Renormalization Group Approaches to Strongly Correlated Electron - PDF document

Renormalization Group Approaches to Strongly Correlated Electron and Electron-Phonon Systems Outline 1. The Application of NRG-DMFT to the Hubbard-Holstein Models 2. Polaron Bands in a a Many-electron System 3. Calculation of Renormalised Parameters from NRG Cal- culations 4. De-renormalisation as a function of Magnetic Field Strength 5. Applications within a Renormalised Perturbation Expan- sion. 4. Spin and Charge Dynamics in an Impurity Model 5. Potential Application to Heavy Fermions and Quantum Critical Points. Collaborators: Winfried Koller Dietrich Meyer, Akira Oguri Ralf Bulla Johannes Bauer Yoshiaki Ono.

Holstein-Hubbard Model t ( c † c † i ↑ c i ↑ c † H = − � iσ c jσ + h.c. ) + U � i ↓ c i ↓ <i,j>,σ i �� � ( b † ω 0 b † + g � σ n i,σ − 1 i + b i ) + � i b i i i Four significant parameters, band-width 2 D = 4 t , local in- teraction U , electron-phonon interaction g , and phonon fre- quency ω 0 . Strong Correlation Physics that can be studied by this model: • Metal-Insulator Transitions at half-filling • Bipolaron Formation • Polaronic Formation • Charge Order • Antiferromagnetism • Superconductivity

DMFT-NRG for Lattice Models Hubbard Model and Metal Insulator Transition Bethe hypercubic 0.4 0.4 U=0.8U U=0.8U 0.3 0.3 c c 0.2 0.2 0.1 0.1 0 0 U=0.99U U=0.99U 0.3 0.3 c c A( ω ) 0.2 0.2 0.1 0.1 0 0 U=1.1Uc U=1.1Uc 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0 −8 −4 0 4 8 −8 −4 0 4 8 ω ω Results for Spectral Density of Half-filled Hubbard Model as a function of U of Ralf Bulla. Critical value U c /D = 2 . 93 ( W = 2 D ).

Results for Half-filled Holstein Model g=0.03 g=0.08 g=0.098 1 g=0.12 ρ(ω) 0.5 0 -1 -0.5 0 0.5 1 ω The effect of increasing g on the interacting density of states. 1 0.8 ME 0.6 Z NRG 0.4 0.2 0 0 0.05 0.1 g The decrease of quasiparticle weight factor z with increase of g .

Phonon Spectra for the Holstein Model at Half-filling g 0.45 0.42 0.40 0.38 0.36 ρ d 0.34 0.30 0.25 0.20 0.10 0.00 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 ω This shows the softening of the phonon spectrum with in- crease of g for the Holstein model (U=0) and the rehardening after the transition to a bipolaronic state. There is a regime near the transition where two peaks in the spectrum can be seen.

Quasiparticle Interactions for the Holstein Model 0 −0.1 −0.2 −0.3 −0.4 1 −0.5 � U 0.8 −0.6 0.6 −0.7 0.4 −0.8 0.2 z from levels z from self energy 0 −0.9 0 0.1 0.2 0.3 0.4 g −1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 g From an analysis of the fixed point we can deduce both z and the local quasiparticle interaction ˜ U . The value of z deduced in in complete agreement with that deduced by dif- The value of ˜ ferentiating the self-energy Σ( ω ). U is new information. Note that the value of ˜ U behaves as expected becoming more negative initially and then less negative as the bipolarons form. Effective interaction due to single phonon exchange: U eff = U − 2 g 2 ω 0

Phase Diagram for Hubbard-Holstein Model 0.9 0.8 bipolaronic 0.7 0.6 0.5 g 0.4 0.3 metallic Mott 0.2 insulator 0.1 0 0 1 2 3 4 5 6 7 8 U The possibility of a broken symmetry state (charge order or antiferromagnetism) is not included.

Density of States for H-H Model U=5 0.5 g=0.00 0.4 g=0.60 d 0.3 0.3 g=0.70 0.2 g=0.72 z 0.1 g=0.75 0 0.2 0 0.2 0.5 0.7 g ρ G 0.1 0 −6 −4 −2 0 2 4 6 ω Here we start with strongly renormalized quasiparticles (small z and narrow quasiparticle peak at the Fermi level). As g increases z increases slowly but for large g there is a first order transition to a bipolaronic state.

The Hole Doped Holstein Model 0.6 0.4 0.2 z 0 −0.2 −0.4 U tilde −0.6 −0.8 −1 −1.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 n Quasiparticle weight z as a function of filling. 0.3 x=0.10 0.3 x=0.20 0.2 0.2 ρ G ρ G 0.1 0.1 0 0 −2 0 2 4 −2 0 2 4 ω ω 0.3 x=0.40 0.3 x=0.70 0.2 0.2 ρ G ρ G 0.1 0.1 0 0 −2 0 2 4 −2 0 2 4 ω ω Fixed g = 0 . 35 and increased doping x . Quasiparticle peak tied to the Fermi level.

Polaronic Quasiparticles in the Holstein-Hubbard Model Do pure polaronic excitations exist in the Holstein-Hubbard model? Polarons have been studied almost exclusively for one or two electrons in models without spin, such as the original Holstein model. In these models there is no effective local interaction inducing bipolaron formation, and no com- plications phase space restrictions due to the other electrons. Conditions favouring polaronic excitations: • Large U to inhibit local bipolaron formation • Away from half-filling where spin fluctuations dominate. We take U = 6 and look near quarter filling, and increase the electron-phonon coupling g 0.7 0.6 0.5 0.4 z 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 g Quasiparticle weight z as a function of g from t self-energy ( ◦ ) and from renormalised parameters ( × ). The quasiparticles are being renormalized due to polaronic effects!

