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)

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.

Acknowledgments

University of Hamburg-CFEL Martin Eckstein University of Fribourg Philipp Werner University of Tokyo Hideo Aoki Takashi Oka University of Augsburg Marcus Kollar University of Geneva Peter Barmettler

Outline

• 1. Prethermalization in the Hubbard model (overview)
• 2. Nonthermal fixed point in the Hubbard model
• 3. Universality of the nonthermal fixed point

“How” and “in which time scale” does an isolated quantum many-body system thermalize?

2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 U T

Hubbard model

Ji j U

Mott transition metal insulator

2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 U T

paramagnetic phase antiferromagnetic phase

H(t) =

• i jσ

Ji jc†

iσcjσ + U

• i

c†

i↑ci↑c† i↓ci↓

➤ DMFT (d=∞) phase diagram at half filling.

Hubbard model

H(t) =

• i jσ

Ji jc†

iσcjσ + U

• i

c†

i↑ci↑c† i↓ci↓

➤ Possible phase diagram in “two dimensions”.

density temperature half filling d-wave superconductivity antiferromagnetic Mott insulator “pseudogap phase”

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).

Schmidt, Monien (2002), Freericks, Turkowski, Zlatić (2006), Aoki, Tsuji et al., arXiv:1310.5329.

Nonequilibrium DMFT

Glatt

loc (t, t) ≡ Gimp[Λ](t, t)

Glatt

loc =

• k

1 it + µ − k − Σlatt

k

Λ(t, t)

DMFT self-consistency DMFT approximation impurity solver (QMC, IPT, NCA, ED, ...) Lattice model Impurity model

➤ DMFT scheme becomes exact in d→∞ limit of lattice models.

Metzner, Volhardt (1989).

t t

Σlatt

k (t, t) ≡ Σimp(t, t)

Prethermalization in d=∞

Eckstein, Kollar, Werner, PRB (2010). Eckstein, Kollar, Werner, PRL (2009).

➤ d(t)=⟨n↑ n↓⟩ ¡: the double occupancy → “mode-integrated” ➤ n(ϵk,t)=⟨ck†(t)ck(t)⟩ : the full momentum distribution → “mode-resolved”

0.13 0.17 0.21 0.25 d(t) U=0.5 U=1 U=1.5 U=2 U=2.5 U=3

a

0.2 0.4 0.6 0.8 1 1 2 3 4 ∆n(t) t U=0.5 U=1 U=1.5 U=2 U=2.5 U=3

c

1 2 3 4 t

0.5 1 n(ε,t) a) U=2 1 2 3 4 5

• 2
• 1

1 2 ε t n(ε,t)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 1 10 n t Short-time approx. DMFT Quantum Boltzmann GGE

Short-time approximation: Moeckel, Kehrein, PRL (2008) Nonequilibrium DMFT : Aoki, Tsuji et al., arXiv:1310.5329. Quantum Boltzmann equation: Stark, Kollar, arXiv:1308.1610 Generalized Gibbs ensemble (GGE): Kollar, Wolf, Eckstein, PRB (2011)

• Short-time approx.

DMFT Quantum Boltzmann GGE

Prethermalization in d=∞

Generalized Gibbs ensemble (GGE)

➤ Weak interaction: ➤ Approximate constants of motion:

Kollar, Wolf, Eckstein, PRB (2011)

➤ Construct GGE with them: ➤ Prethermalization plateau is described by GGE:

with

˜ n = n + g

• 2Vc†

c† cc + h.c.

+ − − + O(g2) ρ

GGE ∝ exp

• −
• α

λα˜ nα

• ˜

nα

GGE !

= ˜ nα0 nα

GGE = nαpretherm + O(g3)

Prethermalization in d=1 and 2

Tsuji, Barmettler, Aoki, Werner, arXiv:1307.5946.

DMRG 2 DMFT DCA Nc2 DCA Nc4 DCA Nc8 DCA Nc16 a 1 2 3 4 5 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 nn b DCA Nc64 1 2 3 4 5 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00

tJ

n

a DMFT 2 DCA Nc22 DCA Nc44 DCA Nc88 DCA Nc1616 2 4 6 8 10 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 nn Π2, Π2 Π, 0 DCA Nc88 DCA Nc1616 2 DMFT b 2 4 6 8 10 0.4 0.5 0.6 0.7 0.8 0.9 1.0

tJ

n ➤ d=1, U/J=0 → 1 ➤ d=2, U/J=0 → 2

Science 337, 1318 (2012)

