Axion mass in antiferromagnetic insulators
Koji Ishiwata
Kanazawa University

Based on arXiv:2103.02848(Accepted by PRD)

Kyoto, June 9, 2021

1. Introduction

Axion in condensed matter physics
A hot topic and relates to• Topological insulator• Axion insulator• Magnetoelectric effect• Ferromagnetism / Antiferromagnetism• Magnon• It may be used in particle axion detectionProposals to detect axion/axion-like particles (ALPs)
5
Resonance Frequency [THz]in Ve↵ can increase linearly with N , with wide band response [56]. With N = 100 (a feasible total number
10 6
for solid state synthesis [94]), the increased sensitivity isstage-Ishown in Fig. 4 (stage-II).
stage-II8
CAST
10
A further increase in Ve↵ can be achieved by surroundstage-III
ing the A-TI samples with a cavity with a volume, Vc .
Long wavelength modes of the cavity E-field can cou10 10
ple to high frequency AP modes resulting in a TM010
type [95] component to the AP, allowing Ve↵ ⇡ Vc evenwith a small sample volume. In Fig. 4 (stage-III) we
10 12
SN 1987A
show the sensitivity benefit of a Ve↵ = (0.1 dB )3 ⇡ADMX
2000(1 meV/ma )3 cm3 . The same stage-III sensitivity
10 14
could be achieved if technology and investment allowedionx
for fabrication of a very large volume of A-TI.
DACQ
In summary, we have shown that A-TIs can host
10 16
10 6
10 5
10 4
10 3
10 2
10 1
dynamical axionic quasiparticles which are resonantly
2
Axion Mass, ma [eV/c ]driven in the presence of DAs with mass of order 1
meV and emit THz photons which can be detected using an SPD, allowing A-TIs to detect dark matter. We
FIG. 4. Axion parameter space. Vertical lines lines show theSensitivity
the magnon to Fe-doped
the ALPDM
thega plane. The oranthat ofantiferromagneticBia vs.
projected sensitivity of our proposal
using Fe doped
BiFigureSe3 at 1: showed
2 Sein3 satisfies
Marshet al.
Chigusa
etmal.
’21
2’19
2
the threeWilczek tocriteria
described earlier,can be⇠5T applied field for 10 s integration time with dark(green)count region
corresponds
the sensitivityof the and-mode(↵-mode) with u~0 v~0 = 1
rate d = 0.001 Hz. Staged designs are described in the
text.the used
to realizelinea DA
in theshows
0.7 to the
3.5 meVrange. of the corresponding mo
while
dot-dashedin detector
each region
sensitivity
Gray shaded regions assume scanning 1 T  B0  10 T. The
Fig.
4 shows the projected reach of three possible schemes 3
with
u
v
~0
~0 = 1. We postulate the target volume V = (10 cm) and the magnetic fie
KSVZ and DFSZ axion models are shown as the red band.
with di↵erent e↵ective volumes. Varying the applied B
scanned
1 T < B0 < 7 T (1 T < B0 < 10 T) for the -mode (↵-mode). For each st
Existing exclusions from ADMX [17, 18], CAST [27], andsu- overfield scans theresonant
frequency, giving sensitivity to2pernova 1987A [23] are shown as coloured regions.
of the scan,
we
use
t
=
10
s
for
an observation,
which requires
axion dark matter in a parameter
space inaccessible
to ⇠ 1 yr for the whole sca
See the text
formethods.
more detailsof work
the material
properties.
Also shown as colored regions aotherFutureon the material
characterisexisting constraints,
the black solid(dashed) can
lineallow
shows
tics (such aswhile
the anisotropy
field strength)
forthe prediction for the KSV(DFSZ)
a wider range of DA mass detection.
given above and setting ! (k = 0) = m wefind model.
10 2

10 1

100

101

Axion Coupling |g | [GeV 1 ]10 3

Topological magnetic insulators are used+a

0.7 meV  ma  3.5 meV (the lower limit is approximately the B0 = 0 spin wave mass). Other materials
for a single photon detector
in the THz regime at the temperature T = 0.05 K [52]. W
ACKNOWLEDGMENTS
with di↵erent anisotropy field strengths can cover a wider
estimate the sensitivity by requiring the signal-to-noise ratio (SNR)
range of masses.
We acknowledge useful discussions with Francesca Day,
Sensitivity to g is computed setting the signal to noise/dt) tscan
signalJoerg Jaeckel, Alexander Millar,(dNJensNiemeyer,Davida static axion field ✓. However, the antiferromagnetic phase also has
amplitude and spin-wave excitations, which can induce fluctuationsof the axion field. The fluctuation of the Néel vector M can be generally written as M = (M0 + Mz (x,t ))ẑ+ Mx (x,t )x̂+ My (x,t )ŷ.
To the linear order, it can be shown from symmetry analysis that the fluctuation of the axion field depends only on
ARTICLES
Mz , because ✓ is a pseudo-scalar. In other words, we have
PUBLISHED ONLINE: 7 MARCH 2010 | DOI: 10.1038/NPHYS1534✓ (x,t ) = m5 (x,t )/g = (2/3)U Mz (x,t )/g , where the coefficient g can be determined from equation (3). The dispersion of the
Dynamical axion field in topologicalamplitude mode Mz (x,t ) can be obtained in the standard randommagnetic insulators
phase approximation, leading to a massive axion field ✓ (x, t ).
Considering the coupling term ✓E·B of the axion withµ⌫
the elecRundong Li1 , Jing Wang1,2 , Xiao-Liang Qi1 and Shou-Cheng Zhang1 *tromagnetic field, the effective action describing µ⌫
the axion–photon
ARTICLES
Axions are weakly interacting particles of low mass, and were postulated more than 30 years ago in the framework of theStandard Model of particle physics. Their existence could explain the missing dark matter of the Universe. coupled
However, despitesystem is given byt of aDynamical axion is predicted in topological magnetic insulators✓F F̃

intensive searches, axions have yet to be observed. Here we show that magnetic fluctuations of topological insulators couple
ed byto the electromagnetic fields exactly like the axions, and propose several experiments to detect this dynamical axion field. Inof theparticular, we show that the axion couplingAenables a nonlinear modulation of the electromagnetic field, leading to attenuatedd up tototal reflection. We propose a new optical-modulator device based on this principle.
totzationBrom ahe electromagnetic response of a three-dimensional insulator
refer to such an antiferromagnetic insulator as a ‘topologicalR 3
is described by Maxwell’s action SC0 =Quintuple(8/⇡) layerd xdt magnetic insulator’ (TMI). We propose a minimal model in which(✏E2 (1/µ)B2 ), with material-dependent dielectric con- the antiferromagnetic order breaks the time-reversal symmetry(2)stant ✏ and magnetic permeability µ, where E and B are the spontaneously and the magnetic fluctuations couple linearly to theAelectromagnetic fields inside the insulator. However,generally, axion field, thus realizing the dynamic axion field in condensedit is possible to include anothereffective matter systems. Compared with its high-energy version, theR 3 quadratic term in theaction S✓ = (✓ /2⇡)(↵/2⇡) d xdt E·B, whereB ↵ = e 2 /h̄c is the axion proposed here has the advantage that it can be observed
cosky ),fine-structure constant and ✓ is a parameter describing the insulator in controlled experimental settings5 . With an externally appliednd thein question. In the field theory literature, thisC effective action is magnetic field, the axion field couples linearly to light, resulting in⌦ sx , xknown as the axion electrodynamics1 , where ✓ has thethe axionic polariton. By measuring the attenuated total reflection
z role of the1+(ATR), the gap in the axionic polariton dispersion can be observed.
z ,#i,axion field. Under the periodic boundary condition, the partitionyfunction and all physical quantities are invariant
if ✓ is shifted
An attractive feature is that the axionic polariton gap is tunable byA
by integer multiples of 2⇡. Therefore, all time-reversal invariant changing the external electric or magnetic fields. The control of theinsulators fall into two distinct classes described by either ✓ =x 0 light transmission through the material enables a new type of opticaletry T or ✓ = ⇡ (ref. 2). Topological insulators areB defined by ✓ = ⇡ modulator. We also propose another experiment to detect thet2
spatialand can be connected
continuously
to trivial insulators, defined dynamic axion by microcantilever torque magnetometry, where thet1C(withby ✓ = 0, only by time-reversal-breaking
perturbations.
The form double-frequency response of the cantilever is a unique signature oft3⌦ I2⇥2 ,of the effective action implies that an electric field can induce the dynamic axion field.
la (1),a magnetic polarization, whereas a magnetic field can induce
We propose several materials that may realize the topologicalA
eters2 ,an electric polarization. This effect is known as the topological magnetic insulator with dynamic axion field. One possibility is theider amagneto-electric effect and ✓ is related to the magneto-electric topological insulators Bi2 Te3 , Bi2 Se3 and Sb2 Te3 doped with 3dBi(Fe)on
viationpolarization P3 by P3 = ✓ /2⇡. Physically the parameter✓ depends
transition metal elements such as Fe (refs 6, 7). Another possible
B
paritythe band structure of the insulator and has a microscopic expression class of material is the 5d transition metal compound Ax BOy with
Se1
ationsof the momentum-space Chern–Simons form2 C
B and A standing for a 5d transition metal and an alkali metal,
show
respectively. Electrons in the 5d orbital can have both strong

