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Effective model for
pyrochlore iridates
Leon Balents, KITP UCSB
Stanford, November 2013
$$:NSF, DOE
People
Lucile Savary
Eun-Gook Moon
Cenke Xu
Yong-Baek Kim
Imported Author Today, 6:43 PM
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Ru Chen
Outline
• Overview of pyrochlore iridates
• Electronic structure - quadratic band
touching = nodal Fermi point?
• Paramagnetic phase, Coulomb forces,
perturbations
• Quantum criticality
J / t2 /U , rather than t. As a result, the vertical boundary shifts to the left for large U/t.
that there is an intermediate regime in which insulating states are obtained only from the co
influence of SOC and correlations – these may be considered spin-orbit assisted Mott ins
Here we are using the term “Mott insulator” to denote any state which is insulating by
of electron-electron interactions. In Sec. IV, we will remark briefly on a somewhat philo
debate as to what should “properly” be called a Mott insulator.
Motivations
Terminology aside, an increasing number of experimental systems have appeared in
years in this interesting correlated SOC regime. Most prolific are a collection of iridates,
conducting or insulating oxides containing iridium, primarily in the Ir4+ oxidation state.
cludes a Ruddlesdon-Popper sequence of pseudo-cubic and planar perovskites Srn+1 Irn O3n
1, 2, 1),8–15 hexagonal insulators (Na/Li)2 IrO3 ,16–21 a large family of pyrochlores, R2 Ir2 O7 ,2
some spinel-related structures.25,26 Close to iridates in the periodic table are several osmium
such as NaOsO3 27 and Cd2 Os2 O7 ,28 which experimentally display MITs. Apart from thes
• Ultimate goal: new collective
phenomena in strongly SOC correlated
systems
6
TABLE I. Emergent quantum phases in correlated spin-orbit coupled materials. All phases have U(1)
particle-conservation symmetry – i.e. superconductivity is not included. Abbreviations are as follows:
TME = topological magnetoelectric e↵ect, TRS = time reversal symmetry, P = inversion (parity),
(F)QHE = (fractional) quantum Hall e↵ect, LAB = Luttinger-Abrikosov-Beneslavskii.57 Correlations
are W-I = weak-intermediate, I = intermediate (requiring magnetic order, say, but mean field-like), and S
= strong. [A/B] in a material’s designation signifies a heterostructure with alternating A and B elements.
Phase
Symm.
Topological Insulator
TRS
Axion Insulator
P
WSM
TRS or P
(not both)
W-I
Dirac-like bulk states, surface R2 Ir2 O7 ,
Fermi arcs, anomalous Hall
HgCr2 Se4 ,
...
LAB Semi-metal
cubic
+
TRS
broken
TRS
W-I
Non-Fermi liquid
R2 Ir2 O7
Bulk gap, QHE
Sr[Ir/Ti]O3 ,
R2 [B/B 0 ]2 O7
broken
TRS
I-S
Bulk gap, FQHE
Sr[Ir/Ti]O3
I-S
Several possible phases. Charge R2 Ir2 O7
gap, fractional excitations
Chern Insulator
Fractional Chern
Insulator
Fractional Topological TRS
Insulator, Topological
Mott Insulator
Quantum spin liquid any
Multipolar order
various
II.
Correlation Properties
W-I
I
I
Bulk gap, TME, protected
surface states
Magnetic insulator, TME,
no protected surface states
Proposed
materials
many
R2 Ir2 O7 ,
A2 Os2 O7
S
Several possible phases. Charge (Na,Li)2 IrO3 ,
gap, fractional excitations
Ba2 YMoO6
S
Suppressed or zero magnetic mo- A2 BB 0 O6
ments. Exotic order parameters.
WEAK TO INTERMEDIATE CORRELATIONS
W. Witczak-Krempa et al, ARCMP 2013
Following the discovery of topological insulators4,5 (TIs), it is now recognized that SOC is
an essential ingredient in forming certain topological phases. The TI is characterized by a Z2
topological invariant, which may be obtained from the band structure, in the presence of time-
Ln2Ir2O7
Ir4+:
107
106
Localized moment
Magnetic frustration
(4f)n
5d5
106
Dy
104
Conduction electrons
Ir[t2g]+O[2p]
105
Ho
Pyrochlores
Tb
105
103
conduction band
! (m" cm)
Ln3+:
pyrochlore oxides
Itinerant electron system
on the pyrochlore lattice
BO
IrO66
104
Dy
60
80 100
300
Ho
103
Tb
Gd
http://www.shapeways.com/shops/cmm.html
Eu
Sm
102
Nd
101
Ln2Ir2O7
Pr
100
A
Ln
0
50
100
150
200
250
300
T(K)
O!
Metal Insulator Transition
(Ln=Nd, Sm, Eu, Gd, Tb, Dy, Ho)
A2B2O7
K. Matsuhira et al. : J. Phys. Soc. Jpn. 76 (2007) 043706.
(Ln=Nd, Sm, Eu)
1
A (B) sublattice
Gardner, Gingras, and Greedan: Magnetic pyrochlore oxides
63
Possible A-site elements
and B site elements
(A3+)
2
(B4+)
2
(O2-)
FIG. 6. !Color online" Elements known to produce the
!3 + , 4 + " cubic pyrochlore oxide phase.
From Gardner,
Gingras,
Greedan, RMP
2010.
7
probably the most studied, followed by the zirconates,
While the polycrystalline samples have been fairly well
Ln2Ir2O7
Ir4+:
107
106
Localized moment
Magnetic frustration
(4f)n
5d5
106
Dy
104
Conduction electrons
Ir[t2g]+O[2p]
105
Ho
Pyrochlores
conduction band
Itinerant electron system
on the pyrochlore lattice
BO
IrO66
Tb
105
103
! (m" cm)
Ln3+:
pyrochlore oxides
104
Dy
60
80 100
300
Ho
103
Tb
Gd
http://www.shapeways.com/shops/cmm.html
Eu
Sm
102
Nd
101
Ln2Ir2O7
Pr
100
A
Ln
0
50
100
150
200
250
300
T(K)
O!
