download.uni-mainz.de€¦ · E k 3D (massive) Dirac fermions in crystals = spin = orbital (parity) Equations for PPT slides Ion Garate D´epartement de Physique, Universit´e de

Interplay between electronic topology and phonons in Dirac materials

Ion GarateKush Saha (postdoc à UC Irvine )Katherine Légaré (undergrad intern)

Prelude: Dirac materials

A recent review:

Ando, JPSJ 82, 102001 (2013).

E

k

3D (massive) Dirac fermions in crystals

= spin

= orbital (parity)

Equations for PPT slides

Ion Garate

Departement de Physique, Universite de Sherbrooke, Sherbrooke, Quebec, Canada J1K 2R1

(Dated: February 26, 2015)

PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|

�Ekn

'X

qn

0

hkn|hep|k� qn0i|2

Ekn

� Ek�qn

0(2)

⇧(q, !) =

1

V

X

k

X

nn

0

|hkn|Ue�ph|k� qn0i|2 fkn

� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(3)

h0ep(r) = u0(r) 12 ⌦ 12 (4)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(5)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(6)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(7)


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧

�Ekn

'X

qn

0

hkn|hep|k� qn0i|2

Ekn

� Ek�qn

0(2)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(3)

h0ep(r) = u0(r) 12 ⌦ 12 (4)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(5)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(6)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(7)


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧

�Ekn

'X

qn

0

hkn|hep|k� qn0i|2

Ekn

� Ek�qn

0(2)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(3)

h0ep(r) = u0(r) 12 ⌦ 12 (4)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(5)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(6)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(7)


Ion Garate


(Dated: March 18, 2015)

PACS numbers:

m(k) = m + tk2

h(k)|kni = Ekn

|kniEk = ±pv2k2

+ m2

hep / 12 ⌦ ⌧x

hep / 12 ⌦ ⌧z

hep / 12 ⌦ 12

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

H = vk · �⌧x

+ m ⌧z

(2)

|m(0)|�⌧�

x

/�z

q > !0/vF

2|m(0)| > !0

⇥ = i�yK (3)

⇧ = ⌧z

(4)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (5)

�x

= gz

D(✏F

)|h⌧xi|2 (6)

��

= g�

D(✏F

)|hh�

epi|2 (7)

�q�

' 2⇡X

k

X

nn

0

|hkn|hep|k� qn0i|2(fkn

� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (8)

!⇤q�

' !q�

+

X

k

X

nn

0

|hkn|hep|k� qn0i|2 fkn

� fk�qn

0

Ekn

� Ek�qn

0 � !q(9)


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧�

x

/�z

q > !0/vF

2|m(0)| > !0

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (4)

�x

= gz

D(✏F

)|h⌧xi|2 (5)

��

= g�

D(✏F

)|hh�

epi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0

|hkn|hep|k� qn0i|2

Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)

Time-reversal and inversion symmetry

in k-space

Trivial insulator

E

k

Topological insulator

E

k


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zi

�Ekn

'X

qn

0

hkn|hep|k� qn0i|2

Ekn

� Ek�qn

0(4)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(5)

h0ep(r) = u0(r) 12 ⌦ 12 (6)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(7)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(8)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(9)


Ion Garate



PACS numbers:

mt > 0

mt < 0

m(k) = m + tk2

h(k)|kni = Ekn

|kniEk = ±pv2k2

+ m2

hep / 12 ⌦ ⌧x

hep / 12 ⌦ ⌧z

hep / 12 ⌦ 12

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

H = vk · �⌧x

+ m ⌧z

(2)

|m(0)|�⌧�

x

/�z

q > !0/vF

2|m(0)| > !0

⇥ = i�yK (3)

⇧ = ⌧z

(4)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (5)

�x

= gz

D(✏F

)|h⌧xi|2 (6)

��

= g�

D(✏F

)|hh�

epi|2 (7)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (8)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(9)


Ion Garate



PACS numbers:

mt > 0

mt < 0

m(k) = m + tk2

h(k)|kni = Ekn

|kniEk = ±pv2k2

+ m2

hep / 12 ⌦ ⌧x

hep / 12 ⌦ ⌧z

hep / 12 ⌦ 12

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

H = vk · �⌧x

+ m ⌧z

(2)

|m(0)|�⌧�

x

/�z

q > !0/vF

2|m(0)| > !0

⇥ = i�yK (3)

⇧ = ⌧z

(4)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (5)

�x

= gz

D(✏F

)|h⌧xi|2 (6)

��

= g�

D(✏F

)|hh�

epi|2 (7)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (8)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(9)

Part I:Effect of phonons on band topology

I. Garate, PRL 110, 046402 (2013).

K. Saha and I. Garate, PRB 89, 205103 (2014).

Electron-phonon interaction


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zi

�z

/ D(✏F

)|h⌧zi|2 (4)

�x

/ D(✏F

)|h⌧xi|2 ' D(✏F

)

h1� |h⌧zi|2

i(5)

� / D(✏F

)|hhepi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(12)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(13)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(14)


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zi

�z

/ D(✏F

)|h⌧zi|2 (4)

�x

/ D(✏F

)|h⌧xi|2 ' D(✏F

)

h1� |h⌧zi|2

i(5)

� / D(✏F

)|hhepi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(12)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(13)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(14)

electron

electron

phonon


Ion Garate


(Dated: April 21, 2015)

PACS numbers:

�0

⌧ 0

|h0n|qn0i|2mt > 0

mt < 0

m(k) = m + tk2

h(k)|kni = Ekn

|kniEk = ±pv2k2

+ m2

hep / 12 ⌦ ⌧x

hep / 12 ⌦ ⌧z

hep / 12 ⌦ 12

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

H = vk · �⌧x

+ m ⌧z

(2)

|m(0)|�⌧�

x

/�z

q > !0/vF

2|m(0)| > !0

⇥ = i�yK (3)

⇧ = ⌧z

(4)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (5)

�x

= gz

D(✏F

)|h⌧xi|2 (6)

��

= g�

D(✏F

)|hh�

epi|2 (7)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (8)


Ion Garate



PACS numbers:

�0

⌧ 0

|h0n|qn0i|2mt > 0

mt < 0

m(k) = m + tk2

h(k)|kni = Ekn

|kniEk = ±pv2k2

+ m2

hep / 12 ⌦ ⌧x

hep / 12 ⌦ ⌧z

hep / 12 ⌦ 12

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

H = vk · �⌧x

+ m ⌧z

(2)

|m(0)|�⌧�

x

/�z

q > !0/vF

2|m(0)| > !0

⇥ = i�yK (3)

⇧ = ⌧z

(4)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (5)

�x

= gz

D(✏F

)|h⌧xi|2 (6)

��

= g�

D(✏F

)|hh�

epi|2 (7)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (8)

2

0.2 0.4 0.6 0.8EF (eV)

0

0.5

1

1.5|<τz>ia|

2

|<τx>ia|2

0.2 0.4 0.6 0.8EF(eV)

0

0.5

1

1.5|<τz>ia|

2

|<τx>ia|2

Topological Trivial

c) d)

k k

Topological Triviala) b)

FIG. 1: (Color online) (a) and (b) Expectation value of theelectronic parity operator, ⟨τ z⟩ (represented by arrows), as afunction of momentum. (c) and (d) Fermi surface averages of|⟨τ z⟩|2 and |⟨τx⟩|2 (cf. Eq. (7)), as a function of the Fermienergy, for m = −0.2 eV (c) and m = 0.2 eV (d). The rest ofthe band parameters are the same as those in Ref. [3].

number for the state |ukn⟩ with a Fermi energy ϵF . Also,

gλnn′(k,q) = ⟨ukn|gλ(q)|uk−qn′⟩, (3)

where gλ(q) = gλ(−q)† is the electron-phonon vertexoperator in the low-energy electronic subspace [6, 7].

