Twisted Fermi surface of a thin-film Weyl semimetal

The Fermi surface of a conventional two-dimensional electron gas is equivalent to a circle, up to smooth deformations that preserve the orientation of the equi-energy contour. Here we show that a Weyl semimetal confined to a thin film with an in-plane magnetization and broken spatial inversion symmetry can have a topologically distinct Fermi surface that is twisted into a $\mbox{figure-8}$ $-$ opposite orientations are coupled at a crossing which is protected up to an exponentially small gap. The twisted spectral response to a perpendicular magnetic field $B$ is distinct from that of a deformed Fermi circle, because the two lobes of a \mbox{figure-8} cyclotron orbit give opposite contributions to the Aharonov-Bohm phase. The magnetic edge channels come in two counterpropagating types, a wide channel of width $\beta l_m^2\propto 1/B$ and a narrow channel of width $l_m\propto 1/\sqrt B$ (with $l_m=\sqrt{\hbar/eB}$ the magnetic length and $\beta$ the momentum separation of the Weyl points). Only one of the two is transmitted into a metallic contact, providing unique magnetotransport signatures.


I. INTRODUCTION
The Fermi surface of degenerate electrons separates filled states inside from empty states outside, thereby governing the electronic transport properties near equilibrium. In a two-dimensional electron gas (2DEG) the Fermi surface is a closed equi-energy contour in the momentum plane. It is a circle for free electrons, with deformations from the lattice potential such as the trigonal warping of graphene or the hexagonal warping on the surface of a topological insulator [1]. These are all smooth deformations which do not change the orientation of the Fermi surface: The turning number is 1, meaning that the tangent vector makes one full rotation as we pass along the equi-energy contour.
The turning number defined as the contour integral of the curvature C in units of 2π, identifies topologically distinct deformations of the circle in the plane, socalled "regular homotopy classes" [2]. A theorem going back to Gauss [3] says that a contour Γ with turning number ν has s ≥ |ν| − 1 self-intersections and that the sum |ν| + s must be an odd integer. Fig. 1 shows examples of contours with {ν, s} = {0, 1}, {1, 0}, and {2, 1}. The turning number is preserved by any smooth deformation of the contour. This includes socalled "uncrossing" deformations [2]: As illustrated in Fig. 1, uncrossing breaks up a self-intersecting contour Γ into a collection of nearly touching oriented contours Γ i , with turning numbers ν i . The total turning number ν = i ν i is invariant against uncrossing deformations, which is another result due to Gauss [3].
All familiar 2D electron gases belong to the |ν| = 1 universality class. Here we show that a thin-film Weyl semimetal with an in-plane magnetization M and broken FIG. 1: Three oriented contours (black curves) with turning number ν = 0, 1, 2. The red segments show the uncrossing deformation that removes a self-intersection without changing the total turning number ν = i νi.
spatial inversion symmetry can have ν = 0: if the Fermi level lies in between the two Weyl points the circular Fermi surface is twisted into a figure-8 with zero total curvature [4]. The self-intersection introduced when the Fermi level passes through a Weyl point, to ensure that |ν| + s remains odd, is a crossing of Fermi arcs on the top and bottom surfaces of the thin film (width W ). These have a penetration depth ξ 0 into the thin film that can be much less than the Fermi wavelength of the bulk states, so that we can be in the 2D regime of a single occupied subband [5] without appreciable overlap of the surface states [6][7][8]. The effect of a nonzero surface state overlap is to open up an exponentially small gap δk ∝ e −W/ξ0 in the figure-8, as in Fig. 1a.
In a perpendicular magnetic field B the signed area enclosed by the Fermi surface is quantized in units of 2π/l 2 m , with l m = /eB the magnetic length. A figure-8 Fermi surface of linear dimension k F has a signed area much smaller than k 2 F , because the upper and lower loops have opposite orientation. We find that this twisted Fermi surface produces edge states of width k F l 2 m -much wider than the usual narrow quantum Hall edge states of width l m . The wide and the narrow edge states are counterpropagating: if the wide channel moves parallel to M , the narrow channel moves antiparallel. An applied voltage selectively populates one of the two types of edge states, resulting in a conductance of e 2 /h instead of 2e 2 /h -even though there are two conducting edges.
The outline of the paper is as follows. In the next section we formulate the problem, on the basis of a twoband model Hamiltonian [9,10], and calculate the band structure in a slab geometry. The way in which the Fermi arcs reconnect with the bulk Weyl cones is described exactly by a simple transcendental equation (Weiss equation). The Fermi surface in the thin-film regime is calculated in Sec. III, to show the topological transition from turning number 1 to turning number 0 when the Fermi level passes through a Weyl point. In Sec. IV we calculate the edge states in a perpendicular magnetic field, by semiclassical analytics and comparison with a numerical solution. The implications of the two types of counterpropagating edge channels for electrical conduction are investigated in Sec. V. We conclude with an overview of possible experimental signatures of the twisted Fermi surface.

