Dynamical mean field approach to correlated lattice systems in and - - PowerPoint PPT Presentation

Philipp Werner University of Fribourg, Switzerland

Kyoto, December 2013

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

Dynamical mean field approximation applied to quantum field theory Dynamical mean field theory for bosonic lattice systems Dynamical mean field theory for fermionic lattice systems Nonequilibrium extension of dynamical mean field theory Some applications: Hubbard model in strong electric fields

Overview

in collaboration with: O. Akerlund, P. de Forcrand, A. Georges in collaboration with: P. Anders, L. Pollet, M. Troyer in collaboration with: E. Gull, A. Millis in collaboration with: M. Eckstein, M. Kollar, N. Tsuji,

• T. Oka, H. Aoki

in collaboration with: M. Eckstein, N. Tsuji,

• T. Oka, H. Aoki

Simple example: real, scalar quantum field theory Lagrangian density Spontaneous symmetry breaking for negative (renormalized) After Wick rotation, discretization and variable transformation In the limit (Ising model)

DMFT for quantum field theories

Akerlund, de Forcrand, Georges & Werner (2013)

L[ϕ(x)] = 1 2(∂µϕ(x))2 − 1 2m2

0ϕ(x)2 − g0

4! ϕ(x)4 Z2 m2 S = X

x

−2κ X

µ

ϕx+b

µϕx + ϕ2 x + λ(ϕ2 x − 1)2

! λ → ∞, ϕx = ±1 ϕ4

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 Partition function of the lattice model becomes Self-consistency condition ϕ4 Z = Z D[ϕ] Y

x

exp −ϕ2

x−λ(ϕ2 x − 1)2+2κ d

X

µ=1

ϕxϕx+b

µ

! ZMF = Z ∞

−∞

dϕ exp

• −ϕ2 − λ(ϕ2 − 1)2 + 2κ(2d)vϕ
• Akerlund, de Forcrand, Georges &

Werner (2013)

DMFT for quantum field theories

v ⌘ hϕi = 1 ZMF Z ∞

−∞

ϕ ϕ exp ⇣ ϕ2 λ(ϕ2 1)2 + 2κ(2d)vϕ ⌘ v

Simple example: real, scalar quantum field theory Mean field theory: Phase diagram (d=3+1)

DMFT for quantum field theories

ϕ4

0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 1 2 3 4 5

• Monte Carlo, 324

Mean Field

symmetric phase symmetry-broken phase Akerlund, de Forcrand, Georges & Werner (2013)

Simple example: real, scalar 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: (with conjugate momentum ) Frozen dimensions: (conjugate momenta ) Breaks Lorentz invariance, but let’s try it anyhow ...

DMFT for quantum field theories

ϕ4 t ω x1, . . . , xd−1 k1, . . . , kd−1

Akerlund, de Forcrand, Georges & Werner (2013) This convention allows to study finite-temperature behavior by varying the extent of the dynamical dimension

Simple example: real, scalar 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:

DMFT for quantum field theories

ϕ4

Akerlund, de Forcrand, Georges & Werner (2013) non-zero in the symmetry-broken phase effect of the frozen dimensions on the local dynamics

Z = Z D[ϕ] exp 2κ X

x

X

µ

ϕxϕx+b

µ −

X

x

V (ϕx) ! ZDMFT = Z D[ϕ] exp − X

t,t0

K1(t − t0)ϕtϕt0 − X

t

V (ϕt) + h X

t

ϕt !