Z
1
2
Se2
must
spin–orbital coupling and strong interaction, which is ideal for3
ijk✓=
d k✏ Tr Ai @j Ak + i Ai Aj Ak(1)turbedthe realization of the topological magnetic insulator8 . We propose
4⇡
3 A
a
with
that such a compound with the corundum structure may have), m5 ).where A↵i (k) = ih↵k|@/@ki | ki is the momentum-space non- a topological magnetic insulator phase if the states close to the
Figure1 | Crystalof Bi(Fe)structureof Bi(Fe)xplicitabeliangaugefieldstructureand |↵ki
is the2 Seperiodicof theBloch
3 . Crystalpart
2 Se3 Fermi level are formed by t2g orbitals with total angular momentum
with three primitive
vectors
denotedas t1,2,3
quintuple tolayer
wavefunctionof the lattice↵th bandwithindices↵, . Areferring
thewith Jeff = 1/2 (ref. 9). We also noticed two very recent works on 5d
Se1–Bi(Fe)1–Se2–Bi(Fe)
10 –Se10 is indicatedthe orange rectangle.
occupiedbands. The ✓ parameterhas beenin calculated
explicitlyThefor transition metal compounds with the pyrochlore structure, which
2,3spin-ordering
configuration
giving riseinsulators
to the 05 massby the may also realize the topological magnetic insulator phase10,11 .
severalbasic models
of topological. Inis indicateda topological(3)
black arrow,z direction
andeffects
insulator,
the whichaxionis antiferromagnetic
field gives risealong
to thenewphysicalwithinthe xy plane.
suchferromagnetic
as the image
monopole
and anyonic statistics4 . This field, Effective model for the 3D topological insulator
however, is static in a time-reversal invariant topological insulator. Although all of the physical effects discussed in this article areaDynamical
axionthein theantiferromagnetic
phase order in generic for any system supporting axionic excitation and do not
we consider
antiferromagnetic
long-range
1 da d .Here,founda topological
We have seen
above whichthat theaxiontime-reversal
field ✓ can symmetry
deviate fromrely on a specific model, we would like to start from a simple
insulator,
breaksS

T= SMaxwell + Stopo + Saxion
✓
◆
Z
Z
1
1
↵
=
d3 xdt ✏E2
B2 + 2 d3 xdt (✓0 + ✓)E · B
8⇡
µ
4⇡
Z
+ g 2 J d3 xdt [(@t ✓ )2 (vi @i ✓ )2 m2 ✓ 2 ](4)

where J , vi and m are the stiffness, velocity and mass of thespin-wave mode Mz , E and B are the electric field and the magnetic
field respectively and ✏ and µ are the dielectric constant andmagnetic permeability respectively. The second term describes the
topological coupling between the axion and the electromagnetic
field, with ↵ ⌘ e 2 /h̄c being the fine-structure constant. The third
term describes the dynamics of the massive axion. Within the model
we have adopted, the parameters J and m are given by

Bi2 Se3

Axion mass ~ O(meV)Z
Topological insulatord k d (k)d (k)
J=

3

(2⇡)3

i

i

16|d|5

,

✓

2
2
Jm =
UMz
3

◆2 Z

d3 k 1(2⇡)3 4|d|3

He
wh
an
ob
lin

wh
co
can
the

wi
sp
m
be
the
at
lin
the
po
the
eff
sam
the

ARTICLES
PUBLISHED ONLINE: 7 MARCH 2010 | DOI: 10.1038/NPHYS1534

Dynamical axion field in topologicalmagnetic insulators
1,2

1

ARTICLES

1

Axions are weakly interacting particles of low mass, and were postulated more than 30 years ago in the framework of theStandard Model of particle physics. Their existence could explain the missing dark matter of the Universe. However, despite
t of aintensive searches, axions have yet to be observed. Here we show that magnetic fluctuations of topological insulators couple
ed byto the electromagnetic fields exactly like the axions, and propose several experiments to detect this dynamical axion field. Inof theparticular, we show that the axion couplingAenables a nonlinear modulation of the electromagnetic field, leading to attenuatedd up tototal reflection. We propose a new optical-modulator device based on this principle.
zationBrom ahe electromagnetic response of a three-dimensional insulator
refer to such an antiferromagnetic insulator as a ‘topologicalR 3
is described by Maxwell’s action SC0 =Quintuple(8/⇡) layerd xdt magnetic insulator’ (TMI). We propose a minimal model in which(✏E2 (1/µ)B2 ), with material-dependent dielectric con- the antiferromagnetic order breaks the time-reversal symmetry(2)stant ✏ and magnetic permeability µ, where E and B are the spontaneously and the magnetic fluctuations couple linearly to theAelectromagnetic fields inside the insulator. However,generally, axion field, thus realizing the dynamic axion field in condensedit is possible to include anothereffective matter systems. Compared with its high-energy version, theR 3 quadratic term in theaction S✓ = (✓ /2⇡)(↵/2⇡) d xdt E·B, whereB ↵ = e 2 /h̄c is the axion proposed here has the advantage that it can be observed
cosky ),fine-structure constant and ✓ is a parameter describing the insulator in controlled experimental settings5 . With an externally appliednd thein question. In the field theory literature, thisC effective action is magnetic field, the axion field couples linearly to light, resulting in⌦ sx , xknown as the axion electrodynamics1 , where ✓ has thethe axionic polariton. By measuring the attenuated total reflection
z role of the1+(ATR), the gap in the axionic polariton dispersion can be observed.
z ,#i,axion field. Under the periodic boundary condition, the partitionyfunction and all physical quantities are invariant
if ✓ is shifted
An attractive feature is that the axionic polariton gap is tunable byA
by integer multiples of 2⇡. Therefore, all time-reversal invariant changing the external electric or magnetic fields. The control of theinsulators fall into two distinct classes described by either ✓ =x 0 light transmission through the material enables a new type of opticaletry T or ✓ = ⇡ (ref. 2). Topological insulators areB defined by ✓ = ⇡ modulator. We also propose another experiment to detect thet2
spatialand can be connected
continuously
to trivial insulators, defined dynamic axion by microcantilever torque magnetometry, where thet1C(withby ✓ = 0, only by time-reversal-breaking
perturbations.
The form double-frequency response of the cantilever is a unique signature oft3⌦ I2⇥2 ,of the effective action implies that an electric field can induce the dynamic axion field.
la (1),a magnetic polarization, whereas a magnetic field can induce
We propose several materials that may realize the topologicalA
eters2 ,an electric polarization. This effect is known as the topological magnetic insulator with dynamic axion field. One possibility is theider amagneto-electric effect and ✓ is related to the magneto-electric topological insulators Bi2 Te3 , Bi2 Se3 and Sb2 Te3 doped with 3dBi(Fe)on
viationpolarization P3 by P3 = ✓ /2⇡. Physically the parameter✓ depends
transition metal elements such as Fe (refs 6, 7). Another possible
B
paritythe band structure of the insulator and has a microscopic expression class of material is the 5d transition metal compound Ax BOy with
Se1
ationsof the momentum-space Chern–Simons form2 C
B and A standing for a 5d transition metal and an alkali metal,
show
respectively. Electrons in the 5d orbital can have both strong

Z
1
2
Se2
must
spin–orbital coupling and strong interaction, which is ideal for3
ijk✓=
d k✏ Tr Ai @j Ak + i Ai Aj Ak(1)turbedthe realization of the topological magnetic insulator8 . We propose
4⇡
3 A
a
with
that such a compound with the corundum structure may have), m5 ).where A↵i (k) = ih↵k|@/@ki | ki is the momentum-space non- a topological magnetic insulator phase if the states close to the
Figure1 | Crystalof Bi(Fe)structureof Bi(Fe)xplicitabeliangaugefieldstructureand |↵ki
is the2 Seperiodicof theBloch
3 . Crystalpart
2 Se3 Fermi level are formed by t2g orbitals with total angular momentum
with three primitive
vectors
denotedas t1,2,3
quintuple tolayer
wavefunctionof the lattice↵th bandwithindices↵, . Areferring
thewith Jeff = 1/2 (ref. 9). We also noticed two very recent works on 5d
Se1–Bi(Fe)1–Se2–Bi(Fe)
10 –Se10 is indicatedthe orange rectangle.
occupiedbands. The ✓ parameterhas beenin calculated
explicitlyThefor transition metal compounds with the pyrochlore structure, which
2,3spin-ordering
configuration
giving riseinsulators
to the 05 massby the may also realize the topological magnetic insulator phase10,11 .
severalbasic models
of topological. Inis indicateda topological(3)
black arrow,z direction
andeffects
insulator,
the whichaxionis antiferromagnetic
field gives risealong
to thenewphysicalwithinthe xy plane.
suchferromagnetic
as the image
monopole
and anyonic statistics4 . This field, Effective model for the 3D topological insulator
however, is static in a time-reversal invariant topological insulator. Although all of the physical effects discussed in this article areaDynamical
axionthein theantiferromagnetic
phase order in generic for any system supporting axionic excitation and do not
we consider
antiferromagnetic
long-range
1 da d .Here,founda topological
We have seen
above whichthat theaxiontime-reversal
field ✓ can symmetry
deviate fromrely on a specific model, we would like to start from a simple
insulator,
breaksT

10 6

Axion Coupling |g | [GeV 1 ]Rundong Li , Jing Wang , Xiao-Liang Qi and Shou-Cheng Zhang *1

10 8

Resonance Frequency [THz]

10 3

10 2

10 1

100

stage-Istage-IIstage-III

101

CAST

10 10

10 12

SN 1987A
ADMX

10 14

10 16
10 6

ion

Ax
D
QC
10 5

10 4

10 3

10 2

10 1

2

Axion Mass, ma [eV/c ]FIG. 4. Axion parameter space. Vertical lines lines show the
projected sensitivity of our proposal using Fe doped Bi2 Se3 at⇠5T applied field for 102 s integration time with dark countrate d = 0.001 Hz. Staged designs are described in the text.
Gray shaded regions assume scanning 1 T  B0  10 T. The
KSVZ and DFSZ axion models are shown as the red band.
Existing exclusions from ADMX [17, 18], CAST [27], and supernova 1987A [23] are shown as coloured regions.