A. Two-band model
We consider the two-band model Hamiltonian of a Weyl semimetal [9,10], The Pauli matrices are σ α , α ∈ {x, y, z}, with σ 0 the 2 × 2 unit matrix, acting on a hybrid of spin and orbital degrees of freedom. The momentum k varies over the Brillouin zone |k α | < π of a simple cubic lattice (lattice constant a 0 ≡ 1, and we also set ≡ 1). The two Weyl points are at the momenta k = (0, 0, ±K), K ≈ β, and at energies E = ±E 0 , E 0 ≈ λ sin β, displaced along the k zaxis by the magnetization M = βẑ and displaced along the energy axis by the strain λ. Time-reversal symmetry and spatial inversion symmetry are broken by β and λ, respectively. We take a slab geometry, unbounded in the y-z plane and confined in the x-direction between x = 0 and x = W . The magnetization along z is therefore in the plane of the slab. We impose the infinite-mass boundary condition [11] on the wave function ψ, This boundary condition corresponds to a mass term m 0 (x)σ z in H that vanishes inside the slab and tends to +∞ outside.

B. Dispersion relation
The Schrödinger equation Hψ = Eψ can be solved analytically in the low-energy regime by linearizing in k x and applying the effective mass approximation [12] k x → −i∂/∂x. Integration of the resulting first-order differential equation in x gives To ensure that an eigenstate of H satisfies the boundary condition (3), we require that This reduces to the following dispersion relation for E(k y , k z ): with transverse wave number q given by In the mass term m k we should set k x = 0, as required by the linearization in k x . For imaginary q = iκt x /W the transcendental equation (7) takes the form which is known as the Weiss equation in the theory of ferromagnetism [13]. A unique solution with κ ≥ 0 exists for γ ≥ 1, given by a generalized Lambert function [14,15]: A representative band structure is shown in Fig. 2.

C. Weyl cones and Fermi arcs
In the large-W limit of a thick slab, Eq. (7) can be solved separately for the bulk Weyl cones and the surface Fermi arcs. We thus recover the familiar dispersion relations in the bulk and surface Brillouin zones of a Weyl semimetal [16][17][18][19].
The bulk states have wave number q |m k |, quantized by q = (n + 1 2 )πt x /W , n = 0, 1, 2, . . ., with dispersion The ± distinguishes the upper and lower halves of the Weyl cones. The surface Fermi arcs have a purely imaginary q = im k ⇒ κ = −γ, which solves Eq. (8) in the large-W limit if m k < 0. The corresponding surface dispersion (6) is The ± sign distinguishes the Fermi arcs on opposite surfaces (− at x = 0 and + at x = W ). The trajectory of an electron in a Fermi arc state moves chirally along the surface (see top inset in Fig. 2), spiralling in the direction of the magnetization M = βẑ with velocity v z = λ cos k z . The surface Fermi arc reconnects with the bulk Weyl cone near k z = ±β. This "Fermi level plumbing" [20] is described quantitatively by the Weiss equation (7), as q switches from imaginary to real at a critical k crit z for which γ = 1. The penetration length ξ = 1/Im q of the surface state into the bulk is plotted in Fig. 3, as a function of k z for k y = 0. Its minimal value near the center of the Brillouin zone is The critical wave vector k = (0, 0, k crit z ) at which the Fermi arc terminates because its penetration length diverges is slightly smaller than the position β of the Weyl point,