Simple example: real, scalar quantum field theory Cavity construction: separate action into internal and external degrees of freedom Expand and integrate out the external degrees of freedom

DMFT for quantum field theories

ϕ4

Akerlund, de Forcrand, Georges & Werner (2013)

x = (~ 0, t) x 6= (~ 0, t)

Sext Sint S x t

S = Sint + ∆S + Sext Sint = X

t

h − 2κϕint,t+1ϕint,t + ϕ2

int,t + λ(ϕ2 int,t − 1)2i

∆S = −2κ X

t

X

hint,exti

ϕint,tϕext,t Sext = X

x6=(~ 0,t)

h − 2κ X

⌫

ϕx+b

⌫ϕx + ϕ2 x + λ(ϕ2 x − 1)2i

exp(−∆S)

Simple example: real, scalar quantum field theory Cavity construction: to allow for symmetry breaking, we write Expand

∆S = −κ X

t

2(d − 1)φ†

extδϕint,t +

X

hint,exti

δϕ†

int,tδϕext,t

! + f(ϕext) ≡ S1 + X

t

δS + f(ϕext)

DMFT for quantum field theories

ϕ4

Akerlund, de Forcrand, Georges & Werner (2013)

Sext Sint S x t

ϕext,t = φext + δϕext,t, hϕexti = φext ϕint,t = φint + δϕint,t, hϕinti = φint

can be included in Sint can be included in Sext

exp(−δS)

Simple example: real, scalar quantum field theory Cavity construction: Terms up to second

• rder yield

First order term is zero because proportional to Second order term is non-zero:

DMFT for quantum field theories

ϕ4

Akerlund, de Forcrand, Georges & Werner (2013)

Sext Sint S x t

Z = Zext Z Dϕint exp(Sint S1) ⇥ ⇣ 1 X

t

hδS(t)iext + 1 2 X

t,t0

hδS(t)δS(t0)iext + . . . ⌘ hδϕextiext = 0 hδS(t)δS(t0)iext ⌘ δϕ†

int,t∆(t t0)δϕint,t0

“hybridization function”

Simple example: real, scalar quantum field theory Cavity construction: After re-exponentiation and switching back from to we find the effective single-site action We call it “impurity action” (condensed-matter convention)

DMFT for quantum field theories

ϕ4

Akerlund, de Forcrand, Georges & Werner (2013)

Sext Sint S x t

Simp = X

t,t0

ϕtK1

imp,c(t − t0)ϕt0 + λ

X

t

(ϕ2

t − 1)2 − h

X

t

ϕt e K1

imp,c(ω)

= 1 − 2κ cos(ω) − e ∆(ω) h = 2φext(2κ(d − 1) − e ∆(0)) δϕ ϕ

because action written in terms of phi Fourier transform of nn interaction in time

Simp = X

t,t0

ϕtK1

imp,c(t − t0)ϕt0 + λ

X

t

(ϕ2

t − 1)2 − h

X

t

ϕt e K1

imp,c(ω)

= 1 − 2κ cos(ω) − e ∆(ω) h = 2φext(2κ(d − 1) − e ∆(0)) Simple example: real, scalar quantum field theory Dynamical mean field equations: Hybridization function and are fixed by self-consistency conditions

DMFT for quantum field theories

ϕ4

Akerlund, de Forcrand, Georges & Werner (2013)

Sext Sint S x t

∆ φext

Simple example: real, scalar quantum field theory Dynamical mean field equations: Hybridization function and are fixed by self-consistency conditions Local lattice Green’s function Impurity Green’s function e Gimp(ω) =

1 e K−1

imp,c(ω)+e

Σimp(ω) = 1 1−2κ cos(ω)−e ∆(ω)+e Σimp(ω)

DMFT for quantum field theories

ϕ4

Akerlund, de Forcrand, Georges & Werner (2013)

Sext Sint S x t

∆ φext

GF of the d-dimensional free theory

e Gloc(ω) = P

k 1 e G−1 (k,ω)+e Σ(k,ω)

e G−1

0 (k, ω) = 1 − 2κ Pd i=1 cos(ki)

DMFT approximation: identify lattice and impurity self-energy

Simple example: real, scalar quantum field theory Dynamical mean field equations: Hybridization function and are fixed by self-consistency conditions Local lattice Green’s function Self-consistency equations:

DMFT for quantum field theories

ϕ4

Akerlund, de Forcrand, Georges & Werner (2013)

Sext Sint S x t

∆ φext e Gloc(ω) = X 1 e G−1

imp(ω) + e

∆(ω) − 2κ Pd−1

i=1 cos ki

e Gimp(ω) = e Gloc(ω) hϕiSimp = φext

substitution yields an implicit equation for the hybridization function

Simple example: real, scalar quantum field theory Dynamical mean field equations: Hybridization function and are fixed by self-consistency conditions

