Prethermalization and nonthermal fixed point in the Hubbard model 8 - PowerPoint PPT Presentation

Prethermalization and nonthermal fixed point in the Hubbard model 8 Dec 2013 @ Kyoto Naoto Tsuji (University of Tokyo)

Acknowledgments University of Fribourg University of Tokyo Philipp Werner Hideo Aoki Takashi Oka University of Hamburg-CFEL University of Augsburg Martin Eckstein Marcus Kollar University of Geneva Peter Barmettler This talk is based on: Tsuji, Eckstein, Werner, Phys. Rev. Lett. 110, 136404 (2013). Tsuji, Werner, Phys. Rev. B 88, 125126 (2013). Werner, Tsuji, Eckstein, Phys. Rev. B 86, 205101 (2012). Aoki, Tsuji, Eckstein, Kollar, Oka, Werner, arXiv:1310.5329 (review). Tsuji, Barmettler, Aoki, Werner, arXiv:1307.5946.

Outline “How” and “in which time scale” does an isolated quantum many-body system thermalize? 1. Prethermalization in the Hubbard model (overview) 2. Nonthermal fixed point in the Hubbard model 3. Universality of the nonthermal fixed point

Hubbard model J i j c † c † i ↑ c i ↑ c † � � H ( t ) = i σ c j σ + U i ↓ c i ↓ i j σ i ➤ DMFT (d= ∞ ) phase diagram at half filling. 0.30 0.30 paramagnetic phase 0.25 0.25 0.20 0.20 J i j insulator 0.15 0.15 T T metal 0.10 0.10 U antiferromagnetic Mott transition 0.05 0.05 phase 0.00 0.00 0 0 2 2 4 4 6 6 8 8 U U

Hubbard model J i j c † c † i ↑ c i ↑ c † � � H ( t ) = i σ c j σ + U i ↓ c i ↓ i j σ i ➤ Possible phase diagram in “two dimensions”. “pseudogap phase” temperature d-wave superconductivity antiferromagnetic Mott insulator density half filling

Interaction quench ➤ An abrupt change of the interaction parameter in an isolated quantum system. ➤ Experimentally realized in cold-atom systems: by changing the optical lattice potential depth or by using Feshbach resonance. Greiner, et al., Nature (2002), Bloch, Dalibard, Zwerger, RMP (2008).

Nonequilibrium DMFT Schmidt, Monien (2002), Freericks, Turkowski, Zlati ć (2006), Aoki, Tsuji et al., arXiv:1310.5329. Λ ( t , t � ) t � t Lattice model Impurity model DMFT self-consistency G latt loc ( t , t � ) ≡ G imp [ Λ ]( t , t � ) 1 impurity solver � G latt loc = (QMC, IPT, NCA, ED, ...) i � t + µ − � k − Σ latt k k DMFT approximation Σ latt k ( t , t � ) ≡ Σ imp ( t , t � ) ➤ DMFT scheme becomes exact in d →∞ limit of lattice models. Metzner, Volhardt (1989).

Prethermalization in d= ∞ ➤ d (t)= ⟨ n ↑ n ↓ ⟩ ¡ : the double occupancy → “mode-integrated” ➤ n ( ϵ k , t )= ⟨ c k † ( t ) c k ( t ) ⟩ : the full momentum distribution → “mode-resolved” 1 0.25 a U=0.5 a) U=2 U=0.5 0.8 U=1 n( ε ,t) n( ε ,t) U=1 0.21 U=1.5 1 U=1.5 0.6 ∆ n(t) d(t) 0.17 U=2 U=2 0.4 0.5 -2 U=2.5 U=2.5 0.13 -1 0.2 0 0 c U=3 ε 1 U=3 1 2 3 4 0 0 2 t 5 0 1 2 3 4 0 0 1 2 3 4 0 Eckstein, Kollar, Werner, PRB (2010). t t Eckstein, Kollar, Werner, PRL (2009).

