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Articleshttps://doi.org/10.1038/s41567-020-0967-9
Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTeAdrien Bouhon1,2,12 ✉, QuanSheng Wu 3,4,12 ✉, Robert-Jan Slager 5,6,12 ✉, Hongming Weng 7,8, Oleg V. Yazyev 3,4 and Tomáš Bzdušek 9,10,11
1Nordic Institute for Theoretical Physics (NORDITA), Stockholm, Sweden. 2Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden. 3Institute of Physics, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland. 4National Centre for Computational Design and Discovery of Novel Materials MARVEL, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland. 5TCM Group, Cavendish Laboratory, University of Cambridge, Cambridge, UK. 6Department of Physics, Harvard University, Cambridge, MA, UK. 7Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, China. 8Songshan Lake Materials Laboratory, Guangdong, China. 9Condensed Matter Theory Group, Paul Scherrer Institute, Villigen, Switzerland. 10Department of Physics, University of Zürich, Zürich, Switzerland. 11Department of Physics, McCullough Building, Stanford University, Stanford, CA, USA. 12These authors contributed equally: Adrien Bouhon, QuanSheng Wu, Robert-Jan Slager. ✉e-mail: [email protected]; [email protected]; [email protected]
SUPPLEMENTARY INFORMATION
In the format provided by the authors and unedited.
NATuRe PHYSicS | www.nature.com/naturephysics
Supplementary Information for
Non-Abelian Reciprocal Braiding of Weyl Points and its Manifestation in ZrTe
Adrien Bouhon1,2,∗ QuanSheng Wu3,4,∗ Robert-Jan Slager5,6,∗
Hongming Weng7,8, Oleg V. Yazyev3,4, and Tomas Bzdusek9,10,11
1Nordic Institute for Theoretical Physics (NORDITA), Stockholm, Sweden2Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 21 Uppsala, Sweden3Institute of Physics, Ecole Polytechnique Federale de Lausanne, CH-1015 Lausanne, Switzerland
4National Centre for Computational Design and Discovery of Novel Materials MARVEL,Ecole Polytechnique Federale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
5TCM Group, Cavendish Laboratory, University of Cambridge,J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
6Department of Physics, Harvard University, Cambridge, MA 021387Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China8Songshan Lake Materials Laboratory, Guangdong 523808, China
9Condensed Matter Theory Group, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland10Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland and
11Department of Physics, McCullough Building, Stanford University, Stanford, CA 94305, USA
(Dated: June 8, 2020)
LIST OF CONTENTS
A. Crystal structure of the minimal 2D model. 1
We give the details of the crystal structure of the minimal 2D tight-binding model discussed in
the main text and presented in Methods.
B. Alternative braiding model. 1
We present a 3-band lattice model that exhibits a non-trivial braiding of band nodes without
the nodes moving across the Brillouin zone boundary.
C. Reality condition. 3
We show that the existence of a k-local antiunitary symmetry squaring to +1 (such as C2T )
implies the existence of a basis in which the Bloch Hamiltonian is a real symmetric matrix.
D. Euler form in three-band models. 4
We prove the geometric interpretation of the Euler form in three-band models, which is presented
in Fig. 4(b) of the main text.
E. Singularity of Euler form at principal nodes. 5
We study analytic properties of Euler form near principal nodes. If the calculation is performed
in the eigenstate basis, the Euler form is integrable albeit non-differentiable at principal nodes.
This is an obstruction for the naıve application of Stokes’ theorem.
F. Euler class for manifold with a boundary. 9
We generalize Euler class to manifolds with a boundary, and we show that it detects the capa-
bility of pairs of principal nodes to annihilate.
G. Non-Abelian frame-rotation charge. 10
We review the homotopic derivation of the non-Abelian frame-rotation charge from Ref. [30],
and we prove its relation to the Euler class on manifolds with a boundary.
H. Numerical calculation of the Euler form. 13
We present a numerical algorithm that calculates the Euler class on a manifold with a boundary,
and we present a way to regularize certain numerically induced divergences.
List of references 14
∗ Contributed equally. Correspondence to [email protected],[email protected], and [email protected].
1
A. Crystal structure of the minimal 2D model
The minimal 2D tight-binding model presented inMethods realizes the centered orthorhombic layer groupc222 (LG22), with point groupD2 = {C2z, C2y, C2x} [72].There are three orbitals per unit cell, one s-wave orbitalat Wyckoff position 1a (φA) and one (real) pz-wave or-bital at Wyckoff position 2g (φB , φC), see Fig. S1 wherethe primitive cell is spanned by the Bravais lattice vectorsa1 = (ax+ by)/2 and a2 = (−ax+ by)/2.
Fig. S1. Crystal structure of the minimal 2D tight-binding modeldiscussed in the main text and presented in Methods. The con-ventional unit cell (centered orthorhombic) is spanned by ax andby. There are three orbitals per each primitive cell spanned bythe primitive vectors a1 and a2, one s-wave orbital labeled φA(blue), and two pz-wave orbitals labeled φB and φC (green).The point group is D2 = {C2z, C2(11) ≡ C2y, C2(11) ≡ C2x}with C2z perpendicular to the basal plane.
The locations of the orbitals within the n-th unit cellare {Rn + rα}α=A,B,C , with rA = (0, 0) for 1a, andrB = (0, u) and rC = (0,−u) for 2g (written in theCartesian frame) with u < b/2. The tight-binding modelis written in the Bloch state basis
|φα,k〉 =1√Nα
∑Rn
eik·Rn |w,Rn + rα〉 , α = A,B,C ,
(S1)that is the discrete Fourier transform of the Wannierstates |w,Rn + rα〉 localized at Rn + rα, with Nα thenumber of lattice sites occupied by orbital α.
B. Alternative braiding model
The change of topological charge of band nodes uponexchange in momentum space, i.e. the reciprocal braid-ing, requires a minimum of three bands, correspondingto two distinct species of band nodes, An explicit tight-binding model demonstrating the reciprocal braiding isgiven in Eq. (4) of Methods, and the corresponding braid-ing protocol is illustrated in Fig. 2 of the main text. Thatmodel, while being minimal in terms of hopping processes
x
y
a
b
Fig. S2. Model for the alternative braiding protocol.Schematic illustration of the lattice model leading to the nearest-neighbor tight-binding Hamiltonian in Eq. (S2). We considerpx (blue), py (purple) and s (green) orbital in the middle of arhombic unit cell (gray). The conventional unit cell is a rectanglewith dimensions a and b, indicated by the black frame. We seta = b = 1 for simplicity. We only consider hopping processes(both intra-orbital T2,4,6 and inter-orbital ±T7,8,9) to the fournearest neighbor sites forming the corners of the conventionalunit cell. On-site energies of the three orbitals are T1,3,5.
to be implemented, achieves the reciprocal braiding whilemoving the band nodes around the Brillouin zone (BZ)torus. However, the non-trivial braiding phenomenon ismore general. To demonstrate this, we here provide anexplicit tight-binding model that realizes the reciprocalbraiding locally, i.e. without either of the nodes passingthough the BZ boundary.
The dependence of the topological charge on the ex-change of band nodes in momentum space implies ro-bustness of the phenomenon, especially in contrast tocertain other proposed schemes that flip the topologicalcharge of a band node by relying on the periodicity of themomentum space. In particular, it has been reported bythe insightful Ref. [73] that in certain lattices the topo-logical charge of a band node can be reversed simply bycircumnavigating the BZ torus. Assuming the conven-tion in which the Bloch Hamiltonian H(k) is periodicin reciprocal lattice vectors, such phenomenon may arisewhen the C2T operator explicitly depends on momentum(k). Such a dependence often occurs for non-symmorphiclattices or for orbitals located on the unit cell boundary.Notably, this phenomenon is pertinent also to two-bandmodels [73]. However, as it explicitly relies on the peri-odicity of the momentum space, such a charge reversalcannot be reproduced by a k · p model. In contrast, thereciprocal braiding considered by our work does not relyon the periodicity of the momentum space but solely onthe braided node trajectories, implying that it can bereproduced with a 3-band k · p model.
