A new cubic Hall viscosity in three-dimensional topological semimetals
Iñigo Robredo, Pranav Rao, Fernando de Juan, Aitor Bergara, Juan L. Mañes, Alberto Cortijo, M. G. Vergniory, Barry Bradlyn
AA new cubic Hall viscosity in three-dimensional topological semimetals
I˜nigo Robredo,
1, 2, ∗ Pranav Rao, ∗ Fernando de Juan,
1, 4
Aitor Bergara,
1, 2, 5
Juan L. Ma˜nes, Alberto Cortijo,
6, 7
M. G. Vergniory,
1, 4, † and Barry Bradlyn ‡ Donostia International Physics Center, 20018 Donostia-San Sebastian, Spain Department of Physics, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain Department of Physics and Institute for Condensed Matter Theory,University of Illinois at Urbana-Champaign, Urbana IL, 61801-3080, USA IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013 Bilbao, Spain Centro de F´ısica de Materiales CFM, CSIC-UPV/EHU,Paseo Manuel de Lardizabal 5, 20018 Donostia, Basque Country, Spain Departamento de F´ısica de la Materia Condensada,Universidad Aut´onoma de Madrid, Madrid E-28049, Spain Condensed Matter Physics Center (IFIMAC), Madrid E-28049, Spain (Dated: February 5, 2021)While nondissipative hydrodynamics in two-dimensional electron systems has been extensivelystudied, the role of nondissipative viscosity in three-dimensional transport has remained elusive. Inthis work, we address this question by studying the nondissipative viscoelastic response of threedimensional crystals. We show that for systems with tetrahedral symmetries, there exist new,intrinsically three-dimensional Hall viscosity coefficients that cannot be obtained via a reductionto a quasi-two-dimensional system. To study these coefficients, we specialize to a theoreticallyand experimentally motivated tight binding model for a chiral magentic metal in (magnetic) spacegroup [(M)SG] P . Introduction.
The discovery of hydrodynamic flowin two-dimensional metallic systems[1, 2] has spurred arenewed interest in the study of nondissipative “Hall”viscosity. In two dimensional systems with rotationalsymemetry, there is a single Hall viscosity coefficient, re-lated to the topological properties of the occupied elec-tronic states[3–11]. In very clean systems, the Hall vis-cosity is expected to manifest in width-dependent cor-rections to the Hall conductance of mesoscopic channels,backflow corrections to the local current density nearpoint contacts, and in moments of the semiclassical dis-tribution function[12–15]. Local voltage measurementson graphene samples in magnetic fields have shown sig-natures of the Hall viscosity[16]. Beyond two dimensions,however, the role of nondissipative viscosity in transportremains largely unexplored. Reports of hydrodynamicbehavior in topological semimetals[17], and the grow-ing interest in magnetic topological semimetals[18], raisethe question of how to generalize the Hall viscosity tothree dimensions. Preliminary efforts have focused onquasi-two dimensional transport[19–24], or made use ofpreferred “polar” directions such as the Weyl node sep-aration direction in topological semimetals. It is alsoknown that octahedral symmetry forbids the presenceof a nonzero Hall viscosity[25, 26]. However, magneticcrystals may have nonpolar point group symmetries that are not octahedral; the nondissipative hydrodynamic re-sponse of such systems remains an open question.In this work, we explore this issue for the first timeby examining the viscous response of a threefold degen-erate “spin-1” fermion. We focus on the experimentallyinteresting case of the cubic MSG P . a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b a new, fundamentally three dimensional nondissipativeviscous force, which to our knowledge has not been en-countered before in the literature. For uniaxial flows,this new viscosity gives a force perpendicular to the flowdirection which vanishes when the velocity is constantalong the direction of flow. We discuss the implication ofour work for chiral magnetic topological semimetals suchas the family Mn IrSi, Mn IrGe, Mn Ir − y Co y Si, andMn CoSi − x Ge x [18, 42, 43] in MSG P . Hall viscosity with cubic symmetry.