Prethermalization in d= ∞ 1 0.9 0.8 0.7 0.6 � n 0.5 0.4 0.3 Short-time approx. � 0.2 DMFT Quantum Boltzmann 0.1 GGE 0.1 1 10 t Short-time approx. Short-time approximation: Moeckel, Kehrein, PRL (2008) DMFT Nonequilibrium DMFT : Aoki, Tsuji et al., arXiv:1310.5329. Quantum Boltzmann Quantum Boltzmann equation: Stark, Kollar, arXiv:1308.1610 GGE Generalized Gibbs ensemble (GGE): Kollar, Wolf, Eckstein, PRB (2011)

Generalized Gibbs ensemble (GGE) Kollar, Wolf, Eckstein, PRB (2011) ➤ Weak interaction: ➤ Approximate constants of motion: � c † 2 V ���� c † � c � c � + h . c . � + O ( g 2 ) n � = n � + g ˜ � � + � � − � � − � � ��� ➤ Construct GGE with them: � � ! � with � ˜ n α � � = � ˜ n α � 0 GGE ∝ exp λ α ˜ ρ � − n α GGE α ➤ Prethermalization plateau is described by GGE: GGE = � n α � pretherm + O ( g 3 ) � n α � �

Prethermalization in d=1 and 2 Tsuji, Barmettler, Aoki, Werner, arXiv:1307.5946. ➤ d=1, U/J=0 → 1 ➤ d=2, U/J=0 → 2 0.25 0.25 0.24 � a � 0.24 � a � DMFT DCA � Nc � 2 � 2 � 0.23 � � 2 � 0.23 DCA � Nc � 4 � 4 � 0.22 0.22 DCA � Nc � 8 � 8 � � n � n � � � n � n � � 0.21 0.21 DCA � Nc � 16 � 16 � DCA � Nc � 2 � 0.20 0.20 DMRG DCA � Nc � 4 � 0.19 0.19 � � 2 � DCA � Nc � 8 � 0.18 0.18 DMFT DCA � Nc � 16 � 0 2 4 6 8 10 0 1 2 3 4 5 1.00 1.0 � � 2 � � b � 0.98 DCA � Nc � 64 � � b � 0.9 � Π � 2, Π � 2 � DMFT 0.96 0.8 0.94 � n � n 0.7 0.92 � Π , 0 � 0.6 0.90 DCA � Nc � 8 � 8 � 0.88 0.5 DCA � Nc � 16 � 16 � 0.86 0.4 0 2 4 6 8 10 0 1 2 3 4 5 tJ tJ

Relaxation and Prethermalization in an Isolated Quantum System Science 337, 1318 (2012) M. Gring, 1 M. Kuhnert, 1 T. Langen, 1 T. Kitagawa, 2 B. Rauer, 1 M. Schreitl, 1 I. Mazets, 1,3 D. Adu Smith, 1 E. Demler, 2 J. Schmiedmayer 1,4 *

Dynamical transition in d= ∞ ➤ Sharp change of the relaxation behavoir between weak- and strong-coupling regimes. 0.25 fast thermalization a 0.2 b U=0.5 b) U=3.3 U=8 U=1 d n( ε ,t) n( ε ,t) 0.21 d med d th 0.1 1 U=1.5 d(t) 4 6 8 U 0.17 U=6 U=2 0.5 U=5 U=2.5 -2 0.13 U=4 -1 U=3 U=3.3 0 0 1 1 ε 2 3 4 1 2 t 5 d U=0.5 2 π / U oscillation 0.8 c) U=5 U=1 U=1.5 U=8 n( ε ,t) n( ε ,t) 0.6 ∆ n(t) U=6 1 U=2 U=5 0.4 U=2.5 U=4 0.5 0.2 c U=3.3 U=3 -2 0 -1 0 1 2 3 4 0 1 2 3 0 0 0.5 1 ε 1 t t 1.5 2 2.5 2 3 t Eckstein, Kollar, Werner (2009, 2010).

