Diagrammatic Green s Functions Approach to the Bose-Hubbard Model

Diagrammatic Green s Functions Approach to the Bose-Hubbard Model Matthias Ohliger Institut für Theoretische Physik Freie Universität Berlin 22nd of January 2008

Content OVERVIEW CONSIDERED SYSTEM BASIC IDEA Introduction Experimental realization Bose-Hubbard model Motivation

Content OVERVIEW CONSIDERED SYSTEM BASIC IDEA Introduction Experimental realization Bose-Hubbard model Motivation Imaginary-time Green s functions Hopping expansion Cumulant decomposition Diagrammatic rules

Content OVERVIEW CONSIDERED SYSTEM BASIC IDEA Introduction Experimental realization Bose-Hubbard model Motivation Imaginary-time Green s functions Hopping expansion Cumulant decomposition Diagrammatic rules Time-of-flight

Content OVERVIEW CONSIDERED SYSTEM BASIC IDEA Introduction Experimental realization Bose-Hubbard model Motivation Imaginary-time Green s functions Hopping expansion Cumulant decomposition Diagrammatic rules Time-of-flight Phase diagram Resummation Results and comparison with QMC

Content OVERVIEW CONSIDERED SYSTEM BASIC IDEA Introduction Experimental realization Bose-Hubbard model Motivation Imaginary-time Green s functions Hopping expansion Cumulant decomposition Diagrammatic rules Time-of-flight Phase diagram Resummation Results and comparison with QMC Excitation spectrum

Content OVERVIEW CONSIDERED SYSTEM BASIC IDEA Introduction Experimental realization Bose-Hubbard model Motivation Imaginary-time Green s functions Hopping expansion Cumulant decomposition Diagrammatic rules Time-of-flight Phase diagram Resummation Results and comparison with QMC Excitation spectrum Summary and outlook

Experimental realization OVERVIEW CONSIDERED SYSTEM BASIC IDEA Optical lattice produced by counter-propagating lasers V sin 2 (2πx/λ) Relative strength of hopping and interaction controllable (Quasi) one-, two-, and three-dimensional configurations possible 2 1.5 1 0.5 0-2 -1 0 1 2-2 -1 0 1 2

Bose-Hubbard model OVERVIEW CONSIDERED SYSTEM BASIC IDEA Bose-Hubbard Hamiltonian: Ĥ BHM =Ĥ 0 + Ĥ 1 Ĥ 0 = [ ] U 2 ˆn i(ˆn i 1) µˆn i i Ĥ 1 = J <i,j> â iâj = i,j J i,j â iâj ˆn i =â iâi { J if i,j nearest neighbors J ij = 0 otherwise

Bose-Hubbard model OVERVIEW CONSIDERED SYSTEM BASIC IDEA Bose-Hubbard Hamiltonian: Ĥ BHM =Ĥ 0 + Ĥ 1 Ĥ 0 = [ ] U 2 ˆn i(ˆn i 1) µˆn i i Ĥ 1 = J <i,j> â iâj = i,j J i,j â iâj Ĥ 0 site-diagonal: Ĥ 0 n = N S E n n ˆn i =â iâi { J if i,j nearest neighbors J ij = 0 otherwise E n = U n(n 1) µn 2 Perturbative expansion in Ĥ 1

Motivation OVERVIEW CONSIDERED SYSTEM BASIC IDEA Green s functions contain many important information about the system: Quantum phase diagram Time-of-flight pictures Excitation spectra Thermodynamic properties

Motivation OVERVIEW CONSIDERED SYSTEM BASIC IDEA Green s functions contain many important information about the system: Quantum phase diagram Time-of-flight pictures Excitation spectra Thermodynamic properties Motivated by: F. Nogueira s Primer to the Bose-Hubbard model Diagrammatic calculations for Fermions by W. Metzner, PRB 43, 8549 (1993)

Imaginary-time Green s function IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES Definition: G 1 (τ,j τ,j) = 1 Z Tr {e βĥ ˆT with Z = Tr{e βĥ } Heisenberg operators in imaginary time ( = 1): Ô H (τ) = eĥτ Ôe Ĥτ [ ]} â j,h (τ)â j,h (τ )

