Spatial and temporal propagation of Kondo correlations. Frithjof B. Anders Lehrstuhl für Theoretische Physik II - Technische Universität Dortmund

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1 Spatial and temporal propagation of Kondo correlations Frithjof B. Anders Lehrstuhl für Theoretische Physik II - Technische Universität Dortmund

2 Collaborators

3 Collaborators Benedikt Lechtenberg

4 Collaborators Benedikt Lechtenberg Fabian Güttge

5 Collaborators Benedikt Lechtenberg Avi Schiller Fabian Güttge Time-dependent NRG FBA, A. Schiller PRL 95, (2005), PRB 74, (2006)

6 Collaborators Benedikt Lechtenberg Fabian Güttge Time-dependent NRG FBA, A. Schiller PRL 95, (2005), PRB 74, (2006) Avi Schiller ( )

7 Outline 1. Kondo model spatial correlation: length scale ξk (R, t) =h ~ S imp ~s c (~r)i(t) 2. Real-time dynamics of quantum impurity systems TD numerical renormalization group 3. Mapping on a two-impurity problem for each distance R 4. Results equilibrium: Kondo cloud propagation of spin correlations 5. Conclusion

8 1. Spatial Kondo correlations

9 Kondo model: drosophila of solid state theory host H = X Z D D d""c " c " J. Kondo, Prog. Theor. Phys 32, 37 (1964)

10 Kondo model: drosophila of solid state theory host ~s c (~r) H = X Z D D d""c " c " J. Kondo, Prog. Theor. Phys 32, 37 (1964)

11 Kondo model: drosophila of solid state theory impurity spin host ~s c (~r) H = X Z D D d""c " c " J. Kondo, Prog. Theor. Phys 32, 37 (1964)

12 Kondo model: drosophila of solid state theory impurity spin host ~s c (~r) H = X Z D D d""c " c " J. Kondo, Prog. Theor. Phys 32, 37 (1964)

13 Kondo model: drosophila of solid state theory impurity spin J host ~s c (~r) H = X Z D D d""c " c " + J ~ S imp ~s c (0) J. Kondo, Prog. Theor. Phys 32, 37 (1964)

14 Kondo model: drosophila of solid state theory impurity spin J host ~s c (~r) H = X Z D D d""c " c " + J ~ S imp ~s c (0) J. Kondo, Prog. Theor. Phys 32, 37 (1964)

15 Kondo model: drosophila of solid state theory impurity spin J host distance R ~s c (~r) H = X Z D D d""c " c " + J ~ S imp ~s c (0) J. Kondo, Prog. Theor. Phys 32, 37 (1964)

16 Kondo model: drosophila of solid state theory impurity spin correlation function: (R, t) =h ~ S imp ~s c (~r)i(t) J host distance R ~s c (~r) H = X Z D D d""c " c " + J ~ S imp ~s c (0) J. Kondo, Prog. Theor. Phys 32, 37 (1964)

17 Kondo model: drosophila of solid state theory impurity spin correlation function: (R, t) =h ~ S imp ~s c (~r)i(t) J equilibrium: 1(R) = lim t!1 (R, t) host distance R ~s c (~r) H = X Z D D d""c " c " + J ~ S imp ~s c (0) J. Kondo, Prog. Theor. Phys 32, 37 (1964)

18 Kondo model: drosophila of solid state theory impurity spin J host ~s c (~r)

19 Kondo model: drosophila of solid state theory impurity spin J host ~s c (~r) experimental evidence

20 Kondo model: drosophila of solid state theory impurity spin J host ~s c (~r) experimental evidence Fe in Cu de Haas, de Boer, van den Berg! Physica 1,1115 (1934)

21 Kondo model: drosophila of solid state theory impurity spin J host ~s c (~r) experimental evidence Fe in Cu Kondo scale TK T K = De 1 J de Haas, de Boer, van den Berg! Physica 1,1115 (1934)

