Dynamical mean field approach to correlated lattice systems in and out of equilibrium

Transcription

1 Dynamical mean field approach to correlated lattice systems in and out of equilibrium Philipp Werner University of Fribourg, Switzerland Kyoto, December 2013

2 Overview Dynamical mean field approximation applied to quantum field theory in collaboration with: O. Akerlund, P. de Forcrand, A. Georges Dynamical mean field theory for bosonic lattice systems in collaboration with: P. Anders, L. Pollet, M. Troyer Dynamical mean field theory for fermionic lattice systems in collaboration with: E. Gull, A. Millis Nonequilibrium extension of dynamical mean field theory in collaboration with: M. Eckstein, M. Kollar, N. Tsuji, T. Oka, H. Aoki Some applications: Hubbard model in strong electric fields in collaboration with: M. Eckstein, N. Tsuji, T. Oka, H. Aoki

3 DMFT for quantum field theories Simple example: real, scalar Lagrangian density ' 4 Akerlund, de Forcrand, Georges & Werner (2013) quantum field theory L['(x)] = 1 2 (@ µ'(x)) m2 0'(x) 2 g 0 4! '(x)4 Spontaneous Z 2 symmetry breaking for negative (renormalized) m 2 After Wick rotation, discretization and variable transformation S = X 2apple X! ' x+bµ ' x + ' 2 x + (' 2 x 1) 2 x µ In the limit!1, ' x = ±1 (Ising model)

4 DMFT for quantum field theories Simple example: real, scalar quantum field theory Mean field theory: mapping to a 0-dimensional effective model replace all interactions by an interaction with a constant background field v Partition function of the lattice model Z Z = D['] Y exp ' 2 x (' 2 x 1) 2 +2apple x becomes Z MF = Z 1 1 ' 4 Akerlund, de Forcrand, Georges & Werner (2013)! dx ' x ' x+bµ µ=1 d' exp ' 2 (' 2 1) 2 +2apple(2d)v' Self-consistency condition v h'i = 1 Z 1 ''exp Z MF 1 ' 2 (' 2 1) 2 +2apple(2d)v'

5 DMFT for quantum field theories Simple example: real, scalar ' 4 Akerlund, de Forcrand, Georges & Werner (2013) quantum field theory Mean field theory: Phase diagram (d=3+1) Monte Carlo, 32 4 Mean Field symmetry-broken phase symmetric phase

6 DMFT for quantum field theories Simple example: real, scalar ' 4 Akerlund, de Forcrand, Georges & Werner (2013) quantum field theory Dynamical mean field theory: mapping to a (0+1)-dimensional effective model explicitly treat fluctuations in one dimension, freeze fluctuations in the (d-1) other dimensions Dynamical dimension: t (with conjugate momentum! ) Frozen dimensions: x 1,...,x d 1 (conjugate momenta k 1,...,k d 1 ) This convention allows to study finite-temperature behavior by varying the extent of the dynamical dimension Breaks Lorentz invariance, but let s try it anyhow...

7 DMFT for quantum field theories Simple example: real, scalar ' 4 Akerlund, de Forcrand, Georges & Werner (2013) quantum field theory Dynamical mean field theory: mapping to a (0+1)-dimensional effective model explicitly treat fluctuations in one dimension, freeze fluctuations in the (d-1) other dimensions Schematically: Z Z = D[']exp 2apple X x Z Z DMFT = D[']exp! X X ' x ' x+bµ V (' x ) µ x X K 1 (t t 0 )' t ' t 0 t,t 0 X t non-zero in the symmetry-broken phase V (' t )+h X t ' t! effect of the frozen dimensions on the local dynamics

8 DMFT for quantum field theories Simple example: real, scalar ' 4 Akerlund, de Forcrand, Georges & Werner (2013) Cavity construction: separate action into internal x =(~0,t) and external x 6= (~0,t) degrees of freedom S = S int + S + S ext S int = X h t S = 2apple X t X h S ext = x6=(~0,t) quantum field theory 2apple' int,t+1 ' int,t + ' 2 int,t + (' 2 int,t 1) 2i X ' int,t ' ext,t hint,exti 2apple X ' x+b ' x + ' 2 x + (' 2 x 1) 2i S int S ext t S x Expand exp( S) and integrate out the external degrees of freedom