Let us examine thesymmetry properties of the Hall viscosity tensor in sys-tems with tetrahedral symmetry. The tetrahedral pointgroup (denoted 23) is the simplest cubic group, and isgenerated by twofold rotations about the ˆ x , ˆ y and ˆ z axes,as well as a threefold rotation about the ˆ x + ˆ y + ˆ z cu-bic body diagonal. For a detailed exposition of the irre-ducible representations (irreps) of the point group 23, werefer the reader to the Supplementary Material, as well asto the group theory tables on the Bilbao CrystallographicServer[44–46]. We define the Hall viscosity tensor as theantisymmetric (and therefore nondissipative) componentof the viscosity tensor η i kj ‘ , [4, 11, 19, 25, 47]( η H ) i kj ‘ ≡ (cid:0) η i kj ‘ − η k i‘ j (cid:1) . (1)where i, j, k, ‘ are indices that run over the three spa-tial directions. The Hall viscosity is explicitly odd un-der time-reversal symmetry[48], and for a fluid with anonuniform velocity field v ‘ , leads to a viscous stress[49] δτ ij = ( η H ) i kj ‘ ∂ k v ‘ (2)Our goal is to identify the independent symmetry-allowedHall viscosity coefficients. Since these coefficients are de-fined to be scalars, we can determine them by finding allrank-4 antisymmetric tensors invariant under 23. Intro-ducing the irreducible tensors,Θ aij = ( √ ( δ i δ j + δ i δ j − δ i δ j ) a = 1 δ i δ j − δ i δ j a = 2 (3)Λ ijk = | (cid:15) ijk | (4)with (cid:15) ijk the Levi-Civita symbol, we can form two invari-ant tensors, and thus identify two viscosity coefficientscompatible with tetrahedral symmetry: ( η H ) i kj ‘ = − η (cid:15) ab Θ aij Θ bk‘ + η √ mij (cid:15) km ‘ − Λ mk‘ (cid:15) im j )= η ( λ ∧ λ ) i kj ‘ + iη √ λ ∧ λ + λ ∧ λ − λ ∧ λ )) i kj ‘ , (5) where (cid:15) ab is the two-dimensional Levi-Civita symbol. Inthe second line we have reexpressed the antisymmetricproduct of irreducible tensors in terms of the Gell-Mannmatrices λ (defined explicitly in the SM). This makes clear that the η term is the antisymmetric dot prod-uct of matrices transforming in the two 3-dimensional T(vector) irreps L ≡ T ( λ , − λ , λ ) ˜ L ≡ T ( λ , λ , λ ) (6)each spanned by a triplet of Gell-Mann matrices.Crucially, neither of the coefficients η , require a pre-ferred spatial direction. This is in direct contrast to thefamiliar “quasi-2D” Hall viscosities which must be pro-portional to a pseudovector (such as a magnetic field).Thus, η and η are new, essentially three-dimensionalHall viscosities, which can be nonzero in systems withbroken rotational symmetry. By contrast, octahedralsymmetry requires η = η = 0, as the two tensors inEq. (5) do not transform in the trivial representationof the group 432 ( O ). Note also that η and η canbe nonzero in centrosymmetric point groups such as T h ( m ¯3).Next, we compute the viscous force density that is pro-duced by these Hall viscosities, f ηj = − ∂ i δτ ij = ( η H ) i kj ‘ ∂ i ∂ k v ‘ , (7)where f η is the force density and δτ ij is the viscous stresstensor. We find that η and η contribute additively to f η : f ηj = η + η √ mik ∂ i ∂ k (cid:0) (cid:15) mj‘ v ‘ (cid:1) . (8)Thus we see that the fully symmetric tensor Λ, whichis only invariant in systems with tetrahedral symmetry,plays a key role in generating the nondissipative forces.This should be contrasted with quasi-2D Hall viscousforces, which take the form f η, Dj = η D B m ∇ ( (cid:15) mj‘ v ‘ ) (9)and require a symmetry-breaking pseudovector B .Finally, since only the sum η + η appears in theviscous forces, there must exist a divergenceless contactterm which shifts between η and η in the bulk. Thisterm is δτ ij = C (cid:15) mik Λ mj‘ ∂ k v ‘ , (10)which shifts η → η + √ C η → η − √ C , (12)analogous to the bulk redundancy between Hall viscosityand odd pressure in two-dimensional systems[11, 50]. Weshow the effects of η , in Fig. 1a. Tight-binding model.
Let us now consider a model fora cubic chiral magnetic system, and compute the valuesof η , . We consider a tight binding model of the form H = X nm, r , r c † n r t r , r nm c m r (13)consisting of s -type orbitals at 4a Wyckoff position ofSG P2 n and m label thefour orbitals, and r , r index the unit cells of the crystalwith lattice spacing a . The nearest-neighbor hopping t r , r nm has uniform magnitude t , and we break time-reversalsymmetry with a cubic-symmetric magnetic flux φ via aPeierls substitution[51]. Shifting to momentum space,and suppressing the orbital n and m indices, we write H = X nm k c † k f ( k ) c k (14)An explicit form of f ( k ) can be found in the SM, andwe show the spectrum in Fig. 1b. As a simple exampleof where to find nonvanishing cubic Hall viscosity, wefocus on the physics near the Γ = ( k x , k y , k z ) = (0 , , f ( k ) ≈ t cos( φ ) iv F e iφ k T − iv F e − iφ k h ! , (15)where v F = ta . The lower-right block corresponds tothe spin-1 bands, and can be expressed in terms of thevectors L and ˜ L as, h = v F (cid:16) cos( φ ) k · L + sin( φ ) k · ˜ L (cid:17) − t cos( φ ) λ (16)When φ = 0, h describes an SO (3)-invariant spin-1fermion. The vector ˜ L in Eq. (6) parameterizes thebreaking of SO (3) to the discrete point group 23.From Eq. (15), we see that when φ is small there is agap of order t separating the spin-0 and spin-1 fermionsat Γ. Thus for small φ and k , transitions from the three-fold to onefold degeneracies mediated by the off-diagonalelements of f ( k ) are parametrically small, and we can re-strict our attention to the spin-1 fermion. Furthermore,in the continuum limit a → v F = ta fixed, we seethat the gap becomes infinitely large. With this in mind,we focus specifically on the threefold fermion. Stress response.
