Spectroscopy of strongly correlatedmaterials: beyond DMFT
Evgeny Gorelov
European XFEL

Alexander Lichtenstein
University of Hamburg
Hiroshima, 05.07.2017

Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

Outline!

Spectroscopy of strongly correlated materials: realistic DMFT!

Non-equilibrium dynamics of correlated materials: XFEL-ultra-strong pulses!

Non-local correlations: cluster and dual scheme

2

Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

Strongly correlated materials!

Interplay of degrees of freedom!

Coupling of control parameters!

Complicated ordering phenomena

OrbitalSpin

ChargeLattice

H. Ehrke et al.,PRL 106, 217401 (2011)

E. Pavarini, E. Koch, and A. L. PRL 101, 266405 (2008)

Spaldin & Fiebig, Science 309:391 (2005)

3

Spectroscopy of strongly correlated materials: beyond DMFT

4Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

Continuous Time Quantum Monte Carlo
Partition function:

Stochastic expansion – “Diagrammatic Monte Carlo” (N. Prokof’ev)

Σ

Σ

Σ

Σ

U

Σ

Σ

Σ

Σ

τ’

U

τ

G(τ−τ’)

E. Gull, A. Millis, A.L., A. Rubtsov, M. Troyer, Ph. Werner, Rev. Mod. Phys. 83, 349 (2011)Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

Fermi surface for Ruthenites: LDA+DMFT
Sr2RuO4

Sr3Ru2O7

A. Damascelli et al.,PRL 85, 5194 (2000)

A. Tamaiet et al.,PRL 101, 026407 (2008)

Sr2RuO4

Sr3Ru2O7

ARPES

TheoryRu
SrG. Zhang, E. Gorelov, E. Sarvestani,and E. Pavarini, accepted to PRL (2016)

E. Gorelov, G. Zhang,and E. Pavarini (unpublished)

OP
OA

5

Figure 4(a) exhibits the azimuthal angle dependence forx ! 0:5 at Q ! #0 2 6$ at 300 K, in which the periodicity is
0
90
180
270
360
360& with the phase shift of 180& against that for x ! 0. In
Azimuthal angle (degree)
the inset of Fig. 4(a), the energy profiles for the interference term at " ! 270& are shown at 300 and 6 K. Based on
FIG. 2. Azimuthal angle dependences of the interference term
Spectroscopy of strongly correlatedmaterials:
beyondDMFT
EvgenyGorelov,
Alexander
Hiroshima,
05.07.2017 forasimilaranalysisLichtenstein,of the azimuthal
angle dependence
for a main edge peak at 305 and 6 K at Q ! #0 2 6$. The origin
x
!
0,
it
is
revealedthat
theorbitalpolarization
with dyz=zx
of " is defined when the a axis parallels the direction of the !
polarization of the x-ray beam. The thick and the thin lines arecharacter is favored for x ! 0:5 at 300 K [!f ! 1:3#1$].
analysis results at 305 and 6 K, respectively.
The RXS intensity for x ! 0:5 slightly decreases withlowering temperature [Fig. 4(b)]. This decrease in orbital
2 polarization
4 is attributable to the electron transfer from the
0
1
0
fk 0
dyz=zx orbital to the dxy orbital at low temperatures.
(3)f~ ! @ 0 fk 0 A;
Notably, the ferromagnetic (FM) instability has been re0 0 f?
vealed at about x ' 0:5 at low temperatures, which is likely
due
to the van Hove singularity of the dxy orbital band (#where fk (f? ) represents the susceptibility tensor element
2g
band)[6,9,22,24]. The susceptibility atxyx ! 0:5 shows
for the direction to which the (un)occupied 4d t2g orbital
significant enhancement below 20 K [9], where the orbital
extends. fk " f? #! !f$ means an anisotropic strength ofHigh-temperature phase: L-Pbca
polarization intensity saturates at its minimum in Fig. 4(b).
an atomic scattering factor, which PHYSICAL
corresponds
to BthedeO. FRIEDT et al.
REVIEW
63 174432
gree of an orbital polarization.
%
100 FmeV!" in formula (2) mainly contributes to the " depenCa2RuO4
dence of the interference signal. The observed " depen0.30
dence shows the minimum and the maximum around" ! 90& and 270& , respectively, while the intensity apxy orbital ordering
Q=(0 2 6)
0.25
proaches zero at " ! 0& and 180&. These features are well
explained by the analysis for a ferro-type dxy ordering, as
0.20
shown in Fig. 2. In addition, we analyzed the " dependence of the resonant signal at Q ! #0 2 14$, which is also
consistent with the behavior of the dxy orbital. Comparing
0.15
the interference signal with the fundamental signal at
305 K,distances
it is found that the !f is about 0.8(1), which is of
FIG. 2. Temperature dependence of the Ru-O!1"-bond
0.10
Low-temperatureedge lengths of thetheoctahedronin Ca2RuO4.0 and that of thephase:orderbasalof 10"2 for a conventional structure factor at Q !
plane.
no orbital ordering
S-Pbca#0 2 6$. We also measured the " dependence of the resonant
signal
0.05
and in particular free refinement of the two
distinct
O!1"at 6 K. The " dependence at 6 K with !f !
meVdisplacements along c led to significant 300
differencewithone the same periodicity as that at 305 K.
1:4#1$
hasdisplacement almost vanishing, leading to aConsequently,
LTT equivalent
it is revealed that the local symmetry at atilt scheme. The structure was finally refined with one O!1"
0
Ru
site
of
orbitalordering does not change between 305
site being fixed at z!0. In case of the stoichiometric com0
50 100 150 200 250 300 350
pound at 400 K this is definitely not an appropriate
In observed dxy ferro-type ordering is consistentand 6model.
K. TheTemperature ( K )space group Pbca one obtains an R value of 5.39% which
withonethez-O!1"
prediction by Refs. [21,22].
increases to 5.62% for the P2 1 /c model with
fixed to zero. Refining the P2 1 /c phase independently
Figure still
3 shows the temperature dependence of the RXS
FIG. 3. Temperature dependence of the interference term at
FIG. 3. Temperature dependence of the lattice parameters ingives a lower R value than Pbca, but this signal.
difference Below
is not '200
K,
the
RXS
signalfortheferro-type
Q ! #0 2 6$ with " ! 270& in Ca2 RuO4 .
Ca2RuO4.0. Open symbol designate the results obtained from thesignificant any more. Therefore we conclude that the high-3