22 Kondo model: drosophila of solid state theory impurity spin J host ~s c (~r) experimental evidence Heavy Fermions Fe in Cu Kondo scale TK T K = De 1 J de Haas, de Boer, van den Berg! Physica 1,1115 (1934) Onuki et al (1987)

23 Kondo model: drosophila of solid state theory impurity spin J host ~s c (~r) experimental evidence Heavy Fermions G(V) in Ta-I-Al Kondo scale TK T K = De 1 J Wyatt, PRL 13,401 (1964) Onuki et al (1987)

24 Kondo model: drosophila of solid state theory impurity spin host J ~s c (~r) Quantum-Mirage:! Co on Cu experimental evidence G(V) in Ta-I-Al Kondo scale TK T K = De 1 J Wyatt, PRL 13,401 (1964) Manoharan et al,! Nature 403, 512 (2000)

25 Kondo model: drosophila of solid state theory T= : free spin K. Wilson, RMP 1975

26 Kondo model: drosophila of solid state theory T= : free spin T=0: Kondo singlet K. Wilson, RMP 1975

27 Kondo model: drosophila of solid state theory T= : free spin crossover scale TK T=0: Kondo singlet K. Wilson, RMP 1975

28 Kondo model: drosophila of solid state theory T= : free spin two length scales: crossover scale TK T=0: Kondo singlet K. Wilson, RMP 1975

29 Kondo model: drosophila of solid state theory T= : free spin crossover scale TK T=0: Kondo singlet two length scales: Fermi wave length : 1/k F K. Wilson, RMP 1975

30 Kondo model: drosophila of solid state theory T= : free spin crossover scale TK T=0: Kondo singlet two length scales: Fermi wave length : 1/k F Kondo length: ξk = 1/TK ξk K. Wilson, RMP 1975

31 Kondo model: drosophila of solid state theory T= : free spin crossover scale TK T=0: Kondo singlet two length scales: Fermi wave length : 1/k F Kondo length: ξk = 1/TK ξk K. Wilson, RMP 1975

32 Kondo model: drosophila of solid state theory T= : free spin crossover scale TK T=0: Kondo singlet two length scales: Fermi wave length : 1/k F Kondo length: ξk = 1/TK ξk K. Wilson, RMP 1975

33 Kondo model: drosophila of solid state theory T= : free spin crossover scale TK T=0: Kondo singlet two length scales: Fermi wave length : 1/k F Kondo length: ξk = 1/TK ξk RKKY interaction K. Wilson, RMP 1975

34 Kondo model: drosophila of solid state theory T= : free spin crossover scale TK T=0: Kondo singlet two length scales: Fermi wave length : 1/k F Kondo length: ξk = 1/TK ξk KRKKY cos(2kfr)/r d RKKY interaction K. Wilson, RMP 1975

35 spatial correlations in 1D Borda PRB 2007

36 spatial correlations in 1D Borda PRB 2007 numerical renormalization group (NRG) calculations

37 spatial correlations in 1D h ~ SImp~s (R)i Borda PRB J =0.1 0 J = J =0.4 0 J =0.5 0 J = k F R/ Lechtenberg, FBA, arxiv: numerical renormalization group (NRG) calculations

38 spatial correlations in 1D h ~ SImp~s (R)i Borda PRB 2007 ferro and antiferromagnetic correlations J =0.1 0 J = J =0.4 0 J =0.5 0 J = k F R/ Lechtenberg, FBA, arxiv: numerical renormalization group (NRG) calculations

39 spatial correlations in 1D h ~ SImp~s (R)i Borda PRB 2007 ferro and antiferromagnetic correlations Z 1 exact sum rule: J =0.1 0 J = J =0.4 0 J =0.5 0 J = drr d 1 1(R) = 3 4 numerical renormalization group (NRG) calculations k F R/ Lechtenberg, FBA, arxiv:

40 spatial correlations in 1D χ (R)Vu(kFR/π) ρ 0 J =0.05 ρ 0 J =0.075 ρ 0 J =0.1 K RKKY in a.u k F R/π R< ξk Affleck et al Borda 2007 Lechtenberg, FBA, arxiv:

