dynamical mean field theory
TRANSCRIPT
Copyright A. J. Millis 2013 Columbia University
Dynamical Mean Field Theory
Antoine Georges, Gabriel Kotliar, Werner Krauth, and Marcelo J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996)
Thomas Maier, Mark Jarrell, Thomas Pruschke, and Matthias H. Hettler, Rev. Mod. Phys. 77, 1027 (2005)
K. Held, II. A. Nekrasov, G. Keller, V. Eyert, N. Bluemer, A. K. McMahan, R. T Scalettar, T.h. Pruschke, V. I. Anisimov, and D. Vollhardt, D.; Phys. Status Solidi 243, 2599-2631 (2006)
G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Rev. Mod. Phys. 78, 865 (2006)
Review articles
http://phys.columbia.edu/~millis/MillisTLH.pdf
Copyright A. J. Millis 2013 Columbia University
Dynamical Mean Field TheoryMany-body formalism: analogous to Hohenberg-Kohn
H =�
α,β
Eαβψ†αψβ +
�
αβγδ
Iαβγδψ†αψ†
βψγψδ + ...
G0: Green function of noninteracting reference problem (contains atomic positions)
F[{Σ}] = Funiv[{Σ}]−Trln[G−10 −Σ]
=>Luttinger-Ward functional
Funiv: determined (formally) from sum ofdiagrams. Depends only on interactions Iαβγδ
Copyright A. J. Millis 2013 Columbia University
ExampleSecond order termHubbard model
12
ΦLW [{G}] defined as sum of allvacuum to vacuum diagrams(with symmetry factors)
δΦδG
= Σ
Funiv = ΦLW[{G}]−Tr [ΣG]
Copyright A. J. Millis 2013 Columbia University
More formalities
δFuniv
δΣ= G
Diagrammatic definition of Funiv =>
Thus stationarity conditionδFδΣ
= 0 => G =�G−1
0 −Σ�−1
F[{Σ}] = Funiv[{Σ}]−Trln[G−10 −Σ]
Difficulties with this formulation:--dont know Funiv (exc. perturbatively)--cant do extremization
Copyright A. J. Millis 2013 Columbia University
1992 BreakthroughKotliar and Georges found analogue of Kohn-Sham steps: useful approximation for F and way to carry out minimization via auxiliary problem
Key first step: infinite d limit Metzner and Vollhardt, PRL 62 324 (1987)
Copyright A. J. Millis 2013 Columbia University
Modern interpretation of Kotliar and Georges idea
Parametrize self energy in terms of small number N of functions of frequency
Σαβ(ω) =�
ab
fαβab Σab
DMFT(ω)
α = 1...M; a = 1...N << M
parametrization function f determines ‘flavor’ of DMFT (DFT analogue: LDA, GGA, B3LYP, ....)
Also must truncate interactionIαβγδ “appropriately”
Copyright A. J. Millis 2013 Columbia University
Approximation to self energy + truncated interactions imply approximation to Funiv
Approximated functional Fapproxuniv is functional of finite (small) number of functions of frequency , thus is the universal functional of some 0 (space) +1 (time) dimensional quantum field theory. Derivative gives Green function of this model:
Σab(ω)
δFapproxuniv
δΣba(ω)= Gab
QI(ω)
Copyright A. J. Millis 2013 Columbia University
Specifying the quantum impurity model
Need
--Interactions. These are the ‘appropriate truncation’ of the interactions in the original model
--a noninteracting (‘bare’) Green function G0
Then can compute the Green function and self energy
GQI =�G−10 −Σ
�−1
Copyright A. J. Millis 2013 Columbia University
Analogy:
Density functional <=> ‘Luttinger Ward functional
Kohn-Sham equations <=> quantum impurity model
Particle density <=> electron Green function
Copyright A. J. Millis 2013 Columbia University
Quantum impurity model is in principle nothing more than a machine for generating self energies (as Kohn-Sham eigenstates are artifice for generating electron density)
Useful to view auxiliary problem as ‘quantum impurity model’ (cluster of sites coupled to noninteracting bath)
As with Kohn Sham eigenstates, it is tempting (and maybe reasonable) to ascribe physical signficance to it
Copyright A. J. Millis 2013 Columbia University
In Hamiltonian representation
Impurity Hamiltonian
Coupling to bath+
�
p,ab
�V
pabd
†acpb + H.c
�. + Hbath[{c†pacpa}]
Important part of bath: ‘hybridization function’
∆ab(z) =�
p
Vpac
�1
z− εbathp
�Vp,†
cb
HQI =�
ab
d†aE
abQIdb + Interactions
G−10 = ω −Eab
QI −∆ab(ω)
Copyright A. J. Millis 2013 Columbia University
Thus
F→ Fapproxuniv [{Σab}]−Trln[G−1
0 −�
ab
fαβab Σab]
and stationarity implies
δFδΣba
= GabQI −Trαβfαβ
ab
�G−1
0 −�
cd
fαβcd Σcd
�−1
= 0
�G−10
�ab = Σab +
Trαβfαβab
�G−1
0 −�
cd
fαβcd Σcd
�−1
−1
ab
so G0 is fixed
or, using GQI =�G−10 −Σab
�−1from ‘impurity solver”
Copyright A. J. Millis 2013 Columbia University
In practice
Guess hybridization function
Solve QI model; find self energy
Use extremum condition to update hybridization function
Continue until convergence is reached.