6

Orbital ordering in Ca RuO
Structural transition

RXS signal for the d ordering

Intensity (arb. units)

Crystal-field orbitalsof Ru 4d t manifold

high-flux patterns and closed symbols those from the high resolutemperature structure in Ca2RuO4.0 has the same space group
026401-3
tion studies.
Pbca as the low-temperature structure. We differentiatetheseM.
phasesas L- PbcaT.
andWehling,
S- Pbca, respectively,
due toforces inO.
the Friedt
La-O or Sr-O
layerPRB
where onestrongly
E. Gorelov,
Karolak,
F. Lechermann,
et al,
63,distance174432(2001)
their different c parameters. The large amount of excess oxydecreases due to the opposite displacements of an apical oxyA. L, E.gen
Pavarini,
104,for226401
seems to bePRL
responsible
the distinct(2010)
diffraction patgen and a neighboring La/Sr site. The stretching of the latticetern in Ca2RuO4.07.
perpendicular to the tilt axis reduces the pronounced shrink8
ing of this bond and may be seen in the orthorhombic splitCombining the new and the previous results we get theting. One should hence consider this behavior as the normal
entire picture of the phase transition in Ca2RuO4.0. Figure 2
one arising from the structural arrangement. Nevertheless thepresents the Ru-O bond lengths, which show a discontinuous
elongation of the octahedron basal plane might influence thechange at the L- Pbca-S- Pbca transition: the in-planeelectronic and magnetic properties, see the discussion below.
bonds become elongated and the out-of-plane one is shrinkThe orthorhombic lattice in Ca2RuO4.0, however, is eloning upon cooling. Close to the transition—but in the S- Pbca
gated along the tilt axis at low temperature much more thanphase—the octahedron is still elongated along c in the temwhat might be expected for a rigid octahedron tilt; the latticeperature range 300–330 K. Upon further cooling one obconstants are given in Fig. 3. It is the stretching of the octaserves a continuous and even larger change in the same

M. Kubota et al., PRL 95, 026401 (2005)

xz

yz

xy

Spectroscopy of strongly correlated materials: beyond DMFT

Metal-Insulator transition in Ca2RuO4

Ru

Simplified Coulombinteraction

OCa

Ru 4d t2g orbitals

xz

no orbital ordering

T = 290 K

Insulating phasexy orbital
ordering

Wrong
multipletstructure!

-6 -4

xy

Full Coulombinteraction
T = 390 K

Metallic phase

DOS

:

7

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017xy
xz, yz-20yz

E. Gorelov, M. Karolak, T. O. Wehling, F. Lechermann, A. L. , and E. Pavarini, PRL 104, 226401 (2010)

2

-6 -4

E (eV)-2

0

2

Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

Spin state transition in LaCoO3
Co 3d states
Low spin(S=0)

High spin
(S=2)

La
Co
O

eg
2 eV

Valence-band XPS
t2g

High spin state, density of states for Co 3d orbitals
T ~ 4TMIT

T ~ 2TMIT

t2g

G. Zhang, E. Gorelov, E. Koch, and E. Pavarini, PRB 86, 184413 (2012)

eg

t2g

M. Abbate et al., PRB 47, 16124 (1993)

8

Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

Spin and Charge modulations in LaCoO3
Density of states for Co 3d orbitals

Phase diagram as a function of the lattice
temperature and the Hund’s rule coupling
J

La
Co
O

Co #1 Co #2
M. Karolak, M. Izquierdo, S. L. Molodtsov, and A. L., PRL 115, 046401 (2015)9

Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

Cluster model for calculation of XAS spectra in LaCoO3!

