Non-Abelian tensor Berry connections in multi-band topological systems
NNon-Abelian tensor Berry connections in multi-band topological systems
Giandomenico Palumbo
School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland (Dated: February 25, 2021)Here, we introduce and apply non-Abelian tensor Berry connections to topological phases in multi-band systems. These gauge connections behave as non-Abelian antisymmetric tensor gauge fieldsin momentum space and naturally generalize Abelian tensor Berry connections and ordinary non-Abelian (vector) Berry connections. We build these novel gauge fields from momentum-space Higgsfields, which emerge from the degenerate band structure of degenerate-band models. Firstly, weshow that the conventional topological invariants of two-dimensional topological insulators andthree-dimensional Dirac semimetals can be derived from the winding number associated to the Higgsfield. Secondly, through the non-Abelian tensor Berry connections we construct higher-dimensionalBerry-Zak phases and show their role in the topological characterization of several gapped and gap-less systems, ranging from two-dimensional Euler insulators to four-dimensional Dirac semimetals.Importantly, through our new theoretical formalism, we identify and characterize a novel class ofmodels that support space-time inversion and chiral symmetries. Our work provides an unifyingframework for different multi-band topological systems and sheds new light on the emergence ofnon-Abelian gauge fields in condensed matter physics, with direct implications on the search fornovel topological phases in solid-state and synthetic systems.
Introduction:
Non-Abelian Berry connections playa central role in multi-band systems with degeneratespectra. These connections behave like non-Abelianvector gauge fields in momentum/parameter space andhave been applied in different research areas rangingfrom quantum computation to topological phases ofmatter . The topology of several multi-band sys-tems is naturally encoded in non-Abelian Berry con-nections which are responsible for the quantum spinHall effect in 2D , the electric polarization in 3D and for the second Chern number in 4D . Non-Abelian Berry connections give rise to Wilson loops,which are a powerful tool of investigation in topologicalmatter . Moreover, topological Bloch oscillationsin topological crystalline insulators and higher-ordertopological insulators are naturally related to the exis-tence of non-Abelian Berry connections, which influencethe dynamics of wave-packets . Importantly, a gener-alization of Abelian Berry connections have been recentlyproposed , where the new connections behave likeAbelian antisymmetric tensor (Kalb-Ramond ) gaugefields in momentum/parameter space. These tensorBerry connections have been employed to characterisethe topology of 3D chiral topological insulators and4D topological semimetals where the Dixmier-Douady (DD) invariant replaces the Chern number. Thetheoretical developments recently let to the experimen-tal measurement of the DD invariant in 4D syntheticsystems .The main goal of our paper is to unveil the exis-tence of novel non-Abelian Berry connections, coined non-Abelian tensor Berry connections , which behave likenon-Abelian antisymmetric gauge fields in the momen-tum space of multi-band systems with degenerate spec-tra. These types of tensor fields naturally appear in high-energy physics where they are defined in real space-time, while in mathematical literature they are known as non-Abelian gerbe connections . Here, we buildthese new connections by combining the conventionalnon-Abelian Berry curvature together with non-Abelianscalar fields that behave as momentum-space Higgs fields.Within the framework of topological phases of matter,considering several gapless and gapped systems in differ-ent dimensions, we will show that these Higgs fields andnon-Abelian tensor Berry connections emerge from bandstructures, hence allowing us to derive their topologicalbulk invariants.Finally, we will show the existence of a novel class oftopological phases characterized by topological invariantsassociated to non-Abelian real bundle gerbes. These sys-tems defined for gapped (2 n + 1)-D and gapless (2 n + 2)-D models are characterised by space-time inversion andchiral symmetries, which give rise to degenerate spec-tra with real Bloch wavefunctions. We will provide twoexplicit models in 3D and 4D, respectively, where theirtopological invariants are directly related to SO (4) tensorBerry connections. Importantly, through these new ten-sor connections, novel higher-dimensional models (withan eventual higher number of bands) in this class can beeasily identified.Our work provides an unifying theoretical framework fordifferent multi-band topological systems and our resultsshed light on the existence of novel gauge structures andgeometric phases in quantum matter, with important im-plications on the search for novel topological phases insolid-state and synthetic setups. Momentum-space Higgs field:
We start by gener-alizing the construction of momentum-space Higgs fieldsin multi-band models. These non-Abelian scalars willplay a central role in the definition of non-Abelian tensorBerry connections as will show in the next section. Wenote that non-Abelian complex scalar fields in param-eter space have been previously considered in the con-text of fictitious ’t Hooft-Polyakov monopoles . How- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b FIG. 1. (Left) Band structure of the topological phase in theBHZ model at m = 1 .
