SLIDE 1 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.
SLIDE 2 Holstein-Hubbard Model H = −
t(c†
iσcjσ + h.c.) + U
c†
i↑ci↑c† i↓ci↓
+g
i + bi) +
ω0b†
ibi
Four significant parameters, band-width 2D = 4t, local in- teraction U, electron-phonon interaction g, and phonon fre- quency ω0. Strong Correlation Physics that can be studied by this model:
- Metal-Insulator Transitions at half-filling
- Bipolaron Formation
- Polaronic Formation
- Charge Order
- Antiferromagnetism
- Superconductivity
SLIDE 3
DMFT-NRG for Lattice Models
Hubbard Model and Metal Insulator Transition
−8 −4 4 8
ω
0.0 0.1 0.2 0.3 0.1 0.2 0.3
A(ω)
0.1 0.2 0.3 0.4 −8 −4 4 8
ω
0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3 0.4
Bethe hypercubic
U=1.1Uc U=0.99U U=0.8U c c U=0.8U U=0.99U U=1.1Uc c c
Results for Spectral Density of Half-filled Hubbard Model as a function of U of Ralf Bulla. Critical value Uc/D = 2.93 (W = 2D).
SLIDE 4 Results for Half-filled Holstein Model
0.5 1 ω 0.5 1 ρ(ω)
g=0.03 g=0.08 g=0.098 g=0.12
The effect of increasing g on the interacting density of states.
0.05 0.1 g 0.2 0.4 0.6 0.8 1 Z NRG ME
The decrease of quasiparticle weight factor z with increase
SLIDE 5
Phonon Spectra for the Holstein Model at Half-filling
−0.1 −0.05 0.05 0.1 0.15 0.2 0.25 0.00 0.10 0.20 0.25 0.30 0.34 0.36 0.38 0.40 0.42 0.45 g ω ρd 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.
SLIDE 6 Quasiparticle Interactions for the Holstein Model
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1
0.1 0.2 0.3 0.4 0.2 0.4 0.6 0.8 1 z from levels z from self energy
g 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- ferentiating the self-energy Σ(ω). The value of ˜ 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: Ueff = U − 2g2 ω0
SLIDE 7
Phase Diagram for Hubbard-Holstein Model
1 2 3 4 5 6 7 8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 U g metallic bipolaronic Mott insulator The possibility of a broken symmetry state (charge order or antiferromagnetism) is not included.
SLIDE 8 Density of States for H-H Model U=5
−6 −4 −2 2 4 6 0.1 0.2 0.3 ω ρG g=0.00 g=0.60 g=0.70 g=0.72 g=0.75
0.2 0.5 0.7 0.1 0.2 0.3 0.4 0.5 g d z
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
- rder transition to a bipolaronic state.
SLIDE 9 The Hole Doped Holstein Model
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 U tilde
z
n
Quasiparticle weight z as a function of filling.
−2 2 4 0.1 0.2 0.3 ω ρG x=0.10 −2 2 4 0.1 0.2 0.3 ω ρG x=0.20 −2 2 4 0.1 0.2 0.3 ω ρG x=0.40 −2 2 4 0.1 0.2 0.3 ω ρG x=0.70
Fixed g = 0.35 and increased doping x. Quasiparticle peak tied to the Fermi level.
SLIDE 10 Polaronic Quasiparticles in the Holstein-Hubbard Model
Do pure polaronic excitations exist in the Holstein-Hubbard model? Polarons have been studied almost exclusively for
- ne or two electrons in models without spin, such as the
- riginal 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
0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 g z
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!
SLIDE 11 Polaronic Quasiparticles
−4 −2 2 4 6 8 10 0.05 0.1 0.15 0.2 0.25 0.3 ω ρG U = 6.0 x = 0.5
Quarter Filled Hubbard for U = 6 and g = 0.
−1 1 2 0.1 0.2 0.3 g=0.00 ω ρG −1 1 2 0.1 0.2 0.3 g=0.30 ω ρG −1 1 2 0.1 0.2 0.3 g=0.40 ω ρG −1 1 2 0.1 0.2 0.3 g=0.55 ω ρG −1 1 2 0.1 0.2 0.3 g=0.60 ω ρG −1 1 2 0.1 0.2 0.3 g=0.65 ω ρG
Narrow polaronic band develops at the Fermi level for U = 6 and increasing values of g.