Marsh et al. ’19

O(meV)

given above and setting !+ (k = 0) = ma we find
0.7 meV  ma  3.5 meV (the lower limit is approxi-

in Ve↵ cansponse [56for solid st
shown in F
A furtheing the ALong wave
ple to hightype [95] cwith a sm
show the2000(1 meV
could be afor fabricat
In summdynamical
driven inmeV and e
ing an SPD
showed tha
the three W
used to rea
Fig. 4 showwith di↵erfield scans
axion darkother methtics (such aa wider ran

Today, I would like to address• What is topological insulator?
• How does antiferromagnetism play a role?
• How is axion in condensed matter described?
Method: Path integral (partition function)
Results: Axion mass depends on the state of insulators

Plan to talk
1. Introduction
2. Brief review of condensed matter physics (related to axion)
3. Axion in antiferromagnetic topological insulators
4. Conclusions and discussion

2. Brief review of condensed matter physics(related to axion)

Topics related to axion in condensed matter physicsa). Insulatorsb). Anomalous quantum Hall effectc). Topological insulatorsd). Magnetoelectric effecta). Insulators

Minimum basics
Energy band

Ekenergy levels

band gapk
1st BZBZ (Brillouin zone)

The region where there is no energy level is called band gap(important for insulators)

a). Insulators

Insulators (and metals)
Insulator

Metal

All electrons are tightly
bound by nucleus

Free electrons exist

free electron

a). Insulators

Insulator

Metal

Ek

Ek

EF
k
Fermi energy (level) is in theband gapEFkFermi energy (level) is in theband

EF : Fermi energy

Electrons around the Fermi energy determines
the features of the materials

a). Insulators

Insulator

Metal

Ek

Ek

EF
k
Fermi energy (level) is in theband gap

EF

Main topic todaykFermi energy (level) is in theband

EF : Fermi energy

Electrons around the Fermi energy determines
the features of the materials

b). Anomalous quantum Hall effect

Hall effecte.g., semiconductor

I+ + + +

B

y
z

x

e

−
−
−
−

−

Lorentz force

VH

Electromotive force VH is induced in the direction
perpendicular to both the electric current and magnetic field(In metal, VH is too small to observe)b). Anomalous quantum Hall effect

Quantum Hall (QH) effect
QH effect appears insemiconductor tooe.g., 2D insulator

BE : electric fieldyzxHall current hjx i

Quantized electric current is induced in x direction
hjx i : current density in x direction

b). Anomalous quantum Hall effect

It seems weird but there was theoretical prediction:

Thouless, Kohmoto, Nightingale, den Nijs ’82

Hall conductivity:

XZ

e
=⌫
h

d2 k
⌫⌘
[rk ⇥ an (k)]z
2⇡
BZ
n
@
an (k) ⌘ i hunk | |unk i@k

TKNN formula|unk i : Bloch statexy ⌘ hjx i /E
2

n : label of band

b). Anomalous quantum Hall effectFor insulators
Z
X
d2 k
⌫⌘
[rk ⇥ an (k)]z2⇡BZnintegral over 1st BZ

EkEF
1st BZν is given by an integer“(Integer) QH effect”kb). Anomalous quantum Hall effect

e.g.,

H=

3
X

d

a a

with

a=1

d1 = sin kx
d2 = sin ky
d3 = m + 2

v
u 3
uX
Two eigenvalues: ±t (da )2
a=1

H'

✓

m
kx + ikykx

around k = 0

iky
m

◆

cos kxcos kyb). Anomalous quantum Hall effect

The band structurem = 0.1

m=0

m=

0.1

b). Anomalous quantum Hall effect

The band structurem = 0.1⌫=0Normal insulator

m=0Band inversion
around k=0

m=

0.1

⌫=1
QH insulator

b). Anomalous quantum Hall effect

The band structurem = 0.1

m=

m=0

⌫=0

H'

⌫=1

Band inversion
around k=0

Normal insulator✓mkx + ikykx

around k = 0

0.1

QH insulator

iky
m

◆

b). Anomalous quantum Hall effectSingular at k = 0

a(k)m = 0.1

m=

0.1
10

1.2

8

1.0

6

0.8

0.6

4

0.4

2

Integer ⌫Note: ν is half-integer if the dispersion
relation is Dirac typeb). Anomalous quantum Hall effect

The current flows at the edge

hjx i

B

E

b). Anomalous quantum Hall effect

The band structure (bulk)
m = 0.1⌫=0Normal insulator

m=0Band inversion
around k=0

m=

0.1

⌫=1
QH insulator

b). Anomalous quantum Hall effect

The band structure (bulk + edge)
m = 0.1⌫=0Normal insulator

m=0Band inversion
around k=0

m=

0.1

⌫=1
QH insulator
Gapless state at edgeE ⇠ ±k“Dirac point”

b). Anomalous quantum Hall effectAnomalous quantum Hall (AQH) effectSame effect by magnetization M, not magnetic field

Anomalous QH effect
QH effectB

AQH effect

E
hjx i

M

E
hjx i
“AQH insulators”“Chern insulators”

c). Topological insulators

Topological insulators (TIs)
Idea: combination of two QH insulators

Kane, Mele ’05

zspin current

EHall current is zero, but
spin current exists

Such a system can be realized due to SOC(without magnetic field)
SOC: spin orbit couplingc). Topological insulators

Spin-orbit coupling (SOC)e.g.,

HSO = V (r)L · S

V(r)Electrons with spin up or down are scattered off
to the opposite directions

c). Topological insulators

Keywords for topological insulators• Time reversal invariance ( T )• Strong spin-orbit coupling (SOC)

c). Topological insulators

Keywords for topological insulators• Time reversal invariance ( T )• Strong spin-orbit coupling (SOC)
T
T

spin current

TE
B breaks T but the combination of B and −B keeps Tc). Topological insulators

Keywords for topological insulators• Time reversal invariance ( T )• Strong spin-orbit coupling (SOC)

Strong SOC is crucial for the realizationc). Topological insulators

The band structure (bulk)
Same as (A)QH insulators
SOC

-3

-2

2

2

2

1

1

1

1

-1

2

3

-3

-2

1

-1

2

3

-3

-2

1

-1

-1

-1

-1

-2

-2

-2

Normal insulator(NI)

Band inversion
around k=02

3Topological insulator(TI)c). Topological insulators

Topological insulators are classified by Z2
The number of band inversions (Dirac points) is• Even• Odd“Normal insulator”“Topological insulator”

c). Topological insulators

Example of 2D TI: HgTe/(Hg , Cd)

CdTe HgTe CdTe

König et al. ’07

CdTe

HgTe

CdTe

Thicker HgTe layer(stronger SOC)

NI

TI

Band inversion happens in the energy band of HgTe

hree-dimensional band insulators with large spin–orbit coupling. So
or is Bix Sb1−x , which is an alloy with complex surface states. Here, we
e calculations of the layered, stoichiometric crystals Sb2 Te3 , Sb2 Se3 ,Bi2 Te3 and Bi2 Se3 are topological insulators, whereas Sb2 Se3 is not.
ce states consisting of a single Dirac cone at the ! point. In addition,
ergy gap of 0.3 eV, which is larger than the energy scale of room
ntinuum model that captures the salient topological features of thisgicalmenf the
onal
e by
ness,
rom
ates
nism
Dirac
sys-

Example of 3D TI: Bi2Se3

Zhang et al. ’09

ARTICLES
a

b
z

y
x

t3
Bi
Se1
Se2

¬
P1x,y,zP1+x,y

P1z+

Bi

A

Se1

B

Bi1

C

Se2

A

Bi1’

B

Se1’

C

Se1

Figure 1 | Crystal structure. a, Crystal structure of Bi2 Se3 with threeprimitive lattice vectors denoted as t1,2,3 . A quintuple layer with
Se1–Bi1–Se2–Bi1! –Se1! is indicated by the red square. b, Top view along
the z-direction. The triangle lattice in one quintuple layer has three different
positions, denoted as A, B and C. c, Side view of the quintuple layer
structure. Along the z-direction, the stacking order of Se and Bi atomic
layers is ···–C(Se1! )–A(Se1)–B(Bi1)–C(Se2)–A(Bi1! )–B(Se1! )–C(Se1)–···.
The Se1 (Bi1) layer can be related to the Se1! (Bi1! ) layer by an inversion
operation in which the Se2 atoms have the role of inversion centres.

Cristal structure

P2z¬¬
P2x,y,zSe

P1
P2¬z,↑,P2¬z,↓P2+x,y,zP2¬x,yP2¬P2x+iy,↑, P2¬x¬iy,↓¬
P2¬x+iy,↓, P2x¬iy,↑¬
P0x,y,z

(I)

+
P1z,↑
, P1+z,↓

¬0.1

¬0.3

(II)

¬0.5
0

(III)

Figure 3 | Band sequence. a, Schematic diagram of the evolution from the atomic px,y,z orbitals of B
Bi2 Se3 at the ! point. The three different stages (I), (II) and (III) represent the effect of turning on crespectively (see text). The blue dashed line represents the Fermi energy. b, The energy levels |P1+zrescaled atomic SOC λ(Bi) = xλ0 (Bi) = 1.25x eV,λ(Se) = xλ0 (Se) = 0.22x eV (see text). A level cro

Energy levels

Ab initio calculations for Sb2 Te3 , Sb2 Se3 , Bi2 Te3 and Bi2 Se3 arecarried out in the framework of the Perdew–Burke–Ernzerhoftype14 generalized gradient approximation of the density functional
theory. The BSTATE package15 with the plane-wave pseudopotential method is used with a k-point grid taken as 10 × 10 × 10
and the kinetic energy cutoff fixed to 340 eV. For Sb2 Te3 , Bi2 Te3and Bi2 Se3 , the lattice constants are chosen from experiments,
whereas for Sb2 Se3 , the lattice parameters are optimized in
the self-consistent calculation for rhombohedral crystal structure(a = 4.076 Å, c = 29.830 Å), owing to the lack of experimental data.
Our results are consistent with the previous calculations16,17 . In