Metal Insulator Transition
(Ln=Nd, Sm, Eu, Gd, Tb, Dy, Ho)
A2B2O7
K. Matsuhira et al. : J. Phys. Soc. Jpn. 76 (2007) 043706.
(Ln=Nd, Sm, Eu)
1
A (B) sublattice
R2Ti2O7,
R2Sn2O7, R2Zr2O7
insulating, f-electron spins, J ~ 1K
R2Ir2O7
semi-metallic, d electrons itinerant,
W ~ 1eV, + (sometimes) f electron spins
Ln2Ir2O7
Ir4+:
107
106
Localized moment
Magnetic frustration
(4f)n
5d5
106
Dy
104
Conduction electrons
Ir[t2g]+O[2p]
105
Ho
Pyrochlores
conduction band
Itinerant electron system
on the pyrochlore lattice
BO
IrO66
Tb
105
103
! (m" cm)
Ln3+:
pyrochlore oxides
104
Dy
60
80 100
300
Ho
103
Tb
Gd
http://www.shapeways.com/shops/cmm.html
Eu
Sm
102
Nd
101
Ln2Ir2O7
Pr
100
A
Ln
0
50
100
150
200
250
300
T(K)
O!
Metal Insulator Transition
(Ln=Nd, Sm, Eu, Gd, Tb, Dy, Ho)
A2B2O7
K. Matsuhira et al. : J. Phys. Soc. Jpn. 76 (2007) 043706.
(Ln=Nd, Sm, Eu)
1
A (B) sublattice
R2Ti2O7,
R2Sn2O7, R2Zr2O7
quantum spin ice may realize Coulomb =
U(1) quantum spin liquid
R2Ir2O7
gapless nodal normal state supports NFL
and unusual quantum criticality
Pyrochlore iridates
11
• Series of compounds exhibiting metalinsulator transitions and magnetism
J. Phys. Soc. Jpn. 80 (2011) 094701
Non⇥Metal
Non
Metal
200
T K⇥
150
FULL PAPERS
Metal
Y
Ho Dy Tb
⇥
Gd Eu
⇥
Lu⇥Yb
⇥
Sm
⇥
⇥
⇥
⇥
100
50
Magnetic Ins.
Magnetic Ins.
Nd
⇥
0
Pr
⇥
100
105
110
115
R3 ionic radius pm⇥
Yanagashima+Maeno, JPSJ 2001
K. Matsuhira et al, JPSJ 2011
W. Witczak-Krempa et al, ARCMP 2013
(b)
d field cooled (FC) and zero field cooled (ZFC)
K. Matsuhira et al, JPSJ 2011
Fig. 2. (Co
Nd, Sm, Eu,
Sm, Eu, G
# and
slower
fluctua
also re
paralle
fractio
η = A
FIG. 3. (a) Representative short-time asymmetry data taken from
orderin
GPS for Y-227 with curves offset for clarity and solid lines
and 0.
representing fits to Eq. (1). (b) FFT of the asymmetry data smoothed
1.8 K.
to highlight evolution of the peak.
system
suitabl
of the data [Fig. 1(c)] showed excellent agreement with
by fixi
previously reported values; we did not observe any evidence
For
of long-range order or structural transitions down to T = 3 K.
from
fi
However, neutron studies of small-moment iridates often
The
on
require single-crystal measurements
to resolve
correlated
spin
FIG. 1. (Color
online)
(a) All-in-all-out
magne
the pre
scattering. Conservatively,
measurement
places
an
allthis
fourpowder
magnetic
moments on
the vertices
of each
upper limit of the Ir4+ ordered
moment
to be
belowindicate
0.5 µB the Ir4+ m
inward or
outward.
Arrows
based on the collected (b)
statistics
andmagnetic
the resolution
of thefour magne
Coplanar
order where
J. Phys. Soc. Jpn. 81 (2012) 034709
FULLdiffractometer.
PAPERS
TOMIYASU
al.
tetrahedron lieK.in
the et(001)
plane and are eith
µSR provides an excellent
means of further probing the
orthogonal.
local
magnetic
order
due
to
the
large gyromagnetic ratio of
(a)
(c) O
(b) all-in all-out structure
(d)
the muon (γµ /2π = 135.5
MHz/T)
which
makes
it possible
25
Nd only 113
surface with
edges
about
0.5 mmto
long [Fig. 2
[010]
Nd & Ir + 222
83
detect static fields
of 97
a few Gauss or less[111]
while simultaneously
20
to a copper plate with varnish14and moun
Ir
probing spin dynamics and correlations
in the MHz regime.
15
220
cycle 4 He refrigerator. The [011] of the sam
Sample asymmetry curves showing the early time behavior
10 133
Nd
perpendicular
to the
420
from PSI are shown in Fig.
3(a) for Y-227
andscattering
Fig. 4(a) plane
for [Fig. 2(
5
135
crystal
was
used
to
analyze
the
polarization
o
200
Yb-227. The most significant feature is the appearance
of
0
′
400
ray.
The
linearly
polarized
σ
(perpendicular
422
=
150
and
spontaneous
muon
spin
precessions
below
T
-5 111
LRO
plane)respectively.
and π ′ (parallel
to the
scattering pla
130 K for Y-227 and Yb-227,
These data
signify
-5 0 5 10 15 20 25
Ir
[100]
Ir kagome
can magnetic
be detected
by⟨Brotating
the
Mo crystal ab
Calculated intensity (barn/cell)
the presence of a static local
fieldplane
muon
loc ⟩ at the
(200) reflection of
beam. Theare2θwell
value
of theindicating
site. Furthermore, the oscillations
defined
Fig. 3. (Color online) Magnetic structure modelling. (a) Comparison between
experimental and best-fit
reflection
intensitiesguarantees
given
in
commensurate
ordercalculated
with11.230
amagnetic
singlekeV,
magnetically
unique
muonwhich
a polarization p
2
2
7
3þ
2%
Table I. (b) All-in all-out magnetic structure. The filled circles represent Nd ions, and the arrows represent the Nd moments. (c) Trigonally distorted O
stopping
site.
The
resultant
asymmetry
curves
were
by structur
4þ
4þ
!
change
in fit
crystal
ligand around Ir ion. All the Ir–O bonds are of the same length (2.01 A). (d) Relation between magnetic momentsTo
(blueinvestigate
thin arrows) of Ir the
ions (blue
small
balls) and magneticMAGNETIC
moments (red thick
arrows) of IN
Nd3þTHE
ions (red
large balls).
theIRIDATES
Ir and
the Nd moments
the all-in all-out
structures.