41 spatial correlations in 1D χ (R)Vu(kFR/π) ρ 0 J =0.05 ρ 0 J =0.075 ρ 0 J =0.1 K RKKY in a.u k F R/π χ (R)Vu(kFR/π) 2 (ρj) ρ 0 J χ (0)Vu k F R/π ρ 0 J =0.5 ρ 0 J =0.6 ρ 0 J =0.7 R< ξk ξk<r Affleck et al Borda 2007 Lechtenberg, FBA, arxiv:

42 spatial correlations in 1D χ (R)Vu(kFR/π) ρ 0 J =0.05 ρ 0 J =0.075 ρ 0 J =0.1 K RKKY in a.u k F R/π χ (R)Vu(kFR/π) 2 (ρj) ρ 0 J χ (0)Vu k F R/π ρ 0 J =0.5 ρ 0 J =0.6 ρ 0 J =0.7 R< ξk ξk<r crossover from 1/R d to 1/R d+1 around ξk Affleck et al Borda 2007 Lechtenberg, FBA, arxiv:

43 spatial correlations in 1D χ (R)Vu(kFR/π) ρ 0 J =0.05 ρ 0 J =0.075 ρ 0 J =0.1 K RKKY in a.u k F R/π χ (R)Vu(kFR/π) 2 (ρj) ρ 0 J χ (0)Vu k F R/π ρ 0 J =0.5 ρ 0 J =0.6 ρ 0 J =0.7 R< ξk ξk<r crossover from 1/R d to 1/R d+1 around ξk correlation follow the RKKY interaction Affleck et al Borda 2007 Lechtenberg, FBA, arxiv:

44 2. Real-time dynamics in quantum impurity systems FBA, A. Schiller, PRL 95, (2005) FBA, A. Schiller, PRB 74, (2006)

45 Time-evolution of states i t (t) = H (t)

46 Time-evolution of states i t (t) = H (t) (t) = e iht 0 H=const.

47 Time-evolution of states i t (t) = H (t) (t) = e iht 0 H=const. eigenstates of H: H n = E n n

48 Time-evolution of states i t (t) = H (t) (t) = e iht 0 H=const. eigenstates of H: H n = E n n time evolution: (t) = n e ie nt c n n

49 Time-evolution of states i t (t) = H (t) (t) = e iht 0 H=const. eigenstates of H: H n = E n n time evolution: (t) = n e ie nt c n n initial conditions: c n = n 0

50 Time-evolution of states i t (t) = H (t) (t) = e iht 0 H=const. eigenstates of H: H n = E n n time evolution: (t) = n e ie nt c n n initial conditions: dynamics c n = n 0 e ie nt

51 Time-evolution of states i t (t) = H (t) (t) = e iht 0 H=const. eigenstates of H: H n = E n n time evolution: (t) = n e ie nt c n n initial conditions: dynamics c n = n 0 e ie nt Can we calculate such a complete basis set?

52 Non-equilibrium dynamics quantum impurity systems System

53 Non-equilibrium dynamics quantum impurity systems system: small finite number of DOF System

54 Non-equilibrium dynamics quantum impurity systems system: small finite number of DOF System environment: infinitely large Bosonic or Fermionic baths environment

55 Non-equilibrium dynamics quantum impurity systems system: small finite number of DOF System environment: infinitely large Bosonic or Fermionic baths environment coupling between system and bath: entanglement Kondo effect decoherence

56 Wilson s numerical renormalisation group Impurity bath/environment: continuum

57 Wilson s numerical renormalisation group Impurity bath/environment: continuum quantum impurity model impurity coupled to a energy-continuum

58 Wilson s numerical renormalisation group t m m/2 Impurity reason: Kondo logarithm contribute equal at each interval I n = Z n (n+1) d" " = log( )