Includes Co-d and O-p orbitals!

Electron Hopping between Co-d and Ligands is allowed!

Full Coulomb vertex at Co-d orbitals is included

Wannier orbitals

Co L2,3 XAS spectra with fixed structureand different electronic temperatures
L3, 300K structure
0.45

300K
350K
400K
450K
500K
550K
600K

0.4
0.35

Co and O bands (green)

E-EF (eV)XAS (a.u.)

0.3
0.25
0.2
0.15
0.1
0.05
0

776

778

780

782
E (eV)!

E. Gorelov, to be published

784

786

10

Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

Spectral properties of correlated materials: CPO-27-Ni XES!

!

Catalytic metal-organic framework CPO-27-Ni!
Ni valence to core K-edge XES!
Ni d-d excitations reflects Ni environment, where catalytic activity takes place!
Study of catalytic activity in-situ is possible

E. Gallo, E. Gorelov, A. A. Guda, F. Bonino, E. Borfecchia,
D. Gianolio, S. Chavan, and C. Lamberti, submitted to Inorg. Chem.
!
!

Full Coulomb interaction for Co-d orbitals
Cluster spectrum codes: XTLS by Arata Tanaka (Hiroshima);
Quanty by Maurits W. Haverkort (Heidelberg)

11Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

Correlated materials in strong time-dependent electric fields!

Investigation of behaviour of correlated materials in presence of strong electric fields!
Hubbard model is used to model correlated materials!
IPT on Keldysh contour with non-equilibrium dynamical mean-field theory!
Electric field is included as Pierls substitution for hoppings

Hubbard model in Electric fields

A. Joura, J. K. Freericks, A.L., Phys. Rev. B 91, 245153 (2015)12

Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

Correlated materials in strong time-dependent electric fields!

Investigation of behaviour of correlated materials in presence of strong electric fields!
We study single-band Hubbard model with anisotropic hoppings!
Arbitrary field shape and strength can be considered!
Multi-orbital extension allowing realistic calculations is feasible
Spectral function
Vector potential

U=4 eV

U=6 eV!

Viktor N. Valmispild, Martin Eckstein et al.

13

Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

Correlated materials in strong time-dependent electric fields!

Infrared vs X-ray
Vector potential

U=4 eV

IR

Xray

!

Viktor N. Valmispild, Martin Eckstein et al.

U=6 eV

14

Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

Correlated materials in strong time-dependent electric fields!

Polarization dependence on rectangular lattice!
We study anisotropic Hubbard model, ty = 2tx!
We use infrared pulse, ω = 1.4 eV!
Linear polarization along X or Y axis is used

tx

ty

15

Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

Lattice distortion and time-dependent electric fields
Small Field

E || x

E || y!

Viktor N. Valmispild, Martin Eckstein et al.

Large Field

16

Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

Non-Local correlations: AFM and d-wave in HTSC

A.L. and M.Katsnelson, PRB 62, R9283 (2000)

