1. Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University Collaboration: Frithjof B. Anders, Dortmund University F.B. Anders and AS, Phys. Rev. Lett. 95,  (2005) F.B. Anders and AS, Phys. Rev. B 74,  (2006)

2. Avraham Schiller / Seattle 09 Outline Confined nano-structures and dissipative systems: Time-dependent Numerical Renormalization Benchmarks for fermionic and bosonic baths Spin and charge relaxation in ultra-small dots Non-perturbative physics out of equilibrium Group (TD-NRG)

3. Avraham Schiller / Seattle 09 Coulomb blockade in ultra-small quantum dots

4. Avraham Schiller / Seattle 09 Quantum dot Coulomb blockade in ultra-small quantum dots

5. Avraham Schiller / Seattle 09 Leads Coulomb blockade in ultra-small quantum dots

6. Avraham Schiller / Seattle 09 Lead Coulomb blockade in ultra-small quantum dots

7. Avraham Schiller / Seattle 09 Lead Coulomb blockade in ultra-small quantum dots

8. Avraham Schiller / Seattle 09 U Lead Coulomb blockade in ultra-small quantum dots

9. Avraham Schiller / Seattle 09 U Lead Coulomb blockade in ultra-small quantum dots

10. Avraham Schiller / Seattle 09 U Lead Coulomb blockade in ultra-small quantum dots  i +U U

11. Avraham Schiller / Seattle 09 U Lead Coulomb blockade in ultra-small quantum dots

12. Avraham Schiller / Seattle 09 U Lead Conductance vs gate voltage Coulomb blockade in ultra-small quantum dots

13. Avraham Schiller / Seattle 09 U Lead Conductance vs gate voltage Coulomb blockade in ultra-small quantum dots

14. Avraham Schiller / Seattle 09 U Lead Conductance vs gate voltage dI/dV (e 2 /h) Coulomb blockade in ultra-small quantum dots

15. Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots

16. Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots

17. Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots Tunneling to leads

18. Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots Inter-configurational energies  d and U+  d

19. Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots Inter-configurational energies  d and U+  d

20. Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots Inter-configurational energies  d and U+  d

21. Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots Inter-configurational energies  d and U+  d Hybridization width

22. Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots Inter-configurational energies  d and U+  d Hybridization width

23. Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots Inter-configurational energies  d and U+  d Condition for formation of local moment: Hybridization width

24. Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots Inter-configurational energies  d and U+  d Condition for formation of local moment: Hybridization width

25. Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots

26. Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots EFEF dd  d +U TKTK

27. Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots EFEF dd  d +U TKTK A sharp resonance of width T K develops at E F when T<T K

28. Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots EFEF dd  d +U Abrikosov-Suhl resonance TKTK A sharp resonance of width T K develops at E F when T<T K

29. Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots EFEF dd  d +U TKTK A sharp resonance of width T K develops at E F when T<T K Unitary scattering for T=0 and =1

30. Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots EFEF dd  d +U TKTK A sharp resonance of width T K develops at E F when T<T K Unitary scattering for T=0 and =1 Nonperturbative scale:

31. Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots EFEF dd  d +U TKTK A sharp resonance of width T K develops at E F when T<T K Unitary scattering for T=0 and =1 Nonperturbative scale: Perfect transmission for symmetric structure

32. Avraham Schiller / Seattle 09 Electronic correlations out of equilibrium

33. Avraham Schiller / Seattle 09 Electronic correlations out of equilibrium dI/dV (e 2 /h) Differential conductance in two-terminal devices Steady state van der Wiel et al.,Science 2000

34. Avraham Schiller / Seattle 09 Electronic correlations out of equilibrium dI/dV (e 2 /h) Differential conductance in two-terminal devices Steady state ac drive Photon-assisted side peaks Kogan et al.,Science 2004van der Wiel et al.,Science 2000

35. Avraham Schiller / Seattle 09 Electronic correlations out of equilibrium dI/dV (e 2 /h) Differential conductance in two-terminal devices Steady state ac drive Photon-assisted side peaks Kogan et al.,Science 2004van der Wiel et al.,Science 2000  

36. Avraham Schiller / Seattle 09 Nonequilibrium: A theoretical challenge

37. Avraham Schiller / Seattle 09 Nonequilibrium: A theoretical challenge The Goal: The description of nano-structures at nonzero bias and/or nonzero driving fields

38. Avraham Schiller / Seattle 09 Nonequilibrium: A theoretical challenge The Goal: The description of nano-structures at nonzero bias Required: Inherently nonperturbative treatment of nonequilibrium and/or nonzero driving fields