Band inversion due to strong SOC

conduction and the valence bands have the opposite parity, anda ‘band inversion’ occurs when the strength of some parameter,
say the spin–orbit coupling (SOC), is tuned. For systems withinversion symmetry, a method based on the parity eigenvalues ofband states at time-reversal-invariant points can be applied13 . On

0.1

+
+
P1x+iy,↓, P1x¬iy,↑

Se1’

C

b

+
P1x,y,z

x

c

t2

+
+
P1x+iy,↑, P1x¬iy,↓y

Quintuple
layer

t1

a

A siteB siteC siteE (eV)

gical
atter
ions
e or
by
onal
sless
case.
the
ulk.
rsalDiracomeme. In
3D)
m of
are
ture
=π
and
mber

c). Topological insulators

at the ! point. T(I), (II) and (III)are mainly cominorbitals and start(4s2 4p4 ). In stage
and Se atoms wit
scale in the curre
in a single unit ce
states (two odd,
(one odd, one e
chemical bondin

d). Magnetoelectric effectpredicted by Landau&Lifshitz

Magnetoelectric (ME) effectdiscovered by Dzyaloshinskii ’60- Electric field ( E ) induces magnetization M- Magnetic field (B) induces electric polarization P

F =Mj = ↵ij Ei
Pi = ↵ij Bj

Mi =
Pi =

1
µ0 c

Z

d3 x ↵ij Ei Bj

1 @F
V @Ei B=0
1 @F
V @Bi E=0
↵ij : constantd). Magnetoelectric effectLet’s consider
TI coated with magnetization directing outsidez
Half-integer AQH current
Hall current
TI

The current induces
magnetization in z direction

↵
M =±
E
µ0 c
magnetization↵ : fine-structure constantd). Magnetoelectric effectThis ME effect can be understood from a free energy:

F✓ =

1
µ0

Z

↵
d x
✓ E·B
with
c⇡
↵
µ⌫
✓Fµ⌫ F̃4⇡3

θ = ± π is called static axion( θ = 0 in NI )✓ = ±⇡

Quick summary
bands around the Fermi energy are crucial to• Energy
determine the characteristics of the materials• SOC and T are crucial for TI• ME effect in TI is described by “static axion”

3. Axion in antiferromagnetic
topological insulators

e-dimensional band insulators with large spin–orbit coupling. So
is Bix Sb1−x , which is an alloy with complex surface states. Here, wealculations of the layered, stoichiometric crystals Sb2 Te3 , Sb2 Se3 ,Te3 and Bi2 Se3 are topological insulators, whereas Sb2 Se3 is not.
states consisting of a single Dirac cone at the ! point. In addition,
gy gap of 0.3 eV, which is larger than the energy scale of room
nuum model that captures the salient topological features of this

al
nhe
al
by
s,
m
es
m
ac
s-

H. Zhang et al. ’09

ARTICLES
a

b
z

y
x

t3
Bi
Se1
Se2

¬
P1x,y,zP1+x,y

P1z+

Bi

A

Se1

B

Bi1

C

Se2

A

Bi1’

B

Se1’

C

Se1

Figure 1 | Crystal structure. a, Crystal structure of Bi2 Se3 with threeprimitive lattice vectors denoted as t1,2,3 . A quintuple layer with
Se1–Bi1–Se2–Bi1! –Se1! is indicated by the red square. b, Top view along
the z-direction. The triangle lattice in one quintuple layer has three different
positions, denoted as A, B and C. c, Side view of the quintuple layer
structure. Along the z-direction, the stacking order of Se and Bi atomic
layers is ···–C(Se1! )–A(Se1)–B(Bi1)–C(Se2)–A(Bi1! )–B(Se1! )–C(Se1)–···.
The Se1 (Bi1) layer can be related to the Se1! (Bi1! ) layer by an inversion
operation in which the Se2 atoms have the role of inversion centres.

Cristal structure

P2z¬¬
P2x,y,zSe

P1
P2¬z,↑,P2¬z,↓P2+x,y,zP2¬x,yEF

P2¬
P2x+iy,↑, P2¬x¬iy,↓¬
P2¬x+iy,↓, P2x¬iy,↑¬
P0x,y,z

(I)

+
P1z,↑
, P1+z,↓

¬0.1

¬0.3

(II)

¬0.5
0

(III)

Figure 3 | Band sequence. a, Schematic diagram of the evolution from the atomic px,y,z orbitals of B
Bi2 Se3 at the ! point. The three different stages (I), (II) and (III) represent the effect of turning on crespectively (see text). The blue dashed line represents the Fermi energy. b, The energy levels |P1+zrescaled atomic SOC λ(Bi) = xλ0 (Bi) = 1.25x eV,λ(Se) = xλ0 (Se) = 0.22x eV (see text). A level cro

Energy levels

Ab initio calculations for Sb2 Te3 , Sb2 Se3 , Bi2 Te3 and Bi2 Se3 arecarried out in the framework of the Perdew–Burke–Ernzerhoftype14 generalized gradient approximation of the density functional
theory. The BSTATE package15 with the plane-wave pseudopotential method is used with a k-point grid taken as 10 × 10 × 10
and the kinetic energy cutoff fixed to 340 eV. For Sb2 Te3 , Bi2 Te3and Bi2 Se3 , the lattice constants are chosen from experiments,
whereas for Sb2 Se3 , the lattice parameters are optimized in
the self-consistent calculation for rhombohedral crystal structure(a = 4.076 Å, c = 29.830 Å), owing to the lack of experimental data.
Our results are consistent with the previous calculations16,17 . In

Band inversion due to strong SOI

conduction and the valence bands have the opposite parity, anda ‘band inversion’ occurs when the strength of some parameter,
say the spin–orbit coupling (SOC), is tuned. For systems withinversion symmetry, a method based on the parity eigenvalues ofband states at time-reversal-invariant points can be applied13 . On

0.1

+
+
P1x+iy,↓, P1x¬iy,↑

Se1’

C

b

+
P1x,y,z

x

c

t2

+
+
P1x+iy,↑, P1x¬iy,↓y

Quintuple
layer

t1

a

A siteB siteC siteE (eV)

al
er
ns
or
by
al
ss
e.
he
k.
lac
me
en
D)
of
re
re
π
d
er

We consider 3D TI, Bi2Se3

at the ! point. T(I), (II) and (III)are mainly cominorbitals and start(4s2 4p4 ). In stage
and Se atoms wit
scale in the curre
in a single unit ce
states (two odd,
(one odd, one e
chemical bondin

x+iy,↑¬
P1x,y,zP1+x,yx¬iy,↓

0.1

Two bands near the Fermi energy are important++
P1x+iy,↓, P1x¬iy,↑+
P1x,y,zP2¬z,↑,P2¬z,↓EF¬P2x,y,zP2+x,y,zH0 (k)¬
P0x,y,z

0

P2z¬

+
P1z,↑
, P1+z,↓

¬0.1

P1z+

4 by 4 matrices

E (eV)

P1z+

P2z¬

¬0.3

P2¬x,y¬P2x+iy,↑, P2¬x¬iy,↓¬
P2¬x+iy,↓, P2x¬iy,↑

¬0.5
0

xc

0.2

0.4

0.6
x

1

✏(II)
0
iA1 sin kz
A2 (sin kx i sin ky )0 (k) + M(k)
(III)
B
C
0
✏
(k)
+
M(k)
A2 (sin kx + i sin ky )iA1 sin kz
0
B
C
=@
A
iA1 sin kz
A2 (sin kx i sin ky )✏0 (k) M(k)0nce. a, Schematic diagram of the evolution from the atomic px,y,z orbitals of Bi and Se into the conduction andA2 (sin kx + i sin ky )iA1 sin kz
0✏0 (k) M(k)The three different stages (I), (II) and (III) represent the effect of turning on chemical bonding, crystal-field sp. The blue dashed line represents the Fermi energy. b, The energy↵levels |P1+
! and |P2−↵zz !↵of Bi2 Se3 at↵ the ! po++P 1ztext).
, " ,A Plevel
1z ,crossing# , P 2occurs
P 2z , # these) twoz , " ,between
λ(Bi) = xλ0 (Bi) = 1.25x eV,λ(Se) = xλ0 (Se) =basis:0.22x eV((see(I)

tions for Sb2 Te3 , Sb2 Se3 , Bi2 Te3 and Bi2 Se3 are at the ! point. This is summarized schemat
framework of the Perdew–Burke–Ernzerhof(I), (II)“Effective Hamiltonian
forand3D(III)TI”in Fig. 3a. As the states neradient approximation of the density functional are mainly coming from p orbitals, we will nThe effective Hamiltonian is written by Gamma matrices:

H0 (k) = ✏0 (k)14⇥4 +5Xda (k) aa=1(d1 , d2 , d3 , d4 , d5 ) = (A2 sin kx , A2 sin ky , A1 sin kz , M(k), 0)
M(k) = M

1

2

3

=
=
=

✓

✓

✓

0

x

x

0

0

y

y

0

0
i1

◆

◆

◆
i1
0

2B1

4B2 + 2B1 cos kz + 2B2 (cos kx + cos ky )

4

5

=
=

✓

✓

◆

1
0

0
1

0

z

z

0

◆

{ a,

b

} = 2 ab 14⇥4

tr( a b ) = 4 abtr( a ) = 0

ϵ0 = 0A1 = A2 = 1
B1 = B2 = − 0.5
ky = kz = 0

The band structure (bulk)

-3

-2

2

2

2

1

1

1

1

-1

2

3

-3

-2

1

-1

2

3

-3

-2

1

-1

-1

-1

-1

-2

-2

-2

M = 0.3

NI

M=0

2

3

kx

M = − 0.3

Phase transition

TI(M/B1, M/B2 describe SOI)

Partition function (given by path integral)

S0 =

Z

4

d x

†

(x) [i@t

Z0 =Z

H0 ] (x)

