Band geometry from position-momentum duality at topological band crossings
BBand geometry from position-momentum duality at topological band crossings
Yu-Ping Lin and Wei-Han Hsiao Department of Physics, University of Colorado, Boulder, Colorado 80309, USA Independent Researcher (Dated: March 3, 2021)We show that the position-momentum duality offers a transparent interpretation of the bandgeometry at the topological band crossings. Under this duality, the band geometry with Berryconnection is dual to the free-electron motion under gauge field. This identifies the trace of quantummetric as the dual energy in momentum space. The band crossings with Berry defects thus inducethe dual energy quantization in the trace of quantum metric. For a Z nodal-point or nodal-surfacesemimetal, a dual Landau level quantization occurs owing to the Berry charge. Meanwhile, a nodal-loop semimetal exhibits a Berry vortex line, leading to the quantized dual rotational energy aboutthe nodal loop. A Z monopole brings about another dual rotational energy, which originates fromthe link with an additional nodal line. Nontrivial band geometry generically induces finite spreadin the Wannier functions. While the spread manifests a quantized lower bound in a Z nodal-pointor nodal-surface semimetal, logarithmic divergence occurs in a nodal-loop semimetal. The bandgeometry at the band crossings may be probed experimentally by a periodic-drive measurement. The modern study of gapless topological systems haveuncovered various unconventional band crossings withfascinating phenomena. A band crossing in three dimen-sions (3D) may occur at a nodal point [1–11], a nodalline [1, 12–18], or a nodal surface [15, 17, 19, 20] (Fig. 1).These band crossings are realized in the semimetallicphases of solid-state materials [1], the topological su-perconductors [21–26], and the spin liquids [27]. Recentstudies of synthetic quantum matter present another setof platforms for the band crossings, including the ultra-cold atomic systems [28, 29], the photonic systems [30],and the superconducting circuits [31, 32]. Distinct band-topology classifications are defined based on the sym-metries and the band-crossing structures [33]. For the3D gapless topological systems, the topological classi-fications involve the Z and Z classifications. Varioustypes of integer topological invariants are proposed asthe indicators of these classifications. For example, a Z nodal point [Fig. 1(a)] carries a Berry monopole, leadingto a quantized Chern number under an enclosing-surfaceintegration [34]. The inflation of a Z nodal point mayrealize a Z nodal surface [Fig. 1(b)] [23, 24], where theBerry-charge scenario still apply [19]. A nodal line in acombined parity and time-reversal P T symmetric systemcan carry a Z monopole [14, 16]. The according topo-logical invariant corresponds to the linking number withadditional nodal lines [Fig. 1(d)] [18].The band crossings also bring about nontrivial quan-tum geometry of the bands [35]. The band geometryis characterized by the quantum metric, which measuresthe state variation under momentum change [36–38]. Thecomponents of quantum metric may be probed experi-mentally, for example, by a periodic drive [39, 40]. Re-cent studies have uncovered various manifestations ofthe quantum metric. The integrated trace of quantummetric defines a lower bound of the spread of Wannierfunctions [41–44], which can be measured directly in the experiments [39, 45–50]. The finite spread can triggeranomalous superfluid stiffness on flat bands, leading tothe geometric enhancement of flat-band superconductiv-ity [35, 51–55]. It can also lead to finite current noise evenin the insulating phase [56]. Other works have adoptedthe quantum metric as an indicator of phase transitions[57–59], fractional Chern insulators [60–62], excitons [63],and orbital susceptibilities [64, 65].While the bulk of the literature focuses on the extrin-sic manifestations of the quantum metric, its intrinsiccharacter has not received sufficient investigation. Thisdirection is recently explored in the context of nodal-point semimetals. The trace of quantum metric re-ceives a transparent interpretation in the chiral multifoldsemimetals [35]. Under the position-momentum duality,the trace of quantum metric is dual to the kinetic energyon the Haldane sphere [66], thereby acquiring a dual Lan-dau level quantization from the Berry monopole. On theother hand, the integrated determinant of quantum met-ric over an enclosing surface is proposed as a measureof the Berry defect [67, 68]. These results exemplify thenature of the quantum metric at the nodal points. Anatural question then arises as whether these scenariosare