Dynamical mean field approach in interacting electron-electron and electron phonon systems: an introduction with some selected applications (II) S. Ciuchi Uni/INFM Aquila Collaborations M. Capone - ISC CNR Rome P. Carta - Cassa Depositi e Prestiti Rome G. Sangiovanni - MPI Stuttgart S. Paganelli - Uni Rome I P. Paci - SISSA/ISAS Tieste E. Cappelluti - ISC CNR Rome Summary Spinless Polaronic Crossover vs Spinful MIT in Holstein Model Half filling T = DMFT Holstein Model Half filling T > DMFT Mott-Hubbard vs Bipolaronic MITs in the Holstein-Hubbard Model Normal Phase T =
The Holstein-(Hubbard) Model H = Σ i a i a i Einstein phonons t 2 z Σ ijc i c j tight binding electrons gσ i (c i,σ c i,σ n)(a i +a i) local el-ph interaction +UΣi n i, n i, U 2 (n i, + n i, ) local el-el interaction Adiabatic ratio γ = /D: ratio of phonon energy to halfbandwidth (D) e-ph Coupling constant - Adiabatic γ < : λ = 2g 2 / D: ratio of bipolaron binding energy (spinful), (2) polaron energy (spinless) (ɛ p = g 2 / ) to halfbandwidth (D) - Adiabatic γ < : α = g/ : ratio of polaron energy to phonon energy ( ) e-e Coupling constant U/D: ratio of Hubbard repulsion to half-bandwidth (D) Spinless and Spinful cases allows to disentagle Polaron and Bipolaron formation and stability
DMFT Solution for classical phonons [Millis Mueller Shraiman PRB 96] but also briefly in [Freericks Jarrel Scalapino PRB 93] P(X) 4.5 4 3.5 3 2.5 2.5.5 Phonon PDF (spinless) > -3-2 - 2 3 X λ=.5 λ=.3 λ=.5 λ=2.5 > DOS.8.6.4.2 electron DOS (spinless) Polaron binding energy λ=.5 λ=.3 λ=.5 λ=2.5-3 -2-2 3 (Bi)Polaron xover as bimodality of phonon PDF (λ X =.8(.59)) MIT as vanishing of Quasi particle spectral weight (λ MIT =.328(.664))
Holstein model: Adiabatic Phase diagram at half-filling T g T/D.. Normal CDW (Bi)Polarons T c (Bi)Polarons Charge Ordering 2 3 4 5 6 We consider the normal Phase up to T = Spinless Polarons at strong coupling Spinful Bipolarons at strong coupling λ
How to include quantum phonon fluctuations? Solve Scröd. eq. in the adiabatic potential i.e. add phonon kinetic energy term P 2 /2M Take the phonon ground state wave function Ψ(X) 2 as an adiabatic phonon PDF. The local Green function is G() = dxp(x) G () + gx.8 Adiabatic (Bi)Polaron Quantum Flucts. binding energy DOS.6.4.2-3 -2-2 3 Spinless and Spinful Holstein model give the same results provided the adiabatic scaling of e-ph λ No quasi particle peak at intermediate coupling enhancing of the bandwidth of the lower and upper Holstein band due to phonon quantum fluctuations
Spectral function for spinful system results from an approximate theory ρ().4.3.2 λ=.5 λ=.25 λ=.36 λ=.375 λ=.4 λ=2. λ=4... 7. 3.5. 3.5 7. from [Benedetti Zeyer PRB 98] results from NRG g=.3 g=.8 g=.98 g=.2 ρ().5 - -.5.5 from [Meyer Hewson Bulla PRL 22] Peak at low energy Upper and lower Holstein bands at high energy quite well described by adibatic approximation
Polarons and Bipolarons behaves different even without Coulomb repulsion Question Quantum Fluctuations of phonons are able to stabilize metallic behaviour up to strong coupling? Answer Yes for all coupling and for all phonon frequency in the spinless case Answer Not for all coupling and for all phonon frequency in the spinful case Polaron Formation Is ruled by the formation of a definite polarization (multimodal PDF at half filling) around a given electron Pairs can be formed due to phonon mediated attraction even without a definite polarization associated to the pair Pairs and Bipolarons are not the same thing