To compute the Hall viscosity, we em-ploy the stress-stress form of the Kubo formula( η H ) µ λν ρ = 12 ω + V Z ∞ dte iω + t (cid:0) h [ T µν ( t ) , T λρ (0)] − [ T λρ ( t ) , T µν (0)] i (cid:1) . (17)To do so we first must define the stress tensor T µν = X nm k c † n k T µν ( k ) c m k , (18) corresponding to Eq. (13). We can do this in two ways:either indirectly by considering an electron-phonon cou-pling ansatz[38, 39, 52] and perturbing the backgroundlattice, or by perturbing the electronic degrees of freedomdirectly via coupling to background geometry[10, 11]. Werefer to the resulting stress tensors the phonon and con-tinuity stress, respectively.In the phonon method, strain is introduced into themodel through small displacements of the orbital posi-tions, modifying the hopping parameters t r , r nm as t r , r → e − ( δ r ) t r , r + O ( δ r ) . (19)Above, δ r is the change in distance between orbitals givenby the applied strain as u µν = ∂ µ δr ν (Note that we do not symmetrize the strain tensor, and though this ob-ject is sometimes called the distortion tensor, we choosethis convention to be consistent with[10, 11]). Applyingthis prescription to Eq. (13) we define the phonon stresstensor as T ( p ) µν = δH ( u µν ) δu µν (20)Given the structure of the viscosity tensor Eq. (5) andthe fact that antisymmetric strains enter only at higherorders in δ r in Eq. (19), it suffices to consider “diagonal”strains (i.e. u xx , u yy and u zz ) [53]. We find that to firstorder (see SM for explicit form) t , t → t + ( u xx + u yy ) tt , t → t + ( u yy + u zz ) tt , t → t + ( u zz + u xx ) t (21)The diagonal phonon stress tensor around the Γ point isthen given by T ( p ) xx ( k ) = v F cos( φ ) ( k x L x + k y L y )+ v F sin( φ ) (cid:16) k x ˜ L x + k z ˜ L z (cid:17) (22) T ( p ) yy ( k ) = v F cos( φ ) ( k y L y + k z L z )+ v F sin( φ ) (cid:16) k x ˜ L x + k y ˜ L y (cid:17) (23) T ( p ) µν transforms as a tensor in the point group 23, whichis the point group describing both the underlying latticeand the Γ point. Note that even when φ = 0, althoughthe Hamiltonian h is invariant under SO (3), T ( p ) µν is co-variant only under the discrete group 23.In the continuity method, the stress tensor T ( c ) µν is de-fined via a lattice analog of the momentum continuityequation (See SM)[11], resulting in T (c) µν ( k ) = (cid:18) k ν ∂ µ f ( k ) + i (cid:15) µνρ (cid:2) f ( k ) , L int ρ (cid:3)(cid:19) . (24) T (c) µν contains contributions from “kinetic” strains (spatialdeformations) and from “spin” strains due to the internalangular momentum L int . The overall stress generalizesthe Belinfante (improved) stress in field theory[11, 54, 55]In our model, we have L Γ int = 0 ⊕ L describing the spin-0 and spin-1 excitations. Using this, we can compute thestress tensor near the Γ point restricted to the spin-1fermion to find (See SM) T (c) µν = v F cos( φ )2 ( k µ L ν + k ν L µ )+ v F sin( φ )2 (cid:18) k ν ˜ L µ − k µ ˜ L ν + X aρλ (cid:15) µνρ Θ aρλ k λ (cid:15) ab v b (cid:19) , (25)where a, b = 1 , T (c) µν = T ( p ) µν . In the continuity approach,antisymmetric stress (caused by anisotropy) enters atorder φ and T (c) µν matches the symmetries of the Blochhamiltonian at the Γ point - when φ = 0 the continuitystress is SO (3)-covariant. By contrast, when φ = 0 thephonon stress is anisotropic. The distinction betweenthe two stress tensors stems from their different physi-cal interpretations: The phonon method is sensitive tothe nonzero orbital positions of the 4 a Wyckoff position(see SM), which result in anisotropy in T ( p ) µν when φ = 0.On the other hand, the continuity method provides along-wavelength stress tensor that averages over intra-unit cell momentum transport, and so is sensitive onlyto the symmetries of the effective Hamiltonian. Below,we will compute the viscous response for both T ( p ) µν and T (c) µν . Hall viscosity.
Next we compute the Hall viscosity co-efficients η and η from Eq. (5), and the physical re-sponse η tot = η + η , for both the phonon and continuitymethods. Focusing on the spin-1 fermion, we can simplifythe Kubo formula Eq. (17) in terms of eigenstates | n i of h as η H ijkl = − Z d k X n = m O nm ∆ (cid:15) nm Im ( h n | T ij | m i h m | T kl | n i )(26)where ∆ (cid:15) nm = (cid:15) n − (cid:15) m , and the relative occupation factoris O nm = n ( (cid:15) n − µ, T ) − n ( (cid:15) m − µ, T ) with n ( (cid:15), T ) = (1 + e (cid:15)/T ) − the Fermi distribution with chemical potential µ and temperature T . We now specify to T = 0.For the phonon method, the stress tensor Eq. (20) isexplicitly symmetric under µ ↔ ν and therefore η = η is zero. The total viscosity in this case is entirelydue to η = η , and given by (to first order in φ ): η ( p )tot = η ( p )1 = v F ( β (cid:0) − + 60 µ (cid:1) φ, µ > β (cid:0) − + 42 µ (cid:1) φ, µ < β = π ≈ . φ is entirely due to η . Given the expressions for the energies and states of the threefold inthe SM, the integrand for η in Eq. (26) has an energy de-nominator that is odd in k z at order φ , which suppressesthe zeroth order contribution from the numerator. Whenthe states are taken to zeroth order in φ , the numeratoris odd in k x and k y , and when the states are taken tofirst order in φ the only nonvanishing matrix elements inthe numerator are odd in k z , all of which leads to η = 0.[56]. The total viscosity is η ( c )tot = η ( c )2 = v F ( β (cid:0) Λ − µ (cid:1) φ µ > − β (cid:0) Λ + µ (cid:1) φ µ < β = π ≈ . µ = 0, the viscosityis discontinuous. This discontinuity arises from the factthat, since the antisymmetric part of the coninuity stressis linear in φ , we must consider the unperturbed bandstructure in the energy denominators in Eq. (26). When φ = 0, the band structure for a spin-1 fermion has a flatband bisecting two linealry dispersing bands. The fillingof the flat band when µ passes through zero then causesthe discontinuity in η , which we can attribute to thecontribution of this band to the Hall viscosity. We plot η (c ,p )tot in Fig. 1d.Similar to the Hall viscosity for Dirac fermions in twodimensions[11, 39, 57], we see that both η ( p )tot and η (c)tot consist of two terms, one of which depends explicitly onthe cutoff Λ. We can interpret the cutoff-independentcontribution (or, more properly, its derivative with re-spect to chemical potential) as the Fermi surface contri-bution to the Hall viscosity, while the cutoff-dependentterm parametrizes unknown contributions to the viscos-ity from occupied states at large momenta. Using thecontinuity form of the stress tensor, we can go beyondthis approximation to compute the Hall viscosity for thefull tight-binding model numerically. We give the detailsof this calculation in the SM. Conclusion.