III. THIN-FILM FERMI SURFACE
For Fermi energies a single two-dimensional (2D) subband is occupied at the Fermi level, formed out of hybridized bulk and surface states. This two-dimensional electron gas (2DEG) regime exists for thin films of width The Fermi surface of the 2DEG, defined by the equienergy contour E(k y , k z ) = E F , is plotted in Fig. 4 for several parameter values. As discussed in the introduction, the turning number ν is a topological invariant of the equi-energy contour [2]. We see from passes through a Weyl point must introduce a crossing in the Fermi surface [21].
The crossing of the equi-energy contour for small E F is possible since the intersecting states are spatially separated on the top and bottom surfaces of the slab. For a finite ratio W/ξ 0 of slab width and penetration length (12) the crossing is narrowly avoided because of the exponentially small overlap of the states at opposite surfaces. From the Weiss equation (8) we calculate that the δk z gap in the figure-8 is given by When W W c the gap in the figure-8 is exponentially small if W c ξ 0 , so for To make contact with some of the older literature [22][23][24], we note that the figure-8 Fermi surface of a Weyl semimetal is essentially different from the figure-8 equienergy contour of a conventional metal with a saddle point in the Fermi surface. In that case the figure-8 requires fine tuning of the energy to the saddle point, while here the figure-8 persists over a range of energies between two Weyl points. Moreover, the orientation of the two lobes of the figure-8 is the same in the case of a saddle point, while here it is opposite.

IV. QUANTUM HALL EDGE CHANNELS
A. Semiclassical analysis A magnetic field B in the x-direction, perpendicular to the thin film, introduces Landau levels in the energy spectrum: For a gauge A = (0, 0, By) the momentum k z is still a good quantum number, we seek the dispersion E n (k z ) of the n-th Landau level.
Semiclassically, the n-the Landau level is determined by the quantization of the signed area S(E) = k y dk z enclosed by the oriented equi-energy contour [25], with l m = ( /eB) 1/2 the magnetic length and γ ∈ [0, 1) a B-independent offset. Depending on the clockwise or anti-clockwise orientation of the contour, the enclosed area is negative or positive. Note that the signed area enclosed by the figure-8 Fermi surface of Fig. 4a equals zero. The phase shift γ = 0 in a bulk Weyl semimetal, when the equi-energy contour encloses a gapless Weyl point [26][27][28][29]. For the thin film the numerical data indicates γ = 1/2. If the thin film is confined to the strip 0 < y < W y , with W y l m , the spectrum within the strip remains dispersionless, but at the boundaries y = 0 and y = W y propagating states appear. In the quantum Hall effect these are chiral edge channels, moving in opposite directions on opposite edges [30,31]. The electrical conductance of the strip, for a current flowing in the z-direction, equals the number of edge channels N moving in the same direction times the conductance quantum e 2 /h.
The classical skipping orbits that form the edge channels in a magnetic field can be directly extracted from the zero-field Fermi surface: The cyclotron motion in momentum space follows the equi-energy contour is the cyclotron effective mass. (The figure-8 has m c ≈ β/t y .) Becausek = eṙ × B, the cyclotron motion in real space is obtained from the momentum space orbit by rotation over π/2 and rescaling by a factor l 2 m . Specular reflection at the edge (with conservation of k z ) then gives for the figure-8 Fermi surface the skipping orbits of The real-space counterpart of the quantization rule (18) is that the Aharonov-Bohm phase e A · dl picked up in one period of the cyclotron motion equals 2π(n+γ). For the skipping orbits this Bohr-Sommerfeld quantization rule still applies if the contour is closed by a segment along the edge, with an additional contribution to γ from reflection at the edge [33,34].
For small n the skipping orbit should enclose a flux of the order of the flux quantum h/e, which divides the edge channels into two types, designated narrow and wide: The narrow edge channel propagates along the edge in the direction opposite to the magnetization [35]. It is tightly bound to the edge over a distance of order l m , so that the enclosed area of order l 2 m encloses a flux of order h/e. The wide edge channel propagates in the direction of the magnetization and extends further from the edge over a distance of order βl 2 m . It still encloses a small flux of order h/e because contributions to A · dl from the two sides of the crossing point have opposite sign.
The gap δk z at the crossing point has no effect on the quantization if l m δk z 1, which is satisfied for l m W when Because the exponent wins it is sufficient that W ξ 0 to ensure that the figure-8 is effectively unbroken: The field-induced tunneling through the gap then occurs with near-unit probability, so to a good approximation the wave packet propagates in an unbroken figure-8.
The presence of counterpropagating edge channels at each edge requires a Fermi energy in between the Weyl points, |E F | < λ sin β, for a twisted Fermi surface. When the Fermi surface is a simple contour without selfintersections the edge channels are chiral, propagating in opposite directions on opposite edges as in Fig. 6.

