1. Cedric Weber - KCL
Dynamical Mean Field Theory: a
dynamical quantum embedding and its
application to oxide materials
School of natural science
Department of Physics
Cedric Weber
Nijmegen, Hamburg and Uppsala,
October 2021
2. Background
2
EPFL, PhD, Quantum magnetism,
superconductivity, F. Mila
Rutgers, DMFT,
Kotliar
Cambridge, Linear scaling DFT, oxide interfaces,
Peter Littlewood, Mike Payne
3. 2012’-current: KCL (virtual) group
Top-bottom, left-right:
Araf Haque, C Weber,Yao Wei, Francois Jamet
Zelong Zhao, Evgeny Plekahnov, Carla Lupo, Evan Sheridan
Elena Chachkarova, Hovan Lee,Terence Tse, Debalina Banerjee
4. Collaborators
4
Theory
Laser spectroscopy
STM
Dr Swagata Acharya, Radboud University, NL
Dr Francois Jamet, UK National Physics Laboratory
Prof MarkVan Schilfgaarde, NREL, US
Prof Edoardo Baldini, Austin University, US
Dr Mostafa Shalaby, PSI & Beijing Key Laboratory
ProfYayu Wang,Tsinghua, China
5. Softwares (UK) :
5
DFT and DFT+DMFT : forces
GW and GW+DMFT
www.castep.org www.onetep.org
www.questaal.org
CASTEP/ONETEP now free for all academics world wide
Siesta/Smeagol+DMFT+transport
www.smeagol.tcd.ie
6. Outlines
Software Developments
The canonical phase diagram of high Tcs
…. and recently revisited by Matsumoto
QSGW + DMFT
2-particle response functions
superconducting temperature
Oxide interfaces
Pressure control of Tc
Conclusion / Outlook 6
7. Density functional theory
7
Simple idea : Transform
a problem of interacting
electrons (N-body problem) to a
problem of individual electron
interacting with a medium
1.3 The Hartree-Fock approximation.
Suppose that Ψ0 (the ground state wave function) is approximated as an antisymmetrized pr
of N orthonormal spin orbitals ψi(⃗
x), each a product of a spatial orbital φk(⃗
r) and a spin fun
σ(s) = α(s) or β(s), the Slater determinant
Ψ0 ≈ ΨHF =
1
√
N!
!
!
!
!
!
!
!
!
ψ1(⃗
x1) ψ2(⃗
x1) ... ψN (⃗
x1)
ψ1(⃗
x2) ψ2(⃗
x2) ... ψN (⃗
x2)
.
.
. .
.
. .
.
.
ψ1(⃗
xN ) ψ2(⃗
xN ) ... ψN (⃗
xN )
!
!
!
!
!
!
!
!
The Hartree-Fock approximation is the method whereby the orthogonal orbitals ψi are
that minimize the energy for this determinantal form of Ψ0:
EHF = min(ΨHF →N)E [ΨHF ]
2.1 The electron density.
The electron density is the central quantity in DFT. It is defined as the integral
coordinates of all electrons and over all but one of the spatial variables (⃗
x ≡ ⃗
r, s)
ρ(⃗
r) = N
!
...
!
|Ψ(⃗
x1, ⃗
x2, ..., ⃗
xN)|2
ds1d⃗
x2...d⃗
xN.
ρ(⃗
r) determines the probability of finding any of the N electrons within volumen
Walter Kohn : Nobel prize 1998
8. 8
The following figure shows the number of publications where the phrase“densit
theory”appears in the title or abstract (taken from the ISI Web of Scien
1980 1984 1988 1992 1996 2000
Year
0
500
1000
1500
2000
2500
3000
Number
of
Publications
publications related
to DFT
W. Kohn (Nobel Lecture):
“In the intervening decades
enormous progress has been
made in finding approximate
solutions of Shrodinger’s
wave equation for systems
with several electrons [...].
D F T i s a n a l t e r n a t i v e
approach to the theory of
electronic structure, in which
the electron density, rather
than the many-body electron
wave-function plays a central
role.“
9. 9
The following figure shows the number of publications where the phrase“densit
theory”appears in the title or abstract (taken from the ISI Web of Scien
1980 1984 1988 1992 1996 2000
Year
0
500
1000
1500
2000
2500
3000
Number
of
Publications
publications related
to DFT
W. Kohn (Nobel Lecture):
“In the intervening decades
enormous progress has been
made in finding approximate
solutions of Shrodinger’s
wave equation for systems
with several electrons [...].
D F T i s a n a l t e r n a t i v e
approach to the theory of
electronic structure, in which
the electron density, rather
than the many-body electron
wave-function plays a central
role.“
Any limiting cases where DFT needs improvement ?
Yes! for localised d- of f- atoms where Coulomb
repulsion between electrons is large! Particularly 3d and
4f are close to the nucleus for orthogonality reasons.
10. Strongly correlated systems
transitio
n metal
ion+oxygen cage
=transition metal
oxide
LixCoO2, NaxCoO2
Battery materials
Thermoelectrics
10
La1-xSrxCuO4
High temperature
superconductor
11. Hubbard model : Coulomb repulsion U
One band crossing the Fermi level
tunneling/transfer integral “t”
Hilbert space 4N, simple theory, but hard to solve.