SLIDE 12 −3 −2 −1 1 2 3 4 −2 −1.5 −1 −0.5 0.5 1 1.5 2 ω ρG (εk)
Plot of ρk(ω) = − 1
πIm Gk,σ(ω + iδ) as ǫ(k) is varied,
where Gk,σ(ω) = 1 ω + µ − ǫ(k) − Σσ(ω)
−2 −1.2 −0.4 0.4 1.2 2 −1 −0.5 0.5 1 1.5 2 2.5 3
ε(k) ω
U=6.0, g=0.6 U=6.0, g=0.0
Plot of position of maximimum as a function of ǫ(k).
SLIDE 13
−0.1 −0.05 0.05 0.1 0.15 0.2 0.25 0.00 0.10 0.20 0.30 0.40 0.50 0.55 0.60 0.65 ω
g ρd
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.
SLIDE 14 Luttinger’s Theorem
Fermi surface of non-interacting system: ǫ(kF) = µ0 Fermi surface of interacting system: ǫ(kF) = µ − Σ(0), For same number of particles: µ0 = µ − Σ(0), We can check the theorem by calculating n, in two ways: (i) From µ0 = µ−Σ(0) we can calculate n from the volume
- f the non-interacting Fermi surface.
(ii) We can deduce n from the expectation value ni from the ground state of the interacting system. 1 2 3 0.5 0.6 0.7 0.8 0.9 1 µ n U=6.0, g=0.0 1 2 3 0.5 0.6 0.7 0.8 0.9 1 µ n U=6.0, g=0.5 From (i)-circles and from (ii) full lines.
SLIDE 15 Polaronic Quasiparticle Band
Expanding the self-energy about the Fermi level, ω = 0, the retaining only the first two terms, Σ(0) + ωΣ′(0), ˜ Gk,σ(ω) = 1 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 ˜ ρ0(ω) for the non- interacting quasiparticles is given by ˜ ρ0(ω) = 2 π ˜ D2
D2 − (ω + ˜ µ)2 where ˜ D = zD plays the role of a renormalized band width. Luttinger’s theorem is satisfied as ˜ µ = zµ0 = z(µ−Σ(0)) and the interacting and quasiparticle Fermi surface (˜ ǫ(k) = ˜ µ) is the same as that of the non-interacting system.
0.1 0.2 0.3 0.4 0.5 0.6 0.5 1 1.5 g U tilde
Renormalised interaction ˜ U between the quasiparticles.
SLIDE 16 Fermi Liquid Theory and the Anderson Model
Quasiparticles in the Anderson model: HAM =
σdσ + Und,↑nd,↓ +
(Vkd†
σck,σ+Vk ∗c† k,σdσ) +
ǫk,σc†
k,σck,σ,
energy of the impurity level ǫd, interaction at the impurity site U, hybridization matrix element Vk, conduction electron energy ǫk. Gd(ω) = 1 ω − ǫd − V 2g0(ω) + Σ(ω) where g0(ω) = −iπρ0 for a wide conduction band, so Gd(ω) = 1 ω − ǫd + i∆ + Σ(ω) where ∆ = iV 2g0(ω) = πρ0V 2. We expand Σ(ω) = Σ(0) + ωΣ′(0) + .... Gd(ω) = z ω − ˜ ǫd + i ˜ ∆ where ˜ ǫd= z(ǫd + Σ(0)) ˜ ∆= z∆ and z = 1/(1 − Σ′(0)). Rescale the fields so that ˜ Gd(ω) = 1 ω − ˜ ǫd + i ˜ ∆ so this is the quasiparticle Green’s function corresponding to a renormalized non-interacting Anderson model.
SLIDE 17
Calculation of ˜ ǫd and ˜ ∆ using the NRG
Impurity conduction electron chain V
λn
n n+1 Given an ǫd and hybridization V , the one-particle excitations ω = ǫn are given by the poles of G(0)
d (ω) =
1 ω − ǫd − V 2g0(ω) ie, solutions of ω − ǫd − V 2g0(ω) = 0 In the inverse case, given the solutions ǫn, we can use this equation to deduce the parameters ǫd and V . We can apply this to the NRG results. If Ep(N), Eh(N) are the lowest energy single particle ex- citations of the interacting Anderson model, then from the equation ωΛ−(N−1)/2 − ˜ ǫd(N) = Λ(N−1)/2 ˜ V (N)2g0(ω), we can deduce the effective parameters, ˜ ǫd(N) and ˜ ∆(N) = π ˜ V (N)2/D. If these can be described by a non-interacting Anderson model, then they should be independent of N.