D D

=

Z

dt H0

† iS0

e

(x) : wavefunction of electron⇠ ( P 1+
z ,"

↵

,

P 1+
z ,#

↵

↵
↵T
, P 2z , " , P 2z , # )ARTICLES
Sb2 Te3 . The low-energy bands of these materials consist of abonding and an antibonding state of pz orbitals, labelled by|P2z ," (#)i and |P1+
z ," (#)i, respectively. The generic form of theeffective Hamiltonian describing these four bands is obtained up toquadratic order of momentum k in ref. 12. As a lattice regularizationis necessary for computing the axion field ✓ , here we start from alattice version of this model, with the HamiltonianA

Now we introduce antiferromagnetism (AFM) by hand
H0 (k) = ✏0 (k)I4⇥4 +

5
X

B
C

da (k)0a

Explicit T breaking

(2)

a=1

A

d1,2,...,5 (k) = (A2 sinkx ,A2 sinky ,A1 sinkz , M(k),0)B

where ✏0 (k) = C + 2D1 + 4D2 2D1 coskz 2D2 (coskx + cosky ),M(k) = M 2B1 4B2 + 2B1 coskz + 2B2 (coskx + cosky ) and the
NATURE PHYSICS
DOI: 10.1038/NPHYS1534
Dirac 0 matrices have the representation 0 (1,2,3,4,5) = ( x ⌦ sx , x+⌦ sy , y ⌦ I2⇥2 , z ⌦ I2⇥2 , x ⌦ sz ) in the basis of (|P1+
z ,"i,|P1z ,#i,
|P2z , "i, |P2
z , #i).
Sb Te . The low-energy
bandsof these materials consist of a
2

R.CARTICLES
Li et
al. ’10z

Suppose electrons at Bi
are AFM orderbonding and an
antibonding
state of pz orbitals, labelled byP,T-breaking terms+|P2z ," (#)i andThe|P1aboverespectively.
Thebothgeneric
form of symmetry
thez ," (#)i,
Hamiltonian
preservestime-reversal
T
effective Hamiltonian
describing
four
bands
up to
and parity
P . In the theseeffectivemodel(2),istheobtained
time-reversal
and spatial
quadratic order ofmomentumk in ref. 12.
a latticeinversiontransformation
are Asdefinedas Tregularization= i(I2⇥2 ⌦ sy ) · K (withbeing the thecomplexoperator)
andfrom
P =a z ⌦ I2⇥2 ,is necessary for K
computingaxionconjugation
field ✓ , here
we start
field ✓ can be calculated by formula (1),lattice version ofrespectively.
this model,The
withaxion
the Hamiltonian2

which gives 0 or ⇡ depending on the value of the parameters ,5
as expected from time-reversal
symmetry. Now we consider aXaperturbation
to theHamiltonianthat can lead (2)to deviation
H0 (k) = ✏0 (k)Ida (k)0(2)4⇥4 +of ✓ from 0 or ⇡. As a=1✓ is odd under time-reversal and parity
operation,
only,Atime-reversaland parity-breaking perturbations
d1,2,...,5 (k)= (A2 sinkx
2 sinky ,A1 sinkz , M(k),0)
can induce a change of ✓ . By simple algebra, one can showthat to the leading order the
perturbation must
P T , P -breakingitheform H2D(k)1 cosk= zi=1,2,3,5
they ),perturbed
i 0 . Thus,
where ✏0 (k) = Chave
+ 2D
2Dm
Pcosk
1 + 4D2
2 (coskx +5astill be written as H (k) = a=1 dthea (k)0 withM(k) = M 2BHamiltonian
4B2 + 2Bcan
1
1 coskz + 2B2 (coskx + cosky ) andda (k) = (A2 sin kx + m1 , A2 sin ky (1,2,3,4,5)+ m2 , A1 sin kz + m3 , M(k), m5 ).
Dirac 0 matrices
havetherepresentation0= ( x ⌦ tosx , xFor this model, equation (1) can be +reduced+ an explicit⌦ sy , y ⌦ I2⇥2 , expression⌦✓:sz ) in the basis of (|P1z ,"i,|P1z ,#i,z ⌦ I2⇥2 , xfor
|P2z , "i, |P2z , #i).
Z
12|d| + d4✓=
d3 k✏ ijkl di @kx dj @ky dk @kz dl(3)
2
3
P,T -breaking terms
4⇡(|d| + d4 ) |d|
where i, j, k, l take values from 1, 2, 3, 5 and |d| =da d a .

yA

3

The above Hamiltonian preserves both time-reversal symmetry T
qP
and parity P . In the effective model (2), the time-reversal and spatial
5

Quintuple layer

x

A B
t1

t2
t3

C

B
A

C

Quintuple layer
B

AC

Bi(Fe)
Se1
Se2

B
A

C
z
Figure 1 | Crystal structure of Bi(Fe)2 Se3 . Crystal structureof Bi(Fe)2 Se3with three primitive lattice vectors denoted as t1,2,3 . A quintuple layer withy The
A orange rectangle.
Se1–Bi(Fe)1–Se2–Bi(Fe) 10 –Se10 is indicated in the
spin-ordering configuration giving rise to the 05 mass is indicated by thex
black arrow, which is antiferromagnetic along the z direction andBferromagnetic within the xy plane.
t2tDynamical
1 axion in the antiferromagnetic phase1534This corresponds to introducing the Hubbard terms:
ARTICLES

H = H0 + Hint

aterials consist of aorbitals, labelled byhe generic form of theands is obtained up toa lattice regularization
here we start from a
nian

0

A

where

a

(2)

z , M(k),0)UV
Hint =
N

2D2 (coskx + cosky ),skx + cosky ) and the
1,2,3,4,5)= ( x ⌦ sx , x+of (|P1z ,"i,|P1+
z ,#i,

onsist of aabelled byormof symmetry
thereversal
T
ained
up
to
me-reversal and spatialularizationi(I2⇥2 ⌦ sy ) · K (with) andfrom
P =a z ⌦ I2⇥2 ,tartated by formula (1),e of the parameters2 ,Now we consider aan lead (2)to deviation
e-reversal and parity)eaking perturbations
ebra, one can show
g perturbation must
Thus,
they ),perturbed
Pcosk
x +
5
a
a (k)0 with
a=1 dthek=y ) andkz + m3 , M(k), m5 ).
(uced
sx , anx explicitx ⌦ to+i,|P1z ,#i,

Z

B
C

Quintuple layer

d3 x (nA" nA# + nB" nB# )
A

B
C

:
parameter
to
give
AFM
U
ARTICLES
z

y

A

x

A B
t2

t1

t3

N : number of site

A
C

B

A

C

Quintuple layer
B

A

B
C

Bi(Fe)
Se1
Se2

B
A

C
z
Figure 1 | Crystal structure of Bi(Fe)2 Se3 . Crystal structureof Bi(Fe)2 Se3with three primitive lattice vectors denoted as t1,2,3 . A quintuple layer with00

V : volume
nA =
nB =

†
A
†
B

A
B

This corresponds to introducing Hubbard terms:

H = H0 + Hint

where

UV
Hint =NZ

d3 x (nA" nA# + nB" nB# )U : parameter to give AFM

Difficult to handle since
they’reA four-body int.
A

V : volume
N : number of site

B

C

A

B

Quintuple layer

B

Bi(Fe)
Se1
Se2

nA =
nB =

†
A
†
B

A
B

Hubbard-Stratonovich (HS) transformation

UV
Hint =NZ

d3 x (nA" nA# + nB" nB# )HS transformation

Hubbard-Stratonovich (HS) transformation

UV
Hint =NZ

d3 x (nA" nA# + nB" nB# )HS transformation• A dynamical scalar that gives5d5 ( d5 = )ARTICLES= ( x ⌦ sx , x+
1z ,"i,|P1+
z ,#i,t of aed byof symmetry
the
rsal
T
dversalup toand spatial
zation⌦ sy ) · K (with⇥2
dfrom
P =a z ⌦ I2⇥2 ,by formula (1),he parameters2 ,we consider aead (2)to deviation
ersal and parity
ng perturbations
one can show
rturbation must
they ),perturbed
cosk
5aa (k)0 witha=1
nddthem3 , M(k), m5 ).
⌦ tosx , x+ an explicit

1z ,#i,dk @kz dl

(3)

z

y

A

x

A B
t1

t2
t3

A
C

B

A

C

Quintuple layer
B

A

B
C

Bi(Fe)

h i ⇠ hSA i =

hSB i

Se1
Se2

B
A

C
z
Figure 1 | Crystal structure of Bi(Fe)2 Se3 . Crystal structureof Bi(Fe)2 Se3with three primitive lattice vectors denoted as t1,2,3 . A quintuple layer withy The
A orange rectangle.
Se1–Bi(Fe)1–Se2–Bi(Fe) 10 –Se10 is indicated in the
spin-ordering configuration giving rise to the 05 mass is indicated by thex
black arrow, which is antiferromagnetic along the z direction andBferromagnetic within the xy plane.