In a centrosymmetric crystal, lattice vibrations are ei-ther even or odd under spatial inversion. Each of thesemodes couples to electrons and can, as we shall see, in-herit signatures of the underlying band topology. For themodel of Eq. (1) and for q ≃ 0 optical phonons, inversionand time-reversal symmetries dictate [6]

geven(q) ≃ g0(q) + gz(q)τz

godd(q) ≃ gx(q)τx + g′(q) · στy, (4)

where “even” (“odd”) denotes the coupling of elec-trons to parity-even (parity-odd) phonon modes, with[geven, τz] = 0 and {godd, τz} = 0. Inversion symmetryguarantees that geven and godd will not be mixed in a sin-gle phonon mode. Also, q = q/q, and the coefficients gi(i = 0, x, z) and g′i (i = x, y, z) can be obtained from theatomic displacements in the particular phonon mode [6].Physically, g0 and gz lead to phonon-induced modula-tions of the chemical potential and the Dirac mass, re-spectively. Next, we identify ways in which gλ can trans-

fer the information about electronic band topology to thephonon sector.

Intraband phonon damping.– The main electronicmechanism contributing to phonon linewidths is the scat-tering of phonons off electron-hole pairs. The rate ofthis process is γλ(q) ≡ −ImΠλ(q,ωqλ). In this work,we focus on long-wavelength optical phonons and on lowtemperatures.

We begin by considering the commonly realized case inwhich the phonon frequency is smaller than the bandgapof the insulator. In this case, the “insulator” must bedoped in order for carriers to absorb phonons and in-duce a linewidth γλia. The subscript “ia” is shorthand for“intraband” and makes it explicit that phonons decayinto particle-hole pairs in the vicinity of the Fermi sur-face. Assuming that the distance from the Fermi levelto the bulk band edge is large compared to the phononfrequency, we have

γλia(q ≃ 0) ≃ πηD(ϵF )|gλia(kF , q)|2δ(vF · q− η), (5)

where D(ϵF ) is the electronic density of states at theFermi level, kF is the Fermi wave vector, vF = vF vF

is the Fermi velocity, η ≡ ω0λ/(qvF ), δ(x) is the Diracdelta, and |gλia(k, q)|2 denotes the sum of |gλnn′ |2 overthe two degenerate bands at momentum k and energyEk (hence the label “intraband”). In addition, O =!

k Oδ(Ek − ϵF )/(VD(ϵF )).Equation (5) contains information about the electronic

band topology. The simplest way to see this is to imag-ine a parity-even phonon mode and a parity-odd phononmode that couple to electrons purely through gz ≡ gzτz

and gx ≡ gxτx, respectively. More general couplings withg0 = 0 and g′i = 0 will be discussed below. From Eqs. (1),(3) and (5), the linewidths of these two phonon modesare

γjia(q ≃ 0) ≃ |gj(q)|2D(ϵF )|⟨τ j⟩ia|2πη

4Θ(1− η), (6)

where j ∈ {x, z}, Θ(x) is the Heaviside function and

|⟨τz⟩ia|2 = 1− |⟨τx⟩ia|2 = M2kF

/(α2k2F +M2kF

). (7)

Note that |⟨τ j⟩ia|2 ∈ [0, 1] (cf. Fig. 1). In particular,when MkF

= 0, |⟨τz⟩ia|2 = 0 and |⟨τx⟩ia|2 = 1. Com-bining Eqs. (6) and (7) with Fig. 1, it follows that γjiareflects the orbital texture, and therefore the topology,of the bulk bands. In order clarify this point, we elimi-nate the non-topological features coming from D(ϵF ) byconsidering the ratio γxia/γ

zia ≃ (g2x/g

2z)|⟨τx⟩ia|2/|⟨τz⟩ia|2.

For fixed bandgap, Eq. (6) predicts a strong maximumfor γxia/γ

zia as a function of ϵF in the topological phase

(but not in the trivial phase) becauseMkFcrosses zero as

a function of ϵF in the topological phase (but not in thetrivial phase). This difference in behavior between thetrivial and topological phases is significant for a sizeable|m|, but becomes gradually weaker as the energy gap

2

0.2 0.4 0.6 0.8EF (eV)

0

0.5

1

1.5|<τz>ia|

2

|<τx>ia|2

0.2 0.4 0.6 0.8EF(eV)

0

0.5

1

1.5|<τz>ia|

2

|<τx>ia|2

Topological Trivial

c) d)

k k




















4Θ(1− η), (6)


|⟨τz⟩ia|2 = 1− |⟨τx⟩ia|2 = M2kF

/(α2k2F +M2kF

). (7)



zia ≃ (g2x/g

2z)|⟨τx⟩ia|2/|⟨τz⟩ia|2.





2

0.2 0.4 0.6 0.8EF (eV)

0

0.5

1

1.5|<τz>ia|

2

|<τx>ia|2

0.2 0.4 0.6 0.8EF(eV)

0

0.5

1

1.5|<τz>ia|

2

|<τx>ia|2

Topological Trivial

c) d)

k k





where gλ(q) = gλ(−q)† is the electron-phonon vertexoperator in the low-energy electronic subspace [6, 7].In a centrosymmetric crystal, lattice vibrations are ei-

ther even or odd under spatial inversion. Each of thesemodes couples to electrons and can, as we shall see, in-herit signatures of the underlying band topology. For themodel of Eq. (1) and for q ≃ 0 optical phonons, inversionand time-reversal symmetries dictate [6]




fer the information about electronic band topology to thephonon sector.Intraband phonon damping.– The main electronic

mechanism contributing to phonon linewidths is the scat-tering of phonons off electron-hole pairs. The rate ofthis process is γλ(q) ≡ −ImΠλ(q,ωqλ). In this work,we focus on long-wavelength optical phonons and on lowtemperatures.We begin by considering the commonly realized case in

which the phonon frequency is smaller than the bandgapof the insulator. In this case, the “insulator” must bedoped in order for carriers to absorb phonons and in-duce a linewidth γλia. The subscript “ia” is shorthand for“intraband” and makes it explicit that phonons decayinto particle-hole pairs in the vicinity of the Fermi sur-face. Assuming that the distance from the Fermi levelto the bulk band edge is large compared to the phononfrequency, we have








4Θ(1− η), (6)


|⟨τz⟩ia|2 = 1− |⟨τx⟩ia|2 = M2kF

/(α2k2F +M2kF

). (7)



zia ≃ (g2x/g

2z)|⟨τx⟩ia|2/|⟨τz⟩ia|2.