In this work, we have highlighted amanifestly three dimensional cubic Hall viscosity, whichappears with tetrahedral symmetry. We have shownthrough explicit calculations how this viscosity can befound by looking at the threefold fermion at the Γ pointin MSG P . a) b)c) d) η ( � ) η ( � ) - ��� - ��� � ��� ��� - ��� - ���������� μ η ℏ � � Γ X M Γ R X - ����� � � Λ - Λ - � - � - � � � � � - ��� - ���������� � � Figure 1. a) Schematic of η and η . Dynamic strains (yellow)and viscosity give rise to stresses (orange). In response to adynamic strain that elongates the length and width of a cubicparcel of fluid while compressing the depth, η produces adiagonal shear stress. In response to a dynamic rotation ofthe parcel, η produces an off-diagonal shear stress. b) Bandstructure of the full tight binding model Eq. (15). The gap atthe Γ point scales with the hopping strength t . c) Effectivethree band description of the Γ point for µ near the threefolddegeneracy, with Λ a momentum cut-off chosen to regulatethe Hall viscosity, which is plotted in d) for the phonon “(p)”and continuity “(c)” methods.
21] or thermoelectric transport coefficients[17] in mag-nets in the tetrahedral SGs (Nos. 195–206) would re-veal the signatures of our 3D viscosity. For a three-dimensional cubic magnet with approximate Galileansymmetry at low energies, we can relate the force tensorEq. (7) to a contribution to the wavevector dependentHall conductivity[10, 47, 58] ω δσ Hij ∝ η tot Λ mk‘ q k q ‘ (cid:15) mij ≡ η tot V ‘ ( q ) (cid:15) ‘ij , (29)where we have introduced the vector V ( q ) to highlightthe structural parallel with the expression for the naturaloptical activity of a crystal[30]. We can decompose thevector V into longitudinal and transverse components as V k = ˆ q ( ˆq · V ) = 6 η tot q x q y q z / | q | and V ⊥ = V − V k . Wethen see that V k indeed gives a q -dependent correctionto natural optical activity, while V ⊥ leads to a Hall cur-rent response proportional to the longitudinal componentof the electric field. Note, crucially, that V k vanishes forplane waves at normal incidence to the sample. Analo-gous considerations for flow in narrow channels suggestthat η tot may play a role in interaction-dominated trans-port in narrow channels[12, 13].Chiral magnets such as the family of Mn IrSi[42, 43]are promising platforms to study these effects. As wasshown in Ref. 18, this compound has a noncollinear mag-netic configuration preserving the size of the unit cell; group theory analysis showed further that the groundstate magnetic order preserved all of the unitary symme-try operations consistent with MSG P
3. Another inter-esting candidate is MnTe in MSG P a ¯3 (No. 205.33)[59–61]. It has a reported noncollinear magnetic structure,with the magnetic moments of the four inequivalent man-ganese ions pointing along the cubic body diagonals. Al-though naturally a semiconductor, Ag-doping could in-crease the carrier concentration[62]. Our findings showthat nondissipative hydrodynamics in three-dimensionalcrystals holds exciting physics beyond what can be in-ferred from two-dimensional materials.
Acknowledgements.
P.R. and B.B. acknowledge sup-port from the Alfred P. Sloan Foundation, and the Na-tional Science foundation under grant DMR-1945058.M.G.V. and I.R. acknowledge the Spanish Ministe-rio de Ciencia e Innovacion (grant number PID2019-109905GB-C21). A.C. acknowledges financial supportthrough European Union structural funds, the Comu-nidad Autonoma de Madrid (CAM) NMAT2D-CM Pro-gram (S2018-NMT-4511) and the Ramon y Cajal pro-gram through the grant RYC2018-023938. A.B. ac-knowledges financial support from the Spanish Min-istry of Science and Innovation (PID2019-105488GB-I00). The work of J.L.M. has been supported by SpanishScience Ministry grant PGC2018-094626-B-C21 (MCI-U/AEI/FEDER, EU) and Basque Government grantIT979-16. ∗ These authors contributed equally to this work † [email protected] ‡ [email protected][1] A. Lucas and K. C. Fong, Hydrodynamics of electronsin graphene, Journal of Physics: Condensed Matter ,053001 (2018).[2] J. A. Sulpizio, L. Ella, A. Rozen, J. Birkbeck, D. 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Volume 47), 1975.343 p. , Vol. 47 (1975).[49] We use the notation τ for the stress tensor and T for theintegrated stress tensor T µν = R τ µν d x .[50] A. G. Abanov and G. M. Monteiro, Free-surface varia-tional principle for an incompressible fluid with odd vis-cosity, Physical review letters , 154501 (2019).[51] D. R. Hofstadter, Energy levels and wave functions ofbloch electrons in rational and irrational magnetic fields,Physical review B , 2239 (1976).[52] T. Matsushita, S. Fujimoto, and A. P. Schnyder, Topolog-ical piezoelectric effect and parity anomaly in nodal linesemimetals, arXiv preprint arXiv:2002.11666 (2020).