39. Avraham Schiller / Seattle 09 Nonequilibrium: A theoretical challenge The Goal: The description of nano-structures at nonzero bias Required: Inherently nonperturbative treatment of nonequilibrium and/or nonzero driving fields Problem: Unlike equilibrium conditions, density operator is not known in the presence of interactions

40. Avraham Schiller / Seattle 09 Nonequilibrium: A theoretical challenge The Goal: The description of nano-structures at nonzero bias Required: Inherently nonperturbative treatment of nonequilibrium and/or nonzero driving fields Problem: Unlike equilibrium conditions, density operator is not Most nonperturbative approaches available in equilibrium known in the presence of interactions are simply inadequate

41. Avraham Schiller / Seattle 09 Nonequilibrium: A theoretical challenge Two possible strategies Work directly at steady state e.g., construct the many- particle Scattering states Evolve the system in time to reach steady state

42. Avraham Schiller / Seattle 09 Time-dependent numerical RG

43. Avraham Schiller / Seattle 09 Time-dependent numerical RG Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which a sudden perturbation is applied at time t = 0

44. Avraham Schiller / Seattle 09 Time-dependent numerical RG Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which a sudden perturbation is applied at time t = 0 Lead VgVg t < 0

45. Avraham Schiller / Seattle 09 Time-dependent numerical RG Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which a sudden perturbation is applied at time t = 0 Lead VgVg t > 0 Lead VgVg t < 0

46. Avraham Schiller / Seattle 09 Time-dependent numerical RG Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which a sudden perturbation is applied at time t = 0

47. Avraham Schiller / Seattle 09 Time-dependent numerical RG Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which a sudden perturbation is applied at time t = 0 Perturbed Hamiltonian Initial density operator

48. Avraham Schiller / Seattle 09 Wilson’s numerical RG

49. Avraham Schiller / Seattle 09 Wilson’s numerical RG -1 1 --1--1 --2--2 --3--3 -1-1 -2-2 -3-3 /D/D Logarithmic discretization of band:

50. Avraham Schiller / Seattle 09 Wilson’s numerical RG -1 1 --1--1 --2--2 --3--3 -1-1 -2-2 -3-3 /D/D Logarithmic discretization of band:     imp After a unitary transformation the bath is represented by a semi-infinite chain

51. Avraham Schiller / Seattle 09 Why logarithmic discretization? Wilson’s numerical RG

52. Avraham Schiller / Seattle 09 Why logarithmic discretization? Wilson’s numerical RG To properly account for the logarithmic infra-red divergences

53. Avraham Schiller / Seattle 09 Why logarithmic discretization? Wilson’s numerical RG To properly account for the logarithmic infra-red divergences     imp Hopping decays exponentially along the chain:

54. Avraham Schiller / Seattle 09 Why logarithmic discretization? Wilson’s numerical RG     imp Hopping decays exponentially along the chain: Separation of energy scales along the chain To properly account for the logarithmic infra-red divergences

55. Avraham Schiller / Seattle 09 Why logarithmic discretization? Wilson’s numerical RG     imp Hopping decays exponentially along the chain: Exponentially small energy scales can be accessed, limited by T only To properly account for the logarithmic infra-red divergences Separation of energy scales along the chain

56. Avraham Schiller / Seattle 09 Why logarithmic discretization? Wilson’s numerical RG     imp Hopping decays exponentially along the chain: Iterative solution, starting from a core cluster and enlarging the chain by one site at a time. High-energy states are discarded at each step, refining the resolution as energy is decreased. To properly account for the logarithmic infra-red divergences Exponentially small energy scales can be accessed, limited by T only Separation of energy scales along the chain

57. Avraham Schiller / Seattle 09 Equilibrium NRG: Geared towards fine energy resolution at low energies Discards high-energy states Wilson’s numerical RG

58. Avraham Schiller / Seattle 09 Equilibrium NRG: Problem: Real-time dynamics involves all energy scales Geared towards fine energy resolution at low energies Discards high-energy states Wilson’s numerical RG

59. Avraham Schiller / Seattle 09 Equilibrium NRG: Problem: Real-time dynamics involves all energy scales Resolution: Combine information from all NRG iterations Geared towards fine energy resolution at low energies Discards high-energy states Wilson’s numerical RG

60. Avraham Schiller / Seattle 09 Time-dependent NRG   imp Basis set for the “environment” statesNRG eigenstate of relevant iteration