2

0.2 0.4 0.6 0.8EF (eV)

0

0.5

1

1.5|<τz>ia|

2

|<τx>ia|2

0.2 0.4 0.6 0.8EF(eV)

0

0.5

1

1.5|<τz>ia|

2

|<τx>ia|2

Topological Trivial

c) d)

k k




















4Θ(1− η), (6)


|⟨τz⟩ia|2 = 1− |⟨τx⟩ia|2 = M2kF

/(α2k2F +M2kF

). (7)



zia ≃ (g2x/g

2z)|⟨τx⟩ia|2/|⟨τz⟩ia|2.






Electron-phonon coupling Phonon parity

Even

Even

Odd

Odd

Long wavelength + local-in-space

2

h⌧zihz

ep

/ 12

⌦ ⌧z

hx

ep

/ 12

⌦ ⌧x

��

= g�

Djoint

(!0

)|hck|hep

|vki|2 (10)

�x

= gz

D(✏F

)|h⌧xi|2 (11)

�z

= gz

D(✏F

)|h⌧zi|2 (12)

�x

= gz

D(✏F

)|h⌧xi|2 (13)

��

= g�

D(✏F

)|hh�

ep

i|2 (14)

�q�

' 2⇡X

k

X

nn

0

|hkn|hep

|k� qn0i|2(fkn

� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (15)

!⇤q�

' !q�

+

X

k

X

nn

0

|hkn|hep

|k� qn0i|2 fkn

� fk�qn

0

Ekn

� Ek�qn

0 � !q(16)

E⇤kn

' Ekn

+

X

qn

0

|hkn|hep

|k� qn0i|2Ekn

� Ek�qn

0(17)

⇧(q, !) =

1

V

X

k

X

nn

0

|hkn|Ue�ph

|k� qn0i|2 fkn

� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+

(18)

h0

ep

(r) = u0

(r) 12

⌦ 12

(19)

hx

ep

(r) = ux

(r) 12

⌦ ⌧x

(20)

hz

ep

(r) = uz

(r) 12

⌦ ⌧z

(21)

hiy

ep

(r) = uiy

(r) �i ⌦ ⌧y

(22)

2

h⌧zihz

ep

/ 12

⌦ ⌧z

hx

ep

/ 12

⌦ ⌧x

��

= g�

Djoint

(!0

)|hck|hep

|vki|2 (10)

�x

= gz

D(✏F

)|h⌧xi|2 (11)

�z

= gz

D(✏F

)|h⌧zi|2 (12)

�x

= gz

D(✏F

)|h⌧xi|2 (13)

��

= g�

D(✏F

)|hh�

ep

i|2 (14)

�q�

' 2⇡X

k

X

nn

0

|hkn|hep

|k� qn0i|2(fkn

� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (15)

!⇤q�

' !q�

+

X

k

X

nn

0

|hkn|hep

|k� qn0i|2 fkn

� fk�qn

0

Ekn

� Ek�qn

0 � !q(16)

E⇤kn

' Ekn

+

X

qn

0

|hkn|hep

|k� qn0i|2Ekn

� Ek�qn

0(17)

⇧(q, !) =

1

V

X

k

X

nn

0

|hkn|Ue�ph

|k� qn0i|2 fkn

� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+

(18)

h0

ep

(r) = u0

(r) 12

⌦ 12

(19)

hx

ep

(r) = ux

(r) 12

⌦ ⌧x

(20)

hz

ep

(r) = uz

(r) 12

⌦ ⌧z

(21)

hiy

ep

(r) = uiy

(r) �i ⌦ ⌧y

(22)

2

h⌧zihz

ep

/ 12

⌦ ⌧z

hx

ep

/ 12

⌦ ⌧x

��

= g�

Djoint

(!0

)|hck|hep

|vki|2 (10)

�x

= gz

D(✏F

)|h⌧xi|2 (11)

�z

= gz

D(✏F

)|h⌧zi|2 (12)

�x

= gz

D(✏F

)|h⌧xi|2 (13)

��

= g�

D(✏F

)|hh�

ep

i|2 (14)

�q�

' 2⇡X

k

X

nn

0

|hkn|hep

|k� qn0i|2(fkn

� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (15)

!⇤q�

' !q�

+

X

k

X

nn

0

|hkn|hep

|k� qn0i|2 fkn

� fk�qn

0

Ekn

� Ek�qn

0 � !q(16)

E⇤kn

' Ekn

+

X

qn

0

|hkn|hep

|k� qn0i|2Ekn

� Ek�qn

0(17)

⇧(q, !) =

1

V

X

k

X

nn

0

|hkn|Ue�ph

|k� qn0i|2 fkn

� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+

(18)

h0

ep

(r) = u0

(r) 12

⌦ 12

(19)

hx

ep

(r) = ux

(r) 12

⌦ ⌧x

(20)

hz

ep

(r) = uz

(r) 12

⌦ ⌧z

(21)

hiy

ep

(r) = uiy

(r) �i ⌦ ⌧y

(22)

2

h⌧zihz

ep

/ 12

⌦ ⌧z

hx

ep

/ 12

⌦ ⌧x

��

= g�

Djoint

(!0

)|hck|hep

|vki|2 (10)

�x

= gz

D(✏F

)|h⌧xi|2 (11)

�z

= gz

D(✏F

)|h⌧zi|2 (12)

�x

= gz

D(✏F

)|h⌧xi|2 (13)

��

= g�

D(✏F

)|hh�

ep

i|2 (14)

�q�

' 2⇡X

k

X

nn

0

|hkn|hep

|k� qn0i|2(fkn

� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (15)

!⇤q�

' !q�

+

X

k

X

nn

0

|hkn|hep

|k� qn0i|2 fkn

� fk�qn

0

Ekn

� Ek�qn

0 � !q(16)

E⇤kn

' Ekn

+

X

qn

0

|hkn|hep

|k� qn0i|2Ekn

� Ek�qn

0(17)

⇧(q, !) =

1

V

X

k

X

nn

0

|hkn|Ue�ph

|k� qn0i|2 fkn

� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+

(18)

h0

ep

(r) = u0

(r) 12

⌦ 12

(19)

hx

ep

(r) = ux

(r) 12

⌦ ⌧x

(20)

hz

ep

(r) = uz

(r) 12

⌦ ⌧z

(21)

hiy

ep

(r) = uiy

(r) �i ⌦ ⌧y

(22)

Effect of phonons on energy gap

Intraband transitions

= intraband + interband

Typing Equations for Slides

Ion Garate

May 7, 2014

�E0n =

Xn0q

g2q|hn0|n0qi|2E0n � Eqn0

(1)

E⇤g 6= 2m⇤

Eg = 2mAn(R) = ihn,R|rR|n,Ri� = 0

� = 1

d|Eg|/dT < 0

d|Eg|/dT > 0

0nkn0

�so < 1mK

nK = nK0

nK = �nK0

nChern = nK + nK0= sgn(mH)

|+i|�inK =

12 sgn(vx

KvyKmK)

dz(q) = mS dz(q) = mH⌧z

|dz(0)|h(q) = v ⌧z�x qx + v �yqy + dz(q)�z

h(d) = d · �|±iE± = ±|d|!c = eB/(mc)c py/(eB)

�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS

Fn(R) · dS = nChern ⇥ e (2)

An(R) = ihun(R)|rR|un(R)ih(k, t) = h(k, t + ⌧)

1

E

k

E

k

Interband transitions

Electron-phonon matrix elements

Trivial insulator

E

k

Arrows = orbital pseudospin q

Interband

Intraband

1

0


Ion Garate



PACS numbers:

|h0n|qn0i|2mt > 0

mt < 0

m(k) = m + tk2

h(k)|kni = Ekn

|kniEk = ±pv2k2

+ m2

hep / 12 ⌦ ⌧x

hep / 12 ⌦ ⌧z

hep / 12 ⌦ 12

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

H = vk · �⌧x

+ m ⌧z

(2)

|m(0)|�⌧�

x

/�z

q > !0/vF

2|m(0)| > !0

⇥ = i�yK (3)

⇧ = ⌧z

(4)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (5)

�x

= gz

D(✏F

)|h⌧xi|2 (6)

��

= g�

D(✏F

)|hh�

epi|2 (7)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (8)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(9)

Electron-phonon matrix elements


E

k

Arrows = orbital pseudospin

Intraband

Interband

q

1

0


Ion Garate



PACS numbers:

|h0n|qn0i|2mt > 0

mt < 0

m(k) = m + tk2

h(k)|kni = Ekn

|kniEk = ±pv2k2

+ m2

hep / 12 ⌦ ⌧x

hep / 12 ⌦ ⌧z

hep / 12 ⌦ 12

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

H = vk · �⌧x

+ m ⌧z

(2)

|m(0)|�⌧�

x

/�z

q > !0/vF

2|m(0)| > !0

⇥ = i�yK (3)

⇧ = ⌧z

(4)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (5)

�x

= gz

D(✏F

)|h⌧xi|2 (6)

��

= g�

D(✏F

)|hh�

epi|2 (7)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (8)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(9)

E

k

E

k

Stronger electron-phonon coupling

m

E

km

Topologically trivial Critical point Topologically nontrivial

Higher temperature

Thermal topological transition (crossover).

Phonon-induced topological insulation

Part II:Back action of band topology on phonons

K. Saha, K. Légaré and I. Garate

arXiv: 1506.02621

Phonon linewidth

E

k


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧�

x

/�z

q > !0/vF

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (4)

�x

= gz

D(✏F

)|h⌧xi|2 (5)

��

= g�

D(✏F

)|hh�

epi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧�

x

/�z

q > !0/vF

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (4)

�x

= gz

D(✏F

)|h⌧xi|2 (5)

��

= g�

D(✏F

)|hh�

epi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧�

x

/�z

q > !0/vF

2|m(0)| > !0

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (4)

�x

= gz

D(✏F

)|h⌧xi|2 (5)

��

= g�

D(✏F

)|hh�

epi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)

EF

Case I: Phonon frequency < Bandgap

Electronic DOS at Fermi levelElectron-phonon coupling


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zi

�z

= gz

D(✏F

)|h⌧zi|2 (4)

�x

= gz

D(✏F

)|h⌧xi|2 (5)

��

= g�

D(✏F

)|hhepi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(12)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(13)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(14)

Raman-active phonon IR-active phonon


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zi

�z

= gz

D(✏F

)|h⌧zi|2 (4)

�x

= gz

D(✏F

)|h⌧xi|2 (5)

��

= g�

D(✏F

)|hhepi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(12)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(13)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(14)


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zi

�z

= gz

D(✏F

)|h⌧zi|2 (4)

�x

= gz

D(✏F

)|h⌧xi|2 (5)

��

= g�

D(✏F

)|hhepi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(12)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(13)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(14)


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (4)

�x

= gz

D(✏F

)|h⌧xi|2 (5)

��

= g�

D(✏F

)|hh�

epi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(12)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(13)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(14)


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (4)

�x

= gz

D(✏F

)|h⌧xi|2 (5)

��

= g�

D(✏F

)|hh�

epi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(12)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(13)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(14)

Small (but nonzero) q

Fermi-surface average

Matrix elements (I): topological phase


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zi

�z

/ D(✏F

)|h⌧zi|2 (4)

�x

/ D(✏F

)|h⌧xi|2 ' D(✏F

)

h1� |h⌧zi|2

i(5)

� / D(✏F

)|hhepi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(12)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(13)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(14)

Bulk band edge

1

Bulk conduction band


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zi

�z

/ D(✏F

)|h⌧zi|2 (4)

�x

/ D(✏F

)|h⌧xi|2 ' D(✏F

)

h1� |h⌧zi|2

i(5)

� / D(✏F

)|hhepi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(12)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(13)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(14)

Bulk band edge

1

EF EF

Matrix elements (II): trivial phase


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zi

�z

/ D(✏F

)|h⌧zi|2 (4)

�x

/ D(✏F

)|h⌧xi|2 ' D(✏F

)

h1� |h⌧zi|2

i(5)

� / D(✏F

)|hhepi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(12)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(13)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(14)

Bulk band edge

1

Bulk conduction band


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zi

�z

/ D(✏F

)|h⌧zi|2 (4)

�x

/ D(✏F

)|h⌧xi|2 ' D(✏F

)

h1� |h⌧zi|2

i(5)

� / D(✏F

)|hhepi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(12)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(13)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(14)

Bulk band edge

1

EF EF

3

decreases, ultimately disappearing whenm → 0. In otherwords, γxia/γ

zia contains no signatures of band topology

near the topological quantum critical point. Figure 2aconfirms our analytical statements in a lattice model, forwhich Eq. (5) is solved numerically.

Alternatively, in a sample with fixed carrier density,γxia/γ

zia shows a pronounced maximum as a function of m

in the topological phase only. The maximum takes placeatm∗ ≃ −βk2F , whereMkF

undergoes a sign change. Mo-tivated by recent claims of pressure-induced band inver-sions in Sb2Se3 and Pb1−xSnxSe [8, 9], in Fig. 2b we plotγxia/γ

zia as a function of pressure, using a lattice model.

This corroborates the emergence of a “topology-induced”maximum in γxia/γ

zia.

