The Hubbard model out of equilibrium - Insights from DMFT -
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1 The Hubbard model out of equilibrium - Insights from DMFT - t U Philipp Werner University of Fribourg, Switzerland KITP, October 212
2 The Hubbard model out of equilibrium - Insights from DMFT - In collaboration with: Naoto Tsuji (Fribourg / Tokyo) Takashi Oka, Hideo Aoki (Tokyo University) Martin Eckstein (Hamburg) KITP, October 212
3 Motivation Explore nonequilibrium properties of correlated electron systems Tune material properties by external fields e. g. photo-doping S. Iwai et al. (23), H. Okamoto et al. (27),... Create long-lived transient states with novel properties e. g. light-induced room temperature superconductivity temperature Mott insulator metal superconductor hole doping D. Fausti et al. (21)
4 Motivation Explore nonequilibrium properties of correlated electron systems Tune material properties by external fields e. g. photo-doping S. Iwai et al. (23), H. Okamoto et al. (27),... Create long-lived transient states with novel properties e. g. light-induced room temperature superconductivity temperature Mott insulator metal stripe order superconductor hole doping D. Fausti et al. (21)
5 Model and method Dynamical mean field theory DMFT: mapping to an impurity problem Metzner & Vollhardt (1989); Georges & Kotliar (1992) lattice model impurity model t G latt G imp t k latt imp Impurity solver: computes the dynamics on the correlated site QMC: Werner et al. (29), Perturbation theory: Eckstein et al. (29, 21) Formalism can be extended to nonequilibrium systems Schmidt & Monien (22); Freericks et al. (26)
6 Model and method Dynamical mean field theory DMFT: mapping to an impurity problem Metzner & Vollhardt (1989); Georges & Kotliar (1992) lattice model M latt M eff single-site effective model J h eff Impurity solver: computes the dynamics on the correlated site QMC: Werner et al. (29), Perturbation theory: Eckstein et al. (29, 21) Formalism can be extended to nonequilibrium systems Schmidt & Monien (22); Freericks et al. (26)
7 Model and method Dynamical mean field theory DMFT: mapping to an impurity problem Metzner & Vollhardt (1989); Georges & Kotliar (1992) lattice model impurity model t G latt G imp t k latt imp Impurity solver: computes the dynamics on the correlated site QMC: Werner et al. (29), Perturbation theory: Eckstein et al. (29, 21) Formalism can be extended to nonequilibrium systems Schmidt & Monien (22); Freericks et al. (26)
8 Model and method Muehlbacher & Rabani (28) Werner, Oka & Millis (29) Continuous-time QMC 1 i hoi(t) =Trh Z e H U(,t)OU(t, ) h 1 = Tr Z e H T e R d H I ( ) i Te R t dsh I (s) O(t) Te i R t dsh I (s) i ih I ih I t O i H I ih I H I ih I Expand time evolution operators in powers of H I
9 Model and method Muehlbacher & Rabani (28) Werner, Oka & Millis (29) Continuous-time QMC: weak-coupling formalism 1 i hoi(t) =Trh Z e H U(,t)OU(t, ) h 1 = Tr Z e H T e R d H I ( ) i Te R t dsh I (s) O(t) Te i R t dsh I (s) i i G i t O i 1 1 i i Expand time evolution operators in powers of the interaction term
10 Model and method Georges & Kotliar (1992) Perturbative weak-coupling formalism: Generate a subset of all weakcoupling diagrams by approximating the self-energy Truncation at second order: Iterated Perturbation Theory (IPT) = U t t G = G G = G + G G + + +
11 Model and method Freericks & Turkowski (25) Field E(t) in the body diagonal, applied at t= Choose gauge with pure vector potential: E(t) t A(t) t U Peierls substitution: "(k)! "(k Lattice: hybercubic, infinite-d limit ea(t)) (") = 1 p W exp( " 2 /W 2 )
12 Periodic fields Tsuji, Oka, Werner & Aoki (211) AC-field quench in the Hubbard model Electric field (amplitude E, frequency ) applied at t= t U
13 Periodic fields Tsuji, Oka, Werner & Aoki (211) AC-field quench in the Hubbard model.35.3 E/ =4 E/ =3 attractive (>.25) double occupation E/ =2.5 E/ =1 repulsive (<.25).1 E/ = time [inverse hopping]