Relaxation and Prethermalization in an Isolated Quantum System

• 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. Schmiedmayer1,4*

Dynamical transition in d=∞

0.13 0.17 0.21 0.25 d(t) U=0.5 U=1 U=1.5 U=2 U=2.5 U=3 U=3.3 U=4 U=5 U=6 U=8

0.1 0.2 4 6 8 d U dth dmed

0.2 0.4 0.6 0.8 1 1 2 3 4 ∆n(t) t U=0.5 U=1 U=1.5 U=2 U=2.5 U=3 1 2 3 t U=3.3 U=8 U=6 U=5 U=4

a d c b

0.5 1 n(ε,t) b) U=3.3 1 2 3 4 5

• 2
• 1

1 2 ε t n(ε,t) 0.5 1 n(ε,t) c) U=5 0.5 1 1.5 2 2.5 3

• 2
• 1

1 2 ε t n(ε,t)

➤ Sharp change of the relaxation behavoir between weak- and strong-coupling regimes.

fast thermalization 2π/U oscillation 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,

➤ Validity of the equation:

quasiparticle thermalization

• rder parameter

Γ∂m ∂t = δFGL δm = am + bm3 c 2M 2m

2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 U T

Hubbard model with AFM

paramagnetic phase antiferromagnetic phase “Slater” “Heisenberg”

➤ Interaction quench in the Hubbard model with AFM order:

Ji j H(t) =

• i jσ

Ji jc†

iσc jσ + U(t)

• i

c†

i↑ci↑c† i↓ci↓

➤ : AFM order parameter

Quench: AFM → PM

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 U T

Tsuji, Eckstein, Werner, PRL (2013)

➤ The initial Ui is fixed. ➤ The final Uf (< Ui) is systematically changed.

: Higgs amplitude mode

ω ≈ 2∆

Ui = 2.0, U f = 1.0, 1.1, . . . , 1.9

m(t) = |ni↑(t) ni↓(t)|

50 100 150 200 0.0 0.1 0.2 0.3 0.4 t mHtL

Higgs Amplitude Mode in the BCS Superconductors Nb1-xTixN 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 Shimano1

1Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan 2National Institute of Information and Communications Technology, 588-2 Iwaoka, Nishi-ku, Kobe 651-2492, Japan 3National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

(Received 2 May 2013; published 29 July 2013)

PRL 111, 057002 (2013) P H Y S I C A L R E V I E W L E T T E R S

week ending 2 AUGUST 2013

(a) Re Im

➤ Higgs mode ➤ Nambu-Goldstone mode

8 6 4 2

• 2
• 4

1 nJ/cm

2

9.6 8.5 7.9 7.2 6.4 5.6 4.8 4.0 3 2 1 10 5 0.8 0.6 0.4 0.2 0.0 (a) (b) 2 (c)

pump/

=0.57 Pump Energy (nJ/cm2) b f (THz) tpp (ps) Eprobe(tgate=t0) (arb. units) f

τ τ

a Ui 2.5

etΤdeph etΤth

50 100 150 106 105 104 103 102 101 t m

• nonth. critical

behavior thermal critical behavior

Two step relaxation

Φ(t)

➤ 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) Ui = 2.5, U f = 1.6, 1.7, 1.8, 1.9

τnth

50 100 150 200 0.0 0.1 0.2 0.3 0.4 t mHtL

m(t)

: Nonthermal relaxation time. : Frequency of the Higgs mode. : Thermal value of the order parameter.

τ−1

th

Nonthermal criticality

Uc Fth w tnth-1

Ui = 2

0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 0.1 0.2 0.3 0.2 0.4 0.6 Uf w, t-1 F

Tsuji, Eckstein, Werner, PRL (2013)

tnth-1 w Fth

m

mth

mth

ω

τ−1

nth

This implies...

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 U T

Nonthermal (quasi)critical point! thermal critical point nonthermal ordered state

Momentum distribution

• cf. T=0 static mean-field:

➤ The momentum distribution shows a power-law

decay: (in this case) : non-universal?

nk(t) = N−1

ij

eik·(Ri−Rj)c†

iσ(t)cjσ(t)

nk = 1 2 − k 2

• 2

k + ∆2

∼ −2

k

(k → ∞)

Ui = 2 → U f = 1.4

Quench: PM → AFM

20 40 60 80 100 120 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t FHtL

0.0 0.5 1.0 1.5 2.0 0.00 0.02 0.04 0.06 0.08 0.10 U T

Tsuji, Werner, PRB (2013)

➤ The initial Ui is fixed. ➤ The final Uf (> Ui) is systematically changed.