AlternativePHYSICAL
ORDER
PYROCHLORE
A . . . form
REVIEW we
B
theBoth
simple
depolarization
function
for
a
magnetically
ordered
transition, x-ray oscillation photographs
H. SAGAYAMA et al.
AIAO order
•
RAPID COMMUNICATIONS
2
6
2
3
TMI = 120K
8
σ−π'
σ−σ'
2
1
0
(d)
FC
0.02
2
10
ZFC
0
10
0
0
9.99 10.00 10.01
H of (H 0 0) (r.l.u.)
Nd Ir O
K. Tomiyasu et al, 2012
2
(10 0 0)
E = 11.230 keV
4
10
6
4
0
(b)
10
50
100
Temperature [K]
Eu Ir O
H. Sagayama
et al, 2013
0.01
M/H (emu/mol)
4
σ−π'
4 (c)
Experimental intensity (barn/cell)
σ−σ'
(a)
0
Counts (arb. unit)
(10 0 0)
E=11.23 keV
ρ /ρ (300 K)
Counts (arb. unit)
4
Integrated intensity (arb. unit)
PHYSICAL REVIEW B 87, 100403(R) (2013)
[001]
Growing evidence in many materials
0.00
150
FIG. 3. (Color online) (a), (b) Profiles
2 of2 the7 (10 0 0) reflections of Eu2 Ir2 O7 scanned along (h00) in reciprocal space with a
photon energy of 11.230 keV at several temperatures through the
(a) σ -σ ′ and (b) σ -π ′ channels. (c) Temperature dependence of the
integrated intensities of the (10 0 0) reflection detected though the
σ -σ ′ (solid circles) and σ -π ′ (open circles) channels. (d) Temperature
dependence of the electrical resistivity (solid line) and magnetic
susceptibility (circles). The magnetic susceptibilities were measured
for zero-field-cooling (ZFC; solid circles) and field-cooling (FC; open
directions of Nd moments in the case of a ferromagnetic Nd–Ir interaction are depicted: when the moments are antiferromagnetic, the directions of all the red
arrows are reversed, and the all-in all-out type structure is retained.
Figure 2(b) shows the inelastic scattering intensity
distributions in ðQ; EÞ space measured below TMI . The
horizontally spreading excitation mode is observed at around
1.3 meV; it corresponds to the splitting of the Nd3þ ground
doublet. The scattering intensity decreases with increasing
Q, confirming that the excitations are magnetic. However,
the mode appears slightly dispersive. To verify the
dispersion, we compared the constant–E scans measured at
1.2 and 1.4 meV in Fig. 2(b); the comparison results are
shown in Fig. 2(c). The Q dependences at these energies
are clearly different, indicating that the excitations are not
completely flat but dispersive, with a bandwidth of $0:1
meV (1 K).
large cylindrical imaging plate installed at BL
polycrystal,
as used
in Ref.of5 35
for Eu-227:
Japan,
at a photon
energy
keV, which
2
β
absorption
edges
of
the
constituent
A(t)
= absolute
A1 exp[−(#t)
φ) + A2 e
The remaining fitting parameter
is the
value of ] cos(ωt +elements.
3þ
sample
with
a
dimension
of
0.05
mm
was
c
magnetic moments of Nd (m). Therefore, we evaluated it
The
first
describes(Ref.
the 26)
oscill
by the least-squares method,
by employing
thecomponent
magnetic SHELXL
w
He-gas
spray
refrigerator.
27)
spin
precessing
about a spontaneo
form factor of Nd3þ calculated bymuon
Freeman
and Desclaux.
the
structural
parameters.
The
magnetic
susce
Since the sample was cylindrical
in shape,
reflection- ωµ /2π = ⟨B loc ⟩γµ /2π
field
with the
frequency
single
crystals
(0.49
mg),
thewith
onech
u
angle dependence of the absorption
factor wasby
ignored.
The including
described
a stretched
exponential
evaluation results showed that the all-in all-out model had
# and
β<
second
component describ
the best-fit calculated intensities,
with
mð91.KÞThe
¼ 1:3
'
longitudinal
relaxation
due to spin-latt
0:2!B (Table I); these intensities
are in agreement
with
the
(a) slower
(b)
Detector
experimental ones, as shown in Fig.
3(a).
fluctuations
of the local moments. Because
Further, from Fig. 1(d), the ratio
magnetic
scattering
alsoofreflects
muons
for which the initial muo
intensity at 0.7 K to that at 9 K is estimated to be about 3.2.
parallel to theIinternal
field at the stoppingFIG
sit
Since the magnetic intensity is proportional to the σ’
square of
from
fi
fraction
of
slow
relaxing
asymmetry
to
the
m,28)
the
value
of
mð0:7
KÞ
is
estimated
to
be
2:3
'
0:4
Iπ’
pffiffiffiffiffiffiffi
GPS, w
4. Analyses
η
=
A
/(A
+
A
)
=
1/3
at
temperatures
½¼ 2
3:2 )data
mð9taken
KÞ*!Bfrom
. ThisAnalyzer
value is in(Mo)
2 good1 agreement
2
2 2 short-time
7
2
FIG. 3. (a) Representative
asymmetry
4. value
(a)7Representative
short-time
dataand
takenindeed
from wethe
solit
We analyzed the magnetic structure on the basis of the FIG.
with the
of 2.37!B estimated
by theasymmetry
crystalline field
ordering
temperature,
find
GPS for Y-227 with curves offset 21)for
clarity
and
solid
lines
21)
for Yb-227 with curves and
offset
for
clarity,
and
solid
line
#,
solid
q0 intensities given in Table I. Crystalline field analysis GPSanalysis.
-ray
0.35(1)
π’ for Y-227 and Yb-227, respecti
representing
to Eq.
(1). (b)toFFT
of the
asymmetry
datatosmoothed
representing
fits
Eq.
(1). x4,
(b) we
FFT
of
EM
revealed the magnitude
of fits
Nd3þ
moments
be about
To summarize
x3 and
found
that the antiferro1.8
K.asymmetry
The
phasedata.
factor φ was found and
to be
r
4þ evolution of the peak.
to
highlight
2.37!B , whereas that of Ir moments is expected to be 1!B magnetic long-range structure with q0 grows with decreasingσ’
(c)
system
described
by
a
single
oscillatory
freq
at most. Therefore, as the first approximation, we assumed temperature below TNd ¼ 15 ' 5 K. The structure can be
014428-3
fitstype
for of
both
samples at all
temperatur
that the q0 intensities
described
by the suitable
all-in all-out
model
only 1(c)]
the Nd showed
moments. excellent
In approximately
of theconsist
data of[Fig.
agreement
with
◦ K.