17

00 isolated
1
The1Hamiltonian of theplaquettereads
tum
criticalpoint.
The00=ĥ
0 crystal,µthat
plaquettehas
crucialimportµtttof the
is
very
close
to
the
optimaltttplaquettehquette of the crystal, that is very close to the optimal
X
X
0 C has beenstatesinvestigB
0
2strong-coup
2 -wavestrong-couplingd†
B t valueCtµtt
3.4 electrons.
0
x
y
of
3.4
electrons.
µ
t
t
18
B
C
Hp =
ci 0 cj +Uni" ni# ,(2)ĥ0 =hij
.
B
C
ĥ
=
.
@
A
[20]
and
themany
plaq
portanceofof alsotheseinthree0
t t portanceµ
tofof theisolatedTheory
plaquette
reads
Hamiltonian
oft0thetThe
isolated
plaquette
reads@Plaquette
Valence
Bond
HTSC
i=1..4µHamiltoniant A(i,j)=1..4
0 tte CDMFT has been
tte CDMFT
discuss
t
t
t0 µ
0
1
of
a
quantum
criticalp
XX
X t t0 † t µ X†
0t i" nti#[19].
In thai epW=n, that(2)
In
case
an
addition
Hcpim=ehc eht he0ijdcui lccn
U ni"hniji#c,i cj +(2)µ tU[19].
j H+
discussed
in Refs.
[23,
0 C
B
0
sectorappetµtt
i=1..4(i,j)=1..4
sectorappeared
withoutforWe. jinclude
thechemicalpotential
inthediagonal
partof
i=1..4
rtcep(i,j)=1..4
s yg r e n e e h T
h
B
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ĥ
=
.
i
Plaquette
Valence
Bond
Theoryofd-waveSuperconductivity
01
00bottom-up
approach,t@Atum
critica
0 h0in. 0 The
1diagonal
0 of
t
t
µ
t
chemicalpotential
the
part
tum
criticalpoint.
The
cross
energyspectrumoftheisolated
plaquettenearµ t t t Quantum
3 Critical Point“„Plaquette
si gnillfi nortceµle-t3ij tehtt 1
0states hasb
0 Ct statesconsistent
minimal
pict
B
t
t
µ
1.0
has
been
investigated
t
µ
t
t
the
3-electron
fillingisvery
unusual.
We
presentinFig.
1
ypsspectrumoftheisolated
plaquette
near0C.
2 ĥ0 = Bµ Uĥ0eh=t B
nit snµoitgetr C[20] and als
0
B
C
@
A
N
=
2
.
t
t
µ
t
[20]
and
also
in
the
plaquetteinthe U0.8 inµ Fig.
space,1 whose groundstates havean cs,
@ t0 regions
A3present
filling
is very unusual.
We
Ifweapproach
this
t
µ
t
of
a
quantuNµ= 3 of ain
lp eerht fo noitaWe
p0 ucinclude
co 4 thechemicalthe
diagonal
partof ant plus-minus
t0 t potential
quantum
criticalpoint
occupation
of
three
one
electrons.
The
onee
discussed
0.6 have ant ht†0 groundt Theµ 5energy
U µ space, whosestatesN =plaquette
3 groundstain
N of= 4thethe.
spectrum
isolatednearbrijtcele spectrum
discussed
in Refs.
[23,
31, 32]
esso p murtce ps noelectron
possesses fourstates withthe energies:
bottom-up
6
the0 chemical
potential
in unusual.
the diagonalof in Fig.
-! We include
0.4
three plus-minus
electrons.
The
oneof
states
(DOS)
at
the
V
the
3-electron
fillingisveryWe
present
1
0 part0 one
bottom-up
approach,
that
±2t
t the
µdiagonal
and double-degenerate
t µ.nearAt zerointer- thm
7spectrumlbutheodchemical
tThe
t
2
±
consistentude
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in
part
of
1dna µh0ij
2.
energyoftheisolated
plaquette
†
in8=
the
Uthereµ isspace,
whose
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an of
um
possesses
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0.2energies:
U t fourcregions
ke energy
consistent
minimalpictureactionU0,nostablegroundthree
spectrumoftheisolated
plaquettenear
si ereht ,0t´=
noitc0 a fillingtheU
3-electronvery unusual. We present in Fig. 1
If we app
9 ofis threeoccupationplus-minus
oneelectrons.
Theonend double-degenerate
t
µ.
At
zero
interfolddegenerate
N
==33
0.0
electrons,
in
the
sense
that
one
can
add
or
removeone
ectronfillingisvery
unusual.
We
present
in
Fig.
1
If
we
approachthiscritical!
!
regions
intheUµspace,
whose
groundstates
have
an
the
N
10
rtcele spectrum possesses four states with the energies:- esnes eht ni ,snoelectron
11
electron
without
thermodynamic
potential.
there
ground
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and
= 4the(in the is
U noµ stable
space,
whose
groundstates
have
anthe
the
N0 The=side)
3onegroundstate,
0.2changing
occupationofthreeplus-minus
oneelectrons.
ofN
states(D0-gnahc tuohtiw no±2t
2
3
45t
6 At zero
7
8
rtspectrum
coneelet electrons.
12 µ and double-degenerateµ.
inter4
electronpossesses
fourstateswiththeenergies:
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strength
U
⇡Ferm
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3
ion
of
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plus-minus
The
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states
(DOS)
at
the
he
that one canaddorremoveoneenergy.
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correspond13
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0actionUa=0, thereis nostableground
state
with
three
emos possesses
m-!or±2t
f gnfour
ti0trastatesµtS
anddouble-degenerate
t
µ.
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interfolddegene
there
is
small
region,inthattheplaquette
groundstate
nutspectrumwiththeenergies:
electronandholesides,due t
14!
Periodic
BC:
t->2t
and
U->2U
potential.
izationofthe-functio- changing the thermodynamic
electrons,
the
sense
that
one
candegenerateaddorremoveone
015 in ist=-1
t‘=0.3
action
U
=
0,
there
no
stableground
state
with
three
(hole
side)
Fig
µ
and
double-degenerate
t
µ.
At
zero
interfold
N
=
3
groun
Fig.
1:
(coloronline).
Zero-temperaturephase
diagramwithN=
3electronsisseparatedbyenergy
gaps
from
,
n
o
i
g
e
r
l
l
a
m
s
a
s
i
e
r
e
h
t
ysome critical interaction16strength U ⇡ 3holeandelectronsides,
electron
withoutchanging
thethermodynamic
potential.
electrons,
in
the
sense
that
one
can
add
or
removeoneenergy.
Th
M.
Harland,
M.
Katsnelson,
A.L.
PRB
94,
125133(2016)chaoftheisolated
plaquetteasfunctionof
Hubbard
U
and
Ug =
0,
there
is
no
stableground
state
with
three
(hole
side)
and
N
=
4
(electro
thestates
with
N
=
2
and
N
=
4,
see
Fig.
1.
Ims
n
o
r
t
c
e
l
e
3
=
N
h
t
i
w
(a)
(b)
17 chemical potential µ in proximity of the quantum critelectron
withoutchanging
thecritical
thermodynamic
potential.
izationtregion,inthattheplaquette
groundstate
Startingfromsomeinteraction
strength
U ⇡ 3ofspin
animportantelectronns,
in
the
sense
that
one
can
add
or
removeoneenergy.
The
correspondingsp
portantly,
this
N
=
3
groundstate
isfourfolddegener18
d = N h t i w s e ta t s e h t
0
poi
ical
point
(circle)
for
t
/tthe
= plaquette
0.3. U ⇡ groundStartingfromsomecriticalinteraction
strength
3
hole
and
el
there
is
a
small
region,
in
that
state
19
ectrons is separated by energy gaps fromthat this plaquette quµSpectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