Thermalization w/ long-range order ➤ How does the fermionic condensed-matter system prethermalize and thermalize after the interaction quench in the presence of a long-range order (classical fluctuations)? ➤ The order parameter dynamics has been described by a macroscopic (sometimes phenomenological) Ginzburg-Landau equation, � Γ ∂ m ∂ t = δ F GL c = am + bm 3 � 2 M � 2 m δ m ➤ Validity of the equation: quasiparticle thermalization order parameter

Hubbard model with AFM ➤ Interaction quench in the Hubbard model with AFM order: J i j c † c † i ↑ c i ↑ c † � � H ( t ) = i σ c j σ + U ( t ) i ↓ c i ↓ i j σ i 0.30 0.25 paramagnetic phase 0.20 J i j 0.15 T 0.10 “Heisenberg” antiferromagnetic 0.05 phase “Slater” 0.00 0 2 4 6 8 U

Quench: AFM → PM Tsuji, Eckstein, Werner, PRL (2013) 0.4 ➤ : AFM order parameter m ( t ) = � | n i ↑ ( t ) � n i ↓ ( t ) | � : Higgs amplitude mode ω ≈ 2 ∆ ➤ The initial U i is fixed. 0.3 ➤ The final U f (< U i ) is systematically changed. m H t L 0.2 0.14 0.12 0.10 0.1 0.08 T 0.06 0.04 0.02 0.0 0.00 0 50 100 150 200 0.0 0.5 1.0 1.5 2.0 2.5 3.0 t U U i = 2 . 0 , U f = 1 . 0 , 1 . 1 , . . . , 1 . 9

week ending P H Y S I C A L R E V I E W L E T T E R S PRL 111, 057002 (2013) 2 AUGUST 2013 Higgs Amplitude Mode in the BCS Superconductors Nb 1 - x Ti x N Induced by Terahertz Pulse Excitation Ryusuke Matsunaga, 1 Yuki I. Hamada, 1 Kazumasa Makise, 2 Yoshinori Uzawa, 3 Hirotaka Terai, 2 Zhen Wang, 2 and Ryo Shimano 1 1 Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan 2 National Institute of Information and Communications Technology, 588-2 Iwaoka, Nishi-ku, Kobe 651-2492, Japan 3 National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan (Received 2 May 2013; published 29 July 2013) 0.8 (a) pump / =0.57 τ τ (a) 0.6 (b) f (THz) E probe ( t gate = t 0 ) (arb. units) 0.4 2 nJ/cm 1 9.6 0.2 8.5 7.9 f 7.2 2 6.4 0.0 Im 5.6 Re 3 4.8 (c) 4.0 2 b ➤ Higgs mode 1 0 0 ➤ Nambu-Goldstone mode 5 10 -4 -2 0 2 4 6 8 t pp (ps) Pump Energy (nJ/cm 2 )

Two step relaxation ➤ Relaxation crossovers from the nonthermal critical behavior in the intermediate time scale to the thermal critical behavior in the long time scale. Tsuji, Eckstein, Werner, PRL (2013) 0.4 � a � U i � 2.5 10 � 1 thermal critical 0.3 behavior 10 � 2 e � t � Τ deph 10 � 3 τ nth m ( t ) Φ ( t ) m H t L 0.2 m 10 � 4 nonth. critical 0.1 10 � 5 e � t � Τ th behavior 10 � 6 0.0 0 50 100 150 0 50 100 150 200 t t U i = 2 . 5 , U f = 1 . 6 , 1 . 7 , 1 . 8 , 1 . 9

Nonthermal criticality Tsuji, Eckstein, Werner, PRL (2013) 0.3 0.6 U i = 2 F th w ω t nth - 1 0.2 0.4 τ − 1 w , t - 1 th τ − 1 F nth m 0.1 0.2 m th U c 0 0 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 U f t nth - 1 : Nonthermal relaxation time. : Frequency of the Higgs mode. w : Thermal value of the order parameter. F th m th

This implies... nonthermal ordered state 0.14 Nonthermal 0.12 (quasi)critical point! 0.10 0.08 T 0.06 0.04 thermal critical point 0.02 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 U

Momentum distribution n k = 1 � k n k ( t ) = N − 1 � e i k · ( R i − R j ) � c † cf. T =0 static mean-field: i σ ( t ) c j σ ( t ) � 2 − � � 2 k + ∆ 2 2 ij ∼ � − 2 ( � k → ∞ ) k U i = 2 → U f = 1 . 4 ➤ The momentum distribution shows a power-law decay: (in this case) : non-universal?