Dirac interaction picture IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES Time evolution of operators determined only by Ĥ 0 : Ô D (τ) = eĥ0τ Ôe Ĥ 0 τ Dirac time evolution operator calculated by Dyson series: Û D (τ,τ 0 ) = τ ( 1) n n=0 = ˆTexp ( τn 1 dτ 1... dτ n Ĥ 1D (τ 1 )... Ĥ 1D (τ n ) τ 0 τ 0 ) dτ 1 Ĥ 1D (τ 1 ) τ 0 τ

Partition function IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES Full partition function: Z = Tr { } e βĥ 0 Û D (β,0)

Partition function IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES Full partition function: nth order contibution: Z (n) = 1 n! Z(0) Z = Tr i 1,j 1,...,i n,j n J i1j 1... J inj n { } e βĥ 0 Û D (β,0) β Unperturbed n-particle Green s function: 0 dτ 1 β 0 β dτ 2... dτ n 0 G (0) n (τ 1, j 1 ;...;τ n, j n τ 1, i 1 ;...;τ n, i n ) (0) G (0) n (τ 1, i 1 ;...;τ n, i n τ 1, i 1 ;... ; τ n, i n ) = ˆTâ i 1(τ 1 )â i 1 (τ 1 )...â i (τ n n )â i n (τ n )

Cumulant decomposition IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES Decompose G (0) n (τ 1,i 1 ;... ;τ n,i n τ 1,i 1 ;... ;τ n,i n ) into simple parts Ĥ 0 not harmonic Wick s theorem not applicable

Cumulant decomposition IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES Decompose G (0) n (τ 1,i 1 ;... ;τ n,i n τ 1,i 1 ;... ;τ n,i n ) into simple parts Ĥ 0 not harmonic Wick s theorem not applicable But: Decomposition into cumulants Ĥ 0 site-diagonal cumulants local. Example: G (0) 2 (τ 1, i 1; τ 2, i 2 τ 1, i 1 ; τ 2, i 2 ) = δ i1,i 2 δ i 1,i 2 δ i 1,i 1 C(0) 2 (τ 1, τ 2 τ 1, τ 2 ) + δ i1,i 1 δ i 2,i 2 C(0) 1 (τ 1 τ 1)C (0) 1 (τ 2 τ 2) + δ i1,i 2 δ i 2,i 1 C(0) 1 (τ 2 τ 1)C (0) 1 (τ 1 τ 2)

Cumulant decomposition IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES Decompose G (0) n (τ 1,i 1 ;... ;τ n,i n τ 1,i 1 ;... ;τ n,i n ) into simple parts Ĥ 0 not harmonic Wick s theorem not applicable But: Decomposition into cumulants Ĥ 0 site-diagonal cumulants local. Example: G (0) 2 (τ 1, i 1; τ 2, i 2 τ 1, i 1 ; τ 2, i 2 ) = δ i1,i 2 δ i 1,i 2 δ i 1,i 1 C(0) 2 (τ 1, τ 2 τ 1, τ 2 ) + δ i1,i 1 δ i 2,i 2 C(0) 1 (τ 1 τ 1)C (0) 1 (τ 2 τ 2) + δ i1,i 2 δ i 2,i 1 C(0) 1 (τ 2 τ 1)C (0) 1 (τ 1 τ 2) Denote contributions diagrammatically: Points for cumulants, lines for hopping matrix elements Perturbation theory in number of lines

Calculation of cumulants IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES Generating functional: ( β C (0) 0 [j, j ] = log ˆTexp dτj (τ)â(τ) + j(τ)â (τ) Cumulants calculated by functional derivatives: C (0) n (τ 1,...,τ n τ 1,..., τ n ) = 0 ) (0) δ 2n δj(τ 1 )...δj(τ n)δj (τ 1 )...δj (τ n ) C(0) 0 [j, j ] j=j =0