fields acting on the bath fermions b†i , bj , i.e. dx2 y2 wave pairing, singlet magnetic states on the bonds, andthe conventional Neel antiferromagnetic state:
Plaquette susceptibility: one level bathXhd =( 1)i d (bi, bi+1, + h.c.),(3)
Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017=",#,i=1..4

hs =

X

†
(b
s i, bi+1,( 1)i+ h.c.),(4)=",#,i=1..4

X
1
hm =
( 1)i
ilbert space size of 216 , 2
without any†bm i, bi, .

(5)=",#,i=1..4) with di↵erent symmetry breaking
assumeperiodic boundary conditions, thatth fermionsHereb†i , bj we, namelyd-waveetic statesmeanson the forbonds,
and4 thei =we define i + 1 = 1. We switch on
iferromagnetic
smallstate:fields=== 0.01t simultaneously and
d

( 1)i

( 1)i

X

s

m

calculatebi+1, numerically+ h.c.) (4) their associated susceptibilities ofthe plaquette. The hybridization V between the fermionsc†i and b†j breaks the sixfold degeneracy of the plaquette’squantum†+ h.c.)critical(5) point and without external fieldss (bi, bi+1,
it results in a singlet ground state, see Fig. 2a. The
dx2 y2 -wave superconducting (Eq. (3)) and the magnetic
d (bi,

for V  0.2.
for 19the clust
rameters costate near t
dx2 y2 -wave
dering. Thi
calculations
3
Plaquette
more realist
sider a plaq
in quadrup
a plaquette
CDMFT benectivity ofobtain simil
a two-site cthis model rĜ 1

Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

Singlet State of SC-plaquette with bath
Isolated Plaquete:

Plaquete in SC-Bath: V=0.2, Δd=0.05

6x
|ψ0>

N=3

N=2

+0.06

|ψ0> =0.05

<4x>S

N=4

+0.08

<8x>S

<2x>S

20

to39infinity.
obtainsimilarresults
as for the double
A. P. Kampf,Weand A.
A. Katanin,
Phys. Rev.
B 67 125104
41,43(2003).
Bethe
model
for
a
two-sitecluster. The bath Green’s
40
W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden, and21functionin this model
K. Schönhammer,
Rev. Mod.reads
Phys. 84, 299 (2012).
Plaquette-Bethe
lattice41
B 62
H. Hafermann, M. I. Katsnelson,
and
A.
I.
Lichtenstein,
FIG.
4:(Coloronline)Sketch of a plaquette-Bethe lattice⇣ , 37006 (2009). ⌘ 1
Europhys. Lett. 85
with connectivity q=2. Only one plaquette is shown for sim1
42ttler,
J.
and ĥ
A.0 I. Lichtenstein,Ĝ Otsuki,(i!)H.=Hafermann,
i! + µplicity.
tb z ĜPhys.
(i!) z tb
(6)

psen,
01).
berg.

Rev.

007).
t, G.
009).
t, G.
009).
A. J.

n, J.
B 81,

Lett.

