ J !N/2 exp ✓ 1 2 ˜ J ◆ ~ ~ ~ ~
82. Different Kondo Scales • scaling invariant: • zero susceptibility: • various definitions of Kondo scales in the literature • cross-over quantity → hard to derive accurately ˜ J = ZJ, Z = 1/(1 + d d⇤ ) spin relaxation rate Z d ˜ J d⇤ = 1 ⇤ + 2 ˜ J2 (1 N ˜ J ) TK = (⇤ + ) N ˜ J 1 N ˜ J !N/2 exp ✓ 1 2 ˜ J ◆ (T = 0, V = 0) ⌘ 0 = S(S + 1) 3T0 difficult quantity
83. Different Kondo Scales • scaling invariant: • zero susceptibility: • half conductance: • various definitions of Kondo scales in the literature • cross-over quantity → hard to derive accurately ˜ J = ZJ, Z = 1/(1 + d d⇤ ) spin relaxation rate Z d ˜ J d⇤ = 1 ⇤ + 2 ˜ J2 (1 N ˜ J ) TK = (⇤ + ) N ˜ J 1 N ˜ J !N/2 exp ✓ 1 2 ˜ J ◆ accessible by experiments n G(T = T⇤ K , V = 0) = G0 2 G(T = 0, V = T⇤⇤ K ) = G0 2 (T = 0, V = 0) ⌘ 0 = S(S + 1) 3T0 difficult quantity
84. • for S=1: • use G(V) to measure Kondo temperature Differential Conductance (2/3 G0 for S=½) [PRL 108 (2012)] G(T) < G(V ) G(V = T⇤ K ) ⇡ 0.574 G0 [A. V. Kretinin et al., PRB 85 (2012)]
85. • for S=1: • use G(V) to measure Kondo temperature • comparison with NRG: reasonable agreement Differential Conductance (2/3 G0 for S=½) [PRL 108 (2012)] [Costi et al., PRL 102 (2009)] charge fluctuations G(T) < G(V ) G(V = T⇤ K ) ⇡ 0.574 G0 [A. V. Kretinin et al., PRB 85 (2012)]
86. Differential Conductance • by Fermi liquid approach ➡ NEW: NRG: cB, cT N-channel: Hanl et al., PRB 89 (2014) SIAM: Merker et al., PRB 87 (2013) G(T, V ) = G0 " 1 c0 T ✓ T T0 ◆2 c0 V ✓ eV T0 ◆2 # c0 V c0 T = c⇤ V c⇤ T = 3 2⇡2 4 + 10S 5 + 8S
87. • maybe T0 causes large error in 3rd c'T ➡ zero energy quantity, sensitive to truncation order • ratios reliable Differential Conductance • by Fermi liquid approach ➡ NEW: NRG: cB, cT N-channel: Hanl et al., PRB 89 (2014) SIAM: Merker et al., PRB 87 (2013) G(T, V ) = G0 " 1 c0 T ✓ T T0 ◆2 c0 V ✓ eV T0 ◆2 # RTRG S = ½ 2nd 5.02 (18%) 0.34 (122%) 3rd 18.04 (196%) 0.21 (38%) S = 1 2nd 7.40 (16%) 0.25 (52%) 3rd 31.77 (261%) 0.18 (9%) T0 strong dependence on truncation order c'T (error) c*V/c*T (error) c0 V c0 T = c⇤ V c⇤ T = 3 2⇡2 4 + 10S 5 + 8S
88. Static Susceptibility • reasonable agreements with low T results from Bethe Ansatz • asymptotic from PT: • log-order corrections out of scope ➡ require addit. loop for full • for S=½: deviations due to higher orders O(J3) O(J3) (T) ! 1/(4T) (T) T ⌧T0 = (0) " 1 p 3⇡3 8 ✓ T T0 ◆2 + O ✓ T T0 ◆4 # (T) T T0 = 1 4T " 1 1 ln[T/(wT0)] 1 2 ln ⇥ ln[T/(wT0)] ⇤ ln2[T/(wT0)] +O ✓ 1 ln3[T/(wT0)] ◆
89. • weak dependence on 2S = N Static Susceptibility • reasonable agreements with low T results from Bethe Ansatz • asymptotic from PT: • log-order corrections out of scope ➡ require addit. loop for full • for S=½: deviations due to higher orders O(J3) O(J3) (T) ! 1/(4T) (T) T ⌧T0 = (0) " 1 p 3⇡3 8 ✓ T T0 ◆2 + O ✓ T T0 ◆4 # (T) T T0 = 1 4T " 1 1 ln[T/(wT0)] 1 2 ln ⇥ ln[T/(wT0)] ⇤ ln2[T/(wT0)] +O ✓ 1 ln3[T/(wT0)] ◆ ⇤ S = S(S + 1)/(3T)
90. • a'T again strong dependence on truncation order (3rd) • expect reliable ratios (as for differential conductance) • Bethe Ansatz results for a'T → prediction for a'V Static Susceptibility S=½: [Tsvelick and Wiegmann, J. Stat. Phys. 38 (1995)] [Mel’nikov, JETP Lett. 35 (1982)] a0 T = ⇡2S2 ( S + 1) 18 ( S ) 3 (3 S ) 3 3S = (p 3 ⇡3/ 8 for S = 1 / 2 3 ⇡2/ 2 for S = 1 S=1: agrees with recent NRG data (T, V ) = 0 " 1 a0 T ✓ T T0 ◆2 a0 V ✓ eV T0 ◆2 # a0 V ⇡ ( 0 . 8 for S = 1 / 2 1 . 4 for S = 1
91. • used single framework for different Kondo models ➡ overscreened Kondo model: • recover characteristic power-law • differential conductance channel couple via magnetic field ➡ screend Kondo model: • linear and differential conductance • qualitative behavior of susceptibility / correlation functions • ratios for susceptibility and conductance • How far did we get with this? • some crossover features not captured (hard) • and thus also at zero energy a0 V /a0 T c0 V /c0 T Summary 0 = (T = V = E = 0) T0
92. • in Aachen • Herbert Schoeller • Frank Reininghaus • Dante Kennes, Stefan Göttel • in Utrecht • Dirk Schuricht • Cristiane Morais Smith • people from ITF • in Paris • Christophe Mora Thank You! [James Chapman http://chapmangamo.tumblr.com]