Topological piezoelectric effect and parity anomaly in nodal line semimetals

Taiki Matsushita,1 Satoshi Fujimoto,1 and Andreas P. Schnyder2

1Department of Materials Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan2Max-Planck-Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany

(Dated: February 27, 2020)

Lattice deformations act on the low-energy excitations of Dirac materials as effective axial vectorfields. This allows to directly detect quantum anomalies of Dirac materials via the response to axialgauge fields. We investigate the parity anomaly in Dirac nodal line semimetals induced by latticevibrations, and establish a topological piezoelectric effect; i.e., periodic lattice deformations generatetopological Hall currents that are transverse to the deformation field. The currents induced by thispiezoelectric effect are dissipationless and their magnitude is completely determined by the lengthof the nodal ring, leading to a semi-quantized transport coefficient. Our theoretical proposal can beexprimentally realized in various nodal line semimetals, such as CaAgP and Ca3P2.

Introduction.— Over the last few years a number ofnew types of topological semimetals have been discov-ered [1–7]. Among them are Weyl semimetals and Diracsemimetals with point nodes, around which the bandshave linear dispersion in all directions. The low-energyphysics of these point node semimetals is described byrelativistic field theories with quantum anomalies, i.e.,by quantum field theories that break symmetries of theclassical action. For instance, two-dimensional Dirac ma-terials, such as graphene, are described by quantum fieldtheories with parity anomalies, that break space-time in-version symmetry [8, 9]. The low-energy theories of Weylsemimetals, on the other hand, exhibit chiral anomalies,which violate conservation of axial charge [10–14]. Thechiral anomaly in Weyl semimetals give rise to numerousexperimental phenomena [15–19], for example, negativemagnetoresistance, which has been observed in recent ex-periments [19, 20]. Lattice strain, which generates ax-ial magnetic fields, can also be used to probe the chiralanomaly [21–34].

At the same time, recent research has focused on Diracmaterials with line nodes [35–43]. These nodal-line semi-metals (NLSMs) can be viewed as three-dimensional gen-eralizations of graphene. They exhibit Dirac band cross-ings along a one-dimensional line in a three-dimensionalBrillouin zone, with low-energy excitations that are lin-early dispersing in the two directions perpendicular to theband-crossing line. NLSMs possess a number of interest-ing properties, e.g., topological surface charges, drum-head surface states [43, 44], and quasitopological electro-magnetic responses [45, 46]. The low-energy excitationsaround the nodal ring of these semimetals are describedby one-parameter families of (2+1)-dimensional quantumfield theories with parity anomalies [47]. That is, theelectromagnetic responses of these nodal rings are givenby Chern-Simons actions, which break parity symmetry.These Chern-Simons terms lead to transverse Hall effects,where electrons from opposite sides of the nodal ring flowto opposite surfaces, when an electric field is applied [47].Unfortunately, due to time-reversal symmetry, the totalcurrent generated by the Chern-Simons action vanishes,

once the sum over all momenta is taken. Therefore, theelectric-field induced Hall currents can only be measuredby special devices, that filter electrons based on theirmomenta [47].

In this letter, we propose to use pseudo electricfields, induced by lattice vibrations, to probe the par-ity anomaly of NLSMs. As opposed to external electricfields, pseudo electric fields are axial, as they couple withopposite sign to electrons with opposite momenta. Thispermits to directly probe the parity anomaly of NLSMs,via the response to axial electric fields. We derive a low-energy description of NLSMs in the presence of strain,and show that periodic lattice deformations generate atopological piezoelectric effect (TPEE), which originatesfrom the parity anomaly. This piezoelectric effect mani-fests itself by dissipationless Hall currents that are trans-verse to the deformation field. We show that the TPEEcan be interpreted as a polarization current and that ithas a semi-quantized transport coefficient, given by thelength of the nodal ring. Furthermore, we discuss exper-imental considerations for the observation of the TPEEin the NLSM materials CaAgP and Ca3P2.

Model.— First, we introduce a lattice model for aNLSM with a single nodal ring, and discuss its topolog-ical properties. We consider the following tight-bindingHamiltonian on the cubic lattice

H(p) = t [2 + cos p0a− cos pxa− cos pya− cos pza] τz

+ v sin pza τy + ∆τx, (1)

where τi (i = 1, 2, 3) are Pauli matrices acting in orbitalspace. For simplicity, we assume t, v, p0 > 0. To discussthe parity anomaly and the electric polarization, we haveintroduced a small parity-breaking term ∆τx. In the ab-sence of ∆τx, the lattice Hamiltonian is parity-time (PT )symmetric with the PT operator PT = τzK. This tight-binding Hamiltonian describes the low-energy dispersionnear the Fermi level of CaAgP and Ca3P2 [36, 43]. Thesymmetry-breaking term ∆τx can be induced by apply-ing uniaxial pressure, or an electric field [47].