Metal to insulator transition:
U<<1: paramagnetic Metal
U>>1 : Mott insulator
too simple but contains most of the
physics 11
93
U
t
12. Hubbard model : Coulomb repulsion U
One band crossing the Fermi level
tunneling/transfer integral “t”
Hilbert space 4N, simple theory, but hard to solve.
12
93
U
t
13. 13
spectral function A (ω) (DMFT)
Metal
insulator
increasing
U
1e/atom
Dynamical mean-field theory
Lattice Hubbard model
Anderson impurity model
A. Georges and G. Kotliar, PRB (1992)
A. Georges et al., RMP (1996)
14. DMFT: provides a realistic local self energy
An “effective atom”
approach
“Replace the full solid by
an effective atom
hybridized to an energy
dependent environment,
in a self-consistent
manner” A.Georges
14
hybridized, in a self-consistent manner,
to an energy-dependent environment
(effective medium)
Hubbard model :
impurity can have 4
states
We describe the history
of the fluctuations
between those states
A simple example: the Hubbard model
Focus on a given lattice site:
Atom can be in 4 possible configurations:
Describe ``history’’ of fluctuations between those configurations
namical mean-field theory
Bath
model mapped onto a single-site model
bath is self-consistently determined):
model
Anderson impurity model
A. Georges and G. Kotliar, PRB (1992)
A. Georges et al., RMP (1996)
Z ¯
G0
Dynamical mean field theory (DMFT)
DMFT maps a lattice many-body system onto
multi-orbital Anderson-like impurity model (AIM)
15. single site DMFT
Lattice Dyson equation:
GF Matrix reprensentation:
Local projected Green’s function:
DMFT
solver
Projectors,
localised
orbitals
16. Step 6: Self energy embedding
Green’s function written in the basis of a set of NGWFs :
Projected Green’s function:
DMFT - projection on a set of atomic wave-function {f}:
= Wm↵G↵
(i!n) V m0 ,
where m and m0
run over the five iron 3d SNGWF
tor functions (in real cubic-harmonic notation:
d3z2 r2 , dyz, dxz, dxy), ↵ and are the ind
the NGWFs, and the matrices NGWF-project
lap matrices are defined as V
(I)
↵m = h ↵|'
(I)
m i and
h'
(I)
m | ↵i.
In practice, in order to imbue the SNGWF H
projectors with a more plausible physical interpr
a real-space rotation of the functions was carried
in order to better align their lobes. The subsp
W(I)
m↵ = h'(I)
m | ↵i
G↵
(i!n) = ((i!n + µ)S↵ H↵ ⌃↵ )
1
G̃0mm0 (i!n) = Wm↵G↵
(i!n) V m0
Projected Self energy:
˜
⌃mm0 (i!n) = Wm↵⌃↵
(i!n) V m0
17. Step 7: Self consistent equations
DMFT AIM local problem Hybridization of the AIM is given by:
The DMFT
e correlated
maller local
AIM), which
is projected
self consis-
ction of the
larger space
level of the
nected to a
. The bath
ment of the
and the hy-
are dynam-
he impurity.
part of the
decay proportional to 1/i!n. We tested that the
limiting condition Õ = lim!!1
h
G̃ 1
(i!)
00
/!
up to a high precision, ensuring that the self-e
physically meaningful. It can also be straightfo
obtained by doing a high frequency expansion
Green’s function that:
Eimp
= ÕW S 1
HS 1
VÕ.
The self-energy ˜
⌃ is thus obtained by solv
Anderson impurity model (AIM) defined by
bridization (9) and the interaction Hamiltonian (
a finite-temperature Lanczos DMFT algorithm
The Lanczos solver uses a finite discretization o
bridization (9), su↵ering finite size e↵ects, yet a
4
33–35], the metric tensor on the SNGWFs is
Õ = WS 1
V
1
. (10)
is in general non-trivial, i.e., Õ 6= 1 and
WFs are nonorthogonal and not identical to
even if their parent atomic orbitals form an
set, if the trial impurity subspace does not
per subspace of the converged Kohn-Sham
e. However, for this particular case of study,
(i!n) = (i!n + µ) Õ ˜
⌃ Eimp
G̃ 1
with :
Obtain the self-energy from the local problem, and upfold back to NGWF
space. How can we upfold ? It should be the inverse operation :
⌃upfolded = V˜
⌃W
˜
⌃(! = 1) = ÕW S 1
⌃upfolded(! = 1)S 1
VÕ
⇣
ÕWS 1
⌘
V = 1
W
⇣
S 1
VÕ
⌘
= 1
Causal ! But this simplication is only for G=0 ! The k
dependence of the overlap matrix complicates everyting.
18. Summary: the DFT+ DMFT cycle
18
Impurity
Solver
Self-consistency
Condition
Gimp=Gloc
Ne( )=N0
= G - G
G = + G
L
(k)
H
DFT
r
fνν (k)
k
imp
imp
o S am
S loop
upfolding
downfolding
19. Three Tier approach
19
Overwiew of QSGW+DMFT+BSE
QSGW
non local charge fluctuation
DMFT
local spin fluctuation
Bethe Salpeter equation
Compute response function
●Magnetic/charge susceptibility
●Superconducting order parameter
Bethe Salpeter Equation
21. Electronic correlations
21
Figure 1: The hierarchy of ab initio first
principle approaches. Depending on the level of
Trilex, dGa, …
But what about the structural properties?
