## Abstract

We introduce spectral flow techniques to explain why the Fermi arcs of Weyl semimetals are topologically protected against boundary condition changes and perturbations. We first analyse the topology of a certain universal space of self-adjoint half-line massive Dirac Hamiltonians, and then exploit its non-trivial and homotopy invariant spectral flow structure by pulling it back to generic Weyl semimetal models. The homological perspective of using Dirac strings/Euler chains as global topological invariants of Weyl semimetals/Fermi arcs, is thereby analytically justified.

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## Notes

Whereas translations in Euclidean space are denoted by \({\mathbb {R}}^d\), the Fourier transform, or momentum space, variable is denoted \(\widehat{{\mathbb {R}}}^d\).

Here \(H^1({\mathbb {R}})\) is standard notation for the space of locally absolutely continuous functions with \(L^2\) (weak) derivative. Later, we encounter (co)homology groups, similarly denoted \(H^n, H_n\), but it should be clear from context what is meant.

Recall that a self-adjoint operator

*D*is*Fredholm*if it has finite-dimensional kernel and cokernel, while its*essential*spectrum comprises those \(\lambda \in {\mathbb {R}}\) such that \(D-\lambda \) fails to be Fredholm.We will move freely between \(\gamma \) as a real number modulo \(2\pi \), and \(\gamma \) as a phase, i.e. \(e^{i\gamma }\).

By elliptic regularity, we only need to find smooth eigenfunctions.

A physical way to understand why boundary conditions must be constrained as above, is given by Witten, §1.10 of [31]. Namely, the

*helicity*, or*chirality*, of Weyl spinors is constrained such that the angular momentum is parallel to the direction of motion. So if the linear momentum \(p_z\) is reversed after impinging on the boundary (to conserve probability), then the*z*-angular momentum must likewise be reversed, which is Eq. (10).The important condition is really that \(H(\varvec{k})\) maintains a spectral gap about zero whenever \(\varvec{k}\ne \varvec{k}^*\), or equivalently, \(|a(\varvec{k})|<|\varvec{b}(\varvec{k})|\) needs to hold away from \(\varvec{k}^*\). Given this, the rest of this section can easily accommodate an additional continuous

*a*without modification of the spectral flow values, cf. Sect. 4.3.The 2D Chern insulator has, by definition, its negative eigenprojection in the class of \(P_{\mathrm{Chern}}\), thus the name. There are sign conventions at various stages, such as how \({\mathbb {C}}{\mathbb {P}}^1\) is identified with the unit vectors of \(S^2\) in the Bloch sphere construction, and also the Chern number \(\pm 1\) of the tautological (Hopf) bundle.

Observe that \(g=\mathrm{id}\) gives the right-handed Weyl Hamiltonian, while \(g(p_x,p_y)=p_x-ip_y\) gives some spin-rotated version of the left-handed Weyl Hamiltonian.

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## Acknowledgements

The author thanks G. De Nittis and K. Yamamoto for stimulating discussions, and A. Carey for clarifying some technicalities of spectral flow.

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This work was supported by the Australian Research Council via Grants DE170900149 and DP200100729.

## Spectrum of Half-Line Dirac Hamiltonians

### Spectrum of Half-Line Dirac Hamiltonians

### 1.1 Deficiency indices of half-line Dirac Hamiltonians

It is well-known that (minus) the momentum operator \(i\frac{d}{dz}\) on the half-line cannot be made self-adjoint. Physically, the dynamics generated by \(i\frac{d}{dz}\) on \(L^2({\mathbb {R}})\) is that of left translation with unit speed, and this is unitary (conserves probability) on the full line Hilbert space \(L^2({\mathbb {R}})\). But once a boundary at \(z=0\) is introduced, there are no right-movers to reflect into, and the left-movers keep getting absorbed by the boundary.

To compensate for this problem, one adds as a direct sum, the operator \(-i\frac{d}{dz}\), so that the left/right movers exactly compensate. However, another issue arises in the *uniqueness* of this procedure. Namely, a left-moving wave \(e^{ip(z+t)}\) could be superposed with a right-moving \(e^{ip(z-t)}\) (having the same energy *p*) with some \(\mathrm{U}(1)\) phase shift constituting a choice of boundary condition—this is another way of understanding the spin polarisation condition in Eq. (3).

These issues can be systematically addressed in von Neumann’s theory of deficiency indices [24]. Roughly speaking, these indices count the number of \(\pm i\) eigenvalues that need to be eliminated when constructing self-adjoint extensions that generate genuine unitary dynamics. For example, \(z\mapsto e^{-z}\) is a normalisable eigenfunction for \(\pm i\frac{d}{dz}\) with imaginary eigenvalue \({\mp } i\).

We start with the massless case, \(\widetilde{H}^{\mathrm{1D}}(0,0)=-i\frac{d}{dz}\oplus i\frac{d}{dz}\), which is first defined as a symmetric operator on \(C_0^\infty ({\mathbb {R}}_+)^{\oplus 2}\), the smooth functions with compact support away from the boundary. This is easily seen to have deficiency indices (1, 1). Accordingly, there is a \(\mathrm{U}(1)\)-family of possible self-adjoint extensions, and they are explicitly given by \(\widetilde{H}^{\mathrm{1D}}(0,0;\gamma )\).