applicable to the other topological systems.In this Letter, we show that the position-momentumduality [35] offers a unified framework of the band geom-etry at the topological band crossings. Under this dual-ity, the band geometry with Berry connection is dual tothe free-electron motion under gauge field. This identi-fies the trace of quantum metric as the dual energy inmomentum space. The band crossings with Berry de-fects thus induce the dual energy quantization in thetrace of quantum metric. For a Z nodal-point or nodal-surface semimetal, a dual Landau level quantization oc-curs owing to the Berry charge. Meanwhile, a nodal-loopsemimetal exhibits a Berry vortex line, leading to thequantized dual rotational energy about the nodal loop. a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r (a) (b) (c)(d) FIG. 1. Band crossings in the 3D Brillouin zone. (a) Nodalpoint. (b) Nodal surface. (c) Z -trivial nodal loop. (d) Z -nontrivial nodal loop. A Z monopole brings about another dual rotational en-ergy, which originates from the link with an additionalnodal line. The spread of Wannier functions manifestsa quantized lower bound in a Z nodal-point or nodal-surface semimetal, while logarithmic divergence occursin a nodal-loop semimetal. Our duality-based analysispaves the way for advanced comprehension of nontrivialband geometry at the topological band crossings.We begin with an introduction to the band geometry.Consider a band with the Bloch eigenstate | u k (cid:105) , whichgenerically evolves under the variation of momentum k .Such an evolution manifests the variations in the phaseand the state. A quantitative measure is provided by thequantum geometric tensor [69] T ab k = (cid:104) ∂ k a u k | (1 − | u k (cid:105)(cid:104) u k | ) | ∂ k b u k (cid:105) (1)with a, b = x, y, z . The real part g ab k = Re[ T ab k ]is the quantum or Fubini-Study metric [36–38], whichcaptures the quantum distance under state variation1 − |(cid:104) u k | u k + d k (cid:105)| = g ab k dk a dk b . On the other hand, theimaginary part measures the phase variation and the ac-cording Berry flux B a k = − (cid:15) abc Im[ T bc k ] [34, 70]. Definethe Berry connection A k = (cid:104) u k | i ∇ k | u k (cid:105) as a momentum-space gauge field under nontrivial band geometry [34].The quantum metric and the Berry flux become [71] g ab k = 12 (cid:104) u k |{ r a , r b }| u k (cid:105) , B k = ∇ k × A k , (2)where the position r = i ∇ k − A k is a momentum-space covariant derivative. The effect of Berry flux as amomentum-space “magnetic field” has been studied ex-tensively [34]. Here we focus on the quantum metric andsearch for useful indications to the band geometry.The individual components of quantum metric are usu-ally complicated and hard to interpret. Nevertheless, thetrace of quantum metric takes a profound form [35]Tr g k = (cid:104)| r | (cid:105) , (3)where the expectation value (cid:104)· · ·(cid:105) = (cid:104) u k | · · · | u k (cid:105) ofmomentum-space Laplacian | r | is measured. Signifi-cantly, this quantity captures the “dual energy” of a free particle in momentum space, which carries a half-unity“dual mass” and experiences the Berry connection. Wethus establish a position-momentum duality between theband geometry and the free-particle motion under gaugefield. The duality presents a feasible solution to the inter-pretation of the band geometry. If the duality leads to asimple free-particle model with well-studied solution, thetrace of quantum metric can be understood under a directinference. Similar concept of position-momentum dualitywas proposed previously in the context of harmonic traps[72–74] and fractional Chern insulators [61, 62]. Theseworks adopted the confinement potential V ( r ) ∼ | r | to insert the band-geometry contribution into the actualelectronic energy. In comparison, our analysis directlyidentifies the trace of quantum metric as the true dualenergy in momentum space. This deepens the compre-hension of the band geometry as the dual free-particletheory of general bands.We now introduce how the duality operates on the non-trivial band geometry. Here we focus on the 3D gaplesstopological systems, where approximate rotation sym-metries can occur near the band crossings. Under suchsymmetries, simple models become available to the dualfree-particle theories. A band crossing can host a defectof Berry connection and induce nontrivial band geom-etry. We will show that this feature is encoded in the“dual energy quantization” in the trace of quantum met-ric. This scenario is applicable to both Z and Z gaplesstopological systems.We first study the Z gapless topological systems withthe duality. As a paradigmatic