To obtain an explicit tight-binding model for such a“local” braiding, we consider a 2D lattice belonging tothe cmm wallpaper group, and we assume px, py and s
2
a b
Fig. S3. Node trajectories for the alternative braiding protocol. a. Trajectory of band nodes of the 2D tight-binding model inEq. (S2) in the (kx, ky)-plane as a function of adiabatic parameter t ∈ [π
2, π]. The red (blue) lines indicate trajectories of band
nodes in the upper (lower) band gaps, and the indicated orientations follow the convention from Refs. [30] and [31]. The green frameindicates the Brillouin zone of the tight-binding model, and the two diagonal planes indicate the mirror-invariant lines of the Brillouinzone. b. Analogous plot for the k · p model in Eq. (S4). The node trajectories are almost identical (they are very slightly closer tokx = ky = 0). However, the periodic momentum space of the original tight-binding model is now replaced by the Euclidean R2.
orbitals located in the center of each unit cell. The con-ventional unit cell is a rectangle with dimensions a andb, and the Brillouin zone is a rhombus with horizontaldiagonal of length 2π/a, and vertical diagonal of length2π/b. For simplicity, we set a = 1 = b throughout thediscussion. The lattice is symmetric under mirror mx :x 7→ −x represented by mx = diag (−1,+1,+1), mirrormy : y 7→ −y represented by my = diag (+1,−1,+1),their composition C2 : (x, y) 7→ (−x,−y) represented by
C2 = diag (−1,−1,+1), as well as under time-reversal Trepresented by complex conjugation K. The C2T oper-
ator is rotated to the canonical form K [see Sec. C be-low] through a unitary rotation of the Hilbert space withV = diag (+i,+i,+1). The mirror symmetries are notessential for the node braiding, but they pin the bandnodes to high-symmetry lines, simplifying the analysis.Mirror-breaking perturbations compatible with the C2Tsymmetry preserve the non-trivial braiding characteristicof the model.
Setting the energy of the three orbitals to T1,3,5,the intra-orbital hopping amplitudes between the fournearest-neighbor (NN) sites to T2.4.6, and the inter-orbital hopping amplitudes for NN sites to ±T7,8,9, weobtain Bloch Hamiltonian
HTB(k) =
T1 + 4T2 cos kx cos ky 4T7 sin kx sin ky 4iT8 sin kx cos ky4T7 sin kx sin ky T3 + 4T4 cos kx cos ky −4iT9 cos kx sin ky−4iT8 sin kx cos ky 4iT9 cos kx sin ky T5 + 4T6 cos kx cos ky
. (S2)
To realize the reciprocal braiding, we assume the following dependence of the parameters on time t ∈ [π2 , π],
T1 = 3.5 + 3 cos t T2 = −0.1 + 0.2 cos t T3 = 0.25− 0.25 cos t
T4 = 0.1 cos t T5 = −1.5− cos t T6 = 0.25− 0.5 cos t (S3)
T7 = +0.15 T8 = −0.5 cos t T9 = −0.2.
The node trajectories as a function of time are plotted in Fig. S3a. We observe that the pair of “blue nodes”, whichcorrespond to the lower/principal band gap and are pairwise created at time t ≈ 16
28π, fail to annihilate when they
meet at a later time t ≈ 2228π, after being braided with a “red node” (corresponding to the upper/adjacent band gap).
The orientations of the trajectories displayed in Fig. S3 follow the convention explained in Refs. [30] and [31]. Amore detailed illustration of the spectrum along the two high-symmetry lines as a function of parameter t is shownin Fig. S4.
Although the Hamiltonian in Eq. (S2) is periodic in reciprocal lattice vectors, the change of the topological chargeof the nodes after reciprocal braiding does not depend on the periodicity of momentum space. One can confirm thisby explicitly expanding the Hamiltonian to the second order in kx,y, which implicitly replaces the BZ torus by R2.
3
n = 0 n = 1 n = 2 n = 3 n = 4
n = 5 n = 6 n = 7 n = 8 n = 9
n = 10 n = 11 n = 12 n = 13 n = 14
Fig. S4. Spectrum of the alternative braiding Hamiltonian along the Brillouin zone diagonals. Spectrum of the tight-bindingHamiltonian in Eq. (S2) for times t = π
2+ n π
28, plotted along the kx-axis (solid orange lines), and along the ky-axis (dashed purple
lines). The blue circles (red squares) indicate band nodes formed inside the lower/principal (upper/adjacent) band gap. The graydotted lines indicate the spectrum along either momentum axis when the orbital mixing terms T7,8,9 are turned off.
The resulting k · p Hamiltonian
Hk·p(k) =
T1 + 4T2
(1− k2x+k
2y
2
)4T7kxky 4iT8kx
4T7kxky T3 + 4T4
(1− k2x+k
2y
2
)−4iT9ky
−4iT8kx 4iT9ky T5 + 4T6
(1− k2x+k
2y
2
))
. (S4)
leads to node trajectories shown in Fig. S3b, whichare just imperceptibly narrower than those of the tight-binding model in Eq. (S2).
C. Reality condition
In the main text we assumed that C2T symmetry im-plies reality of the Bloch Hamiltonian. While this is nottrue in general, the conclusions presented in the maintext still apply. More precisely, the presence of antiu-nitary operator A that obeys (i) A2 = +1 and (ii)∀k : AH(k)A−1 = H(k), implies the existence of aHilbert-space basis, in which the Hamiltonian H(k) isreal. In this section we justify this claim by two differ-ent methods. We remark that the antiunitary A thatfulfills the two conditions can be realized as C2T in two-dimensional spinful or spinless systems [33], or as PTin spinless systems of arbitrary dimension [30]. There-fore, for such symmetry settings, both the frame-rotationcharge resp. the Euler class can be defined if the rightHilbert-space basis has been adopted.
We first prove this statement formally, before providing
a physical insight in the next paragraph. Every antiuni-tary operator A can be represented as some unitary op-erator U composed with the complex conjugation K [74],i.e. A = UK . Unitarity means that UU† = 1, whileA2 = +1 implies that UU∗ = 1. It follows that U = U>.The Autonne-Takagi factorization [75] then guaranteesthat U = VDV> for some unitary V and a diagonalmatrix D = diag{eiϕj}nj=1. Constructive and finite al-gorithms exist that find the Autonne-Takagi decompo-sition of a symmetric unitary matrix [76]. Rotation ofthe Hilbert-space by unitary matrix
√D∗V † ≡ W then
transform the antiunitary operator to WAW † = K, i.e. tothe form assumed in the main text.
From a physical point of view, it is well known thatan antiunitary symmetry squaring to +1 (rather than−1) does not imply Kramers degeneracies [77]. In otherwords, eigenstates of A form one-dimensional irreduciblerepresentations. By taking these eigenstates to form thebasis of the Hilbert space, the antiunitary operator is rep-
resented by D′K , where D′ = diag{eiϕ′j}nj=1 is a diagonal
matrix of phase factors. Rotating the Hilbert space basisby W ′ =
√D′∗ then transforms the antiunitary opera-
tor to the complex conjugation K . While the absence ofKramers doubling for such a symmetry is well accepted
4
in the solid-state community, the formal proof of thisstatement actually follows from the Autonne-Takagi fac-torization, as discussed in the previous paragraph.
D. Euler form in three-band models.
In this section, we elaborate on the analogy betweenthe Euler class and the first Chern number by brieflyfocusing on the minimal models. More specifically, weshow that both the Berry curvature F(k) of one bandobtained from a two-band complex Hamiltonian, as wellas the Euler form Eu(k) of two bands obtained from athree-band real Hamiltonian, can be understood by con-sidering geometry on a 2-sphere (S2). The discussionbelow thus proves the geometric interpretation of Eulerform presented in Fig. 4(b) of the main text.
Let us first recall the mathematics behind the firstChern number of a complex Hamiltonian, defined inEq. (21) of Methods, for the case of two bands. Her-mitian two-band Hamiltonians can be decomposed usingthe Pauli matrices and the unit matrix as
H(k) = h(k) · σ + h0(k)1 (S5)
where h0,x,y,z(k) are real functions of momentum. Spec-tral flattening brings eigenvalues of the Hamiltonian to±1 without changing the band topology [6], and is asso-ciated with replacing h0 7→ 0 and h 7→ h/||h|| ≡ n. Theband topology of the two-band complex Hamiltonian isthus completely captured by the three-component unitvector n(k) ∈ S2. It is known [2, 3] that the Berry cur-vature of one of the two bands can be expressed as
Fij = 12n · (∂kin × ∂kjn), (S6)
such that Fij dki dkj corresponds to one half of the (ori-ented) solid angle spanned by n on the S2 as the momen-tum argument is varied over a rectangle of size dki×dkj .For a closed two-dimensional base manifold, the vectorn has to wrap around the unit sphere an integer numberof times, hence the integral of Fij dki dkj , i.e. the total(oriented) solid angle spanned by n, must be quantizedto integer multiples of 2π. Therefore, c1(E) defined inEq. (21) of Methods is an integer in the case of two-bandmodels. This simple argument does not generalize tomodels with more than two bands, in which case one hasto follow the proof outlined in the paragraph Quantiza-tion of Euler class in Methods.
