Dynamical mean field approach to correlated lattice systems in and out of equilibrium Philipp Werner University of Fribourg, Switzerland Kyoto, December 2013
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
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)) 2 1 2 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)
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'
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) 0.14 0.13 0.12 0.11 0.1 Monte Carlo, 32 4 Mean Field symmetry-broken phase 0.09 0.08 0.07 symmetric phase 0 1 2 3 4 5
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...
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
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
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
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 + 1 2 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
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)
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))
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
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
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
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): 0.14 0.13 0.12 0.11 0.1 Monte Carlo, 32 4 Mean Field DMFT, L=75 symmetry-broken phase 0.09 0.08 0.07 symmetric phase 0 1 2 3 4 5
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
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)
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
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 + ]
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
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
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
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,
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,...
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,
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
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 c2 0.01 metal Mott insulator U c1 5 5.5 U c2 6 6.5 U (µ µ 1 ) /t
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 c2 0.01 metal Mott insulator U c1 5 5.5 U c2 6 6.5 U (µ µ 1 ) /t
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
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
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
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
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
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
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
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 ( )
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) 0.56 0.55 0.54 t=40, B t=40, C t=20, B t=20, C n 0.53 0.52 0.51 B C B C 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 µ/t
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) 0.56 0.55 0.54 t=40, B t=40, C t=20, B t=20, C Pseudogap phase as a momentumselective Mott state n 0.53 0.52 B 0.51 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 µ/t C
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
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 )
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
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 0.004 0.003 0.002 F=1.0 F=0.8 F=0.6 F=0.4 a d(t) 0.017 0.015 b F=1.0 F=0.8 F=0.6 F=0.4 j/f 0.002 c 0.6 0.3 0 A( ) -5 0 5 0.001 0.013 0.001 dh t /F j t /F 0 0.011 0 5 10 15 20 0 5 10 15 20 0 0 0.3 0.6 0.9 1.2 time time F build-up of a polarization quasi-steady state quasi-steady current follows a threshold-law: j/f / exp( F th /F )
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 0.004 0.003 0.002 0.001 F=1.0 F=0.8 F=0.6 F=0.4 a 0.017 0.015 d(t) 0.013 b F=1.0 F=0.8 F=0.6 F=0.4 j/f 0.002 0.001 c 0.6 0.3 0 dh t /F j t /F A( ) -5 0 5 0 0.011 0 5 10 15 20 time build-up of a polarization quasi-steady state 0 5 10 15 20 time 0 0 0.3 0.6 0.9 1.2 quasi-steady current follows a threshold-law: j/f / exp( F th /F ) F
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 0.004 0.003 0.002 0.001 0 F=1.0 F=0.8 F=0.6 F=0.4 a d(t) 0.017 0.015 0.013 A < (,t) 0.011 time-dependent b occupation function ccompared to a A( ) rescaled spectral function (black) 0.006 0.005 0.004 0.003 0.002 0.001 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 0.002 A < (,t) 0.006 0.005 0.004 0.001 0.003 0.002 0.001 0.6 0.3 b0 dh t /F j t /F -5 0 5 F=0.3 t=5 t=10 t=15 t=20 t=25 t=30 0 5 10 15 20 time Doublons (and holes) quickly heat up to infinite temperature! no contribution to the current 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 5 10 15 20 time 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.3 0.6 0.9 1.2 F
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
Periodic electric fields AC-field quench in the Hubbard model (metal phase) Tsuji, Oka, Werner and Aoki (2011) 0.35 0.3 E/ =4 E/ =3 attractive (>0.25) double occupation 0.25 0.2 0.15 E/ =2.5 E/ =1 repulsive (<0.25) 0.1 E/ =1.5 0.05 0 1 2 3 4 5 6 7 8 time [inverse hopping]
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
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/
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
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
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
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
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/ )
Experimental realization? Potentially interesting material: Sn doped In2O3 Transparent conductor Single s-band crossing the Fermi level band structure by Bernard Delley (PSI)
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