[100]
fact, the statistical errors of our data would be too large to for Nd moments with a magnitude of 2:3 ' 0:4! at 0.7
Y Ir O , Yb Ir O
S. Disseler et al, 2012
ε
ε
Heisenberg
• Symmetry constrains form of
generic Hamiltonian for
Kramer’s doublets
z
HzzH = =JzzHzz S+iz SH
j ± + Hz± + H±±
i,j⇥
H±
=
Hz±
H±±
J±
+
=
Si+ Sj + Si Sj+
⇤i,j⌅
⇧⇤
Jz±
Siz
+
ij Sj
+
⇥
⇥
ij Sj
⌅i,j⇧
+J
=
⇤
±±
⇤
⌅i,j⇧
S. Curnoe, 2008
+ +
ij Si Sj
+
⇥
ij Si
Sj
⇥
+i
⇥
j
⌅
local z
axes
Heisenberg
• Symmetry constrains form of
generic Hamiltonian for
Kramer’s doublets
local z
axes
z
HzzH = =JzzHzz S+iz SH
j ± + Hz± + H±±
i,j⇥
Jzz > 0 : spin ice
Jzz < 0 : all-in/all-out order
Pyrochlore iridates
11
• Series of compounds exhibiting metalinsulator transitions and magnetism
Non⇥Metal
Non
Metal
200
T K⇥
150
Beyond Heisenberg
• Metallic state
• Topological phases?
• Weyl fermions?
• Quantum criticality?
Metal
Y
Ho Dy Tb
⇥
Gd Eu
⇥
Lu⇥Yb
⇥
Sm
⇥
⇥
⇥
⇥
100
50
Magnetic Ins.
Magnetic Ins.
Nd
⇥
0
Pr
⇥
100
105
110
115
R3 ionic radius pm⇥
Yanagashima+Maeno, JPSJ 2001
K. Matsuhira et al, JPSJ 2011
W. Witczak-Krempa et al, ARCMP 2013
(b)
d field cooled (FC) and zero field cooled (ZFC)
Topological phases?
• Simple model Hamiltonians can exhibit
topological insulator, and perhaps more
exotic interacting topological phases
PYROCHLORE ELECTRONS UNDER PRESSURE, HEAT, . . .
AF ( AIO)
0.1
1.5
m AF
M
0.0
− 1.2
TI
SM
− 1.0
− 0.8
tσ
D. Pesin + LB, 2010
AIO′
TWS
0.2
T
PHYSICAL R
U
1.0
FIG. 2. (Colo
field phase diagra
ping (toxy = 1).
Tc for the contin
magnetic order m
correspond to A
or a type related
solid/dashed lines
quantum phase tr
lines in the AIO
signaling a Lifshi
metallic AF (mA
− 0.6
hoppings in the self-consistent treatment. Indeed, the previous
orange (larger) markers in Fig.
W.
Witczak-Krempa,
A. Go,
Y-Bthe
Kim,
2013
paper was
mainly concerned with the self-consistent
analysis
magnetization
m grows abruptly a
of the NN Hamiltonian: The NNN hoppings were added to the
final spectrum to establish that the line nodes at EF can give
rise to a Weyl phase when the ordering is of the AIO type.
becomes a gapped AF insulator: En
it preferable to preempt the conti
points of opposite chirality via a di
Pyrochlore iridates
11
• Series of compounds exhibiting metalinsulator transitions and magnetism
Non⇥Metal
Non
Metal
200
T K⇥
150
Metal
Y
Ho Dy Tb
⇥
Gd Eu
⇥
Lu⇥Yb
⇥
Sm
⇥
⇥
⇥
⇥
100
50
Magnetic Ins.
Magnetic Ins.
start with paramagnetic
structure
Nd
⇥
0
Pr
⇥
100
105
110
115
R3 ionic radius pm⇥
Yanagashima+Maeno, JPSJ 2001
K. Matsuhira et al, JPSJ 2011
W. Witczak-Krempa et al, ARCMP 2013
(b)
d field cooled (FC) and zero field cooled (ZFC)
Minimal model
HYSICAL REVIEW B 82, 085111 !2010"
• Quadratic band touching
(b) te / t a=2.2 (c) te / t a=2.5
4-dim
irrep
Γ X ! L Γ Κ X Γ X ! L Γ Κ X
B.J. Yang + Y.B. Kim, 2010
e" Dispersions of jeff = 1 / 2 bands under
X.we
Wan
al, 2011
ta " te. Here
haveetfixed
%SO = 4.0, ta
increases the energy gap between the two
ces. When te / ta # 2.3 band inversion oce system becomes metallic.
ent first-principles calculation on
e have used a simplified tight-binding
4 Ir’s per unit cell
2 J=1/2 states per Ir
8/2 = 4 two-fold degenerate bands
Minimal model
HYSICAL REVIEW B 82, 085111 !2010"
• Quadratic band touching
(b) te / t a=2.2 (c) te / t a=2.5
4-dim
irrep
k·p expansion:
5 2
(kx2 Jx2 + ky2 Jy2 + kz2 Jz2 )
(k · J)2
k2
4k
H0 (k) =
+
2m
2Mc
2M̃0
Γ X ! L Γ Κ X Γ X ! L Γ Κ X
da (k) a
k2
d4 (k) 4 + d5 (k) 5
=
+
+
2m
2M
2Mc
0
B.J. Yang + Y.B. Kim, 2010
e" Dispersions of jeff = 1 / 2 bands under
X.we
Wan
al, 2011
ta " te. Here
haveetfixed
%SO = 4.0, ta
increases the energy gap between the two
ces. When te / ta # 2.3 band inversion oce system becomes metallic.
ent first-principles calculation on
e have used a simplified tight-binding
Same Luttinger Hamiltonian which
describes inverted band gap
semiconductors like HgTe
Minimal model
HYSICAL REVIEW B 82, 085111 !2010"
• Quadratic band touching
(b) te / t a=2.2 (c) te / t a=2.5
4-dim
irrep
Γ X ! L Γ Κ X Γ X ! L Γ Κ X
B.J. Yang + Y.B. Kim, 2010
e" Dispersions of jeff = 1 / 2 bands under
X.we
Wan
al, 2011
ta " te. Here
haveetfixed
%SO = 4.0, ta
increases the energy gap between the two
ces. When te / ta # 2.3 band inversion oce system becomes metallic.
ent first-principles calculation on
e have used a simplified tight-binding
HgTe
Minimal model
HYSICAL REVIEW B 82, 085111 !2010"
• Quadratic band touching
(b) te / t a=2.2 (c) te / t a=2.5
4-dim
irrep
Γ X ! L Γ Κ X Γ X ! L Γ Κ X
B.J. Yang + Y.B. Kim, 2010
e" Dispersions of jeff = 1 / 2 bands under
X.we
Wan
al, 2011
ta " te. Here
haveetfixed
%SO = 4.0, ta
increases the energy gap between the two
ces. When te / ta # 2.3 band inversion oce system becomes metallic.