DMFT for quantum field theories

ϕ4

Akerlund, de Forcrand, Georges & Werner (2013)

∆ φext

impurity solver momentum average Σlatt

k

≡ Σimp DMFT approximation DMFT self-consistency

S[∆, φext] Glatt

k

= [G−1

0,k + Σlatt k

]−1 Glatt

loc

Glatt

loc ⌘ Gimp, hϕi ⌘ φext

Gimp, Σimp, hϕi

Lattice model impurity model

Simple example: real, scalar quantum field theory Dynamical mean field phase diagram (d=3+1):

DMFT for quantum field theories

ϕ4

Akerlund, de Forcrand, Georges & Werner (2013) symmetry-broken phase symmetric phase

0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 1 2 3 4 5

• Monte Carlo, 324

Mean Field DMFT, L=75

Simple example: Bose-Hubbard model Mott-Insulator to superfluid transition at low temperature

DMFT for Bosons

Fisher, Grinstein, Weichmann & Fisher (1989)

H = −t X

hi,ji

b†

ibj + U

2 X

i

ni(ni − 1) − µ X

i

ni

3U t U

Simple example: Bose-Hubbard model Derivation of DMFT formalism analogous to the case Split action of the lattice model into To allow for symmetry breaking (condensation), write

DMFT for Bosons

Anders, Pollet, Gull, Troyer & Werner (2011)

ϕ4 S = Sint + ∆S + Sext bext = φext + δbext, bint = φint + δbint Sint = Z β dτ h − µb†

intbint + U

2 nint(nint − 1) i ∆S = −t Z β X

hint,exti

(b†

intbext + b† extbint)

Sext = Z β dτ h − µb†

extbext + U

2 next(next − 1) i

Simple example: Bose-Hubbard model Derivation of DMFT formalism analogous to the case Expand , integrate out external degrees of freedom End up with an impurity model

DMFT for Bosons

Anders, Pollet, Gull, Troyer & Werner (2011)

ϕ4 exp(−∆S)

U ()

• Hybridization function

(hopping of normal bosons) exchange of particles with the condensate

Simp = −1 2 Z β b†(τ)∆(τ − τ 0)b(τ 0) + Z β dτ h − µn(τ) + U 2 n(τ)[n(τ) − 1 i −κΦ† Z β dτb(τ)

Simple example: Bose-Hubbard model Derivation of DMFT formalism analogous to the case Expand , integrate out external degrees of freedom End up with an impurity model and fixed by the DMFT self-consistency condition

DMFT for Bosons

Anders, Pollet, Gull, Troyer & Werner (2011)

ϕ4 exp(−∆S)

U ()

• Hybridization function

(hopping of normal bosons) exchange of particles with the condensate

Φ = hb(τ)iSimp, Glatt

loc = Gimp,c

[= hTb(τ)b†(0)iSimp + ΦΦ†]

approximation: identification of lattice and impurity self-energy

∆ Φ

Nambu notation

Simple example: Bose-Hubbard model Derivation of DMFT formalism analogous to the case Results: Phase diagram (d=3+1)

DMFT for Bosons

Anders, Pollet, Gull, Troyer & Werner (2011)

ϕ4

Mott insulator n=1 Mott insulator n=2 superfluid T=0 superfluid Mott insulator normal

Simple example: Bose-Hubbard model Derivation of DMFT formalism analogous to the case Results: Connected density-density correlation functions (d=3+1)

DMFT for Bosons

Anders, Pollet, Gull, Troyer & Werner (2011)

ϕ4

DMFT lattice QMC

Simple example: single-band Hubbard model Describes Mott transition, magnetic and superconducting transitions

DMFT for Fermions

Gutzwiller, Kanamori, Hubbard (1963)