B. Numerical simulation
To go beyond the semiclassical analysis we have diagonalized the model Hamiltonian (2) numerically, using the Kwant tight-binding code [36]. Fig. 7 shows the dispersion relation with four edge states at E F = 0, two counterpropagating at each edge. The corresponding density profile for each edge state is shown in Fig. 8. The two types of edge channels, one wide and the other narrow, are clearly visible.
In Fig. 9 we show the Landau levels in an infinite system as a function of the flux Φ through a unit cell. The Landau fan is fitted to corresponding to the semiclassical formula (18). The resulting offset γ is consistent with γ = 1/2. We checked that the fitted value of S E is close (within 2%) of the signed area enclosed by the figure-8 equienergy contour. We also checked that the same γ = 1/2 is obtained when the equienergy contour is a slightly deformed circle, rather than a figure-8.

V. MAGNETOCONDUCTANCE
To determine the magnetotransport through the Weyl semimetal strip we connect it at both ends z = 0 and z = L to a metal reservoir. Following a similar approach used for graphene [37], it is convenient to take the same model Hamiltonian (2) throughout the system, with the addition of a z-dependent chemical potential term −µ(z)σ 0 . (Physically, this potential could be controlled by a gate voltage.) We set µ(z) = 0 in the semimetal region 0 < z < L and take µ(z) E 0 in the metal reservoirs (x < 0 and x > L). This corresponds to n-type doping of the reservoir. (For p-type doping we would take µ(z) −E 0 .) We distinguish n-type and p-type edge channels in the Weyl semimetal depending on whether they reconnect at FIG. 10: Undoped Weyl semimetal (chemical potential µ ≈ 0) connected to heavily doped metal reservoirs (µ E0 for ntype doping). Edge channels in a perpendicular magnetic field are shown in red, with arrows indicating the direction of propagation. The L± edge channels are n-type and can enter into the reservoirs, while the R± edge channels are p-type and remain confined to the semimetal region (dotted lines). The current I flows along the n-type edge in the semimetal, irrespective of the sign of the applied voltage V .
large |E| with the upper Weyl cones (n-type) or with the lower Weyl cones (p-type). Referring to the dispersion of Fig. 7, the channels L ± at the y = 0 edge are n-type, while the channels R ± at the y = W y edge are p-type. The distinction is important, because only the n-type edge channels can be transmitted into the n-type reservoirs. As indicated in Fig. 10, the p-type channels are confined to the semimetal region, without entering into the reservoirs.
Upon application of a bias voltage V between the two n-type reservoirs a current I will flow along the n-type edge, with a conductance determined by the backscattering probability T y=0 along the edge at y = 0, so G = e 2 /h without impurity scatter- ing -see Fig. 12. This is not the usual edge conduction of the quantum Hall effect: As shown in Fig. 11, the current flows along the same edge when we change the sign of the voltage bias (switching source and drain), while in the quantum Hall effect the current switches between the edges when V changes sign. The only way to switch the edge here is to change the sign of the magnetic field, so that the n-type edge is at y = W y rather than at y = 0.