example, we present theanalysis of the chiral multifold semimetals (CMS) [35],which constitute a large family of Z nodal-point semimet-als. The band crossings in these systems occur at distinctpoints [Fig. 1(a)] [1–4, 75]. At each nodal point, the low-energy theory manifests the chiral fermions with multi-fold degeneracy [Fig. 2(a)]. A minimal model exhibits arotationally symmetric spin- s Hamiltonian with integeror half-integer spin s = 1 / , , / , . . . H CMS , k = v k · S . (4)Here v is the effective velocity, k = k ˆk with mag-nitude k and direction ˆk , and S = ( S x , S y , S z ) isthe spin- s representation. There are 2 s + 1 bandsin this model, carrying the dispersion energies ε sn k = vkn with n = − s, − s + 1 , . . . , s . In accordance withthe Berry flux B sn k = − ( n/k ) ˆk , the nodal point at k = carries a quantized Berry monopole q sn = − n .This monopole charge corresponds to the Chern num-ber C sn = (1 / π ) (cid:72) d S k · B sn k = 2 q sn = − n under anenclosing-surface integration, which is the topological in-variant. The rotationally symmetric structure aroundthe monopole indicates the duality to a Haldane sphere[66, 76, 77]. Analogous to the quantum Hall effect on theHaldane sphere, the “dual Landau level quantization”(a)(b) (c)(d) ε k k x FIG. 2. The band structures in the minimal models forthe (a) nodal-point, (b) nodal-surface, (c) Z -trivial nodal-loop, and (d) Z -nontrivial nodal-loop semimetals. The bandcrossings are indicated by the same colors as in Fig. 1. Thepoints belonging to the same band crossing are connected bya dashed curve. occurs in the trace of quantum metric herein [35]. An al-ternative form of the Hamiltonian ˜ H CMS , k = vk ˆk · S indi-cates that the wave functions are independent of k . Dueto the absence of radial contribution, the trace of quan-tum metric Tr g sn CMS , k = (cid:104)| Λ sn | /k (cid:105) measures the dualrotational energy of the dynamical angular momentum Λ sn = r sn × k . Note that the angular momentum un-der rotation symmetry takes the form L sn = Λ sn + q sn ˆk .With | Λ sn | = | L sn | − ( q sn ) , we obtain the quantizedtrace of quantum metricTr g sn CMS , k = 1 k [ s ( s + 1) − ( q sn ) ] , (5)labeled by the angular momentum s and the monopolecharge q sn . This exemplifies the quantized band geom-etry from the dual Haldane sphere in the Z nodal-pointsemimetals.The duality can also be adopted to the Z nodal-surface (NS) semimetals [19], which accommodate two-band crossings on closed surfaces [Fig. 1(b)]. The Z nodal surface may be realized, for example, from a pairof degenerate chiral multifold semimetallic points P ± .When the degeneracy is broken by an energy splitting∆ ε = ε + − ε − > Z topo-logical structures of the original nodal points. Considerthe low-energy theory of a nodal surface below charge neutrality, which is formed by the n + -th and n − -th orig-inal bands. For the lower band, the band eigenstate cor-responds to the n ∓ -th original band inside(outside) thenodal surface. This implies the occurrence of differentBerry fluxes B in/out k = B sn ∓ k and according Chern num-bers C in/out = C sn ∓ in the two regions. A topologi-cal invariant is defined by the change of Chern numberacross the nodal surface ∆ C = C in − C out [19]. Thequantum metric can also be inferred directly, with theindividual components in the spin-1 / q = C in /
2, the spherical shell carries the totalcharge Q = ∆ C/ g in/outNS , k = Tr g sn ∓ CMS , k . (6)Analogous results hold for the upper band. Notably, theband geometry exhibit different structures on differentsides of the nodal surface. This feature is attributed tothe Berry charge at the nodal surface and is generic inthe Z nodal-surface semimetals.We next turn to the analysis of Z gapless topologi-cal systems. Before diving into the 3D systems, we dis-cuss how the dual energy quantization occurs at the two-dimensional (2D) Dirac point (2DDP). This example willserve as an important hint to the 3D Z gapless topolog-ical systems. A minimal model at the 2D Dirac pointtakes a rotationally symmetric form [79] H l , k = v ( k l + σ − + k l − σ + ) (7)with k ± = k x ± ik y , l = 1 / , , / , . . . , and Pauli ma-trices σ ± = ( σ x ± iσ y ) /