VEV ofis the order parameter of AFM•h i=0hi=
60•etry TqP
t2
spatial5at=Dynamical
phase
1 axion in the antiferromagneticC
a=1 da d .
(withon can be found We have seen above that t3the axion field ✓ can deviate from⌦,urI2⇥2independent
0 or ⇡ if the electrons are coupled to the antiferromagneticlanly(1),
m5 leads to order parameter. Such a staggered field
can be induced if anA
2netersthe ,following antiferromagnetic long-range order is established in the system.
generality,
and Physically, an antiferromagnetic ordered phase can be obtainediderameaning of the naturally if the screened Coulomb interactionof electrons
becomes
Bi(Fe)
viationB#i,|B, "i, |B, #i) strong. For example, in the Bi2 Te3 family doped with magnetic
parityimpurities such as Fe, the substitution of p electrons of Se1
Bi or Te byz , " (#)i) andbations
Ci). We see that d electrons of the magnetic impurities effectively enhanced the onshow(#)i stand site repulsion of electrons. In the basis (|A,"i,|A,#i,|B,"i,|B,#i)|B,"Se2
ell,must
for example, the Hamiltonian with interaction can be written as
turbedfted away fromA
X
X
ansformingwith 0 5 to
H = H0 + U(niA" niA# + niB" niB# ) + V
niA niBed
Zeeman
field
), m
5 ).
i
i
sublattice. Figure
This 1 | Crystal structure of Bi(Fe)2 Se3 . Crystal structure of Bi(Fe)2 Se3
xplicitntiferromagnetic
in primitive
which thelatticefirst termis thekinetic
by equation(2)
with threevectors
denotedasenergy
t1,2,3 . Agiven
quintuple
layer withz directions on and the other two
terms
represent the on-site repulsion U and00

PM orderAFM order
PM (paramagnetic)Hubbard-Stratonovich (HS) transformation

UV
Hint =NZ

d3 x (nA" nA# + nB" nB# )HS transformation• A dynamical scalar that gives5d5 ( d5 = )Hubbard-Stratonovich (HS) transformation

UV
Hint =NZ

d3 x (nA" nA# + nB" nB# )HS transformation• A dynamical scalar that gives• Mass term of5d5 ( d5 = )missed in

Sekine, Nomura ’16
Sekine, Nomura ’20
Schütte-Engel ’21(confirmed by private communication
with Sekine-san)This can be understood fromHS transformation⇠ Inverse of integrating out a scalar

HS transformation
Integrate out ϕnA" nA# , nB" nB#††( " " # #)Four Fermi int.

†
"

",

†
#

Yukawa int.

Thus the mass term should appear#

Partition function (TI + AFM)

Z=
S=

Z

Z

†

D D
4

†

D e

5

iS+iS massd x (x) [i@t H] (x)
Z
mass
4
2 2
S
=
d xM

H = H0 + H
2

M =

Z

d3 k 2(2⇡)3 U

Partition function (TI + AFM)

Z=
S=

Z

Z

†

D D
4

D e

†

d x (x) [i@t H] (x)
Z
mass
4
2 2
S
=
d xM
Integrate out

5

iS+iS mass,

H = H0 + H
2

M =

†

Effective action forZ

d3 k 2(2⇡)3 U

Effective potential for ϕ
KI ’21V =2Z

d3 k p
2+
(
|d
|
0
(2⇡)3

2

|d0 |) + M 2 2|d0 |2 =

M2 =

4
X

a=1

Z

|da |2d3 k 2(2⇡)3 U

Effective potential for ϕ
KI ’21V =2Z

d3 k p
2+
(
|d
|
0
(2⇡)3

2

|d0 |) + M 2 2

Negative potential
The mass term stabilizes the potential|d0 |2 =

M2 =

4
X

a=1

Z

|da |2d3 k 2(2⇡)3 U

KI ’21

4

2

A1 = A2 = 1

0

B1 = B2 = − 0.5
-2
-2

-1

0

1

2

KI ’214

2PM

Large U gives VEV of ϕ
A1 = A2 = 1

0

B1 = B2 = − 0.5

AFM
-2
-2

-1

0

1

2

Phase transition from PM to AFM* M 2 / 1/U

KI ’21

M dependence

9

4

4

4

2

2

2

0

0

0

-2

-2

-2

-2

-1

0

1

2

-2

-1

0

1

2

-2

4

4

4

2

2

2

0

0

0

-2

-2

-2

-2

-1

0

1

2

-2

-1

0

1

2

-2

-1

0

1

2

-1

0

1

2

FIG. 6. V as function of for various values of M in the e↵ective model for 3D topological insulators. A1 , A2 , B1 , B2 and M
are taken the same as Fig. 4.

KI ’21

M dependence

TI9

4

4

4

2

2

2

0

0

0

-2

-2

-2

-2

-1

0

1

2

-2

-1

0

1

2

-2

4

4

4

2

2

2

0

0

0

-2

-2

-2

-2

-1

0

1

2

-2

-1

0

1

2

-2

-1

0

1

2

-1

0

1

2

The difference between TI and NI is not clear

NI

FIG. 6. V as function of for various values of M in the e↵ective model for 3D topological insulators. A1 , A2 , B1 , B2 and M
are taken the same as Fig. 4.

Recall ME effect in TI is described by

L✓ =

↵
4⇡

Z

d4 x ✓Fµ⌫ F̃ µ⌫with✓ = ±⇡

Namely,• ✓ = ±⇡ for TI• ✓ = 0 for NI✓ is the order parameter ofphase transition between TI andNI

We need potential for ✓✓ can be computed from Hamiltonian:

•

1
✓=
4⇡

Z

4
2|d|
+
d
ijkl i
j
k
l
d3 k
✏
d
@
d
@
d
@
d
kx
ky
kz
4
2
3
(|d| + d ) |d|

R. Li et al. ’10• Approximately given by chiral anomaly (Fujikawa method)✓ can be computed from Hamiltonian:

•

1
✓=
4⇡

Z

4
2|d|
+
d
ijkl i
j
k
l
d3 k
✏
d
@
d
@
d
@
d
kx
ky
kz
4
2
3
(|d| + d ) |d|

R. Li et al. ’10• Approximately given by chiral anomaly (Fujikawa method)

Derivation as chiral anomaly
H(k) =

5Xda (k) aa=1(d1 , d2 , d3 , d4 , d5 ) = (A2 sin kx , A2 sin ky , A1 sin kz , M(k), )M(k) = M

2B1

4B2 + 2B1 cos kz + 2B2 (cos kx + cos ky )

Derivation as chiral anomaly
H(k) =

5Xda (k) aa=1(d1 , d2 , d3 , d4 , d5 ) = (A2 sin kx , A2 sin ky , A1 sin kz , M(k), )M(k) = M

2B1

4B2 + 2B1 cos kz + 2B2 (cos kx + cos ky )- expand around k = 0- redefine kH(k) = kx

1

+ ky

2

+ ky

“Dirac model”

3

+M

4

+

5

H(k) = kx

1

+ ky

2

+ ky

3

+M

4

+

5

Unitary transformation of the basis†

Ũ H(k)Ũ = ( · k + M +S=

Z

d4 x ¯[i µ (@µieAµ )

M
5

5)

i

5]

reduces to i

5

i

5

term can be rotated away, which gives rise to ✓ term:

S⇥ =

↵
4⇡

⇡
⇥ = [12Z

d4 x ⇥Fµ⌫ F̃ µ⌫sgn(M )]sgn( ) + tan

1

M

it is consistent with
1
✓=
4⇡

Z

4
2|d|
+
d
ijkl i
j
k
l
d3 k
✏
d
@
d
@
d
@
d
k
k
k
x
y
z
(|d| + d4 )2 |d|3

KI ’21
0.6

m5 =

0.4
0.2
0.0

- 0.2
- 0.4
- 0.6
- 1.0

- 0.5

0.0

0.5

1.0

M
calculation for Diracmodel is done by
Zhang ’19

KI ’21
0.6

m5 =

0.4
0.2

✓ = ±⇡( ! 0)

0.0

✓=0

- 0.2

( ! 0)

- 0.4
- 0.6
- 1.0

- 0.5

0.0

M

0.5

1.0

KI ’21
0.6

m5 =

0.4
0.2

✓ = ±⇡( ! 0)

0.0

✓=0

- 0.2

( ! 0)

- 0.4
- 0.6
- 1.0

- 0.5

0.0

0.5

1.0

M

=0
(T )

PM order✓ = ±⇡
✓=0

M <0
M >0

(TI)
(NI)

KI ’21
0.6

m5 =

0.4
0.2
0.0

- 0.2
- 0.4
- 0.6
- 1.0

- 0.5

0.0

0.5

1.0

M

6 0
=
(T )

AFM order✓ takes continuum variable

Effective potential in terms of ✓NI phase
4

2

0

-2

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Potential minimum:• ✓=0• ✓ 6= 0(small U , i.e., PM)(large U , i.e., AFM)

Effective potential in terms of ✓TI phase
4

2

0

-2

0.0

0.1

0.2

0.3

0.4

0.5

Potential minimum:• ✓=⇡• ✓ 6= 0(small U , i.e., PM)(large U , i.e., AFM)

axion• static• dynamical axion

Axion in condensed matter

TI phase420fluctuation around the VEV“dynamical axion”

-2

0.0

0.1

“static axion”

0.2

0.3

0.4

0.5

axion• static• dynamical axion

Axion in condensed matter

NI phase420“dynamical axion”

-2

0.00

0.02

0.04

0.06

“static axion”

0.08

0.10

0.12

Axion in condensed matter

4

4

2

2

0

0

-2

-2

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.0

0.1

0.2

0.3

Dynamical axion exits in both TI and NI phases

0.4

0.5

Axion in condensed matter“dynamical axion”=h i+“static axion”“dynamical axion” = amplitude mode of magnon

magnon: quantum of magnetization
Note: magnon can be dynamical axion
see Chigusa, Nakayama, Moroi ’21