But with strong
correlations
ent first-principles calculation on
e have used a simplified tight-binding
HgTe
paramagnetic, GGA+SO, Wien2k
yiro atom 0
size 0.20
Bands
priro atom 0 size 0.20
0.2 eV
EF
0.0
EF
0.0
Energy (eV)
Energy (eV)
Imported Author Today, 6:43 PM
correlations may also renormalize
these bands
tends to push occupied states down
-1.0
-2.0
W
L
Λ
Γ
Δ
Y2Ir2O7
X Z W K
-1.0
-2.0
W
L
Λ
Γ
Δ
X Z W K
Pr2Ir2O7
paramagnetic, MBJLDA+SO, Wien2k
yiro atom 0
size 0.20
1.0
1.0
Energy (eV)
Y2Ir2O7
EF
0.0
Energy (eV)
EF
0.0
-1.0
Bands
priro atom 0 size 0.20
-1.0
Pr2Ir2O7
Modified Becke-Johnson exchange
potential
(known
to give better estimate of
W
L
Λ
Γ
∆
X Z W K
W
L
Λ
Γ
∆
X Z W K
gaps) reduces band overlap and predicts
Pr2Ir2O7 is a zero gap semimetal
Imported Author Today, 6:43 PM
experiment will decide!
-2.0
-2.0
Minimal model
HYSICAL REVIEW B 82, 085111 !2010"
• Quadratic band touching + interactions
(b) te / t a=2.2 (c) te / t a=2.5
Non-Fermi liquid normal state
4-dim
irrep
Large anomalous Hall response
Γ X ! L Γ Κ X Γ X ! L Γ Κ X
B.J. Yang + Y.B. Kim, 2010
e" Dispersions of jeff = 1 / 2 bands under
X.we
Wan
al, 2011
ta " te. Here
haveetfixed
%SO = 4.0, ta
increases the energy gap between the two
ces. When te / ta # 2.3 band inversion oce system becomes metallic.
ent first-principles calculation on
e have used a simplified tight-binding
Novel quantum criticality
Correlations
+ Coulomb?
Model
+ Coulomb = NFL
SOVIET PHYSICS JETP
VOLUME 32, NUMBER 4
APRIL, 1971
POSSIBLE EXISTENCE OF SUBSTANCES INTERMEDIATE BETWEEN METALS AND
DIELECTRICS
A. A. ABRIKOSOV and S.D. BENESLAVSKII
QFT
L. D. Landau Institute of Theoretical Physics
SL =
Submitted April 13, 1970
Zh. Eksp. Teor. Fiz. 59, 1280-1298 (October, 1970)
The question of the possible existence of substances having an electron spectrum without any energy
gap and, at the same time, not possessing a Fermi surface is investigated. First of all the question
of the possibility of contact of the conduction band and the valence band at a single point is investigated within the framework of the one-electron problem. It is shown that the symmetry conditions for
the crystal admit of such a possibility. A complete investigation is carried out for points in reciprocal lattice space with a little group which is equivalent to a point group, and an example of a more
complicated little group is considered. It is shown that in the neighborhood of the point of contact the
spectrum may be linear as well as quadratic.
The role of the Coulomb interaction is considered for both types of spectra. In the case of a linear
dispersion law a slowly varying (logarithmic) factor appears in the spectrum. In the case of a quadratic spectrum the effective interaction becomes strong for small momenta, and the concept of the
one-particle spectrum turns out to be inapplicable. The behavior of the Green's functions is determined by similarity laws analogous to those obtained in field theory with strong coupling and in the
neighborhood of a phase transition point of the second kind (scaling). Hence follow power laws for
the electronic heat capacity and for the momentum distribution of the electrons.
1. INTRODUCTION
0
NE of the basic assumptions of the Landau lll theory
of a Fermi liquid is the relationship, according to which
the limiting momentum of the excitations in an isotropic
Fermi liquid is determined in the same way as in a gas,
by the density of the atoms in the liquid, i.e.,
Po = (3JT1l) 113 (where n is the density of particles). According to the work of Luttinger and Ward, laJ with a
certain amount of alteration this theorem is also applicable to the electronic liquid in metals. Namely, it
gap is present. One can represent such a substance as
the limiting case of a metal with a point Fermi surface.
In the language of the theory of noninteracting particles
in a periodic field, this corresponds to the case when a
completely filled valence band touches a completely
empty conduction band. Indications of the possible existence of such substances have recently appeared in the
literature (gray tinl 3 l and mercury telluridel 4 l).
In the present article the conditions under which a
point Fermi surface may appear in a model of noninteracting electrons will be investigated, and the question of
Z
d
n
†
h
d d x ⇥ ⌅
stable, NFL
fixed point
“Luttinger-Abrikosov-Beneslavskii” phase
i
o
c0
2
ie ⇤ + Ĥ0 ⇥ + (⌅i ⇤)
2
C
I
O
Mass structure
• Following Murakami, 2004
5 2
(k · J)2
k2
4k
H0 (k) =
+
2m
2M̃0
da (k)
=
2m
a
(kx2 Jx2 + ky2 Jy2 + kz2 Jz2 )
2Mc
k2
d4 (k) 4 + d5 (k)
+
+
2M0
2Mc
particle-hole
asymmetry
5
cubic
anisotropy
Mass structure
• Following Murakami, 2004
5 2
(k · J)2
k2
4k
H0 (k) =
+
2m
2M̃0
da (k)
=
2m
(kx2 Jx2 + ky2 Jy2 + kz2 Jz2 )
2Mc
k2
d4 (k) 4 + d5 (k)
+
+
2M0
2Mc
a
5
emergent
isotropy + p/h
symmetry
irrelevant under RG
d
d
d
d
✓
✓
m
M0
m
Mc
◆
◆
=
=
8
15 u
✓
m
M0
0.152u
✓
◆
m
Mc
◆
m e2
u= 2
8 c0 4
d
⇠
1
aB
Model
+ Coulomb = NFL
scaling,
e.g.