This actually works
Copyright A. J. Millis 2013 Columbia University
Technical noteFrom your ‘impurity solver’ you need G at ‘all’ interesting frequencies. Solution ~uniformly accurate over whole relevant frequency range.
This is challenging.
Severe sign problem if �∆, E
��= 0
=>reasonably high symmetry desirable so hybridization function commutes with impurity energies
For models with rotationally invariant multiplet interactions, cost grows exponentially with number of orbitals. 5 do-able with great effort more with Ising approximation (typically bad)
For simple 1 band Hubbard model, can access up to 16 orbitals at interesting temperatures.
Copyright A. J. Millis 2013 Columbia University
advantages of method
*‘Moving part’ Trp [φa(p)Glattice(Σapprox)]
some sort of spatial average over electron spectral function--but still a function of frequency
*Computational task: solve quantum impurity model: not necessarily easy, but do-able
=>releases many-body physics from twin tyrannies of --focus on coherent quasiparticles/expansion about
well understood broken symmetry state--emphasis on particle density and ground state properties
Copyright A. J. Millis 2013 Columbia University
In formal terms:
--Approximation to full M x M self energy matrix in terms of N(N+1)/2 functions determined from solution of auxiliary problem specified by self-consistency condition.
--Auxiliary problem: find (at all frequencies) Green functions of N-orbital quantum impurity model.
Question (practical): for feasible N, can you get the physics information you want?
``Exponential wall’’ remains:
Copyright A. J. Millis 2013 Columbia University
Questions (not addressed here)
•What choices of parametrization function f are admissible
•What kinds of impurity models
•What can be solved• Where is the exponential wall (how large can N be?)• what kinds of interactions can be included
•What else (besides electron self energy) can be calculated
•Quality of approximation
Copyright A. J. Millis 2013 Columbia University
Impurity model is just the Anderson model, with conduction band density of states fixed by self-consistency condition.
Regimes of Anderson model correspond to different physical behavior
Single-site (N=1) DMFT of Hubbard modelimpurity model: One non-degenerate d orbital per unit
cell
A =�
dτdτ��d†
σ(τ)G−10 (τ − τ
�)dσ(τ
�) + U
�dτn↑(τ)n↓(τ)
�
DMFT impurity model
H = −�
ij
ti−jc†iσcjσ + U
�
i
ni↑ni↓
Copyright A. J. Millis 2013 Columbia University
Phase diagram and spectral function at n=1
X. Y. Zhang, M. J. Rozenberg, and G. KotliarPhys. Rev. Lett. 70, 1666 (1993
Copyright A. J. Millis 2013 Columbia University
interaction driven MIT via pole splitting
-4 -2 0 2 4Energy
-15
-10
-5
0
5
Im Σ
U=0.85Uc2
-6 -4 -2 0 2 4 6Energy
0
0.5
1
Im G
U=0.85Uc2
P. Cornaglia dataU=0.85 Uc2
’mass’ renormalizationdiverges as approach insulator
µ renormalization0 if p-h symmetry
Σ(z) ≈ ∆21
z − ω1+
∆22
z − ω2
ω1,2 → 0 as U → Uc2
Σ(z → 0) = −∆21ω2 + ∆2
2ω1
ω1ω2− z
∆21ω
22 + ∆2
2ω21
ω21ω2
2
Copyright A. J. Millis 2013 Columbia University
doping driven transition:also coexistence regime;
2 pole structure
-6 -4 -2 0 2 4 6ω
0
5
10
15
20
25
30
35
40
ImΣ(ω)
δ=0.041152δ=0.056402δ=0.106226δ=0.13505
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
0
2
4
6
8
10
Copyright A. J. Millis 2013 Columbia University
Many regimes of behavior
From A Georges College de France lecture spring 2010 16 June 2010
Key Point: theory captures incoherent dynamics at high frequency and crossover to coherent low frequency behavior
Copyright A. J. Millis 2013 Columbia University
Regimes of behaviorconfigurations of impurity |0>, | >, | >, |2>
Strong correlationsgeneric occupancy
T
All configs; equal prob.
