Long-range Coulomb interaction in nodal-ring semimetals
LLong-range Coulomb interaction in nodal ring semi-metals
Yejin Huh, Eun-Gook Moon,
2, 3 and Yong Baek Kim
1, 4 Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada Kadanoff Center for Theoretical Physics and Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea Canadian Institute for Advanced Research, Toronto, Ontario, M5G 1Z8, Canada (Dated: September 17, 2018)Recently there have been several proposals of materials predicted to be nodal ring semi-metals,where zero energy excitations are characterized by a nodal ring in the momentum space. Thisclass of materials falls between the Dirac-like semi-metals and the more conventional Fermi-surfacesystems. As a step towards understanding this unconventional system, we explore the effects of thelong-range Coulomb interaction. Due to the vanishing density of states at the Fermi level, Coulombinteraction is only partially screened and remains long-ranged. Through renormalization groupand large- N f computations, we have identified a non-trivial interacting fixed point. The screenedCoulomb interaction at the interacting fixed point is an irrelevant perturbation, allowing controlledperturbative evaluations of physical properties of quasiparticles. We discuss unique experimentalconsequences of such quasiparticles: acoustic wave propagation, anisotropic DC conductivity, andrenormalized phonon dispersion as well as energy dependence of quasiparticle lifetime. I. INTRODUCTION
Tremendous efforts have been made to understandthe symmetry-protected gapped topological phases af-ter the discovery of topological insulators . Follow-ing this progress, various theoretical and experimen-tal studies have begun to explore the gapless analogsof symmetry-protected topological phases such as theDirac and Weyl semi-metal , where low energy ex-citations possess Dirac-like spectra. Recently, three-dimensional materials with symmetry-protected Fermiline nodes have also been theoretically proposed and ex-perimentally synthesized . These systems have nodalrings in momentum space protected by various combina-tions of time reversal invariance, inversion, chiral andother lattice symmetries. These non-trivial systems arepredicted to host topologically protected surface states.However, so far no efforts have been made to study theeffects of interactions.In this study, we investigate the effects of the long-range Coulomb interaction in nodal ring semi-metals.This is known in various other Fermion systems. Inthe best-studied system, the Fermi liquid metal, 1 /r -long range interaction is marginal, but Fermi liquid sur-vives due to the strong Thomas-Fermi screening whichmakes the Coulomb interaction effectively short-ranged.This is caused by metals having an extended Fermi sur-face and a constant density of states at the Fermi level.The results are known in the other limit, where the en-ergy vanishes only at isolated points of the Brillouinzone. Graphene (in two-dimensions), Weyl semi-metals(in three-dimensions) and double Weyl semi-metals re-ceive logarithmic corrections due to the Coulomb inter-action that remains marginal . In the quadratic bandtouching case, a non-Fermi liquid phase was found .For anisotropic Weyl fermions, the Coulomb interactionbecomes anisotropic and irrelevant .Nodal ring semi-metals lie in between these two well-studied limits. The energy gap closes on a one-dimensional line node, on which the density of statesvanishes. Because of this, short-range interaction wasfound to be irrelevant . Screening of the Coulombinteraction is expected to be much weaker comparedto the Fermi liquid metal, because fewer states areavailable to participate. Nonetheless, we show belowthat the Coulomb interaction is relevant at the non-interacting fixed point. Through renormalization group(RG) analysis and large- N f computations, we identifya non-trivial interacting fixed point where the partially-screened Coulomb interaction becomes irrelevant, mak-ing the fermions asymptotically free in the low energylimit. This allows us to treat the partially-screenedCoulomb interaction as a perturbation and calculate thelifetime of the quasiparticles. It is found that the quasi-particle scattering rate vanishes as E at low energieseven though the partially-screened Coulomb interactionis still long-ranged. II. MODEL