! ⇠ kz
z ⇡ 1.8
cv ⇠ T d/z ⇡ T 1.7
(!) ⇠ ! 1/z
“Luttinger-Abrikosov-Beneslavskii” phase
Perturbations
LAB
strain
TOPOLOGICAL INSULATORS WITH INVERSION SYMMETRY
PHYSICAL REVIEW B 76, 045302 !2007"
FIG. 5. Schematic representation of band energy evolution of
Bi1−xSbx as a function of x. Adapted from Ref. 43.
FIG. 6. !a" Band structure of $-Sn near the % point, which
describes zero gap semiconductor due to the inverted %+8 and %−7
bands. !b" In the presence of uniaxial strain, the degeneracy at % is
lifted, opening a gap in the spectrum. The parity eigenvalues remain
unchanged.
= 0.09 the T valence band clears the Ls valence band, and the
alloy is a direct-gap semiconductor at the L points. As x is
increased further, the gap increases until its maximum value
of order 30 meV at x = 0.18. At that point, the valence band at
H crosses the Ls valence band. For x ! 0.22, the H band
crosses the La conduction band, and the alloy is again a
semimetal.
Since the inversion transition between the Ls and La bands
occurs in the semimetal phase adjacent to pure bismuth, it is
clear that the semiconducting Bi1−xSbx alloy inherits its to-
degenerate set of states with %+8 symmetry, which can be
derived from p states with total angular momentum j = 3 / 2.
The fourfold degeneracy at the %+8 point is a consequence of
the cubic symmetry of the diamond lattice. Applying uniaxial
strain lifts this degeneracy into a pair of Kramers doublets
and introduces an energy gap into the spectrum.53 For pressures of order 3 & 109 dyn/ cm2, the induced energy gap is of
order 40 meV. We now argue that this insulating phase is, in
fact, a strong topological insulator.
Table V shows the symmetry labels for unstrained $-Sn
Fu-Kane 2007
TI
Perturbations
Pressure/Strain effect on Pyrochlore
•
Pressure/Strain along (111) direction
•
Cubic -> Rhombohedral symmetry
•
Compressive pressure (in-plane tensile strain)
LAB
priro atom, a=1.05a
0 size 0.20
c=0.96c
0
0
EF
Energy (eV)
0.0
! Pressure/Strain effect
Full lattice relaxation: GGA
• Compressive
pressure (tensile strain)
!
mBJ+SO
calculation
c=0.96c0,
a=1.05a0
(111)
compression
Insulator
gap=0.08eV
on Pyrochlore
strain
gap
= 80
meV
Parity at time
reversal
invariant
momenta:
-1.0
-2.0
L
F
Z
Γ
L
Z
F
+
-
-
-
Z2 invariant (1; 111)
Compressive pressure/ tensile strain along (111) direction"
will make Pr2Ir2O7 a topological insulator!
TI
Perturbations
M
??
LAB
Anomalous Hall Effect
LETTERS
Tf
a
TH
0.8
B = 0.05 T
// [111]
FC
–10
ZFC
0.6
–8
"H (Ω–1 cm–1)
• Striking Pr Ir O
–6
"H
2 2
!
–4
ZFC
–2
0
0.4
7
experiments
0.2
FC
0
" (103 Ω–1 cm–1)
b
"H(B = 0) (Ω–1 cm–1)
–10
"H
–5
1.78
B = 0.05 T
// [111]
–0.02
1.76
FC and ZFC
0.1
T (K)
1
M
–0.01
M(B = 0) ( !B per Pr atom)
B=0T
!3 (104 e.m.u. per mol Pr T–3)
0
–4
0
c
2
Cm
–3
!
–2
–1
0
3
1
B=0T
0.1
1
T (K)
10
Cm (J per mol Pr K–1)
large
χ 3!
–12
! (e.m.u. per mol Pr)
ets because the spontaneous
pically even in the absence of
rt of the AHE in moderately
d by the band-intrinsic mechons under an electric field E
e of the relativistic spin-orbit
n. This phase acts as a macbends the orbital motion of
a real magnetic field B. Thus,
Hall conductivity sH at B 5 0.
e fictitious magnetic field b,
the AHE at B 5 0, is not
the macroscopically broken
al operation cannot be comations of the crystal (Supthe scalar spin chirality in
al spin glasses can also proE4,5,12,13,20, as indeed has been
refs 6, 7), and MnSi (refs 9,
, the spin chirality is not the
mpanies a chiral spin texture
y the applied magnetic field.
pen issue to find a possible
he macroscopically broken
field.
-broken phase in the absence
eezing in the thermodynamic
iquid state. In particular, we
he absence of uniform magin the metallic cooperative
ng temperature, as indicated
Both the experiment and the
ase is induced by melting of a
ions of the Pr 4f magnetic
r spin-ice systems14,15.
n antiferromagnetic Curie–
nly due to the correlations
nts of Pr31 ions, which point
centre of the Pr tetrahedron
ctrons are weakly correlated
e22. They mediate the RKKY
a the Kondo coupling. The
ting conventional magnetic
eat, magnetic susceptibility,
nals strong geometrical frused in the magnetic susceptiers of magnitude lower than
0
Figure 2 | Temperature dependence of the magnetic and transport
properties of Pr2Ir2O2. a, Temperature dependence of the Hall conductivity
sH (left axis) and the direct-current susceptibility x 5 M/H (right axis)
under a magnetic field of B 5 0.05 T along the [111] direction. e.m.u.,
electromagnetic unit. Here, Hall conductivity is given by sH 5 2rH/
Hall conductivity with
“zero” magnetization
Anomalous Hall Effect
• Hall conductivity is a pseudo-vector
H
⇥µ⇥
e2
=
h
µ⇥
K
• Hall vector K has exactly the same
symmetries as uniform magnetization M,
once SOC is included (and it is strong here)
• In general, these should be linearly coupled
in Landau theory: K ∝ M
Anomalous Hall Effect
LETTERS
Tf
a
TH
0.8
B = 0.05 T
// [111]
FC
–10
ZFC
0.6
–8
"H (Ω–1 cm–1)
• Striking Pr Ir O
–6
"H
2 2
!