59 Wilson s numerical renormalisation group t m m/2 Impurity discretization of the energy-continuum reason: Kondo logarithm contribute equal at each interval I n = Z n (n+1) d" " = log( )

60 Wilson s numerical renormalisation group t m m/2 Impurity discretization of the energy-continuum separation of energy scales: tm Λ -m/2 reason: Kondo logarithm contribute equal at each interval I n = Z n (n+1) d" " = log( )

61 Wilson s numerical renormalisation group t m m/2 Impurity iterative diagonalization

62 Wilson s numerical renormalisation group t m m/2 Impurity iterative diagonalization: approx. eigenbasis

63 Wilson s numerical renormalisation group t m m/2 Impurity iterative diagonalization: approx. eigenbasis

64 Wilson s numerical renormalisation group t m m/2 Impurity iterative diagonalization: approx. eigenbasis

65 Wilson s numerical renormalisation group t m m/2 Impurity iterative diagonalization: approx. eigenbasis

66 Wilson s numerical renormalisation group t m m/2 Impurity iterative diagonalization: approx. eigenbasis kept states discarded states

67 Wilson s numerical renormalisation group t m m/2 Impurity iterative diagonalization: approx. eigenbasis kept states discarded states RG: discarding + rescaling

68 Wilson s numerical renormalisation group t m m/2 Impurity iterative diagonalization: approx. eigenbasis kept states discarded states RG: discarding + rescaling

69 Wilson s numerical renormalisation group t m m/2 Impurity iterative diagonalization: approx. eigenbasis kept states discarded states RG: discarding + rescaling

70 Wilson s numerical renormalisation group t m m/2 Impurity iterative diagonalization: approx. eigenbasis kept states discarded states RG: discarding + rescaling

71 complete basis set FBA, A. Schiller, PRL 95, (2005) FBA, A. Schiller, PRB 74, (2006)

72 complete basis set the sum of all discarded high-energy states in the NRG iteration form a complete basis set { l,e;m>} ˆ1 = NX X m l e X l, e; mihl, e; m FBA, A. Schiller, PRL 95, (2005) FBA, A. Schiller, PRB 74, (2006)

73 complete basis set the sum of all discarded high-energy states in the NRG iteration form a complete basis set { l,e;m>} ˆ1 = NX X m l e an approximate eigenbasis X l, e; mihl, e; m H l, e; mi E l l, e; mi FBA, A. Schiller, PRL 95, (2005) FBA, A. Schiller, PRB 74, (2006)

74 Real-time evolution of observables Ô (t) = m l or l discarded l,l l Ô l e i(e l E l )t red l l (m) FBA, A. Schiller, PRL 95, (2005) FBA, A. Schiller, PRB 74, (2006)

75 Real-time evolution of observables Ô (t) = m l or l discarded l,l l Ô l e i(e l E l )t red l l (m) reduced density matrix: red ll (m) = contains information on dissipation entanglement with the environment e l, e; m ˆ0 l, e; m FBA, A. Schiller, PRL 95, (2005) FBA, A. Schiller, PRB 74, (2006)

76 Real-time evolution of observables Ô (t) = m l or l discarded l,l l Ô l e i(e l E l )t red l l (m) reduced density matrix: red ll (m) = contains information on dissipation entanglement with the environment e l, e; m ˆ0 l, e; m RG concept upside down: discarded states contain information on the real-time evolution FBA, A. Schiller, PRL 95, (2005) FBA, A. Schiller, PRB 74, (2006)

77 spin-boson model: decoherence by dephasing time-depended NRG ρ 01 (t)/ρ 01 (0) 1 0,8 0,6 0,4 s=1.5 s=1.0 s=0.8 s=0.6 s=0.4 s=0.2 0,2 0 0,001 0,01 0, t*t X H = x + z 2 q q(b q + b q )+ X q b qb q q Δ=0 FBA, A. Schiller, PRL 95, (2005) FBA, A. Schiller, PRB 74, (2006)