We start with a non-interacting effective Hamiltonianfor the nodal ring semi-metal. This can be written as H = k x + k y − k F m σ x + γk z σ y ≡ (cid:15) a ( k ) σ a . a = x, y , (1)where the Pauli matrices σ x and σ y describe the orbitalor pseudo-spin degrees of freedom. This Hamiltonian issimilar to that of Ref 18. This system has a nodal Fermiring in the k x − k y plane of radius k F , and a linear dis-persion in the k z -direction. Its energy spectrum is E ± ( k ) = ± (cid:115)(cid:18) k x + k y − k F m (cid:19) + ( γk z ) , (2) a r X i v : . [ c ond - m a t . s t r- e l ] J a n for the empty (+) and filled ( − ) bands. In order to de-scribe the effects of Coulomb interaction, we use the Eu-clidean path integral formalism for the action in 3 + 1dimensions. S = (cid:90) dτ d x ψ † [ ∂ τ − ieφ + H ] ψ + 12 (cid:90) dτ d x ( ∂ i φ ) (3)The bosonic field φ represents the instantaneousCoulomb interaction introduced by the Hubbard-Stratonovich transformation.To study how important the interaction is at low en-ergies, we start with finding the engineering dimensionof the coupling constant. The non-trivial Fermi surface(ring) in the system affects the scaling dimensions of bothfermionic and bosonic fields.Here we use an RG scheme where a momentum cutoffis applied in the directions around the Fermi ring. Wescale the fermion momentum towards the Fermi ring ; k F is fixed and scaling is done only in the Dirac dimen-sions in which there are linear dispersions. Using def-initions k r = (cid:113) k x + k y and ˜ k r ≡ k r − k F , ˜ k r and k z are scaled. However there is no scaling in the angular( φ ≡ cos − ( k x /k r )) direction since this represents thegapless degree of freedom. Because of this anisotropy,it is easier to calculate the scaling dimensions from anaction written in momentum space rather than in theform given in Eq. 3. Here we generalize the expressionto general d -spacial dimensions and write the Coulombinteraction as a 4-fermion term. S ∼ (cid:90) ω, k ψ † ( − iω + H ) ψ + e (cid:90) ω ,ω ,ω , k , k (cid:48) , q q ψ † ( k + q ) ψ ( k ) ψ † ( k (cid:48) − q ) ψ ( k (cid:48) )(4)We have used the notation (cid:82) ω = (cid:82) dω , (cid:82) k = k F (cid:82) d d − k (cid:82) dφ and (cid:82) q = (cid:82) d d q . The constants thathave no scaling dimensions such as k F and π have beendropped for clarity. Note that while k and k (cid:48) are scaledonly in the Dirac directions with d − q isscaled in all d dimensions. This is because the importantcontribution arises from when the momentum carried bythe Coulomb interaction is small and when the fermionsare close to the Fermi ring. The scaling dimensions canbe found to be [ ˜ k r ] = 1, [ k z ] = 1, [ ω ] = 1, [ q i ] = 1,[ ψ ] = − ( d + 1) /
2, and [ e ] = 3 − d . Therefore the criticaldimension is the physical dimension d = 3. From thiswe would conclude that the Coulomb interaction to bemarginal. III. RG ANALYSIS
The energy scales of this problem are the Coulombenergy E c = e mv F , the kinetic energy E k = mv F , and ~q, ! ! ~q, q n ~p, ! (a)$ (b)$ ~q, q n ~p, ! FIG. 1: Diagrammatic representations of (a) Boson self en-ergy and (b) Fermion self energy. Straight arrowed lines rep-resent the fermion propagators and the wiggly lines the bosonpropagators. the energy cutoff E Λ = v F Λ. We also define a velocityanisotropy parameter η = γ/v F , where v F = k F /m is thefermion radial velocity in the k z = 0 plane. The followingdimensionless ratios determine the scaling behaviors. α = E c E k = e v F , β = E c E Λ = e k F v F Λ , η = γv F (5)To allow for anisotropic Coulomb interaction, we use asthe action for the boson, S φ = 12 (cid:90) dτ d x (cid:20) a (cid:0) ( ∂ x φ ) + ( ∂ y φ ) (cid:1) + 1 a ( ∂ z φ ) (cid:21) . (6)We perform a 1-loop momentum shell RG around theFermi ring by calculating the boson and fermion self en-ergies to find the RG flow for various parameters. TheFeynman diagrams for these self energies are shown inFig. 1. The boson self energy is,Π( q, iω ) = − e (cid:90) k Tr[ G ( k + q, Ω + ω ) G ( k, Ω)] , (7)where G ( k, i Ω) = ( − i Ω + H ) − is the bare Green’sfunction of the fermions.For ω = 0, this givesΠ( q ,