SLIDE 18 10 20 30 40 50 N 2 4 6 8 10 12 25 30 35 40 45 N
0.01 0.02 0.03 0.04
The colour black curve is the value of ˜ ǫd(N). The colour red curve is the value of ˜ ∆(N) = π ˜ V (N)2/D. The values of the ’bare’ parameters are π∆ = 0.05, ǫd = −0.2, U = 0.3.
SLIDE 19
Quasiparticle Interactions
We can also define a local renormalized interaction ˜ U If Epp(N) is the energy of a two-quasiparticle excitation, we can define an N-dependent renormalized interaction ˜ Upp(N) via, Epp(N) − 2Ep(N) = ˜ U(N)Λ(N−1)/2|ψ∗
p,1(−1)|2|ψ∗ p,1(−1)|2,
where |ψp,1(−1)|2 is given by |ψp,1|2 = 1 1 − ˜ V 2(N)Λ(N−1)g′0(Ep(N)), Similar equations for ˜ Uhh(N) and ˜ Uph(N) We find a unique limit, lim
N→∞
˜ Upp(N) = lim
N→∞
˜ Uhh(N) = lim
N→∞
˜ Uph(N) = ˜ U We can define a quasiparticle density of states: ˜ ρ(ω) = ˜ ∆/π (ω − ˜ ǫd)2 + ˜ ∆2 For strong coupling it can be shown that ˜ U ˜ ρ(0) → 1
SLIDE 20 10 20 30 40 50 N 2 4 6 8 10 12 25 30 35 40 45 50 N
0.01 0.02 0.03 0.04
The colour black curve is the value of ˜ ǫd(N). The colour red curve is the value of ˜ ∆(N) = π ˜ V (N)2/D. The colour green and turquoise curves give the values of ˜ Upp, ˜ Uhh and ˜ Uph. We are in the Kondo regime so there is only one energy scale ˜ U = π ˜ ∆ = 4TK .
SLIDE 21 0.25 0.5 0.75 1 1.25 1.5 1.75 2
nd
0.25 0.5 0.75 1
∆/∆
U/π∆=0.5 U/π∆=1.0 U/π∆=1.5 U/π∆=2.0 U/π∆=2.5 U/π∆=3.0
~
U > 0: Gives wavefunction renormalization factor z = ˜ ∆/∆ (quasiparticle weight) as a function of filling nd. The max- imum renormalization is at half-filling giving the minimum value of z.
0.25 0.5 0.75 1 1.25 1.5 1.75 2
nd
0.2 0.4 0.6 0.8 1
U/U ~
The ratio of the quasiparticle interaction ˜ U to the bare in- teraction U as a function of filling nd. The minimum value
Uis at half-filling rising to the bare value at nd = 0 and nd = 1.
SLIDE 22 0.25 0.5 0.75 1 1.25 1.5 1.75 2
nd
0.25 0.5 0.75 1
∆/∆
~
U < 0: z is flat as a function of nd for a broad range near nd ∼ 1.
0.25 0.5 0.75 1 1.25 1.5 1.75 2
nd
0.5 1 1.5 2 2.5
U/U ~
˜ U is also flat as a function of nd for a broad range near nd ∼ 1. In contrast to the case with U > 0 there is a range where | ˜ U| > |U| before it approaches the bare value at nd = 0 and nd = 1.
SLIDE 23 0.25 0.5 0.75 1 1.25 1.5 1.75 2
nd
0.2 0.4 0.6 0.8 1
εd/εd
~ _
U < 0: Ratio ˜ ǫd/¯ ǫd, where ¯ ǫd = ǫd + U/2, is always < 1.
0.25 0.5 0.75 1 1.25 1.5 1.75 2
nd
1 2 3 4 5
εd/εd
_ ~
U < 0: Ratio ˜ ǫd/¯ ǫd is always > 1 and has maxima as nd → 0 and nd → 1.
SLIDE 24
1 2 3
U/π∆
0.2 0.4 0.6 0.8 1
∆/∆ ~ U/π∆ ~ 4TK This plot shows how the renormalized parameters vary as the value of U varies for the symmetric model ǫd = −U/2.