22. Electronic correlations
22
Figure 1: The hierarchy of ab initio first
principle approaches. Depending on the level of
Trilex, dGa, …
But what about the structural properties?
23. Electronic correlations
23
Figure 1: The hierarchy of ab initio first
principle approaches. Depending on the level of
Trilex, dGa, …
But what about the structural properties?
24. The Hubbard model
24
Sir Nevill Mott, Nobel Prize in Physics
(1977); Professor 1954-1971 in Cambridge
93
U
t
The “standard model” of strongly
correlated materials
Kotliar(and(Vollhardt(2004(
25. Doping : Phase diagram assymetry
Striking properties: Very similar structure?
Why so different phase diagram? 25
Nd2CuO2 La2CuO4
27. o Strong repulsion Cu, Ud=8eV
o Parent compound : one hole / Cu
o tpp , tdp : hybridization, +/- signs
o Competition : Charge transfer energy / Coulomb repulsion
o Mapping to 1 band reliable for large charge transfer energies
o Cuprates have intermediate charge transfer energies
o Typical LDA values for (Ed-Ep)/tdp are : LSCO=1.95, NCCO=1.38
o ZSA paper : strength of correlation is determined by the
charge transfer energy (for large Ud) (Zaanen et al PRL 55, 418‘85)
o LSCO more correlated than NCCO at the LDA level
Refined theory: d-p model
28. Doping : Phase diagram assymetry
Striking properties: Very similar structure?
Why so different phase diagram? 28
Nd2CuO2 La2CuO4
29. Doping : Phase diagram assymetry
Striking properties: Very similar structure?
Why so different phase diagram? 29
Nd2CuO2 La2CuO4
Undoped with NCCO Tc=30
•NCCO fully reduced samples (no residula apical oxygens) is
superconducting at zero doping (Matsumoto et al., condmat/
0805.4463)
•Tc=30K
•Supporting the idea that NCCO
is not a Mott insulator
•Superconductivity is killed at integer
•filling by the formaiton of the Mott gap
•or magnetic pseudo-gap
•Fully reduced samples
NCCO Experiments : Matsumoto et al., Physica C, 469, 15 ‘09
NCCO Experiments : Matsumoto et al., Physica C, 469, 15 ‘09
35. Predicting trends, chemical design
35
Materials dependent properties
C. Weber, C.-H. Yee, K. Haule, and G. Kotliar, ArXiv e-prints (2011), 1108.30
Materials dependent properties
C. Weber, C.-H. Yee, K. Haule, and G. Kotliar, ArXiv e-prints (2011), 1108.3028
C. Weber et al., Europhysics letters 100, 37001 , 2012
charge-transfer energy
EXP Theory
ZSA
boundary
charge
transfer
insulator
36. Tc increase in bi-layer cuprates
Comparaison with INS and RIXS
Inelastic Neutron Scattering Resonant Inelastic X-ray Scattering
Experiment Theory
F Jamet et al., arXiv 2012.04897
37. 37
FIG. 3. Evolution of the Fermi surface against applied strain,
as computed from QSGW ++. The Fermi surface (full lines)
is composed of a vertical line originating from the CuO chain,
and other lines from the bilayer coupling as explained in the
text. Strain increases the separation between antibonding and
bonding Fermi sheets for both (a) ideal strain (SS) and (b)
fully relaxed strain (SO). SS and SO di↵er in the degree that
dx2 y2 and dz2 are coupled, as discussed in the text. This is
manifest by the colorbar, which shows the o↵-diagonal compo-
nent |G(Q, != 0)z2,x2 y2 |, an indicator of the hybridization
between dz2 and dx2 y2 on the Fermi surface.
38. FIG. 3. Evolution of the Fermi surface against applied strain,
as computed from QSGW ++. The Fermi surface (full lines)
is composed of a vertical line originating from the CuO chain,
and other lines from the bilayer coupling as explained in the
text. Strain increases the separation between antibonding and
bonding Fermi sheets for both (a) ideal strain (SS) and (b)
fully relaxed strain (SO). SS and SO di↵er in the degree that
dx2 y2 and dz2 are coupled, as discussed in the text. This is
manifest by the colorbar, which shows the o↵-diagonal compo-
nent |G(Q, != 0)z2,x2 y2 |, an indicator of the hybridization
between dz2 and dx2 y2 on the Fermi surface.
Compressive strain:
inter Cu layer distance reduction
increases bonding / anti-bonding splitting
nesting improved
Upon internal coordinates relaxation:
Ba-O distance increases, octahedral cage tilt
Cu-dz gets closer to Fermi level
two-particle response and spin fluctuations reduced
39. 39
FIG. 3. Evolution of the Fermi surface against applied strain,
as computed from QSGW ++. The Fermi surface (full lines)
is composed of a vertical line originating from the CuO chain,
and other lines from the bilayer coupling as explained in the
text. Strain increases the separation between antibonding and
bonding Fermi sheets for both (a) ideal strain (SS) and (b)
fully relaxed strain (SO). SS and SO di↵er in the degree that
dx2 y2 and dz2 are coupled, as discussed in the text. This is
manifest by the colorbar, which shows the o↵-diagonal compo-
nent |G(Q, != 0)z2,x2 y2 |, an indicator of the hybridization
between dz2 and dx2 y2 on the Fermi surface.