U t

H = −t X

hi,ji,σ

c†

iσcjσ + U

X

i

ni,"ni,# − µ X

i,σ

ni,σ

Simple example: single-band Hubbard model Dynamical mean field theory: Mapping to a quantum impurity model Hybridization functions fixed by DMFT self-consistency condition

DMFT for Fermions

Gutzwiller, Kanamori, Hubbard (1963) Georges & Kotliar (1992)

• U

()

Hybridization function (describes hopping of electrons into the lattice and back)

∆σ Glatt

loc,σ = Gimp,σ

approximation: identification of lattice and impurity self-energy

Simple example: single-band Hubbard model Dynamical mean field theory: Mapping to a quantum impurity model Various numerical approaches to solve the impurity problem: weak-coupling perturbation theory, strong-coupling perturbation theory, exact diagonalization, NRG, QMC, ...

DMFT for Fermions

Gutzwiller, Kanamori, Hubbard (1963) Georges & Kotliar (1992)

Σlatt ≡ Σimp Glatt ≡ Gimp

lattice model impurity model

• t

Simple example: single-band Hubbard model Dynamical mean field theory: Mapping to a quantum impurity model Single-site DMFT can treat two-sublattice order (e. g. AFM) Pure Neel order:

DMFT for Fermions

Gutzwiller, Kanamori, Hubbard (1963) Georges & Kotliar (1992)

t

Σlatt ≡ Σimp Glatt ≡ Gimp

lattice model impurity model

• BathB,σ[GA,σ],

BathA,σ[GB,σ] BathB,σ = BathA,¯

σ

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

DMFT for Fermions

“Mott” insulator metal U T

µ /t

1

µc1 U T (µµ ) 6.5 6 5.5 5 0.01 0.05

c2

µ

• Uc1

Uc2

Equilibrium DMFT phase diagram Paramagnetic calculation: Metal - Mott insulator transition at low T Doping-driven metallization upon increasing metal “Mott” insulator

DMFT for Fermions

µ

µ /t

1

µc1 U T (µµ ) 6.5 6 5.5 5 0.01 0.05

c2

µ

• Uc1

Uc2

Equilibrium DMFT phase diagram Paramagnetic calculation: Metal - Mott insulator transition at low T Doping-driven metallization upon increasing

DMFT for Fermions

µ “Mott” insulator metal

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

DMFT for Fermions

“Mott” insulator metal U T

paramagnetic Mott insulator has large entropy: metal-insulator transition upon increasing T

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

DMFT for Fermions

“Mott” insulator metal U T

curvature of the phase boundary changes if short-range spatial correlations are taken into account

+

“plaquette singlet” state dominates low-T insulator

Equilibrium DMFT phase diagram (half-filling) With 2-sublattice order: Antiferromagnetic insulator at low T Smooth crossover at high T

DMFT for Fermions

“Mott” insulator metal U T AFM insulator

Equilibrium DMFT phase diagram (half-filling) Transformation maps repulsive model onto attractive model U

DMFT for Fermions

metal ci↑ → c†

i↑

(i ∈ A), ci↑ → −c†

i↑

(i ∈ B) AFM insulator superconductor T repulsive attractive

Equilibrium DMFT phase diagram (half-filling) Half-filling: transformation maps repulsive model onto attractive model U

DMFT for Fermions

metal ci↑ → c†

i↑

(i ∈ A), ci↑ → −c†

i↑

(i ∈ B) AFM insulator superconductor T “BCS” “BEC” “Slater” “Mott” repulsive attractive

Low-dimensional systems DMFT is exact in Neglect of spatial fluctuations problematic in Hubbard model is believed to describe the physics of high-Tc (cuprate) superconductors

DMFT for Fermions

d = ∞

Metzner & Vollhardt (1989)

d < 3 d = 2

hole doping superconductor AFM Mott insulator pseudogap T metal

Low-dimensional systems DMFT is exact in Neglect of spatial fluctuations problematic in Hubbard model is believed to describe the physics of high-Tc (cuprate) superconductors