9 DMFT for quantum field theories Simple example: real, scalar quantum field theory Cavity construction: to allow for symmetry breaking, we write ' 4 Akerlund, de Forcrand, Georges & Werner (2013) S int S ext ' ext,t = ext + ' ext,t, h' ext i = ext ' int,t = int + ' int,t, h' int i = int S = apple X t 2(d 1) ext ' int,t + X t ' int,t x ' ext,t! S + f(' ext ) hint,exti S 1 + X t S + f(' ext ) Expand exp( S) can be included in Sint can be included in Sext

10 DMFT for quantum field theories Simple example: real, scalar Cavity construction: Terms up to second order yield ' 4 Akerlund, de Forcrand, Georges & Werner (2013) Z = Z ext Z D' int exp( S int S 1 ) 1 quantum field theory X h S(t)i ext t X t x S int S S ext t,t 0 h S(t) S(t 0 )i ext +... First order term is zero because proportional to h ' ext i ext =0 Second order term is non-zero: hybridization function h S(t) S(t 0 )i ext ' int,t (t t 0 ) ' int,t 0

11 DMFT for quantum field theories Simple example: real, scalar ' 4 Akerlund, de Forcrand, Georges & Werner (2013) quantum field theory S int S ext Cavity construction: After re-exponentiation and switching back from ' to ' we find the effective single-site action t x S S imp = X t,t 0 ' t K 1 imp,c (t t0 )' t 0 + X t (' 2 t 1) 2 h X t ' t ek 1 imp,c (!) = 1 2apple cos(!) e (!) Fourier transform of nn interaction in time h = 2 ext (2apple(d 1) e (0)) because action written in terms of phi We call it impurity action (condensed-matter convention)

12 DMFT for quantum field theories Simple example: real, scalar ' 4 Akerlund, de Forcrand, Georges & Werner (2013) Dynamical mean field equations: quantum field theory S int S ext Hybridization function and ext are fixed by self-consistency conditions t x S S imp = X t,t 0 ' t K 1 imp,c (t t0 )' t 0 + X t (' 2 t 1) 2 h X t ' t ek 1 imp,c (!) = 1 2apple cos(!) e (!) h = 2 ext (2apple(d 1) e (0))

13 DMFT for quantum field theories Simple example: real, scalar Dynamical mean field equations: Hybridization function and ' 4 Akerlund, de Forcrand, Georges & Werner (2013) are fixed by self-consistency conditions Local lattice Green s function eg loc (!) = P 1 k quantum field theory Impurity Green s function 1 eg imp (!) = ek = 1 1 imp,c (!)+e imp (!) 1 2apple cos(!) e (!)+ imp e (!) ext eg 1 0 (k,!)+e (k,!) eg 1 0 (k,!) =1 2apple P d i=1 cos(k i) t x S int S GF of the d-dimensional free theory S ext DMFT approximation: identify lattice and impurity self-energy

14 DMFT for quantum field theories Simple example: real, scalar Dynamical mean field equations: Hybridization function and are fixed by self-consistency conditions Local lattice Green s function ' 4 Akerlund, de Forcrand, Georges & Werner (2013) quantum field theory ext eg loc (!) = X 1 eg 1 imp (!)+ e (!) 2apple P d 1 i=1 cos k i t x S int S S ext Self-consistency equations: eg imp (!) = e G loc (!) h'i Simp = ext substitution yields an implicit equation for the hybridization function

15 DMFT for quantum field theories Simple example: real, scalar ' 4 quantum field theory Akerlund, de Forcrand, Georges & Werner (2013) Dynamical mean field equations: Hybridization function and ext are fixed by self-consistency conditions Lattice model impurity model momentum average G latt loc DMFT self-consistency G latt loc G imp, h'i ext S[, ext] impurity solver G latt k =[G 1 0,k + latt k ] 1 latt k imp G imp, imp, h'i DMFT approximation

16 DMFT for quantum field theories Simple example: real, scalar ' 4 Akerlund, de Forcrand, Georges & Werner (2013) quantum field theory Dynamical mean field phase diagram (d=3+1): Monte Carlo, 32 4 Mean Field DMFT, L=75 symmetry-broken phase symmetric phase

17 DMFT for Bosons Simple example: Bose-Hubbard model Fisher, Grinstein, Weichmann & Fisher (1989) H = t X b i b j + U 2 hi,ji X i n i (n i 1) µ X i n i 3U t U Mott-Insulator to superfluid transition at low temperature