As shown in Fig. 3, the Fermi surface is formed of three
bands. The two curved lines correspond to the bond-
ing and antibonding dx2 y2 bands noted above. The
interlayer hybridization is strongest at the two antin-
Compressive strain:
inter Cu layer distance reduction
increases bonding / anti-bonding
splitting
nesting improved
Upon internal coordinates
relaxation:
Ba-O distance increases,
octahedral cage tilt
Cu-dz gets closer to Fermi level
4.0 3.32 3.28 2.15 2.22 0.22 0.29
TABLE I. Interlayer Cu-Cu spacing (first column), co
to apical oxygen distance (second column), and vertical c
ponent of AO to Ba distance (third column). We report
tances for the pristine material (first row), and under unia
strain (✏z). Parameters for ideal (SS) and fully relaxed (
structures are shown (see text).
unfavorable for the superconducting order.
a)
b)
40. Metallisation of a Mott
insulator by ultra-fast laser
spectroscopy
40
Baldini et al, Proceedings of the National Academy of Sciences 117 (12), 6409-6416 (2020)
41. Laser pumping
41
results open an avenue toward the phonon-driven control of the
IMT in a wide class of insulators in which correlated electrons
are strongly coupled to fully symmetric lattice modes.
2
planes. As a consequence, over an energy scale of 3.50 eV,
charge excitations in equilibrium are mainly confined within each
CuO2 plane.
1.2
0.8
0.4
0
4
3
2
1
Energy (eV)
3
2
1
0
1.6
a
b
c
1a
(10
3
-1
cm
-1
)
1c
(10
3
-1
cm
-1
)
Exp.
The.
UHB
Density of states Density of states
Energy
(eV)
Energy
(eV)
LHB
UHB
QP
O 2p
O 2p
LHB
CT
A C
B
Fig. 1. (A) Crystallographic structure of La2CuO4 in its low-temperature orthorhombic unit cell. The Cu atoms are depicted in black, the O atoms
in red, and the La atoms in violet. The brown shading emphasizes the CuO6 octahedron in the center. (B) Schematic representation of the interact-
ing density of states in undoped insulating (Left) and photodoped metallic (Right) La2CuO4. The O-2p, lower Hubbard band (LHB), upper Hubbard
band (UHB), and quasiparticle (QP) peak are indicated. In the insulating case, the optical charge-transfer gap ( CT) is also specified. The blue arrow
indicates the 3.10-eV pump pulse, which photodopes the material and creates particle–hole pairs across the charge-transfer gap. The multicolored
arrow is the broadband probe pulse, which monitors the high-energy response of the material after photoexcitation. (C) Real part of the optical
conductivity at 10 K, measured with the electric field polarized along the a axis (violet solid curve) and the c axis (brown solid curve). The shaded
area represents the spectral region monitored by the broadband probe pulse in the nonequilibrium experiment. The theory data for the in-plane
response are shown as a violet dashed curve. The a-axis response comprises a well-defined peak in correspondence to the optical charge-transfer
gap around 2.20 eV and a tail extending toward low energies down to 1.00 eV. In contrast, the c-axis response is featureless and increases mono-
tonically with increasing energy, as expected from a particle–hole continuum. Exp. and The. in C refer to the experimental and theoretical results,
respectively.
6410 | www.pnas.org/cgi/doi/10.1073/pnas.1919451117 Baldini et al.
Downloaded
by
guest
on
February
21,
2021
Inducing apex displacement via electron-phonon coupling
42. Ultra-fast spectroscopy
42
PHYSICS
ity in response to in-plane photoexcitation. Transient spectra
at representative time delays are displayed in Fig. 2 C and D.
These data are obtained from the measured transient reflectivity
through a differential Lorentz analysis (23, 24), which avoids the
systematic errors of Kramers–Kronig transformations on a finite
energy range.
thermalizes to equilibrium. The response is featureless and one
order of magnitude smaller than its in-plane counterpart. Here
we show that this suppressed background is key to unraveling
invaluable information on the intricate dynamics of LCO.
First, we compare the temporal evolution of 1 along the
two crystallographic axes and focus on the dynamics close to
2.4
2.2
2.0
1.8
)
V
e
(
y
g
r
e
n
E
4
3
2
1
0
Time delay (ps)
50
0
-50
2.5
2.3
2.1
1.9
)
V
e
(
y
g
r
e
n
E
4
3
2
1
0
Time delay (ps)
5
0
-5
Pump || a
Probe || c
Pump || a
Probe || a
A C
B D
Energy (eV)
0.10 ps
0.17 ps
1.50 ps
E
F
-60
-20
20
2.4
2.2
2.0
1.8
Energy (eV)
0.10 ps
0.17 ps
1.50 ps
-3
-1
1
2.5
2.3
2.1
1.9
-50
0
4
3
2
1
0
Time delay (ps)
-3
-2
-1
0
4
3
2
1
0
Time delay (ps)
0.4
0.2
0.0
-0.2
Time delay (ps)
1.6
1.2
0.8
0.4
Time delay (ps)
Fig. 2. (A and B) Color-coded maps of the differential optical conductivity ( 1) at 10 K with in-plane pump polarization and (A) in-plane and (B) out-of-
plane probe polarization, as a function of probe photon energy and pump–probe time delay. The pump photon energy is 3.10 eV and the excitation photon
density is xph ⇠ 0.06 photons per copper atom. For in-plane probe polarization (A), we observe a significantly reduced 1 above the optical CT edge at
1.80 eV, due to spectral weight redistribution to lower energies. For out-of-plane probe polarization (B), the depletion in 1 is considerably weaker and
rather featureless. Oscillatory behavior is visible in the color-coded map, hinting at coherently excited phonon modes. (C and D) Snapshots of the same data
43. Driven phonon modes
43
PHYSICS
A B
C Ag
(1) Ag
(2) Ag
(3) Ag
(4) Ag
(5)
al reflectivity change (normalized to the largest amplitude) after subtraction of the recovering background, exhibiting coherent oscil-
ective bosonic modes. (B) Fast Fourier transform of data in A. The data in A and B refer to different pump and probe polarizations as
e traces have been selected in the probe spectral region that maximizes the oscillatory response (2.00 to 2.20 eV for the violet curve
V for the brown and the green curves). Different polarizations show the presence of a set of totally symmetric (Ag) phonon modes of
crystal structure. The asterisks in B indicate the phonon energy measured by spontaneous Raman scattering (25). a.u., arbitrary units.