0) = e (cid:90) Λ k (cid:16) − (cid:15) a ( k + p/ (cid:15) a ( k − p/ E k + q/ E k − q/ (cid:17) × − E k + q/ + E k − q/ , (8)where E k = E + ( k ) is defined by the dispersion relationshown in Eq. 2. We define the momentum shell integralas (cid:82) Λ k = π ) (cid:82) π dφ ( (cid:82) Λ µ + (cid:82) − µ − Λ ) k F d ˜ k r (cid:82) ∞−∞ dk z with µ =Λ e − d(cid:96) .The resulting integral can be done after expanding theintegrand to second order in q r and q z . We findΠ( q r , q z ) ≈ − e (2 π ) (cid:18) q z m γ k F + q r k F γ (cid:19) d(cid:96) Λ r = − β (cid:48) (cid:18) aq r aη + 1 a q z aη (cid:19) d(cid:96) , (9)where β (cid:48) = β π ) . This is infrared (IR) divergent asΛ r → p = ( k F + p x , , p z ).Σ f ( p ) = − e (cid:90) Λ q H ( p + q ) E ( p q ) 1 a ( q x + q y ) + 1 /a q z ≡ − α (2 π ) ( σ x v F p x F ( aη ) + σ y γp z F ( aη )) d(cid:96) (10)The momentum shell integral is defined as (cid:82) Λ q = π ) ( (cid:82) Λ µ + (cid:82) − µ − Λ ) dq x (cid:82) ∞−∞ dq y dq z . The detailed calcula-tion and expressions for F and F are given in Ap-pendix A. This scaling of the fermion self energy is con-sistent with the marginal engineering dimension of thebare Coulomb interaction. The final RG flow equationsfor α , β (cid:48) , and aη are dαd(cid:96) = α (cid:20) − β (cid:48) (cid:18) aη + 2 aη (cid:19) − α (2 π ) F ( aη ) (cid:21) dβ (cid:48) d(cid:96) = β (cid:48) + β (cid:48) (cid:20) − β (cid:48) (cid:18) aη + 2 aη (cid:19) − α (2 π ) F ( aη ) (cid:21) d ( aη ) d(cid:96) = aη (cid:34) β (cid:48) (cid:18) aη − aη (cid:19) + α (2 π ) ( F ( aη ) − F ( aη )) (cid:35) (11)There are two fixed points: the non-interacting fixedpoint at α = 0 , β (cid:48) = 0 ( aη is arbitrary) is unstable andthe interacting one at α = 0 , β (cid:48) = 1 , aη = 1 / α is marginally ir-relevant and β is relevant. The non-zero value of β (cid:48) at thenon-trivial interacting fixed point shows a strong renor-malization of the Coulomb interaction while α = 0 showsthat the renormalized Coulomb interaction is irrelevantto the fermions.After a step of eliminating high energy degrees of free-dom, the boson propagator D ( q ) can be written as D − ( q )= a (cid:18) β (cid:48) aη d(cid:96) (cid:19) ( q x + q y ) + 1 a (1 + β (cid:48) aηd(cid:96) ) q z . (12)Therefore the anomalous dimension is 1 which arises fromthe existence of a k F scale. The renormalized propagatorat the new interacting fixed point satisfies D − ( q ) ∼ q − r + | q z | − = q r + | q z | . (13)This will be confirmed by a direct calculation below. IV. LARGE N f CALCULATION
The screened Coulomb interaction in d = 3 can also bedirectly calculated using the random phase approxima-tion. This can be viewed as a large N f calculation where N f is the number of fermion flavors. The physical case is N f = 2 for the spin states. After introducing a sum overfermion flavors and modifying the coupling constant to e √ N f , the same Eqs. 7 and 8 are calculated without the q expansions or the k cutoffs. The result isΠ( q r , q z , ω = 0) = − e (2 π ) (cid:18) k F q r γ C + 2 m | q z | C (cid:19) , (14)where C = 6 . C = 7 .
28 are calculated numerically.(Details of the calculation are presented in the Supple-mental Material.) Therefore for a small | q | , the screenedCoulomb potential is V s ( q ) ∼ C k F γ q r + 2 mC | q z | . (15)Notice that the screened Coulomb interaction still hasalgebraic momentum dependence 1 / | q | in sharp contrastto that of Fermi liquids. The presence of k F in nodal ringexcitation is not enough to make the Coulomb interactionshort-ranged. Furthermore, the directional dependence isqualitatively the same even though the nodal ring spec-trum is strongly anisotropic. It is important to note thatthis result is independent of choice in RG scheme sinceno cutoff has been imposed. The RG calculation is aweak coupling analysis whereas this is a strong couplinganalysis with 1 /N f as a control parameter. However thisresult is still consistent with the RG result presented inEq. 13, which provides validity to both.The imaginary part of the bosonic self energy deter-mines the decay. This can be calculated by performing aWick rotation. This gives the resultsImΠ( q r = 0 , q z , ω + i + ) ∼ θ ( ωγq z − q r , q z = 0 , ω + i + ) ∼ ω k F q r . (16)(Full expressions are presented in the Supplmental Ma-terial.) Therefore there is no damping in the direc-tion perpendicular to the ring, while the boson with in-plane momentum shows damping less than that of theFermi liquid. Landau damping, for comparison, givesImΠ( q, ω ) ∼ ω/q .The vertex correction vanishes at the one loop level.This can be easily checked by setting all the externalmomenta and frequency to 0. This is as required by theWard identity because the fermion self energy (Fig. 1(b))has no frequency dependence. ~q, ! ! ~q, q n ~p, ! (a)$ (b)$ ~q, q n ~p, ! FIG. 2: The straight line represents the fermion propagatoras in Fig. 1 and the double wiggly line represents the renor-malized boson propagator.