Axion mass
6.5

AFM
5

6.0

4

3

5.5

2

PM

5.0

TI

NI

- 0.4

- 0.2

0.0

0.2

0.4

M

Axion mass is O(eV)1a static axion field ✓. However, the antiferromagnetic phase also has
amplitude and spin-wave excitations, which can induce fluctuationsof the axion field. The fluctuation of the Néel vector M can be generally written as M = (M0 + Mz (x,t ))ẑ+ Mx (x,t )x̂+ My (x,t )ŷ.
To the linear order, it can be shown from symmetry analysis that the fluctuation of the axion field depends only on
ARTICLES
Mz , because ✓ is a pseudo-scalar. In other words, we have
PUBLISHED ONLINE: 7 MARCH 2010 | DOI: 10.1038/NPHYS1534✓ (x,t ) = m5 (x,t )/g = (2/3)U Mz (x,t )/g , where the coefficient g can be determined from equation (3). The dispersion of the
Dynamical axion field in topologicalamplitude mode Mz (x,t ) can be obtained in the standard randommagnetic insulators
phase approximation, leading to a massive axion field ✓ (x, t ).
Considering the coupling term ✓E·B of the axion withµ⌫
the elecRundong Li1 , Jing Wang1,2 , Xiao-Liang Qi1 and Shou-Cheng Zhang1 *tromagnetic field, the effective action describing µ⌫
the axion–photon
ARTICLES
Axions are weakly interacting particles of low mass, and were postulated more than 30 years ago in the framework of theStandard Model of particle physics. Their existence could explain the missing dark matter of the Universe. coupled
However, despitesystem is given byt of aDynamical axion is predicted in topological magnetic insulators✓F F̃

intensive searches, axions have yet to be observed. Here we show that magnetic fluctuations of topological insulators couple
ed byto the electromagnetic fields exactly like the axions, and propose several experiments to detect this dynamical axion field. Inof theparticular, we show that the axion couplingAenables a nonlinear modulation of the electromagnetic field, leading to attenuatedd up tototal reflection. We propose a new optical-modulator device based on this principle.
totzationBrom ahe electromagnetic response of a three-dimensional insulator
refer to such an antiferromagnetic insulator as a ‘topologicalR 3
is described by Maxwell’s action SC0 =Quintuple(8/⇡) layerd xdt magnetic insulator’ (TMI). We propose a minimal model in which(✏E2 (1/µ)B2 ), with material-dependent dielectric con- the antiferromagnetic order breaks the time-reversal symmetry(2)stant ✏ and magnetic permeability µ, where E and B are the spontaneously and the magnetic fluctuations couple linearly to theAelectromagnetic fields inside the insulator. However,generally, axion field, thus realizing the dynamic axion field in condensedit is possible to include anothereffective matter systems. Compared with its high-energy version, theR 3 quadratic term in theaction S✓ = (✓ /2⇡)(↵/2⇡) d xdt E·B, whereB ↵ = e 2 /h̄c is the axion proposed here has the advantage that it can be observed
cosky ),fine-structure constant and ✓ is a parameter describing the insulator in controlled experimental settings5 . With an externally appliednd thein question. In the field theory literature, thisC effective action is magnetic field, the axion field couples linearly to light, resulting in⌦ sx , xknown as the axion electrodynamics1 , where ✓ has thethe axionic polariton. By measuring the attenuated total reflection
z role of the1+(ATR), the gap in the axionic polariton dispersion can be observed.
z ,#i,axion field. Under the periodic boundary condition, the partitionyfunction and all physical quantities are invariant
if ✓ is shifted
An attractive feature is that the axionic polariton gap is tunable byA
by integer multiples of 2⇡. Therefore, all time-reversal invariant changing the external electric or magnetic fields. The control of theinsulators fall into two distinct classes described by either ✓ =x 0 light transmission through the material enables a new type of opticaletry T or ✓ = ⇡ (ref. 2). Topological insulators areB defined by ✓ = ⇡ modulator. We also propose another experiment to detect thet2
spatialand can be connected
continuously
to trivial insulators, defined dynamic axion by microcantilever torque magnetometry, where thet1C(withby ✓ = 0, only by time-reversal-breaking
perturbations.
The form double-frequency response of the cantilever is a unique signature oft3⌦ I2⇥2 ,of the effective action implies that an electric field can induce the dynamic axion field.
la (1),a magnetic polarization, whereas a magnetic field can induce
We propose several materials that may realize the topologicalA
eters2 ,an electric polarization. This effect is known as the topological magnetic insulator with dynamic axion field. One possibility is theider amagneto-electric effect and ✓ is related to the magneto-electric topological insulators Bi2 Te3 , Bi2 Se3 and Sb2 Te3 doped with 3dBi(Fe)on
viationpolarization P3 by P3 = ✓ /2⇡. Physically the parameter✓ depends
transition metal elements such as Fe (refs 6, 7). Another possible
B
paritythe band structure of the insulator and has a microscopic expression class of material is the 5d transition metal compound Ax BOy with
Se1
ationsof the momentum-space Chern–Simons form2 C
B and A standing for a 5d transition metal and an alkali metal,
show
respectively. Electrons in the 5d orbital can have both strong

Z
1
2
Se2
must
spin–orbital coupling and strong interaction, which is ideal for3
ijk✓=
d k✏ Tr Ai @j Ak + i Ai Aj Ak(1)turbedthe realization of the topological magnetic insulator8 . We propose
4⇡
3 A
a
with
that such a compound with the corundum structure may have), m5 ).where A↵i (k) = ih↵k|@/@ki | ki is the momentum-space non- a topological magnetic insulator phase if the states close to the
Figure1 | Crystalof Bi(Fe)structureof Bi(Fe)xplicitabeliangaugefieldstructureand |↵ki
is the2 Seperiodicof theBloch
3 . Crystalpart
2 Se3 Fermi level are formed by t2g orbitals with total angular momentum
with three primitive
vectors
denotedas t1,2,3
quintuple tolayer
wavefunctionof the lattice↵th bandwithindices↵, . Areferring
thewith Jeff = 1/2 (ref. 9). We also noticed two very recent works on 5d
Se1–Bi(Fe)1–Se2–Bi(Fe)
10 –Se10 is indicatedthe orange rectangle.
occupiedbands. The ✓ parameterhas beenin calculated
explicitlyThefor transition metal compounds with the pyrochlore structure, which
2,3spin-ordering
configuration
giving riseinsulators
to the 05 massby the may also realize the topological magnetic insulator phase10,11 .
severalbasic models
of topological. Inis indicateda topological(3)
black arrow,z direction
andeffects
insulator,
the whichaxionis antiferromagnetic
field gives risealong
to thenewphysicalwithinthe xy plane.
suchferromagnetic
as the image
monopole
and anyonic statistics4 . This field, Effective model for the 3D topological insulator
however, is static in a time-reversal invariant topological insulator. Although all of the physical effects discussed in this article areaDynamical
axionthein theantiferromagnetic
phase order in generic for any system supporting axionic excitation and do not
we consider
antiferromagnetic
long-range
1 da d .Here,founda topological
We have seen
above whichthat theaxiontime-reversal
field ✓ can symmetry
deviate fromrely on a specific model, we would like to start from a simple
insulator,
breaksS

T= SMaxwell + Stopo + Saxion
✓
◆
Z
Z
1
1
↵
=
d3 xdt ✏E2
B2 + 2 d3 xdt (✓0 + ✓)E · B
8⇡
µ
4⇡
Z
+ g 2 J d3 xdt [(@t ✓ )2 (vi @i ✓ )2 m2 ✓ 2 ](4)

where J , vi and m are the stiffness, velocity and mass of thespin-wave mode Mz , E and B are the electric field and the magnetic
field respectively and ✏ and µ are the dielectric constant andmagnetic permeability respectively. The second term describes the
topological coupling between the axion and the electromagnetic
field, with ↵ ⌘ e 2 /h̄c being the fine-structure constant. The third
term describes the dynamics of the massive axion. Within the model
we have adopted, the parameters J and m are given by

Bi2 Se3

J=

Z

Axion mass ~ O(meV)

d3 k di (k)d i (k),(2⇡)3 16|d|5

✓

2
2
Jm =
UMz
3

◆2 Z

d3 k 1(2⇡)3 4|d|3

He
wh
an
ob
lin

wh
co
can
the

wi
sp
m
be
the
at
lin
the
po
the
eff
sam
the

a static axion field ✓. However, the antiferromagnetic phase also has
amplitude and spin-wave excitations, which can induce fluctuationsof the axion field. The fluctuation of the Néel vector M can be generally written as M = (M0 + Mz (x,t ))ẑ+ Mx (x,t )x̂+ My (x,t )ŷ.
To the linear order, it can be shown from symmetry analysis that the fluctuation of the axion field depends only on
ARTICLES
Mz , because ✓ is a pseudo-scalar. In other words, we have
PUBLISHED ONLINE: 7 MARCH 2010 | DOI: 10.1038/NPHYS1534✓ (x,t ) = m5 (x,t )/g = (2/3)U Mz (x,t )/g , where the coefficient g can be determined from equation (3). The dispersion of the
Dynamical axion field in topologicalamplitude mode Mz (x,t ) can be obtained in the standard randommagnetic insulators
phase approximation, leading to a massive axion field ✓ (x, t ).
Considering the coupling term ✓E·B of the axion withµ⌫
the elecRundong Li1 , Jing Wang1,2 , Xiao-Liang Qi1 and Shou-Cheng Zhang1 *tromagnetic field, the effective action describing µ⌫
the axion–photon
ARTICLES
Axions are weakly interacting particles of low mass, and were postulated more than 30 years ago in the framework of theStandard Model of particle physics. Their existence could explain the missing dark matter of the Universe. coupled
However, despitesystem is given byt of aDynamical axion is predicted in topological magnetic insulators✓F F̃