For the massive half-line Dirac Hamiltonians, \(\widetilde{H}^{\mathrm{1D}}(m,\theta )\), they also have deficiency indices (1, 1), due to the stability of the latter against the bounded perturbation (the mass term) [14]. In fact, deficiency indices are stable against relatively bounded perturbations with bound smaller than 1, and even more generally, see [4]. Thus the \(\widetilde{H}^{\mathrm{1D}}(m,\theta ;\gamma )\) of Eq. (3) give the most general self-adjoint half-line Dirac Hamiltonians.

### 1.2 Essential spectrum of half-line Dirac Hamiltonians

Due to the finite deficiency indices, for any boundary condition \(\gamma \), and \(\lambda \in {\mathbb {C}}\setminus {\mathbb {R}}\), the resolvents \((\widetilde{H}^{\mathrm{1D}}(m,\theta ;\gamma )-\lambda )^{-1}\) and \((\widetilde{H}^{\mathrm{1D}}(m,\theta ;0)-\lambda )^{-1}\) only differ by a finite-rank, thus compact operator ([25] XIII.4, Example 5). Consequently, Weyl’s essential spectrum theorem ([25] Theorem XIII.14) says that

Together with the unitary equivalence \(\widetilde{H}^{\mathrm{1D}}(m,\theta ;\gamma )\cong \widetilde{H}^{\mathrm{1D}}(m,\theta -\gamma ;0)\), it therefore suffices to consider \(\gamma =0=\theta \) as far as \(\sigma _{\mathrm{ess}}\) is concerned.

It is convenient to carry out a further unitary spin rotation \(\frac{1}{\sqrt{2}}\begin{pmatrix} 1 &{} 1 \\ i &{} -i\end{pmatrix}\) in the \({\mathbb {C}}^2\) degrees of freedom, which effects

and transforms the \(\gamma =0\) boundary condition to read \(\psi (0)\propto \begin{pmatrix} 1 \\ 0\end{pmatrix}\). Notice that the latter entails a Dirichlet condition on the second component of the spinor, and no condition on the first component. Explicitly, the domain of self-adjointness is \(\mathring{H}^1({\mathbb {R}}_+)\oplus H^1({\mathbb {R}}_+)\), where \(\mathring{H}^1({\mathbb {R}}_+)\) denotes \(H^1({\mathbb {R}}_+)\) subject to the vanishing condition at \(z=0\). For the first component, we can still use the odd (sine) Fourier transform, so the spectral problem for \(\widetilde{H}^{\mathrm{1D}}(m,0;0)\) may be analysed in momentum space. Similarly to the boundaryless case, we find that the full spectrum is

which therefore completely comprises essential spectrum. Then Eq. (5) follows.

### 1.3 Relatively compact perturbations

Let us abbreviate \(\widetilde{H}^{\mathrm{1D}}(m,0;0)\) to \(H_0\). The resolvent \((H_0-\lambda )^{-1}\) at \(\lambda \in {\mathbb {C}}\setminus \sigma (H_0)\) can be computed explicitly to have the integral kernel (see pp. 4 of [9])

where \(\mu =(\lambda ^2-m^2)^{1/2}\) has \(\mathrm{Im}\,\mu >0\). This kernel is square-integrable on bounded subsets, so if \(V=V(z)\) is a \(2\times 2\) Hermitian matrix-valued continuous potential on \({\mathbb {R}}_+\) that vanishes at infinity, we can, by truncating kernels, approximate \(V(H_0-\lambda )^{-1}\) in norm using Hilbert–Schmidt operators; thus \(V(H_0-\lambda )^{-1}\) is compact. This argument to establish the \(H_0\)*-relatively compactness* of *V* is of a fairly standard form, and works for potentials which are merely \(L^2\) up to a \(\epsilon \)-essentially bounded piece, see e.g. [25] XIII.4 Example 6.

It follows that the resolvent difference

is compact. Then Weyl’s essential spectrum theorem applies, saying that \(H_0\) and \(H_0+V\) have the same essential spectrum, i.e.,

Finally, since \(\widetilde{H}^{\mathrm{1D}}(m,\theta ;\gamma )\) and \(\widetilde{H}^{\mathrm{1D}}(m,0;\gamma +\theta )\) are unitarily equivalent, and the resolvents of \(\widetilde{H}^{\mathrm{1D}}(m,0;\gamma +\theta )\) and \(\widetilde{H}^{\mathrm{1D}}(m,0;0)\) only differ by a compact operator (Sect. A.2), we also see that *V* is also compact relative to any \(\widetilde{H}^{\mathrm{1D}}(m,\theta ;\gamma )\). Thus for the perturbed half-line Dirac Hamiltonians \(\widetilde{H}^{\mathrm{1D}}_V(m,\theta ;\gamma )\equiv \widetilde{H}^{\mathrm{1D}}(m,\theta ;\gamma )+V\),

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Thiang, G.C. On Spectral Flow and Fermi Arcs.
*Commun. Math. Phys.* **385**, 465–493 (2021). https://doi.org/10.1007/s00220-021-04007-z

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DOI: https://doi.org/10.1007/s00220-021-04007-z