Equivalent Anderson Hamiltonian The Weiss Green Function G can be obtained as the Green function of an Anderson Hamiltonian introducing a set of auxiliary fermions c k H = Σ k,σ E k c k,σ c k,σ + k,σ V k c σc k,σ + c k,σ c σ Equivalent Hamiltonian is H AIM = H + V where V is the local interaction (e.g. Holstein and/or Hubbard) Weiss Field where Σ h () = de (E) E (E) = k V k 2 δ(e E k )
Adiabatic Regime T=: Born Oppenheimer scheme In adiabatic regime for U = the functional integral is gaussian and can be done analitically Phonon PDF P(X) = Z exp( βv ad(x)) = Z e βkx2 /2 βgn 2 Adiabatic potential G Π n (i n ) + gx i n s V ad (X) = 2 kx2 s β n log(g (i n ) + gx) + const. BO step-: Solve Scröd. eq. in the adiabatic potential i.e. add phonon kinetic energy term P 2 /2M BO step-2: Change phonon harmonic oscillator basis to the BO basis BO step-3: Project the equivalent Anderson Impurity Model onto the lowest Born-Oppenheimer phonon states BO step-4: Solve projected Anderson Impurity model numerically (BO-ED) or with an analitic approximation (BO-CPA) BO step-5: Determine new bath Green function G (i n ) and new adiabatic potential, goto step-
Born Oppenheimer approximation: Two State Phonon Model (TSPM) H AIM = E k c k,σc k,σ + ( V k f σ c k,σ + c ) k,σf σ + bath + Hybridiz k,σ k,σ σ x + Phonon tunneling ɛ σ f σ + f σ σ z + 2 el-ph interaction +Un n U 2 (n + n ) el-el interaction Low-lying phonon states Weak e-ph coupling Strong e-ph coupling Parameters ɛ = (g s/2) < + a + a > = ( /2)(< + a a + > < a a >) Limits local el-ph interaction Phonon tunneling Adiabatic limit: the phonon tunneling go to zero. Recover the adiabatic solution Weak e-ph coupling: Phonon transitions involves a single phonon exchange, truncation is qualitatively correct Strong e-ph coupling: ɛ λ gives the correct scale of the coupling; freezing the phonon quantum flucts.
Excitations of the Two State Phonon Model (TSPM) U= Weak e-ph coupling Levels of TSPH V k = are strongly hibridized with electron s bath Strong e-ph coupling Levels of TSPH V k = are weakly hibridized with electron s bath Weak e ph coupling Strong e ph coupling η Spinless > > η = ɛ 2 + 2 Polaronic States with and electron are degenerate Hybridization leads to = resonance Weak e ph coupling Strong e ph coupling Spinful u> v> v> η u> Bipolaronic states ( u >) with and 2 electron are degenerate Polaronic states ( v >) with one electron are excited states they are associated at smaller lattice polarization Single electron excitations must pay a deformation energy Hybridization is effective to stabilize a metallic solution only up to a critical value of the e-ph coupling
Born-Oppenheimer Coherent Potential Approximation During each hybridization process the BO-phonon quantum number remains constant G() = 2 α=± G a,α() Σ h () G a,α () Propagator in absence of hybridization with a given BOphonon state α Σ h () is the Weiss field due to the action of hybridization V k Spinless Spinful.8.8 ρ el.6.4 ρ el.6.4.2.2 ρ el.8.6.4.5.4.3.2. -Im Σ -.4 -.2.2.4 ρ el.8.6.4 -Im Σ.4.3.2. -.4 -.2.2.4.2.2-2 -.5 - -.5.5.5 2-2 -.5 - -.5.5.5 2 Spinless CPA is qualitatively correct at any energy scale, quasiparticle peak at any e-ph coupling strenght Spinful Gives a qualitative picture of the high energy scale, no quasiparticle peak at strong e-ph coupling
Adiabatic regime: DOS γ =. Spinless (upper), γ =.2 Spinful (lower)..8 ρ el.6.4.2..8 ρ el.6.4.2. -... -... -... -... At a given e-ph coupling using ED we have more spectral weight located at low energy in the spinless case High energy spectral properties are well described by BO-CPA Spikey features of ED due to discretization of bath energies (N s up to depending on the e-ph coupling)