–4
ZFC
–2
0
0.4
7
experiments
0.2
FC
0
" (103 Ω–1 cm–1)
b
"H(B = 0) (Ω–1 cm–1)
–10
"H
–5
1.78
B = 0.05 T
// [111]
–0.02
1.76
FC and ZFC
0.1
T (K)
1
M
–0.01
M(B = 0) ( !B per Pr atom)
Hall conductivity with
“zero” magnetization
B=0T
!3 (104 e.m.u. per mol Pr T–3)
0
–4
2
Cm
–3
!
–2
–1
0
small
0
c
3
1
B=0T
0.1
1
T (K)
10
Cm (J per mol Pr K–1)
large
χ 3!
–12
! (e.m.u. per mol Pr)
ets because the spontaneous
pically even in the absence of
rt of the AHE in moderately
d by the band-intrinsic mechons under an electric field E
e of the relativistic spin-orbit
n. This phase acts as a macbends the orbital motion of
a real magnetic field B. Thus,
Hall conductivity sH at B 5 0.
e fictitious magnetic field b,
the AHE at B 5 0, is not
the macroscopically broken
al operation cannot be comations of the crystal (Supthe scalar spin chirality in
al spin glasses can also proE4,5,12,13,20, as indeed has been
refs 6, 7), and MnSi (refs 9,
, the spin chirality is not the
mpanies a chiral spin texture
y the applied magnetic field.
pen issue to find a possible
he macroscopically broken
field.
-broken phase in the absence
eezing in the thermodynamic
iquid state. In particular, we
he absence of uniform magin the metallic cooperative
ng temperature, as indicated
Both the experiment and the
ase is induced by melting of a
ions of the Pr 4f magnetic
r spin-ice systems14,15.
n antiferromagnetic Curie–
nly due to the correlations
nts of Pr31 ions, which point
centre of the Pr tetrahedron
ctrons are weakly correlated
e22. They mediate the RKKY
a the Kondo coupling. The
ting conventional magnetic
eat, magnetic susceptibility,
nals strong geometrical frused in the magnetic susceptiers of magnitude lower than
0
Figure 2 | Temperature dependence of the magnetic and transport
properties of Pr2Ir2O2. a, Temperature dependence of the Hall conductivity
sH (left axis) and the direct-current susceptibility x 5 M/H (right axis)
under a magnetic field of B 5 0.05 T along the [111] direction. e.m.u.,
electromagnetic unit. Here, Hall conductivity is given by sH 5 2rH/
Anomalous Hall Effect
LETTERS
Tf
a
TH
0.8
B = 0.05 T
// [111]
FC
–10
ZFC
0.6
–8
"H (Ω–1 cm–1)
• Striking Pr Ir O
–6
"H
2 2
!
–4
ZFC
–2
0
0.4
7
experiments
0.2
FC
0
" (103 Ω–1 cm–1)
b
"H(B = 0) (Ω–1 cm–1)
–10
"H
–5
1.78
B = 0.05 T
// [111]
–0.02
1.76
FC and ZFC
0.1
T (K)
1
M
–0.01
M(B = 0) ( !B per Pr atom)
Hall conductivity with
“zero” magnetization
B=0T
!3 (104 e.m.u. per mol Pr T–3)
0
–4
2
Cm
–3
!
–2
–1
0
small
0
c
3
1
B=0T
0.1
1
T (K)
10
Cm (J per mol Pr K–1)
large
χ 3!
–12
! (e.m.u. per mol Pr)
ets because the spontaneous
pically even in the absence of
rt of the AHE in moderately
d by the band-intrinsic mechons under an electric field E
e of the relativistic spin-orbit
n. This phase acts as a macbends the orbital motion of
a real magnetic field B. Thus,
Hall conductivity sH at B 5 0.
e fictitious magnetic field b,
the AHE at B 5 0, is not
the macroscopically broken
al operation cannot be comations of the crystal (Supthe scalar spin chirality in
al spin glasses can also proE4,5,12,13,20, as indeed has been
refs 6, 7), and MnSi (refs 9,
, the spin chirality is not the
mpanies a chiral spin texture
y the applied magnetic field.
pen issue to find a possible
he macroscopically broken
field.
-broken phase in the absence
eezing in the thermodynamic
iquid state. In particular, we
he absence of uniform magin the metallic cooperative
ng temperature, as indicated
Both the experiment and the
ase is induced by melting of a
ions of the Pr 4f magnetic
r spin-ice systems14,15.
n antiferromagnetic Curie–
nly due to the correlations
nts of Pr31 ions, which point
centre of the Pr tetrahedron
ctrons are weakly correlated
e22. They mediate the RKKY
a the Kondo coupling. The
ting conventional magnetic
eat, magnetic susceptibility,
nals strong geometrical frused in the magnetic susceptiers of magnitude lower than
0
Figure 2 | Temperature dependence of the magnetic and transport
properties of Pr2Ir2O2. a, Temperature dependence of the Hall conductivity
sH (left axis) and the direct-current susceptibility x 5 M/H (right axis)
under a magnetic field of B 5 0.05 T along the [111] direction. e.m.u.,
electromagnetic unit. Here, Hall conductivity is given by sH 5 2rH/
How much Hall conductivity
does small M produce?
Scaling
• Exchange/Zeeman field couples to
† z
⇠
J
†
12
• Standard scaling argument
K
•
Essentially
b
1
⇠ Hz
H
F(Hbz
1
12
12
)
⇡ H .51
⇠ M 1/2
Hall effect strong even when M is small
How does this work?
