# Quantum description of Fermi arcs in Weyl semimetals in a magnetic field

Tim Bauer  Institut für Theoretische Physik, Heinrich-Heine-Universität,
D-40225 Düsseldorf, Germany  Francesco Buccheri  Dipartimento Scienza
Applicata e Tecnologia, Politecnico di Torino, Corso Duca degli Abruzzi 24,
10129 Torino, Italy  Alessandro De Martino  Department of Mathematics, City,
University of London, Northampton Square, EC1V OHB London, United Kingdom
Reinhold Egger  Institut für Theoretische Physik, Heinrich-Heine-Universität,
D-40225 Düsseldorf, Germany

###### Abstract

For a Weyl semimetal (WSM) in a magnetic field, a semiclassical description of
the Fermi-arc surface state dynamics is usually employed for explaining
various unconventional magnetotransport phenomena, e.g., Weyl orbits, the
three-dimensional Quantum Hall Effect, and the high transmission through
twisted WSM interfaces. For a half-space geometry, we determine the low-energy
quantum eigenstates for a four-band model of a WSM in a magnetic field
perpendicular to the surface. The eigenstates correspond to in- and out-going
chiral Landau level (LL) states, propagating (anti-)parallel to the field
direction near different Weyl nodes, which are coupled by evanescent surface-
state contributions generated by all other LLs. These replace the Fermi arc in
a magnetic field. Computing the phase shift accumulated between in- and out-
going chiral LL states, we compare our quantum-mechanical results to
semiclassical predictions. We find quantitative agreement between both
approaches.

##  I Introduction

Two hallmark features of topological electronic systems are their anomalous
magnetotransport properties and the existence of robust boundary states. In
the rich material class of Weyl semimetals (WSMs) [1, 2, 3, 4], these distinct
features manifest themselves in the chiral anomaly and in Fermi-arc surface
states, respectively. WSMs are three-dimensional (3D) semimetals characterized
by touching points of non-degenerate bands near the Fermi energy which are
separated in the Brillouin zone. These so-called Weyl nodes are effectively
described by massless relativistic Weyl fermions with conserved chirality. In
the presence of electromagnetic fields, Weyl fermions exhibit the chiral
anomaly which, on the level of the electronic band structure, implies the
formation of a chiral zeroth Landau level (LL). This chiral LL state has a
gapless linear dispersion along a direction determined by the chirality which
is parallel or antiparallel to the magnetic field [5, 6, 7, 8]. On the other
hand, Weyl nodes act as sources or sinks of Berry curvature and give rise to
non-trivial band topology [3]. Correspondingly, topological surface states
emerge near the boundary of a WSM. Since the Nielsen-Ninomiya theorem requires
Weyl nodes to come in pairs of opposite chirality [9], the energy contour of
these surface states must terminate at the projection of the bulk cones of two
Weyl nodes on the surface Brillouin zone and form an open disjoint curve — the
Fermi arc.

Given the experimental observation of signatures for both the chiral anomaly
and Fermi arcs [10, 11, 12, 13], it is natural to ask how both phenomena
conspire near the boundary of a semi-infinite WSM in a homogeneous magnetic
field oriented perpendicular to the surface. In a semiclassical picture, the
presence of the Lorentz force implies that fermions slide along the Fermi arc
connecting two Weyl node projections. Due to the open nature of the Fermi-arc
energy contour, no closed cyclotron orbit can form on the surface.
Accordingly, fermions have to tunnel into the bulk upon reaching the chiral
termination point of the Fermi arc [14], see Fig. 1 for a schematic
illustration in a half-space geometry. Since the only available bulk states at
low energies are provided by the chiral LL states, semiclassics predicts that
fermions will then move through the bulk and thereby escape from the surface.
Consequently, Fermi-arc states acquire a finite lifetime in a perpendicular
magnetic field $B$, and thus ultimately become unstable. Indeed, for the model
studied below, no stable surface state solutions exist if $B\neq 0$.

Figure 1: Schematic sketch of the half-space WSM geometry (defined by $z\geq
0$) in a homogeneous magnetic field $\mathbf{B}=B\hat{\mathbf{e}}_{z}$
perpendicular to the surface. The 3D WSM has two bulk Weyl nodes at momenta
$\mathbf{k}=(\pm k_{0},0,0)^{T}$, corresponding to the red and blue circles
for their surface projections. For $B=0$, the surface projections are
connected by a Fermi-arc surface state (green curve). For $B\neq 0$, the low-
energy bulk physics is dominated by the $n=0$ chiral LLs which have opposite
chirality near different Weyl nodes (pink and light blue arrows show the
respective propagation direction). Fermions incoming from a bulk chiral LL
enter the arc at the surface. After sliding along the arc to the opposite Weyl
node surface projection (in a semiclassical picture), they tunnel into the
outgoing chiral LL [14].

Furthermore, in a WSM slab geometry (or in similar confined nanostructures),
fermions in a chiral LL state move through the bulk and eventually tunnel into
the opposite surface. There, they will traverse the corresponding opposite
Fermi arc (in the semiclassical picture). In the simplest scenario, the
fermion subsequently occupies the chiral LL state with opposite chirality and
travels back to the initial Fermi arc state. This closed trajectory resembles
an exotic cyclotron orbit, commonly referred to as “Weyl orbit”. Such orbits
are predicted to cause unconventional quantum oscillations in the
magnetoconductivity, with a strong dependence on the sample thickness [14,
15]. The initial prediction of Weyl orbits sparked excitement in the WSM
community and led to a variety of subsequent theoretical proposals [16],
including non-local transport experiments [17, 18, 19], the detection of
chiral separation [20], and the chiral magnetic effect [21]. Weyl orbits are
also important ingredients for an unconventional so-called 3D Quantum Hall
Effect (QHE) [22, 23]. Both Shubnikov-de Haas oscillations due to Weyl orbits
and the 3D QHE were thoroughly investigated in transport experiments for the
Dirac semimetal Cd3As2 [24, 25, 26, 27, 28, 29], see also the review [16].

In a Dirac semimetal (DSM), Weyl nodes of opposite chirality share the same
position in momentum space but are stabilized by space group symmetries of the
crystal [3]. While the band structure is topologically trivial, pairs of Dirac
cones can be connected by Fermi-arc surface states nevertheless. The
semiclassical argument for the surface-bulk hybridization of Weyl orbits can
be adapted to DSMs despite of the formally closed energy contour of surface
states [14]. While clear experimental evidence for the predicted signatures
has been collected for Cd3As2 [24, 25, 26, 27, 28, 29], their interpretation
in terms of Weyl orbits remains debated [16]. In particular, thin films of
Cd3As2 show an intricate dependence of the QHE on sample thickness [28, 30].
Moreover, energy quantization due to Weyl orbits is difficult to distinguish
from the trivial size quantization of confined bulk states [31]. Similar
arguments might also apply to the Weyl orbits reported in the non-
centrosymmetric WSMs NbAs [32], TaAs [33] and WTe2 [34]. So far, no Weyl
orbits have been observed in magnetic WSMs with broken time-reversal symmetry
[16]. However, recent experimental work on magnetic WSMs [35] and progress in
quasiparticle interference experiments [36] render near-future advances in
Weyl-orbit physics for this class of materials likely. These developments also
motivated us to perform the present study.