V. FATE OF THE QUASIPARTICLES
We have seen above that the bosons are strongly renor-malized. However since the screened Coulomb interac-tion is still long-ranged, we must check whether the in-teraction destroys the fermi liquid or not. The fate of thequasiparticles can be determined from the self energy ofthe fermions. Using the renormalized boson propagatoras shown in Fig. 2, the self energy isΣ f ( p, iω n ) = (cid:90) q , q n − iω n − iq n + H ( p + q ) × − e q − Π( q, iq n ) . (17)We find that this is both UV and IR convergent andtherefore the screened Coulomb interaction is an ir-relevant perturbation to the fermions. Therefore thefermions remain as valid quasiparticles of the system andare effectively decoupled. The result is again consistentwith the RG analysis presented earlier.The lifetime of these fermions can be found from theimaginary part of the self energy after analytic contin-uation by the relation 1 /τ = − f . The channelthat has the largest contribution is the one that satis-fies the Fermi’s Golden rule. Focusing on this channel,for a fermion with initial momentum p close to the linenode and energy E p , we have1 τ ∼ e (cid:90) k , q q, (cid:18) − (cid:15) a ( k ) (cid:15) a ( k + q ) E k E k + q (cid:19) × δ ( E k + E k + q + E p + q − E p ) . (18)Leading order contributions only come from the regionwhere the intermediate wave vector k is very close to thenodal ring. In fact, k needs to be closer to the ring than q is to the origin. We find that τ ∼ mk F E p C ( χ p ) where χ p controls the in-plane component versus the out-of-planecomponent of p . C ( χ p ) is a numerical factor that canbe numerically calculated for any χ p . (Details of thiscalculation can be found in the Appendix.) It is identi-cally zero when the in-plane component disappears. Thisis because there is no decay channel that satisfies theenergy-momentum conservation. This can be seen aboveby ImΠ( q r = 0 , q z , ω ) = 0 when q z is small. Overall, 1 /τ ∼ E p and therefore the quasiparticles are long-lived.Interestingly it has the same energy dependence as theFermi liquid case. While the density of states of thissystem is vanishing at the Fermi level ( ∼ ω ), this is com-pensated by the partially screened Coulomb potential. VI. EXPERIMENTAL SIGNATURES
Similar to surface acoustic wave propagation experi-ments in two-dimensional materials , a bulk soundwave propagation measurement in a periodic potentialwith wavelength λ ∼ /q can be used to probe themomentum dependence of the dielectric function. Thesound velocity shift and attenuation is found as∆ v s v s = α f ( q ) , κ = α q f ( q )1 + f ( q ) (19)where f ( q ) = C (cid:16) v s v F (cid:17) k F q , C = π e γ(cid:15) and α is the cou-pling constant between the piezoelectrics and mediumwhich depends on the geometry. In contrast, in the Fermiliquid metal, f ( q ) = C (cid:48) v s v F (cid:16) k F q (cid:17) , C (cid:48) = e (cid:15)v F .A related physical observable is the DC conductivity.Using the Kubo formula and taking the limits q → ω →
0, we find the DC conductivity in the cleanlimit to be finite (due to the underlying particle-holesymmetry) and anisotropic: σ xx = σ yy = e (cid:126) k F v F γ and σ zz = e (cid:126) k F γ v F with restored units. These results areconsistent with a previous compuatation for a similarsystem . The characteristic screening of the Coulombinteraction also affects the phonon dispersion. Longitu-dinal acoustic phonon dispersion follows ω ( q ) = Ω p /(cid:15) ( q )where Ω p is the plasma frequency of the ions. This showsan unusual ω ( q ) ∼ √ q dependence for small q . VII. CONCLUSION
It is shown that the long-range Coulomb interaction innodal ring semi-metals leads to a non-trivial fixed pointwhere the screened Coulomb interaction acquires ananomalous dimension. On the other hand, the screenedCoulomb interaction becomes irrelevant at the interact-ing fixed point while remaining long-ranged. Hence thequasi-particles are asymptotically free and physical prop-erties can be computed using a perturbation theory. Weshow that the quasi-particles have a long life time eventhough the screening of charged impurity potential wouldfollow an unusual power-law form due to the anoma-lous dimension. Sound wave propagation and acousticphonon dispersion show unique momentum dependences.Anisotropic DC conductivity is found and is proportionalto the size of the nodal ring. These properties could betested in future experiments. Interesting future direc-tions include studies of the coupling to critical bosonicmodes and impurity/disorder effects.
Acknowledgments
We thank Jun-Won Rhim, Hae-Young Kee, and YigeChen for helpful discussions. This work was supported by the NSERC of Canada, Canadian Institute for Ad-vanced Research, and Center for Quantum Materials atthe University of Toronto.
