NUMERICAL METHODS FOR QUANTUM IMPURITY MODELS

Transcription

1 NUMERICAL METHODS FOR QUANTUM IMPURITY MODELS March 2015 Anrew Mitchell Utrecht University

2 Quantum impurity problems Part 1: Quantum impurity problems an theoretical bacgroun Part 2: Kono effect an RG. 1 chain formulation an iterative iagonalization Part 3: Logarithmic iscretization an truncation. The RG in NRG Part 4: Physical quantities. Results an iscussion. Anrew Mitchell Quantum impurity problems

3 General References: K. G. Wilson Rev. Mo. Phys (1975) A. C. Hewson The Kono problem to Heavy Fermions (Cambrige University Press 1997) H. R. Krishnamurthy J. W. Wilins an K. G. Wilson Phys. Rev. B (1980); ibi (1980) R. Bulla T. Costi T. Prusche Rev. Mo. Phys (2008) Anrew Mitchell Quantum impurity problems

4 NUMERICAL METHODS FOR QUANTUM IMPURITY MODELS Part 1: Quantum Impurity Problems an theoretical bacgroun March 2015 Anrew Mitchell Utrecht University

5 Overview: Part 1 Impurities in metals Quantum ots Dynamical Mean Fiel Theory Non-interacting limit Green functions The problem of interactions Anrew Mitchell Quantum impurity problems: Part 1

6 Impurities in metals Defects Potential scattering centres Magnetic impurities Anrew Mitchell Quantum impurity problems: Part 1

7 Impurities as probes Brea translational symmetry of host Cause scattering of quasiparticles FT-STS quasiparticle interference

8 Impurities in metals Quantum impurity problem Hamiltonian: H H host H imp H Host consists of non-interacting conuction electrons: hyb H host b b ( iagonal or -space basis) Impurity part: a few local interacting egrees of freeom Anrew Mitchell Quantum impurity problems: Part 1

9 Impurities in metals Kono moel: a spin-½ impurity S imp couple by antiferromagnetic exchange to conuction electron spin ensity of the host at impurity location (0) H K H host J S imp s 0 b b ' b ' b ' ' 2 Anrew Mitchell Quantum impurity problems: Part 1

10 Impurities in metals Kono moel: spin-flip scattering J conuction ban impurity Anrew Mitchell Quantum impurity problems: Part 1

11 Impurities in metals Anerson moel: a single quantum level with onsite Coulomb repulsion tunnel-couple to conuction electrons of host hyb imp host AIM H H H H U H imp host b b H H.c. hyb b V H Anrew Mitchell Quantum impurity problems: Part 1

12 Impurities in metals Anerson moel: V E E imp 20 U conuction ban impurity Anrew Mitchell Quantum impurity problems: Part 1

13 Impurities in metals Anerson moel: V essentially singly-occupie local moment for U 0 conuction ban impurity Anrew Mitchell Quantum impurity problems: Part 1

14 Semiconuctor Quantum Dots S u r f a c e e l e c t r o e s ot! O h m i c c o n t a c t -V Ga x Al 1-x As ( n - o p e ) 2 D E G GaAs Anrew Mitchell Quantum impurity problems: Part 1

15 Quantum ots Golhaber-Goron et al. Nature (1998) s V s V g Anrew Mitchell Quantum impurity problems: Part 1

16 Coulomb Blocae ot E s ev s s Anrew Mitchell Quantum impurity problems: Part 1

17 Coulomb Blocae ot E ev s 0 s Anrew Mitchell Quantum impurity problems: Part 1

18 Coulomb Blocae E s Anrew Mitchell Quantum impurity problems: Part 1

19 Coulomb Blocae Coulomb repulsion blocs sequential tunneling E s 2 0 f U f Active level carries a spin-½ local moment Anrew Mitchell Quantum impurity problems: Part 1

20 Coulomb Blocae But: gate voltage controls ot occupancy V g s V g Anrew Mitchell Quantum impurity problems: Part 1

21 Coulomb Blocae But: gate voltage controls ot occupancy V g s Sequential tunneling at points of ot valence fluctuation Effective level with renormalize by interactions V g Anrew Mitchell Quantum impurity problems: Part 1

22 Coulomb Blocae Conuctance peas as gate voltage is swept: G c s V g Vg Anrew Mitchell Quantum impurity problems: Part 1

23 Coulomb Blocae: experiment Conuctance peas as gate voltage is swept: High-T Van er Wiel et al Science (2000) Anrew Mitchell Quantum impurity problems: Part 1