In the absence of ∆τx, Hamiltonian (1) exhibits a nodalring within the pz = 0 plane, centered around Γ. This

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nodal ring is topologically protected by the following Z2

invariant ν[S1] [48],

ν[S1] =1

π

∑α∈occ. states

∮S1

dp · Aα,α(p) mod 2, (2)

where the integration path is along the closed loop S1.Here, Aα,βi (p) = i〈uα(p)|∂pi |uβ(p)〉 and |uα(p)〉 are theBerry connection and the Bloch eigenstates of Eq. (1),respectively. PT symmetry restricts Eq. (2) to the valuesν[S1] = 0, 1 [49]. When the loop S1 encircles the nodalring, we obtain ν[S1] = 1, otherwise ν[S1] = 0.

The topological protection of the nodal ring is linkedto a bulk electric polarization. To see this, let us de-compose the three-dimensional Hamiltonian (1) into one-dimensional subsystems parametrized by the inplane mo-menta p⊥ = (px, py). The electric polarization of eachsubsystem is given by the Zak’s phase [50],

Pz(p⊥) =∑

α∈occ. states

∫ π

−π

dpz2πAα,αz (p) = 0,

1

2, (3)

and the total polarization is the summation of thesephases over the inplane momenta

Pz =

∫dp⊥(2π)2

Pz(p⊥). (4)

From Eqs. (2)and (3), we find that the Zak’s phase is12 for a region of inplane momenta p⊥ that is boundedby the nodal ring. By the bulk-boundary correspon-dence [44], this leads to midgap surface states at the (001)face of the NLSM, whose fillings determine the surfacecharge. Due to PT symmetry, the surface states at thetop and bottom (001) faces are degenerate, thus the elec-tric polarization is determined only up to a multiple ofthe elementary charge. To unambiguously determine thebulk polarization, it is necessary to include an infinites-imal PT symmetry breaking term ∆τx, which opens abulk gap and removes the degeneracy of the midgap sur-face states. With the inclusion of ∆τx, we find that thebulk polarization is semi-quantized and given by [46],

Pz =S

8π2sign(∆), (5)

where S is the area encircled by the nodal ring projectedonto the surface Brillouin Zone.

Parity anomaly and Chern-Simons action.— Next, weuse a family of (2+1)-dimensional quantum field theo-ries to derive the topological responses due to externaland pseudo electromagnetic fields. For small p, Eq. (1)reduces to the low-energy continuum Hamiltonian

Heff(p) =p2 − p2

0

2mτz + λpzτy + ∆τx, (6)

where 1/(2m) = ta2/2 and λ = va. Eq. (6) has rotationalsymmetry around the pz axis. Thus, we introduce cylin-drical coordinates (pρ, φ, pz) with pρ ∈ (−∞,∞), and

FIG. 1: Schematics of the nodal ring (red) within the pz = 0plane. Using the cylindrical coordinates (pρ, φ, pz), we de-compose the NLSM into two-dimensional subsystems (blue)parametrized by φ. Each subsystem contains two Dirac pointswith opposite sign of Berry curvature χ.

φ ∈ [0, π), see Fig. 1. With these cylindrical coordinates,we can decompose the three-dimensional system into afamily of two-dimensional subsystems labeled by the az-imuth angle φ ∈ [0, π). Each of these subsystems containstwo Dirac points that are related by time-reversal sym-metry, and which have opposite sign of Berry curvatureχ = sign(∆) = ±1.