|0>, |2> freeze outlocal momentincoherent transport
T
All configs; equal prob
|0>, |2> freeze outlocal momentgap opens
Mott insulator
resonance forms
fermi liquid transport
Strong correlationsinteger occupancy
Copyright A. J. Millis 2013 Columbia University
Orbital degeneracy: t2g orbitals
Chan, Werner, Millis, Phys. Rev. B 80, 235114 (2009)
J=U/6
Mott, magnetic and orbitally ordered phases.
Also ``spin freezing’’
Copyright A. J. Millis 2013 Columbia University
Spin freezing line: associated with larger spin. =>power law self energy
Werner, Gull, Troyer, Millis, Phys. Rev. Lett. 101, 166405 (2008)
Copyright A. J. Millis 2013 Columbia University
The power law self energy does not extend to very lowest energies
Georges, Medici, Mravlje, Ann Rev Cond Mat Phys (2012)
Higher spin: extraordinarily low quasiparticle scale
Copyright A. J. Millis 2013 Columbia University
Phys. Rev. Lett. 100 016404 (2008)
Third example: nickelate superlattices
Idea:
Bulk LaNiO3 Ni [d]7
(1 electron in two degenerate eg bands). In correctly chosen superlattice structure, split eg bands, get 1 electron in 1 band--”like” high-Tc
Chaloupka-Khalliulin
Copyright A. J. Millis 2013 Columbia University
Pseudocubic LaNiO3
Relevant orbitals: eg symmetry Ni-O antibonding combinations
3z2-r2
x2-y2
Hybridizes strongly along zHybridizes weakly in x-y
Hybridizes strongly along x-yHybridizes very weakly in z
2 orbitals transform as doublet in cubic symmetry
Copyright A. J. Millis 2013 Columbia University
One electron (band theory) physics
Pseudocubic LaNiO3
Hamada J Phys. Chem Sol 54 1157
Two-fold degenerate eg band complex
Hamada’s result:
Copyright A. J. Millis 2013 Columbia University
Superlattice
MJ Han, X Wang, C. Marianetti and A. J. Millis, arXiv:1105.0016
Superlattice: La2AlNiO6
LaAlO3 layer: insulating barrier
Question (Khalliulin): how much orbital polarization can we get?
P =nx2−y2 − n3z2−r2
nx2−y2 + n3z2−r2
Copyright A. J. Millis 2013 Columbia University
Band theory of La2NiAlO6 heterostructure:
Hansmann et al PRL 103 016410
degeneracy of eg bands lifted: 3z2-r2 band moves up
La2NiAlO6
LaNiO3
Copyright A. J. Millis 2013 Columbia University
Band theory of La2NiAlO6 heterostructure:
Hansmann et al PRL 103 016410
La2NiAlO6 LaNiO3
2d fermi surface 3d fermi surface
No qualitative difference
Hamada J Phys. Chem Sol 54 1157
Copyright A. J. Millis 2013 Columbia University
DFT+DMFT: Correlate frontier orbitals
P. Hansmann, X. Yang, A. Toschi, G. Khaliullin, O. K. Andersen,and K. Held, Phys. Rev. Lett. 103, 016401 2009 and Phys. Rev. B 82, 235123 2010
Map conduction bands to (multiple-orbital) Hubbard model. Solve with single-site DMFT
Interaction (phenomenological)
Disperson: fit band theory
Copyright A. J. Millis 2013 Columbia University
Result: Proximity to Mott insulator=>enhanced polarization
Phys. Rev. Lett. 103, 016401 2009
Interactions enhance ‘orbital polarization’
U=0
Interactions drive ‘orbitally selective metal-insulator transition
U=6eV
Phys. Rev. B 82, 235123 2010
Copyright A. J. Millis 2013 Columbia University
SummaryDynamical mean field theory:
--approximation to electron self energy obtained from solution of auxiliary quantum impurity problem
--can treat high temperature and incoherent phases; metal insulator transition and low scales
--limits of accuracy of approximation: subject of different set of lectures
--meshes localized and k-space physics
?how to go beyond phenomenologically defined models?