envectors of the five modes of Ag symmetry. Black atoms refer to Cu, red atoms to O, and violet atoms to La. Modes Ag(1) and Ag(2)
rotations of CuO6 octahedra. Modes Ag(3) and Ag(4) present large c-axis displacements of the La atom, which in turn modify the La–apical
ly difference between them lies in the displacement of the apical O: While its out-of-plane motion is the same, its in-plane motion occurs
ection. Mode Ag(5) is the breathing mode of the apical O. The phonon spectrum has been computed using density-functional theory.
ent involves coherent displacements of the api-
La atoms along the c axis, i.e., an oscillating
ely follows the destabilization of the Jahn–Teller
on–Phonon Coupling in the Insulator-to-Metal Tran-
of relevant Raman-active modes. While this adiabatic method
can provide information only on the electron–phonon coupling
in the electronic ground state, it represents a first important
step to elucidate how specific atomic motions affect the elec-
tronic properties of this correlated insulator. Fig. 4 A C shows
some representative results, whereas SI Appendix, Figs. S13–
A B C
Fig. 4. (A C) Many-body calculations of the in-plane optical conductivity for the La2CuO4 unit cell.
structure (brown curve) and the response for the structure displaced by 0.04 Å along the phonon coo
along totally symmetric modes (an example is shown in A), a metallic state emerges and gives rise to
displacements along Bg modes (examples are given in B and C), there is no metallization and hence n
optical charge-transfer gap.
the trends reported for the Ag modes, establishing that displace- tors [i.e., devoid o
44. THz pumping - VO2
Weber et al, PRR 2, 023076 (2019)
Mostafa Shalabi
Beijing Advanced Innovation Center
45. 20 30 40 50 60 70 80 90
Temp [°C]
0
10
20
30
40
50
60
M1
Phase
Rutile Phase
10 15
Frequency [THz]
1
3
5
7
9
25°C
35°C
48°C
51°C
°
54 C
57°C
60°C
67°C
(a) (c)
(b)
200
300
mission
[arb.
units]
short 23
o
C
short 48o
C
short 57o
C
short 67o
C
short 77o
C
long 23o
C
long 48o
C
long 57o
C
long 67o
C
long 77o
C
d) e)
0.7
0.8
0.9
1
rb.
units]
Ethreshold
ld
Phonon
amplitude
[arb.
units]
Relative
Transmission
[arb.
units]
20 30 40 50 60 70 80 90
Temp [°C]
0
10
20
30
40
50
60
M1
Phase
Rutile Phase
10 15
Frequency [THz]
1
3
5
7
9
25°C
35°C
48°C
51°C
°
54 C
57°C
60°C
67°C
(c)
(b)
30 40 50 60
0.1
0.3
0.5
0.7
0.9
Phonon
amplitude
[arb.
units]
.5 .7 .9
E
2
[arb. units]
0
100
200
300
Relative
Transmission
[arb.
units]
short 23
o
C
short 48o
C
short 57o
C
short 67o
C
short 77o
C
long 23o
C
long 48o
C
long 57o
C
long 67o
C
long 77o
C
d) e)
20 30 40 50 60 70 80
Temp [ o
C]
0.4
0.5
0.6
0.7
0.8
0.9
1
E
2
[arb.
units]
Ethreshold
T
threshold
T
c
Phonon
amplitude
[arb.
units]
Temp [ o
C]
(a) (b)
w
[eV]
Z B Y C D A E -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
d [0.1 Å]
-0.2
0
0.2
0.4
0.6
c
[eV]
INFLATED
DEFLATED
Metal
Insulator
Ag - II
Ag - II
46. Tuning the strength of
many-body correlations by
epitaxial engineering:
magnetic switching
46
47. From Slater to Mott physics: epitaxial engineering of electronic correlations in oxide
interfaces
Random structure search: prediction of a magnetic bound
state at the FeO2 interface
48. re 1: Epitaxial engineering of robust high-spin ferrous oxides. a Illustrative
tituent materials LaTiO3, LaFeO3 and the superlattice. The dashed black lines are
black arrow indicates the direction of transferred charge when forming the superlat
gram predicted by spin assisted ab-initio random structure searches of the LaTiO3/
ped to the LaAlO3 substrate. The reference structure is the fully relaxed bulk LaA
O3/LaFeO3 superlattice in the high spin configuration. The phase space is categori
netic configurations and is ordered by energy. c Orbital resolved density of states fo
rity (positive y-axis) and minority (negative y-axis) spin-species at the LaTiO3/LaF
tive energy (ER = Esub Ebulk) stability with respect to the bulk superlattice of th
2+
2
51. Figure S4: Density of slates of the charge transfer high spin (CT HS) configuration across
all the substrates considered in our work with the Fermi level being shifted to the top of
the valence band.The value of the band gap in function of the substrate is also shown.