Adiabatic regime: The quasi particle spectral weight Spinless /D=. Z...8 Ren. ph. freq...6.4..2.5.5 2 e-5.5.5 2 λ ED Spinful /D=.. Z...8 Ren. ph. freq...6 e-5 e-6.4.2.2.4.6.8 e-7.2.4.6.8 λ ED
Adiabatic regime: Phonon PDF γ =. Spinless (upper), γ =.2 Spinful (lower) P(X) P(X).6.5.4.3.2...5.4.3.2.. -3-2 - 2 3-3-2-2 3-3-2-2 3-3-2-2 3 X X X X BO approximation works well in both weak and strong coupling Electron phonon states entangled across the (Bi)polaron crossover
Adiabatic regime: Optimized BO phonon basis λ =.5 γ =.2 Spinless.8 2.6.4.2.8.6.4.2 3 2.5 2.5.5-2 -.5 - -.5.5.5 N ph =2 N ph =4 N ph =8 N ph =6 N ph =3-2 -.5 - -.5.5.5 N ph =2 (BO) N ph =4 (BO) N ph =3 At a given number of discretized bath energies (N s ) using BO phonon basis increases the convengency Low energy convergence is obtained even with few BO states
Antiadiabatic regime: Lang Firsov canonical transfomation Perform Lang-Firsov transformation in the Anderson impurity equivalent model S = α σ (f σf σ 2 )(a a). exp(s)a exp( S) = a + σ (f σf σ 2 ) exp(s)f σ exp( S) = f σ exp(α(a a)) exp(s)h exp( S) = k,σ e α(a a) V k (c k,σf σ + h.c.) + + k,σ E k c k,σc k,σ 2 g2 (s )n n g2 σ ( 2 f σ f σ) + a a, interaction term simplyfied spinless simple polaronic shift (ɛ P ) spinful Negative U attraction (U = 2ɛ P ) hybridization term complicated by the polaron motion
Antiadiabatic regime: Holstein-Lang Firsov approximation Anti-adiabatic regime: phonons are much faster than electron el. density n is almost a constant of motion move phonon coordinate X at given n displaced oscillator parametrically dependent on n average exp(s)h exp( S) on the new phonon ground state ( >) to approximate the hybridization term in H AIM e α(a a) V k (c k,σf σ + h.c.) e α2 /2 V k (c k,σf σ + h.c.) k,σ k,σ spinless non interacting impurity hybridized with a bath of conduction electrons trough an exponentially reduced hybridization spinful case the impurity has an on site interaction term of the negative U type (U = 2g 2 / ) G() = e α2 G p () + e α2α2 n 2 n n! G p( n ). G p () polaron impurity Green function Self consistency condition spinless G p () = e α2 t 2 4 G(). spinful case CPA to solve the negative U G p () = 2 e α2 t 2 4 G() U 2 + e α2 t 2 4 G() + U 2
Antiadiabatic regime: DOS γ = 2. Spinless (upper), γ = 4. Spinful (lower).2 ρ el.8.4..2 ρ el.8.4. -2-2 -2-2 -2-2 -2-2 Spinless /D=. Spinful /D=.. Ren. Ph. freq.. Ren. Ph. freq. Z Z.... 2 3 4 5. 2 3 4 5 6 λ 6 ED..4.8.2.6..2.4.6.8.2.4.6.8 λ ED
Holstein Model: ground state at half-filling (no Symmetry Breaking) 5 4 8 Pairs 3 Normal 6 Normal γ 2 4 Polarons 2 Bipolarons 2 3 4 5 6 7 8 2 λ λ Spinless Polaron Xover depends strongly on adiabaticity Spinless No Polaron without an induced polarization Spinful Pairs MIT depends weakly on adiabaticity 3 4 Spinful Pairs without a definite induced polarization in the antiadiabatic regime
Holstein-(Hubbard) Model at T>: the action S = 2 β dτ ẋ2 (τ) 2 + x 2 (τ) + Einstein phonons + β dτ β dτ σ c σ(τ)g (τ τ )c σ (τ ) + electrons λ β dτx(τ) (n(τ) ) el-ph interaction ( (U/2) β dτ (n (τ) n(τ) ) 2) el-el interaction Solve by QMC thru Blackenbecker-Sugar-Scalapino algorithm (integration of electrons), No sign problem
Holstein Model at T>: adiabatic vs anti-adiabatic regime Adiabatic X Classical field Antiadiabatic n classical field, S e ph = λx β dτ(n(τ) ) S e ph = λ(n ) β dτx(τ) X Statonovich-Hubbard field non-retarded el-el interaction (spinful) Charge fuctuations are coupled to the n = component of the fluctuating phonon field the Centroid or Classical Coordinate X c = β β X(τ)