Each subsystem, labeled by φ, is described by the fol-lowing (2+1)-dimensional quantum field theory

Sφ =⊕χ=±1

Sφ,χ (7)

Sφ,χ =

∫d2x dt ψ

[iχγµ

(∂µ + iAµ + iχA5

µ) + ∆)]ψ,

where ψ is a two-component Dirac spinor, ψ = ψ†γ0,{γµ, γν} = ηµν , and ηµν = diag(1,−1,−1). The Diracspinors interact with the total gauge fieldAχµ = Aµ+χA5

µ,which contains both an external gauge field Aµ and anaxial gauge field A5

µ, respectively. We note that the axialgauge field couples with opposite sign χ to the two Diracpoints of the subsystem. The physical origin of A5

µ dueto lattice strain will be discussed later. Upon regular-ization [8], we obtain from Eq. (8) the parity breakingChern-Simons term

Sφ,χCS =χ

4π

∫d2x dt εµνλAχµ∂νA

χλ, (8)

which is a manifestation of the parity anomaly. Varyingthe Chern-Simons action with respect to Aχµ, gives theanomalous transverse current

jφ,χµ = −δSφ,χCS

δAµ=

χ

4πεµνλ(∂νAλ + χ∂νA

5λ), (9)

3

for a single Dirac point with chirality χ in subsystem φ.We observe that transverse currents induced by exter-nal electromagnetic fields cancel out, since contributionsfrom opposite sides of the nodal ring have opposite signχ = ±1. Currents induced by axial gauge fields, however,do not cancel, since they have the same sign everywherealong the nodal ring. This remarkable feature originatesfrom the axial nature of the strain-induced gauge fieldA5µ, which couples oppositely to Dirac fermions with op-

posite momenta.Strain-induced axial gauge field.— We now discuss the

physical origin of the axial gauge field. The basic ideais to incorporate lattice strain into the tight-bindingmodel (1), which acts on the low-energy excitations aseffective gauge fields. Strain shifts the lattice sites Rby the displacement vector u(R), as R + u(R), therebymodifying and introducing new overlaps between atomicorbitals. In our tight binding model this changes thehopping parameters as [51, 52]

t(ax)τz ' t(1− uxx)τz + ivuxzτy, (10a)

t(ay)τz ' t(1− uyy)τz + ivuyzτy, (10b)

t(az)τz ' t(1− uzz)τz, (10c)

iv(az)τy ' iv(1− uzz)τy + t∑i6=z

uziτz, (10d)

where t(aµ)(µ = x, y, z) represents the hopping ampli-tudes along the bond direction aµ, and uµν = (∂µuν(R)+∂νuµ(R))/2 is the symmetrized strain tensor. The firstterms in Eqs. (10) describe changes in the hopping am-plitudes between two like orbitals, when the bond lengthsare modified by strain. The second terms originate fromnew hopping processes between different orbitals, whichare symmetry forbidden in the unstrained lattice. In thefollowing, we focus on the gauge fields induced by the uzνcomponents of the stress tensor, as these are the onesthat probe the parity anomaly. The other componentsof uµν only renormalize the Fermi velocity, which is notimportant for our purpose. Using Eq. (10) we find thatthis lattice strain generates additional terms in the tight-binding Hamiltonian (1), H(p)→ H(p) + δH(p), whichare of the form [53]

δH(p) ' −tuzz cos pza τz (11)

+v(uxz sin pxa+ uyz sin pya)τy,

These modifications change the low-energy Hamilto-nian (6) to

Heff(p) + δHeff(p) (12a)

' vF(qr −A5

r

)τz + λ(pz −A5

z(ϕ))τy + ∆τx,

with the pseudo gauge potentials

A5r = uzz/(p0a

2), (12b)

A5z(ϕ) =

∑i

fi(ϕ)Ai,5z ' −p0 (uxz cosϕ+ uyz sinϕ) ,

along the r- and z-directions, respectively, where fx(ϕ) =p0 cosϕ, fy(ϕ) = p0 sinϕ, and Ai,5z = −uiz (i = x, y).Here, we have introduced the Fermi velocity vF = p0/m,the radial momentum qr = pr − p0, and the cylin-drical coordinates (pr, pz, ϕ) with pr ∈ [0,∞), pz ∈(−∞,∞), ϕ ∈ [0, 2π).

We conclude that in NLSMs with a nodal ring in thepz = 0 plane, the strain field components uzν act onthe low-energy excitations like effective gauge potentials.Interestingly, these effective gauge potentials are axial,since they couple with opposite sign to the excitationsat opposite sides of the nodal ring, i.e., at (p0, 0, ϕ) and(p0, 0, ϕ + π). From Eq. (12), we see that A5

r concentri-cally shrinks or expands the nodal ring, while Ai,5z tiltsthe nodal ring out of the pz = 0 plane, see Figs. 2(a)and 2(b), respectively.