A related exciting topic concerns twisted WSM interfaces [37, 38, 39, 40] and
tunnel junctions [41]. Upon twisting interfaces with respect to each other,
theory predicts a Fermi-arc reconstruction, implying the existence of “homo-
chiral” Fermi arcs connecting Weyl nodes of equal chirality [38, 39]. In a
magnetic field perpendicular to the interface, incoming electrons in a chiral
LL may traverse the homo-chiral Fermi arc and tunnel back into bulk states on
the other side of the junction, thereby achieving (almost) perfect
transmission. Numerical transport simulations show good agreement with this
semiclassical picture [41, 42, 43].

Adopting the half-space geometry, we here study the fate of Fermi-arc surface
states in WSMs in a magnetic field. We construct a full quantum solution,
going beyond semiclassics. To this end, we study a four-band low-energy
continuum model for a magnetic WSM. While we find analytical solutions of the
eigenproblem for the DSM limit of two degenerate Weyl nodes, we develop a
numerical approach (with a controlled cut-off procedure) to obtain the
eigenstates for the WSM scenario. We find that the eigenstates with lowest
energy are composed of in- and out-going chiral zeroth-order LLs which are
coupled by evanescent states localized near the surface. These are generated
by all remaining higher-order LLs and cause a phase shift between in- and
outgoing chiral LLs. In a slab geometry, this phase shift is experimentally
observable through magnetoconductivity oscillations [14]. We compare our
numerical results for the phase shift to semiclassical predictions by varying
the energy $\varepsilon$ and a boundary parameter $\alpha$ encoding the arc
curvature in the surface momentum plane. In addition, the energy derivative of
the phase shift determines the Fermi-arc lifetime which is finite for $B\neq
0$. We show how the lifetime depends on key parameters such as $\alpha$,
$\varepsilon,$ and $B$, and compare it to the semiclassical traversal time
across the Fermi arc.

The remainder of this paper is structured as follows. In Sec. II we discuss
the continuum WSM model employed here, derive boundary conditions for the
half-space geometry, and present the surface state spectrum at zero magnetic
field. We include the magnetic field in Sec. III and construct the eigenstates
in the half-space geometry. In addition, we consider the limit of a DSM and
obtain analytical solutions in several limiting cases. Subsequently, we derive
the corresponding semiclassical predictions in Sec. IV and compare them with
our quantum-mechanical results for the phase shift and for the Fermi-arc
lifetime. The paper closes with concluding remarks in Sec. V. Details of our
calculations can be found in several appendices. A derivation of $B=0$ Fermi-
arc surface states is given in App. A. Their spin and current structure is
discussed in App. B. We validate our numerical approach for finite magnetic
fields in App. C, and discuss the Goos-Hänchen effect for the present setup in
App. D. Throughout this paper, we use units with Fermi velocity $v_{F}=1$ and
put $\hbar=1$.

##  II WSM in half space

In this section, we discuss the four-band WSM model employed in our study. The
3D model (in the absence of a magnetic field) is introduced in Sec. II.1. We
then discuss the half-space geometry in terms of boundary conditions in Sec.
II.2. The Fermi-arc surface states for $B=0$ are specified in Sec. II.3, see
also App. A and App. B for further details.

###  II.1 Model

We study a four-band continuum WSM model which in 3D space, with conserved
momentum $\mathbf{k}=(k_{x},k_{y},k_{z})^{T}$, is defined by the Hamiltonian
[3]

$H(\mathbf{k})=\mathbf{k}\cdot\bm{\sigma}\tau^{z}+k_{0}\sigma^{x}\tau^{0},$
(1)

where $k_{0}\geq 0$ is the only free parameter. This parameter determines the
distance between the Weyl nodes in momentum space. In Eq. (1), $\sigma^{\mu}$
and $\tau^{\mu}$ are Pauli matrices acting on effective spin and orbital
degrees of freedoms, respectively, where $\mu=0$ refers to the identity and
$\mu=x,y,z$ otherwise. We use
$\bm{\sigma}=(\sigma^{x},\sigma^{y},\sigma^{z})$. While the limit $k_{0}=0$
describes a degenerate Dirac cone centered at $\mathbf{k}=\mathbf{0}$, i.e., a
Dirac semimetal, the model exhibits two separated Weyl nodes at momenta
$\mathbf{k}=\pm k_{0}\hat{\mathbf{e}}_{x}$ for $k_{0}>0$. Due to the block-
diagonal structure of $H$, these Weyl nodes are decoupled. Their conserved
chirality $\chi$ is associated with the orbital degree of freedom, namely the
eigenvalues $\chi=\pm 1$ of $\tau^{z}$.

We note that adding a mass term $H^{\prime}=m\sigma^{0}\tau^{x}$ in Eq. (1)
couples the Weyl nodes. However, for $m<k_{0}$, the Weyl nodes are thereby
only shifted in momentum space and the low-energy description is not changed
in an essential manner [3]. We thus put $m=0$ throughout this work. The two
Weyl nodes are then fully decoupled _in the bulk_. This key simplification
allows us to obtain explicit results in a finite magnetic field. Importantly,
in our approach, the boundary conditions will couple both Weyl nodes.
Furthermore, while Eq. (1) formally describes a type-I WSM with broken time-
reversal symmetry and the minimum number of two Weyl nodes, we expect our
arguments to apply to any WSM involving a pair of two isolated type-I Weyl
nodes.

Below, we use the standard representation of Pauli matrices. States are
written in the eigenbasis of $\sigma^{z}$ and $\tau^{z}$, i.e.,
$\ket{\chi}_{\bm{\tau}}$ for chirality $\chi=\pm 1$ and
$\ket{\sigma}_{\bm{\sigma}}$ for spin $\sigma\in\\{\uparrow,\downarrow\\}$,
with the spinor representations

$\displaystyle\ket{+}_{\bm{\tau}}$ $\displaystyle=\begin{pmatrix}1\\\
0\end{pmatrix}_{\bm{\tau}},\quad\ket{-}_{\bm{\tau}}$
$\displaystyle=\begin{pmatrix}0\\\ 1\end{pmatrix}_{\bm{\tau}},$
$\displaystyle\ket{\uparrow}_{\bm{\sigma}}$ $\displaystyle=\begin{pmatrix}1\\\
0\end{pmatrix}_{\bm{\sigma}},\quad\ket{\downarrow}_{\bm{\sigma}}$
$\displaystyle=\begin{pmatrix}0\\\ 1\end{pmatrix}_{\bm{\sigma}}.$ (2)

###  II.2 Half-space geometry and boundary conditions

We next proceed to the half-space geometry defined by $z\geq 0$, where we have
a planar boundary at $z=0$ as illustrated in Fig. 1. Since the surface
projections of the two Weyl nodes are separated, topological Fermi-arc surface
states with an open energy contour connecting the projected nodes arise [3].
Before solving for the Fermi arcs, we first derive the general boundary
condition from the constraint of Hermiticity of the Hamiltonian. For
relativistic continuum models, such an approach typically allows for a few
free parameters with physical implications on the surface state dispersion
[44, 45, 46, 47].