[53] Off-diagonal yet symmetric strains do not contribute to η .[54] F. J. Belinfante, On the current and the density of theelectric charge, the energy, the linear momentum and theangular momentum of arbitrary fields, Physica , 449(1940). [55] J. M. Link, D. E. Sheehy, B. N. Narozhny, andJ. Schmalian, Elastic response of the electron fluid inintrinsic graphene: The collisionless regime, Physical Re-view B , 195103 (2018).[56] There are no zeroth order contributions to the Hall vis-cosity as the model gains an effective rotational symme-try as φ = 0.[57] T. L. Hughes, R. G. Leigh, and E. Fradkin, Torsionalresponse and dissipationless viscosity in topological insu-lators, Physical review letters , 075502 (2011).[58] C. Hoyos and D. T. Son, Phys. Rev. Lett. , 066805(2012).[59] L. Elcoro, B. J. Wieder, Z. Song, Y. Xu, B. Brad-lyn, and B. A. Bernevig, Magnetic Topological Quan-tum Chemistry, arXiv e-prints , arXiv:2010.00598 (2020),arXiv:2010.00598 [cond-mat.mes-hall].[60] Y. Xu, L. Elcoro, Z.-D. Song, B. J. Wieder, M. G.Vergniory, N. Regnault, Y. Chen, C. Felser, andB. A. Bernevig, High-throughput calculations of mag-netic topological materials, Nature , 702 (2020).[61] P. Burlet, E. Ressouche, B. Malaman, R. Welter, J. P.Sanchez, and P. Vulliet, Noncollinear magnetic structureof mnte , Phys. Rev. B , 14013 (1997).[62] Y. Xu, W. Li, C. Wang, Z. Chen, Y. Wu, X. Zhang, J. Li,S. Lin, Y. Chen, and Y. Pei, Mnte2 as a novel promisingthermoelectric material, Journal of Materiomics , 215(2018). upplementary Material for “A new cubic Hall viscosity in three-dimensionaltopological semimetals” I˜nigo Robredo,
1, 2, ∗ Pranav Rao, ∗ Fernando de Juan,
1, 4
Aitor Bergara,
1, 2, 5
Juan L. Ma˜nes, Alberto Cortijo,
6, 7
M. G. Vergniory,
1, 4, † and Barry Bradlyn ‡ Donostia International Physics Center, 20018 Donostia-San Sebastian, Spain Department of Physics, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain Department of Physics and Institute for Condensed Matter Theory,University of Illinois at Urbana-Champaign, Urbana IL, 61801-3080, USA IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013 Bilbao, Spain Centro de F´ısica de Materiales CFM, CSIC-UPV/EHU,Paseo Manuel de Lardizabal 5, 20018 Donostia, Basque Country, Spain Departamento de F´ısica de la Materia Condensada,Universidad Aut´onoma de Madrid, Madrid E-28049, Spain Condensed Matter Physics Center (IFIMAC), Madrid E-28049, Spain (Dated: February 5, 2021)
I. PROPERTIES OF THE POINT GROUP In the main text we consider the effects of strain on the threefold fermion found in MSG P2 .
9) at the Γpoint, where the little (co)group is isomorphic to point group 23 [1–3]. In this section, we describe the representationsof 23 and their products. This will allow us to identify the possible Hall viscosity components, and also provide a wayof writing down the strained Hamiltonian necessary for the phonon-stress approach.A set of generators of the point group is, in the vector representation V , V ( C z ) = − − , V ( C y ) = − − , V ( C − ) = (1)A character table for the representations of 23 can be found below: (23) (T) E C i C − j C +3 j A E w w ∗ E w ∗ wT TABLE I: Character table for Point Group 23. w = e πi/ . Notice that the vector representation subduces to the T irreducible representation.In this paper we use the following irreducible tensors for 23: the Kronecker delta δ ij , the Levi-Civita symbol (cid:15) ijk and the two tensors Θ and Λ also defined in the main text:Θ aij = ( √ ( δ i δ j + δ i δ j − δ i δ j ) a = 1 δ i δ j − δ i δ j a = 2Λ ijk = ( i = j = k else (2) ∗ These authors contributed equally to this work † [email protected] ‡ [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Note that Θ carries an index a = 1 , E E . Indices i, j, k representcoordinates in 3-dimensional space. By means of these tensors, we can write explicitly the Kronecker Product tableof the group: A ( x ) ⊗ A ( y ) = A ( xy ) A ( x ) ⊗ E E ( y a ) = E E ( xy a ) A ( x ) ⊗ T ( y i ) = T ( xy i ) E E ( x a ) ⊗ E E ( y b ) = A ( δ ab x a y b ) ⊕ A ( (cid:15) ab x a y b ) ⊕ E E ( − x y + x y , x y + x y ) E E ( x a ) ⊗ T ( y i ) = T ( x y i ) ⊕ T ( x y i ) T ( a i ) ⊗ T ( b j ) = A ( δ ij a i b j ) ⊕ E E (cid:0) Θ aij a i b j (cid:1) ⊕ T (Λ ij a i b j ) ⊕ T ( (cid:15) ijk a j b k ) (3)One can use the Gell-Mann matrices λ = , λ = , λ = − i i ,λ = − , λ = , λ = − i i ,λ = , λ = − i i , λ = 1 √ − . (4)to parametrize the Hamiltonian near the Γ point Hamiltonian for states spanning the spin-1 fermion consideredin the text. At Γ, the basis states for this degeneracy transform in the irreducible representation T of 23. In thenoninteracting picture, the Bloch Hamiltonian is a matrix, which is thus a bilinear in the ¯ T ⊗ T = T ⊗ T representation.Thus, the