78 spin-boson model: decoherence by dephasing time-depended NRG plus analytic solution ρ 01 (t)/ρ 01 (0) s=1.5 s=1.0 s=0.8 s=0.6 s=0.4 s= t*t exact solution and TD-NRG: excellent agreement FBA und A. Schiller, PRB 74, (2006)

79 3. Spatial correlations: mapping to a two-impurity Kondo Problem

80 mapping of the Kondo problem substrate

81 mapping of the Kondo problem impurity spin substrate -R/2

82 mapping of the Kondo problem impurity spin substrate ~s c (~r) -R/2 R/2

83 mapping of the Kondo problem impurity spin substrate ~s c (~r) -R/2 R/2 correlation function: (R, t) =h ~ S imp ~s c (~r)i(t)

84 mapping of the Kondo problem impurity spin substrate ~s c (~r) -R/2 R/2 correlation function: (R, t) =h ~ S imp ~s c (~r)i(t) Borda, PRB 2007, Jones, Varma PRB 1987, Lechtenberg, FBA 2014

85 mapping of the Kondo problem impurity spin substrate ~s c (~r) -R/2 R/2 correlation function: (R, t) =h ~ S imp ~s c (~r)i(t) mirror symmetry: even and odd parity! Borda, PRB 2007, Jones, Varma PRB 1987, Lechtenberg, FBA 2014

86 mapping of the Kondo problem impurity spin substrate ~s c (~r) -R/2 R/2 correlation function: (R, t) =h ~ S imp ~s c (~r)i(t) mirror symmetry: even and odd parity! two band, two impurity model Borda, PRB 2007, Jones, Varma PRB 1987, Lechtenberg, FBA 2014

87 mapping of the Kondo problem impurity spin substrate ~s c (~r) -R/2 R/2 correlation function: e(") = o(") = 1 p 2Ne (") 1 p 2No (") (R, t) =h ~ S imp ~s c (~r)i(t) R ( ~! R 2 )+ ( ~ 2 ) ( ~ R 2 ) ( ~ R 2 )! two band, two impurity model Borda, PRB 2007, Jones, Varma PRB 1987, Lechtenberg, FBA 2014

88 mapping of the Kondo problem impurity spin substrate ~s c (~r) H = X X + J 8 =e,o Z D D d""c ", c ", X Ne (R)f 0,e + N o (R)f 0,o 0 X [~ ] 0 Ne (R)f 0,e + N o (R)f 0,o ~ Simp Borda, PRB 2007, Jones, Varma PRB 1987, Lechtenberg, FBA 2014

89 mapping of the Kondo problem impurity spin substrate ~s c (~r) H = X X + J 8 =e,o ~s( ~ R 2 )= 1 8V u X Z D D d""c ", c ", X Ne (R)f 0,e + N o (R)f 0,o [~ ] 0 Ne (R)f 0,e + N o (R)f 0,o Simp ~ 0 0 Ne (R)f 0,e N o (R)f 0,o [~ ] 0 Ne (R)f 0,e No (R)f 0,o Borda, PRB 2007, Jones, Varma PRB 1987, Lechtenberg, FBA 2014

90 mapping of the Kondo problem impurity spin substrate ~s c (~r) H = X X + J 8 =e,o ~s( ~ R 2 )= 1 8V u X Z D D d""c ", c ", X Ne (R)f 0,e + N o (R)f 0,o [~ ] 0 Ne (R)f 0,e + N o (R)f 0,o Simp ~ 0 0 Ne (R)f 0,e N o (R)f 0,o [~ ] 0 Ne (R)f 0,e No (R)f 0,o for each distance R one independent two band NRG calculation Borda, PRB 2007, Jones, Varma PRB 1987, Lechtenberg, FBA 2014

91 4. Results

92 equilibrium: antiferromagnetic coupling χ (R)Vu(kFR/π) ρ 0 J =0.05 ρ 0 J =0.075 ρ 0 J =0.1 K RKKY in a.u k F R/π 1D: Lechtenberg, FBA arxiv:

