Skip to main content
Log in

On Spectral Flow and Fermi Arcs

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  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.

References

  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 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Guo Chuan Thiang.

Ethics declarations

Conflict of interest

The author declares that he has no conflict of interest.

Additional information

Communicated by M. Salmhofer.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

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

Navigation