24 Kono Effect Low temperature: quantum many boy effects! Van er Wiel et al Science (2000) Anrew Mitchell Quantum impurity problems: Part 1

25 Anerson Impurity Moel Real quantum ot evices: Moel as a single active interacting quantum level Tunnel-couple to source an rain leas hyb ot leas AIM H H H H s b b U H.c. b V Anrew Mitchell Quantum impurity problems: Part 1

26 Anerson Impurity Moel Equilibrium (zero bias): Combine leas into a single conuction electron channel H.c. hyb b V H f V H.c. 2 2 V V 1 s s b V b V V f Anrew Mitchell Quantum impurity problems: Part 1

27 Anerson Impurity Moel Equilibrium (zero bias): Combine leas into a single conuction electron channel Other bath egrees of freeom ecouple s leas b b H σ f f 1 s s b V b V V g Anrew Mitchell Quantum impurity problems: Part 1

28 Two-lea evice: single channel conuction ban ot conuction ban Anrew Mitchell Quantum impurity problems: Part 1

29 Couple Quantum Dots R. Poto et al. Nature (2007) H. Jeong et al. Science (2001) N. Craig et al. Science (2004) A. Vian et al. App. Phys. Lett (2004)

30 Nanotube Quantum Dots Dot source gates rain Nygar et al Nature (2000) Anrew Mitchell Quantum impurity problems: Part 1

31 Dynamical Mean Fiel Theory DMFT A. Georges G. Kotliar W. Krauth M. Rozenberg Rev. Mo. Phys (1996) For correlate lattice problems (Hubbar moel etc) Local self-energy approximation (exact in the limit of infinite imensions) Map onto a single-impurity Anerson impurity moel in a bath that must be etermine self-consistently Anrew Mitchell Quantum impurity problems: Part 1

32 Dynamical Mean Fiel Theory Kotliar & Vollhart Physics Toay (2004) Anrew Mitchell Quantum impurity problems: Part 1

33 Dynamical Mean Fiel Theory Anrew Mitchell Quantum impurity problems: Part 1

34 Dynamical Mean Fiel Theory Hubbar Moel in = R. Bulla PRL (1999) Mott Metal-Insulator transition at T=0 Anrew Mitchell Quantum impurity problems: Part 1

35 Quantum impurity problems Why are quantum impurity problems har to solve? strong electron correlations Before we try to solve the impurity problem let s loo again at the easy bit: representations of the non-interacting host that we will nee later Anrew Mitchell Quantum impurity problems: Part 1

36 Bul metal: Host metal: real-space representation STM image Anrew Mitchell Quantum impurity problems: Part 1

37 Bul metal: Host metal: real-space representation Non-interacting tight-bining moel j i j i ij host c c t H c T c ; c c c c 22 * t t t t T Anrew Mitchell Quantum impurity problems: Part 1

38 Bul metal: Host metal: iagonal representation where b D b c T c H host c A b b b ' ' ' A T A D Anrew Mitchell Quantum impurity problems: Part 1

39 Bul metal: is the ispersion: For example 2 square lattice: x a 0 / Anrew Mitchell Quantum impurity problems: Part 1

40 Bul metal: is the ispersion: For example triangular lattice: x a 0 / Anrew Mitchell Quantum impurity problems: Part 1

41 Bul metal: is the ispersion: For example honeycomb (graphene) lattice: x a 0 / Anrew Mitchell Quantum impurity problems: Part 1

42 Density of states: Total ensity of states: E tot E E Anrew Mitchell Quantum impurity problems: Part 1

43 Density of states: Local ensity of states (LDOS): As measure locally at a given point in real space Obtaine experimentally by STM E a i E 2 i Expansion coefficients of real-space A a i site i in terms of -space orbitals i Anrew Mitchell Quantum impurity problems: Part 1

44 Density of states: Local ensity of states (LDOS): Relate to local (real-space) Green function 1 Im i G i i G i i ci ; ci FT G t i t c t c 0 i i i i Anrew Mitchell Quantum impurity problems: Part 1

45 Free host Green functions But what is the local host Green function? G i i ci ; ci ' ' a a i ' * a a G i i Diagonal representation! i ' G * ' b ' ; b ' i0 i0 ' Anrew Mitchell Quantum impurity problems: Part 1

46 Free host Green functions But what is the local host Green function? 0 2 ' i i i i a G i i i i a G 2 Im 1 Anrew Mitchell Quantum impurity problems: Part 1

47 Free host Green functions Dyson representation: 1 G G ~ 1 isolate -level Green function: 1 i0 hybriization with rest of system G 1 i0 Anrew Mitchell Quantum impurity problems: Part 1