If we consider time-dependent lattice strain, i.e., lat-tice vibrations, we can also generate axial electric fields.That is, the time dependence of the strain tensor uµz(t)produces axial electric fields via

E5r = −∂A

5r

∂t, Ei,5z = −∂A

i,5z

∂t, (13)

where the axial electric fields Ei,5z are defined by theangnular independent parts of the axial vector potentials,and fi(ϕ) are absorbed into the axial charge couplingconstants.

Topological piezoelectric effect.— Next, we demon-strate that axial electric fields in NLSMs induce net topo-logic currents that flow in the direction perpendicular tothe axial fields. For that purpose we use linear responsetheory to compute the axial conductivity tensors σµr(ω)and σxµr(ω), which are defined as

〈jµ〉(ω) = σµr(ω)E5r (ω) + σxµz(ω)Ex,5z (ω), (14)

with the current density operator j. By use of Kubo’s for-mula we compute the axial Hall conductivities σzr(T, ω)and σxz(T, ω) [54]. In the DC limit ω → 0, they are givenby

σDCzr (T ) = − 1

V

∑p,α

f(εαp)Bαzr(p), (15a)

σx,DCxz (T ) = − 1

V

∑p,α

f(εαp)p0 cosϕBαxz(p), (15b)

where εαp , f(εαp) and Bαµν(p) = −2Im〈∂pµuα(p)|∂pνuα(p)〉are the energy of the Bloch electrons, the Fermi function,and the Berry curvature, respectively.

Thus, it follows that axial electric fields produce trans-verse Hall currents, whose magnitude is determined bythe Berry curvature. These Hall currents are perpendic-ular to both the axial electric field and the Berry curva-ture, see Figs. 2(b) and 2(c). For instance, axial electricfields along the r direction lead to electric currents in

4

px

py

pz

px

py

pz

(a) (b)

px

py

pz

(c)

px

pz

Side view

py=0

uzz(t) uzz(t)(d)

px

py

pz

uxz(t)

FIG. 2: (a),(c) The axial gauge potential A5r(t) changes the size of the nodal ring, while A5

z(t) tilts it out of plane, as indicatedby the dashed lines. (b),(d) The strain induced currents jφ,χ (pink) are perpendicular to both the Berry curvature B−

zr(p)(green) and the axial electric field E5 (blue).

the z direction, since the direction of the Berry curva-ture is within the pz = 0 plane. Similarly, axial electricfields along the z direction produce currents in the pz = 0plane. Because the axial electric fields are generated bylattice vibrations, we refer to this type of Hall responseas a topological piezoelectric effect.

Interestingly, in the low-frequency regime |ω/∆| � 1,the axial Hall conductivities become semi-quantized, i.e.,their magnitude depends only on the length of the nodalring L = 2πp0. That is, in the limit |ω/∆| � 1 we find

σDCzr (T = 0) ' − L

8π2sign(∆), (16a)

σx,DCxz (T = 0) ' L

16π2sign(∆), (16b)

where |∆/(vF pcut)|, |∆/(λpcut)| � 1 is assumed, withsome cut-off momentum pcut. This is confirmed by nu-merical evaluations of σzr(T = 0, ω), see Fig. 3. Weobserve in Figs. 3(a) and 3(b) that the axial Hall con-ductivity asymptotically approaches its semi-quantizedvalue for ω → 0, once ∆ becomes sufficiently small andpcut sufficiently large, respectively. As displayed in theinset of Fig. 3(a) the semi-quantized value of the DC ax-ial Hall conductivity scales linearly with the size of thenodal ring.

Before concluding, we show that the TPEE is related tothe polarization current of NLSMs. As discussed above,the axial electric field E5

r (t) periodically shrinks and ex-pands the nodal ring. This leads to a periodic fluctua-tion of the bulk electric polarization, which is determinedby the size of the nodal ring. Hence, the axial electricfield generates a polarization current, which according toEq. (5), takes the form

jpolz =

dPz(t)

dt=

1

8π2

dS(t)

dt' − L

8π2sign(∆)E5

r (t), (17)

where S(t) = π(p0 +A5r(t))

2 is the area of the nodal ring.Since Eq. (17) coincides with Eq. (16b), we conclude thatthe TPEE is linked to the polarization current of NLSMsand that its quantization arises from the semi-quantizedelectric polarization.