3. (Right) Plot of the integrand of thewinding number w ( T ) for the BHZ model at m = 1 .
3. Themaximum of the function is at the Γ point. ever, their formulation in multi-band topological modelsand their role in topological phases of matter have neverbeen analyzed in detail. The main goal of this sectionis thus to fill this gap. Given a degenerate Bloch state | u ( k ) (cid:105) = ( | u ( k ) (cid:105) , | u ( k ) (cid:105) , ..., | u N ( k ) (cid:105) ) T associated to N degenerate bands with energy E ( k ), with k referring tothe momenta of a generic lattice system, we define thematrix components ˜Φ ab of a non-Abelian scalar field ˜Φas follows ˜Φ ab = (cid:104) u a | G | u b (cid:105) , (1)where G is a suitable matrix that depends on a givenlattice model and { a, b } = 1 , ..., N . Here, ˜Φ = ˜ φ i T i ,where ˜ φ i are the vector components of ˜Φ and T i are thegenerators of a U ( N ), SU ( N ) or SO ( N ) Lie algebra. Wethen define the corresponding normalized quantity givenby Φ = ˜Φ (cid:112) ˜Φ . ˜Φ , Φ . Φ = I , (2)with I is the identity matrix. Here, Φ represents the momentum-space Higgs field that we will employ in therest of the paper. However, before introducing the non-Abelian tensor Berry connections, we first propose toshow that the topology of the Higgs field has relevantimplications in topological phases. In quantum field the-ory, the topology of the Higgs field in three-dimensionalspace is given by the winding number w ( S ) = 116 πi ˆ S dS i (cid:15) ijk tr (Φ ∂ j Φ ∂ k Φ) , (3)which represents the magnetic charge related to theresidual U(1) gauge invariance of ’t Hooft-Polyakovmonopoles. In a similar way, we can construct thewinding number on the two-dimensional torus T (i.e.the first Brillouin zone of a lattice system). To givea first concrete application of the winding number as-sociated to the Higgs field, we consider the Bernevig-Hughes-Zhang (BHZ) model for 2D quantum spin Hall(QSH) insulators . The corresponding momentum spaceHamiltonian is given by H BHZ = γ sin k x + γ sin k y + γ ( m − cos k x − cos k y ) , (4) where γ = − σ y ⊗ σ z , γ = − σ y ⊗ σ x and γ = σ x ⊗ σ and σ i the Pauli matrices (here, we have adopted a chiral-invariant basis). Its degenerate spectrum is given by E ± = ± (cid:113) m − m (cos k x + cos k y ) + 2 cos k x cos k y , (5)and at half filling, the time-reversal-invariant topologicalphase holds for | m | < m (cid:54) = 0) for which the Z invari-ant is 1. We can now build the Higgs field for this modelby taking G = iγ γ , with γ = σ z ⊗ σ and γ = − σ y ⊗ σ y and | u (cid:105) the doubly degenerate Bloch state associated tothe lower occupied bandsΦ = f i σ i | E | , (6)with f x = sin k x , f y = m − cos k x − cos k y and f z =sin k y . By applying the formula for the winding numberin Eq.(3) on T , we obtain w ( T ) = ˆ T d k cos k x + cos k y − m cos k x cos k y π | E | == − sign( m ) , | m | < , (7)with w ( T ) = 0 for | m | >
2. Thus we recognize theabsolute value of the winding number | w ( T ) | as the Z invariant in 2D QSH insulators. Notice that the inte-grand of the above winding number has a maximum atthe Γ point, i.e. in correspondence with the gap closingof the BHZ model as shown in Fig.(1).The winding number on the sphere w ( S ) has also rele-vant applications in gapless topological phases. In thiscase, we consider the linearized momentum-space Hamil-tonian of a 3D Dirac semimetal given by H D = γ k x + γ k y + γ k z , (8)with γ , γ and γ the same Dirac matrices defined inEq.