M. S.

anke,

Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

Rev. B 90, 235132 (2014).
G. Moeller, V. Dobrosavljevic, and A. E. Ruckenstein,Phys. Rev. B 59, 6846 (1999).
Optimal Bethe-hopping: tp =0.25
44
where
Ĝ
(i!)
is
a
8
⇥
8
matrixthe superconducting
M. Hashimoto, I. M. Vishik, R.-H. He, T.of
P. Devereaux,and Z.-X.
Shen, Nature
Physics
10,483 (2014). with opposite spinGreen’s
functionfortheplaquette
45
R. H. Zadik, Y. Takabayashi, G. Klupp, R. H. Colman, A.
Nambu
spinors
in thebath.
WeP. Matus,
discretize
the bath
Y. Ganin,
A. Potočnik,
P. Jeglič,D. Arčon,
K.
Kamarás,
Y. Kasahara,
Y. Iwasa,
A. fourN. Fitch,
Y. Ohishi,
Green’sfunctionwith
onlystatessimilartothe
G. Garbarino, K. Kato, M. J. Rosseinsky, and K. Prassides,
DMET
approachSci. Adv.
1, 1500059 using(2015). the Lanczos scheme in order to46t‘ Greiner, and D. S. Jin, Phys. Rev. Lett.
C.
A.
Regal,
M.
find92,
theGreen’s function of the superconducting
t matrix
040403(2004).
28,29
47
states. J.Asmentioned
above,
there isM. a sixfold deE. Gull, A.
Millis,A. I. Lichtenstein,
A. N. Rubtsov,
Troyer, and Ph. Werner, Rev. Mod. Phys. 83, 349 (2011).
48
P. Seth, I. Krivenko, M. Ferrero, and O. Parcollet, Comp.
tPhys. Comm.p 200, 274 (2016).
49
B. Dalla Piazza, M.M.Mourigal,
M.Katsnelson,
Guarise, H.
Berger,T. 125133 (2016)
Harland, M.
A.L.
PRB 94,
Schmitt, K. J. Zhou, M. Grioni, and H. M. Ronnow, Phys.
FIG. 5: (Colort online) Local part of normal Green function
43

Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

DMFT-functional

Start from
Correlated Lattice

Find the optimal
Reference System
Bath hybridization

Expand around
DMFT solution

22

Vk )

⌦

⌦

⇣

⌦

⌘

of strongly correlatedmaterials:†⇤ Spectroscopy⇤† beyond DMFT
⌘
f
f
+
⌘
f
!⌦ ⌦ !+⌦ !
!⌦ ⌦ ! f!+⌦

+ 41

P

†
f!+⌦
f!† 0 ⌦ f! f!0 + ...

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017!! 0 ⌦!! 0 ⌦

23

Dual Fermion: Beyond DMFT

< c!+⌦ c!0 ⌦ c†! c†!0 >imp g! g!0 ( ⌦+! !0!! 0 ⌦ =tk
Ug!+⌦ g!0 ⌦ g! g!0

⌦)
~
G
ij

~
G ii

!⌦ =

< c!+⌦ c†!6⇢i⌦ >imp

< ⇢ >imp g! !i⌦4t
gi aii‘
g g

!

~
G

ji

~
Gij

!j 4

+...

! !+⌦ ⌦

~
Gki
⌦! =

+

4
i

1
⌦

X

1

!0

0

!! 0 ⌦ g! 0 s
0˜ k! g! ) 1 +Gk! = [(g! + g! ⌃Xq⌦ = [( ⌦ +˜⌦ ⇧q⌦ ⌦ )!

A. Rubtsov,! et al, PRB 77, 033101 (2008)00

g! 0 ⌦(

tk ] 1(
1+ ⇤⌦Vk ] 1

Spin-Polaron near van Hove singularity in real Material: NaxCoO242
Spectroscopy of strongly correlated materials: beyond DMFT

n=1.75DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

n=1.3
DMFT

DF

DF

QP

SP

DF-spectral function NaxCoO2
SP
QP

n=1.75

QP UH
A. Wilhelm, F. Lechermann, H. Hafermann,
M. Katsnelson, A. L. Phys. Rev. B 91, 155114 (2015)

Spin-Polaron physics for n=1.75

2t (t − J cosk )E (k ) = −t +J
1d t-J model, M. Katsnelson (1982)

0.1

Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

P H Y Ssingularity:
I C A L R E V I E-8triangular
W -6L E T-4T E Rlattice-2S
0
Extended
PRL 112, 070403(2014) van Hove

25

0

0.3

2

2

ω/t
tb
DMFT
LDFA

DOS
DOS(ω)

DOS(ω)

0.2

0.1

0
-8

PRL
112, 070403(2014)
Spectral
Function

-6

-4

-2

0

2

ω/t
week ending
P HD.
Y SYudin,I C A L D.
RE
V I E W L E T et
T Eal.,R S PRL 112, 070403
Hirschmeier,(2014)2014
21 FEBRUARY

FIG. 3 (color online). Spectral function in dual fermionFigure 5.13: Density of states for different param
approach at U=t
N(k)¼ 8 and T=t ¼ 0.05. Local maxima correspond-Consistency
FS check for the DOS at T /t = 0.05 an
ing to the lower band are indicated by a white line. In the vicinityof the data close to the Fermi energy and for tof the Fermi level, this lower band perfectly matches thethe upper Hubbard band is overestimated by ana
prediction εk − μ ¼ T ln½ð1 − nk Þ=nk & following from the Fermidue toFIG.
4 (color cut-off
online).dependence.
Broadened Top
Fermisu
its artificial
righcondensate hypothesis (thick black line). The bare dispersion iselectronsfor U=tas¼the
8 and
T=t ¼
0.1. Thethe data
is consistentobtained
self-energy
U=8shown for comparison (blue, dashed).
LDFA U=8t
the noninteracting
result.
panel:shows
Comparedto the DMFT
DOS the peak a
Fermion Condensate