SLIDE 25
1
εd/π∆
0.5 1 1.5 2
R nimp U/π∆ ∆/∆ εd/π∆ ~ ~ ~
This plot shows how the renormalized parameters vary as the impurity level is moved from well below the Fermi level to well above for a fixed value of U = 2π∆.
SLIDE 26
0.5 1 1.5 2 2.5
U/π∆
0.5 1 1.5 2
R nimp ∆/∆ ~ U/π∆ ~ εd/π∆ ~
This plot shows how the renormalized parameters vary as U varies for a fixed impurity level ǫd = π∆.
SLIDE 27
1 2
ω/π∆
1 2
U/π∆
1 2
ω/π∆
1
εd/π∆
(i) (ii)
This plot shows how the position of the renormalized level relates to the peaks in the spectral density of the impurity Green’s function, for the two cases presented.
SLIDE 28
Renormalized Perturbation Theory for the Anderson Model
No divergences associated with lack of cut-off in this case. Gd(ω) = 1 ω − ǫd + i∆ + Σ(ω) We eliminate Σ(0) and Σ′(0) explicitly, Σ(ω) = Σ(0) + ωΣ′(0) + Σrem(ω) Gd(ω) = z ω − ˜ ǫd + i ˜ ∆ + ˜ Σ(ω) where ˜ ǫd = z(ǫd + Σ(0)) ˜ ∆ = z∆ ˜ Σ(ω) = zΣrem(ω) and z = 1/(1 − Σ′(0)). Rescale the fields so that ˜ Gd(ω) = 1 ω − ˜ ǫd + i ˜ ∆ + ˜ Σ(ω) U is replaced by the renormalized one, ˜ U = z2Γ↑,↓(0, 0, 0, 0) We have used the Luttinger result ImΣ′(0) = 0.
SLIDE 29 Renormalized Perturbation Theory
L(ǫd, ∆, U) = L(˜ ǫd, ˜ ∆, ˜ U) + Lcounter(λ1, λ2, λ3) The expansion is carried out in powers of ˜ U, with λ1, λ2, λ3 determined by the renormalization conditions: (i) ˜ Σ(0) = 0 (ii) ˜ Σ′(0) = 0 (iii) ˜ Γ↑,↓(0, 0, 0, 0) = ˜ U Renormalizability is not an issue because we have no ultra- violet divergences due to a lack of an upper cut-off. Note that the renormalized form of L is the same as that
- f the original so the corresponding Hamiltonian is simply a
renormalized version of the Anderson model. i.e. ǫd → ˜ ǫd, ∆ → ˜ ∆ and U → ˜ U.
SLIDE 30 Low order results are asymptotically exact as T → 0 All the leading order exact low temperature results, which correspond to a local Fermi liquid theory, can be calculated by working to second order in ˜ U only. Zero Order nd,σ = ˜ nd,σ = 1 2 − 1 π tan−1
˜
ǫd,σ ˜ ∆
γimp = 2π2 3 ˜ ρ(0) with ˜ ρ(0) = ˜ ∆/π ˜ ǫ2
d + ˜
∆2 First Order χimp = (gµB)2 2 ˜ ρ(0)(1 + ˜ U ˜ ρ(0)), χc,imp = 2˜ ρ(0)(1 − ˜ U ˜ ρ(0)) Second Order σimp(T) = σ0
1 + π2
3
T
˜ ∆
2 1 + 2 ˜
U π ˜ ∆
2 + O(T 4)
SLIDE 31 THE KONDO LIMIT—-ONLY ONE RENORMALIZED PARAMETER TK In the localized limit, nd → 1, ˜ ǫd → 1 and χc,imp → 0, ˜ U = π ˜ ∆ = 4TK γimp = π2 6TK χimp = (gµB)2 4TK σimp(T) = σ0
1 + π2
16
T
TK
2
+ O(T 4)
N-fold Degenerate Anderson Model ˜ ∆ = TK N2sin2(π/N) π(N − 1) ˜ ǫd = TK N2sin(2π/N) 2π(N − 1) ˜ U = TK
N − 1
2
The n-Channel Anderson Model with n=2S ˜ U = π ˜ ∆ = 4TK ˜ J = −8 3TK In all cases TK is defined such that χimp = (gµB)2S(S + 1)/3TK.