14 Periodic fields Tsuji, Oka, Werner & Aoki (211) AC-field quench in the Hubbard model Sign inversion of the interaction: repulsive attractive Dynamically generated high-tc superconductivity? temperature metal superconductor (attractive interaction) superconductor (repulsive interaction) hole doping
15 Origin of the attractive interaction Periodic E-field leads to a population inversion E(t) Gauge with pure vector potential E(t) =E cos( t) = t A(t) A(t) = (E/ ) sin( t) Peierls substitution Renormalized dispersion k = 2 2 / k k A(t) dt k A(t) = J (E/ ) k J(E/ ) E/
16 Origin of the attractive interaction Periodic E-field leads to a population inversion J (E/ )=1 Renormalized dispersion k = 2 2 / dt k A(t) = J (E/ ) k
17 Origin of the attractive interaction Periodic E-field leads to a population inversion J (E/ )=.8 Renormalized dispersion k = 2 2 / dt k A(t) = J (E/ ) k
18 Origin of the attractive interaction Periodic E-field leads to a population inversion J (E/ )=.3 Renormalized dispersion k = 2 2 / dt k A(t) = J (E/ ) k
19 Origin of the attractive interaction Periodic E-field leads to a population inversion J (E/ )=.2 Renormalized dispersion k = 2 2 / dt k A(t) = J (E/ ) k
20 Origin of the attractive interaction Inverted population = negative temperature State with T <, J < T e = U>,T < is equivalent to state with U<,T > T J > / exp = exp 1 T 1 T e " X k " X k J k n k k n k + U X i n i" n i# #! #! + U X n i" n i# J i Effective interaction of the T e > state U e = U J (E/ )
21 Summary I Tsuji, Oka, Werner & Aoki (211) Controlling the Coulomb interaction by ac fields Advantages Interaction continuously tunable U U e = J (E/ ) Reversible Disadvantages Need high frequency ac field (interband transitions?) Effect lasts only during irradiation Strong heating effect
22 Pulsed field Tsuji, Oka, Aoki & Werner (212) Shift the population using an asymmetric mono-cycle pulse e Consider a physical pulse with = Z E(t)dt = Interacting electrons: e 6= (depends sensitively on pulse-shape) By combining fast and slow half-cycle pulses can achieve =, e
23 Pulsed field Tsuji, Oka, Aoki & Werner (212) Shift the population using an asymmetric mono-cycle pulse pulse asymmetry
24 Pulsed field Tsuji, Oka, Aoki & Werner (212) Shift the population using an asymmetric mono-cycle pulse example of a realistic pulse Christoph Hauri (PSI) pulse asymmetry
25 Pulsed field Tsuji, Oka, Aoki & Werner (212) Shift the population using an asymmetric mono-cycle pulse Desired material properties: Metallic system with weak to moderate correlations Single band crossing the Fermi level Large gaps to other bands Desired properties of the field pulse: ~1 fs monocycle pulse peak asymmetry ~7:3 field strength V/m Proposed measurements: Time-resolved ARPES (negative) optical conductivity
26 Pulsed field Tsuji, Oka, Aoki & Werner (212) Shift the population using an asymmetric mono-cycle pulse Potentially interesting material: Sn doped In2O3 Transparent conductor Single s-band crossing the Fermi level band structure by Bernard Delley (PSI)
27 Pulse excited Mott insulator Photo-excitation of carriers across the Mott gap Eckstein & Werner (211) Question: How quickly does the electronic system thermalize? T weakly correlated regime metal insulator Mott regime U
28 Pulse excited Mott insulator Photo-excitation of carriers across the Mott gap Eckstein & Werner (211) Question: How quickly does the electronic system thermalize? pulse form total energy T e 4.1 E(t) gap 2 x gap energy(t) t t
29 Pulse excited Mott insulator Photo-excitation of carriers across the Mott gap Eckstein & Werner (211) Question: How quickly does the electronic system thermalize? thermal value.15 d(t) gap 2 x gap t
30 Pulse excited Mott insulator Photo-excitation of carriers across the Mott gap Eckstein & Werner (211) Question: How quickly does the electronic system thermalize? -1 log 1 d(t)-d(t eff ) U=5 U=3 U=2.5 U=1.5 U= t