Appendix A: RG Calculation
The fermion self energy isΣ f ( k F + p x , , p z ) = − e (cid:90) Λ q H ( p + q ) E ( p + q ) 1 a ( q x + q y ) + 1 /a q z ≈ − e (cid:90) Λ q (cid:32) aq z γ ( a ( q x + q y ) + q z )( v F q x + γ q z ) / v F p x σ x + aq x v F ( a ( q x + q y ) + q z )( v F q x + γ q z ) / γp z σ y (cid:33) (A1)Using local Cartesian coordinates for q , we integrate from −∞ to ∞ in q y and q z , and from µ = Λ e − d(cid:96) to Λ in | q x | .This gives usΣ f ( k F + p x , , p z ) = − α (2 π ) (cid:32) σ x v F p x a η K (cid:0) − a η (cid:1) − E (cid:0) − a η (cid:1) a η − σ y γp z (cid:0) E (cid:0) − a η (cid:1) − K (cid:0) − a η (cid:1)(cid:1) a η − (cid:33) d(cid:96) ≡ − α (2 π ) ( σ x v F p x F ( aη ) + σ y γp z F ( aη )) d(cid:96) , (A2)where E ( x ) is the complete elliptic integral of the second kind defined by E ( x ) = (cid:82) π/ (1 − x sin θ ) / dθ and K ( x )is the complete elliptic integral of the first kind defined by K ( x ) = (cid:82) π/ (1 − x sin θ ) − / dθ . The energy ratio α andanisotropy parameter η are defined in the main text.Including the self energies calculated, we can write the effective action as the following. S = S + (cid:90) d xψ † ( − Σ) ψ + 12 (cid:90) d xφ ( − Π) φ = (cid:90) d x (cid:34) ψ † (cid:18) ∂ τ − ieφ + σ x v F (cid:18) α (2 π ) F ( aη ) d(cid:96) (cid:19) ∂ r + σ y γ (cid:18) α (2 π ) F ( aη ) d(cid:96) (cid:19) ∂ z (cid:19) ψ + 12 a (cid:18) β (cid:48) aη d(cid:96) (cid:19) (cid:0) ( ∂ x φ ) + ( ∂ y φ ) (cid:1) + 12 a (1 + β (cid:48) aηd(cid:96) ) ( ∂ z φ ) (cid:35) (A3)RG equations for various parameters are d ln v F d(cid:96) = α (2 π ) F ( aη ) d ln ηd(cid:96) = α (2 π ) ( F ( aη ) − F ( aη )) d ln e d(cid:96) = − β (cid:48) (cid:18) aη + 2 aη (cid:19) d ln ad(cid:96) = 12 β (cid:48) (cid:18) aη − aη (cid:19) (A4)Combining these, we can find RG flow equations for α , β , and aη . These are presented in the main text. Appendix B: Large N f calculation of the screened Coulomb interaction By scaling ˜ k r ≡ k r − k F → | q x | r , k F → | q x | κ and k z → | q z | z , we can write the boson self energy with ω = 0 as thefollowing. We further define ξ ≡ q x m κ/ ( γq z ).Π( q x , q z ) = − e (2 π ) (cid:90) r,θ,z k F | q x || q z | (cid:32) − ξ (2 r + cos θ )(2 r − cos θ ) + ( z + )( z − ) (cid:112) ξ (2 r + cos θ ) + ( z + 1 / (cid:112) ξ (2 r − cos θ ) + ( z − / (cid:33) × (cid:112) ξ (2 r + cos θ ) + ( z + 1 / + (cid:112) ξ (2 r − cos θ ) + ( z − / γ | q z | = − e (2 π ) k F | q x || q z | γ | q z | f ( ξ )= − e (2 π ) k F | q x || q z | q x m κ f ( 1 ξ ) (B1)The second line is better suited to see the behavior of ξ (cid:29) ξ (cid:28) f ( ξ ) can becalculated numerically and fitted by a C + C ξ curve which yields C = 6 .
86 and C = 7 .