SLIDE 32
Quasiparticles in the limit z → 0
If U → ∞ which corresponds to z → 0, ˜ ∆ → 4TK/π → 0 ˜ ǫf → 0 and ˜ ρ(ω) → δ(ω) but ˜ Uρ(0) → 1 Hence, the results for the spin and charge susceptibility at T = 0, χimp = (gµB)2 2 ˜ ρ(0)(1 + ˜ U ˜ ρ(0)), χc,imp = 2˜ ρ(0)(1 − ˜ U ˜ ρ(0)) in this limit give χimp → ∞, χc,imp → 0 What is more if we consider T = 0 and include the Fermi factors we find χimp → (gµB)2 4T , χc,imp → 0 The Fermi liquid theory describes a local moment! The factor 1+ ˜ U/π ˜ ∆ → 2 ensures the correct Curie constant.
SLIDE 33
Quasiparticles in a Magnetic Field
We can generalized ˜ ǫdσ ˜ ∆ and ˜ U, such that they become functions of the magnetic field H. Particle-hole symmetric model: ˜ ǫd,σ(h) = z(h)(−hσ + Σσ(0, h)), ˜ ∆(h) = z(h)∆ where z(h) = 1/(1 − Σ′
↑(0, h)) with h = gµBH/2.
The impurity magnetization M(h) at T = 0 is then given simply by M(h) = gµB π tan−1
˜
ǫd(h) ˜ ∆(h)
where ˜ ǫd(h) = −σ˜ ǫdσ(h). Exact expression for susceptibility: ˜ ρ(ω, h) = ˜ ∆(h) (ω − ˜ ǫd(h))2 + ˜ ∆(h)2 in terms of quasiparticle density of states, χimp(h) = (gµB)2 2 ˜ ρ(0, h)(1 + ˜ U(h)˜ ρ(0, h)) We can follow the de-renormalization of the quasiparticles as a function of h. As h → ∞, ˜ ǫd(h) h → 1 ˜ ∆(h) → ∆ ˜ U(h) → U
SLIDE 34
2 4 6
Ln(h/T*)
1 2 3 4 5
R(h) ∆(h)/∆ εd(h)/εd U(h)/U ~ ~ ~ _
This plot shows how the renormalized parameters vary for the symmetric model at strong coupling (U/π∆ = 3) as a function of magnetic field h = gµBH/2. The slow derenor- malization of the quasiparticles with magnetic field can be
∆/4 and T ∗ → TK as U → ∞. The parameters are not independent: 1 + ˜ U(h)˜ ρd(0, h) = ∂˜ ǫd(h) ∂h − ˜ ǫd(h) ˜ ∆(h) ∂ ˜ ∆(h) ∂h . ˜ ǫd(h) = h+UmMF(h), ˜ ∆(h) = ∆, where mMF(h) is the mean field magnetisation. We can deduce the value of ˜ U(h) which gives ˜ U(h) = U 1 − U ˜ ρdMF(0, h)
SLIDE 35
2 4 6 8 log(h/TK) 1 10 100 η(h) ∆(h)/∆ U(h)/U R(h) ~ ~ ~ ~ ~
Plot on a logscale of the ratios of the renormalized parame- ters for h compared to their values at h = 0 for U/π∆ = 4.
SLIDE 36
2 4 6 8 ln(h/T*) 5 10 15 20 25 30 U(h)/π∆(h)
U/π∆=5.0 U/π∆=3.0
~ ~ The ratio ˜ ǫd(h)/ ˜ ∆(h) for U/π∆ = 5.0 plotted as a function
- f the logarithm of the magnetic field
SLIDE 37
2 4 6 Ln(h/T*) 0.1 0.2 0.3 0.4 0.5
R(h)/4
m(h) The impurity magnetisation m(h) for the symmetric model with U/π∆ = 3.0, together with R(h)/4, where R(h) is the Wilson ratio, plotted as a function of the logarithm of the magnetic field. Also shown for comparison are the corre- sponding Bethe ansatz results for the field induced magneti- sation for the Kondo model m(h) = M(h) gµB = 1 πtan−1
˜
ǫd(h) ˜ ∆(h)
SLIDE 38 Applications
Susceptibility: χs(0, h) − χs(T, h) = −π2 12T 2∂2˜ ρd(0, h) ∂h2 = cχ(h)
T
T ∗
2
Impurity contribution to conductivity: σ(h, T) = σ(h, 0)
1 + σ2(h) πT
˜ ∆(h)
2
+ O(T 4)
Conductance of quantum dot: G(T, h) = G(0, h)
1 − G2(h) πT
˜ ∆(h)
2
Differential conductance as a function of Vds: dI dVds = G0(h)
ds + O(V 4 ds)
- All have a change of sign of second order term at h = hc,
where hc lies in the range 0 < hc < T ∗
SLIDE 39
Spin and Charge Dynamics
Repeated Quasiparticle Scattering We look at the contribution from the repeated scattering of a quasiparticle ↑ and a quasihole ↓ to the transverse suscep- tibility.