eight Gell-Mann matrices will transform under T ⊗ T . Following the Kronecker Product table, we can writethe representation and symmetry adapted coordinates of the Gell-Mann matrices in the point group 23. We find ρ λ = A ( λ ) ⊕ E E ( λ , λ ) ⊕ T ( λ , λ , λ ) ⊕ T ( λ , − λ , λ ) (5)As in the main text we label the T representations L ↔ T ( λ , − λ , λ ) ˜L ↔ T ( λ , λ , λ ) (6)We also use the letter indices a, b, c = 1 , v a , an element of the two dimensional representation E E ( λ , λ ).We can use this decomposition to analyze the symmetries of the spin-1 Hamiltonian at the Γ point given by Eq. (16)in the main text, which we recall has the general form H Γ = cos φ k · L + sin φ k · ˜L . (7)The contribution L · k , with k = ( k x , k y , k z ), has full SO (3) invariance, while ˜L · k is invariant only under point group23. We understand this as follows: the Gell-Mann matrices in Eq. (4) belong to T ⊗ T , which can be obtained viasubduction from the ( l = 1) ⊗ ( l = 1) representation of SO (3). Denoting the irreps of SO (3) by their dimension2 l + 1, the rules of angular momentum composition give 3 ⊗ → L = ( λ , − λ , λ ) are antisymmetric and belong to 3, the vectorrepresentation of SO (3). On the other hand, ˜L = ( λ , λ , λ ) are symmetric and together with the two tracelesssymmetric matrices ( λ , λ ) form a basis for the irrep 5 ( l = 2) of SO(3). Thus ˜L can not be promoted to a vectorof SO (3) and, as a consequence, ˜L · k can not be invariant under SO (3). Instead we have the subduction rule 5( λ , λ , λ , λ , λ ) → T ( λ , λ , λ ) + E E ( λ , λ ). Thus, we see directly why φ = 0 implies a breaking of continuousrotational symmetry. II. TIGHT BINDING MODEL
We build a tight binding model for MSG P .
9) by placing s -like spinless orbitals at the 4a Wyckoff position: q = ( x, x, x ) q = (1 / x, / − x, − x ) q = ( − x, / x, / − x ) q = (1 / − x, − x, / x ) , (8)For simplicity, we take x = 0 in this work. However, we must take care that simply setting x = 0 for this choice oforbitals leads to a lattice with the symmetries of space group F m ¯3 m (225) rather than P . x , and then take the limit x →
0. Consideringhopping processes for nearest neighbor sites, the Hamiltonian matrix reads [4–6]: H = 2 t e − ikya cos (cid:0) k x a + φ (cid:1) e − ikza cos (cid:16) k y a + φ (cid:17) e − ikxa cos (cid:0) k z a + φ (cid:1) e ikya cos (cid:0) k x a + φ (cid:1) e − ikxa cos (cid:0) k z a − φ (cid:1) e ikza cos (cid:16) k y a − φ (cid:17) e ikza cos (cid:16) k y a + φ (cid:17) e ikxa cos (cid:0) k z a − φ (cid:1) e − ikya cos (cid:0) k x a − φ (cid:1) e ikxa cos (cid:0) k z a + φ (cid:1) e − ikza cos (cid:16) k y a − φ (cid:17) e ikya cos (cid:0) k x a − φ (cid:1) (9)where t is the overlap integral parameterizing hopping from neighbor to neighbor. The phase φ represents a time-reversal symmetry breaking magnetic flux, introduced via a Peierls substitution. A. Symmetry
This Hamiltonian satisfies the symmetry constraints of the MSG P . ρ (cid:0) { C − | } (cid:1) = , ρ (cid:18) { C x |
12 12 0 } (cid:19) = e − i ( k x − ky ) , ρ (cid:18) { C y | } (cid:19) = e − i ( k y − k z ) (10)so that ρ ( g ) † H ( k , Q ) ρ ( g ) = H ( g k , g Q ) (11)where k is crystal momentum and Q is any external field. When exactly at the Γ point, Γ = ( k x , k y , k z ) = (0 , , U = 12 − − − − − − . (12) B. Threefold at the Γ point: energies and states We can expand the Hamiltonian Eq. (18) around the Γ = (0 , ,
0) point, and apply the unitary transformationEq. (12) to obtain the low-energy model H = X nm k c † n k f ( k ) c m k ,f ( k ) = t cos( φ ) iv F e iφ k T − iv F e − iφ k h ! (13)shown in the main text. As in the main text the Fermi velocity is v F = ta . The lower-right block corresponds to thespin-1 bands (threefold fermion) with h = v F (cid:16) cos( φ ) k · L + sin( φ ) k · ˜ L (cid:17) − t cos( φ ) λ . (14)If we take θ = φ + π , the energies for general φ are given by [5] E n = 2 | k |√
13 arccos √ k x k y k z | k | cos(3 θ ) ! − π ( n − ! (15)Above n ∈ { , , } indexes the three states [7], for small φ , the threefold dispersion takes a form that matches Fig.1c in the main text, E = | k | + 3 φ k x k y k z | k | + O ( φ ) E = − φ k x k y k z k + O ( φ ) E = −| k | + 3 φ k x k y k z | k | + O ( φ ) (16)which have normalized eigenfunctions given by ψ n = 1 p (3 E n − | k | )( E n − k z ) E n − k z E n k x e − iθ + k y k z e iθ E n k y e iθ + k x k z e − iθ (17) III. PHONON METHOD FOR VISCOSITY