28. This gives the leadingorder behavior of the boson self energy at ω = 0 provided in the main text.Effects of finite ω are seen mainly in the imaginary part of the boson self energy which is 0 when ω = 0. This canbe calculated by performing a standard Wick rotation.ImΠ( q, ω + i + ) = − πe (cid:90) k Tr (cid:0) P + ( k + q/ − ( k − q/ (cid:1)(cid:16) δ ( − ω + E k + q/ + E k − q/ ) − ( ω → − ω ) (cid:17) (B2)P α ( k ) are operators that project the states on to the lower and upper bands.P α ( k ) = 12 (cid:16) α H ( k ) | E ( k ) | (cid:17) ( α = ± ) (B3)For a positive frequency, the integral becomesImΠ( q, ω + i + ) = − πe (cid:90) k Tr (cid:0) P + ( k + q/ − ( k − q/ (cid:1) δ ( − ω + E k + q/ + E k − q/ )= − e π (2 π ) (cid:90) dk ⊥ k ⊥ (2 π ) (cid:90) dk z
24 (1 − (cid:15) a ( k + q/ (cid:15) a ( k − q/ E k + q/ E k − q/ ) δ ( − ω + E k + q/ + E k − q/ ) . (B4)The integral is only non-trivial when q (cid:54) = 0. For convenience, we separate it into two cases: one where the externalmomentum lies in the ring plane and the other where it is perpendicular to the plane.ImΠ( q z , ω + i + ) = − πe (2 π ) mπ (cid:114)(cid:16) ωγp z (cid:17) − | q z | Θ( (cid:18) ωγq z (cid:19) −
1) (B5)ImΠ( q x , ω + i + ) = − e (2 π ) k F | q x | γ π (cid:16) − E ( Ω ) + K ( Ω ) (cid:17) if | Ω | < − e (2 π ) k F | q x | γ π √ Ω − (cid:16) − (Ω − E ( − − ) + Ω K ( − − ) (cid:17) if | Ω | > , (B6)where Ω ≡ mk F | q x | ω is the dimensionless frequency. E ( x ) ( K ( x )) is the complete elliptic integral of the second (first)kind defined earlier. Asymptotic behavior of this isImΠ( q r , ω + i + ) = (cid:40) − e (2 π ) k F q r γ π Ω if | Ω | < − e (2 π ) k F q r γ π if | Ω | (cid:29) . (B7) Appendix C: Fermion self energy correction
Here we show the irrelevance of screened Coulomb interaction to the fermions. As a representative example, weonly present the renormalization of the fermion dispersion in the p z -direction. For simplicity, we fix the fermionmomentum to p = ( k F , , p z ) such that it is only slightly off the line node in the p z -direction and set the frequencyto 0. We also fix the internal frequency to 0 (take the Coulomb interaction to be instantaneous) as the effects of anon-zero frequency in the real part of boson propagator is quite small. In the limit where the momentum transfer | q | is small, the bare term of the boson propagator is less important than the self energy correction, and we can setΣ f ( p, → (cid:90) q H ( p + q ) E ( p + q ) e N f Π( q, . (C1)Here we are only interested in the σ y component of the self energy. Imposing a momentum cutoff in the q x and q y direction, we getΣ f (0 , p ) σ y = − N f γp z (cid:90) Λ − Λ dq x (cid:90) Λ − Λ dq y (cid:90) ∞−∞ dq z (cid:0) m (2 k F q x + q x + q y ) (cid:1) (cid:16)(cid:0) m (2 k F q x + q x + q y ) (cid:1) + γ q z (cid:17) / C k F γ | q y | + C m | q z |≈ − N f γp z C z Λ k F , (C2)where C z = G , , C124C22 | , , , , , , π / C1 ∼ .
72. Since the self energy is linear in cutoff, screened Coulomb interaction isirrelevant to the fermions. Alternatively, the full integral without cutoffs can be carried out numerically which givesthe same conclusion.
Appendix D: Lifetime of the fermions