93 equilibrium: antiferromagnetic coupling χ (R)Vu(kFR/π) ρ 0 J =0.05 ρ 0 J =0.075 ρ 0 J =0.1 K RKKY in a.u k F R/π 1D: 1/R Lechtenberg, FBA arxiv:

94 equilibrium: antiferromagnetic coupling χ (R)Vu(kFR/π) ρ 0 J =0.05 ρ 0 J =0.075 ρ 0 J =0.1 K RKKY in a.u k F R/π χ (R)Vu(kFR/π) k F R/π ρj =0.05 ρj =0.1 ρj =0.15 RKKY in a.u. 1D: 1/R 2D: Lechtenberg, FBA arxiv:

95 equilibrium: antiferromagnetic coupling χ (R)Vu(kFR/π) ρ 0 J =0.05 ρ 0 J =0.075 ρ 0 J =0.1 K RKKY in a.u k F R/π χ (R)Vu(kFR/π) k F R/π ρj =0.05 ρj =0.1 ρj =0.15 RKKY in a.u. 1D: 1/R 2D: 1/R 2 Lechtenberg, FBA arxiv:

96 equilibrium: antiferromagnetic coupling χ (R)Vu(kFR/π) ρ 0 J =0.05 ρ 0 J =0.075 ρ 0 J =0.1 K RKKY in a.u k F R/π χ (R)Vu(kFR/π) k F R/π ρj =0.05 ρj =0.1 ρj =0.15 RKKY in a.u. 1D: 1/R 2D: 1/R 2 Z 1 0 drr d 1 1(R) = 3 4 Lechtenberg, FBA arxiv:

97 equilibrium: ferromagnetic coupling χ (R)VukFR/π ρ 0 J = 0.5 ρ 0 J = 0.1 ρ 0 J k F R/π χ (0)Vu Z 1 0 drr d 1 1(R) =0 Lechtenberg, FBA arxiv:

98 equilibrium: ferromagnetic coupling χ (R)VukFR/π ρ 0 J = 0.5 ρ 0 J = 0.1 ρ 0 J k F R/π χ (0)Vu Z 1 0 drr d 1 1(R) =0 Lechtenberg, FBA arxiv:

99 non-equilibrium dynamics (R, t) =h ~ S imp ~s c (~r)i(t) 0 J =0.3 Lechtenberg, FBA arxiv:

100 non-equilibrium dynamics (R, t) =h ~ S imp ~s c (~r)i(t) 0 J =0.3 light-cone physics R=v Ft Lechtenberg, FBA arxiv:

101 non-equilibrium dynamics (R, t) =h ~ S imp ~s c (~r)i(t) 0 J =0.3 light-cone physics R=v Ft surprice: buildup of correlations outside the light-cone Lechtenberg, FBA arxiv:

102 non-equilibrium dynamics (R, t) =h ~ S imp ~s c (~r)i(t) 0 J =0.3 light-cone physics R=v Ft surprice: buildup of correlations outside the light-cone fast short time scale and thermalisation inside the lightcone Lechtenberg, FBA arxiv:

103 perturbation theory Lechtenberg, FBA arxiv:

104 perturbation theory second order perturbation theory no Kondo effect Lechtenberg, FBA arxiv:

105 perturbation theory second order perturbation theory no Kondo effect TD-NRG Kondo effect included Lechtenberg, FBA arxiv:

106 perturbation theory second order perturbation theory no Kondo effect TD-NRG Kondo effect included qualitative very good agreement: correlations outside the light cone persist Lechtenberg, FBA arxiv:

107 correlations outside of the light cone? Medvedyeva et al PRB 2013: intrinsic correlations of the Fermi sea

108 correlations outside of the light cone? Medvedyeva et al PRB 2013: intrinsic correlations of the Fermi sea s(0) s(r) V u 2 (kfr/π) k F R/π NRG theory spatial spin-density correlation function: NRG vs analytic