8
53. Beyond oxides -
high pressure and
temperature superconductivity
Towards many-body structural relaxation and
electron-phonon coupling
54. High pressure superconductors
54
01 High-pressure scientific development
Pressure is a basic thermodynamic parameter to describe the state of matter which plays an extremely important role in the
research of condensed matter.
01 Research significance of hydrogen-rich compounds
Binary hydrides
55. Calculating DMFT forces with pseudo-potentials
Plekhanov, Bonini,Weber arXiv 2102.04756
Forces with all-electron calculations, K Haule PRB 94,
195146 (2016)
Limits the scope of applicability to PPT DFT
Quantum Espresso EPW, Samuel Ponce Comp Phys Com
209, 116 (2016)
Eliashberg equations, electron phonon coupling
Inter-operability / modularity
56. CASTEP/DMFT - forces with ultra soft and
norm-conserving PP
56
Vanderbilt formalism - PRB 47, 10142 (1993)
Luttinger Ward functional, functional expression for Free
energy, and its derivative forces:
Here we used the fact that qn,m =
R
Qn,m(r)dr and
the definition of S from Ref. 31.
3. USSP DFT+DMFT forces
Variating with respect to Rµ, and using the above def-
initions, we obtain:
FDMFT
µ = Tr
X
k,⌫,⌫0
e
"⌫⌫0 (k, i!n)
Rµ
G⌫0⌫(k, i!n)
+ Tr
✓
⇢
Rµ
(VH + Vxc)
◆
@U
@Rµ
(3.9)
+ Tr
✓
Gloc
Rµ
⌃ V DC
◆
,
where e
"⌫⌫0 (k, i!n) ⌘ "k,⌫ ⌫⌫0 + ⌃B
⌫⌫0 (k, i!n) and
Green function, density and self-energy are expresse
the KS basis.
Therefore,
e
"⌫⌫0 (k, i!n)
Rµ
= ⌫,⌫0
"k,⌫
Rµ
+
⌃B
Rµ
= ⌫,⌫0
⌧
k,⌫
H
Rµ
k,⌫ ⌫,⌫0 "k,⌫
⌧
k,⌫
S
Rµ
+
P?
⌫,L(k)
Rµ
⌃ V DC
L,L0 PL0,⌫0 (k)
+ P?
⌫,L(k) ⌃ V DC
L,L0
PL0,⌫0 (k)
Rµ
+ P?
⌫,L(k)
Rµ
⌃ V DC
L,L0 PL0,⌫0 (k).
The last term in this expression, when substituted
FDMFT
µ cancels out the last term in Eq.(3.9), and we
!
ne,
which is identical to Eq.(3.4). Here, the occupancies are
defined according to the definition (2.3) (except for the
omitted summation on ⌫) as: ok,⌫ = TrG⌫,⌫(k, i!n).
Let us see how the number of particles is calculated in
the Vanderbilt’s pseudo-potential formalism:
(r)|
2
+
X
n,m
Qn,m(r)
⌦
k,⌫| I
n
↵ ⌦ I
m| k,⌫
↵
)
⌦
k,⌫| I
n
↵ ⌦ I
m| k,⌫
↵
)
1| k,⌫i} =
X
k,⌫
ok,⌫ h k,⌫ |S| k,⌫i =
X
k,⌫
ok,⌫.
nd where e
"⌫⌫0 (k, i!n) ⌘ "k,⌫ ⌫⌫0 + ⌃B
⌫⌫0 (k, i!n) and the
Green function, density and self-energy are expressed in
the KS basis.
Therefore,
e
"⌫⌫0 (k, i!n)
Rµ
= ⌫,⌫0
"k,⌫
Rµ
+
⌃B
Rµ
⌧
H
⌧
S
N = ⇢(r)dr =
k,⌫
ok,⌫ dr | k,⌫(r)|
2
+
n,m
Qn,m(r) k,⌫| I
n
I
m| k,⌫
=
X
k,⌫
ok,⌫
(
h k,⌫| k,⌫i +
X
n,m
qn,m
⌦
k,⌫| I
n
↵ ⌦ I
m| k,⌫
↵
)
=
X
k,⌫
ok,⌫ {h k,⌫| k,⌫i + h k,⌫ |S 1| k,⌫i} =
X
k,⌫
ok,⌫ h k,⌫ |S| k,⌫i =
X
k,⌫
ok,⌫.
Here we used the fact that qn,m =
R
Qn,m(r)dr and
the definition of S from Ref. 31.