For a derivation that is also valid in the presence of a finite magnetic
field, we switch to real space by using the substitution
$\mathbf{k}\rightarrow-i\nabla_{\mathbf{r}}$ in Eq. (1). Following standard
arguments [44, 45, 46], we impose
$\braket{\Psi_{1}}{H\Psi_{2}}-\braket{H\Psi_{1}}{\Psi_{2}}=0$ for arbitrary
states $\Psi_{1}$ and $\Psi_{2}$ in the half-space geometry to infer a
_sufficient_ boundary condition,

$\Psi_{1}^{\dagger}(\mathbf{r}_{\perp},z=0)j^{z}\Psi_{2}(\mathbf{r}_{\perp},z=0)=0.$
(3)

Here, $\mathbf{r}_{\perp}=(x,y)^{T}$ is the in-plane position and
$j^{z}=\sigma^{z}\tau^{z}$ is the $z$-component of the relativistic fermion
particle current operator, $\mathbf{j}=\bm{\sigma}\tau^{z}$. Physically, Eq.
(3) thus prohibits any local current flowing through the surface. This
condition is ensured for states that satisfy boundary conditions of the form

$M\Psi(\mathbf{r}_{\perp},z=0)=\Psi(\mathbf{r}_{\perp},z=0),$ (4)

where $M$ is an operator with the properties

$j^{z}M=-M^{\dagger}j^{z},\quad M^{2}=\mathbbm{1},$ (5)

where $\mathbbm{1}=\sigma^{0}\tau^{0}$ is the identity. To parameterize all
possible choices of $M$, we define the operators

$M^{\bm{\tau}}_{\gamma}=\tau^{x}\cos{\gamma}+\tau^{y}\sin{\gamma},\quad
M^{\bm{\sigma}}_{\delta}=\sigma^{x}\cos{\delta}+\sigma^{y}\sin{\delta},$ (6)

in orbital and spin space, respectively. The most general Hermitian
parameterization then involves four real parameters
($\alpha,\beta,\gamma,\delta)$,

$\displaystyle M_{\alpha\beta\gamma\delta}$ $\displaystyle=$
$\displaystyle\cos{\alpha}\left(\sigma^{z}M^{\bm{\tau}}_{\gamma}\cos{\beta}+\sigma^{0}M^{\bm{\tau}}_{\gamma-\pi/2}\sin{\beta}\right)$
(7) $\displaystyle+$
$\displaystyle\sin{\alpha}\left(\tau^{0}M^{\bm{\sigma}}_{\delta}\cos{\beta}+\tau^{z}M^{\bm{\sigma}}_{\delta-\pi/2}\sin{\beta}\right).$

An equivalent parameterization was found in Refs. [48, 49] in the context of
graphene monolayers. On general grounds, the number of free parameters in the
boundary condition increases with the number of higher-energy bands in the
model Hamiltonian [50, 51]. Consequently, the low-energy spectrum is not
expected to change significantly when varying parameters within certain
submanifolds of the full parameter space. Below, we do not exploit the
complete parameter freedom in Eq. (7) but instead focus on a simple one-
parameter boundary matrix $M$ allowing us to describe curved Fermi arcs.

###  II.3 Boundary spectrum at zero magnetic field

Figure 2: Surface-state spectrum of the four-band model (1) in a half space
for $B=0$. (a) Zero-energy Fermi-arc contours in the $k_{x}$-$k_{y}$ plane as
described by Eq. (9) for different boundary parameters $\alpha$, see also Fig.
1. (b) Surface-state contour-plot in the $k_{x}$-$k_{y}$ plane for a DSM
($k_{0}\to 0$) with $\alpha/\pi=0.1$ for different energies
$\varepsilon(\mathbf{k}_{\perp})$ as indicated by the color bar, using a fixed
scale $p_{0}$ as reference. The surface-state termination points result from
the condition in Eq. (14).

Given the boundary condition (4) with the general parameterization (7), we
next construct physical Fermi arc solutions for $B=0$, which are labeled by
the conserved in-plane momentum $\mathbf{k}_{\perp}=(k_{x},k_{y})^{T}$. The
simplest approach is to choose a block diagonal matrix $M$, e.g., the
parametrization $M_{\frac{\pi}{2},0,0,\delta}$ in Eq. (7), which allows one to
solve the problem for both Weyl nodes separately. The Fermi arc of a single
Weyl node, which after a shift of $k_{x}$ is effectively described by
$H_{\chi}=\chi\mathbf{k}\cdot\bm{\sigma}$, then becomes a semi-infinite line
which terminates at the Weyl cone projection at an angle determined by the
parameter $\delta$ [44, 45, 46]. The resulting surface-state spectrum of the
four-band WSM model thus yields two semi-infinite arcs, in contrast to
physical Fermi arcs which are open curves connecting both Weyl cone
projections. Here, we will use boundary conditions which couple different Weyl
nodes. This approach is especially convenient for $B\neq 0$.

To this end, we consider off-diagonal boundary matrices $M$ satisfying
$[M,\sigma^{0}\tau^{z}]\neq 0$. The resulting boundary conditions couple the
Weyl nodes _at the surface_. This picture is analogous to the one in Ref.
[49], where armchair edges in graphene monolayers are modeled by boundary
conditions that are non-diagonal in the valley degree of freedom. Below, we
assume the boundary condition (4) with

$M_{\alpha}=M_{\alpha,0,0,0}=\sigma^{z}\tau^{x}\cos{\alpha}+\sigma^{x}\tau^{0}\sin{\alpha}.$
(8)

The parametrization (8) is a simple choice that allows us to construct
physical Fermi arcs with a curvature in the surface momentum plane controlled
by the single parameter $\alpha$. A curved arc is then symmetric under mid-
point reflection, and a straight arc is found for $\alpha=0$ (mod $\pi$). As
discussed in App. A, the particular choice of $M_{\alpha}$ in Eq. (8) is
motivated by the observation that a straight arc requires $[M,j^{y}]=0$, where
$j^{y}=\sigma^{y}\tau^{z}$ is the in-plane current operator along
$\hat{\mathbf{e}}_{y}$. For a microscopic analysis of a specific material, one
may instead employ boundary matrices $M$ containing more parameters (while
still coupling both Weyl nodes), possibly guided by numerical calculations for
lattice models. For simplicity, however, we focus on the one-parameter family
of matrices in Eq. (8). In App. A, we derive the corresponding $B=0$ surface-
state spectrum presented next.