109 correlations outside of the light cone? Medvedyeva et al PRB 2013: intrinsic correlations of the Fermi sea s(0) s(r) V u 2 (kfr/π) k F R/π NRG theory χ(r, t)vu(kfr) 2 /(tdπ) order 2. order order spatial spin-density correlation function: NRG vs analytic k F R/π perturbation theory

110 correlations outside of the light cone? Medvedyeva et al PRB 2013: intrinsic correlations of the Fermi sea s(0) s(r) V u 2 (kfr/π) k F R/π NRG theory χ(r, t)vu(kfr) 2 /(tdπ) order 2. order order spatial spin-density correlation function: NRG vs analytic k F R/π perturbation theory time scale 1/D

111 correlations outside of the light cone? Medvedyeva et al PRB 2013: intrinsic correlations of the Fermi sea s(0) s(r) V u 2 (kfr/π) k F R/π NRG theory χ(r, t)vu(kfr) 2 /(tdπ) order 2. order order spatial spin-density correlation function: NRG vs analytic k F R/π perturbation theory time scale 1/D

112 correlations outside of the light cone? Medvedyeva et al PRB 2013: intrinsic correlations of the Fermi sea s(0) s(r) V u 2 (kfr/π) k F R/π NRG theory χ(r, t)vu(kfr) 2 /(tdπ) order 2. order order spatial spin-density correlation function: NRG vs analytic k F R/π perturbation theory time scale 1/D maxima and minima coincide: AF coupling

113 ferromagnetic coupling 0 J = 0.1 TD-NRG

114 ferromagnetic coupling 0 J = 0.1 TD-NRG second order perturbation theory

115 conclusion

116 real-time dynamics of the spin-correlation χ(r,t) using the TD-NRG conclusion

117 conclusion real-time dynamics of the spin-correlation χ(r,t) using the TD-NRG equilibrium: improved mapping: AF and FM correlations

118 conclusion real-time dynamics of the spin-correlation χ(r,t) using the TD-NRG equilibrium: improved mapping: AF and FM correlations correct sum-rule

119 conclusion real-time dynamics of the spin-correlation χ(r,t) using the TD-NRG equilibrium: improved mapping: AF and FM correlations correct sum-rule non-equilibrium AF(FM): (anti)ferromagnetic wave propagation, speed vf

120 conclusion real-time dynamics of the spin-correlation χ(r,t) using the TD-NRG equilibrium: improved mapping: AF and FM correlations correct sum-rule non-equilibrium AF(FM): (anti)ferromagnetic wave propagation, speed vf correlations outside the light cone: entanglement of the Fermi sea

121 conclusion real-time dynamics of the spin-correlation χ(r,t) using the TD-NRG equilibrium: improved mapping: AF and FM correlations correct sum-rule non-equilibrium AF(FM): (anti)ferromagnetic wave propagation, speed vf correlations outside the light cone: entanglement of the Fermi sea excellent agreement with perturbation theory on short time scales

122 conclusion real-time dynamics of the spin-correlation χ(r,t) using the TD-NRG equilibrium: improved mapping: AF and FM correlations correct sum-rule non-equilibrium AF(FM): (anti)ferromagnetic wave propagation, speed vf correlations outside the light cone: entanglement of the Fermi sea excellent agreement with perturbation theory on short time scales fast initial time scale: 1/D, thermalisation

123 conclusion real-time dynamics of the spin-correlation χ(r,t) using the TD-NRG equilibrium: improved mapping: AF and FM correlations correct sum-rule non-equilibrium AF(FM): (anti)ferromagnetic wave propagation, speed vf correlations outside the light cone: entanglement of the Fermi sea excellent agreement with perturbation theory on short time scales fast initial time scale: 1/D, thermalisation outlook: hybrid method: NRG+DMRG (Güttge, FBA, Schollwöck, Eidelstein,Schiller, PRB 87, (2013)) pulses: (Eidelstein, Schiller, Güttge, FBA, PRB 85, (2012)) general Hamiltonians by Costi and coworkers (PRB 2014) Thank you for your attention!