Starting from the Euclidean fermion self energy, we can calculate the imaginary part of the self energy by analyticcontinuation. Σ f ( p, iω ) = e N f (cid:90) q , Ω i (Ω + ω ) + H ( p + q )(Ω + ω ) + E p + q q, Ω) ≈ − e N f (cid:90) q , k π Π( q, (cid:18) − (cid:15) a ( k ) (cid:15) a ( k + q ) E k E k + q (cid:19) iω + E k + E k + q + E p + q E p + q H p + q ( E k + E k + q + E p + q ) + ω (D1)This can be analytically continued by taking iω → ω + iη , and the imaginary part of this would beImΣ f ( p, E p ) = − e N f (cid:90) q , k π Π( q, (cid:18) − (cid:15) a ( k ) (cid:15) a ( k + q ) E k E k + q (cid:19) δ ( E k + E k + q + E p + q − E p ) (cid:18) (cid:15) x ( p + q ) E p + q σ x + (cid:15) y ( p + q ) E p + q σ y (cid:19) , (D2)where we have used the approximation 1Π( q, Ω) ≈ Π( q, Ω)Π( q, . (D3)To proceed, we divide the integral into two regions, one where k is farther from the Fermi ring than q is to the origin( E k > E q ) ( E q is defined in the main text below Eq. 19.) and the opposite case ( E k < E q ). For the former case, wecan expand the energies up to linear order in q . E p + q − E p ∼ E q sin χ q sin χ p + E q cos χ q cos χ p cos θE k + E k + q ∼ E k + E q sin χ q sin χ k + E q cos χ q cos χ k cos φδ ( E k + E k + q + E p + q − E p ) ∼ δ (2 E k + E q sin χ q (sin χ k + sin χ p ) + E q cos χ q (cos χ k cos φ + cos χ p cos θ )) (D4)Here we have defined φ , θ , and χ k (and similarly χ p and χ q ) such that γk z = E k sin χ k k F m ˜ k r = E k cos χ k (cid:126)k r · (cid:126)q r = | k r || q r | cos φγq z = E q sin χ q k F m q r = E q cos χ q (cid:126)p r · (cid:126)q r = | p r || q r | cos θ (D5)where (cid:126)k r is the k projected on to the k x − k y plane. However it is impossible to satisfy the δ -function in Eq. D4 and E k > E q simultaneously. Therefore there is no phase space that conserves momentum and energy in this case, leadingto a 0 contribution to ImΣ f ( p, E p ).For the case where E k < E q , the second line in Eq. D4 needs to be modified. Instead of expanding terms in powersof q , we expand in powers of k r − k F and k z . However q is still assumed to be small compared to k F . E k + E k + q = E k χ k sin χ q + cos χ q cos χ k cos φ (cid:113) cos χ q cos φ + sin χ q + E q (cid:113) cos χ q cos φ + sin χ q + O ( E k E q ) (D6)In the limit of E k /E q < − (cid:15) α ( k ) (cid:15) α ( k + q ) E k E k + q = 1 − sin χ k sin χ q + cos χ q cos χ k cos φ (cid:113) cos χ q cos φ + sin χ q + O ( E k E q ) (D7)which to leading order, has no energy dependence. Combining everything, the integral givesImΣ f ( p, E p ) = − π N f mk F E p C ( χ p ) . (D8)The largest contributions come from the in-plane scattering where the momentum transfer is opposite to the externalmomentum. The angle integrals can be done numerically for any given χ p . For χ p = π/
2, which is when the externalmomentum is only off the Fermi ring in the z -direction, this integral is zero, meaning the lifetime is longer than E p . Appendix E: Propagation of Acoustic Waves
From linear response theory we have, (cid:104) ρ ( q, ω ) (cid:105) = − χ ( q, ω ) φ ext ( q, ω ) = Π ( q, ω ) φ tot ( q, ω ) . (E1)The Π here differs from the RPA calculated in the main text (Π) by a factor of ( ie ) . − χ ( q, ω ) = 1Π ( q, ω ) + V ( q ) (E2) σ xx = − iωq Π ( q, ω ) (E3) − χ ( q, ω ) = 4 πe (cid:15)q − iωq σ xx ( q, ω ) (E4)Using results already obtained, this gives − χ ( q = q ˆ z, ω = v s q ) = (cid:15)q πe − iσ m /σ xx (E5)where σ m = (cid:15)ω πe and R eσ xx ≈ v s k F γ ( v s /v F ) .Induced energy per unit area is δU = − χ | φ ext | (E6)= 12 (cid:15)q πe − i σ m σ xx | φ ext | (E7)Measure the energy shift with respect to the shift for σ xx → ∞ .∆ U = δU − δU ( σ xx = ∞ ) (E8)= 12 (cid:15)q πe − − i σ xx σ m | φ ext | (E9)Acoustic wave with energy density proportional to q . Therefore the energy U per unit surface area is U = q C H .∆ UU = ∆ qq = − ∆ v s v s + iκq (E10)∆ v s v s − iκq = α /