Calculation of cumulants IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES One- and two-particle cumulants: C (0) 1 (τ, τ) = ˆTâ (τ )â(τ) (0) = 1 Z (0) = 1 Z (0) n=0 n ˆTâ (τ )â(τ) n e βen n=0 [ Θ(τ τ )(n + 1)e (En En+1)(τ τ ) +Θ(τ τ)ne (En En 1)(τ τ) ] e βen C (0) 2 (τ 1, τ 2 τ 1, τ 2 ) = ˆTâ (τ 1)â (τ 2)â(τ 1 )â(τ 2 ) (0) C (0) 1 (τ 1, τ 1 )C (0) 1 (τ 2, τ 2 ) C (0) 1 (τ 1, τ 2 )C (0) 1 (τ 2, τ 1 )

Diagrammatic rules for Z (n) IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES 1 Draw all possible combinations of vertices with total n entering and n leaving lines 2 Connect them in all possible ways and assign time variables and hopping matrix elements onto the lines 3 Sum over all site indices and integrate all time variables from 0 to β =J ij τ i τ =C (0) 1 (τ τ) =C (0) 2 (τ 1, τ 2 τ 1, τ 2 )

Calculation of free energy IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES Grand-canonical free energy: F = 1 β log Z, F(0) = N S β log Z(0), Z (0) = n e βen

Calculation of free energy IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES Grand-canonical free energy: F = 1 β log Z, F(0) = N S β log Z(0), Z (0) = n e βen Contains only connected vacuum diagrams

Calculation of free energy IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES Grand-canonical free energy: F = 1 β log Z, F(0) = N S β log Z(0), Z (0) = n e βen Contains only connected vacuum diagrams First correction: F (2) = 1 2β i τ 1 τ 2 = N S2DJ 2 UZ (0)2 j = 1 2β n,k β β J i,j J j,i dτ 1 i,j 0 [ ] (n + 1)k n(k + 1) + e β(en+e k) k n + 1 n k + 1 0 dτ 2 C (0) 1 (τ 2 τ 1 )C (0) 1 (τ 1 τ 2 )

IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES Diagrammatic rules for Green s functions G 1 (τ, i τ, i) = 1 } {e Z Tr βĥ 0 ˆTâ i (τ )â i (τ)û D (β, 0) G (n) 1 (τ, i τ, i) = Z(0) 1 J i1j Z n! 1... J inj n i 1,j 1,...,i n,j n β 0 β dτ 1... dτ n 0 G (0) n+1 (τ 1, j 1 ;... ; τ n, j n ; τ, i τ 1, i 1 ;... ; τ n, i n, τ, i)

IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES Diagrammatic rules for Green s functions G 1 (τ, i τ, i) = 1 } {e Z Tr βĥ 0 ˆTâ i (τ )â i (τ)û D (β, 0) G (n) 1 (τ, i τ, i) = Z(0) 1 J i1j Z n! 1... J inj n i 1,j 1,...,i n,j n β 0 β dτ 1... dτ n 0 G (0) n+1 (τ 1, j 1 ;... ; τ n, j n ; τ, i τ 1, i 1 ;... ; τ n, i n, τ, i) Diagrams have external lines with fixed time and site variables

IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES Diagrammatic rules for Green s functions G 1 (τ, i τ, i) = 1 } {e Z Tr βĥ 0 ˆTâ i (τ )â i (τ)û D (β, 0) G (n) 1 (τ, i τ, i) = Z(0) 1 J i1j Z n! 1... J inj n i 1,j 1,...,i n,j n β 0 β dτ 1... dτ n 0 G (0) n+1 (τ 1, j 1 ;... ; τ n, j n ; τ, i τ 1, i 1 ;... ; τ n, i n, τ, i) Diagrams have external lines with fixed time and site variables Disconnected diagrams cancelled by Z (0) /Z

IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES Diagrammatic rules for Green s functions G 1 (τ, i τ, i) = 1 } {e Z Tr βĥ 0 ˆTâ i (τ )â i (τ)û D (β, 0) G (n) 1 (τ, i τ, i) = Z(0) 1 J i1j Z n! 1... J inj n i 1,j 1,...,i n,j n β 0 β dτ 1... dτ n 0 G (0) n+1 (τ 1, j 1 ;... ; τ n, j n ; τ, i τ 1, i 1 ;... ; τ n, i n, τ, i) Diagrams have external lines with fixed time and site variables Disconnected diagrams cancelled by Z (0) /Z Zeroth and first order: G (0) 1 (τ, i τ, j) = G (1) 1 (τ, i τ, j) = i τ τ = δ i,jc (0) 1 (τ τ) i j τ τ 1 τ = Jδ d(i,j),1 β 0 dτ 1 C (0) 1 (τ τ 1 )C (0) 1 (τ 1 τ)