p h
U
~
This gives χs(ω) = 1 2 ˜ Πp↑
h↓(ω)
1 − ˜ U p↑
h↓ ˜
Πp↑
h↓(ω)
, where U p↑
h↓ is the irreducible particle-hole vertex.
We determine this from the condition: χs(0) = 1 2 ˜ ρ(0) 1 − ˜ U p↑
h↓ ˜
ρ(0) = ˜ ρ(0) 2 (1 + ˜ U ˜ ρ(0)), which gives the result, ˜ U p↑
h↓ =
˜ U 1 + ˜ U ˜ ρ(0)
SLIDE 40
p p
U ~
Conributions to the charge susceptibility from repeated scat- tering of a quasiparticle ↑ and a quasiparticle ↓, χc(ω) = 1 2 ˜ Πp↑
p↓(ω)
1 − ˜ U p↑
p↓ ˜
Πp↑
p↓(ω)
where ˜ U p↑
p↓ =
˜ U 1 − ˜ U ˜ ρ(0)
0.5 1 1.5 2 2.5 3
U/π∆
0.5 1 1.5 2 2.5
The irreducible vertices, ˜ U p↑
h↓ (black) and ˜
U p↑
p↓ (red) as a func-
tion of U/π∆ for the symmetric Anderson model.
SLIDE 41
- 0.1
- 0.08 -0.06 -0.04 -0.02
0.02 0.04 0.06 0.08 0.1
ω
0.5 1 1.5 2 2.5
Imχ(ω)
NRG results (black-dashed) for Im χs(ω) for (U = 0) com- pared with exact results (blue).
0.005 0.01
ω
20 40 60
Imχ(ω)/π
Im χs(ω)/π in the Kondo regime U/π∆ = 3.0. The dashed curve (black) is the NRG results and the full line (red) the RPT result.
SLIDE 42
0.003 0.006 0.009
ω
50 100 150 200 250 300
Reχ(ω)
Real part χs(ω) for the symmetric model in the Kondo regime U/π∆ = 3.0. NRG results (black) and RPT (blue).
- 0.15 -0.12 -0.09 -0.06 -0.03
0.03 0.06 0.09 0.12 0.15
ω
2 4 6
Imχ(ω)/π
Imχs(ω) and Imχc(ω) for U/π∆ = 1.0. NRG: spin (black dashed) and charge (black). RPT: spin (blue) and charge (red).
SLIDE 43
0.02 0.04
ω
1 2 3 4
Imχ(ω)π
Im χs(ω) for U/π∆ = 0.5. T (i) NRG calculations (black), (ii) RPT (blue) and (iii) RPA (red).
0.02 0.04 0.06 0.08 0.1
ω
0.5 1 1.5 2
Imχ(ω)/πωχ
2(0) U/π∆=0.0 U/π∆=0.5 U/π∆=1.0 U/π∆=3.0
Imχs(ω)/πχ2
s(0) calculated using the RPT for U/π∆ =.
Area under curve = 1 χs(0) = 2π ˜ ∆ (1 + ˜ U/π ˜ ∆)
SLIDE 44 Spin and Charge Dynamics in a Magnetic Field
χs,(0, h) = 1 2 ˜ ρ(0, h)(1 + ˜ U(h)˜ ρ(0, h)) we deduce ˜ U p↑
h↑(h) =
˜ U(h) (1 + ˜ U(h)˜ ρ(0, h)) χs,⊥(0, h) = m(h) 2h = 1 2πhtan−1
˜
h(h) ˜ ∆(h)
,
we find ˜ U p↑
h↓(h) =
πh(˜ η(h) − 1) tan−1(˜ h(h)/ ˜ ∆(h)).
1 2 3 4 5 6 7
ln(h/TK)
0.1 0.2 0.3 0.4 0.5
Irreducible vertices, ˜ U p↑
h↓(h) (black), ˜
U p↑
h↑(h) (blue), and ˜
U p↑
p↓(h)
(red) as a function of ln (h/TK) for U/π∆ = 3.0.