VI. DISCUSSION
We have discussed the unusual magnetic response of a two-dimensional electron gas with a twisted Fermi surface. The topological transition from turning number ν = 1 (the usual deformed Fermi circle) to turning number ν = 0 (the figure-8 Fermi surface) happens when the Fermi level passes through the Weyl point of a thinfilm Weyl semimetal with an in-plane magnetization and broken spatial inversion symmetry. We discuss several transport properties that could serve as signatures for the topological transition from ν = 1 to ν = 0.
In a magnetic field the figure-8 Fermi surface supports counterpropagating edge channels, see Fig. 10. At E F = 0, with an equal number of left-movers and right-movers at each edge, the Hall resistance will vanish. This is the first magnetotransport signature. If we vary the Fermi level and enter the regime of chiral edge channels, we should see the appearance of a voltage difference between the edges in response to a current flowing along the edges.
The second signature is the edge-selectivity: although both edges support counterpropagating states, the current flows entirely along one of the two edges, determined by the direction of M × B. This edge-selective current flow might be detected directly, or indirectly by introducing disorder on one edge only and measuring a difference between the conductance G for positive and negative B. Note that G(B) = G(−B) does not violate Onsager reciprocity, since for that we would need to change the sign of both magnetic field B and magnetization M .
A third signature is in the cyclotron resonance condition for the optical conductivity σ. As explained by Koshino [38] in the context of a type-II Weyl semimetal (which has a figure-8 cyclotron orbit at a specific energy where electron and hole pockets touch [39]), the resonance frequency is twice as small for an electric field oriented along the long axis of the figure-8, than it is for an electric field oriented along the short axis. In the geometry of Fig. 5, the resonance frequency equals eB/m c for σ yy and 2eB/m c for σ zz .
In our analysis we have not included disorder effects. The counterpropagating edge channels can be coupled by disorder, and this would reduce the conductance below the quantized value of G = e 2 /h seen in Fig. 12. There is no symmetry to protect this quantization, like there is for the helical edge channels in the quantum spin Hall effect, but there is a spatial separation of wide and narrow edge channels (see Fig. 8), which may provide some robustness against backscattering by disorder.
We have focused here on Fermi surfaces with turning number ν = 0 and ν = 1. It would be of interest to compare with other values of ν. A model Hamiltonian for ν = 2, that could be a starting point for such a study, is given in the Appendix.
For simplicity we have set t x = t y = t ≡ 1. Since the λ term is a scalar, we can set it to zero for now and then add it at the end of the calculation. After the unitary transformation H → U † HU with U = e iπσz/4 e iπσy/4 we have The square H 2 is block-diagonal in the σ index, The two W × W matrices Z and Z have the same eigenvalues ζ, given by The low-energy spectrum is therefore given which evaluates to For M k 1 we have simply ζ 0 ≈ M 2W k . The corresponding effective low-energy Hamiltonian takes the form H eff = σ x ζ 0 + σ y sin k y + λσ 0 sin k z , where we have reinsterted the λ term. A comparison of the energy spectrum of the effective Hamiltonian with the result from an exact numerical diagonalization of the full Hamiltonian is shown in Fig. 13. In closing, we note that a simple modification of this effective 2D Hamiltonian can be used to describe Fermi surfaces with turning number greater than unity. As an example, the Hamiltoniañ H eff = H eff + µ (2 − cos k z − cos k y )σ 0 (A10) has the ν = 2 Fermi surface shown in Fig. 14.