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On Spectral Flow and Fermi Arcs

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

  2. 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.

  3. 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.

  4. We will move freely between \(\gamma \) as a real number modulo \(2\pi \), and \(\gamma \) as a phase, i.e. \(e^{i\gamma }\).

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

  6. 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).

  7. 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.

  8. 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.

  9. 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.


  1. Armitage, N.P., Mele, E.J., Vishwanath, A.: Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod Phys. 90(1), 015001 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  2. Atiyah, M.F., Singer, I.M.: Index theory for skew-adjoint Fredholm operators. Publ. Math. IHES 37, 305–326 (1969)

    Article  Google Scholar 

  3. Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry III. Math. Proc. Camb. Philos. Soc. 79, 71–99 (1976)

    Article  MathSciNet  Google Scholar 

  4. Behncke, H., Focke, H.: Stability of deficiency indices. Proc. R. Soc. Edinb. 78A, 119–127 (1977)

    Article  MathSciNet  Google Scholar 

  5. Burello, M., Guadagnini, E., Lepori, L., Mintchev, M.: Field theory approach to the quantum transport in Weyl semimetals. Phys. Rev. B 100, 155131 (2019)

    Article  ADS  Google Scholar 

  6. Booss-Bavnbek, B., Lesch, M., Phillips, J.: Unbounded Fredholm operators and spectral flow. Can. J. Math. 57(2), 225–250 (2005)

    Article  MathSciNet  Google Scholar 

  7. Braverman, M.: Spectral flows of Toeplitz operators and bulk-edge correspondence. Lett. Math. Phys. 109, 2271–2289 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  8. Carey, A., Thiang, G.C.: The Fermi gerbe of Weyl semimetals. arXiv:2009.02064

  9. Enblom, A.: Resolvent estimates and bounds on eigenvalues for Dirac operators on the half-line. J. Phys A Math. Theor. 51, 165203 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  10. Graf, G.M., Jud, H., Tauber, C.: Topology in shallow-water waves: a violation of bulk-edge correspondence. arXiv:2001.00439

  11. Gruber, M.J., Leitner, M.: Spontaneous edge currents for the Dirac equation in two space dimensions. Lett. Math. Phys. 75, 25–37 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  12. Hashimoto, K., Kimura, T., Wu, X.: Boundary conditions of Weyl semimetals. Prog. Theor. Exp. Phys. 2017, 053I01 (2017)

  13. Joachim, M.: Unbounded Fredholm operators and \(K\)-theory. In: Farrell, F.T., Lück, W. (eds.) High-Dimensional Manifold Topology, pp. 177–199. World Scientific Publishing, Singapore (2003)

  14. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)

    Book  Google Scholar 

  15. Ludewig, M., Thiang, G.C.: Cobordism invariance of topological edge-following states. arXiv:2001.08339

  16. Lv, B.-Q., et al.: Experimental discovery of Weyl semimetal TaAs. Phys. Rev. X 5, 031013 (2015)

    Google Scholar 

  17. Mathai, V., Thiang, G.C.: Global topology of Weyl semimetals and Fermi arcs. J. Phys. A Math. Theor. 50, 11LT01 (2017)

    Article  MathSciNet  Google Scholar 

  18. Mathai, V., Thiang, G.C.: Differential topology of semimetals. Commun. Math. Phys. 355, 561–602 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  19. Milnor, J.: Topology from the Differentiable Viewpoint. Princeton University Press, NJ (1997)

    MATH  Google Scholar 

  20. Morali, N., et al.: Fermi-arc diversity on surface terminations of the magnetic Weyl semimetal Co\(_{3}\)Sn\({}_2\)S\({}_2\). Science 365, 1286–1291 (2019)

  21. Phillips, J.: Self-adjoint Fredholm operators and spectral flow. Can. Math. Bull. 39(4), 460–467 (1996)

    Article  MathSciNet  Google Scholar 

  22. Prodan, E., Schulz-Baldes, H.: Bulk and Boundary Invariants for Complex Topological Insulators. Mathematics Physics Studies. Springer, Berlin (2016)

    Book  Google Scholar 

  23. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. I. Academic Press, San Diego (1980)

    MATH  Google Scholar 

  24. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. II. Academic Press, San Diego (1975)

    MATH  Google Scholar 

  25. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. IV. Academic Press, San Diego (1978)

    MATH  Google Scholar 

  26. Souma, S., et al.: Direct observation of nonequivalent Fermi-arc states of opposite surfaces in the noncentrosymmetric Weyl semimetal NbP. Phys. Rev. B 93, 161112(R) (2016)