3. USSP DFT+DMFT forces
Variating with respect to Rµ, and using the above def-
initions, we obtain:
FDMFT
µ = Tr
X
k,⌫,⌫0
e
"⌫⌫0 (k, i!n)
Rµ
G⌫0⌫(k, i!n)
+ Tr
✓
⇢
Rµ
(VH + Vxc)
◆
@U
@Rµ
(3.9)
+ Tr
✓
Gloc
Rµ
⌃ V DC
◆
,
where e
"⌫⌫0 (k, i!n) ⌘ "k,⌫ ⌫⌫0 + ⌃B
⌫⌫0 (k, i!n) and the
Green function, density and self-energy are expressed in
the KS basis.
Therefore,
e
"⌫⌫0 (k, i!n)
Rµ
= ⌫,⌫0
"k,⌫
Rµ
+
⌃B
Rµ
= ⌫,⌫0
⌧
k,⌫
H
Rµ
k,⌫ ⌫,⌫0 "k,⌫
⌧
k,⌫
S
Rµ
k,⌫
+
P?
⌫,L(k)
Rµ
⌃ V DC
L,L0 PL0,⌫0 (k)
+ P?
⌫,L(k) ⌃ V DC
L,L0
PL0,⌫0 (k)
Rµ
+ P?
⌫,L(k)
Rµ
⌃ V DC
L,L0 PL0,⌫0 (k).
The last term in this expression, when substituted into
FDMFT
µ cancels out the last term in Eq.(3.9), and we note
Cancellation of derivatives of self-energy, only for free
energy functional
57. CASTEP / QE inter-operability
02 Structure prediction method
The chemical and physical properties of substances depend on their crystal structure, so effective methods are increasingly
needed to predict the crystal structure of materials.
Global minimal point
e to discover the
lculate the relevant
perties of materials
ng strong correlation
calculation for
ting temperature
cture by the force
d by the strong
8
0.02 0.04 0.06
A)
Fint
EDMFT
T=0.01eV
-0.020
-0.010
0.000
0.010
0.020
0.030
0.040
0.050
0.060
-0.06 -0.04 -0.02 0 0.02 0.04 0.06
Energy
(eV)
∆z (A)
Fint
EDMFT
Ce: a=3.81, T=0.01eV
0.02 0.04 0.06
A)
Fint
EDMFT
T=0.01eV
-0.020
-0.010
0.000
0.010
0.020
0.030
0.040
0.050
0.060
-0.06 -0.04 -0.02 0 0.02 0.04 0.06
Energy
(eV)
∆z (A)
Fint
EDMFT
O: a=3.81, T=0.01eV
s) in Ce2O3 when displacing Ce (top row) and O (bottom row) along z direction. Left
while right column corresponds to a = 3.81Å. The red curves correspond to the energy
nalytical DFT+DMFT forces.
udization procedure.
the DFT+Embedded
cular P
G = 0, and,
t to calculate cancel
wave basis, employed
simplifies the formal-
ugmentation charges.
in the DMFT frame-
, which in the past al-
gy calculations within
general and suitable
ra-soft pseudopoten-
specific DMFT solver
well with all solvers.
of our approach on
wed excellent agree-
lly calculated within
ined from numerical
at very low temper-
ared the total energy
es profiles which also
alyzed the differences
of atomic forces within DFT, one-shot DFT+DMFT and
full charge self-consistent DFT+DMFT on the example
of Ce2O3. Our approach allows for quick and reliable
force calculations within fully self-consistent pseudopo-
tential DFT+DMFT and paves the way to the structural
optimization, phonon and molecular dynamics calcula-
tions within DFT+DMFT.
ACKNOWLEDGMENTS
This work was performed using resources provided by
the Cambridge Service for Data Driven Discovery (CSD3)
operated by the University of Cambridge Research Com-
puting Service (www.csd3.cam.ac.uk), provided by Dell
EMC and Intel using Tier-2 funding from the Engineer-
ing and Physical Sciences Research Council (capital grant
EP/P020259/1), and DiRAC funding from the Science
and Technology Facilities Council (www.dirac.ac.uk),
Project cs085. In addition, this work used the com-
putational support from the Cirrus UK National Tier-
ing temperature. We report (a) the superconducting temperature
ectron-phonon coupling strength and (c) log(!), as a function of
level is obtained at different levels of approximation: i) DFT PBE
ody corrections obtained by one-shot dynamical mean-field theory
ly charge self-consistent formalism (DFT+DMFT+CSC, filled red
e the DFT Tc in the direction of the experimental value (horizontal
crease of the superconducting temperature, overshooting largely the
ncrease of log(!). The charge self-consistency mitigates this effect—
reduces the Tc. The physical value of the Hund’s coupling for Ce
panels. All calculations were performed in the P6(3)/mmc phase of
re-
ces
a
n-
at
ec-
x-
m-
ns
mic
w-
th
em
19]
ues
ll-
(a)
(c)
P6(3)/mmc
CeH9-DMFT
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
80
100
120
140
160
180
T
c
(K)
J(eV)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.75
0.90
1.05
1.20
1.35
λ
J(eV)
log
ω
Tc,l~Jh
58. Enhancement of l by many-body effects
Plekhanov et al., arXiv 2107.12316
4
H
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
0.0
0.2
0.4
0.6
0.8
1.0
α
2
F(ω)
DFT
DFT+DMFT
DFT+DMFT+CSC
ω(eV)
(b)
DOS(a.u.)
at high pressure.(a) Phonon dispersion relation, (b) Eliashberg function ↵2
F(!) and
(3)/mmc phase of CeH9 at 200 GPa. The phonon density of states is resolved in the Ce
ontribution to the Eliashberg function is due to the hydrogen vibrational modes located
FIG. 3. Lattice dynamics of CeH9 at high pressure.(a) Phonon dispersion relation, (b) Eliashberg function ↵2
F(!) and
(c) phonon density of states for the P6(3)/mmc phase of CeH9 at 200 GPa. The phonon density of states is resolved in the Ce
and H contributions. The dominant contribution to the Eliashberg function is due to the hydrogen vibrational modes located
above the phonon gap (! > 750cm 1
).