We find that a very similar geometric interpretationalso applies to Euler form of two bands obtained from athree-band real Hamiltonian. In this case, spectral flat-tening brings the Hamiltonian with two occupied bands
and one unoccupied band to
H(k) = 2n(k) · n(k)> − 1, (S7)
where n(k) = u1(k) × u2(k) ∈ S2 is the (normal-ized) column vector representing the unoccupied state,which can be represented as cross product of the (nor-malized) occupied states u1(k) and u2(k). Note that,because of the reality condition, the left (“bra”) and theright (“ket”) eigenstates are componentwise equal to eachother. The quadratic dependence of the Hamiltonian onthe unit vector, manifest in Eq. (S7), implies that vec-tors ±n represent the same Hamiltonian. Therefore thespace of unique spectrally flattened 3-band Hamiltoni-ans is S2/Z2 ≡ RP 2 [30]. However, if the vector bundledefined by H(k) is orientable (which is a necessary condi-tion to define Euler form), then there are no closed pathsγ ⊂ B in the base manifold which would be mappedby the Hamiltonian to the non-contractible path in RP 2.Therefore, Euler form of an orientable rank-2 bundle ob-tained from a three-band real Hamiltonian, is related togeometry on S2. In fact, we show below that
Euij = n ·(∂kin× ∂kjn
), (S8)
which [besides the altered interpretation of n(k)] quali-tatively differs from Eq. (S6) only by the absence of theprefactor 1
2 . Following the same arguments as for thefirst Chern number, we find that for three-band modelsthe Euler class χ(E) defined in Eq. (20) of Methods mustbe an even integer. This agrees with the known fact,that odd values of the Euler class (corresponding to anon-trivial second Stiefel-Whitney class) require modelswith at least two occupied and with at least two unoccu-pied bands [33]. We also remark that the cross-productdefinition of n(k) in terms of the two occupied statesmakes the expression in Eq. (S8) invariant only underthe proper SO(2) gauge transformations of the occupiedstates, reminding us of the importance of orientability ofthe vector bundle.
The remainder of this section contains a proof ofEq. (S8). While the logic of the proof is straightforward,some of the expressions are rather lengthy. We employthe Einstein summation convention, and we write
na = εabcu1bu
2c (S9)
where ε is the Levi-Civita symbol. The right-hand sideof Eq. (S8) can be expressed as
(S8) = εabcu1bu
2cεade∂ki(εdfgu
1fu
2g)∂kj (εehiu
1hu
2i ). (S10)
Using the identity εabcεade = δbdδce − δbeδcd, and per-forming the summation over indices b and c, we obtain
(S8) = εdfgεehi(u1du
2e − u1eu2d)∂ki(u1fu2g)∂kj (u1hu2i ). (S11)
To get rid off the remaining Levi-Civita symbols, we use
5
εdfgεehi = δde(δfhδgi − δfiδgh)− δdh(δfeδgi − δfiδge) + δdi(δfeδgh − δfhδge). (S12)
This long identity has to be substituted into Eq. (S11). Note that the combinations of Kronecker symbols containingδde trivially lead to zero after the substitution, because δde(u
1du
2e − u1eu2d) = 0. The remaining terms in Eq. (S12),
after summing over indices e, h, and i, lead to
(S8) = −(u1du2f − u1fu2d)∂ki(u1fu2g)∂kj (u1du2g) + (u1du
2g − u1gu2d)∂ki(u1fu2g)∂kj (u1du2f )
+(u1du2f − u1fu2d)∂ki(u1fu2g)∂kj (u1gu2d)− (u1du
2g − u1gu2d)∂ki(u1fu2g)∂kj (u1fu2g) (S13)
Performing the derivatives by parts, Eq. (S13) expands into 32 individual terms
(S8) = −u1du2f (∂kiu1f )u2g(∂kju
1d)u
2g − u1du2f (∂kiu
1f )u2gu
1d(∂kju
2g)− u1du2fu1f (∂kiu
2g)(∂kju
1d)u
2g − u1du2fu1f (∂kiu
2g)u
1d(∂kju
2g)
+u1fu2d(∂kiu
1f )u2g(∂kju
1d)u
2g + u1fu
2d(∂kiu
1f )u2gu
1d(∂kju
2g) + u1fu
2du
1f (∂kiu
2g)(∂kju
1d)u
2g + u1fu
2du
1f (∂kiu
2g)u
1d(∂kju
2g)
+u1du2g(∂kiu
1f )u2g(∂kju
1d)u
2f + u1du
2g(∂kiu
1f )u2gu
1d(∂kju
2f ) + u1du
2gu
1f (∂kiu
2g)(∂kju
1d)u
2f + u1du
2gu
1f (∂kiu
2g)u
1d(∂kju
2f )
−u1gu2d(∂kiu1f )u2g(∂kju1d)u
2f − u1gu2d(∂kiu1f )u2gu
1d(∂kju
2f )− u1gu2du1f (∂kiu
2g)(∂kju
1d)u
2f − u1gu2du1f (∂kiu
2g)u
1d(∂kju
2f )
+u1du2f (∂kiu
1f )u2g(∂kju
1g)u
2d + u1du
2f (∂kiu
1f )u2gu
1g(∂kju
2d) + u1du
2fu
1f (∂kiu
2g)(∂kju
1g)u
2d + u1du
2fu
1f (∂kiu
2g)u
1g(∂kju
2d)
−u1fu2d(∂kiu1f )u2g(∂kju1g)u
2d − u1fu2d(∂kiu1f )u2gu
1g(∂kju
2d)− u1fu2du1f (∂kiu
2g)(∂kju
1g)u
2d − u1fu2du1f (∂kiu
2g)u
1g(∂kju
2d)
−u1du2g(∂kiu1f )u2g(∂kju1f )u2d − u1du2g(∂kiu1f )u2gu
1f (∂kju
2d)− u1du2gu1f (∂kiu
2g)(∂kju
1f )u2d − u1du2gu1f (∂kiu
2g)u
1f (∂kju
2d)
+u1gu2d(∂kiu
1f )u2g(∂kju
1f )u2d + u1gu
2d(∂kiu
1f )u2gu
1f (∂kju
2d) + u1gu
2du
1f (∂kiu
2g)(∂kju
1f )u2d + u1gu
2du
1f (∂kiu
2g)u
1f (∂kju
2d)
Most of these 32 terms vanish. To see this, we use orthonormality u1au2a = 0 on the vector coordinates displayed in
red. Furthermore, the normalization u1au1a = 1 = u2au
2a implies that u1a(∂kβu
1a) = 0 = u2a(∂kβu
2a), which we indicate
in blue. Only two terms remain, in which we further use the normalization to 1 on vector components displayed ingreen. After renaming the repeated indices, we obtain
Euij = (∂kiu1f )(∂kju
2f )− (∂kju
1f )(∂kiu
2f ) ≡
⟨∂kiu
1|∂kju2⟩−⟨∂kju
1|∂kiu2⟩. (S14)
The last expression exactly corresponds to the components of the the Euler form over two bands as defined in Eq. (19)of Methods. This completes the proof of Eq. (S8).
E. Analytic properties of Euler form at principalnodes
E1. General remarks
In the main text we consider the Euler form Eu(k)defined by the two principal bands. Note that adjacentnodes pose problems for the mathematical construction.This is because circumnavigating an adjacent node re-verses the orientation of one of the principal Bloch states(the one that participates in the formation of the adja-cent node), but not the other one. Therefore, paralleltransport around an adjacent node is associated with animproper gauge transformation X = σz /∈ SO(2). Sincesuch vector bundle is not orientable, its Euler curvaturecannot be defined. This is the reason why we only con-sider calculations over regions with no adjacent nodes.