SLIDE 45 Non-interacting Case U = 0
- 0.15 -0.12 -0.09 -0.06 -0.03
0.03 0.06 0.09 0.12 0.15
ω
1 2 3 4 5
Imχ(ω)/π
Exact results for Im χs,⊥(ω, h) (blue) and Im χs,(ω, h) (red) for U = 0 compared with the NRG results (black) for h = 0.4π∆.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ω
1 2 3 4 5 6
Imχ(ω)/π
Im χs,⊥(ω, h) forU = 0; NRG (black) and exact results (blue) for h = 5π∆.
SLIDE 46
0.003 0.006 0.009
ω
20 40 60 80
Imχ(ω)π
The RPT results for Im χs,⊥(ω, h) (blue) and Im χs,(ω, h) (red) for h = 0.4TK for U/π∆ = 3.0 compared with the NRG results (black).
0.01 0.02 0.03 0.04 0.05
ω
20 40 60 80 100
Imχ(ω)/π
The RPT results for Im χs,⊥(ω, h) (blue) for h = 2TK for U/π∆ = 3.0 compared with the NRG results (black).
SLIDE 47 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
ω
10 20 30 40 50
Imχ(ω)π
The RPT results (blue) for Im χs,⊥(ω, h) for h = 20TK, (ln h/TK = 3.0) for U/π∆ = 3.0 compared with the NRG (black).
0.5 1 1.5 2 2.5 3 3.5 4
ω
1 2 3 4 5 6
Imχ(ω)/π
The RPT results for the imaginary parts of χs,⊥(ω, h) (blue) for h = 103TK (ln h/TK = 6.93) as a function of ω for U/π∆ = 3.0 compared with the NRG results (black). In this regime the peak is at ωp = 2h, and all many-body effects have disappeared.
SLIDE 48
0.003 0.006 0.009
ω
0.5 1 1.5 2
Imχ(ω)/πωχ
2(0) h=0.1TK h=0.4TK h=TK
Results for Imχs,(ω, h)/πχ2
s,(0, h) calculated using the RPT
equations for a range of values of the magnetic field h. The Shiba-Korringa relation for the parallel susceptibility is sat- isfied in a magnetic field. The area under the curve is equal to 1/χ(h), and increases indefinitely as h → ∞.
Contribution to Self-energy from Scattering with Spin Fluctuations U U ~ ~
p h h p
Dashed line is χperp(ω, h). Exact to order ω2. ρ(ω, h) = ˜ ∆ ∆ ˜ ρ(ω, h) agrees well with lowest energy peak as a function of h from NRG calculations,
SLIDE 49 Lattice Case in Dynamical Mean Field Theory Hubbard Model at half-filling :
1 2 3 4 5 6
U
0.5 1 1.5 2 z from self-energy z from levels U from levels
~
Complete agreement with z calculated from the self-energy and from the renormalized parameters. ˜ U does not mono- tonically increase with U. Ratio ˜ U ˜ ρ(0) → 0.84 as U → Uc
1 2 3 4 5 6
U
0.5 1 1.5 2 2.5 3
U/z
~
Unique energy scale in strong coupling limit.
SLIDE 50
Lattice Case in Dynamical Mean Field Theory Hubbard Model with increasing doping U = 6 :
0.4 0.5 0.6 0.7 0.8 0.9 1
band filling n
0.5 1 1.5 2 z from self-energy z from levels U from levels ~
Again complete agreement with z calculated from the self- energy and from the renormalized parameters. ˜ U monotoni- cally increase with doping. Again the ratio of ˜ U ˜ ρ(0) → 0.84 as n → 1, indicating a single energy scale in the strong cor- relation limit. As (1 + ˜ U ˜ ρ(0)) = 1.84 < 2, reduced moment?
SLIDE 51
Spin and Charge Dynamics for Lattice Models
χs(q, ω) = 1 2 ˜ Πp↑
h↓(q, ω)
1 − ˜ U p↑
h↓ ˜
Πp↑
h↓(q, ω)
, χc(q, ω) = 1 2 ˜ Πp↑
p↓(q, ω)
1 − ˜ U p↑
p↓ ˜
Πp↑
p↓(q, ω)
where U p↑
h↓ and U p↑ p↓ the irreducible vertices, which in DMFT
are local quantities. Contribution to Self-energy from Scattering with Spin Fluc- tuations
U U ~ ~
p h h p
q
Contribution to self-energy Σ(ω, k) from scattering with a transverse spin fluctuations. Note the self-energy now has a k-dependence because it goes beyond DMFT.