2. This system contains twobands with dispersion energies ε ± k = ± vk l , which be-come degenerate at the Dirac point k = . The Berryconnection forms a vortex structure around the Diracpoint, thereby driving a 2 πl phase winding in the bandeigenstates. Accordingly, the Berry flux experiences asingularity at the Dirac point and vanishes everywhereelse. The trace of quantum metric is determined by thisBerry vortex structure. According to the Hamiltonian H l , k = vk l (ˆ k l + σ − +ˆ k l − σ + ), the wave functions are k -independent. Without the radial contribution, the traceof quantum metric measures the dual rotational energyTr g ln , k = (cid:104) (Λ lnz ) /k (cid:105) . Here Λ lnz = L lnz serves as theangular momentum since the Berry monopole is absent.The magnitude | L lnz | = l corresponds to the 2 πl phasewinding around the Dirac point. We arrive at the dualenergy quantization in the trace of quantum metricTr g ln , k = l k , (8)consistent with a direct calculation for graphene when l = 1 / Z gapless topological systems. A natural gener-alization occurs in the nodal-loop semimetals, where theband crossings take place along closed loops [Fig. 1(c)][12–18]. Consider a minimal two-band model of thenodal-loop semimetals [22, 80] H NL , k (∆) = k ⊥ − ∆2 m σ x + v z k z σ y , (9)where k ⊥ = ( k x + k y ) / and ∆ >
0. This systemshows a nodal loop in the k x - k y plane at charge neu-trality [Fig. 2(c)], which is defined by the major radius k R = √ ∆. Assume an isotropic dispersion around thenodal loop v ⊥ = k R /m = v z = v . A linearization at lowenergy leads to the l = 1 / H l NL , k = v ( k lr, + σ − + k lr, − σ + ) , (10)where k r, ± = ( k ⊥ − k R ) ± ik z are defined around the nodalloop. This model obeys two rotation symmetries, whichare about the nodal loop and the k ⊥ = 0 line at loop cen-ter, respectively. The two bands exhibit the dispersionenergies ε ± k = ± vk lr , where k r = [( k ⊥ − k R ) + k z ] / isthe minor radius from the nodal loop. Importantly, themodel serves as a “stacking” of the 2D Dirac-point sys-tems along the nodal loop. This identifies the nodal loopas a Berry vortex line, a generalization from the Berryvortex at the 2D Dirac point. The 2 πl phase winding nowoccurs around the nodal loop. Accordingly, the Berryflux experiences a singularity along the nodal loop andvanishes everywhere else. The Berry vortex line inducesthe dual rotational energy of the axial angular momen-tum | L lnφ | = l , where ˆ φ is the tangent unit vector alongthe nodal loop. The trace of quantum metric thus ex-hibits the dual energy quantizationTr g ln NL , k = l k r , (11)similar to the result at the 2D Dirac point. Note that thenodal loop in H NL , k (∆) shrinks to a point and vanisheswhen ∆ decreases and turns negative. Since the nodalloop can be annihilated by itself, it is Z -trivial in thetopological classification [14].There also exist nodal loops which are Z -nontrivial[14, 16–18]. These nodal loops are realized under com-bined parity and time-reversal P T -symmetry, which im-poses the reality condition on the systems. Consider aminimal model of the Z -nontrivial nodal loops [14] H Z NL , k = vk x σ x + vk y τ y σ y + vk z σ z + mτ x σ x , (12)where the Pauli matrices σ a and τ a are defined. Thismodel accommodates four bands with the dispersion en-ergies ε ±± k = ± v [( k ⊥ ± k R ) + k z ] / for k R = | m | /v . At finite | m | >
0, the two middle bands form a nodal loopat charge neutrality with major radius k R [Fig. 2(d)].These two bands obey the same rotation symmetries asthe model of Z -trivial nodal loops. Similar to the Z -trivial nodal loop with l = 1 /
2, the nodal loop hereinserves as a Berry vortex line for a π phase winding. Animportant difference is the interplay with an additionalnodal line k ⊥ = 0 at finite energy [18]. For the up-per(lower) band, the additional nodal line occurs at theband crossing with the highest(lowest) band. This addi-tional nodal line penetrates the center of the nodal loop,leading to a topologically nontrivial “link” [Fig. 1(d)].The link with the nodal line defines a Z monopole atthe nodal loop. This monopole drives the nodal loop Z -nontrivial, which shrinks to a point (when m = 0) atmost under changing m . The Z -nontrivial nodal loopscan only be annihilated through the pair annihilation.The link of the Z monopole is encoded in the bandgeometry. We calculate the quantum metric for the twomiddle bands in the minimal model. The individual com-ponents of quantum metric are complicated, with thespecial cases derived at k z = 0 in Ref. 78. Neverthe-less, the trace of quantum metric is amiable according tothe duality. It can be shown that the wave functions areindependent of the radial components k r and k R . This in-dicates the absence of radial contributions in the trace ofquantum metric, leaving only the angular contributions.We further uncover that each wave function is a tensorproduct of two Z -trivial wave functions, which are at thenodal loop and the additional nodal line, respectively. Adirect calculation obtains the according dual rotationalenergy in the trace of quantum metricTr g Z NL , k = 14 k r + 14 k ⊥ . (13)The first term is the dual rotational energy about thenodal loop, which is identical to the Z -trivial case with l = 1 /