31 Pulse excited Mott insulator Photo-excitation of carriers across the Mott gap Eckstein & Werner (211) Strong correlation regime: Relaxation time depends exponentially on U -1 3 log 1 d(t)-d(t eff ) U=5 U=3 U=2.5 log 1 relax U=1.5 U= t U
32 Pulse excited Mott insulator Photo-excitation of carriers across the Mott gap Eckstein & Werner (211) Strong correlation regime: Relaxation time depends exponentially on U -1 3 log 1 d(t)-d(t eff ) U=5 U=3 U=2.5 log 1 relax U=1.5 U= Different t relaxation pathway at intermediate U! U
33 Nonthermal symmetry-broken states Photo-doped antiferromagnetic Mott insulator (U-quench) Werner, Tsuji & Eckstein (212) U-quench into the strongly correlated regime freezes doublons / holes.25 U =8: T e 1.2 T quench PM AFM U
34 Nonthermal symmetry-broken states Photo-doped antiferromagnetic Mott insulator (U-quench) Werner, Tsuji & Eckstein (212) Magnetization does not vanish, even if thermal state PM 1.8 magnetization trapped state u=4 6 u=4 7 u= thermal value t
35 Nonthermal symmetry-broken states Photo-doped antiferromagnetic Mott insulator (U-quench) Werner, Tsuji & Eckstein (212) Trapped state similar to chemically doped Mott insulator A(,t) t= t=6 t=12 t=18 thermal trapped timed-dependent spectral function (minority spin)
36 Nonthermal symmetry-broken states Photo-doped antiferromagnetic Mott insulator (U-quench) Werner, Tsuji & Eckstein (212) Trapped state similar to chemically doped Mott insulator Interpretation: Trapped state is a t-j model state with fixed doublons / holes This state is protected by the slow decay of doublons Effective temperature below the Neel temperature of the t-j state entropy cooling due to AFM background
37 Nonthermal symmetry-broken states Photo-doped antiferromagnetic Mott insulator (U-quench) Werner, Tsuji & Eckstein (212) Cooling effect is evident from the time-evolution of the occupation A < (, t) t=2 t=4 t=6 t=8 t=1 initial reduction of the magnetization (relaxation to trapped state) occupied part of the minority-spin spectral function doped carriers
38 Nonthermal symmetry-broken states Photo-doped antiferromagnetic Mott insulator (U-quench) Werner, Tsuji & Eckstein (212) Cooling effect is evident from the time-evolution of the occupation A < (, t) AFM, t=4 t=6 t=8 t=1 PM, t=4 t=6 t=8 t=1 paramagnetic case: occupation almost time-independent antiferromagnet: occupation relaxes and accumulates at lower band edge
39 Nonthermal symmetry-broken states Real photo-doping simulation pulse 12 work in progress Cooling effect is evident from the time-evolution of the occupation t=3 t=6 t=9 t=12... A < (,t) antiferromagnet: occupation relaxes and accumulates at lower band edge
40 Nonthermal symmetry-broken states Weak-coupling regime Tsuji, Eckstein & Werner (212) Slow ramp from (Slater-)Antiferromagnet to Paramagnet
41 Nonthermal symmetry-broken states Weak-coupling regime Tsuji, Eckstein & Werner (212) Time-evolution of the magnetization for different final U arrows indicate thermal magnetization U=1.4: Oscillations around a nonthermal value (thermal magnetization=)
42 Nonthermal symmetry-broken states Weak-coupling regime Tsuji, Eckstein & Werner (212) Evidence for a nonthermal critical point (GL-description fails) diverging timescales (period of amplitude mode, dephasing time,...)
43 Summary Nonequilibrium dynamical mean field results for Hubbard model Metallic system: Population inversion by an asymmetric mono-cycle pulse Interaction conversion: U! U Dynamically generated superconductivity? Antiferromagnetic insulator: Nonthermal symmetry-broken states Thermalization delayed by slow decay of doublons Similar effect expected in superconductors experiment by Fausti et al.? Trapped states also in the weak-coupling regime Short-time dynamics controlled by nonthermal critical points
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