DMFT for Fermions

d = ∞

Metzner & Vollhardt (1989)

d < 3

to describe the physics of the cuprates, we need a cluster extension of DMFT, with at least 4 sites hole doping superconductor AFM Mott insulator pseudogap T metal

d = 2

Low-dimensional systems Cluster DMFT self-consistently embeds a cluster of sites into a fermionic bath If cluster is periodized: coarse-graining of the momentum-dependence “Tiling” of the Brillouin zone

DMFT for Fermions

Nc Σ(p, ω) =

• a

φa(p)Σa(ω)

Hettler, Prushke, Krishnamurthy & Jarrell (1998)

Low-dimensional systems Cluster DMFT self-consistently embeds a cluster of sites into a fermionic bath Doping driven insulator-metal transition in the 8-site cluster DMFT first 8% of dopants go into the B sector

DMFT for Fermions

Nc

B C

0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.2 0.4 0.6 0.8 1 1.2 1.4 n µ/t C B t=40, B t=40, C t=20, B t=20, C

Hettler, Prushke, Krishnamurthy & Jarrell (1998) Werner, Gull, Parcollet & Millis (2010)

Low-dimensional systems Cluster DMFT self-consistently embeds a cluster of sites into a fermionic bath Doping driven insulator-metal transition in the 8-site cluster DMFT first 8% of dopants go into the B sector

DMFT for Fermions

Nc

0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.2 0.4 0.6 0.8 1 1.2 1.4 n µ/t C B t=40, B t=40, C t=20, B t=20, C

Werner, Gull, Parcollet & Millis (2010) Pseudogap phase as a momentum- selective Mott state Hettler, Prushke, Krishnamurthy & Jarrell (1998)

Nonequilibrium DMFT

DMFT accurately treats time-dependent fluctuations can be directly applied to nonequilibrium systems Equilibrium: solve DMFT equations on the imaginary-time interval Nonequilibrium: solve DMFT equations on the Kadanoff-Baym contour

Freericks et al. (2006)

−iβ t

vertical branch: initial equilibrium state propagate and converge the DMFT solution step by step along the real-time axis Time-translation invariance: can use Fourier transformation Integral-differential equations of Volterra type

Field E(t) applied at t=0 Choose gauge with pure vector potential: Peierls substitution: Lattice: hybercubic, infinite-d limit

Nonequilibrium DMFT

U t

ε(k) → ε(k − eA(t)) ρ(ε) =

1 √πW exp(−ε2/W 2)

E(t) = −∂tA(t)

Freericks et al. (2006)

Dielectric breakdown of the Mott insulator

DC electric fields

Electric field (amplitude F) applied at t=0 in the body-diagonal

Eckstein, Oka, &Werner (2011); Eckstein & Werner (2013)

Dielectric breakdown of the Mott insulator Time evolution of the current and double occupancy

DC electric fields

build-up of a polarization

0.001 0.002 0.003 0.004 5 10 15 20 j(t)/F time F=1.0 F=0.8 F=0.6 F=0.4 a 0.011 0.013 0.015 0.017 5 10 15 20 d(t) time F=1.0 F=0.8 F=0.6 F=0.4 b 0.001 0.002 0.3 0.6 0.9 1.2 j/F F dht/F jt/F c 0.3 0.6

• 5

5

• A()

Eckstein, Oka, &Werner (2011); Eckstein & Werner (2013) quasi-steady state quasi-steady current follows a threshold-law:

j/F ∝ exp(−Fth/F)

Dielectric breakdown of the Mott insulator Time evolution of the current and double occupancy

DC electric fields

build-up of a polarization

0.001 0.002 0.003 0.004 5 10 15 20 j(t)/F time F=1.0 F=0.8 F=0.6 F=0.4 a 0.011 0.013 0.015 0.017 5 10 15 20 d(t) time F=1.0 F=0.8 F=0.6 F=0.4 b 0.001 0.002 0.3 0.6 0.9 1.2 j/F F dht/F jt/F c 0.3 0.6

• 5

5

• A()

Eckstein, Oka, &Werner (2011); Eckstein & Werner (2013) quasi-steady state quasi-steady current follows a threshold-law:

j/F ∝ exp(−Fth/F)

constant production of doublon-hole pairs

Dielectric breakdown of the Mott insulator Why is there a steady-state current if number of carriers increases? Doublons (and holes) quickly heat up to infinite temperature! no contribution to the current