Calculations in Matsubara space IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES Translational invariance in time suggests Matsubara transform: C (0) 1 (ω m) = 1 [ ] (n + 1) n e βen, ω Z (0) m = 2π E n+1 E n iω m E n E n 1 iω m β m n

Calculations in Matsubara space IMAGINARY-TIME GREEN S FUNCTION CUMULANT DECOMPOSITION DIAGRAMMATIC RULES AND EXAMPLES Translational invariance in time suggests Matsubara transform: C (0) 1 (ω m) = 1 [ ] (n + 1) n e βen, ω Z (0) m = 2π E n+1 E n iω m E n E n 1 iω m β m n In rule 3 integration over τ replaced by summation over ω m under consideration of frequency conservation on vertices: G (1) 1 (ω; i, j) = i j ω ω ω = Jδ d(i,j),1c (0) 1 (ω)2 G (2) 1 (ω; i, j) = i k j ω 1 ω 1 + ω ω ω ω ω i ω =J 2 (δ d(i,j),2 + 2δ d(i,j), 2 + 2Dδ i,j )C (0) 1 (ω)3 + J 2 2Dδ i,j 1 (ω 1)C (0) 2 (ω, ω 1 ω, ω 1 ) ω 1 C (0) k

Equal-time correlations TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Momentum space density: ˆn k = ˆψ (k) ˆψ(k) = w(k) 2 e ik(ri rj) â iâj, â iâj = lim G 1(τ, i 0, j) τ ր0 i,j }{{} S(k)

Equal-time correlations TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Momentum space density: ˆn k = ˆψ (k) ˆψ(k) = w(k) 2 e ik(ri rj) â iâj, â iâj = lim G 1(τ, i 0, j) τ ր0 i,j }{{} Quasi-momentum distribution: S(k) J(k) S(k) = K 0 + K 1 U + K J 2 (k) 2 U 2 +..., J(k) = 2J 3 cos(k l a) l=1

Equal-time correlations TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Momentum space density: ˆn k = ˆψ (k) ˆψ(k) = w(k) 2 e ik(ri rj) â iâj, â iâj = lim G 1(τ, i 0, j) τ ր0 i,j }{{} Quasi-momentum distribution: S(k) J(k) S(k) = K 0 + K 1 U + K J 2 (k) 2 U 2 +..., J(k) = 2J Lowest orders: G (0) 1 (0, i j, 0) = δ ij Z (0) G (1) 1 (0, i j, 0) =J2δ d(i,j),1 UZ (0)2 n ne βen n,k 3 cos(k l a) l=1 n(n + 1) (n k + 1)(k n + 1) e β(en+e k)

Time-of-flight pictures TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM first-order calculation (T = 0) second-order calculation (T = 0) experimental absorption pictures (a) V 0 = 8E R, (b) V 0 = 14E R, (c) V 0 = 18E R, and (d) V 0 = 30E R

Phase transition TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Phase transition requires diverging Green s function. Not possible in perturbation theory

Phase transition TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Phase transition requires diverging Green s function. Not possible in perturbation theory Solved by resummation: G 1 (ω; i, j) = ω i ω + i j ω ω ω + i k j ω ω ω ω + i k h j ω ω ω ω ω +...