We find that a physical Fermi-arc contour at energy $\varepsilon$ is given by
$k_{y}=q_{\alpha}(\varepsilon,k_{x})$, where

$q_{\alpha}(\varepsilon,k_{x})=\frac{(\varepsilon\sin{\alpha}-k_{0})(\varepsilon-
k_{0}\sin{\alpha})-k_{x}^{2}\sin{\alpha}}{\cos{\alpha}\sqrt{(\varepsilon\sin{\alpha}-k_{0})^{2}-k_{x}^{2}\sin^{2}{\alpha}}}.$
(9)

At zero energy, the contour terminates at both Weyl node surface projections
($k_{x}=\pm k_{0}$, $k_{y}=0$). The termination points $k_{x}=\pm
k_{\varepsilon\alpha}$ for $\varepsilon\neq 0$ are implicitly given by

$\varepsilon^{2}={\left(|k_{x}|-k_{0}\right)^{2}+[q_{\alpha}(\varepsilon,k_{x})]^{2}}.$
(10)

Assuming $|\varepsilon|\ll k_{0}$ and expanding Eq. (9) to lowest order in
$\varepsilon/k_{0}$, we estimate

$k_{\varepsilon\alpha}=k_{0}-\frac{4\varepsilon\sin{\alpha}}{3-\cos{(2\alpha)}}.$
(11)

Moreover, we find the low-energy dispersion relation

$\displaystyle\varepsilon(\mathbf{k}_{\perp})$
$\displaystyle\simeq\frac{k_{0}^{2}-k_{x}^{2}\sin^{2}{\alpha}}{k_{0}^{2}-k_{x}^{2}\sin^{4}{\alpha}}\bigg{(}\frac{k_{0}^{2}-k_{x}^{2}}{k_{0}}\sin{\alpha}$
$\displaystyle-\sqrt{k_{0}^{2}-k_{x}^{2}\sin^{2}{\alpha}}\,\frac{k_{y}}{k_{0}}\cos{\alpha}\bigg{)}.$
(12)

For $\alpha=0$, the above expressions are exact and describe a straight arc
with $\varepsilon(\mathbf{k}_{\perp})=-k_{y}$. For $\alpha\neq 0$, the arc is
curved in the surface momentum plane as illustrated in Fig. 2(a). We note that
the Fermi arc for $\alpha\to\pi-\alpha$ with the same energy $\varepsilon$
follows by reflection with respect to the $k_{x}$-axis. In effect, this
transformation yields the Fermi arc for the same boundary condition but in the
opposite half-space $z\leq 0$, see App. A. We briefly discuss the in-plane
spin and current densities associated with Fermi-arc states in App. B.

Next, we turn to the limit $k_{0}\to 0$ describing the low-energy theory of a
DSM with a single degenerate cone. While the band structure is now topogically
trivial, surface states may nonetheless exist. Such states become important
for $B\neq 0$, see Sec. IV. Deferring technical details to App. A, we find
topologically trivial surface states with the dispersion relation

$\varepsilon_{\pm}(\mathbf{k}_{\perp})=\pm\sqrt{k_{x}^{2}+k^{2}_{y}\cos^{2}{\alpha}},$
(13)

which only exist if the condition

$k_{y}\sin{\alpha}>0$ (14)

is satisfied. In particular, there neither are surface state solutions for
$\alpha=0$ (mod $\pi$), corresponding to a straight arc for finite $k_{0}$,
nor for $\varepsilon=0$. Below, we focus on those two cases for analytical
results. For finite $\alpha$ or $\varepsilon$, however, surface states emerge
which form open energy contours. These contours shrink with decreasing energy,
see Fig. 2(b).

##  III Half space in a magnetic field

In this section, we include the magnetic field
$\mathbf{B}=B\hat{\mathbf{e}}_{z}$ with $B>0$ and study the WSM model in Sec.
II for the half-space geometry using the boundary condition (4) defined by the
matrix $M_{\alpha}$ in Eq. (8). The parameter $\alpha$ determines the
curvature of the $B=0$ Fermi arc solutions. In Sec. III.1, we briefly review
the eigenstates for the infinite 3D problem. In Sec. III.2, we then turn to
the half-space problem and construct the low-energy quantum-mechanical
eigenstates.

###  III.1 Landau quantization

We start with the free-space WSM model and we incorporate the homogeneous
magnetic field $\mathbf{B}$ by minimal coupling,
$\mathbf{k}\to-i\nabla_{\mathbf{r}}+\frac{e}{c}\mathbf{A}$, where $e>0$ is the
(absolute value of the) electron charge and $c$ is the speed of light. For
convenience, we choose the Landau gauge, $\mathbf{A}=-By\hat{\mathbf{e}}_{x}$,
where Eq. (1) gives

$H=-i\nabla_{\mathbf{r}}\cdot\bm{\sigma}\tau^{z}-\frac{y}{\ell_{B}^{2}}\sigma^{x}\tau^{z}+k_{0}\sigma^{x}\tau^{0}=\begin{pmatrix}H_{+}&0\\\
0&H_{-}\end{pmatrix}_{\bm{\tau}}$ (15)

with the magnetic length $\ell_{B}=\sqrt{c/eB}$. Note that the chosen gauge
retains translation invariance along $\hat{\mathbf{e}}_{x}$. The momentum
component $k_{x}$ therefore remains a good quantum number.

When solving for LL solutions, it is convenient to consider the Weyl nodes
separately. A single Weyl node with chirality $\chi=\pm 1$ and momentum
$\mathbf{k}=-\chi k_{0}\hat{\mathbf{e}}_{x}$ is described by the Hamiltonian
$H_{\chi}$ in spin space using the block diagonal form in Eq. (15). For given
$\chi$ and $k_{x}$, we define the bosonic ladder operator

$a^{\dagger}_{\chi}=\frac{\ell_{B}}{\sqrt{2}}\left(\frac{y}{\ell_{B}^{2}}-k_{x}-\chi
k_{0}-\partial_{y}\right),$ (16)

with the commutator $[a_{\chi},a_{\chi}^{\dagger}]=1$. The transverse part of
$H_{\chi}=-i\chi\partial_{z}\sigma^{z}+H^{\perp}_{\chi}$ is thereby written as

$H_{\chi}^{\perp}=-\frac{\chi}{\sqrt{2}\ell_{B}}\left[\left(a_{\chi}+a_{\chi}^{\dagger}\right)\sigma^{x}+i\left(a_{\chi}-a_{\chi}^{\dagger}\right)\sigma^{y}\right].$
(17)

In the infinite 3D system (without boundary), the momentum component $k_{z}$
is also conserved. It is then straightforward to obtain the well-known
relativistic LLs $\varepsilon^{\chi}_{nk_{z}}$ labeled by non-negative integer
$n\in\mathbbm{N}_{0}$ [3],

$\varepsilon^{\chi}_{0,k_{z}}=-\chi
k_{z},\quad\varepsilon^{\chi}_{\pm,n>0,k_{z}}=\pm\chi\sqrt{\frac{2n}{\ell_{B}^{2}}+k_{z}^{2}}.$
(18)

Here, $\varepsilon^{\chi}_{0,k_{z}}$ is the dispersion of the gapless chiral
LL, while $n>0$ correspond to higher-order gapped LL states. Eigenstates are
expressed in terms of harmonic oscillator eigenfunctions,

$\varphi_{n}(y)=\frac{H_{n}(y/\ell_{B})}{\sqrt{2^{n}n!\sqrt{\pi}\,\ell_{B}}}\,{\mathrm{e}}^{-\frac{1}{2}{(y/\ell_{B})^{2}}},$
(19)

where $H_{n}$ is the $n$th-order Hermite polynomial. Writing
$a_{\chi}^{\dagger}a_{\chi}\varphi_{n}^{\chi}=n\varphi_{n}^{\chi}$, the wave
functions

$\varphi_{n}^{\chi}(y)=\varphi_{n}(y-\ell_{B}^{2}k_{x}-\chi\ell_{B}^{2}k_{0})$
(20)

incorporate a shift with respect to the Weyl node position. In anticipation of
the half-space geometry, we label the solutions
$\ket{\psi^{\chi}_{n\varepsilon}}$ of
$H_{\chi}\ket{\psi^{\chi}_{n\varepsilon}}=\varepsilon\ket{\psi^{\chi}_{n\varepsilon}}$
in terms of energy $\varepsilon$ instead of $k_{z}$. The chiral LL with $n=0$
is then described by

$\psi^{\chi}_{0,\varepsilon}(y,z)=\frac{\mathrm{e}^{-i\chi\varepsilon
z}}{\sqrt{2\pi}}\begin{pmatrix}0\\\
\varphi_{0}^{\chi}(y)\end{pmatrix}_{\bm{\sigma}}.$ (21)

Since $k_{x}$ is conserved, we keep plane-wave factors $e^{ik_{x}x}$ and the
$k_{x}$-dependence of observables implicit below. Similar expressions as Eq.
(21) hold for the wave functions of $n>0$ bulk LLs [5].