In contrast, understanding the behavior of the Eulerform near a principal node is more subtle. First, circum-navigating a principal node reverses the sign of both prin-cipal Bloch states, which corresponds to a proper gaugetransformation X = −1 ∈ SO(2), such that Euler formof the bundle is well-defined on an annulus around thenode. On the other hand, the eigenstates of the Hamil-tonian are not continuous functions of k at the location
of principal nodes, suggesting that the derivatives of theeigenstates are not well-defined at these points. Nev-ertheless, the rank-2 vector bundle spanned by the twoprincipal bands is actually continuous and differentiableat principal nodes. The last two statements are not incontradiction! Indeed, a discontinuous basis of the bun-dle does not imply discontinuity of the bundle. Theremay be (and we show that there really is) a differentbasis which is perfectly continuous at principal nodes.
However, one has to consider the relevance of the twobases for physical observables. Since the two principalbands are separated by a band gap away from the prin-cipal nodes, the discontinuous basis of the bundle corre-sponding to the principal eigenstates has a special phys-ical significance. Especially, we show in Sec. F that thiscanonical (although discontinuous) basis encodes observ-able features, such as the path-dependent ability of bandnodes to annihilate. This information is lost once we al-low for mixing of the two principal eigenstates by a gen-eral SO(2) gauge transformation – such as when going tothe basis that reveals the continuity of the bundle.
The physical importance of the eigenstate basis canbe naturally built into the mathematical description byintroducing the notion of a flag bundle [78, 79], and bycontrasting to the more familiar notion of a vector bun-
6
dle. On the one hand, the two principal bands span atwo-dimensional vector space that varies continuously asa function of k. In other words, the two principal bandsconstitute a rank-two vector bundle. In this description,the two-dimensional vector space admits arbitrary SO(2)gauge transformations, and the physical eigenstate basisis not treated as being special. Therefore, in this descrip-tion it is natural to adopt the continuous basis, wherethere are no singularities – and thus also no topologi-cal information. This is compatible with the well-knownfact that any smooth vector bundle is trivializable overa patch (disc).
In contrast, a flag bundle keeps information about thesubdivision of the two-dimensional vector space into twoone-dimensional vector spaces, and thus preserves in-formation about the two physical eigenstates. In thislanguage, the admissible (orientation-preserving) gaugetransformations of the two dimensional vector space arejust S[O(1)×O(1)] ∼= Z2 (corresponding to X = ±1, ex-cluding X = ±σz). The flag bundle exhibits a singularityat the principal node, which is unremovable by admissi-ble gauge transformations, and which can be describedusing a topological invariant – the Euler class. Note thatthe rank-2 flag bundle exhibits a singularity and topo-logical invariant on a patch (disc), even though the cor-responding rank-2 vector bundle is continuous and trivi-alizable on the patch. This is because the vector bundledescription carries less information, which is insufficientto describe the principal nodes.
In the remainder of this text, we avoid the notion offlag bundle. Instead, we speak of the computation in theeigenstate basis, which (based on the remark in the previ-ous paragraphs) we except to convey topological informa-tion. This is contrasted to the computation in the contin-uous (or rotated) basis, where no topological informationis expected. The robustness of the invariant derived inthe eigenstates basis cannot be rigorously justified withinthe vector bundle description, and only comes from therestricted S[O(1)×O(1)] ∼= Z2 gauge transformation ofthe flag bundle formalism.
E2. Computation in the eigenstate basis
We begin our discussion by presenting the most generalHamiltonian near a principal node to the linear order inmomentum. We treat the obtained Hamiltonian pertur-batively, and we consider the eigenstate basis to revealthe structure of the Euler form near the principal node.
We first consider a two-band model that exhibits anode at k = 0 at zero energy. To linear order, the Hamil-tonian near the node must take the form
H2-band(k) =
2∑i=1
3∑j=0
kihijσj (S15)
where k1,2 are the two momentum coordinates, hij are8 real coefficients, and {σj}3j=0 is the unit matrix and
the three Pauli matrices. It is known that by a suitableproper rotation and by a linear rescaling of the momen-tum coordinates to κ(k), we can bring the Hamiltonianto the form
H2-band(κ) =
(κ1 + ε(κ) ±κ2±κ2 −κ1 + ε(κ)
), (S16)
where ε(κ) = v1κ1+v2κ2 describes the tilting of the bandnode [43]. Since the considered coordinate transforma-tion is linear, the associated Jacobian Jij = ∂κi/∂kj is aconstant matrix. The ± sign corresponds to nodes withpositive vs. negative winding number w ∈ π1[SO(2)] = Z,which we keep unspecified throughout the whole section.We assume det J > 0, i.e. the change of coordinates pre-serves the orientation of the bundle.
If there are additional bands, then the same rotationof the two basis vectors brings the corresponding n-bandlinear-order Hamiltonian to the form
Hn-band(κ)=
κ1+ε(κ) ±κ2 f>(κ)±κ2 −κ1+ε(κ) g>(κ)f(κ) g(κ) E+h(κ)
, (S17)
where f(κ) and g(κ) are κ-linear (n − 2)-componentcolumn vectors with real components {fc(κ)}nc=3 and{gc(κ)}nc=3, E is a non-degenerate diagonal matrix of(n−2) non-zero band energies, and h(κ) is κ-linear Her-mitian matrix of size (n − 2) × (n − 2). In Eq. (S17)we explicitly assume that the additional (n − 2) basisvectors of the Hilbert space are given by the additionaleigenstates of the Hamiltonian at the node, thereforeh(κ=0) = 0. Therefore, after adopting the properly ro-tated and rescaled momentum coordinates and the rightHilbert-space basis, the model in Eq. (S17) representsthe most general n-band real Hamiltonian near a princi-pal node to linear order in momentum.
To proceed, we split the Hamiltonian in Eq. (S17) intoH0 = diag(ε(κ), ε(κ), E), and a “perturbation” H′ thatcontians all the terms linear in κ, that is
H′(κ) =
κ1 ±κ2 f>(κ)±κ2 −κ1 g>(κ)f(κ) g(κ) h(κ)
. (S18)
The first step of the pertubation theory requires us tofind states that diagonalize the matrix H′ab = 〈a|H′ |b〉with a, b ∈ {1, 2} representating the degenerate states atthe principal nodes. We take |1〉 = (1 0 0 . . .)> and|2〉 = (0 1 0 . . .)>, in which case this matrix correspondssimply to the upper-left 2 × 2 block of Eq. (S18). If wefurther decompose κ using polar coordinates as κ1 =κ cosφ and κ2 = κ sinφ, this block is diagonalized by
∣∣∣1(0)⟩ =
± sin φ2
− cos φ20
and∣∣∣2(0)⟩ = ζ
+ cos φ2± sin φ
20
(S19)
where ζ = ±1 corresponds to two different orientations of
7
the bundle. Changing the relative sign between the twostates corresponds to orientation-changing gauge trans-formation X = ±σz. On the other hand, increasingφ 7→ φ + 2π flips the sign of both bands, which corre-sponds to a proper gauge transformation X = −1.
The first-order correction to the states in Eq. (S19) isgiven by
|a(1)〉 =
n∑c=3
〈c|H′∣∣a(0)⟩
ε(κ)− Ec|c〉 (S20)
where |c〉 is the cth element of the basis in which weexpressed Eq. (S18). This prescription does not lead toa change in the first two compoments of the principalvectors, while for compoments with c ≥ 3 we find
〈c|1(1)〉 = 1ε(κ)−Ec
[±fc(κ) sin φ
2 − gc(κ) cos φ2
](S21)
〈c|2(1)〉 = ζε(κ)−Ec
[+fc(κ) cos φ2 ± gc(κ) sin φ
2
].(S22)
Note that the expressions inside the square brackets arelinear in κ, while the prefactor can be approximated forκ close to 0 as
1
ε(κ)− Ec≈ − 1
Ec+v1κ1 + v2κ2
E2c. (S23)
Therefore, if we are after terms of the lowest order in κ,we can approximate the prefactor simply by −1/Ec. Fur-thermore, notice that states
∣∣1(0+1)⟩
and∣∣2(0+1)
⟩, which
we obtained by performing the first-order perturbationtheory, are not properly normalized. However, since thelowest-order corrections
∣∣1(1)⟩ and∣∣2(1)⟩ are linear in κ,
the correction from the proper normalization would bequadratic in κ. More explicitly, normalizing the stateswould induce a prefactor of the form
1√1 + ||κN||2
≈ 1− 1
2||κN||2 (S24)
where ||κN|| represents the normalization of the first-order correction. Since we are interested only in correc-tions to the principal states of the lowest order in κ, wesafely ignore the normalization.