Ui = 1.0, U f = 1.05, 1.1, . . . , 1.5

➤ : AFM order parameter

m(t) = |ni↑(t) ni↓(t)|

m(t)

τi

Nonthermal criticality

: Maximum of the first peak in amplitude oscillation. : Minimum of the first peak in amplitude oscillation. : Thermal values of order parameter reached in the long-time limit. : Rate of the initial exponential growth (Φ ∝ ¡et/τi). ti-1H¥4L Fmax Fmin Fth 1.0 1.1 1.2 1.3 1.4 1.5 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Uf Tsuji, Werner, PRB (2013)

Ui = 1.0

mmax mmin mth mmax mmin mth

Summary of critical behavior

Hohenberg, Halperin, RMP (1977)

zν

β ω U f Ui Uc

U∗

τ−1

th

τ−1

nth

Ui Uc

U∗

U f

intermediate time scale longer time scale τ ∝ |U f − U∗|−1 ∝ |U f − Uc|−1 m ∝ |U f − U∗|1 ∝ |U f − Uc|1/2 ω ∝ |U f − U∗|1 —

mth mnth mth

Classical equation of motion

➤ Time-dep. Hartree approximation:

• it + µ − ΣA

−k −k it + µ − ΣB GAA

k

GAB

k

GBA

k

GBB

k

• =
• C

C

• ΣA(t, t) = U(t)nB(t)δC(t, t)

ΣB(t, t) = U(t)nA(t)δC(t, t) ∂t f k(t) = bk(t) × f k(t) bk(t) = (−2k, 0, U(t)m(t)) m(t) =

• k

f z

k(t)

• cf. Time-dep. BCS equation: Barankov, et al. (2004, 2006),

Yuzbashyan et al. (2005, 2006), Warner, Leggett (2005).

➤ Dyson equation: ➤ The classical equation of motion :

Classical equation of motion

a Ui 2

50 100 150 200 0.0 0.1 0.2 0.3 0.4 t m 0.0 0.1 0.2 0.3 0.4

➤ The Hartree results: ➤ The momentum distribution (U=2→1.8):

flat DOS semicircular DOS

Ui = 2.0, U f = 1.0, 1.1, . . . , 1.9

∼ −2 ∼ −3

Hartree doesn’t agree

➤ Hartree results: dashed lines (left panel) 1 2 3 4 0.00 0.05 0.10 0.15 0.20 U T ➤ DMFT results: solid lines (left panel)

20 40 60 80 100 0.0 0.1 0.2 0.3 0.4 0.5 0.6 t FHtL

Ui = 2.0, U f = 1.0, 1.1, . . . , 1.9

m(t)

Hartree DMFT

Σ(t, t) =

Renormalized interaction

➤ Argument: short-time dynamics is governed by quasiparticles having a renormalized

interaction , which has a one-to-one correspondence with the Hartree particles:

˜ U(t)

+ + + …

QMC BdG 1 2 3 4 0.00 0.05 0.10 0.15 0.20 U T

➤ Pairing interaction is effectively reduced due to quantum fluctuations.

e.g. T-matrix theory (Nozières, Schmidtt-Rink, ’85).

• U =

U 1 + UΓ

≈

• U

: non-singular

U → U(U)

Hartree DMFT

50 100 150 200 0.0 0.2 0.4 0.6 0.8 1.0 t FHtLêFH0L

20 40 60 80 100 120 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t FHtL

Renormalized Hartree agrees

➤ We compare the DMFT results (solid curves) with the Hartree solution for the

quasiparticles with the renormalized interaction (dashed).

m(t) m(t)

a = a0(U f − U∗)2

a = a0(U f − Uc)

Nonthermal criticality

where f0(ϵk) is the initial momentum distribution. From this, one can show that which contrasts with the conventional GL theory,

➤ From the argument, it follows that the order parameter satisfies:

with

➤ The constant a is determined from a condition ➤ This evidences that the nonthermal critical point belongs to a universality class different

from the conventional GL.

−∂2m ∂t2 = ∂Fnth ∂m

Fnth = −1 2am2 +

• U2

f

8 m4

Ui = 4 → U f = 3.8, 3.2, 2.6, 2.2, 2.0, 1.8, 1.6

/15 32

Time-dep. Gutzwiller approximation

Sandri, Fabrizio, PRB (2013).

20 40 60 80 100 t 0.2 0.4 0.6 0.8 m 5 10 15 20 Uf 0.2 0.4 0.6 0.8 1 Uf > Ui Uf < Ui

m(t → ∞)

quasiparticle residue Z nonthermal critical point dynamical transition point? Qualitatively similar results have been obtained by other methods.

Summary

nonthermal fixed point thermal state initial state

τ ∝ |U f − Uc|−1 m ∝ |U f − Uc|1/2 ω — τ ∝ |U f − U∗|−1 m ∝ |U f − U∗|1 ω ∝ |U f − U∗|1

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.

Acknowledgments

University of Hamburg-CFEL Martin Eckstein University of Fribourg Philipp Werner University of Tokyo Hideo Aoki Takashi Oka University of Augsburg Marcus Kollar University of Geneva Peter Barmettler