intensive searches, axions have yet to be observed. Here we show that magnetic fluctuations of topological insulators couple
ed byto the electromagnetic fields exactly like the axions, and propose several experiments to detect this dynamical axion field. Inof theparticular, we show that the axion couplingAenables a nonlinear modulation of the electromagnetic field, leading to attenuatedd up tototal reflection. We propose a new optical-modulator device based on this principle.
totzationBrom ahe electromagnetic response of a three-dimensional insulator
refer to such an antiferromagnetic insulator as a ‘topologicalR 3
is described by Maxwell’s action SC0 =Quintuple(8/⇡) layerd xdt magnetic insulator’ (TMI). We propose a minimal model in which(✏E2 (1/µ)B2 ), with material-dependent dielectric con- the antiferromagnetic order breaks the time-reversal symmetry(2)stant ✏ and magnetic permeability µ, where E and B are the spontaneously and the magnetic fluctuations couple linearly to theAelectromagnetic fields inside the insulator. However,generally, axion field, thus realizing the dynamic axion field in condensedit is possible to include anothereffective matter systems. Compared with its high-energy version, theR 3 quadratic term in theaction S✓ = (✓ /2⇡)(↵/2⇡) d xdt E·B, whereB ↵ = e 2 /h̄c is the axion proposed here has the advantage that it can be observed
cosky ),fine-structure constant and ✓ is a parameter describing the insulator in controlled experimental settings5 . With an externally appliednd thein question. In the field theory literature, thisC effective action is magnetic field, the axion field couples linearly to light, resulting in⌦ sx , xknown as the axion electrodynamics1 , where ✓ has thethe axionic polariton. By measuring the attenuated total reflection
z role of the1+(ATR), the gap in the axionic polariton dispersion can be observed.
z ,#i,axion field. Under the periodic boundary condition, the partitionyfunction and all physical quantities are invariant
if ✓ is shifted
An attractive feature is that the axionic polariton gap is tunable byA
by integer multiples of 2⇡. Therefore, all time-reversal invariant changing the external electric or magnetic fields. The control of theinsulators fall into two distinct classes described by either ✓ =x 0 light transmission through the material enables a new type of opticaletry T or ✓ = ⇡ (ref. 2). Topological insulators areB defined by ✓ = ⇡ modulator. We also propose another experiment to detect thet2
spatialand can be connected
continuously
to trivial insulators, defined dynamic axion by microcantilever torque magnetometry, where thet1C(withby ✓ = 0, only by time-reversal-breaking
perturbations.
The form double-frequency response of the cantilever is a unique signature oft3⌦ I2⇥2 ,of the effective action implies that an electric field can induce the dynamic axion field.
la (1),a magnetic polarization, whereas a magnetic field can induce
We propose several materials that may realize the topologicalA
eters2 ,an electric polarization. This effect is known as the topological magnetic insulator with dynamic axion field. One possibility is theider amagneto-electric effect and ✓ is related to the magneto-electric topological insulators Bi2 Te3 , Bi2 Se3 and Sb2 Te3 doped with 3dBi(Fe)on
viationpolarization P3 by P3 = ✓ /2⇡. Physically the parameter✓ depends
transition metal elements such as Fe (refs 6, 7). Another possible
B
paritythe band structure of the insulator and has a microscopic expression class of material is the 5d transition metal compound Ax BOy with
Se1
ationsof the momentum-space Chern–Simons form2 C
B and A standing for a 5d transition metal and an alkali metal,
show
respectively. Electrons in the 5d orbital can have both strong

Z
1
2
Se2
must
spin–orbital coupling and strong interaction, which is ideal for3
ijk✓=
d k✏ Tr Ai @j Ak + i Ai Aj Ak(1)turbedthe realization of the topological magnetic insulator8 . We propose
4⇡
3 A
a
with
that such a compound with the corundum structure may have), m5 ).where A↵i (k) = ih↵k|@/@ki | ki is the momentum-space non- a topological magnetic insulator phase if the states close to the
Figure1 | Crystalof Bi(Fe)structureof Bi(Fe)xplicitabeliangaugefieldstructureand |↵ki
is the2 Seperiodicof theBloch
3 . Crystalpart
2 Se3 Fermi level are formed by t2g orbitals with total angular momentum
with three primitive
vectors
denotedas t1,2,3
quintuple tolayer
wavefunctionof the lattice↵th bandwithindices↵, . Areferring
thewith Jeff = 1/2 (ref. 9). We also noticed two very recent works on 5d
Se1–Bi(Fe)1–Se2–Bi(Fe)
10 –Se10 is indicatedthe orange rectangle.
occupiedbands. The ✓ parameterhas beenin calculated
explicitlyThefor transition metal compounds with the pyrochlore structure, which
2,3spin-ordering
configuration
giving riseinsulators
to the 05 massby the may also realize the topological magnetic insulator phase10,11 .
severalbasic models
of topological. Inis indicateda topological(3)
black arrow,z direction
andeffects
insulator,
the whichaxionis antiferromagnetic
field gives risealong
to thenewphysicalwithinthe xy plane.
suchferromagnetic
as the image
monopole
and anyonic statistics4 . This field, Effective model for the 3D topological insulator
however, is static in a time-reversal invariant topological insulator. Although all of the physical effects discussed in this article areaDynamical
axionthein theantiferromagnetic
phase order in generic for any system supporting axionic excitation and do not
we consider
antiferromagnetic
long-range
1 da d .Here,founda topological
We have seen
above whichthat theaxiontime-reversal
field ✓ can symmetry
deviate fromrely on a specific model, we would like to start from a simple
insulator,
breaksS

T= SMaxwell + Stopo + Saxion
✓
◆
Z
Z
1
1
↵
=
d3 xdt ✏E2
B2 + 2 d3 xdt (✓0 + ✓)E · B
8⇡
µ
4⇡
Z
+ g 2 J d3 xdt [(@t ✓ )2 (vi @i ✓ )2 m2 ✓ 2 ](4)

where J , vi and m are the stiffness, velocity and mass of thespin-wave mode Mz , E and B are the electric field and the magnetic
field respectively and ✏ and µ are the dielectric constant andmagnetic permeability respectively. The second term describes the
topological coupling between the axion and the electromagnetic
field, with ↵ ⌘ e 2 /h̄c being the fine-structure constant. The third
term describes the dynamics of the massive axion. Within the model
we have adopted, the parameters J and m are given by

Bi2 Se3

J=

Z

Axion mass ~ O(meV)?

d3 k di (k)d i (k),(2⇡)3 16|d|5

✓

2
2
Jm =
UMz
3

◆2 Z

d3 k 1(2⇡)3 4|d|3

He
wh
an
ob
lin

wh
co
can
the

wi
sp
m
be
the
at
lin
the
po
the
eff
sam
the

The thing is …

•

h i (= m5 ) = 1 meV is taken(i.e., h i is considered to be a free parameter)• AFM order is assumed

R. Li et al. ’10

The thing is …

•

R. Li et al. ’10

h i (= m5 ) = 1 meV is taken(i.e., h i is considered to be a free parameter)
Axion mass ~ O(meV)• AFM order is assumed(*

22
ma / m5 )The thing is …

•

R. Li et al. ’10

h i (= m5 ) = 1 meV is taken(i.e., h i is considered to be a free parameter)
Axion mass ~ O(meV)(*

But this is not naively possible since22
ma / m5 )m ⇠ U ⇠ eV (in AFM order)• AFM order is assumed
5

Axion mass
6.5

AFM
5

6.0

4

3

5.5

2

PM

5.0

TI
- 0.4

NI
- 0.2

0.0

0.2

0.4

M

Suppressed U

No AFM

1

The thing is …

•

R. Li et al. ’10

h i (= m5 ) = 1 meV is taken(i.e., h i is considered to be a free parameter)
Axion mass ~ O(meV)(*22ma / m5 )But this is not naively possible sincem ⇠ U ⇠ eV (in AFM order)• AFM order is assumed
5

They may have misunderstood theamplitude mode as magnon?

The thing is …

•

R. Li et al. ’10

h i (= m5 ) = 1 meV is taken(i.e., h i is considered to be a free parameter)
Axion mass ~ O(meV)• AFM order is assumed(*

22
ma / m5 )The thing is …

•

R. Li et al. ’10

h i (= m5 ) = 1 meV is taken(i.e., h i is considered to be a free parameter)
Axion mass ~ O(meV)(*• AFM order is assumed
No AFM in TI in the first place

Fe -doped Bi2 Te3 is proposed22
ma / m5 )• Fe-doped Bi Te
23“likely to be AFM”
(by first-principles calculation)

J.M. Zhang et al. ’13

it looks unlikely to be realized …

•

Mn2 Bi2 Te5“rich magnetic topological quantum states”(by first-principles calculation)

J. Zhang et al. ’19
Y. Li et al. ’20

)
=
v
(k
σ
−
k
σ
)
⊗
τ
+
(k
σ
−
k
σ
)
⊗
τ
+
m(k)1
(k
,
k
)
=
v
(k
−
k
σ
)
H
(k
,
k
)
=
v
(k
σ
−
y
F
y
x
x
y
z
F
y
x
x
y
z
film
x
y
F
y
x
x
y
film
x
y
F
y
x
σx − kx σy ) ⊗ τz + m(k)1 ⊗ τx
Y. Li et al. ’20

g
σ
⊗
τ
+
g
σ
⊗
1
+z ⊗
gaτσzz+
⊗
τ
+
g
σ
⊗
1,
+
g
σ
⊗
τ
+
g
+
g
σ
⊗
a
z
z
f
z
f az zaz z+ g fzσz ⊗ 1,
FIG. 4. 2NL M2 BT film. (a)–(e) Band structures with AFM, FM,
and interstate magnetic configurations (denoted as I, II, and III).
The relative total energies are shown, where the total energy of thereference AFM state is set to be zero. The band gap at the ! points

Axion mass
6.5

AFM
5

6.0

4

3

5.5

2

PM

5.0

TI
- 0.4

NI
- 0.2

0.0

M

0.2

0.4

1

Axion mass
6.5

AFM
5

6.0

4

3

5.5

2

PM

5.0

TI
- 0.4

NI
- 0.2

0.0

0.2

1

0.4

M

It can be suppressednear the phase boundary

Rich magnetic topologicalstates in that region?

4. Conclusions and discussion

We have static and dynamical axions in AFM TI consistently
by using path integral

h i is obtained from the effective potential,• Nonzero
which gives rise to AFM order and breaks T• Dynamical axion appears both in TI and NI
.
O(eV)
Axion
mass
is
•

Discussion• How do we describe axion in Mn Bi Te ?
• What about axion in NI ?
axion in ferromagnetic state or other• Dynamicalmagnetic states?
225