Copyright A. J. Millis 2013 Columbia University
Green function:
H0 =�
i
P2i
2me+ Vext(ri) Bare Hamiltonian:
kinetic energy and lattice potential
G(r, r�;ω) =
�{ψ(r),ψ†(r
�)}
�
ω
=�ω1− H0 −Σ(r, r
�;ω)
�−1
DFT: static approximation to self energy
Many body theory: supplement DFT self energy with additional terms expressing ``beyond DFT’’ physics
GDFT (r, r�;ω) =
�ω1− H0 −ΣDFT (r, r
�)�−1
General formalism
Copyright A. J. Millis 2013 Columbia University
Express beyond DFT physics as ‘dynamical’ self energy
Σ(r, r�;ω) = ΣDFT(r, r
�) + Σdyn(r, r
�;ω)
Cannot treat many body physics of all electronic degrees of freedom=>must select subset of states for which dynamical self energy is computed*different ways to do this
Copyright A. J. Millis 2013 Columbia University
Frontier Orbital Approach
Express G in band basis. DFT part is diagonal. Many-body (``dynamical’’) self energy is in general a matrix
GDFT (k, ω) =��
ω − εKSn (k)
�δnm − Σdyn(k, ω)
�−1
Σnmdyn(k, ω) =
�drdr
�ψKS
nk (r)Σdyn(r, r�;ω)ψKS
mk (r�)
*Keep only self-energy and interaction matrix elements within frontier orbital basis.(Interaction: phenomenological U,J or screened coulomb interaction projected onto these states)
Copyright A. J. Millis 2013 Columbia University
Issues:
relevant (mainly d) bands entangled with p bands.
Practical question: how to separate bands.
Principle question: how important are the O states
Red: Ni-d
Blue : O2pσ
Cubic LaNiO3Fat bands
J. Rondinelli prvt comm
``d’’ orbital corresponding to the p-d antibonding bands has large spatial extent. Is it reasonable to ascribe just an on-site interaction U
OK Andersen JPhys. Chem. Solids 56, 1573 ~1995
Copyright A. J. Millis 2013 Columbia University
Atomic Orbital Approach
alternative: define ‘atomic like’ orbital centered on sites iproject self energy onto this basis
ψiat(r)
G(r, r�,ω) =�ω − H
KS − Σdyn(r, r�;ω)�−1
Σdyn(r, r�;ω) ≈
�
ij
|ψiat(r) > Σij
dyn(ω) < ψjat(r
�)|
Interaction: intra-d interactions already discussed
Copyright A. J. Millis 2013 Columbia University
In real materials applications
retain only on-site (but orbitally dependent) terms in self energy and only on-site Slater-Kanamori terms in interaction
Σdyn(r, r�;ω)→
�
iab
|ψiad (r) > Σab
dyn(ω) < ψibd (r
�)|
Σdouble−count(r, r�)→
�
iab
|ψiad (r) > Σab
dc < ψibd (r
�)|
G(r, r�,ω) =�ω − H
KS −Σdc(r, r�)−Σdyn(r, r�;ω)
�−1
Copyright A. J. Millis 2013 Columbia University
Additional step (not often taken in practice): Full charge self consistency
Recall Kohn-Sham potential VKS[{n(r)}] is a functional of density.
=>need to use density computed from interacting G
n(r) =� µ dω
πImG(r, r,ω)
to construct VKS.