significantly smaller due to the non-local self-en−1−1
densityof states
growing on-sitereGω ðkÞ ¼ FIG.
½ðgω3 þgω Σ~online).
ðkÞg
Þ
þ
Δ
−
&
; (5)of thespin(colorSpectralfunctionin εdual
fermionωωωkchannel[28]. with
The observed
tendency
U=0approach at U=t ¼ 8 and T=t ¼ 0.05. Local maxima correspond- of the Fermi level peak and formation of upperordering due to frustration is in line with pre
ing to the lower band are indicated by a white line. In the vicinityof the Fermi level, this lower band perfectly matches the visible. The large self-energy in the vicinity ofSpectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.201726

Dual Boson: non-local interaction + screening
"
#
ZZ
X
1
1 s
1 ⇤
0
c
Simp =d⌧ d⌧
G⌧ ⌧ 0 c⌧ c⌧ 0 + (U + ⇤⌧ ⌧ 0 ) n⌧ n⌧ 0 + ⇤⌧ ⌧ 0 ~s⌧ · ~s⌧ 02
20Simp =

N
X
R

0

d⌧

i,j=0

Ssimp =
1

)

+
1

⌫

1
tHTSC
k]

1 
+ ⇤! ~ VJττk ]' Sτ ⋅ Sτ'

= g⌫

R

0

d⌧

0

c⇤i (⌧ )

N
X
R
i=0

d⌧
0

⇥
R

G

1

0

⇤

0

R

(⌧, ⌧ ) ii ci (⌧ ) + 0 d⌧ ni," (⌧ )Ui,j ni,# (⌧ )

2

0 ⇤
1
0
0
d⌧
c
(⌧
)
[G
(⌧,
⌧
)]
c
(⌧
)
j
i
ij
0

+0.5

PN R
i=1

0

d⌧ U ni," (⌧(14))ni,# (⌧ )µdci

Ge 1 = G 1µdcie = T rG
T r(G)

(15)

(1)
(2)

=

X⇣

⌫ ⇤ †
⌫⇤
†
⌘
f
f
+
⌘
f
⌫+!
⌫
! ⌫ f
!Spectroscopy! ⌫+!
of strongly correlated⌦materials: beyond
DMFT

1 X ! ††
f
f
+ Evgeny Gorelov,
0⌫ 0 + ...
⌫⌫Alexander⌫+!
⌫ 0 Hiroshima,! f⌫ f05.07.2017
Lichtenstein,
4 0

⌫⌫ ⌦
DB-diagrammatic
scheme⌫!

!! 0 ⌦⌘

Bosonic Selfenergy=< c!+⌦ c!0

⌫
! =

† †
⌦ c! c! 0 >imp

27

Fermionic Selfenergy

g! g!0 ( ⌦+! !0g!+⌦ g!0 ⌦ g! g!0⌦)< c⌫+! c⌫ a† ⇢! >imp < ⇢ >imp g⌫ !
g⌫ g⌫+! !

˜ k⌫ g⌫ ) 1 + ⌫ tk ] 1
Gk⌫ = [(g⌫ + g⌫ ⌃Renormalized
˜ k⌫ g⌫ ) 1 + ⌫ tk ] 1
Gk⌫ =![(g⌫ + g⌫ ⌃vertex:
˜ q! ! ) 1 + ⇤! Vk ] 1
X
=
[(
+
⇧
X
q!
!
!
0
0
⌫
1
1
! = !
⌫⌫ 0 ! g⌫ 0 s 0 gX
⌫ 0q!!= [( ! + ! ⇧˜ q! ! ) 1 + ⇤! Vk ] 1
00XFermionic and Bosonic ⌫Green Functions
SCF-conditionGk⌫ = g⌫Xk1
11
˜ k⌫ g⌫ ) + ⌫ tG= )gg⌫⌫ )1 +˜gk⌫
˜gG⌫k⌫k⌫Gk⌫ = [(g⌫ + g⌫ ⌃]= ⌫[(g+⌫ g+⌫ ⌃⌃+ ⌫ ⌫ tkt]k ]k k⌫G=k⌫ [(g⌫k

Xq! = [( ! +˜! ⇧q! ! )

1

X

1
1
X˜q! ⇧
˜=q!!)! )1 ++ ⇤! XVq!
]=
kX
= [(
+
+
+
⇧
⇤!⇤!
q! [(
!
!
! X! q! !
qX

A. Rubtsov, M. Katsnelson, A. L., Annals of Phys. 327, 1320 (2012)q!
q

=

!