18 DMFT for Bosons Simple example: Bose-Hubbard model Derivation of DMFT formalism analogous to the Anders, Pollet, Gull, Troyer & Werner (2011) case Split action of the lattice model into S = S int + S + S ext Z h S int = d µb int b int + U i 2 n int(n int 1) Z S = t 0 Z h S ext = d 0 0 X (b int b ext + b extb int ) hint,exti To allow for symmetry breaking (condensation), write b ext = ext + b ext, b int = int + b int ' 4 µb extb ext + U i 2 n ext(n ext 1)

19 DMFT for Bosons Simple example: Bose-Hubbard model Derivation of DMFT formalism analogous to the Anders, Pollet, Gull, Troyer & Werner (2011) ' 4 case Expand exp( S), integrate out external degrees of freedom End up with an impurity model S imp = 1 2 apple Z 0 Z exchange of particles with the condensate b ( ) ( 0 )b( 0 )+ 0 d b( ) Z 0 U h d ( ) Hybridization function (hopping of normal bosons) µn( )+ U 2 n( )[n( ) 1 i

20 DMFT for Bosons Simple example: Bose-Hubbard model Derivation of DMFT formalism analogous to the Anders, Pollet, Gull, Troyer & Werner (2011) case Expand exp( S), integrate out external degrees of freedom ' 4 End up with an impurity model Nambu notation exchange of particles with the condensate U ( ) Hybridization function (hopping of normal bosons) and fixed by the DMFT self-consistency condition approximation: identification of lattice and impurity self-energy = hb( )i Simp, G latt loc = G imp,c [= ht b( )b (0)i Simp + ]

21 DMFT for Bosons Simple example: Bose-Hubbard model Derivation of DMFT formalism analogous to the Anders, Pollet, Gull, Troyer & Werner (2011) Results: Phase diagram (d=3+1) ' 4 case T=0 normal Mott insulator n=2 superfluid superfluid Mott insulator Mott insulator n=1

22 DMFT for Bosons Simple example: Bose-Hubbard model Derivation of DMFT formalism analogous to the Anders, Pollet, Gull, Troyer & Werner (2011) ' 4 case Results: Connected density-density correlation functions (d=3+1) DMFT lattice QMC

23 DMFT for Fermions Simple example: single-band Hubbard model Gutzwiller, Kanamori, Hubbard (1963) H = t X hi,ji, c i c j + U X n i," n i,# µ X i i, n i, t U Describes Mott transition, magnetic and superconducting transitions

24 DMFT for Fermions Simple example: single-band Hubbard model Gutzwiller, Kanamori, Hubbard (1963) Dynamical mean field theory: Mapping to a quantum impurity model Georges & Kotliar (1992) U ( ) Hybridization function (describes hopping of electrons into the lattice and back) Hybridization functions approximation: identification of lattice and impurity self-energy fixed by DMFT self-consistency condition G latt loc, = G imp,

25 DMFT for Fermions Simple example: single-band Hubbard model Gutzwiller, Kanamori, Hubbard (1963) Dynamical mean field theory: Mapping to a quantum impurity model Georges & Kotliar (1992) lattice model t G latt G imp impurity model latt imp Various numerical approaches to solve the impurity problem: weak-coupling perturbation theory, strong-coupling perturbation theory, exact diagonalization, NRG, QMC,...

26 DMFT for Fermions Simple example: single-band Hubbard model Gutzwiller, Kanamori, Hubbard (1963) Dynamical mean field theory: Mapping to a quantum impurity model Georges & Kotliar (1992) lattice model t G latt G imp impurity model latt imp Single-site DMFT can treat two-sublattice order (e. g. AFM) Bath B, [G A, ], Bath A, [G B, ] Pure Neel order: Bath B, = Bath A,

27 DMFT for Fermions Equilibrium DMFT phase diagram (half-filling) Paramagnetic calculation: Metal - Mott insulator transition at low T Smooth crossover at high T T metal Mott insulator U

28 DMFT for Fermions Equilibrium DMFT phase diagram Paramagnetic calculation: Metal - Mott insulator transition at low T Doping-driven metallization upon increasing µ T 0.05 µ µ c1 µ c metal Mott insulator U c U c U (µ µ 1 ) /t

29 DMFT for Fermions Equilibrium DMFT phase diagram Paramagnetic calculation: Metal - Mott insulator transition at low T Doping-driven metallization upon increasing µ T 0.05 µ µ c1 µ c metal Mott insulator U c U c U (µ µ 1 ) /t