    Article  ADS  Google Scholar 

  27. Thiang, G.C.: Edge-following topological states. J. Geom. Phys. 156, 103796 (2020)

    Article  MathSciNet  Google Scholar 

  28. Thiang, G.C., Sato, K., Gomi, K.: Fu–Kane–Mele monopoles in semimetals. Nucl. Phys. B 923C, 107–125 (2017)

    Article  ADS  Google Scholar 

  29. Turaev, V.G.: Euler structures, nonsingular vector fields, and torsions of Reidemeister type. Math. USSR-Izvestiya 34(3), 627 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  30. Wahl. C.: A New topology on the space of unbounded selfadjoint operators, \(K\)-theory and spectral flow. In: Burghelea, D., Melrose, R., Mishchenko, A.S., Troitsky, E.V. (Eds.) \(C^*\)-Algebras and Elliptic Theory II, Trends in Mathematics, pp. 297–309. Birkhäuser, Basel (2008)

  31. Witten, E.: Three lectures on topological phases of matter. Nuovo Cimento 39, 313–370 (2016)

    Google Scholar 

  32. Xu, S.-Y., et al.: Observation of Fermi arc surface states in a topological metal: a new type of 2D electron gas. Science 347, 294–298 (2015)

    Article  ADS  Google Scholar 

  33. Xu, S.-Y., et al.: Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015)

    Article  ADS  Google Scholar 

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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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Correspondence to Guo Chuan Thiang.

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Communicated by M. Salmhofer.

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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

$$\begin{aligned} \sigma _{\mathrm{ess}}\left( \widetilde{H}^{\mathrm{1D}}(m,\theta ;\gamma )\right) =\sigma _{\mathrm{ess}}\left( \widetilde{H}^{\mathrm{1D}}(m,\theta ;0)\right) ,\qquad \forall \; e^{i\gamma }\in \mathrm{U}(1). \end{aligned}$$

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

$$\begin{aligned} \widetilde{H}^{\mathrm{1D}}(m,0;0)= \begin{pmatrix} -i\frac{d}{dz} &{} m \\ m &{} i\frac{d}{dz} \end{pmatrix} \overset{\cong }{\longrightarrow } \begin{pmatrix} m &{} -\frac{d}{dz} \\ \frac{d}{dz} &{} -m \end{pmatrix}, \end{aligned}$$

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

$$\begin{aligned} \sigma (\widetilde{H}^{\mathrm{1D}}(m,0;0))=(-\infty ,-m]\cup [m,\infty ), \end{aligned}$$

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])

$$\begin{aligned} K(z,z^\prime ;\lambda )&=\frac{1}{2}\begin{pmatrix}-\frac{\lambda +m}{i\mu } &{} \mathrm{sgn}(z-z^\prime ) \\ -\mathrm{sgn}(z-z^\prime ) &{} -\frac{\lambda -m}{i\mu }\end{pmatrix}e^{i\mu |z-z^\prime |}\\&\qquad -\frac{1}{2}\begin{pmatrix}\frac{\lambda +m}{i\mu } &{} 1\\ 1 &{} \frac{\lambda -m}{i\mu }\end{pmatrix}e^{i\mu (z-z^\prime )},\qquad z,z^\prime \in {\mathbb {R}}_+, \end{aligned}$$

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

$$\begin{aligned} (H_0+V-\lambda )^{-1}-(H_0-\lambda )^{-1}&=(H_0+V-\lambda )^{-1}(1-(H_0-\lambda +V)(H_0-\lambda )^{-1})\\&=-(H_0+V-\lambda )^{-1}V(H_0-\lambda )^{-1} \end{aligned}$$

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

$$\begin{aligned} \sigma _{\mathrm{ess}}(\widetilde{H}^{\mathrm{1D}}(m,0;0)+V)=\sigma _{\mathrm{ess}}(\widetilde{H}^{\mathrm{1D}}(m,0;0)). \end{aligned}$$

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 \(\widetilde{H}^{\mathrm{1D}}_V(m,\theta ;\gamma )\equiv \widetilde{H}^{\mathrm{1D}}(m,\theta ;\gamma )+V\),

$$\begin{aligned} \sigma _{\mathrm{ess}}(\widetilde{H}^{\mathrm{1D}}_V(m,\theta ;\gamma ))=\sigma _{\mathrm{ess}}(\widetilde{H}^{\mathrm{1D}}(m,\theta ;\gamma )),\qquad \forall \; (m,\theta ,\gamma )\in \widehat{{\mathbb {R}}}^2\times \mathrm{U}(1)\cong \mathcal {M}. \end{aligned}$$

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Thiang, G.C. On Spectral Flow and Fermi Arcs. Commun. Math. Phys. 385, 465–493 (2021).

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