Following equations (17) and (19) from the main text, and using the tight binding model Eq. (18), we have thatthe strained Hamiltonian takes the following form: H = 2 t f ( a, b ) e − ikya cos (cid:0) k x a + φ (cid:1) f ( b, c ) e − ikza cos (cid:16) k y a + φ (cid:17) f ( a, c ) e − ikxa cos (cid:0) k z a + φ (cid:1) f ( a, b ) e ikya cos (cid:0) k x a + φ (cid:1) f ( a, c ) e − ikxa cos (cid:0) k z a − φ (cid:1) f ( b, c ) e ikza cos (cid:16) k y a − φ (cid:17) f ( b, c ) e ikza cos (cid:16) k y a + φ (cid:17) f ( a, c ) e ikxa cos (cid:0) k z a − φ (cid:1) f ( a, b ) e − ikya cos (cid:0) k x a − φ (cid:1) f ( a, c ) e ikxa cos (cid:0) k z a + φ (cid:1) f ( b, c ) e − ikza cos (cid:16) k y a − φ (cid:17) f ( a, b ) e ikya cos (cid:0) k x a − φ (cid:1) (18)with f ( a, b ) = 1 + a + b and ( a, b, c ) ≡ ( u xx , u yy , u zz ). We now transform tha Hamiltonian into the symmetryadapted coordinate basis using Eq. (12) and expand to first order in the product of strain and momentum. Theresulting perturbed Hamiltonian is t − k x ( a + b ) s φ + ik x ( a + c ) c φ − k y ( b + c ) s φ + ik y ( a + b ) c φ − k z ( a + c ) s φ + ik z ( b + c ) c φ − k x ( a + b ) s φ − ik x ( a + c ) c φ k z ( a + c ) s φ − ik z ( b + c ) c φ k y ( b + c ) s φ + ik y ( a + b ) c φ − k y ( b + c ) s φ − ik y ( a + b ) c φ k z ( a + c ) s φ + ik z ( b + c ) c φ k x ( a + b ) s φ − ik x ( a + c ) c φ − k z ( a + c ) s φ − ik z ( b + c ) c φ k y ( b + c ) s φ − ik y ( a + b ) c φ k x ( a + b ) s φ + ik x ( a + c ) c φ , (19)with ( s φ , c φ ) ≡ (sin( φ ) , cos( φ )) and ( a, b, c ) ≡ ( u xx , u yy , u zz ). When taking into account lab frame effects [8], we get anextra term in the first row (column) of the Hamiltonian that can be expressed as a vector δH = 2 ic φ (0 , ck z , ak x , bk y )( δH = − ic φ (0 , ck z , ak x , bk y ) T ). There is no contribution to the bottom 3x3 block, so the viscosity tensor for the3-fold fermion remains unchanged.When coupling to antisymmetric strain, the distances between orbitals does not change at first order in strain,so the contribution to the stress tensor is zero within our series expansion. There is a coupling, however, whenconsidering lab-frame effects. Just as above, there is no contribution to the 3x3 or 1x1 blocks describing the lowenergy physics of the system, but there is a contribution to the row connecting them. In this case, this contributionsgives δH = 2 ic φ (0 , u k x + u k y , − u k y − u k z , u k x − u k z ). Thus there is no coupling of the spin-1 fermion toanti-symmetric strain in the phonon method. IV. CONTINUITY METHOD FOR VISCOSITY
In this section we describe the continuity method utilized in the main text, which is the approach to long-wavelengthmomentum transport and stress response for lattice systems recently formulated in Ref. 9 for two dimensional systems.First we describe the generalization of the approach in Ref. 9 to three dimensions, focusing specifically on cubic lattices.Next, we present the long-wavelength lattice stress tensor for the full four band tight binding model, which simplifiesto T (c) µν in the main text when considering the threefold fermion at the Γ point. A. Long-wavelength momentum transport
In this approach, the stress tensor is identified through a long-wavelength analog of a continuity equation formomentum density. In the continuum, the momentum continuity equation describes a relationship between momentumdensity and stress: ∂ t g ν ( r ) + ∂ µ τ µν ( r ) = f ext µ ( r ) (20)Above f ext is the density of external forces acting on the continuum system. In a system with Hamiltonian H , internalangular momentum generator L int , and f ext = 0, we can write the integrated stress tensor T µν = R d rτ µν ( r ) in termsof strain generators J µν as [10, 11] T µν = − i [ H, J µν ] (21)The strain generators are made up of two terms, the first below accounting for spatial deformations (which we callthe “kinetic” part) and the second due to rotations of internal angular degrees of freedom: J µν = − { x µ , p ν } − (cid:15) µνρ L ρ int (22)On the lattice, with discrete rather than continuous translation invariance, we no longer have the continuity equationEq. (20), and our notion of momentum transport outlined above must be modified.We start by considering the lattice momentum density operator g Lµ , which can be decomposed into a kinetic partand a contribution due to internal angular momentum L int . The kinetic piece can be written g kin µ ( R ) = i | a µ | X n (cid:16) c † n R + a µ c n R − c † n R − a µ c n R + c † n R c n R − a µ − c † n R c n R + a µ (cid:17) , (23)We note the Bravais lattice vectors for this cubic lattice can be written a µ = a e µ , where e µ denote Cartesian basisvectors. The internal angular momentum contribution is given by [9, 11] g int µ ( R ) = X nmν | a ν | (cid:15) µνρ ( L int ) nmρ (cid:16) c † n R + a ν c m R − c † n R − a ν c m R + c † n R c m R + a ν − c † n R c m R − a ν (cid:17) , (24)If we write the total lattice momentum density g L ( R ) ≡ g kin ( R ) + g int ( R ) in momentum space with coordinate q , wecan expand in powers of q in the long-wavelength limit q →