61. Avraham Schiller / Seattle 09 Time-dependent NRG   imp Basis set for the “environment” statesNRG eigenstate of relevant iteration For each NRG iteration, we trace over its “environment”

62. Avraham Schiller / Seattle 09 Time-dependent NRG Sum over discarded NRG states of chain of length m Matrix element of O on the m-site chain Reduced density matrix for the m-site chain (Hostetter, PRL 2000) Sum over all chain lengths (all energy scales) Trace over the environment, i.e., sites not included in chain of length m

63. Avraham Schiller / Seattle 09 Fermionic benchmark: Resonant-level model

64. Avraham Schiller / Seattle 09 Fermionic benchmark: Resonant-level model

65. Avraham Schiller / Seattle 09 Fermionic benchmark: Resonant-level model We focus on and compare the TD-NRG to exact analytic solution in the wide-band limit (for an infinite system) Basic energy scale:

66. Avraham Schiller / Seattle 09 Fermionic benchmark: Resonant-level model T = 0 Relaxed values (no runaway!)

67. Avraham Schiller / Seattle 09 Fermionic benchmark: Resonant-level model T = 0 T > 0 Relaxed values (no runaway!)

68. Avraham Schiller / Seattle 09 Fermionic benchmark: Resonant-level model T = 0 T > 0 Relaxed values (no runaway!) The deviation of the relaxed T=0 value from the new thermodynamic value is a measure for the accuracy of the TD-NRG on all time scales For T > 0, the TD-NRG works well up to

69. Avraham Schiller / Seattle 09 T = 0 E d (t 0) =   = 2 Source of inaccuracies

70. Avraham Schiller / Seattle 09 T = 0 E d (t 0) =   = 2 Source of inaccuracies

71. Avraham Schiller / Seattle 09 T = 0 E d (t 0) =   = 2 Source of inaccuracies

72. Avraham Schiller / Seattle 09 T = 0 E d (t 0) =   = 2 TD-NRG is essentially exact on the Wilson chain Source of inaccuracies Main source of inaccuracies is due to discretization

73. Avraham Schiller / Seattle 09 Analysis of discretization effects E d (t 0) = 

74. Avraham Schiller / Seattle 09 Analysis of discretization effects E d (t 0) =  E d (t 0) = -10 

75. Avraham Schiller / Seattle 09 Bosonic benchmark: Spin-boson model

76. Avraham Schiller / Seattle 09 Bosonic benchmark: Spin-boson model Setting  =0, we start from the pure spin state and compute

77. Avraham Schiller / Seattle 09 Bosonic benchmark: Spin-boson model Excellent agreement between TD-NRG (full lines) and the exact analytic solution (dashed lines) up to

78. Avraham Schiller / Seattle 09 Bosonic benchmark: Spin-boson model For nonzero  and s = 1 (Ohmic bath), we prepare the system such that the spin is initially fully polarized (S z = 1/2)

79. Avraham Schiller / Seattle 09 Bosonic benchmark: Spin-boson model For nonzero  and s = 1 (Ohmic bath), we prepare the system such that the spin is initially fully polarized (S z = 1/2) Damped oscillations

80. Avraham Schiller / Seattle 09 Bosonic benchmark: Spin-boson model For nonzero  and s = 1 (Ohmic bath), we prepare the system such that the spin is initially fully polarized (S z = 1/2) Monotonic decay

81. Avraham Schiller / Seattle 09 Bosonic benchmark: Spin-boson model For nonzero  and s = 1 (Ohmic bath), we prepare the system such that the spin is initially fully polarized (S z = 1/2) Localized phase

82. Avraham Schiller / Seattle 09 Anderson impurity model t < 0t > 0

83. Avraham Schiller / Seattle 09 Anderson impurity model: Charge relaxation Charge relaxation is governed by t ch =1/  1 TD-NRG works better for interacting case! Exact new Equilibrium values

84. Avraham Schiller / Seattle 09 Anderson impurity model: Spin relaxation

85. Avraham Schiller / Seattle 09 Anderson impurity model: Spin relaxation

86. Avraham Schiller / Seattle 09 Anderson impurity model: Spin relaxation Spin relaxes on a much longer time scale Spin relaxation is sensitive to initial conditions! Starting from a decoupled impurity, spin relaxation approaches a universal function of t/t sp with t sp =1/T K

87. Avraham Schiller / Seattle 09 Conclusions A numerical RG approach was devised to track the real-time dynamics of quantum impurities following a sudden perturbation Works well for arbitrarily long times up to 1/T Applicable to fermionic as well as bosonic baths For ultra-small dots, spin and charge typically relax on different time scales