In the preceding discussion of γxia, we have assumed aparity-odd phonon mode that couples to electrons purelythrough τx (g′i = 0 in Eq. (4)). In general, such a phononcan also couple to electrons through the term g′ · στy.However, we have verified that this coupling producesqualitatively similar features as τx.Similarly, when discussing γzia, we have imagined a

parity-even phonon mode that couples to electrons purelythrough τz (g0 = 0 in Eq. (4)). Nonetheless, symmetryallows a mixture of τz and the identity matrix 1 [11].The latter produces intraband matrix elements that areinsensitive to the orbital texture of the insulator, since⟨ukn|1|ukn⟩ = 1. Consequently, the effect of g0 = 0 is todilute away the topological features of γzia. Although thisconstitutes a problem towards the realization of Fig. 2in real materials, we find that the maximum in γxia/γ

zia

remains pinned to the topological side if |gz| > |g0|.Interband phonon damping.– Thus far, we have consid-

ered the linewidths of phonons with ω0λ < 2|m|. Herein,we investigate the case ω0λ > 2|m|, relevant to Diracinsulators with particularly small bandgaps and/or high-frequency phonon modes. In this case, a phonon is ab-sorbed by an electron in the bulk valence band, whichgets promoted to the bulk conduction band. The asso-ciated phonon linewidth is γλie, where the subscript “ie”is shorthand for “interband”. Assuming for the momentthat ϵF is inside the bulk gap, Eq. (33) yields

γλie(q ≃ 0) ≃ πDjoint(ω0λ)|gλie(k, q)|2, (8)

where Djoint(ω) =!

k δ(Ekc − Ekv − ω)/V is the jointdensity of states, Ekc and Ekv are the bulk conduction(c) and valence (v) band energies. In addition, |gλie|2 =!

n∈v,n′∈c |gλnn′ |2 and |gλie|2 =!

k |gλie|2δ(Ekc − Ekv −ω0λ)/(VDjoint).From Eqs. (1) and (3), we obtain |gzie(k, 0)|2 =

|gxia(k, 0)|2 and |gxie(k, 0)|2 = |gzia(k, 0)|2. Therefore, γλie isas sensitive as γλia to the band topology of the Dirac insu-lator (with x and z interchanged). More so, an importantadvantage of γλie over γλia is that we may effectively takegeven = gz regardless of the value of g0 in Eq. (4), because⟨ukn|1|ukn′⟩ = 0 for interband transitions. Accordingly,

0.1 0.15 0.2 0.25EF(eV)

0

2

4

6

8

10

γ iax /γ

iaz

0 1 2 3 4P(GPa)

0

2

4

6

8

0.4 0.8n0 (1019cm-3)

2

2.5

3

1 2 3

TopologicalTrivial

P*

m<0

m>0

n(1019 cm-3)

P*

Pc

a) b)

FIG. 2: (Color online) (a) γxia/γ

zia as a function of the Fermi

energy and the bulk carrier density, for m = ±0.1 eV. Aprominent maximum emerges in the topological phase only,due to the momentum-space orbital texture of the electroniceigenstates. (b) γx

ia/γzia as a function of pressure P . We use

m = α(P − Pc), where Pc is the critical pressure for a bandinversion and α is a coefficient that can be obtained e.g. fromexperiment [8]. The bulk carrier density is n ≃ n0(1 + P/B),where n0 is the density at P = 0 and B is the bulk modu-lus. The maximum of γx

ia/γzia appears at P = P ∗. Inset: The

dependence of P ∗ on n0. As n0 decreases, P ∗ approachesPc, making it more difficult to identify trivial and topolog-ical phases solely from phonon measurements. Throughoutthis figure, we have used a tetragonal lattice regularizationof Eq. (1). Because α,β, γ are not tabulated for Sb2Se3, wehave replaced them with those of Sb2Te3 [4]. For the bulkmodulus, we have used B = 30GPa [10].

the topological signatures in γλie are more robust thanthose in γλia.In a sample with fixed carrier density, γxie/γ

zie contains

a minimum as a function of m at m∗ ≃ −ω20β/(4α

2),i.e. only in the topological phase [12]. This result hasthe same origin as the maximum of γxia/γ

zia discussed

above, and it holds for doped samples as well so long asαkF /ω0 ≪ 1. Figure 3 confirms this for a lattice model.Discussion.– In sum, there are three reasons why the

linewidths of bulk, long-wavelength optical phonons caninherit distinct signatures of the electronic band topol-ogy. First, phonon linewidths are proportional to thesquare of energy-resolved electron-phonon matrix ele-ments. Second, in a centrosymmetric crystal, the cou-pling of a optical q ≃ 0 phonon to electrons either com-mutes or anticommutes with the electronic parity oper-ator. Third, the momentum-space texture of the expec-tation value of the electronic parity operator determinesthe band topology of a Dirac insulator.

Phonon frequencies, which we have barely mentionedthus far, are much less sensitive than phonon linewidthsto the electronic band topology. This is because the realpart of Eq. (33) contains a sum over electron-phononmatrix elements at multiple energies, with weights thatdepend on non-topological details of the energy bands.

Trivial insulator



Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧�

x

/�z

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (4)

�x

= gz

D(✏F

)|h⌧xi|2 (5)

��

= g�

D(✏F

)|hh�

epi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(12)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(13)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(14)

3



near the topological quantum critical point. Figure 2aconfirms our analytical statements in a lattice model, forwhich Eq. (5) is solved numerically.Alternatively, in a sample with fixed carrier density,

γxia/γzia shows a pronounced maximum as a function of m





zia.



zia




where Djoint(ω) =!





0.1 0.15 0.2 0.25EF(eV)

0

2

4

6

8

10

γ iax /γ

iaz

0 1 2 3 4P(GPa)

0

2

4

6

8

0.4 0.8n0 (1019cm-3)

2

2.5

3

1 2 3

TopologicalTrivial

P*

m<0

m>0

n(1019 cm-3)

P*

Pc

a) b)









zie contains



zia discussed


linewidths of bulk, long-wavelength optical phonons caninherit distinct signatures of the electronic band topol-ogy. First, phonon linewidths are proportional to thesquare of energy-resolved electron-phonon matrix ele-ments. Second, in a centrosymmetric crystal, the cou-pling of a optical q ≃ 0 phonon to electrons either com-mutes or anticommutes with the electronic parity oper-ator. Third, the momentum-space texture of the expec-tation value of the electronic parity operator determinesthe band topology of a Dirac insulator.Phonon frequencies, which we have barely mentioned

thus far, are much less sensitive than phonon linewidthsto the electronic band topology. This is because the realpart of Eq. (33) contains a sum over electron-phononmatrix elements at multiple energies, with weights thatdepend on non-topological details of the energy bands.

3



near the topological quantum critical point. Figure 2aconfirms our analytical statements in a lattice model, forwhich Eq. (5) is solved numerically.Alternatively, in a sample with fixed carrier density,

γxia/γzia shows a pronounced maximum as a function of m





zia.



zia




where Djoint(ω) =!





0.1 0.15 0.2 0.25EF(eV)

0

2

4

6

8

10

γ iax /γ

iaz

0 1 2 3 4P(GPa)

0

2

4

6

8

0.4 0.8n0 (1019cm-3)

2

2.5

3

1 2 3

TopologicalTrivial

P*

m<0

m>0

n(1019 cm-3)

P*

Pc

a) b)









zie contains



zia discussed


linewidths of bulk, long-wavelength optical phonons caninherit distinct signatures of the electronic band topol-ogy. First, phonon linewidths are proportional to thesquare of energy-resolved electron-phonon matrix ele-ments. Second, in a centrosymmetric crystal, the cou-pling of a optical q ≃ 0 phonon to electrons either com-mutes or anticommutes with the electronic parity oper-ator. Third, the momentum-space texture of the expec-tation value of the electronic parity operator determinesthe band topology of a Dirac insulator.Phonon frequencies, which we have barely mentioned

thus far, are much less sensitive than phonon linewidthsto the electronic band topology. This is because the realpart of Eq. (33) contains a sum over electron-phononmatrix elements at multiple energies, with weights thatdepend on non-topological details of the energy bands.