Copyright A. J. Millis 2013 Columbia University
Two ways to define atomic orbital
*A’priori: simply choose a wave function you think is appropriate (e.g. free atom hartree fock d-wave function)
*Wannier function
0.0 0.2 0.4 0.6 0.8 1.0
�0.4
�0.2
0.0
0.2
0.4
0.6
0.8
If isolated band (index n) with wave function ψnk(r)
then real space wave function centered in unit cell i (pos Ri)
φi(r) =�
BZ(dk)eik·Riψnk(r)
Copyright A. J. Millis 2013 Columbia University
Maximally localized Wannier function
If many bands in a energy window, then many ways to construct Wannier functions--parameterized by family of unitary transformations depending on k
J. Rondinelli ANL prvt comm
LaNiO3
φai (r) =
�
BZ(dk)eik·Ri
�
n
�Uan(k)ψnk(r)
�
Marzari-Vanderbilt: showed how to choose unitary transformation to make wannier functions correspond (as closely as Fourier’s theorem allows) to atomic-like functions centered on particular atoms (e.g. O 2p and Ni 3d)
Copyright A. J. Millis 2013 Columbia University
If you have a set of bands reasonably well separated from other bands (e.g. p-d complex in many transition metal oxides)
Write G HKS and self energy as matrices in Wannier basisagain retain only terms in dynamical self energy corresponding to orbitals where correlations are important
G(ω) =�ω1− H
KSn − Σdyn(ω)
�−1
HKSab =
�drdr
�φ∗a(r)
�∇2
2m+ Vlatt + ΣDFT(r, r
�)�
φb(r�)
HKSab is formal definition of tight-binding model corresponding to ``ab-initio’’ wave functions
Copyright A. J. Millis 2013 Columbia University
Note!
All of these procedures assume that the wave functions of band theory have some meaning in the actual correlated material
Copyright A. J. Millis 2013 Columbia University
Note!
atomic orbital approach
Σdyn(r, r�;ω) ≈
�
ij
|ψiat(r) > Σij
dyn(ω) < ψjat(r
�)|
will in general lead to hartree shift which moves atomic (here, d) orbitals relative to the p orbitals
Copyright A. J. Millis 2013 Columbia University
What do people do in practice*DFT+U (Anisimov 1998)
Identify atomic-like d-orbital. Keep on-site d-d self energyarising from Hartree approx to Slater interactions
Va↑ = U < na↓ > +(U− 2J)�
b �=a
< nb↓ > +(U− 3J)�
b �=a
< nb↑ >
This calculation ``knows’’ about correlations in the d-multiplet only via ordering. In non-ordered state, shift is same for all orbitals
As with most Hartree approximations, tendency to order is overestimated
Crucial physical effect: hartree shift of entire d-multiplet by amount of order (U-2.5J)*n/spin/orbital
Copyright A. J. Millis 2013 Columbia University
Hartree term: shifts d-multiplet relative to p-states.this is important to the physics!!
d-d transition: 2dN->dN-1dN+1
E
d-p transition: dNp2->dN+1p1
E
Energy ∆Energy U
∆ > U: Mott-Hubbard ∆ < U: Charge Transfer
Lower energy controls the interesting physics
Shift of d relative to p changes the physics
Copyright A. J. Millis 2013 Columbia University
thus the crucial question:In a beyond-DFT calculation: where is the renormalized d energy, relative to the ligands
Copyright A. J. Millis 2013 Columbia University
thus the crucial question:In a beyond-DFT calculation: where is the renormalized d energy, relative to the ligands
This is sometimes called ``the double counting problem’’, based on the idea that DFT includes some of the intra-d interactions and one does not want to count them twice. But the real question is--how much does the d-level shift relative to band theory.
a large literature exists (see, e.g. Karolak et al, Journal of Electron Spectroscopy and related phenomena, 181 (2010) 11–15) but there is as yet no generally agreed upon prescription. Results are sensitive to choice of double counting correction.
Copyright A. J. Millis 2013 Columbia University
p-d covalency
My point of view (not universally accepted): parametrize the renormalized d energy by d occupancy
key issue: renormalized d energy, relative to the ligands.
Question: what does ‘‘renormalized d energy’’ mean
0.0 0.2 0.4 0.6 0.8 1.0
�0.4
�0.2
0.0
0.2
0.4
0.6
0.8
Unhybridized bands: upper one is pure d. Increase hybridization, increase d-character of occupied states
Copyright A. J. Millis 2013 Columbia University
d-occupancy:* Intuitive notion: e.g. La3+Ni3+O32- =>Ni d7 (Nd=6+1)
*Theoretically needed (if you want to put correlations on d-orbital you need to know what this orbital is and how much it is occupied)
*definition:
In terms of exact Green function G(r, r�;ω)
and predefined d-wave function φd
Nd =�
a,σ
�dω
πf(ω)
�d3rd3r
�Im
�(φa
d(r))∗ Gσ(r, r�,ω)φa
d(r�)�
Copyright A. J. Millis 2013 Columbia University
Notes
*d occupancy depends on how orbital is defined (as does the entire edifice of DFT+....)