(4). The corresponding Higgs field Φ, built from thesame G -matrix as for the previous model, is given byΦ = − k x σ x + k y σ z + k z σ y | k | , (9)such that w ( S ) = 12 π ˆ S dS i k i | k | = 2 . (10)This shows that the 3D Dirac point behaves like amomentum-space monopole and that | w ( S ) | rep-resents its topological charge. We point out that thisresult can be naturally extended to N -degenerate Diraccones where | w ( S ) | = N , such as in double Diracsemimetals with N = 4 . Non-Abelian tensor Berry connections:
Throughthe momentum-space Higgs field Φ we can now buildnovel gauge connections that we coin non-Abelian ten-sor Berry connections B ij . They are defined as follows B ij = Φ F ij , (11)where F ij is the non-Abelian Berry curvature F ij = ∂ i A j − ∂ j A i − i [ A i , A j ] , (12)with A i the non-Abelian Berry connection and ∂ i ≡ ∂ k i .Under gauge transformations | u (cid:105) → U | u (cid:105) we have that F ij → U F ij U − , Φ → U Φ U − , (13)where U is a Lie-algebra-valued matrix, such that B ij also transforms in a gauge-covariant way; we note thatthe trace trace of all these quantities is gauge invariant.The tensor gauge field B ij behaves as a non-AbelianKalb-Ramond field in momentum space and itscurvature tensor is given by H ijk = D i B jk + D j B ki + D k B ij , (14)such that H ijk → U H ijk U − , (15)with D j f = i ∂ j f − [ A j , f ] the covariant derivative (here, f is a generic Lie-algebra valued function). Further-more, a non-Abelian higher-tensor Berry connection canbe built from H ijk as follows C ijk = Φ H ijk , (16)which gives rise to its own higher curvature tensor. Simi-larly to B ij , C ijk also transforms in a gauge-covariantway and its trace is gauge invariant. Antisymmetrictensor fields as B ij and C ijk are known in differentialgeometry and topology as gerbe connections, which arerelated to bundle gerbes and higher bundle gerbes ,respectively. These structures naturally generalize fiberbundles . Moreover, similarly to Wilson loops, it ispossible to build from these gauge connections non-localgauge operators, named Wilson surfaces and Wilsonvolumes. For our purpose, the simplest gauge invariantquantities associated to B ij and C ijk are respectivelygiven by Υ B ( M ) = 12 π ˆ M d k tr B xy , Υ C ( M ) = 12 π ˆ M d k tr C xyz , (17)where M and M are 2D and 3D compact manifolds,respectively. These expressions naturally generalize non-Abelian Berry-Zak phases in higher dimensions.We now revisit some known topological phases andshow that their topological invariants can be formulatedin terms of the higher-dimensional non-Abelian Berry-Zak phases. 2D Euler insulators are the prototypicalexample of topological phases characterized by space-time inversion IT ( C × T ) with time-reversal symme-try T = 1 and an Euler number induced by a SO (2)Berry connection . A simple model with four bandsis given by the BHZ Hamiltonian in Eq.(4), but now withthe Dirac matrices in a real representation of the Cliffordalgebra: γ = σ x ⊗ σ y , γ = σ y ⊗ σ and γ = σ z ⊗ σ . To construct the Higgs field, we consider G = iγ γ suchthat the gauge invariant quantity for the two degeneratelower bands is given byΥ B ( T ) = 12 π ˆ T d k tr B xy = 2 sign( m ) , | m | < , (18)with Υ B ( T ) = 0 for | m | >