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧�

x

/�z

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (4)

�x

= gz

D(✏F

)|h⌧xi|2 (5)

��

= g�

D(✏F

)|hh�

epi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(12)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(13)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(14)

Phonon linewidth

E

k


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧�

x

/�z

q > !0/vF

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (4)

�x

= gz

D(✏F

)|h⌧xi|2 (5)

��

= g�

D(✏F

)|hh�

epi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧�

x

/�z

q > !0/vF

2|m(0)| > !0

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (4)

�x

= gz

D(✏F

)|h⌧xi|2 (5)

��

= g�

D(✏F

)|hh�

epi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)

EF

Case II: Phonon frequency > Bandgap

Joint DOS


Ion Garate



PACS numbers:

�0

⌧ 0

|h0n|qn0i|2mt > 0

mt < 0

m(k) = m + tk2

h(k)|kni = Ekn

|kniEk = ±pv2k2

+ m2

hep

/ 12

⌦ ⌧x

hep

/ 12

⌦ ⌧z

hep

/ 12

⌦ 12

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

H = vk · �⌧x

+ m ⌧z

(2)

|m(0)|�⌧�

x

/�z

q > !0

/vF

2|m(0)| > !0

⇥ = i�yK (3)

⇧ = ⌧z

(4)

h⌧zihz

ep

/ 12

⌦ ⌧z

hx

ep

/ 12

⌦ ⌧x

��

= g�

Djoint

(!0

)|hc|hep

|vi|2 (5)

�x

= gz

D(✏F

)|h⌧xi|2 (6)

�z

= gz

D(✏F

)|h⌧zi|2 (7)

�x

= gz

D(✏F

)|h⌧xi|2 (8)

Interband matrix element

A property of interband matrix elements:

q=0 linewidth:


Ion Garate



PACS numbers:

hep

/ 12

⌦ 12

(1)

�0

hck|1 + ⌧z|vki = hck|⌧z|vki (2)

|hvk|⌧z|cki|2 = |hck|⌧x|cki|2 (3)

|hvk|⌧z|cki|2 = 1� |hck|⌧x|cki|2 (4)

⌧ 0

|h0n|qn0i|2mt > 0

mt < 0

m(k) = m + tk2

h(k)|kni = Ekn

|kniEk = ±pv2k2

+ m2

hep

/ 12

⌦ ⌧x

hep

/ 12

⌦ ⌧z

hep

/ 12

⌦ 12

h(k) = vk · �⌧x

+ m(k) ⌧z

(5)

H = vk · �⌧x

+ m ⌧z

(6)

|m(0)|�⌧�

x

/�z

q > !0

/vF

2|m(0)| > !0

⇥ = i�yK (7)

⇧ = ⌧z

(8)

h⌧zihz

ep

/ 12

⌦ ⌧z

hx

ep

/ 12

⌦ ⌧x

Theoretical prediction 4

-0.04 -0.02 0 0.02m(eV)

0.2

0.4

-0.04 -0.02 0 0.02m(eV)

0

0.2

0.4

γ iex /γ

iez

Sb2Te3

Topological Trivial

Bi2Se3

Topological Trivial

a) b)

FIG. 3: (Color online) γxie/γ

zie as a function of the Dirac

mass m, where the rest of the band parameters correspond toSb2Te3 (a) or Bi2Se3 (b). The minimum of γx

ie/γzie occuring

in the topological side is a direct manifestation of the orbitaltexture in Fig. 1. Throughout this figure, we have used atetragonal lattice regularization of Eq. (1) with the band pa-rameters taken from Ref. [4]. The Fermi level is assumed tobe inside the bulk gap.

This notwithstanding, a recent experiment [8] in Sb2Se3has attributed a kink in the pressure-dependence of thephonon frequency to a band inversion. Our calcula-tions [6] do not support such interpretation.

The main tools to measure q ≃ 0 phonon linewidths areRaman spectroscopy (for parity-even phonons), infraredspectroscopy (for parity-odd phonons) and inelastic neu-tron scattering [13]. In a clean material with ω0λτ ≫ 1(where τ is the disorder scattering time), γλia vanishesunless q > ω0λ/vF [14]. Since ω0λ/vF typically exceedsthe photon wave vector used in optical spectroscopies,γλia should be measured with neutrons. In contrast, γλieremains nonzero at q = 0 and is thus amenable to optics.For Bi2Se3, we estimate γλie, ia ! 1 cm−1, which nears theexperimental resolution [15].

Aside from electron-phonon interactions, anharmoniclattice effects contribute to the phonon linewidth. Toleading order, phonon-phonon interactions contain no in-formation about the electronic band topology and areindependent from the carrier density. Therefore, theanharmonic part can be subtracted by measuring thelinewidths with respect to a baseline carrier density.

In view of our results, it is natural to ask whetherany other physical observable involving Fermi’s goldenrule, such as conductivity, might be sensitive to elec-tronic band topology on the same footing as the phononlinewidths. The answer is generally negative. For ex-ample, the optical conductivity cannot clearly differenti-ate between trivial and nontrivial orbital textures in thebulk bands because the velocity operator mixes 1, τx,and τz [16].

To conclude, we have proven that it is in principlepossible to infer the strong topological invariant of acentrosymmetric Dirac insulator from the linewidths of

bulk, long-wavelength optical phonons. It will be desir-able to complement our theory with first principles elec-tronic structure calculations, and to search for similarinsights in other contexts (cold atoms, photonic crystals,quantum memories) where the interplay between topol-ogy and dissipation may be crucial.Acknowledgements.– We are grateful to K. Pal and U.

Waghmare for sharing useful information about Ref. [8].This work has been funded by Quebec’s RQMP andCanada’s NSERC. The numerical calculations were per-formed on computers provided by Calcul Quebec andCompute Canada.

[1] See e.g. Topological Insulators, Contemporary Conceptsof Condensed Matter Science 6, eds. M. Franz and L.Molenkamp (Elsevier, Amsterdam, 2013).

[2] C.-E. Bardyn, M. A. Baranov, C. V. Krauss, E. Rico,A. Imamoglu, P. Zoller and S. Diehl, New. J. Phys. 15,085001 (2013).

[3] I. Garate, Phys. Rev. Lett. 110, 046402 (2013); K. Sahaand I. Garate, Phys. Rev. B 89, 205103 (2014).

[4] C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, S.-C.Zhang, Phys. Rev. B 82, 045122 (2010).

[5] See e.g. T. Ando, J. Phys. Soc. Jpn 75, 124701 (2006).[6] See the Supplemental Material.[7] We consider only spatially local electron-phonon interac-

tions, which means that gλ(q) is independent of the elec-tronic momentum k. Hence, the entire k−dependence ofgλnn′(k,q) in Eq. (3) originates from |ukn⟩. The neglect ofnonlocal electron-phonon interactions is justifiable whenk is small compared to the size of the Brillouin zone [6].

[8] A. Bera, K. Pal, D. V. S. Muthu, S. Sen, P. Guptasarma,U. V. Waghmare and A. K. Sood, Phys. Rev. Lett. 110,107401 (2013).