Expect: all reasonable definitions give consistent answers (more on this later)
*should focus only on occupancy of ‘relevant’ orbitals
Copyright A. J. Millis 2013 Columbia University
Notes
*Density functional theory only gives you
n(r) =�
a,σ
�dω
πf(ω)
�d3r Im [ Gσ(r, r,ω)]
Thus even exact density functional, solved exactly, is not guaranteed to give you the exact Nd
But: seems plausible that in most cases DFT is a reasonably good approximation (more on this later)
Copyright A. J. Millis 2013 Columbia University
Notes
*Nd: theoretically precisely defined;experimental measures are indirect
--Intensities in resonant absorption spectra--Sizes of moments/knight shifts--peak positions (``many-body ed-ep)
Copyright A. J. Millis 2013 Columbia University
In practice:Available double counting corrections give Nd fairly close to Nd defined from DFT band theory calculations
Example: rare earth nickelates ReNiO3
LuNiO3: insulator. 2 inequivalent Ni sites.
Ni1 Ni2GGA 8.21 8.20GGA+U 8.24 8.22
Charge fixed by electrostatic effects.Very robust
Copyright A. J. Millis 2013 Columbia University
*DFT+DMFT (Georges04,Kotliar06,Held06)
Identify atomic-like d-orbital. Keep on-site d-d self energyarising from single-site DMFT approx using Slater-kanamori interactions
GabQI =< ψia
d (r)|G(r, r�,ω)|ψibd (r
�) >
DMFT self consistency eq:
Same double counting issue as in DFT+U
Copyright A. J. Millis 2013 Columbia University
Another example: High Tc cuprates
Green dots: fully charge self-consistent DFT+DMFT calculations for U=7,10eV. Vertical lines: DFT band theory (Wein2K and GGA)
Copyright A. J. Millis 2013 Columbia University
Nickelate superlattices
Phys. Rev. Lett. 100 016404 (2008)
Phys. Rev. Lett. 103, 016401 2009
U=0
U=6eV
Frontier orbital approach
Copyright A. J. Millis 2013 Columbia University
Atomic-like d orbitalsM. J. Han, X. Wang, C. A. Marianetti, A. J. M, Phys. Rev. Lett. 107 206804 (2011).
-8 -4 0 4 ! (eV)
0
4
8
12
U(e
V)
*LaNiO3/LaXO3 (mainly X=Al)*VASP/Wannier DFT+DMFT (Non-self-consistent) *Computed ‘orbital polarization’*Several combinations of U and ed-ep
M
I
Copyright A. J. Millis 2013 Columbia University
Results 1: Define P by integrating spectral functionfor superlattice
Spectral functions
P=0.05 P=0.09
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-10 -8 -6 -4 -2 0 2 4
DO
S (
sta
tes/e
V)
(a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-10 -8 -6 -4 -2 0 2 4
(b)Op
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-10 -8 -6 -4 -2 0 2 4
DO
S (
sta
tes/e
V)
Energy (eV)
(c)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-10 -8 -6 -4 -2 0 2 4
Energy (eV)
(d) x2-y
2
3z2-r
2
P=0.2 P=0.07
-8 -4 0 4 ! (eV)
0
4
8
12
U(e
V)
Phase Diagram
M
I
Interactions decrease P
Copyright A. J. Millis 2013 Columbia University
Results 2: How many sheets to the Fermi surface
-8 -4 0 4 ! (eV)
0
4
8
12
U(e
V)
Phase Diagram
M
I
P=0.05 P=0.09
P=0.2 P=0.07
-8 -4 0 4 ! (eV)
0
4
8
12
U(e
V)
(a) (b)
(c) (d)
!1.0 !0.5 0.0 0.5 1.0!1.0
!0.5
0.0
0.5
1.0
kx!"
k y!"
!1.0 !0.5 0.0 0.5 1.0!1.0
!0.5
0.0
0.5
1.0
kx!"k y!"
!1.0 !0.5 0.0 0.5 1.0!1.0
!0.5
0.0
0.5
1.0
kx!"
k y!"
!1.0 !0.5 0.0 0.5 1.0!1.0
!0.5
0.0
0.5
1.0
kx!"
k y!"