We have established the lowest-order (linear) correc-tions in κ to the principal Bloch states in Eq. (S19). Wecan use the obtained states to calculate the Euler con-nection and the Euler form in the eigenstate basis,
ai=〈1|∂κi2〉 and Eu=〈∂κ11|∂κ2
2〉−〈∂κ21|∂κ1
2〉 (S25)
where we dropped the superscript “(0+1)” for brevity.However, note that the symbolic computation of thederivatives in Eq. (S25) cannot be fully trusted for k → 0because of the discontinuity of the states in Eq. (S19) atk = 0. Nevertheless, for now we ignore this possible
source of problem. To proceed, we use that
∂κ1=
∂κ
∂κ1∂κ +
∂φ
∂κ1∂φ = cosφ∂κ −
sinφ
κ∂φ (S26)
∂κ2 =∂κ
∂κ2∂κ +
∂φ
∂κ2∂φ = sinφ∂κ +
cosφ
κ∂φ. (S27)
We also rewrite
fc(κ) = αcκ cosφ+ βcκ sinφ (S28)
gc(κ) = γcκ cosφ+ δcκ sinφ. (S29)
With the help of Mathematica, we find the Euler con-nection to the leading order in κ as
a =±ζ2κ
(sinφ,− cosφ) +O(κ) (S30)
which diverges as we approach the node. In contrast, theEuler form to the leading order in κ is
Eu = − ζn∑c=3
1
E2c
[(βcγc − αcδc) (S31)
± 12 (αc cosφ+βc sinφ)2 ± 1
2 (γc cosφ+δc sinφ)2]
(wrong! – read end of Sec. E2)
which does not diverge at the node. All the 1/κ fac-tors, present in some of the previous formulae, cancelout. Substituting back the original coordinates, κ → k,corresponds to a double multiplication by the (constant)Jacobian matrix, which also does not induce a divergence.One observes (wrongly! as explained later) that Eu is dis-continuous at principal nodes (note the dependence onφ for κ→ 0), which prevents the applicability of Stokes’theorem. Nevertheless, the singularity in Eq. (S31) is in-tegrable, which suggests that the definition of Euler classin Eq. (1) of the main text is mathematically meaningful.By integrating over an infinitesimal disc Dε with radiusε around the principal node, we find limε→0
∫Dε Eu = 0
and limε→0
∮∂Dε a = ∓πζ, such that the Euler class on
the disc [cf. Eq. (5) of the main text, and Sec. F below]
χ(Dε) = ∓ ζ2 (S32)
is non-zero, and carries topological information.
However, as we warned before Eq. (S26), the presentedcomputation cannot be fully trusted due to the disconti-nuity of states |1〉 and |2〉 at κ = 0. This means that boththe Euler connection and Euler form are, strictly speak-ing, undefined at κ = 0, and we have attempted to per-form a series expansion around a singular point. It turnsout that our result for Euler connection a in Eq. (S30)is correct. In contrast, as we convincingly show in thenext subsection, the result for the Euler connection Euin Eq. (S31) is wrong! – namely the whole discontinuouscontribution to Eu in the second line of Eq. (S31) shouldnot be there. The Euler form turns out to be continuousat the principal node, and what really prevents the ap-
8
plicability of the Stokes’ theorem is the divergence of afor κ→ 0 observed in Eq. (S30).
E3. Computation in the rotated (continuous) basis
In the previous subsection, we attempted to derive theanalytic properties of Euler connection and Euler curva-ture by performing the computation in the eigenstate ba-sis. However, the discontinuous nature of the eigenstatesat the principal node led to some uncontrolled steps, andthe derived results cannot be fully trusted. Here, weapproach the problem differently. We perform a gaugetransformation to a rotated basis, which reveals the con-tinuity of the vector bundle spanned by the two principalbands and principal nodes. The continuity allows us tokeep the computation of Euler form and Euler connec-tion under control. At the end of the calculation, weperform a large gauge transformation back to eigenstatebasis, and compare to the results obtained in Sec. E2.While this is admittedly a somewhat indirect approachto derive the results, the advantage is that we only haveto deal with the singularity at the very last step when wedo the gauge transformation back to the eigenstate basis.
Recall that a rank-2 bundle is a collection of two-dimensional planes – one plane per every point of thebase space. The specific choice of these planes variescontinuously between the points of the base space. Im-portantly, these planes need not in general be equippedwith any intrinsic basis. The basis vectors that we use tospan these planes are just an auxiliary tool. Performingan SO(2) gauge transformation on the two vectors span-ning the rank-2 bundle does not correspond to a changeof the bundle, just to a change of coordinates that we useto describe the bundle. In particular, it is convenient toconsider the “mixed” (i.e. rotated) states
|A(κ)〉 = ± sin φ2
∣∣∣1(0+1)⟩
+ ζ cos φ2
∣∣∣2(0+1)⟩
(S33)
|B(κ)〉 = −ζ cos φ2
∣∣∣1(0+1)⟩± sin φ
2
∣∣∣2(0+1)⟩
(S34)
which are related to the eigenstates of the perturbedHamiltonian by a proper gauge transformation
X(κ; ζ) =
(± sin φ
2 +ζ cos φ2−ζ cos φ2 ± sin φ
2
)(S35)
Using trigonometric identities, we find that to linear or-der in κ these rotated vectors are
|A(κ)〉=
10
{−fcEc }nc=3
|B(κ)〉=ζ
01
{−gcEc }nc=3
. (S36)
These are manifestly continuous and differentiable atκ → 0, meaning that the vector bundle spanned by thetwo principal bands is continuous and differentiable atthe principal node. We find that the Euler connection
and the Euler form to the leading order in κ in the ro-tated basis are
a=〈A|∇B〉=n∑c=3
ζ (αcκx + βcκy)
E2c(γc, δc) (S37)
and
Eu = 〈∂1A|∂2B〉 − 〈∂2A|∂1B〉
= −ζn∑c=3
1
E2c(βcγc − αcδc) (S38)
which are both perfectly regular for κ → 0. In par-ticular, the result in Eq. (S38) exactly reproduces theφ-independent contribution to Eq. (S31). Moreover, onecan show that Stokes’ theorem applies, i.e. the surfaceintegral of Eu is exactly cancelled by the boundary inte-gral of a. Therefore, in the rotated basis the Euler classχ(D) cannot be meaningfully defined for a manifold Dwith a boundary [see Sec. F for definition]. Euler classdefined this way would always be zero.
We now translate the results in Eqs. (S37) and (S38)to the eigenstates basis, which corresponds to a gaugetransformation using X> from Eq. (S35). We must keepin mind that this is a large gauge transformation, i.e. itintroduces a singular winding at one point (at the prin-cipal node), but it is smooth elsewhere. First, it followsfrom Eq. (18) of Methods that everywhere outside the
principal node we must have Eu = Eu. Since the latteris continuous, the correct functional form of the Eulerform in the eigenstate basis is
Eu = −ζn∑c=3
1
E2c(βcγc − αcδc) , (S39)
which can be analytically continued to the point κ = 0.
A more dramatic change occurs when applyingEq. (13) of Methods to transform the connection a tothe eigenstate basis. First, we derive
X>dX =dφ
2
(0 ∓ζ±ζ 0
)(for κ 6= 0), (S40)
from where Pf(X>dX) = ∓ 12ζdφ everywhere outside the
principal node. From φ = arctan(κyκx
) we obtain
dφ =−κydκx + κxdκy
κ2x + κ2y=− sinφ
κdκx+
cosφ
κdκy. (S41)
Note that the term Pf(X>dX) diverges for κ → 0 (con-sequence of X being a large gauge transformation aroundthe principal node), and therefore it completely over-shadows the expression in Eq. (S37) which vanishes com-pletely for κ → 0. Therefore, according to Eq. (13) ofMethods, the Euler connection in the eigenstate basis to
9
the leading order in κ equals
a = −X>dX =±ζ2κ
(− sinφ, cosφ), (S42)
which agrees with our result in Eq. (S30).