⇡ 300 to 750cm 1
), and high frequency modes dominated
by Hydrogen character (⇡ 750 to 2000cm 1
). The latter
leads to a large weight in the Eliashberg function between
0.1 to 0.25eV, i.e. in the region that mostly contributes
to the electron-phonon coupling strength . The effect
of many-body corrections is indicated in Fig. 3.b, with
an increase in the latter energy region due to the DMFT
corrections (the trend follows the one observed in Fig
1.a).
The changes highlighted above stem directly from a
spectral weight transfer induced by many-body correc-
tions (see Fig. 4.a). In DFT, the Ce system is described
by a two band system in absence of long-range magnetic
order. We note that DFT is a single Slater determinant
approach, and hence can not capture the role of param-
agnetism, with an associated magnetic multiplet (fluctu-
ating magnetic moment). Such effects typically induce a
splitting of spectral features into satellites, as observed in
Figs. 4.b,c, with a resulting large increase of f-character
at the Fermi level. As sharp Ce features occur near the
Fermi level, we emphasize that a high level of theory is
required to capture correctly the superconducting prop-
G A H K G M L H
-8
-6
-4
-2
0
2
4
6
8
E
(eV)
-10 -5 0 5 10
0
2
4
6
8
10
12
14
16
18
DOS(1/eV)
E(eV)
Glatt
Gimp
-10 -5 0 5 10
0
2
4
6
8
10
12
DOS(1/eV)
E(eV)
Glatt
Gimp
-10 -5 0 5 10
0
2
4
6
8
10
12
DOS(1/eV)
E(eV)
Glatt
Gimp
(a) (b)
(c) (d)
FIG. 4. Spectral weight transfer induced by many-
body corrections.(a) Electronic band structure and (b) den-
sity of states obtained by DFT calculations. Glat and Gimp
denote the spectral weight obtained by the imaginary part of
respectively the lattice and f impurity Green’s function, cor-
leads to a large weight in the Eliashberg function between
0.1 to 0.25eV, i.e. in the region that mostly contributes
to the electron-phonon coupling strength . The effect
of many-body corrections is indicated in Fig. 3.b, with
an increase in the latter energy region due to the DMFT
corrections (the trend follows the one observed in Fig
1.a).
The changes highlighted above stem directly from a
spectral weight transfer induced by many-body correc-
tions (see Fig. 4.a). In DFT, the Ce system is described
by a two band system in absence of long-range magnetic
order. We note that DFT is a single Slater determinant
approach, and hence can not capture the role of param-
agnetism, with an associated magnetic multiplet (fluctu-
ating magnetic moment). Such effects typically induce a
splitting of spectral features into satellites, as observed in
Figs. 4.b,c, with a resulting large increase of f-character
at the Fermi level. As sharp Ce features occur near the
Fermi level, we emphasize that a high level of theory is
required to capture correctly the superconducting prop-
erties. For instance, in our calculations the full charge
self-consistent approach (DFT+DMFT+CSC) induces a
small shift of the sharp Ce feature at the Fermi level,
which in turns mitigates the f character increase at the
Fermi level.
As the role of f electronic orbitals is paramount for the
superconducting properties, we study a prototype lan-
thanide clathrates with higher f occupations, by consid-
ering the aliovalent praseodymium hydride PrH9. We
FIG. 4. Spectral weight transfer induced
body corrections.(a) Electronic band structure a
sity of states obtained by DFT calculations. Gla
denote the spectral weight obtained by the imagin
respectively the lattice and f impurity Green’s fun
responding to the spectral weight traced over all o
traced over the f orbitals, respectively. In (c) and (
the energy-resolved spectral weight, obtained resp
the one-shot DFT+DMFT and the full charge sel
DFT+DMFT+CSC. All calculations were perfor
P6(3)/mmc phase of CeH9 at 200 GPa.
with the experimental value obtained at lowe
DFT
DMFT1s DMFT cs
59. LaH16 - ultra high pressure, >250GPas
59
04 Structure prediction and stability
Fig . computed stable structure of LaH16 at 250GPa
04 Characteristics for rich-hydrogen material P6/MMM-LaH16
Fig. Structural relaxation of clathrate lanthanides with many-body corrections. All calculations are performed at 250 GPa.
Internal coordinates are relaxed with DFT+DMFT+CSC, building upon the recent implementation of DFT forces for
ultra-soft pseudo-potentials. We report the forces and total energies obtained during the structural optimization for LaH16.
04 Phonon of P6/MMM-LaH16
Fig. The phonon DOS of LaH16 under 250GPa
60. Conclusions
60
Phase diagram of cuprates - role of structural
properties
Insulator / Metal transition upon small change of
apex oxygen displacements
Calculations within QSGW+DMFT
Two particle response functions
Controlling Tc via strain and structural changes
Ultra-fast spectroscopy
Many-body effects at interfaces
Thank you ! Questions?