• Consider momentum along field direction
2
H(k) =
5 2
4 kz
kz2 Jz2
2m
H(cos Jz + sin Jz3 )
1
-2
-1
Jz=+1/2
1
2
Jz=-3/2
-1
• Crossing is protected and becomes double
-2
Weyl point
⌧ z = | 12 ⇤⇥ 12 |
+
=(
x
+i
|
y
3
2 ⇤⇥
3
2|
)/2 = | 12 ⇤⇥
3
2|
carries ΔJz=2:
couples to (kx+iky)2
Double Weyl Points
⇥µ Bµ = 2 (k
K)
0
2 (k + K)
K ≈ (m H)1/2
kz
edge states propagate in this range of kz
Order of magnitude
e2
h
r
• Scaling ⇥
• Kondo interaction
• estimate from |
• Then
8 q
H
⇥xy ⇠ 10
3
1
cm
easily obtain
1
⇥
xy
<
:
(m ⇠ 20me ,
CW | ⇥ 20K
m
me
q
⇠ 0.1
mH
S(H/ F )
2
~
H ⇠ JK M
q
JK
Ry
p
M , JK M
m p JK
M,
me
F Ry
10
F
1
cm
2
JK
/WIr
1
for
JK M ⌧
M = 0.001
⇠ 10meV, JK ⇠ 100K)
F
,
F
0.01
Perturbations
M
Weyl
LAB
Perturbations
M
Weyl
LAB
strain
TI
Quantum criticality
11
How does magnetic order onset?
Non⇥Metal
Non
Metal
200
T K⇥
150
Metal
Imported Author Today, 6:43 PM
Now I want to move from the PM phase to the
point at which magnetic order onsets
QCPs in metals are of course an old and important
subject. Still, this system offers new aspects.
Y
Ho Dy Tb
⇥
Gd Eu
⇥
Lu⇥Yb
⇥
Sm
⇥
⇥
⇥
⇥
100
50
Magnetic Ins.
Magnetic Ins.
Nd
⇥
0
QCP?
Pr
⇥
100
105
110
115
R3 ionic radius pm⇥
(b)
ws the resistivity, and field cooled (FC) and zero field cooled (ZFC)
LAB+AIAO
+
= ??
Field theory: Ising monopolar order parameter ɸ
S=
Z
3
d⌧ d x

g
a (@⌧ + Ĥ0 ) a + p
N
r
aM a +
2
M = Γ4,5 is completely determined by symmetry
2
LAB+AIAO
+
= ??
Field theory: Ising monopolar order parameter ɸ
S=
Z
3
d⌧ d x

g
a (@⌧ + Ĥ0 ) a + p
N
ie
+p '
N
r 2
aM a +
2
1
2
+
(r')
a a
2
Coulomb is also important
Quantum criticality
Fermi surface (Hertz)
Landau damping
1
⇠ q 2 + |!|/q
mismatch with electron dispersion
electrons scatter weakly
Semimetal
p
1
⇠q+
|!|
matched to electrons
electrons scatter strongly
Quantum criticality
•
Systematic RG
treatment in 1/N
expansion
•
Extreme deviations
from MFT
e.g. h i ⇠ (rc
r)
2
(x logs)
Semimetal
p
1
⇠q+
|!|
matched to electrons
electrons scatter strongly
Quantum criticality
•
Order parameter
fluctuations drive
electron spectrum to
extreme anisotropy
limit - rare example
where a QCP
“remembers” the
crystal symmetry
2 2
2 2
2 2
5 2
2
5 + ky Jy + kz Jz )
J
J)24 + d(k
x
x
dak(k) a 4 kk 2 (kd4·(k)
(k)
5
H0 (k) =
=
++
+
2M0 2m
2Mc
2Mc
22m
M̃0
Semimetal
p
1
⇠q+
|!|
matched to electrons
electrons scatter strongly
Quantum criticality
•
Order parameter
fluctuations drive
electron spectrum to
extreme anisotropy
limit - rare example
where a QCP
“remembers” the
crystal symmetry
h111i
2 2
2 2
2 2
5 2
2
5 + ky Jy + kz Jz )
J
J)24 + d(k
x
x
dak(k) a 4 kk 2 (kd4·(k)
(k)
5
H0 (k) =
=
++
+
2M0 2m
2Mc
2Mc
22m
M̃0
irrelevant
(ultimately leads to N-independent results)
Semimetal
p
1
⇠q+
|!|
matched to electrons
electrons scatter strongly
Quantum criticality
•
Order parameter
fluctuations drive
electron spectrum to
extreme anisotropy
limit - rare example
where a QCP
“remembers” the
crystal symmetry
•
Both long-range
Coulomb and order
parameter fluctuations
are important
=
+
Semimetal
p
1
⇠q+
|!|
matched to electrons
electrons scatter strongly
Picture
Imported Author Today, 6:43 PM
I am sparing you the calculations,
but this is one of if not the most
technically difficult RG calculation I
have seen.
after lots of calculations...
T
Weyl SM
with AIAO
QC
0
LAB
r
Picture
T
Weyl SM
with AIAO
QC
0
EF
LAB
r
200
T K⇥
150
Non⇥Metal
Non
Metal
Metal
Picture
Y
Ho Dy Tb
⇥
Gd Eu
⇥
Lu⇥Yb
⇥
Sm
⇥
⇥
⇥
⇥
100
50
Magnetic Ins.
Magnetic Ins.
Nd
⇥
0
Pr
⇥
100
105
3
R
110
115
ionic radius pm⇥
(b)
eld cooled (FC) and zero field cooled (ZFC)
us muon oscillation frequency. Data adapted
hlore iridates R-227 based on transport and
dified version of the diagram found in Ref. 78.)
nt are emphasized in bold magenta. The only
Weyl SM
manifold into a higher
energy
Je↵ = 1/2
with
AIAO
c picture, since Ir4+ has 5 d-electrons, the
al is involved in the low energy electronic
ed, the Je↵ = 3/2 levels are split and mixed
highest Kramers doublet, whose character
F
een a Je↵ = 1/2 doublet and a S = 1/2 one.
nic picture as we now discuss. If only the
generate bands near the Fermi energy, as
E
T
QC
0
LAB
r
Conclusions
•
The paramagnet electronic structure of A2Ir2O7
may realize a strongly correlated analog of HgTe
•
We explored implications of this for the bulk
phase diagram
•
The analogy suggests interesting possibilities for
strain and confinement studies in films
•
Another next step: include rare earth spins
•
Witczak-Krempa, William, Gang Chen, Yong Baek Kim, and Leon Balents. "Correlated quantum
phenomena in the strong spin-orbit regime." arXiv preprint arXiv:1305.2193 (2013).
•
Eun-Gook Moon, Cenke Xu, Yong Baek Kim, and Leon Balents. "Non-Fermi liquid and
topological states with strong spin-orbit coupling." arXiv preprint arXiv:1212.1168 (2012).
•
•
Lucile Savary, Eun-Gook Moon, and LB, in preparation.
Ru Chen et al, in preparation