It is worth to emphasize that the finite quantized valueof χ(Dε) for the disc enclosing a principal node in theeigenstate basis comes entirely from the connection onthe boundary ∂Dε and is inherited from the large gaugetransformation from the continuous basis. This is com-patible with our claim, mentioned in the beginning ofSec. E, that it is expected that the eigenstate basis car-ries the topological information about the principal nodeson a patch, while the continuous basis does not have ac-cess to this information. Nevertheless, if one considers aclosed base manifold B (e.g. the whole BZ torus), then∂B = ∅, and only the surface integral of the Euler formcontributes to the Euler class χ(B). Since Euler form isthe same both for the eigenstate basis and the continu-ous basis, the value of Euler class on a closed manifold isquantized for both computations.
F. Euler class for manifolds with a boundary
In this section, we generalize the notion of Euler classof a pair of principal bands in two important ways. First,we consider a base manifold with a boundary. Through-out the discussion, we explicitly assume that the manifoldis topologically a disc (denoted D), which has boundaryhomeomorphic to a circle (∂D ' S1). Nevertheless, mostof the presented results readily generalize to arbitrarymanifolds with a boundary as long as the vector bundlespanned by the principal bunds remains orientable. Sec-ond, we admit the occurrence of principal nodes insidethe base manifold. Crucially, to obtain useful informa-tion from such a generalization, it is essential to adoptthe eigenstate basis. We find that the value of the Eulerclass on a disc D, denoted χ(D), indicates whether thecollection of principal nodes can annihilate inside D.
The first generalization is straightforward. If there areno band nodes in D, then the eigenstate basis is contin-uous. Therefore, Stokes’ theorem guarantees that
χ(D)=1
2π
(∫DEu−
∮∂D
a
)= 0 (if no nodes in D)
(S43)is invariant. Let us further assume the presence of prin-cipal nodes in D (but no adjacent nodes as we want thebundle to remain orientable). We proved in the secondpart of Sec. E that the vector bundle spanned by thetwo principal bands is continuous everywhere, includingat the nodes. However one has to depart from the eigen-state basis to reveal this fact, and instead has to considerthe continuous “mixed” basis, cf. Eqs. (S33–S38). In thecontinuous basis, Stokes’ theorem applies, which impliesthe validity of Eq. (S43) even in the presence of princi-
pal nodes. One concedes that the vector bundle spannedby the two principal bands on a disc cannot support anon-trivial topological invariant.
However, one should keep in mind the physical realiza-tion of the vector bundle as a pair of energy bands thatare non-degenerate away from the band nodes. This in-terpretation equips the bundle with a canonical basis,namely the eigenstate basis discussed at length in Sec. E(cf. the comments on flag bundle description in Sec. E1).Therefore, one should only consider deformations of thevector bundle which preserve this additional structure.Indeed, we show below that such a constraint allows fora subtly modified definition of χ(D), which constitutes ahalf-integer topological invariant.
To develop the generalization of the Euler class, oneshould first recognize that each principal node is as-sociated with a Dirac string [32]. Returning back toEqs. (S19–S22), we observe that the continuous realgauge for eigenstates near a principal node is double-valued, namely the overall sign of both states is reversedif we increase φ 7→ φ + 2π. Therefore, any single-valuedgauge must necessarily exhibit a discontinuity – the Diracstring – across which both principal states flip sign. Eachprincipal node must constitute the end-point of a Diracstring. Furthermore, since away from band nodes theeigenstates basis is continuous, there are no other end-points for Dirac strings. Therefore, in the eigenstate basisthere is a one-to-one correspondence between the princi-pal nodes and the end-points of Dirac strings. From thisperspective, the gauge transformation to the continuousbasis, analyzed in Eqs. (S19–S22), can be understood ascreation of a new Dirac string that exactly compensatesthe “physical” Dirac string present in the eigenstate ba-sis. This explains the difference between the Euler con-nection computed in the two different bases in Sec E.
The exact position of the Dirac strings (i.e. besidestheir fixed end-points) is arbitrary, and should not haveany bearing on physical observables. Indeed, we findthat both Euler connection and Euler form are contin-uous at a Dirac string. This readily follows from thetransformation rules presented in the paragraph on topol-ogy of real vector bundles in Methods, namely we seefrom Eqs. (13) and (18) of Methods that the orientation-preserving gauge transformation X = −1 (detX = +1)leaves both Eu(k) and a(k) invariant. We conclude thatthe relation Eu(k) = d a(k) remains valid along Diracstrings, meaning that they are no obstructions for theuse of Stokes’ thorem. Therefore, the only obstructionsfor Stokes’ theorem are the principal nodes themselves,since at their locations the derivatives of the principaleigenstates are not well defined and the Euler connectiondiverges, cf. Sec. E. We thus use Stokes’ theorem to relate
2πχ(D) =
∫D
Eu−∮∂D
a (S44)
=∑α
(∫Dεα
Eu−∮∂Dεα
a
)(S45)
10
D@ D
D1 D3
D2
' '
'
Fig. S5. Relation between Euler class and Dirac strings. Il-lustration of the use of Stokes’ theorem as discussed in the text.Region D is the disc on which we study the real orientable vec-tor bundle, and the blue dots are the principal nodes. We ap-ply Stokes’ theorem to D\ ∪αDεα, i.e. to the region with tinydiscs around the nodes cut out. The blue dashed lines representDirac strings. Only the end-points of Dirac strings are physicallymeaningful. In the case of the eigenstate basis the Dirac stringend-points coincide with principal nodes, whereas a gauge trans-formation that reguralizes the nodes creates a new Dirac stringthat exactly compensates the singular behavior.
where the summation index α indicates the principalnodes inside the region D, and Dεα is a disc with radiusε centered at principal node α.
To simplify Eq. (S45), we proceed as follows. First, wefound in Sec. E that Euler form is bounded near principalnodes, hence the integrals over Dεα converge to 0 in thelimit ε→ 0. Furthermore, it follows from Eq. (S30) thata · dκ = ∓ 1
2ζdφ for ε → 0, which integrates to ∓ζπ on∂Dεα. Plugging this result into Eq. (S45), we find
χ(D) =ζ
2
∑α
wα (S46)
where wα = ±1 describes the winding number of theprincipal node, while ζ = ±1 is the global orientationof the vector bundle. The result in Eq. (S46) provesthat χ(D) of the orientable bundle spanned by the twoprincipal states on region D with a boundary is a half-integer topological invariant if one works in the eigenstatebasis.
Let us conclude the section with several remarks:
1. Note that Eq. (S46) expresses χ(D) as a sum of ± 12
quanta carried by the principal nodes. If the nodeswere able to annihilate inside D, then we would beleft with a nodeless region, for which we proved inEq. (S43) that χ(D) = 0. Therefore, non-trivialvalue χ(D) 6= 0 is an obstruction for annihilatingthe principal nodes inside D.
2. The two admissible values of a winding numberwα = ±1 are reminiscent of the frame-rotation an-
gle associated with the node being either +π or −π.We prove in Sec. G that this intuition is correct,i.e. the two quantities are in an exact one-to-onecorrespondence.
3. One should take into account that Eq. (S46) is notas useful in pratice as it appears! To make sure thatwe take the same orientation of the vector bundle atall principal nodes, it is necessary to know the bun-dle along some trajectories connecting the principalnodes. To avoid this extra work, our numerical al-gorithm for computing χ(D), presented in Sec. His based on directly implementing Eq. (S44).
4. In the presence of additional adjacent nodes, thevector bundle ceases to be orientable, and the rela-tive orientation of two principal nodes depends onthe specific choice of trajectory connecting them.This foreshadows the non-Abelian conversion ofband nodes which we discuss in the main text ofthe manuscript. This “braiding” phenomenon ismore carefully exposed in the next section.
G. Non-Abelian frame-rotation charge
In this section, we review the original derivation of thenon-Abelian exchange of band nodes in k-space, whichwas obtained by Ref. [30] using homotopy theory [61]. Wesubsequently show that the same non-Abelian behavior isreproduced by considering the Euler class on manifoldswith a boundary, as has been defined in Sec. F. Simi-lar observations on a less formal level were also madeby Ref. [32]. The exact correspondence between the twoapproaches provides a proof that the two distinct math-ematical descriptions of the braiding phenomena (homo-topy theory vs. cohomology classes) are two windows intothe same underlying topological structure.