2. Meanwhile, the second term captures the dualrotational energy of the axial angular momentum | L nz | =1 / Z monopole, whichis absent for the Z -trivial nodal loop. We thus arrive ata clear interpretation of the nontrivial band geometry inthe Z nodal-loop semimetals.A direct manifestation of the band geometry occursin the spread of Wannier functions [35, 41–44, 83]. Forthe Wannier functions of a band, the spread functionalΩ = (cid:104) r (cid:105) − (cid:104) r (cid:105) exhibits the gauge-invariant lower boundΩ GI = V Σ with Σ = (cid:82) k Tr g k . Here V is the unit-cellvolume and (cid:82) k = (cid:82) d k/ (2 π ) . As the integrated trace ofquantum metric, the gauge-invariant lower bound orig-inates solely from the band geometry. An estimationΣ SR = (cid:82) SR , k Tr g k ≤ Σ can be further obtained by thefocus on a subregion (SR). We consider this estima-tion in the 3D gapless topological systems, where thestrongest contributions usually arise near the band cross-ings. For the chiral multifold semimetals, the estimatedlower bound Σ sn CMF = (Λ k / π ) G sn is quantized [35].Here G sn = (1 / π ) (cid:72) d S k · ˆk Tr g sn CMS , k = 2[ s ( s +1) − ( q sn ) ]is a quantized enclosing-surface integration, and Λ k isan ultraviolet (UV) cutoff of k . The quantized resultalso applies to the Z nodal-surface semimetals. Mean-while, the nodal-loop semimetals exhibit significantly dif-ferent features. For the Z -trivial nodal loop, the esti-mated lower bound is logarithmically divergent Σ ln NL =(1 / π )(2 πk R ) l log k r | Λ kr , where Λ k r defines the UVcutoff of k r . This result is a direct generalization from thelogarithmic divergence at the 2D Dirac point. The Z -nontrivial nodal loop exhibits an additional logarithmicdivergence ∼ ln k ⊥ | due to the link of the Z monopole.This distinguishes the estimated lower bound of the Z -nontrivial nodal loop from the Z -trivial one. Due to thelogarithmic divergences, the band geometry may havestronger manifestations at the nodal loops than at theother band crossings.The gauge-invariant lower bound can be probed di-rectly with a periodic drive [39]. Under a linear shakealong a direction ˆa , the integrated quantum metric (cid:82) k g aa k of the occupied bands corresponds to the exci-tation rate. The integrated trace of quantum metric isthen obtained from the measurements along all three di-rections a = x, y, z , which determines the gauge-invariantlower bound. Such a measurement has been performedexperimentally in an ultracold atomic Floquet Chern in-sulator [46]. Most of the current measurements apply tothe integrated trace of quantum metric in all occupiedbands. The probe with momentum and band resolutionswill advance the understanding of band geometry in thegapless topological systems [39, 40, 84].In summary, we show that the position-momentum du-ality offers a transparent interpretation of the band ge-ometry at the topological band crossings. The trace ofquantum metric provides a unified framework throughthis duality, which is achieved by encoding the dual ro-tational energy about the band crossings with nontrivialBerry defects. This scenario applies to the Z nodal-pointand nodal-surface semimetals, as well as the Z nodal-loop semimetals. The simple quantization rules may bebroken by the symmetry-breaking perturbations, such asthose far away from the band crossings. Nevertheless,the results in our analysis, such as the integrated traceof quantum metric, may remain close to the quantizedvalues until the band crossing structures completely van-ish. Our duality-based analysis offers a feasible routetoward further understanding of nontrivial band geome-try. The investigations in the other gapless topologicalsystems, such as the gapless topological superconductors[21–26] and spin liquids [27], degenerate bands with non-Abelian Berry connections [58], or higher-dimensionalsystems with tensor monopoles [40, 67], may serve asinteresting topics for future work. The authors thank Nathan Goldman, Ching Hua Lee,and especially Rahul Nandkishore for fruitful discussionsand feedback on the manuscript. YPL was sponsored bythe Army Research Office under Grant No. W911NF-17-1-0482. The views and conclusions contained in thisdocument are those of the authors and should not beinterpreted as representing the official policies, eitherexpressed or implied, of the Army Research Office orthe U.S. Government. The U.S. Government is autho-rized to reproduce and distribute reprints for Govern-ment purposes notwithstanding any copyright notationherein. 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