Phase transition TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Phase transition requires diverging Green s function. Not possible in perturbation theory Solved by resummation: G 1 (ω; i, j) = ω i ω + i j ω ω ω Summed most easily in Fourier space: + i k j ω ω ω ω + i k h j ω ω ω ω ω +... G (1) 1 (ω,k) = C (0) 1 (ω) 1 J(k)C (0) 1 (ω), G (1) 1 (0,0) = C (0) 1 (0) 1 2DJC (0) 1 (0)

Phase transition TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Phase transition requires diverging Green s function. Not possible in perturbation theory Solved by resummation: G 1 (ω; i, j) = ω i ω + i j ω ω ω Summed most easily in Fourier space: + i k j ω ω ω ω + i k h j ω ω ω ω ω +... G (1) 1 (ω,k) = C (0) 1 (ω) 1 J(k)C (0) 1 (ω), G (1) 1 (0,0) = C (0) 1 (0) 1 2DJC (0) 1 (0) Neglected contributions like D k ω 1 ω 1 ω i ω vanish at least as 1/D for

Phase boundary TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Phase boundary given by: 2DJ c = ( n e βen n+1 n e βen E n+1 E n ) T 0 = n E n E n 1 1 n+1 E n+1 E n n E n E n 1

Phase boundary TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Phase boundary given by: 2DJ c = ( n e βen n+1 n e βen E n+1 E n ) T 0 = n E n E n 1 1 n+1 E n+1 E n Same result as obtained by mean-field theory (z = 2D): Jz U 0.2 0.175 0.15 0.125 superfluid k B T/U = 0 k B T/U = 0.02 k B T/U = 0.1 n E n E n 1 0.1 0.075 0.05 0.025 Mott insulator 1 2 3 4 µ U

Beyond mean field TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Replace by sum over all one-particle irreducable diagrams: ( = + + ) +...

Beyond mean field TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Replace by sum over all one-particle irreducable diagrams: ( = + + ) +... Full Green s function obtained by Dyson series: ( ) l+1 G 1 (ω,k) = J(k) l l=0

Beyond mean field TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Replace by sum over all one-particle irreducable diagrams: ( = + + ) +... Full Green s function obtained by Dyson series: ( ) l+1 G 1 (ω,k) = J(k) l Approximated by considering only the first two terms in l=0

Calculation of one-loop diagram TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM k ω 1 ω 1 ( 2Dδ i,j J 2 G (2B) 1 (ω) = ω i ω = 2Dδ i,j 1 Z (0)2 U 2 e β(en+e k) n,k { [ (k + 1)(n 1)n k 2 + 2(n 1) 2 µ 2 + 2k(2 2n µ) ] + β + 7 more terms { n,k C (0) 1 (ω) n,k (k n + 1) 2 (k 2n + µ)(1 n + µ) 2 } C (0) 1 (ω)3 [ ][ (n + 1)k n(k + 1) n + 1 + k n + 1 n k + 1 n µ iω [ ] (n + 1)k n(k + 1) + e β(en+e k) k n + 1 n k + 1 n n 1 µ iω }) ] e β(en+e k)

One-loop corrected phase boundary TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM 2nd order resummed Green s function: G (2) 1 (ω,k = 0) = C (0) 1 (ω) 1 J2DC (0) 1 (ω) + J2 2DG (2B) 1 (ω)/c (0) 1 (ω)

One-loop corrected phase boundary TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM 2nd order resummed Green s function: G (2) 1 (ω,k = 0) = C (0) 1 (ω) 1 J2DC (0) 1 (ω) + J2 2DG (2B) 1 (ω)/c (0) 1 (ω) Phase boundary { } J c = 1/ C (0) 1 (0)D 2DG (2B) 1 (0)/C (0) 1 (0) + D2 C (0)2(0) 1 In T =0 limit same result as E. Santos

TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Comparison with simulations for T = 0 J/U J/U 0.06 0.030 0.025 0.020 0.015 0.010 0.005 D = 3 0.2 0.4 0.6 0.8 1.0 Capogrosso-Sansone et. al. PRB 75, 134302 (2007) n = 1. Black: First order (Mean field) Red: Second order µ/u 0.05 0.04 0.03 0.02 0.01 D = 2 0.2 0.4 0.6 0.8 1 µ/u Capogrosso-Sansone et. al. arxiv:0710.2703v3 (2007) 0.15 0.1 0.05 J/U D = 1 0.2 0.4 0.6 0.8 1 Niyaz, et. al. PRB 44, 7143 (1991) µ/u