Distribution of the n = component of the phonon field vs phonon PDF Phonon PDF becomes bimodal at very large value of the coupling λ.6.4.2 P(x) =8D T=.25D L=64 (D=) λ=. λ=2. λ=2.4 λ=2.8 λ=3.4 λ=3.8.8.6.4.2-4 -3-2 - 2 3 4 5 Centroid PDF becomes bimodal around λ precursor of the pairing MIT 6 5 4 P(centroid) at =8D and β=8 λ=. λ=.9 λ=.8 λ=.7 λ=.6 λ=.5 3 2-4 -3-2 - 2 3 4 x
Antiadiabatic-regime T>: double occupancy < n n >.55.5.45.4.35 T=.25D T=.5D T=D T=2D T=4D T=8D.3.25. U/D Blue circles: Bimodalty of centroid PDF Black squares: Bimodality of phonon PDF Double occupancy weak coupling n n = /4 no pairs strong coupling n n = /2 pairs Bimodality of Centroid PDF occurs when double occupancy reach approximately 7% of the saturation value Bimodality of phonon PDF is not related to pair formation at low temperature Bimodality of Centroid PDF can be a criterion to determine the existence of a strongly correlated state even at nonzero temperature
Antiadiabatic-regime T>: Phase diagram =8D U= Normal T/. Bipolarons. Pairs 2 4 6 8 2 4 6 λ bimodal P(X) bimodal P(X C ) Pairs without polarization in the low temperature antiadiabatic regime Adiabatic extimate of the Bipolaron Crossover and MIT (occuring at about same value of λ and γ) is recovered at large temperature
Holstein-Hubbard Model: ground state at half-filling (ED) λ 3.5 3 2.5 2.5.5 Effective Holstein polaronic crossover Ist order U c U c2 Bipolarons Normal T= =.2 Effective Hubbard.5.5 2 2.5 3 3.5 U/D Mott Insulator data from G. Sangiovanni [Sangiovanni Capone Castellani Grilli PRL 25] see also [Koller Meyer Hewson PRB 24] U U c Weak e-ph: Renormalized Hubbard [Sangiovanni Capone Castellani Grilli PRL 25] λ λ c Weak el-el: Renormalized Holstein First Order polaronic crossover at large U continuous Mott-Hubbard MIT at any value of e-ph coupling λ
Continuous and discontinuous polaron crossover in the Holstein-Hubbard model U small U/2 u> v> u> v> U large u> v> u > doubly occupied states v > singly occupied states u> v> small U Transition to a bipolaronic state occurs smoothly in a way similar to the pure Holstein model. large U Crossing of levels occurs from u > to v > levels when the system is in the strongly correlated metalli case. This triggers an abrupt changing of the phonon part of the ground state which modifies the DOS bath via the self-consistency condition.
Continuous and discontinuous polaron crossover in the Holstein-Hubbard model small U V(x) small λ intermediate λ λ=λ c δ 2.5.5.5.5.5.5 x large U V(x) small λ λ < λ c λ > λ c 2 \N x small U Increase will increase metallic character by decreasing the bimodal character of phonon PDF large U Increase of make the system probe more effectively the lattice displaced configurations leading to a decrease of the metallic properties.
Effective Kondo for Ho-Hu model H AIM = H + J v k ρ z f ρz c (k) + J v k (ρ + f k 2 ρ c (k) + ρ f ρ+ c (k)) k (pseudo) spins impurity (ρ f ) and bath s electrons (ρ c (k)) λ > U/D anisotropic (pseudo-spin) Kondo J, = 8V 2 D e α2 α 2m m (±)m m!(mγ + (λ U/D)/2) effective Holstein λ eff = λ η(α 2 )U λ < U/D isotropic Kondo J, = 8V 2 e α2 α 2m D m m!(mγ + (U/D λ)/2) effective Hubbard (U/D) eff = U/D η(u/ )λ
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Programs available for students A portable version of the DMFT programs with adiabatic phonons is available f77 codes, Unix Makefile Contact me directly or write to Sergio.Ciuchi@aquila.infn.it