2. We identify Υ B withthe topological Euler invariant e , while B xy is math-ematically equivalent to a gauge connection for a realbundle gerbe . As we will see more in detail in thenext section, real bundle gerbes will play a central rolein the characterization of higher-dimensional topologicalphases with real Bloch wavefunctions. To reveal the roleof C connections, we consider the linearized Hamilto-nian of a 4D Dirac semimetal described by the followingmomentum-space Hamiltonian H D = γ k x + γ k y + γ k z + γ k w , (19)where we have employed the same Dirac matrices as forthe 3D Dirac semimetal in the previous section togetherwith γ = − σ y ⊗ σ y . We derive the corresponding Higgsfield Φ with G = iγ γ (here, γ = σ z ⊗ σ ) and introducethe following gauge invariant quantityΥ C ( S ) = 12 π ˆ S dk i ∧ dk j ∧ dk k tr C ijk = 2 , (20)which we thus identify as the Z invariant of the 4DDirac point. So far, we have discussed topological phasesthat were already studied in the literature. In the nextsection, we will employ the non-Abelian tensor Berryconnections to unveil the existence of new topologicalstates in 3D and 4D. Topological phases with space-time inversion andchiral symmetries:
Topological phases with realBloch states appear in spinless models with space-timeinversion IT ( T = 1). The best known examples in thisclass are given by the 2D Euler insulators , 3D realDirac semimetals and Z nodal-line semimetals where SO ( N ) fiber bundles emerge in momentum spacedue to the IT symmetry. Here, Euler, Pontryagin andStiefel-Whitney invariants replace Chern numbers . Wenow consider a novel class of models with real Blochstates that also supports chiral symmetry. In this case, SO ( N ) real bundle gerbes that generalize the SO ( N )fiber bundles become relevant in the description as weshow below. When a quantum system has time-reversal T ( T = 1), inversion I and chiral symmetry S themomentum-space Hamiltonian density H ( k ) satisfies T H ( k ) T − = H ( − k ) , I H ( k ) I − = H ( − k ) , S H ( k ) S − = − H ( k ) . (21)The corresponding topological invariants play a role in(2 n +1)-D gapped and (2 n +2)-D gapless phases with n ≥
1. Their existence relies on the real representation of theClifford algebra in terms of 2 n +2 × n +2 Dirac matrices.We now provide two explicit examples for n = 1. In 3D,the above symmetries are supported, for instance, by thefollowing model H D = γ sin k x + γ sin k y + γ sin k z + (22) γ ( m − cos k x − cos k y − cos k z ) ,γ = σ x ⊗ σ x ⊗ σ x , γ = σ x ⊗ σ x ⊗ σ z , γ = σ x ⊗ σ z ⊗ σ and γ = σ y ⊗ σ z ⊗ σ y . These 8 × SO (2) is replaced by SO (4)and we do not have any well-defined Euler invariant todescribe the 3D bulk. By employing our new theoreticalframework, we can now construct an SO (4) C ijk connec-tion, which is naturally associated to a non-Abelian realbundle gerbe in the first Brillouin zone. For the four-folddegenerate lower band E − = − (3 + m − m cos k x − m cos k y − m cos k z +2 cos k y cos k z + 2 cos k x cos k y + 2 cos k x cos k z ) / , (23)the corresponding Higgs field can be built from the ma-trix G = iγ γ with γ = σ z ⊗ σ ⊗ σ such that thetensor Berry connection is given by C xyz = 4 | E | (cos k y cos k z + cos k x cos k y + cos k x cos k z − m cos k x cos k y cos k z ) . (24)This tensor field allows us to derive the topological in-variant for the 3D gapped bulk through the generalizedBerry-Zak phasesΥ C ( T ) = − , | m | < , Υ C ( T ) = 4 , < | m | < , (25)with Υ C ( T ) = 0 for | m | >