0, finding g L µ ( q ) = P µ + i X ν q ν J L ,νµ + O ( q ) (25)We see that the momentum density in orders is given by the zeroth order contribution – the total momentum P µ (whichis identically g Lµ ( )) – and the first order contribution which is expressed in terms of the lattice strain generators,which generalize the continuum strain generators found in Ref. 10, J L ,µν ≡ − i X k nm c † n k "(cid:26) sin k · a ν | a ν | , ∂∂k µ (cid:27) δ nm + i X ρ (cid:15) µνρ cos k · a µ ( L int ) nmρ c m k (26)The long-wavelength lattice stress can now be calculated from the strain generators Eq. (26) in the same way as inthe continuum: T L ,µν = − i [ H, J L ,µν ] (27) B. Lattice stress tensor for MSG P The general form of the lattice stress for three dimensional cubic lattices (where we can write a µ = a e µ ) is given by T µν = X nm k c † n k ∂ µ f nm ( k ) sin( k · a ν ) | a ν | + i X ρ cos( k · a µ ) (cid:15) µνρ [ f ( k ) , ( L int ) ρ ] nm ! c m k ≡ X k c † k T µν ( k ) c k (28)We now specify to our tight-binding model in MSG P . L int = ⊕ L (29)It’s also possible to view the internal angular momentum generator in the non-symmetry adapted coordinates (whichEq. (18) is written in) where we write the internal angular momentum as L int = U L int U T . For our model in Eq. (18),the stress is therefore given by T µν ( k ) = T kin µν ( k ) + T spin µν ( k ) = sin( k ν a ) a ∂ µ f ( k ) + i cos( k µ a ) (cid:15) µνρ (cid:2) f ( k ) , L ρ (cid:3) (30) C. Lattice viscosity
Equipped with a long-wavelength stress tensor we can compute the Hall viscosity on the lattice using the stress-stress Kubo formula from the main text. Unlike in the main text, where we consider the physics the Γ point at small φ and consider only the threefold fermion, we must take into account the full four band model in Eq. (18) to computethe viscosity. This can be seen in Figure 1a, where at the Γ point the gap between the threefold and the fourth bandis not sizable enough to neglect the fourth band.In contrast to the result for the total viscosity in the main text, we see in Figure 1b that there is no discontinuityacross the µ = 0 point. (a) Band structure along high symmetry points in theBrillouin zone. The red line signifies the Fermi energy (cid:15) . - ��� ��� ��� - �� - �� - ������� μ η ℏ � � (b) Plot of the total viscosity η tot = η + η versus chem-ical potential µ .(c) Plot of the viscosity integrand η ( k ) (d) Plot of the viscosity integrand η ( k ) FIG. 1: Lattice Hall viscosity for the model Eq. (18). We have that φ = . t = 1, and lattice spacing a = 1. (a)shows the band structure with these parameters. We see in (b) that the integrated viscosity η tot is continuous across µ = 0 and is regulated (not divergent) by the cubic lattice. In (c) and (d), we present two dimensional contour plotsof the two relevant viscosity integrands (with η i = R d k η i ( k ), for example) for the k z = 0 plane of the BZ, andsee the symmetries of the point group 23 are satisfied by these viscosity densities. The corner points in (c) and (d)correspond to M = ( ± π, ± π, [1] M. I. Aroyo, J. M. Perez-Mato, D. Orobengoa, E. Tasci, G. de la Flor, and A. Kirov, Bulg. Chem. Commun. , 183(2011).[2] M. I. Aroyo, J. M. Perez-Mato, C. Capillas, E. Kroumova, S. Ivantchev, G. Madariaga, A. Kirov, and H. Wondratschek,Z. Krist. , 15 (2006).[3] M. I. Aroyo, A. Kirov, C. Capillas, J. M. Perez-Mato, and H. Wondratschek, Acta Cryst. A62 , 115 (2006).[4] J. L. Manes, Existence of bulk chiral fermions and crystal symmetry, Physical Review B , 155118 (2012).[5] F. Flicker, F. De Juan, B. Bradlyn, T. Morimoto, M. G. Vergniory, and A. G. Grushin, Chiral optical response of multifoldfermions, Physical Review B , 155145 (2018).[6] G. Chang, S.-Y. Xu, B. J. Wieder, D. S. Sanchez, S.-M. Huang, I. Belopolski, T.-R. Chang, S. Zhang, A. Bansil, H. Lin, et al. , Unconventional chiral fermions and large topological fermi arcs in rhsi, Physical review letters , 206401 (2017).[7] We neglect the constant shift given by the λ term.[8] J. L. Ma˜nes, F. de Juan, M. Sturla, and M. A. H. Vozmediano, Generalized effective Hamiltonian for graphene undernonuniform strain, Phys. Rev. B , 155405 (2013), arXiv:1308.1595 [cond-mat.mes-hall]. [9] P. Rao and B. Bradlyn, Hall viscosity in quantum systems with discrete symmetry: point group and lattice anisotropy,Physical Review X , 021005 (2020).[10] B. Bradlyn, M. Goldstein, and N. Read, Kubo formulas for viscosity: Hall viscosity, ward identities, and the relation withconductivity, Physical Review B , 245309 (2012).[11] J. M. Link, D. E. Sheehy, B. N. Narozhny, and J. Schmalian, Elastic response of the electron fluid in intrinsic graphene:The collisionless regime, Physical Review B98