Copyright A. J. Millis 2013 Columbia University
Results 2: Relation to Nd
-8 -4 0 4 ! (eV)
0
4
8
12
U(e
V)
Phase Diagram
M
I
Nd=2
Nd=1.98
Nd=1.5
Nd=1.43
P=0.05 P=0.09
P=0.2 P=0.07
-8 -4 0 4 ! (eV)
0
4
8
12
U(e
V)
(a) (b)
(c) (d)
!1.0 !0.5 0.0 0.5 1.0!1.0
!0.5
0.0
0.5
1.0
kx!"
k y!"
!1.0 !0.5 0.0 0.5 1.0!1.0
!0.5
0.0
0.5
1.0
kx!"k y!"
!1.0 !0.5 0.0 0.5 1.0!1.0
!0.5
0.0
0.5
1.0
kx!"
k y!"
!1.0 !0.5 0.0 0.5 1.0!1.0
!0.5
0.0
0.5
1.0
kx!"
k y!"
Copyright A. J. Millis 2013 Columbia University
Key point: Ni-O covalency
La2InNiO6
Occupancy per orbitalBand theory: electron moves from O to Ni,
configuration is d8L
d8 is high spin:both orbitals occupied
According to band theory, LaNiO3 and derivatives are close to being negative charge transfer energy materials
Copyright A. J. Millis 2013 Columbia University
d8 (high spin)=> no distortionTwo electrons in high spin state=>no energy gain from distortion
Negligible response to lattice distortion
Over wide range of many-body phase space, Nd is close enough to d8 that low-spin J-T state is disfavored
2-band Hubbard model with 1 el/Ni unitcannot represent d8L physics
Copyright A. J. Millis 2013 Columbia University
Phase diagram: bulk nickelates
Copyright A. J. Millis 2013 Columbia University
Is the metal-insulator transition of the usual Mott/charge transfer kind
4
6
8
10
12
14
16
18
-5 0 5 10
U [e
V]
[eV]
(a) (b)
-5 0 5 10 4
6
8
10
12
14
16
18
U [e
V]
[eV]
(a) (b)
I
M
U
ep-ed
ZSA phase diagramfor cubic LaNiO3
VASP/MLWFDFT+DMFT non-self-consist
Copyright A. J. Millis 2013 Columbia University
Is the metal-insulator transition of the usual Mott/charge transfer kind
4
6
8
10
12
14
16
18
-5 0 5 10
U [e
V]
[eV]
(a) (b)
-5 0 5 10 4
6
8
10
12
14
16
18
U [e
V]
[eV]
(a) (b)
I
M
U
ep-ed
ZSA phase diagramfor cubic LaNiO3
?Where to place the materials?
Copyright A. J. Millis 2013 Columbia University
Is the metal-insulator transition of the usual Mott/charge transfer kind
4
6
8
10
12
14
16
18
-5 0 5 10
U [e
V]
[eV]
(a) (b)
-5 0 5 10 4
6
8
10
12
14
16
18
U [e
V]
[eV]
(a) (b)
I
M
U
ep-ed
ZSA phase diagramfor cubic LaNiO3
Replot in U-Nd space
Copyright A. J. Millis 2013 Columbia University
Is the metal-insulator transition of the usual Mott/charge transfer kind
4
6
8
10
12
14
16
18
-5 0 5 10
U [e
V]
[eV]
(a) (b)
-5 0 5 10 4
6
8
10
12
14
16
18
U [e
V]
[eV]
(a) (b)
I
M
U
ep-ed
ZSA phase diagramfor cubic LaNiO3
0
5
10
15
20
1 1.2 1.4 1.6 1.8 2Nd
U [e
V]M
Replot in U-Nd space
I
Copyright A. J. Millis 2013 Columbia University
Is the metal-insulator transition of the usual Mott/charge transfer kind
4
6
8
10
12
14
16
18
-5 0 5 10
U [e
V]
[eV]
(a) (b)
-5 0 5 10 4
6
8
10
12
14
16
18
U [e
V]
[eV]
(a) (b)
I
M
U
ep-ed
ZSA phase diagramfor cubic LaNiO3
0
5
10
15
20
1 1.2 1.4 1.6 1.8 2Nd
U [e
V]M
Replot in U-Nd space
I GGA Nd nickelate
Copyright A. J. Millis 2013 Columbia University
Interpretation of nickelate MIT as standard ZSA transition is not plausible
Metal-Insulator line occurs at Nd far from GGA Nd=> either band theory is VERY wrong for Nd or standard ZSA picture not applicable: Nickelates not ‘Mott’ materials
0
5
10
15
20
1 1.2 1.4 1.6 1.8 2Nd
U [e
V]M
I GGA Nd nickelate
Copyright A. J. Millis 2013 Columbia University
What is really going on?