3. As shown in Fig.(2), C xyz has maximum at the K points where the gap closes, i.e.at Γ ( m = 1) and X ( m = 3) points, respectively. Sincethe IT symmetry in 3D gapped phases is not related toany 3D strong topological invariant , then we deducethat Υ C ( T ) is related to the presence of the chiral sym-metry. This model is then different from 3D weak Stiefel-Whitney insulators , which do not need the S symme-try and can be seen as stacks of 2D Euler insulators. Inanalogy to conventional 3D chiral-invariant topologicalinsulators with even topological invariant and nonsym-morphic Dirac insulators , our model supports gaplessboundary states given by doubly degenerate (real) Diraccones. In a slab geometry, one can open a boundary gapby introducing boundary terms that break S but pre-serve C × T (here, C is the inversion symmetry on the2D boundary and is also supported by the 3D bulk). Inthis way we obtain an half Euler insulator with e = 1.The existence of this gapped boundary is one of the mainfeatures of the above 3D topological phase.We now define a gapless IT - and S -symmetric systemin 4D. The existence of γ = σ z ⊗ σ ⊗ σ allows us tointroduce a Dirac model for 4D Dirac semimetals in the FIG. 2. (Left-Up) Band structure of the 3D gapped topo-logical phase with IT and S symmetries at m = 0 . k z = 0. (Right-Up) Plot of C xyz associated to Υ C ( T ) = − m = 0 .
1. The maximum of the function is at Γ point.(Left-Down) Band structure of the same model at m = 1 . k z = 0. (Right-Down) Plot of C xyz associated toΥ C ( T ) = 4 at m = 1 .
9. The maximum of the function is at X points. real representation. Formally, the linearized momentum-space Hamiltonian is similar to that in Eq.(19) and theHiggs field is built by employing the same matrix G con-sidered in the previous 3D case. A straightforward cal-culation yields Υ C ( S ) = 4 , (26)which provides the topological charge of the
4D realDirac cone . Also in this case, the chiral symmetry pro-tects the stability of this topological number similarlyto the case of 4D tensor monopoles . In fact, thisfour-dimensional phase can be seen as a stack of the IT - and S -symmetric 3D gapped phases. Although 4Dsystems cannot appear in real solid-state systems, thishigher-dimensional model could be realized in artificialsystems such as topoelectric circuits . Conclusions and outlook:
Summarizing, in this workwe have presented a generalization of non-Abelian Berryconnections built from momentum-space Higgs fields thatallow us to define higher-dimensional versions of the non-Abelian Berry-Zak phases. Through these new fields wehave shown that the topological invariants of several 2D,3D and 4D models such as QSH insulators, Euler insu-lators, 3D and 4D Dirac semimetals can be computedwithin an unified framework. Moreover, our new theo-retical concepts do not only unveil the presence of gen-eralized gauge-theory structures in band theory but alsothe existence of a new class of models characterized by IT (with T = 1) and S symmetries. We have providedtwo explicit models in 3D and 4D although their gen-eralization with higher number of bands and in higherdimensions is possible. Several directions will be consid-ered in future work. In particular, we will extend ourformalism to higher-dimensional non-Hermitian topolog-ical systems, nodal-line and nodal-surface semimetals,higher-spin fermion models and higher-order topo-logical phases . Moreover, via a many-body generaliza-tion of tensor gauge connections we should be able to an-alyze topological phases of 3D interacting models withinour framework. Acknowledgments:
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