The proposed TPEE is testable in the materialsCaAgP or Ca3P2, which exhibit a single nodal ring near

the Fermi energy. In these materials rapid lattice vi-brations can be generated using microwave radiation,which leads to an AC current via the TPEE. To esti-mate the magnitude of the current in Ca3P2, the rele-vant material parameters are vF = 2.72 × 105 m/s, λ =

3.80 × 105 m/s, p0 = 0.206 A−1

, and a = 8.26A [43].If the lattice is vibrating with frequency 100 MHz andamplitude 0.3% of a, and the parity breaking term is as-sumed to be |∆/(vF p0)| = 0.001, then the TPEE currentis estimated to be about jz ' 550 mA/cm2, which isexperimentally detectable.

Conclusion.— We have shown that periodic strainfields lead to the TPEE, which manifests itself by dis-sipationless Hall currents, originating from the parityanomaly. While it would be of fundamental interest toobserve the parity anomaly in NLSMs using lattice vibra-tions, the proposed TPEE could also be useful for futurepiezoelectric devices.

Acknowlegement.— T.M. thanks K. Nomura, A. Ya-makage, and Y. Ominato for invaluable discussion. T.M.was supported by a JSPS Fellowship for Young Sci-entists. S.F. was supported by the Grant-in-Aids forScientific Research from JSPS of Japan (Grants No.17K05517), and KAKENHI on Innovative Areas “Topo-

FIG. 3: Frequency dependence of the axial Hall conductivityσzr(T = 0, ω) for different values of ∆ and pcut. In (a) thered, green, and blue curves correspond to ∆/(vF p0) = 0.1,0.01, and 0.001, respectively, with pcut fixed at pcut = 0.2p0.In (b) the red, green, blue, and pink curves correspond topcut/(p0/10) = 2, 4, 6, and 8, respectively, with ∆ fixed at∆ = 0.1vF p0. The inset shows the p0 dependence of the DCaxial Hall conductivity σDC

zr (T = 0) with pcut = 0.2p0 and∆ = 0.0004. The other parameters in all panels are: vF = 2.0,p0 = 0.2, and λ = 2.5.

5

logical Materials Science” (No. JP15H05852) and ”J-Physics” (No. JP18H04318), and JST CREST GrantNumber JPMJCR19T5, Japan. This research was ini-tiated at the KITP Santa Barbara and supported inpart by the National Science Foundation under GrantNo. NSF PHY-1748958.

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only renormalizes the hopping amplitude v.[54] The details of the calculation are given in the supplemen-

tal materials.

6

Supplemental Materials

Derivation of Eq. (15)

Here, we summarize the derivation of Eq. (20a) andEq. (20b).

To discuss the topological transport, we use the Blocheigenbasis {|u±(p)〉} with

Heff(p)|u±(p)〉 = ε±p |u±(p)〉,

ε±p = ±√v2F q

2r + λ2p2

z + ∆2,(18)

and express the current density operator as

j =∑

p,α=±

∂εαp∂p

c†pαcpα + i∑α6=β

(εαp − εβp

)Aα,β(p)c†pαcpβ ,

(19)where c†pα (cpα) are creation (annihilation) operators.

From this expression and by use of Kubo’s formula weobtain the axial Hall conductivities,

σzr(T, ω) = − limδ→+0

i

V

∑p,α6=β

f(εαp)(εαp − εβp)

×

[Aα,βz Aβ,αr

ω + εαp − εβp + iδ

+Aα,βr Aβ,αz

ω − εαp + εβp + iδ

](20a)

σxxz(T, ω) = − limδ→+0

i

V

∑p,α6=β

f(εαp)p0 cosϕ(εαp − εβp)

×

[Aα,βx Aβ,αz

ω + εαp − εβp + iδ

+Aα,βz Aβ,αx

ω − εαp + εβp + iδ

],(20b)

We note that the factor p0 cosϕ in Eq. (20b) originatesfrom the momentum dependence of the axial electricfield. In the DC limit ω → 0, we obtained Eq. (15).

σDCzr (T ) = − 1

V

∑p,α

f(εαp)Bαzr(p), (21a)

σx,DCxz (T ) = − 1

V

∑p,α

f(εαp)p0 cosϕBαxz(p), (21b)

where Bαµν(p) = −2Im〈∂pµuα(p)|∂pνuα(p)〉 is the Berrycurvature, which is given by

B±zr(p) = ∓ ∆λvF2(v2

F q2r + λ2p2

z + ∆2)3/2, (22a)

B±xz(p) = ± ∆λvF cosϕ

2 (v2F q

2r + λ2p2

z + ∆2)3/2

. (22b)