XX
Gk⌫
Gk⌫
==
g⌫g⌫

VkV

✏E 1 (q = ⇡, ⇡)DOS

X-point, ✏X ✏ = ✏M ✏X = 1. The energy scales as spectral weight is moved to incoherent excitations at
nce and will result in a plasmonic branch [11–13].
tion is expected to be highly localized. As a result, we
ndeed found to be independent of U ⇤ in this regime.
aseehigher
energy.
This leads
to theatformationof Hubbard⇤
such⇤ distances, the long-range interaction is screened
a weaklydispersing
branchanenergyE⇠U.
U = 2.1, the picture has changed
drastically.
The
⇤
⇤
bands
at
energies
Ematerials⇠ ±U/2. Above
critical Uctogether⇠ 2.4,
Plasmonstrongly
correlated
he substrate andthe potential
strength
isin
given
by
Spectroscopyof strongly
correlatedmaterials:
beyondDMFT
EvgenyGorelov, Alexander
Lichtenstein,
Hiroshima,a05.07.2017
28
The
low-energy
plasmonmode
has disappeared
ations2 of the particle-hole continuum are suppressed.
= 2⇡e /, where  is the dielectric constant of thewith the quasiparticle peak.
e strikingly, the dispersion is split into two branches
trate. At short distances, screening e↵ects of the subSimilar
to the single-particle spectra of correlated syspt for small wave vectors. The maximum energy of1e are often negligible. We therefore add a variableU ⇤ =1.1
tems, we observe a spectral weight transfer
in the plasbranches is found at the M-point, where they are⇤
action U to the local partwhich
yieldsane↵ective
U =2.1 4, whereP
mon spectra. This is clearly illustrated in Figure
rated by a gap.⇤ The maximumof the lower branchinteraction U = U + q⇤V0 /|q|.
U ⇤ =2.6
we show the inverse dielectric function for fixed momennsistent with a value of U /2, while the latter is loour calculations, we choose 4t = 1 as the energy
tum. The respective values of U ⇤ are indicated, showing
d at E ⇠ U ⇤ . These features appear concomitantly
and work at fixed temperature T = 0.02 and V0 = 2,
that the spectral weight transfer is associated with thisthe Hubbard bands in the ⇤density of states of Fig. 2.
e varying the parameter U = U + 1.1. In each case,energycan interpret the lower branch to originate mainly
0 scale.
tart from a standard, self-consistent EDMFT calcu-3
-2
-1
0
1
2
3
particle-hole excitations for which the electron is exE
n. The retarded interaction U! as well as the local
fromthe Hubbard band to the quasiparticle-peak⇤Uof
the interaction
treatedon thelevelof theice versa),
whereas theare
upper
branchstems
fromex- Figure1.52. Finite temperature local density of states (DOS)urity
model.
A
hybridization
expansion
continuousU ⇤ = 1.1 Hubbard model with long-rangeU⇤ions between the Hubbard bands [37]. Similar split- of the two-dimensional⇤U⇤
quantum
Monte-Carlosolver[33]
withimprovedesU=
2.1
calculated within EDMFT. The local inhas also been observed in EDMFT+GW calcula- Coulomb
1 interaction⇤⇤⇤ U = 2.6
tors
[34]
is
used
to
computetheimaginary-time
corU/2
teractionU
moves
spectralweight from the quasi-particle
for the extended Hubbard model with short-range
ion functionsof thewithout
at the Fermi energy to the Hubbard bands at E ⇠action[23, 24].
It isimpurity
further model
apparent
that approxthe low peak⇤ 0.5±U /2. For sufficiently large U ⇤ , the system is a Mott inion.
In
the
final
impurity
solverstep,
weadditionallygy, long wavelength excitations are renormalized insulator.
pute theof(reducible)
impurity
function!⌫⌫ 0
vicinity
the point.
This isvertex
expectedfor excita00123from within the quasiparticle peak. We will discuss
Efeature in more detail below.
Single plasmon mode for q->0
gure 3 finally shows the inverse dielectric function
Figure 4. A cross section of the EELS ( Im ✏ 1 ) of Fig. 3
Erik van Loon, et all, PRL 113, 246407 (2014)⇤
at the M point, q = (⇡, ⇡). The interaction causes a transfer
e Mott insulator at U = 2.6. In this state, a twoof spectral density. The arrows indicate the typical energy
cle excitation corresponds to a creation of a doublon⇤⇤
scales
U
and
U
/2.
⇤
a holon, which costs an energy U . Such an excita-

Spectroscopy of strongly correlated materials: beyond DMFT

Evgeny Gorelov, Alexander Lichtenstein, Hiroshima, 05.07.2017

29

Summary!

We developed general methods for spectroscopic calculations of strongly correlated materials!

Accurate calculations of Fermi-surface, metal-to-insulator transition details,
and spectral functions for strongly correlated materials have been done in framework of DFT+DMFT.

!

XAS spectra of real materials are calculated in order to assist in interpretation of experimental data.

!

Non-equilibrium DMFT approach have been used in order to study behavior of correlated materials inpresence of strong electric fields.

!

Non-local correlations beyond DMFT can be well described in dual fermion/boson scheme