In the following, we focus on the _ultra-quantum regime_ ,
$|\varepsilon|<{\sqrt{2}}/{\ell_{B}}$. While $n>0$ bulk LLs do not exist in
this regime, it is possible to construct _evanescent_ solutions in the half-
space geometry by solving the eigenproblem for imaginary momentum
$k_{z}=i\kappa$ with $\kappa=\kappa_{n\varepsilon}>0$. The evanescent solution
for $n>0$ is given by

$\psi^{\chi}_{n\varepsilon}(y,z)=\sqrt{\kappa_{n\varepsilon}}\,{\mathrm{e}^{-\kappa_{n\varepsilon}z}}\begin{pmatrix}\chi\mathrm{e}^{i\chi\gamma_{n\varepsilon}}\varphi_{n-1}^{\chi}(y)\\\
\varphi_{n}^{\chi}(y)\end{pmatrix}_{\bm{\sigma}},$ (22)

with the inverse penetration length

$\kappa_{n\varepsilon}=\sqrt{2n/\ell_{B}^{2}-\varepsilon^{2}}$ (23)

and the phase $\gamma_{n\varepsilon}$ defined by

$\mathrm{e}^{i\gamma_{n\varepsilon}}=-\frac{\ell_{B}}{\sqrt{2n}}(\varepsilon+i\kappa_{n\varepsilon}).$
(24)

One can rationalize the appearance of this complex phase factor by noticing
that evanescent solutions do not carry any current along
$\hat{\mathbf{e}}_{z}$, i.e.,
$\braket{\psi^{\chi}_{n\varepsilon}}{\sigma^{z}}{\psi^{\chi}_{n\varepsilon}}=0$
for $n>0$.

###  III.2 Half-space geometry

####  III.2.1 Coupling of Weyl nodes at the boundary

We now proceed to the half-space geometry $z\geq 0$ sketched in Fig. 1, see
Sec. II for the $B=0$ case. We first rewrite the boundary condition (4) with
the matrix $M_{\alpha}$ in Eq. (8) as

$\mathbbm{V}_{\alpha}(z)\Psi(\mathbf{r})=0,\quad\mathbbm{V}_{\alpha}(z)=\delta(z)\left(\mathbbm{1}-M_{\alpha}\right).$
(25)

Our _Ansatz_ for solving Eq. (25) is a superposition of all eigenstates of $H$
in Eq. (15) with given $\varepsilon$ and $k_{x}$. We focus on the ultra-
quantum regime $|\varepsilon|<\sqrt{2}/\ell_{B}$, where $n>0$ LL states only
contribute through evanescent-state solutions in Eq. (22). Combining the
results of Eqs. (21) and (22) gives

$\ket{\Psi_{\varepsilon}}=\sum_{\chi=\pm}\sum_{n\geq
0}c^{\chi}_{n\varepsilon}\ket{\psi^{\chi}_{n\varepsilon}}_{\bm{\sigma}}\ket{\chi}_{\bm{\tau}},$
(26)

where the $c^{\chi}_{n\varepsilon}$ are complex coefficients which have to be
determined. Equation (25) states that $\ket{\Psi_{\varepsilon}}$ is element of
the kernel of $\mathbbm{V}_{\alpha}$. Matrix elements of this operator,
restricted to the subspace with fixed energy $\varepsilon$, are of the form

$\left[V_{\alpha}(\varepsilon)\right]^{\chi,\chi^{\prime}}_{n,n^{\prime}}=_{\bm{\tau}}\bra{\chi}\,_{\bm{\sigma}}\bra{\psi_{n\varepsilon}^{\chi}}\mathbbm{V}_{\alpha}(z)\ket{\psi^{\chi^{\prime}}_{n^{\prime}\varepsilon}}_{\bm{\sigma}}\ket{\chi^{\prime}}_{\bm{\tau}}.$
(27)

For convenience, we rescale them as

$[\hat{V}_{\alpha}(\varepsilon)]^{\chi,\chi^{\prime}}_{n,n^{\prime}}=\left(\kappa_{n\varepsilon}\kappa_{n^{\prime}\varepsilon}\right)^{-\frac{1}{2}}\left[V_{\alpha}(\varepsilon)\right]^{\chi,\chi^{\prime}}_{n,n^{\prime}}$
(28)

with $\kappa_{0,\varepsilon}=1/2\pi$. Matrix elements between a chiral $n=0$
LL and $n\geq 0$ LLs with equal chirality $\chi$ are given by

$[\hat{V}_{\alpha}(\varepsilon)]^{\chi,\chi}_{0,n}=\delta_{n,0}-\chi\mathrm{e}^{i\chi\gamma_{n\varepsilon}}\delta_{n,1}\sin{\alpha},$
(29)

while for $n,n^{\prime}>0$, we find

$\displaystyle[\hat{V}_{\alpha}(\varepsilon)]^{\chi,\chi}_{n,n^{\prime}}$
$\displaystyle=2\delta_{n,n^{\prime}}-\chi\mathrm{e}^{i\chi\gamma_{n+1,\varepsilon}}\delta_{n,n^{\prime}-1}\sin{\alpha}$
$\displaystyle-\chi\mathrm{e}^{-i\chi\gamma_{n\varepsilon}}\delta_{n,n^{\prime}+1}\sin{\alpha}.$
(30)

Matrix elements for opposite chiralities resemble the coupling of Weyl nodes
in terms of the boundary condition. For $n\geq 0$, we obtain

$[\hat{V}_{\alpha}(\varepsilon)]^{\chi,-\chi}_{0,n}=\braket{\varphi^{\chi}_{0}}{\varphi^{-\chi}_{n}}\cos{\alpha}.$
(31)

Finally, for $n,n^{\prime}>0$, we find

$\displaystyle[\hat{V}_{\alpha}(\varepsilon)]^{\chi,-\chi}_{nn^{\prime}}$
$\displaystyle=\cos{\alpha}\left(\mathrm{e}^{-i\chi(\gamma_{n\varepsilon}+\gamma_{n^{\prime}\varepsilon})}\braket{\varphi^{\chi}_{n-1}}{\varphi^{-\chi}_{n^{\prime}-1}}\right.$
(32)
$\displaystyle\left.+\braket{\varphi^{\chi}_{n}}{\varphi^{-\chi}_{n^{\prime}}}\right).$