SLIDE 52
Can we use temperature dependent running coupling constants?
The relation used in NRG calculations: TN = ηDΛ−(N−1)/2 Allows one to deduce ˜ ǫd(T), ˜ ∆(T) and ˜ U(T) from ˜ ∆(N) and ˜ ǫd(N) and ˜ U(N). χs(T) = (gµB)2 2 ˜ ρd(0, T)(1 + ˜ U(T)˜ ρd(0, T)) with ˜ ρd(0, T) given by ˜ ρd(0, T) = −
∞ −∞ ˜
ρ(ω, T)∂f(ω) ∂ω dω where f(ω) = 1/(eω/T + 1), and ˜ ρ(ω, T) is the free quasipar- ticle density of states with parameters ˜ ∆(T) and ˜ ǫd(T) and ˜ U(T). In mean field or Hartree-Fock theory we have ˜ Umf(T) = U/(1 − U ˜ ρd,mf(0, T)) and substituting this in the expression above gives the mean field result, χs(T)/(gµB)2 = 0.5˜ ρd,mf(0, T)/(1 − U ˜ ρd,mf(0, T))
SLIDE 53
1 2 3 4 5 6 7 ln T/TK
0.2 0.4 0.6 0.8 1
χ(T)/χ(0)
Susceptibility as a function of T
The full curve corresponds to Bethe ansatz results for the s-d model and the stars to results using temperature depen- dent renormalized parameters for the Anderson model for U/π∆ = 5.
SLIDE 54 0.5 1 1.5 2 2.5 h/T*
0.5 1 1.5
U/π∆=3.0 U/π∆=0.5 U/π∆=0.0
`
~ ~
Susceptibility: χs(0, h) − χs(T, h) = −π2 12T 2∂2˜ ρd(0, h) ∂h2 = cχ(h)
T
T ∗
2
SLIDE 55
2 4 6 log(h/T*)
0.2 0.4 0.6 0.8 1 σ2 (h) U/π∆ = 0.5 U/π∆ = 1.0 U/π∆ = 2.0 U/π∆ = 4.0
Impurity contribution to conductivity: σ(h, T) = σ(h, 0)
1 + σ2(h) πT
˜ ∆(h)
2
+ O(T 4)
SLIDE 56 0.5 1 1.5 2 2.5 h/T*
0.2 0.4 0.6 0.8 1 σ2 (h) U/π∆ = 0.5 U/π∆ = 1.0 U/π∆ = 2.0 U/π∆ = 4.0
Impurity contribution to conductivity for range 0 < h < 2.5T ∗. σ2(h) changes sign at h = hc in the local moment regime.
SLIDE 57
5 log(h/T*)
0.2 0.4 0.6 0.8 1 G2(h) U/π∆ = 0.5 U/π∆ = 1.0 U/π∆ = 2.0 U/π∆ = 4.0
Conductance of quantum dot: G(T, h) = G(0, h)
1 − G2(h) πT
˜ ∆(h)
2
SLIDE 58
0.5 1 1.5 2 2.5 h/T* 0.5 1 G2(h) U/π∆ = 0.5 U/π∆ = 1.0 U/π∆ = 2.0 U/π∆ = 4.0
Conductance of quantum dot in range 0 < h < 2.5T ∗ In this case G2(h) changes sign in all cases.
SLIDE 59
0.5 1 1.5 2 2.5 3
eVds/∆
0.5 1 1.5 2
dI/dVds
h/hc= 0.50 h/hc= 0.75 h/hc= 1.00 h/hc= 1.25 h/hc= 1.50 h/hc= 1.75
~
Conductance of quantum dot at T = 0 dI dVds = G0(h)
ds + O(V 4 ds)
- There is a critical field hc for two peaks to be seen in the
differential conductance of a quantum dot at a function of the bias voltage Vds. This occurs when A2(h) changes sign.
SLIDE 60
Differential Conductance through a Quan- tum Dot
In the Kondo regime the differential conductance as a func- tion of the bias voltage Vds shows the evolution of a two-peak structure with increasing magnetic field H. Taken from the paper of S.Amasha, I.J. Gelfand, M.A. Kast- ner and A. Kogan, cond-mat/0411485.