48 Potential scatterer Potential scattering impurity H host i j t ij c i c j g c c Moifie LDOS: efine without impurity ps - 1 (0) G G - 1 g Anrew Mitchell Quantum impurity problems: Part 1

49 T matrix T matrix escribes scattering between iagonal eigenstates of the free system inuce by the impurity: Born approx: Τ (0) ' ' ' (0) (0) ' ' G G G G 1 Τ (0) ' * ' G g g a a g a a Τ ' * ' Anrew Mitchell Quantum impurity problems: Part 1

50 Resonant level Resonant level impurity: (non-interacting U=0 Anerson moel) H RL i j t c H.c. ij c V c i j 0 Anrew Mitchell Quantum impurity problems: Part 1

51 Resonant level Resonant level Green function: G i0 1 local host Green function V 2 G (0) 0 0 Hybriization with rest of system Anrew Mitchell Quantum impurity problems: Part 1

52 Hybriization function V 2 G (0) 0 0 Im V 2 Im G (0) 0 0 V 2 i Local Density of States (LDOS) of host site to which impurity is couple Anrew Mitchell Quantum impurity problems: Part 1

53 Hybriization function Consier wie flat conuction ban: 1 2D D i i D D Anrew Mitchell Quantum impurity problems: Part 1

54 Hybriization function Hybriization function by Kramers-Kronig: V 2 G (0) 0 0 V 2 P i i i For wie flat ban: 0 2 i 0 i V 0 Anrew Mitchell Quantum impurity problems: Part 1

55 Impurity spectral function Flat conuction ban: Spectrum: i i G / Im 1 A G Anrew Mitchell Quantum impurity problems: Part 1

56 Impurity spectral function Spectrum: 0A Lorentzian centre on 1 2 / 1 0 Effective level with Pea height pinne to 0 1/ 0 Quaratic approach to maximum value Anrew Mitchell Quantum impurity problems: Part 1

57 T matrix T matrix for resonant level: Τ (0) ' ' ' (0) (0) ' ' G G G G Τ 2 ' * ' G V a a Impurity Green function Anrew Mitchell Quantum impurity problems: Part 1

58 Interacting problem: The Anerson impurity moel with strong interactions U>0 is MUCH more ifficult! Why?! Anrew Mitchell Quantum impurity problems: Part 1

59 Interacting problem: In the non-interacting case the Hamiltonian can be brought into iagonal form by performing a canonical transformation of operators: with c T c b A c b b Achieve by simply iagonalizing the hopping matrix which is of imension N x N for an N-particle system T Anrew Mitchell Quantum impurity problems: Part 1

60 Interacting problem: For an interacting problem containing terms lie U c c c c (not quaratic in electronic operators) the Hamiltonian itself cannot be brought to iagonal form by transformation of operators. Must construct the Hamiltonian matrix with elements a Hˆ b in the many-particle basis. Fermions: matrix is of imension 4 N x 4 N Anrew Mitchell Quantum impurity problems: Part 1

61 Interacting problem: So: we cannot o exact iagonalization (except for very small systems an at high T) Perturbation theory in the interaction U oes not give information about the strongly correlate regime. Plague by ivergences! Mean fiel approaches completely fail to capture the physics Anrew Mitchell Quantum impurity problems: Part 1

62 Asie: many-particle levels Consier a single interacting quantum level: H imp c c c c U c c c c E 2 U 0 n ; n 1; 1 0; 0 0;1 1; 0 Anrew Mitchell Quantum impurity problems: Part 1

63 Asie: many-particle levels NOT possible to reprouce these many-particle energies just using single-particle levels: H non int c c c c E unless U=0 Anrew Mitchell Quantum impurity problems: Part 1

64 Interacting problem: Large U: Schrieffer-Wolff transformation: Project into singly-occupie (spin) manifol of ot Perturbatively eliminate virtual excitations to empty or oubly-occupie ot states to secon orer in H hyb H eff E H 1ˆ 1 H H hyb 0 imp hyb 1ˆ projector: 1ˆ Anrew Mitchell Quantum impurity problems: Part 1

65 Interacting problem: Large U: Schrieffer-Wolff transformation: Low-energy effective moel is the Kono moel H eff H ~ V 2 U host AF exchange J S s imp 0 impurity spin-½ conuction electron spin ensity at impurity ' c0 c0 ' 2 Anrew Mitchell Quantum impurity problems: Part 1

66 The Kono problem But what is the physics of the Kono moel?! First full solution obtaine by the Numerical Renormalization Group see part 2! Anrew Mitchell Quantum impurity problems: Part 1