In the homotopic description [80], one begins withidentifying the space MN of N -band real symmetric ma-trices that do not exhibit level degeneracy. This cor-responds to Bloch Hamiltonians at momenta lying awayfrom band nodes. With this assumption, we can uniquelyorder all eigenstates of H(k) according to increasing en-ergy into an O(N) matrix {|ua(k)〉}Na=1 ≡ F (k), whichcan be interpreted as a an orthonormal N -frame. Wefurther adjust band energies {εa(k)}Na=1 to some stan-dard values (e.g. εa = a) while preserving their order-ing. The space of such spectrally normalized Hamiltoni-ans is MN = O(N)/ZN2 , where the quotient correspondsto flipping the overall sign of any of the N eigenvec-tors, which leaves the spectrally normalized Hamiltonian
H(k) =∑Na=1 |ua(k)〉 εa 〈ua(k)| invariant. Band nodes
correspond to obstructions for a unique ordering of bandsby energy, and thus induce discontinuities of frame F (k).
To describe the band nodes, we study the topology ofspace MN . Since there are 2N different frames (corre-sponding to the quotient ZN2 ) which all represent thesame Hamiltonian, the following scenario is possible:
11
2
1
3
SO(3)
Fig. S6. Commutativity of π rotations in three dimensions.If we represent clockwise rotation by angle 0 ≤ α ≤ π
2around
unit vector n by point with position r = αn, then SO(3) lookslike a ball with radius π and with antipodal points on the surfaceidentified. The orange vs. the pink trajectory represent two waysof composing a π rotation around axis 1 with a π rotation aroundaxis 3. The two trajectories cannot be continuously deformedinto one another, i.e. they are topologically distinct. From thisobservation we deduce that topological charges associated withband nodes in consecutive band gaps do not commute.
We start at some point k0, and we follow the continu-ously rotating frame F (k) that encodes the HamiltonianH(k) along k ∈ γ, until we reach again k0 as the fi-nal point. Comparing the initial vs. the final frame atk0, we may find that they are two different of the 2N
frames representing the same Hamiltonian H(k0). Wesay that the Hamiltonian underwent a non-trivial framerotation, which represents a non-trivial closed path in-side MN . For example, we know from Secs. E and Fthat a band node leads to a π-rotation of the frame byX = (1, . . . , 1,−1,−1, 1, . . . , 1), where the two negativeentries correspond to the two bands forming the node.More generally, since the handedness of the frame can-not change under the continuous evolution along γ, only12 · 2
N = 2N−1 of the elements with detX = +1 can ac-tually be reached. Since the frame rotation is quantizedto a discrete set of elements, it constitutes a topologicalinvariant of the Hamiltonian H(k) along path γ, whichcannot change under continuous deformations as long asthe spectrum along γ remains non-degenerate.
To explain the origin of the non-Abelian exchange ofband nodes, let us briefly focus on models with N = 3bands. The same discussion also applies to any threeconsecutive bands in models with N ≥ 3 bands. A nodeformed by the lower (upper) two bands corresponds to aπ-rotation in the first (last) two coordinates, i.e. X12 =diag(−1,−1,+1) [X23 = diag(+1,−1,−1)]. A path thatencloses one of each species of nodes is associated withtotal frame rotation X13 = diag(−1,+1,−1). However,while the geometric transformations
X12 ·X23 = X23 ·X12 (S47)
clearly commute, the continuous paths in SO(3) that re-alize the left-hand vs. the right-hand side of Eq. (S47)are topologically distinct. To see this, recall that SO(3)can be visualized as a solid three-dimensional ball withradius π and with antipodal points on the surface beingpairwise identified. This relation is achieved by mappingR(α;n) (i.e. a clockwise rotation by angle 0 ≤ α ≤ πaround axis given by unit vector n), with a point insidethe ball at position r = αn. Then rotating first by πaround axis 1 and then by π around axis 3 traces thepink path in Fig. S6, while performing the two rotationsin reverse order produces the orange path in Fig. S6,which follows from
R(α; e3) ·R(π; e1) = R(π; + cos α2 e1 + sin α2 e2) (S48)
R(α; e1) ·R(π; e3) = R(π; + cos α2 e3 − sin α2 e2) (S49)
where ei indicates the unit vector directed along axisi ∈ {1, 2, 3}. The two paths in Fig. S6 both connectthe center of the ball to the π-rotation around axis 2.However, these paths cannot be continuously deformedinto each other, i.e. they are topologically distinct. Thisultimately follows from the fact that SO(3) is not a simplyconnected space. As a consequence, the ordering of thegroup elements from the ZN2 quotient matters, and thetopological charges associated with a pair of band nodesinside consecutive band gaps do not commute!
A careful analysis [30] reveals that the algebra of closedpaths in space MN = O(N)/ZN2 is governed by groupQN (called “Salingaros group” [81]) which is uniquelycharacterized by the following four conditions [31]. Weuse +1 to indicate the identity element.
(i) There is a unique element −1 6= +1 which has theproperty (−1)2 = +1.
(ii) For each band gap 1 ≤ G ≤ (N − 1) there is anassociated element gG such (gG)2 = −1.
(iii) gG · gG′ = εgG′ · gG, where ε = −1 (anticommute)if |G−G′| = 1 and ε = +1 (commute) otherwise.
(iv) All elements of QN can be expressed by composingelements gG.
The element −1 corresponds to a 2π rotation (aroundany axis), and corresponds to the generator of the funda-mental group π1[SO(N)] = Z2 [38]. Most interestingly,condition (iii) states that band nodes in consecutive bandgaps carry anticommuting charges. This corresponds tothe fact, visible in Fig. S6, that
R(π, e3) ·R(π, e1) = R(2π, e2)︸ ︷︷ ︸“−1”
·R(π, e1) ·R(π, e3) (S50)
if the rotations are interpreted as paths in SO(3).The group Q3 coincides with the quaternion group{±1,±i,±j,±k}, therefore QN for N ≥ 3 has beendubbed “generalized quaternions” by Ref. [30].
We finally show that the same anticommuting be-haviour follows by studying the Euler form on manifolds
12
DL
DR
D.S.R
ÍG.T.
D.S.L
Fig. S7. Relation of the non-commutative topological chargeto Dirac strings. We consider two principal nodes (blue dots)near an adjacent node (red dot). Principal nodes are end-pointsof Dirac strings associated with a proper gauge transformationX = −1 (dashed blue lines), which does not affect the conti-nuity of Euler connection and Euler form of the vector bundlespanned by the two principal bands. On the other hand, theadjacent node is an end-point of a Dirac string associated withan improper transformation X = ±σz (dashed red line), whichreverses the orientation of the bundle. The yellow resp. the greendisc DL,R correspond to two topologically different coverings ofthe principal nodes, for which the bundle can be made orientableby choosing an appropriate position of the red Dirac string (in-dicated by D.S.L resp. D.S.R). The two gauges are related by agauge transformation which has an orientation-reversing discon-tinuity along γG.T. (solid brown line). This gauge transformationreverses the orientation of the bundle near one of the nodes, thusflipping the relative contribution of the two nodes to the sum inEq. (S46).
with a boundary as defined in Sec. F. Our proof thus suc-cessfully bridges the homotopic description of Ref. [30]with the cohomological description proposed by Ref. [32]and further elaborated by the present work. To observethe non-trivial exchange, let us consider the situation, il-lustrated in Fig. S7, with two principal nodes (blue dots)near an adjacent node (red dot). We know from Sec. Fthat principal nodes are end-points of Dirac strings car-rying a proper gauge transformation X=−1 on the twoprincipal bands (dashed blue lines). We showed in thesame section that such a gauge transformation is harm-less for the calculation of the Euler class. On the otherhand, adjacent nodes are end-points of Dirac strings car-rying an improper gauge transformation X=±σz, whichflips the sign of just one of the principal bands. There-fore, the bundle spanned by the two principal bandsis non-orientable on regions containing such “adjacent”Dirac strings. Especially, an annulus enclosing the ad-jacent node is traversed by such a Dirac string for anysingle-valued gauge of the eigenstate basis, i.e. the bun-dle spanned by the principal bands is non-orientable onsuch an annulus.