[9] X. Xi, X.-G. He, F. Guan, Z. Liu, R. D. Zhong, J. A.Schneeloch, T. S. Liu, G. D. Gu, X. Du, Z. Chen, X.G. Gong, W. Ku and G. L. Carr, Phys. Rev. Lett. 113,096401 (2014).

[10] I. Efthimiopoulos, J. Zhang, M. Kucway, C. Park, R. C.Ewing and Y. Wang, Sci. Rep. 3, 2665 (2013).

[11] Particle-hole symmetry, which could distinguish between1 and τ z, is broken in real Dirac insulators.

[12] For simplicity, we have assumed that the parity-odd andthe parity-even mode under consideration have a simi-lar frequency ω0. When this assumption fails, our the-ory predicts a minimum of γx

ie/Djoint(w0x) at m∗ ≃−ω2

0xβ/(4α2), and a maximum of γz

ie/Djoint(w0z) atm∗ ≃ −ω2

0zβ/(4α2).

[13] See e.g. P. Y. Yu and M. Cardona, Fundamentals of Semi-conductors (4th ed., Springer, Berlin, 2010).

[14] In practice the condition q > ω0λ/vF is often compatiblewith long-wavelength phonons. For example, if ω0λ ≃

100 cm−1 and vF ≃ 5× 105m/s, ω0λ/vF ! 0.005A−1

.[15] S. M. Shapiro, G. Shirane and J. D. Axe, Phys. Rev.

B 12, 4899 (1975); X. Zhu, L. Santos, C. Howard, R.Sankar, F. C. Chou, C. Chamon and M. El-Batanouny,Phys. Rev. Lett. 108, 185501 (2012); K. M. F. Shalil,M. Z. Hossain, V. Goyal and A. A. Balandin, J. Appl.Phys. 111, 054305 (2012); Y. Kim, X. Chen, Z. Wang,


Ion Garate



PACS numbers:

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧�

x

/�z

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (4)

�x

= gz

D(✏F

)|h⌧xi|2 (5)

��

= g�

D(✏F

)|hh�

epi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

h0ep(r) = u0(r) 12 ⌦ 12 (11)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(12)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(13)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(14)

Experiments?

• Anharmonic contribution

• Linewidths are hard to measure-0.3

-0.15

0

0.15

0.3

Ener

gy (e

V)

1.0 GPa2.0 GPa4.5 GPa

Γ

(<P )

(=P )

(>P )c

c

c

(c)

M1

(a)(b)

FIG. 2: (color online)–(a) Raman shift versus pressure plot. Solid lines are linear fits to the ob-

served frequencies. The numbers next to the straight lines are the fitted values of dωdP in cm−1/GPa.

The error bars, if not seen, are less than the size of the symbol. Fig.(b) shows FWHM of the M1

mode (solid points) as a function of pressure and the dashed line is drawn as guide to the eyes.

Fig.(c) shows the First-principles calculations of electronic structure near the gap as a function of

pressure in the neighborhood of transition (Pc=2 GPa).

11

A. Bera et al., PRL 110, 107401 (2013).

Phonon anomaly

Summary and Conclusions

Interesting interplay between phonons and electronic band topology

Phonons can change the electronic band topology

Phonons can inherit unique signatures of the electronic band topology

E

k

Dirac fermions (spin ½)

m


Ion Garate



PACS numbers:

hep / 12 ⌦ ⌧x

hep / 12 ⌦ ⌧z

hep / 12 ⌦ 12

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

H = vk · �⌧x

+ m ⌧z

(2)

|m(0)|�⌧�

x

/�z

q > !0/vF

2|m(0)| > !0

⇥ = i�yK (3)

⇧ = ⌧z

(4)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (5)

�x

= gz

D(✏F

)|h⌧xi|2 (6)

��

= g�

D(✏F

)|hh�

epi|2 (7)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (8)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(9)

E⇤kn

' Ekn

+

X

qn

0

|hkn|hep|k� qn0i|2Ekn

� Ek�qn

0(10)

Dirac velocity Dirac mass


Ion Garate



PACS numbers:

m(k) = m + tk2

h(k)|kni = Ekn

|kniEk = ±pv2k2

+ m2

hep / 12 ⌦ ⌧x

hep / 12 ⌦ ⌧z

hep / 12 ⌦ 12

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

H = vk · �⌧x

+ m ⌧z

(2)

|m(0)|�⌧�

x

/�z

q > !0/vF

2|m(0)| > !0

⇥ = i�yK (3)

⇧ = ⌧z

(4)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (5)

�x

= gz

D(✏F

)|h⌧xi|2 (6)

��

= g�

D(✏F

)|hh�

epi|2 (7)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (8)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(9)

Jackiw-Rebbi zero mode

“Domain wall”: energy

2m0Zero energy state

m(z)

z-m0

m0

Insensitive to details of the Hamiltonian

Dirac fermions in 1D z[PRD 13, 3398 (1976)]

Topological insulators

m(z)

zBi2Se3 Vacuum

3

Figure S1. Device architecture. (a) Optical image of the device showing two coupled gmon transmons on the top halfof the chip and the two coupled gmons used in this work on the lower half (zoomed-in view in inset). (b) The layout of thetwo-qubit gmon system. We supply bias currents using the lower blue lines to tune the inductance of the coupler junction(middle) and the qubit frequencies (left, right). We apply microwave pulses to each qubit via the gray trace. We read out thestate of the qubits dispersively via readout resonators: each qubit is capacitively coupled to a resonator (green lines; meanderedlines in inset of (a)).

1. THE GMON QUBITS

1.1. The gmon coupling architecture

In this work we implemented an adjustable inductive coupling between two qubits. Adjustable coupling has typicallybeen difficult with superconducting qubits, as fixed capacitive coupling may only be modified by detuning, so it hasthe problems of limited on/off range and crosstalk. Here we use a novel qubit design called the gmon, which allowsa continuous variation of the inter-qubit coupling strength g over nanosecond time scales without any degradation inthe coherence of the constituent Xmon qubits [1, 2]. The adjustable inductive coupling between the transmons allowsg/2⇡ to be varied between �5 MHz and 55 MHz, including zero, without changing the bare qubit frequencies. Thedevice was fabricated using standard optical and e-beam lithography techniques, discussed in recent works of ourgroup such as [3]. The experiment was performed at the base temperature of a dilution fridge (⇠20 mK).

1.2. Basic design principle of the gmon

As shown in Fig. S1, the gmon design is based on the Xmon qubit design. One important feature of the Xmondesign [4] is the single-ended ground in contrast to differential or floating grounds. In the absence of adjustablecoupling, the SQUID loops (Fig. S1(b)) would be directly connected to the ground plane. This design feature givesus the ability to capacitively couple qubits with elements such as the drive lines, the readout resonators, and nearbyqubits. In the gmon architecture, instead of immediately terminating the qubit SQUID to ground, we add a linearinductor (the meandering CPW element colored in purple and labeled "tapping inductance") between the SQUIDand ground(CPW stands for coplanar waveguides). This creates a node (where the purple CPW meets the horizontalblue CPW) that allows us to couple the two qubits. The two qubits then can be connected with a CPW line. Thisconnecting line is interrupted with a Josephson junction, which acts as a tunable inductor that can be used to tunethe inter-qubit coupling strength g, hence the name gmon.