DC electric fields

0.001 0.002 0.003 0.004 5 10 15 20 j(t)/F time F=1.0 F=0.8 F=0.6 F=0.4 a 0.011 0.013 0.015 0.017 5 10 15 20 d(t) time F=1.0 F=0.8 F=0.6 F=0.4 b 0.001 0.002 0.3 0.6 0.9 1.2 j/F F dht/F jt/F c 0.3 0.6

• 5

5

• A()

Eckstein, Oka, &Werner (2011); Eckstein & Werner (2013)

0.001 0.002 0.003 0.004 0.005 0.006 0.5 1 1.5 2 2.5 3 3.5 4 4.5 A<(,t)

• F=0.2

t=5 t=10 t=15 t=20 t=25 t=30 a 0.001 0.002 0.003 0.004 0.005 0.006 0.5 1 1.5 2 2.5 3 3.5 4 4.5 A<(,t)

• F=0.3

t=5 t=10 t=15 t=20 t=25 t=30 b

time-dependent occupation function compared to a rescaled spectral function (black)

AC-field quench in the Hubbard model (metal phase)

DC electric fields

U t

Electric field (amplitude E, frequency ) applied at t=0 Ω

Tsuji, Oka, Werner and Aoki (2011)

AC-field quench in the Hubbard model (metal phase)

Periodic electric fields

attractive (>0.25) repulsive (<0.25) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 1 2 3 4 5 6 7 8 double occupation time [inverse hopping] E/=1 E/=1.5 E/=2.5 E/=3 E/=4 Tsuji, Oka, Werner and Aoki (2011)

AC-field quench in the Hubbard model (metal phase) Sign inversion of the interaction: repulsive attractive Dynamically generated high-Tc superconductivity?

Periodic electric fields

temperature hole doping superconductor (repulsive interaction) superconductor (attractive interaction) metal

Periodic E-field leads to a population inversion Gauge with pure vector potential Peierls substitution Renormalized dispersion

Origin of the attractive interaction

k → k−A(t) k = Ω 2⇥ 2π/Ω dtk−A(t) = J0(E/Ω)k E(t) = E cos(Ωt) = −∂tA(t) ⇒ A(t) = −(E/Ω) sin(Ωt) E(t) J0(E/Ω) E/Ω

Periodic E-field leads to a population inversion Renormalized dispersion

Origin of the attractive interaction

k = Ω 2⇥ 2π/Ω dtk−A(t) = J0(E/Ω)k J0(E/Ω) = 1

Periodic E-field leads to a population inversion Renormalized dispersion

Origin of the attractive interaction

k = Ω 2⇥ 2π/Ω dtk−A(t) = J0(E/Ω)k J0(E/Ω) = 0.8

Periodic E-field leads to a population inversion Renormalized dispersion

Origin of the attractive interaction

k = Ω 2⇥ 2π/Ω dtk−A(t) = J0(E/Ω)k J0(E/Ω) = 0.3

Periodic E-field leads to a population inversion Renormalized dispersion

Origin of the attractive interaction

k = Ω 2⇥ 2π/Ω dtk−A(t) = J0(E/Ω)k J0(E/Ω) = −0.2

Inverted population = negative temperature State with is equivalent to state with Effective interaction of the state

Origin of the attractive interaction

U > 0, T < 0 U < 0, T > 0 Ueff = U J0(E/Ω) Teff > 0 ˜ T < 0, J0 < 0 ⇢ ∝ exp − 1 ˜ T "X

kσ

J0✏knkσ + U X

i

ni↑ni↓ #! Teff = ˜ T J0 > 0 = exp − 1 Teff "X

kσ

✏knkσ + U J0 X

i

ni↑ni↓ #!

Potentially interesting material: Sn doped In2O3 Transparent conductor Single s-band crossing the Fermi level

Experimental realization?

band structure by Bernard Delley (PSI)

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

Summary