The overlap $\braket{\varphi^{\chi}_{n}}{\varphi^{-\chi}_{m}}$ involves
shifted harmonic oscillator eigenfunctions associated with different Weyl
nodes. Performing the integration for $n\geq m$ yields [52]

$\displaystyle\braket{\varphi^{-}_{n}}{\varphi^{+}_{m}}$ $\displaystyle=\int
dy\,\varphi_{n}(y-l_{B}^{2}k_{0})\varphi_{m}(y+l_{B}^{2}k_{0})$ (33)
$\displaystyle=\sqrt{2^{n-m}\frac{m!}{n!}}\lambda_{B}^{n-m}L_{m}^{n-m}(2\lambda_{B}^{2})\mathrm{e}^{-\lambda_{B}^{2}},$

with the dimensionless quantity

$\lambda_{B}=k_{0}\ell_{B},$ (34)

which measures the decoupling of the Weyl nodes by the magnetic field. In Eq.
(33), $L^{m}_{n}$ is a generalized $n$th-order Laguerre polynomial. In App. C,
we describe a recursion relation allowing for the numerically efficient
computation of the overlaps in Eq. (33). The remaining terms follow from the
relation
$\braket{\varphi^{+}_{n}}{\varphi^{-}_{m}}=(-1)^{n-m}\braket{\varphi^{-}_{n}}{\varphi^{+}_{m}}$.
We note that the overlaps allow for a perturbative treatment in the large-
field limit $\lambda_{B}\ll 1$. Let us also mention in passing that similar
expressions appear when computing matrix elements of the bulk mass term
$m\sigma^{0}\tau^{x}$. In that case, the coupling opens a gap in the
dispersion of the chiral LLs of the order of
$\braket{\varphi_{0}^{+}}{\varphi_{0}^{-}}=\mathrm{e}^{-\lambda_{B}^{2}}$.
This result is consistent with the WKB approximation for a two-band WSM model
with two Weyl nodes [53, 54].

In any case, convergence of the overlaps
$\lim_{n\to\infty}\braket{\varphi^{+}_{n}}{\varphi^{-}_{m}}=0$ is ensured for
arbitrary $\lambda_{B}$. This fact justifies the introduction of a cut-off $N$
for the LL index, $n<N$, reducing the numerical solution of the boundary
problem to a linear algebra problem,

$V_{\alpha}(\varepsilon){\mathbf{c}}_{\varepsilon}=\mathbf{0},$ (35)

where $V_{\alpha}(\varepsilon)$ is a $2N\times 2N$ matrix formed by the matrix
elements (27) of the lowest $N$ LLs and $\mathbf{c}_{\varepsilon}$ is a vector
containing the corresponding coefficients $c^{\chi}_{n\varepsilon}$. The
numerical solution of Eq. (35) then determines the eigenstates of the WSM in
the half-space geometry for $B\neq 0$. In App. C, we carefully verify the
controlled nature of the above cut-off procedure and the accuracy of the
boundary condition.

Due to current conservation, coefficients with the same
$(n,\varepsilon,k_{x})$ but different chiralities have the same absolute
value, $|c_{n\varepsilon}^{+}|=|c_{n\varepsilon}^{-}|$. In particular, we are
interested in the phase shift $\theta_{\alpha}(\varepsilon)$ between in- and
outgoing chiral $n=0$ Landau states,

${c_{0,\varepsilon}^{-}}=\mathrm{e}^{i\theta_{\alpha}(\varepsilon)}{c_{0,\varepsilon}^{+}}.$
(36)

We note that all phases below are defined only modulo $2\pi$. The phase shift
$\theta_{\alpha}(\varepsilon)$ depends on the global phase choices for the
basis states in Eq. (26). While for a fixed phase choice,
$\theta_{\alpha}(\varepsilon)$ is formally gauge invariant, observable
quantities must also be independent of the phase choice. Full gauge invariance
is ensured below by only considering phase shift differences,
$\Phi=\theta_{\alpha^{\prime}}(\varepsilon^{\prime})-\theta_{\alpha}(\varepsilon)$.
When combined with the corresponding phase shift on the opposite surface in a
slab geometry, one can infer the magnetoconductivity oscillation period of the
corresponding Weyl orbit from Eq. (36) [14]. We compare our quantum-mechanical
results for $\Phi$ to semiclassical estimates in Sec. IV.

We note that for a straight arc at zero energy, $\alpha=0$ (mod 2$\pi$) and
$\varepsilon=0$, with the basis choice in Eq. (26), one finds

$\theta_{\alpha=0}(\varepsilon=0)=\pi.$ (37)

We verify Eq. (37) by evaluating the boundary condition at
$y=\ell_{B}^{2}k_{x}$, where
$\varphi^{+}_{n}(\ell_{B}^{2}k_{x})=(-1)^{n}\varphi^{-}_{n}(\ell_{B}^{2}k_{x})$.
By virtue of $|c^{+}_{n\varepsilon}|=|c^{-}_{n\varepsilon}|$ and the boundary
condition, we then arrive at $c_{n,0}^{+}=(-1)^{n+1}c_{n,0}^{-}$, and thus at
Eq. (37).

Since eigenstates in the half-space geometry can be written in the form (26),
a nontrivial $y$-dependence arises since the separation between Weyl nodes in
momentum space appears in the argument of Eq. (20). As shown in App. D, this
observation implies that an electron incident on the surface undergoes a shift
(assuming $\varepsilon>0$)

$\delta y=-2\ell_{B}^{2}k_{0}$ (38)

in the $y$-direction. This effect can be interpreted semiclassically in terms
of chiral transport associated to Fermi arcs, see App. B.

####  III.2.2 Dirac semimetal

Figure 3: Gauge invariant phase shift $\Phi^{\rm DSM}_{\alpha}(\varepsilon)$
vs energy $\varepsilon$ for chiral LLs in a DSM ($k_{0}\to 0$) for several
values of $\alpha$, see Eq. (43). The shown results were obtained by a
numerical solution of the quantum-mechanical problem. We use $p_{0}\ell_{B}=3$
with a reference scale $p_{0}$, where $\varepsilon$ is shown in units of
$p_{0}$ (with $v_{F}=1$). The ultra-quantum regime
$|\varepsilon|<\sqrt{2}/\ell_{B}$ is indicated by vertical dotted lines.

In order to identify contributions to the phase shift (36) picked up by
fermions traversing the Fermi arc in Sec. IV, let us briefly consider the
analogous problem in the DSM limit $k_{0}\to 0$. The corresponding linear
system follows from Eq. (35) by inserting diagonal overlaps
$\braket{\varphi^{-}_{n}}{\varphi^{+}_{m}}=\delta_{nm}$ in Eqs. (31) and (32).
For analytical results, we focus on cases without topologically trivial
surface states for $B=0$, i.e., we consider either $\alpha=0$ (mod $\pi$) or
$\varepsilon=0$.

First, for $\alpha=0$ (mod $2\pi$), it is straightforward to show that the
boundary condition (4) with $M_{0}=\sigma^{z}\tau^{x}$ is satisfied by
antisymmetric superpositions of chiral LLs,

$\ket{\Psi_{\varepsilon}}=\frac{1}{\sqrt{2}}\left(\ket{\psi^{+}_{0,\varepsilon}}_{\bm{\sigma}}\ket{+}_{\bm{\tau}}-\ket{\psi^{-}_{0,\varepsilon}}_{\bm{\sigma}}\ket{-}_{\bm{\tau}}\right).$
(39)

With the above basis choice, we then obtain the phase shift
$\theta_{\alpha=0}(\varepsilon)=\theta^{\rm DSM}_{\alpha=0}(\varepsilon)=\pi$
for arbitrary $\varepsilon$. Similarly, one finds $\theta^{\rm
DSM}_{\pi}(\varepsilon)=0$.