The basic operation of the gmon can be understood from a simple linear circuit model. An excitation created in Q1will mostly flow to ground through its tapping inductance, but a small fraction will flow to the tapping inductance ofQ2, generating a flux in Q2. The mutual inductance resulting from the flux in Q2 due to an excitation current in Q1can be calculated from simple current division, and the coupling strength is proportional to this mutual inductanceto high accuracy [1, 2]. The current division ratio, which sets the coupling strength g, can be varied by changing thesuperconducting phase difference across the tunable inductance. This is done by flux biasing its junction, using thecurrent line labeled "coupler tuning" in panel (b).

An important advantage of this architecture is that it prevents crosstalk, a serious hurdle for many other experi-

Topological phases in superconducting circuits

Roushan et al., Nature (2014)

Examples of topological Dirac materials

E

k

E

k

Vary “external parameter”

m

E

km

Topologically trivial Critical point Topologically nontrivial

“External parameter”: doping, pressure...

Topological quantum phase transition

(Experiment: Xu et al., Science 332, 560 (2011))

Can there be a thermal topological transition?


1) Parity-even phonon (Raman active)


Ion Garate



PACS numbers:

hep / 12 ⌦ ⌧x

hep / 12 ⌦ ⌧z

hep / 12 ⌦ 12

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧�

x

/�z

q > !0/vF

2|m(0)| > !0

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (4)

�x

= gz

D(✏F

)|h⌧xi|2 (5)

��

= g�

D(✏F

)|hh�

epi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

2) Parity-odd phonon (infrared active)


Ion Garate



PACS numbers:

hep / 12 ⌦ ⌧x

hep / 12 ⌦ ⌧z

hep / 12 ⌦ 12

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧�

x

/�z

q > !0/vF

2|m(0)| > !0

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (4)

�x

= gz

D(✏F

)|h⌧xi|2 (5)

��

= g�

D(✏F

)|hh�

epi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

We focus on two types:

Experiment in Sb2Se3

Influence of band topology on phonons

-0.3

-0.15

0

0.15

0.3

Ener

gy (e

V)

1.0 GPa2.0 GPa4.5 GPa

Γ

(<P )

(=P )

(>P )c

c

c

(c)

M1

(a)(b)

FIG. 2: (color online)–(a) Raman shift versus pressure plot. Solid lines are linear fits to the ob-

served frequencies. The numbers next to the straight lines are the fitted values of dωdP in cm−1/GPa.

The error bars, if not seen, are less than the size of the symbol. Fig.(b) shows FWHM of the M1

mode (solid points) as a function of pressure and the dashed line is drawn as guide to the eyes.

Fig.(c) shows the First-principles calculations of electronic structure near the gap as a function of

pressure in the neighborhood of transition (Pc=2 GPa).

11

A. Bera et al., PRL 110, 107401 (2013).

Phonon anomaly

Topological phase transition?


Electron-phonon coupling Phonon parity


Ion Garate



PACS numbers:

�Ekn

'X

qn

0

hkn|hep|k� qn0i|2

Ekn

� Ek�qn

0(1)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(2)

h0ep(r) = u0(r) 12 ⌦ 12 (3)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(4)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(5)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(6)


Ion Garate



PACS numbers:

�Ekn

'X

qn

0

hkn|hep|k� qn0i|2

Ekn

� Ek�qn

0(1)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(2)

h0ep(r) = u0(r) 12 ⌦ 12 (3)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(4)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(5)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(6)


Ion Garate



PACS numbers:

�Ekn

'X

qn

0

hkn|hep|k� qn0i|2

Ekn

� Ek�qn

0(1)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(2)

h0ep(r) = u0(r) 12 ⌦ 12 (3)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(4)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(5)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(6)

Even

Even

Odd


Ion Garate



PACS numbers:

�Ekn

'X

qn

0

hkn|hep|k� qn0i|2

Ekn

� Ek�qn

0(1)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(2)

h0ep(r) = u0(r) 12 ⌦ 12 (3)

hx

ep(r) = ux

(r) 12 ⌦ ⌧x

(4)

hz

ep(r) = uz

(r) 12 ⌦ ⌧z

(5)

hiy

ep(r) = uiy

(r) �i ⌦ ⌧y

(6)Odd

Long wavelength + local-in-space

Theoretical issue

Thus far we have assumed that a parity-even phonon mode can couple to electrons purely through


Ion Garate



PACS numbers:

hep / 12 ⌦ ⌧x

hep / 12 ⌦ ⌧z

hep / 12 ⌦ 12

h(k) = vk · �⌧x

+ m(k) ⌧z

(1)

|m(0)|�⌧�

x

/�z

q > !0/vF

2|m(0)| > !0

⇥ = i�yK (2)

⇧ = ⌧z

(3)

h⌧zihz

ep / 12 ⌦ ⌧z

hx

ep / 12 ⌦ ⌧x

�z

= gz

D(✏F

)|h⌧zi|2 (4)

�x

= gz

D(✏F

)|h⌧xi|2 (5)

��

= g�

D(✏F

)|hh�

epi|2 (6)

�q�

' 2⇡X

k

X

nn

0


� fk�qn

0)�(Ekn

� Ek�qn

0 � !q) (7)

!⇤q�

' !q�

+

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � !q(8)

E⇤kn

' Ekn

+

X

qn

0


Ekn

� Ek�qn

0(9)

⇧(q, !) =

1

V

X

k

X

nn

0


� fk�qn

0

Ekn

� Ek�qn

0 � ! � i0+(10)

In general, it could also couple to electrons via


Ion Garate



PACS numbers:

hep

/ 12

⌦ 12

(1)

�0

hck| (↵1 + �⌧z

) |vki = �hck|⌧z|vki (2)

|hvk|⌧z|cki|2 = 1� |hck|⌧z|cki|2 (3)

|hvk|⌧z|cki|2 = 1� |hck|⌧x|cki|2 (4)

⌧ 0

|h0n|qn0i|2mt > 0

mt < 0

m(k) = m + tk2

h(k)|kni = Ekn

|kniEk = ±pv2k2

+ m2

hep

/ 12

⌦ ⌧x

hep

/ 12

⌦ ⌧z

hep

/ 12

⌦ 12

h(k) = vk · �⌧x

+ m(k) ⌧z

(5)

H = vk · �⌧x

+ m ⌧z

(6)

|m(0)|�⌧�

x

/�z

q > !0

/vF

2|m(0)| > !0

⇥ = i�yK (7)

⇧ = ⌧z

(8)

h⌧zihz

ep

/ 12

⌦ ⌧z

hx

ep

/ 12

⌦ ⌧x

The latter is not sensitive to band topology and therefore can mask the effect of the former.

E

k

Trivial insulator

Intraband transitions dominateà phonons decrease bandgap

Dirac insulator with small energy gap

E

k

Interband transitions dominateà phonons increase bandgap


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download.uni-mainz.de€¦ · E k 3D (massive) Dirac fermions in crystals = spin = orbital (parity) Equations for PPT slides Ion Garate D´epartement de Physique, Universit´e de