30 DMFT for Fermions Equilibrium DMFT phase diagram (half-filling) Paramagnetic calculation: Metal - Mott insulator transition at low T Smooth crossover at high T T paramagnetic Mott insulator has large entropy: metal-insulator transition upon increasing T metal Mott insulator U

31 DMFT for Fermions Equilibrium DMFT phase diagram (half-filling) Paramagnetic calculation: Metal - Mott insulator transition at low T Smooth crossover at high T T + plaquette singlet state dominates low-t insulator metal Mott insulator curvature of the phase boundary changes if short-range spatial correlations are taken into account U

32 DMFT for Fermions Equilibrium DMFT phase diagram (half-filling) With 2-sublattice order: Antiferromagnetic insulator at low T Smooth crossover at high T T metal Mott insulator AFM insulator U

33 DMFT for Fermions Equilibrium DMFT phase diagram (half-filling) Transformation c i"! c i" (i 2 A), c i"! c i" (i 2 B) maps repulsive model onto attractive model T metal superconductor AFM insulator attractive repulsive U

34 DMFT for Fermions Equilibrium DMFT phase diagram (half-filling) Half-filling: transformation c i"! c i" (i 2 A), c i"! c i" (i 2 B) maps repulsive model onto attractive model T metal superconductor AFM insulator BEC BCS Slater Mott attractive repulsive U

35 DMFT for Fermions Low-dimensional systems DMFT is exact in d = 1 Metzner & Vollhardt (1989) Neglect of spatial fluctuations problematic in d<3 d =2 Hubbard model is believed to describe the physics of high-tc (cuprate) superconductors T AFM Mott insulator pseudo gap superconductor metal hole doping

36 DMFT for Fermions Low-dimensional systems DMFT is exact in d = 1 Metzner & Vollhardt (1989) Neglect of spatial fluctuations problematic in d<3 d =2 Hubbard model is believed to describe the physics of high-tc (cuprate) superconductors T to describe the physics of the cuprates, we need a cluster extension of DMFT, with at least 4 sites AFM Mott insulator pseudo gap superconductor metal hole doping

37 DMFT for Fermions Low-dimensional systems Cluster DMFT self-consistently embeds a cluster of fermionic bath Hettler, Prushke, Krishnamurthy & Jarrell (1998) sites into a If cluster is periodized: coarse-graining of the momentum-dependence N c (p, ) = Tiling of the Brillouin zone a a(p) a ( )

38 DMFT for Fermions Low-dimensional systems Cluster DMFT self-consistently embeds a cluster of fermionic bath Hettler, Prushke, Krishnamurthy & Jarrell (1998) sites into a Doping driven insulator-metal transition in the 8-site cluster DMFT first 8% of dopants go into the B sector N c Werner, Gull, Parcollet & Millis (2010) t=40, B t=40, C t=20, B t=20, C n B C B C µ/t

39 DMFT for Fermions Low-dimensional systems Cluster DMFT self-consistently embeds a cluster of fermionic bath Hettler, Prushke, Krishnamurthy & Jarrell (1998) sites into a Doping driven insulator-metal transition in the 8-site cluster DMFT first 8% of dopants go into the B sector N c Werner, Gull, Parcollet & Millis (2010) t=40, B t=40, C t=20, B t=20, C Pseudogap phase as a momentumselective Mott state n B µ/t C

40 Nonequilibrium DMFT DMFT accurately treats time-dependent fluctuations can be directly applied to nonequilibrium systems Freericks et al. (2006) Equilibrium: solve DMFT equations on the imaginary-time interval Time-translation invariance: can use Fourier transformation Nonequilibrium: solve DMFT equations on the Kadanoff-Baym contour Integral-differential equations of Volterra type 0 t 0 vertical branch: initial equilibrium state propagate and converge the DMFT solution step by step along the real-time axis i

41 Nonequilibrium DMFT Field E(t) applied at t=0 Choose gauge with pure vector potential: E(t) t A(t) Freericks et al. (2006) t U Peierls substitution: "(k)! "(k Lattice: hybercubic, infinite-d limit ea(t)) (") = 1 p W exp( " 2 /W 2 )

42 DC electric fields Dielectric breakdown of the Mott insulator Eckstein, Oka, &Werner (2011); Eckstein & Werner (2013) Electric field (amplitude F) applied at t=0 in the body-diagonal