Second, for $\varepsilon=0$ but arbitrary $\alpha$, by using
$\gamma_{n,\varepsilon=0}=-\pi/2$, the linear system (35), expressed in terms
of the rescaled coefficients
$\hat{c}^{\chi}_{n}=\sqrt{\kappa_{n,\varepsilon=0}}{c}^{\chi}_{n,\varepsilon=0}$,
simplifies to

$\displaystyle 0$
$\displaystyle=\hat{c}^{\chi}_{0}+\cos{\alpha}\,\hat{c}^{-\chi}_{0}+i\sin{\alpha}\,\hat{c}^{\chi}_{1},$
$\displaystyle 0$
$\displaystyle=2\hat{c}^{\chi}_{n}-i\sin{\alpha}\left(\hat{c}^{\chi}_{n+1}-\hat{c}^{\chi}_{n-1}\right).$
(40)

The physical solution of the recursion relation is (we here assume
$\cos\alpha>0$)

$\hat{c}^{+}_{0}=-\hat{c}^{-}_{0},\quad\hat{c}^{\chi}_{n}=\left(i\tan{\frac{\alpha}{2}}\right)^{n}\hat{c}^{\chi}_{0}.$
(41)

Without need for a cut-off $N$ and up to normalization, we thereby arrive at
the exact solution

$\ket{\Psi_{\varepsilon=0}}\propto\sum_{\chi=\pm}\chi\sum_{n\geq
0}\frac{1}{\sqrt{\kappa_{n,0}}}\left(i\tan{\frac{\alpha}{2}}\right)^{n}\ket{\psi^{\chi}_{n,0}}_{\bm{\sigma}}\ket{\chi}_{\bm{\tau}}.$
(42)

Clearly, the phase shift is again given by $\theta^{\rm
DSM}_{\alpha}(\varepsilon=0)=\pi$. Remarkably, the superposition state (42)
involves evanescent contributions even though no surface state exists for
$\varepsilon=0$ with $B=0$ and $k_{0}=0$, see Eqs. (13) and (14). An analogous
calculation leads to $\theta^{\rm DSM}_{\pi-\alpha}(0)=0$.

For finite $\alpha$ and finite $\varepsilon$, we solve the problem numerically
as described above. As shown in Fig. 3, we then find a finite gauge invariant
phase shift, which we define as

$\Phi^{\rm DSM}_{\alpha}(\varepsilon)=\theta^{\rm
DSM}_{\alpha}(\varepsilon)-\theta^{\rm DSM}_{\pi}(\varepsilon).$ (43)

(The reason for substracting the phase for $\alpha=\pi$ is explained in Sec.
IV.) For small $\alpha$ and $\varepsilon$, this phase shift turns out to be
small compared to the corresponding phase shifts in WSMs, see Sec. IV. Since
the main focus of this work in on the WSM case, we leave a detailed
(semiclassical) discussion of phase shifts in DSMs to future studies.

##  IV Results and comparison to semiclassics

The semiclassical theory for Fermi arcs in WSMs in a magnetic field is well
established [14, 15]. According to this standard picture, fermions in the
chiral LL tunnel into a Fermi-arc state upon reaching the surface. The Lorentz
force then drives the fermion along the arc to the other Weyl cone projection
of opposite chirality, where it can tunnel back into the bulk and thereby
escape from the surface. In a slab geometry, this process is repeated on the
opposite surface, and the semiclassical trajectory forms a closed Weyl orbit
which can be described using semiclassical quantization [14, 15].

In the half-space geometry, the semiclassical trajectory is open and no
quantization is expected. This enables us to disentangle bulk and surface
contributions. The latter are determined by the semiclassical equations of
motion for an electron moving along the Fermi arc (with
$\mathbf{k}=\mathbf{k}_{\perp}$) [55, 14],

$\dot{\mathbf{k}}=-\frac{1}{\ell_{B}^{2}}\mathbf{v}_{\mathbf{k}}\times\hat{\mathbf{e}}_{z},\quad\dot{\mathbf{r}}=\mathbf{v}_{\mathbf{k}}=\nabla_{\mathbf{k}}\varepsilon(\mathbf{k}),$
(44)

where $\mathbf{v}_{\mathbf{k}}$ is the group velocity in the $x$-$y$ plane and
$\varepsilon=\varepsilon(\mathbf{k})$ is the arc dispersion relation. Here, we
neglect the anomalous velocity contribution due to the Berry curvature of
generic Fermi-arc states [55, 56]. This approximation can be justified by
noting that the Berry curvature vanishes for a straight arc and we consider
the small-$\alpha$ case below. As a consequence, $\dot{\mathbf{k}}$ is
tangential to the energy contour.

###  IV.1 Phase shifts accumulated along Fermi-arc curves

Figure 4:  Schematic illustration of closed trajectories in the surface
momentum plane used for the semiclassical calculation of the gauge invariant
phase $\Phi_{\alpha}(\varepsilon)$ in Eq. (47). A curved Fermi arc (green)
with $0<\cos\alpha<1$ is joined with a straight Fermi arc (purple) for
$\alpha=\pi$. Arrows indicate the direction of $\dot{\mathbf{k}}$ as described
by Eq. (44). We show the corresponding closed trajectories (a) for zero energy
($\varepsilon=0$), (b) for $\varepsilon<0$, and (c) for $\varepsilon>0$. For
$\varepsilon\neq 0$, the curved arc termination points, $(k_{x},k_{y})=(\pm
k_{\varepsilon\alpha},q_{\alpha}(\varepsilon,k_{\varepsilon\alpha}))$, differ
from the Weyl node projections $(\pm k_{0},0)$ corresponding to the circle
centers in panels (b) and (c). To match the arc termination points of the
curved arc and the straight reference arc, we employ a rescaling $k_{0}\to
k_{\varepsilon\alpha}$ for the straight arc case. For details, see main text.

We first consider the phase shift $\theta_{\alpha}(\varepsilon)$ between the
chiral LLs in Eq. (36) for a curved Fermi arc with $0<\cos\alpha<1$. In a
semiclassical picture, this phase shift can be estimated by a phase-space
integral of the schematic form

$\theta_{\alpha}(\varepsilon)=\int
d\mathbf{r}\cdot\left(\mathbf{k}-\frac{e}{c}\mathbf{A}\right).$ (45)

For gauge invariant phases, we need closed trajectories in real space. This
issue is closely related to the fact that the quantum-mechanical phase shift
$\theta_{\alpha}(\varepsilon)$ discussed in Sec. III.2 becomes gauge invariant
only after switching to a phase shift difference. For the semiclassical
counterpart, we resolve this issue by introducing a straight reference arc
which reconnects the termination points of the curved Fermi arc. We thereby
obtain a closed trajectory in the surface momentum plane, see Fig. 4, where
the phase $\Phi_{\alpha}(\varepsilon)$ accumulated along the trajectory is
gauge invariant. To ensure that also the corresponding real-space trajectory
is closed, we recall that the transformation $\alpha\to\alpha+\pi$ inverts the
sign of the group velocity component $v_{y}$, and thus of $\dot{k}_{x}$, see
Eq. (44). In effect, this allows for a closed motion in the surface momentum
plane, where the straight reference arc is chosen to have $\alpha=\pi$.