21 + iσ xx ( q, ω ) /σ m (E11)This gives the results presented in the main text. M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010); X.-L. Qi and S.-C. Zhang, ibid. , 1057 (2011). For a review on recent progress, see T. Senthil, AnnualReview of Condensed Matter Physics , 299 (2015) M. Neupane, S.-Y. Xu, R. Sankar, N. Alidoust, G. Bian,C. Liu, I. Belopolski, T.-R. Chang, H.-T. Jeng, H. Lin, A.Bansil, F. Chou, M. Z. Hasan , Nature Commun. 5, 3786(2014). Su-Yang Xu, Chang Liu, Satya K. Kushwaha, RamanSankar, Jason W. Krizan, Ilya Belopolski, Madhab Ne-upane, Guang Bian, Nasser Alidoust, Tay-Rong Chang,Horng-Tay Jeng, Cheng-Yi Huang, Wei-Feng Tsai, HsinLin, Pavel P. Shibayev, Fang-Cheng Chou, Robert J. Cava,M. Zahid Hasan, Science 347, 294-298 (2015). Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng,D. Prabhakaran, S.-K. Mo, Z. X. Shen, Z. Fang, X. Dai,Z. Hussain, Y. L. Chen, Science, , 864 (2014). Xiangang Wan, Ari M. Turner, Ashvin Vishwanath, andSergey Y. Savrasov, Phys. Rev. B , 205101 (2011). Su-Yang Xu, Ilya Belopolski, Nasser Alidoust, MadhabNeupane, Chenglong Zhang, Raman Sankar, Shin-MingHuang, Chi-Cheng Lee, Guoqing Chang, BaoKai Wang,Guang Bian, Hao Zheng, Daniel S. Sanchez, FangchengChou, Hsin Lin, Shuang Jia, M. Zahid Hasan, Science 349,613 (2015). Chenglong Zhang, Su-Yang Xu, Ilya Belopolski, Zhujun Yuan, Ziquan Lin, Bingbing Tong, Nasser Alidoust, Chi-Cheng Lee, Shin-Ming Huang, Hsin Lin, Madhab Neu-pane, Daniel S. Sanchez, Hao Zheng, Guang Bian, Jun-feng Wang, Chi Zhang, Titus Neupert, M. Zahid Hasan,Shuang Jia, arXiv:1503.02630 (2015). Jun Xiong, Satya K. Kushwaha, Tian Liang, JasonW. Krizan, Wudi Wang, R. J. Cava, N. P. Ong,arXiv:1503.08179 (2015). W. Witczak-Krempa and Y. B. Kim, Phys. Rev. B 85,045124 (2012). A. A. Burkov, M. D. Hook, and Leon Balents, Phys. Rev.B , 235126 (2011). Jean-Michel Carter, V. Vijay Shankar, M. Ahsan Zeb, andHae-Young Kee, Phys. Rev. B 85, 115105 (2012). Shengyuan A. Yang, Hui Pan, and Fan Zhang, Phys. Rev.Lett. , 046401 (2015). Yige Chen, Yuan-Ming Lu, and Hae-Young Kee, NatureCommunications , 6593 (2015). R. Schaffer, E. K. H. Lee, Y.-M.Lu, Y. B. Kim, Phys. Rev.Lett. , 116803 (2015). J. W. Rhim and Y. B. Kim, Phys. Rev. B , 045126(2015) Yuanping Chen, Yuee Xie, Shengyuan A. Yang, Hui Pan,Fan Zhang, Marvin L. Cohen, and Shengbai Zhang, NanoLett. , 6974 (2015). Youngkuk Kim, Benjamin J. Wieder, C. L. Kane, and An- drew M. Rappe, Phys. Rev. Lett. 115, 036806 (2015). Lilia S. Xie, Leslie M. Schoop, Elizabeth M. Seibel, QuinnD. Gibson, Weiwei Xie, and Robert J. Cava, APL Mat. 3,083602 (2015). Kieran Mullen, Bruno Uchoa, and Daniel T. Glatzhofer,Phys. Rev. Lett. Minggang Zeng, Chen Fang, Guoqing Chang, Yu-An Chen,Timothy Hsieh, Arun Bansil, Hsin Lin, and Liang Fu,arXiv:1504.03492 (2015). Hongming Weng, Yunye Liang, Qiunan Xu, Yu Rui, ZhongFang, Xi Dai, Yoshiyuki Kawazoe, Phys. Rev. B 92, 045108(2015). Rui Yu, Hongming Weng, Zhong Fang, Xi Dai, Xiao Hu,Phys. Rev. Lett. 115, 036807 (2015) . Hongming Weng, Chen Fang, Zhong Fang, B. AndreiBernevig, and Xi Dai, Phys. Rev. X 5, 011029 (2015) . Valeri N. Kotov, Bruno Uchoa, Vitor M. Pereira, F.Guinea, and A. H. Castro Neto, Rev. Mod. Phys., ,1067 (2012). Gonzalez, J., F. Guinea, and M. A. H. Vozmediano, Phys.Rev. Lett. , 3589. (1996). Hiroki Isobe and Naoto Nagaosa, Phys. Rev. B. , 165127(2012). Hsin-Hua Lai, Phys. Rev. B , 235131 (2015). Shao-Kai Jian and Hong Yao, Phys. Rev. B. , 045121(2015). B. J. Yang and Y. B. Kim, Phys. Rev B. , 085111 (2010). Eun-Gook Moon, Cenke Xu, Yong Baek Kim, and LeonBalents, Phys. Rev. Lett. Bohm-Jung Yang, Eun-Gook Moon, Hiroki Isobe, andNaoto Nagaosa, Nat. Phys.
774 (2014). T. Senthil and R. Shankar, Phys. Rev. Lett. E. K. H. Lee, S. Bhattacharjee, K. Hwang, H.-S.Kim. H.Jin, Y. B. Kim, Phys. Rev. B. , 205132 (2014). A. Liam Fitzpatrick, Shamit Kachru, Jared Kaplan, andS. Raghu, Phys. Rev. B A. R. Huston and Donald L. White, J. Appl. Phys. , 40(1962). B. I. Halperin, Patrick A. Lee, and Nicholas Read, Phys.Rev. B. Steven H. Simon, Phys. Rev. B.39