Nevertheless, the total Euler class of the two principalnodes can still be calculated, provided that one covers thenodes with a disc lying to the side of the adjacent node.We show in Fig. S7 two such discs, labelled DL,R, whichlie to the left (yellow) resp. to the right (green) of the ad-jacent node. Both discs admit a gauge with a well-definedorientation of the bundle, which is achieved by appropri-ately positioning the adjacent Dirac string (dashed redlines D.S.L resp. D.S.R) such that it lies outside of thecorresponding disc. Importantly, these two gauges are re-lated by a gauge transformation that has a discontinuityalong path γG.T. = D.S.L ∪ D.S.R (solid brown path inFig. S7). This gauge transformation rotates D.S.L intoD.S.R (and vice versa), and is simply equal to X = 1 onone side and to X = ±σz on the other side of the pathγG.T.. Such a gauge transformation necessarily reversesthe orientation of the bundle at exactly one of the twoprincipal nodes. It follows that the relative contributionof the two principal nodes to the sum in Eq. (S46) is re-versed due to the reversed orientation ζ near one of thenodes. Therefore, if the contributions of the two nodes tothe Euler class cancel on DL [e.g. χ(DL)= 1
2−12 =0], then
the Euler class is automatically non-trivial on DR [cor-responding to χ(DR) = ±
(12 + 1
2
)= ±1]. Following the
discussion at the end of Sec. F, the two principal nodesannihilate if brought together along a trajectory insideDL, but are incapable to annihilate if brought togetheralong a trajectory inside DR. We thus conclude that theanticommutation relation in Eq. (S50) [resp. in axiom(iii) on the previous page] is exactly reproduced by thebehaviour of Euler class on manifolds with a boundary.
13
c da b
Fig. S8. Euler form and Euler class for the braiding protocol from the main text. a and b. Numerically computed Euler form forthe model in Eq. (4) of Methods for t=−2.5. The considered rectangular region k1∈ [−1, 1] and k2∈ [−1, 1] contains two principalnodes (blue dots) and no adjacent nodes. The two panels differ in the mesh size, pts∈{25, 400}. The color scheme shows positive(negative) values in red (blue) tones, and white values in zero. The code adjusts the color scheme range to fully fit the computedrange of |Eu(k)|. The estimated value of the Euler class is -0.984220 in a, and -0.999934 in b. This is numerical approximationof χ(D) = −1, implying that the two nodes cannot annihilate inside the region. The result is negative, because the positive areaintegral of Eu (note the red tones of the plots) is overcompensated by a larger negative boudnary integral of Euler connection a. cand d. Analogous data for t = −5.5 and region k1∈ [π − 1, π + 1] and k2∈ [π − 1, π + 1]. For both choices of mesh, the computedEuler class is zero within ≈ 10−7, which implies that the nodes can annihilate inside the region.
H. Numerical calculation of the Euler form.
To test the presented theory numerically, we have im-plemented a Mathematica code that takes as input (1)an N -band real-symmetric Bloch Hamiltonian, (2) two(consecutive) band indices, and (3) a rectangular regioncontaining no adjacent nodes. The program outputs theEuler class on the defined region (with a boundary) forthe selected pair of consecutive bands, by implementingEq. (S44) in the eigenstate basis. The code requires set-ting one hyper-parameter, namely the subdivision of thesides of rectangular region into a discrete set of points.The code is briefly described below, and we have madeit available online [65].
The code sequentially implements the following steps.It begins by initializing the input parameters. We de-fine a Bloch Hamiltonian H[k1,k2], two (consecutive)band indices LowerBand and UpperBand that label thetwo principal bands, and a rectangular domain k1 ∈[k1Min, k1Max], k2 ∈ [k2Min, k2Max]. The code automat-ically extracts the total number of bands TotalBands.The labelling of the bands is such that the lowest-energyband is indexed by 1, and the highest-energy band isindexed by the value TotalBands. We further set thevalue of hyper-parameter pts which defines the dis-cretization of the sides of the rectangular region intopts× pts infinitesimal squares of size dk1× dk2, wheredk1 = (k1Max− k1Min)/pts and similarly for dk2.
In the next stage, we prepare the data for the numer-ical calculation of the Euler connection and Euler form.We save the two numerically obtained principal eigen-states of the Hamiltonian at the regular mesh of pointsinto a (pts + 1)× (pts + 1)× 2 array called AllStates.Note that each entry of this array is itself an array ofsize TotalBands × 1 (i.e. a right eigenstate). However,numerical diagonalization of the Hamiltonian finds theprincipal bands with a random +/− gauge, which has to
be smoothed before computing the derivatives. Further-more, we know from Sec. F that each principal node is asource of a Dirac string associated with a X = −1 gaugetransformation. This implies the absence of a single-valued continuous gauge on regions containing principalnodes. To deal with these two issues, the code proceedsas follows. First, it computes the Berry phase [82] oneach of the pts × pts infinitesimal squares of the mesh,and stores the information in an array Fluxes. The de-fault value is +1, while squares containing a node areindicated by value −1. Positions of all the nodes arethen extracted and saved as pairs of numbers in arrayNodes. Knowing the position of all the principal nodesinside the region, the code follows a set of rules to fixthe position of the Dirac strings. The chosen trajecto-ries of the Dirac strings are saved in array Strings. Theinfinitesimal squares traversed by a Dirac string are char-acterized by Strings[[i,j]]=-1, else the default valueis +1. Finally, the two principal states are gauged suchthat they vary smoothly away from the Dirac strings,while both of the states simultaneously flip sign acrosseach Dirac string. This gauge is then used to update allthe states stored in array AllStates.
In the final stage, the code takes the gauged eigen-states saved in AllStates, and uses them to computeEuler form inside the region and Euler connection onthe boundary of the region. Following Eq. (S44), thesetwo quantities are integrated to obtain the Euler class onthe rectangular region. The code computes Euler con-nection along the boundary by implementing Newton’smethod of finite differences to approximate the deriva-tive in the expression for a(k) in Eq. (S25). The nu-merical integral of the Euler connection is then saved asEulerConnectionIntegral.
To integrate the Euler form inside the rectangular re-gion, we use the complexification discussed in the Meth-ods section. The complexified principal bands are stored
14
in an (pts + 1)× (pts + 1) array ComplexifiedStates.As stated by Eq. (28) in Methods, Euler form of thetwo principal bands is exactly reproduced as the Berry
curvature of the complexified band. However, followingRef. [42], we directly compute the flux of the Euler formthrough each infinitesimal square, which exactly obeys
Eu(k) dk1dk2 = arg tr [Pk+dk2Pk+dk1+dk2Pk+dk1Pk] (S51)
= arg [〈ψk|ψk+dk2〉 〈ψk+dk2 |ψk+dk1+dk2〉 〈ψk+dk1+dk2 |ψk+dk1〉 〈ψk+dk1 |ψk〉]
where Pk = |ψk〉 〈ψk| is a projector onto the complexifiedstate at momentum k. With this trick, we elegantly avoidthe accumulation of numerical errors in computing thederivatives of states in the right Eq. (S25). For infinites-imal squares not traversed by a Dirac string we readilycompute the Euler form via Eq. (S51). For infinitesimalsquares traversed by a Dirac string, we perform an extragauge transformation to guarantee a continuous gauge onthe four vertices of the square. Note also that the codeis set to skip the computation of Euler form on infinites-imal squares containing the principal nodes (i.e. thosewith Fluxes[[i,j]] = −1). We numerically integratethe Euler form by summing up the contributions fromall infinitesimal squares, and the obtained approxima-tion of the integral is saved as EulerFormIntegral. Wefinally output EulerClass, which is the difference ofEulerFormIntegral and EulerConnectionIntegral di-vided by 2π, cf. the definition in Eq. (S44).
To demonstrate the performance of the code, we con-sider the braiding protocol in Eq. (4) of Methods for twovalues of t. First, for t = −2.5, the system exhibits twoprincipal nodes near the center of the Brillouin zone. It
is observed that these two nodes fail to annihilate at acollision for t = −2. To analyze the situation, we con-sider a square region k1 ∈ [−1, 1] and k2 ∈ [−1, 1], whichcontains the two principal nodes and no adjacent nodes.The estimated value of Euler class is very close to −1,cf. Fig. S8a and b, consistent with the observed stabilityof the nodes.
We contrast this to the situation with t = −5.5, whenthe region k1∈ [π−1, π+1] and k2∈ [π−1, π+1] containsno adjacent nodes but a pair of principal nodes, whichhave been pairwise created inside the region at an earliertime. In this case, the numerically computed Euler classis zero within ≈ 10−7, cf. Fig. S8c and d, consistent withthe fact that the nodes were pairwise created inside theregion.
It is worth to emphasize that the two principal nodesstudied for t = −2.5 resp. for t = −5.5 are the same pairof nodes, but we have computed their Euler class insidedifferent regions of the Brillouin zone. The Euler class onthe two regions reaches different value, confirming thatthe capability of the principal nodes to annihilate de-pends on the choice of path used to bring them together.
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