43 DC electric fields Dielectric breakdown of the Mott insulator Eckstein, Oka, &Werner (2011); Eckstein & Werner (2013) Time evolution of the current and double occupancy j(t)/f F=1.0 F=0.8 F=0.6 F=0.4 a d(t) b F=1.0 F=0.8 F=0.6 F=0.4 j/f c A( ) dh t /F j t /F time time F build-up of a polarization quasi-steady state quasi-steady current follows a threshold-law: j/f / exp( F th /F )

44 DC electric fields Dielectric breakdown of the Mott insulator Eckstein, Oka, &Werner (2011); Eckstein & Werner (2013) Time evolution of the current and double occupancy constant production of doublon-hole pairs j(t)/f F=1.0 F=0.8 F=0.6 F=0.4 a d(t) b F=1.0 F=0.8 F=0.6 F=0.4 j/f c dh t /F j t /F A( ) time build-up of a polarization quasi-steady state time quasi-steady current follows a threshold-law: j/f / exp( F th /F ) F

45 DC electric fields Dielectric breakdown of the Mott insulator Eckstein, Oka, &Werner (2011); Eckstein & Werner (2013) Why is there a steady-state current if number of carriers increases? j(t)/f F=1.0 F=0.8 F=0.6 F=0.4 a d(t) A < (,t) time-dependent b occupation function ccompared to a A( ) rescaled spectral function (black) F=1.0 F=0.8 F=0.6 a F=0.4 F=0.2 t=5 t=10 t=15 t=20 t=25 t=30 j/f A < (,t) b0 dh t /F j t /F F=0.3 t=5 t=10 t=15 t=20 t=25 t= time Doublons (and holes) quickly heat up to infinite temperature! no contribution to the current time F

46 DC electric fields AC-field quench in the Hubbard model (metal phase) Tsuji, Oka, Werner and Aoki (2011) Electric field (amplitude E, frequency ) applied at t=0 t U

47 Periodic electric fields AC-field quench in the Hubbard model (metal phase) Tsuji, Oka, Werner and Aoki (2011) E/ =4 E/ =3 attractive (>0.25) double occupation E/ =2.5 E/ =1 repulsive (<0.25) 0.1 E/ = time [inverse hopping]

48 Periodic electric fields AC-field quench in the Hubbard model (metal phase) Sign inversion of the interaction: repulsive attractive Dynamically generated high-tc superconductivity? temperature metal superconductor (attractive interaction) superconductor (repulsive interaction) hole doping

49 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 / 0 k k A(t) dt k A(t) = J 0 (E/ ) k J0(E/ ) E/

50 Origin of the attractive interaction Periodic E-field leads to a population inversion J 0 (E/ )=1 Renormalized dispersion k = 2 2 / 0 dt k A(t) = J 0 (E/ ) k

51 Origin of the attractive interaction Periodic E-field leads to a population inversion J 0 (E/ )=0.8 Renormalized dispersion k = 2 2 / 0 dt k A(t) = J 0 (E/ ) k

52 Origin of the attractive interaction Periodic E-field leads to a population inversion J 0 (E/ )=0.3 Renormalized dispersion k = 2 2 / 0 dt k A(t) = J 0 (E/ ) k

53 Origin of the attractive interaction Periodic E-field leads to a population inversion J 0 (E/ )= 0.2 Renormalized dispersion k = 2 2 / 0 dt k A(t) = J 0 (E/ ) k

54 Origin of the attractive interaction Inverted population = negative temperature State with T <0, J 0 < 0 T e = U>0,T <0 is equivalent to state with U<0,T >0 T J 0 > 0 / exp = exp 1 T 1 T e " X k " X k J 0 k n k k n k + U X i n i" n i# #! #! + U X n i" n i# J 0 i Effective interaction of the T e > 0 state U e = U J 0 (E/ )

55 Experimental realization? Potentially interesting material: Sn doped In2O3 Transparent conductor Single s-band crossing the Fermi level band structure by Bernard Delley (PSI)

56 Summary Overview of DMFT. Basic idea: neglect spatial fluctuations, keep dynamical fluctuations Map (d+1) dimensional lattice to a (0+1) dimensional impurity model Phasediagram of the Hubbard model. Importance of short-range spatial correlations in d=2 Relationship to cuprate physics Extension to nonequilibrium problems. Dielectric breakdown of a Mott insulator in strong DC fields Dynamical band-flipping and interaction conversion by AC fields