Copyright A. J. Millis 2013 Columbia University
Nickelates: ``charge order’’Evidence: Ni-O bond length
magnetic moment
J A Alonso et al PRL 82 3871 (1999)
Copyright A. J. Millis 2013 Columbia University
But `charge order’ on Ni implausible
UNi~5-6eV implies energy COST to have different charges on different Ni sites
Our GGA + U for LuNiO3 (GGA+U relaxed structure)
thus--`charge’ order insignificant.
Copyright A. J. Millis 2013 Columbia University
We find
‘Hybridization wave’ =>site selective Mott transition
Undistorted lattice ‘Charge ordered’ lattice
Copyright A. J. Millis 2013 Columbia University
Long-bond site:Weakly coupled to environment=> local moment, ‘Mott’ insulating
‘Charge ordered’ lattice
Copyright A. J. Millis 2013 Columbia University
Long-bond site:Weakly coupled to environment=> local moment, ‘Mott’ insulating
‘Charge ordered’ lattice
Copyright A. J. Millis 2013 Columbia University
Short-bond site:Strongly coupled to environment=> d8L: spin 1 + 2 holes: singlet (‘band insulator’)
‘Charge ordered’ lattice
Energy gain from hybridization (singlet formation)
Copyright A. J. Millis 2013 Columbia University
Key point:
d-charge need not differ between sublattices
Undistorted lattice ‘Charge ordered’ lattice
(of course, some small charge difference allowed by symmetry, but this is not the important physics)
Copyright A. J. Millis 2013 Columbia University
Density Functional+ DMFT calculations substantiate this picture
Band theory: crystal structures appropriate to LaNiO3 and LuNiO3
eg DOS extract Ni d and O p bands (max. loc. wannier).
Copyright A. J. Millis 2013 Columbia University
DFT(+DMFT) in LuNiO3 structure:
Copyright A. J. Millis 2013 Columbia University
DMFT, low T: magnetic order: very different ordered moments, almost same S2, charge
Red: <S(t)2><=>Nd(Nd=1.24,1.20)
Blue: Ordered moment (FM)
Copyright A. J. Millis 2013 Columbia University
DMFT: Paramagnetic phase
Long bond site: Curie Short bond site: constant (spin gap)
Copyright A. J. Millis 2013 Columbia University
Self energy: pole on long-bond (‘Mott’) site
not on short bond (‘singlet’) site
Copyright A. J. Millis 2013 Columbia University
Energetics (preliminary): lattice distortion (hybridization wave) favored by DMFT
``LuNiO3’’
Copyright A. J. Millis 2013 Columbia University
Energetics: LuNiO3 structure favors distortion much more than LaNiO3 structure
LuNiO3 vs LaNiO3 structures(La pseudopotential)
Copyright A. J. Millis 2013 Columbia University
Energetics: LuNiO3 structure favors distortion much more than LaNiO3 structure
Copyright A. J. Millis 2013 Columbia University
Nickelate: summary
•Hybridization wave mechanism
•NOT charge order
•NOT standard charge transfer insulator
•Coupled to structural distortion
•Favored by particular bond-buckled crystal structures
Copyright A. J. Millis 2013 Columbia University
Covalency in perovskitesGenerally important
IM
Band theory or standard double counting indicates that La2CuO4 is not a Mott insulator
Copyright A. J. Millis 2013 Columbia University
Example: NiO
Fit full p-d band complex:Add ‘Slater-Kanamori’U-J interactions on d-only. Solve.
Karolak et al Journal of Electron Spectroscopy and Related Phenomena 181 (2010) 11–15
DFT
many-bodycalc
choice of double counting (covalency) affects results (perhaps less strongly)
Copyright A. J. Millis 2013 Columbia University
Summary
•Wide variety of behaviors•Important role played by instability of d valence•Energetics and state of the art of realistic theory of correlated electron materials•Key (and still ill-understood) role of double counting correction•Mott metal insulator transition only one part (maybe not the most important) aspect of transition metal oxide physics•Important open question: is the ``standard model’’ (atomic d, local correlation) the most correct and useful description?