Polaronic Quasiparticles 0.3 U = 6.0 x = 0.5 0.25 0.2 ρ G 0.15 0.1 0.05 0 −4 −2 0 2 4 6 8 10 ω Quarter Filled Hubbard for U = 6 and g = 0. 0.3 g=0.00 0.3 g=0.30 0.2 0.2 ρ G ρ G 0.1 0.1 0 0 −1 0 1 2 −1 0 1 2 ω ω g=0.40 g=0.55 0.3 0.3 0.2 0.2 ρ G ρ G 0.1 0.1 0 0 −1 0 1 2 −1 0 1 2 ω ω 0.3 g=0.60 0.3 g=0.65 0.2 0.2 ρ G ρ G 0.1 0.1 0 0 −1 0 1 2 −1 0 1 2 ω ω Narrow polaronic band develops at the Fermi level for U = 6 and increasing values of g .

2 1.5 1 0.5 ρ G ( ε k ) 0 −0.5 −1 −1.5 −2 −3 −2 −1 0 1 2 3 4 ω Plot of ρ k ( ω ) = − 1 π Im G k ,σ ( ω + iδ ) as ǫ ( k ) is varied, where 1 G k ,σ ( ω ) = ω + µ − ǫ ( k ) − Σ σ ( ω ) 3 U=6.0, g=0.6 U=6.0, g=0.0 2.5 2 1.5 1 ω 0.5 0 −0.5 −1 −2 −1.2 −0.4 0.4 1.2 2 ε (k) Plot of position of maximimum as a function of ǫ ( k ).

ρ d g 0.65 0.60 0.55 0.50 0.40 0.30 0.20 0.10 0.00 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 ω Softening of the phonon spectrum with increasing value of g . The position of the kink in the quasiparticle dispersion cor- relates with the renormalised phonon frequency.

Luttinger’s Theorem Fermi surface of non-interacting system: ǫ ( k F ) = µ 0 Fermi surface of interacting system: ǫ ( k F ) = µ − Σ(0) , For same number of particles: µ 0 = µ − Σ(0) , We can check the theorem by calculating n, in two ways: (i) From µ 0 = µ − Σ(0) we can calculate n from the volume of the non-interacting Fermi surface. (ii) We can deduce n from the expectation value � n i � from the ground state of the interacting system. 1 1 0.9 0.9 0.8 0.8 n n 0.7 0.7 0.6 0.6 U=6.0, g=0.0 U=6.0, g=0.5 0.5 0.5 0 1 2 3 1 2 3 µ µ From (i)-circles and from (ii) full lines.

Polaronic Quasiparticle Band Expanding the self-energy about the Fermi level, ω = 0, the retaining only the first two terms, Σ(0) + ω Σ ′ (0), 1 ˜ G k ,σ ( ω ) = ω + ˜ µ − ˜ ǫ ( k ) where ˜ ǫ ( k ) = zǫ ( k ), and z is the usual wavefunction renor- malization factor z = (1 − Σ ′ (0)) − 1 , and ˜ µ = z ( µ − Σ(0)) is a renormalized chemical potential. The corresponding density of states ˜ ρ 0 ( ω ) for the non- interacting quasiparticles is given by 2 � D 2 − ( ω + ˜ ˜ µ ) 2 ρ 0 ( ω ) = ˜ π ˜ D 2 where ˜ D = zD plays the role of a renormalized band width. Luttinger’s theorem is satisfied as ˜ µ = zµ 0 = z ( µ − Σ(0)) and the interacting and quasiparticle Fermi surface (˜ ǫ ( k ) = ˜ µ ) is the same as that of the non-interacting system. 1.5 1 U tilde 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 g Renormalised interaction ˜ U between the quasiparticles.

Fermi Liquid Theory and the Anderson Model Quasiparticles in the Anderson model: ∗ c † ǫ k,σ c † σ ǫ d d † ( V k d † H AM = � σ d σ + Un d, ↑ n d, ↓ + � σ c k,σ + V k k,σ d σ ) + � k,σ c k,σ , k,σ k,σ energy of the impurity level ǫ d , interaction at the impurity site U , hybridization matrix element V k , conduction electron energy ǫ k . 1 G d ( ω ) = ω − ǫ d − V 2 g 0 ( ω ) + Σ( ω ) where g 0 ( ω ) = − iπρ 0 for a wide conduction band, so 1 G d ( ω ) = ω − ǫ d + i ∆ + Σ( ω ) where ∆ = iV 2 g 0 ( ω ) = πρ 0 V 2 . We expand Σ( ω ) = Σ(0) + ω Σ ′ (0) + .... z G d ( ω ) = ǫ d + i ˜ ω − ˜ ∆ where ˜ ǫ d = z ( ǫ d + Σ(0)) ˜ ∆= z ∆ and z = 1 / (1 − Σ ′ (0)). Rescale the fields so that 1 ˜ G d ( ω ) = ǫ d + i ˜ ω − ˜ ∆ so this is the quasiparticle Green’s function corresponding to a renormalized non-interacting Anderson model.