Finite-temperature phase diagram TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM J/U 0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.000 0.0 0.5 1.0 1.5 2.0 2.5 3.0 µ/u J/U 0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.000 0.0 0.5 1.0 1.5 2.0 2.5 3.0 µ/u D = 3. Black: 1st order. Red: 2nd Order. Left: T = 0.1 U. Right: T = 0.02 U Temperature effects small at tip of lobe Second-order correction largest at zero temperature

Real-time Green s function TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Dynamic properties determined by retarded Green s function in real-time: G 1 (t, j, t, j) = Θ(t t ) i {e [ ]} Z Tr βĥ â j,h (t), â j,h (t ) Can be obtained by analytic continuation of imaginary-time result. Easily done by replacing Zeroth order: G (0) 1 (ω; i, j) = δ i,j Z (0) n [ iω m ω + iη ] (n + 1) E n+1 E n ω iη n e βen E n E n 1 ω iη

Excitation spectrum TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Excitation spectrum given by poles of real-time Green s function

Excitation spectrum TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Excitation spectrum given by poles of real-time Green s function For T = 0: [ G (1) 1 (ω,k)] 1! = 0 = ω 1,2 = U 2 (2n 1) µ J(k) ± 1 2 U2 UJ(k)(4n + 2) + J(k) 2 Different signs correspond to particle and hole excitations

Excitation spectrum TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Excitation spectrum given by poles of real-time Green s function For T = 0: [ G (1) 1 (ω,k)] 1! = 0 = ω 1,2 = U 2 (2n 1) µ J(k) ± 1 2 U2 UJ(k)(4n + 2) + J(k) 2 Different signs correspond to particle and hole excitations Dispersion relation of pairs: ω ph (k) = ω 1 (k) ω 2 (k) = U 2 UJ(k)(4n + 2) + J(k) 2

Excitation spectrum TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM ω ph /U 1.0 0.8 0.6 0.4 0.2 E gap/u 1 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 n = 1, T = 0. Black: J = 0 Red: 2DJ = 0.13 U Blue: 2DJ = 2DJ c = 0.172 U k/a T/U 0.02 0.03 0.04 0.05 0.06 0.07 0.08 n = 1. Black: J = 0 Red: 2DJ = 0.13 U Blue: 2DJ = 0.17 U

Excitation spectrum TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM ω ph /U 1.0 0.8 0.6 0.4 0.2 E gap/u 1 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 n = 1, T = 0. Black: J = 0 Red: 2DJ = 0.13 U Blue: 2DJ = 2DJ c = 0.172 U k/a T/U 0.02 0.03 0.04 0.05 0.06 0.07 0.08 n = 1. Black: J = 0 Red: 2DJ = 0.13 U Blue: 2DJ = 0.17 U Gap vanishes when hopping reaches critical value. J c : Hopping at tip of lobe Gap experimentally measurable. Could serve as thermometer.

Summary TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Green s function contain necessary information about finite-temperature properties of Bose-Hubbard model Hopping expansion provides perturbative access to Green s function in Mott regime Time-of-flight pictures for deep lattices agree with experiment

Summary TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Green s function contain necessary information about finite-temperature properties of Bose-Hubbard model Hopping expansion provides perturbative access to Green s function in Mott regime Time-of-flight pictures for deep lattices agree with experiment Diagrammatic representation facilitates resummation One-loop corrected phase diagram in good agreement with Quantum Monte-Carlo data

Summary TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Green s function contain necessary information about finite-temperature properties of Bose-Hubbard model Hopping expansion provides perturbative access to Green s function in Mott regime Time-of-flight pictures for deep lattices agree with experiment Diagrammatic representation facilitates resummation One-loop corrected phase diagram in good agreement with Quantum Monte-Carlo data Excitation spectrum shows characteristic vanishing of gap at critical hopping

Outlook TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Transition between Mott insulator and normal gas

Outlook TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Transition between Mott insulator and normal gas Excitation spectrum in 2nd order Excitation spectrum as thermometer

Outlook TIME-OF-FLIGHT PHASE TRANSITION EXCITATION SPECTRUM Transition between Mott insulator and normal gas Excitation spectrum in 2nd order Excitation spectrum as thermometer Green s function within superfluid phase: near phase boundary with Landau expansion, far away with Bogoliubov theory