The above procedure is straightforwardly implemented at zero energy
($\varepsilon=0$), where the arc termination points are at $(k_{x},k_{y})=(\pm
k_{0},0)$ for all values of $\alpha$, see Fig. 4(a). On the quantum level, we
then consider the phase shift difference
$\Phi_{\alpha}(\varepsilon=0)=\theta_{\alpha}(0)-\theta_{\pi}(0)$, where
$\theta_{\pi}(0)=\pi$, see Eq. (37).

The situation becomes more intricate for $\varepsilon\neq 0$ since now the
curved arc termination points, $(k_{x},k_{y})=(\pm
k_{\varepsilon\alpha},q_{\alpha}(\varepsilon,k_{\varepsilon\alpha}))$, differ
from the corresponding Weyl node projections at $(\pm k_{0},0)$. (We recall
that $k_{\varepsilon\alpha}$ follows by solving Eq. (10), see also the
estimate in Eq. (II.3). Moreover, the function $q_{\alpha}(\varepsilon,k_{x})$
parameterizing the Fermi-arc contour at energy $\varepsilon$ has been defined
in Eq. (9).) For the straight reference arc, we therefore consider a system
with rescaled Weyl node separation, $k_{0}\to k_{\varepsilon\alpha}$, at
energy
$\varepsilon\to\bar{\varepsilon}=q_{\alpha}(\varepsilon,k_{\varepsilon\alpha})$.
The arc termination points for the straight reference arc are then located at
$(\pm k_{\varepsilon\alpha},\bar{\varepsilon})$ and match the termination
points of the curved arc, see Fig. 4(b,c). We note that the energy of the
reference arc differs from the energy of the curved Fermi arc. We can ensure
only in this manner that both arc contours connect at their termination points
and enclose a finite area in momentum space. No need for such a construction
would arise for Weyl orbits in a slab geometry, where tunneling processes via
bulk states take care of the corresponding momentum shifts between arc
termination points on opposite surfaces. The advantage of our approach is that
bulk states do not appear explicitly in the semiclassical calculation.

Figure 5:  Comparison of quantum-mechanical results and semiclassical
estimates for the gauge invariant Fermi-arc surface-state phase shift
$\Phi_{\alpha}(\varepsilon)$, see Eqs. (46) and (IV.1), respectively. The
phase $\Phi_{\alpha}(\varepsilon)$ is shown as function of the parameter
$\lambda_{B}=k_{0}\ell_{B}$ and gives the phase accumulated along the closed
trajectories in surface-momentum space illustrated in Fig. 4. Symbols show the
numerical solution of the quantum problem, and curves show the corresponding
semiclassical predictions. Integer multiples of $2\pi$ have been added to
obtain smooth curves. (a) Zero-energy case ($\varepsilon=0$) for different
$\alpha$. (a) Case $\alpha/\pi=0.1$ for different energies $\varepsilon$.

On the quantum level, we then define the gauge invariant phase shift
difference as

$\Phi_{\alpha}(\varepsilon)=\theta_{\alpha}(\varepsilon)-\bar{\theta}_{\pi}\left(-q_{\alpha}(\varepsilon,k_{\varepsilon\alpha})\right)-\Phi^{\rm
DSM}_{\alpha}(\varepsilon),$ (46)

where $\bar{\theta}_{\pi}$ follows by solving the linear system (35) with the
rescaled parameter
$\lambda_{B}\to\bar{\lambda}_{B}=\ell_{B}^{2}k_{\varepsilon\alpha}$. For a
comparison to semiclassical results, in Eq. (46), we also subtract the phase
shift difference $\Phi_{\alpha}^{\rm DSM}(\varepsilon)$, see Eq. (43), for the
corresponding DSM case as shown in Fig. 3.

On the semiclassical level, the above gauge invariant phase shift takes the
form

$\displaystyle\Phi_{\alpha}(\varepsilon)$ $\displaystyle=\oint
d\mathbf{r}\cdot\left(\mathbf{k}-\frac{e}{c}\mathbf{A}\right)$
$\displaystyle=-\ell_{B}^{2}\int_{-k_{\varepsilon\alpha}}^{k_{\varepsilon\alpha}}dk_{x}\,\left(q_{\alpha}(\varepsilon,k_{x})-q_{\alpha}(\varepsilon,k_{\varepsilon\alpha})\right),$
(47)

As illustrated in Fig. 4, the phase $\Phi_{\alpha}(\varepsilon)$ in Eq. (47)
corresponds to the momentum-space area enclosed by the curved Fermi arc and
the straight reference arc. Assuming $|\varepsilon|\ll k_{0}$, we find

$\displaystyle\Phi_{\alpha}(\varepsilon)$
$\displaystyle\simeq\ell_{B}^{2}k_{0}^{2}\frac{2\alpha\cot{(2\alpha)}-1}{\sin{\alpha}}$
$\displaystyle+2\ell_{B}^{2}\varepsilon
k_{0}\left(1+\alpha\tan{\alpha}-\frac{2\cos^{2}{\alpha}}{3-\cos{(2\alpha)}}\right).$
(48)

In Fig. 5, for small energies $\varepsilon$, we compare quantum-mechanical
results for $\Phi_{\alpha}(\varepsilon)$ obtained numerically from Eq. (46) to
the corresponding semiclassical predictions in Eq. (IV.1). We find
quantitative agreement both for different arc curvatures, see Fig. 5(a), and
for different energies, see Fig. 5(b). It is worth noting that the
semiclassical description remains accurate even for large magnetic fields with
$\lambda_{B}<1$.

###  IV.2 Fermi-arc lifetime and semiclassical traversal time

Figure 6:  Quantum-mechanical results for the Fermi-arc lifetime
$\tau_{\alpha}(\varepsilon)$ [symbols] and for the semiclassical arc traversal
time $t_{\alpha}(\varepsilon)$ [curves], see Eqs. (49) and (51), respectively,
for several values of $\alpha$ and given in units of $k_{0}^{-1}$. (a)
Dependence of $\tau_{\alpha}$ and $t_{\alpha}$ on $\lambda_{B}$ for
$\varepsilon=0$. (b) $\tau_{\alpha}-\tau_{0}$ vs $\varepsilon$ in the ultra-
quantum regime (delimited by vertical dotted lines) for $\lambda_{B}=3$.
Different symbols are for different $\alpha$ as in panel (a). We also show the
corresponding semiclassical traversal time differences $t_{\alpha}-t_{0}$. The
inset shows the straight-arc lifetime $\tau_{\alpha=0}$ vs $\varepsilon$,
again for $\lambda_{B}=3$.

As discussed in Sec. I, one expects that Fermi-arc surface states acquire a
finite _lifetime_ $\tau_{\alpha}(\varepsilon)$ in a finite magnetic field
$B\neq 0$. The lifetime describes the time scale for escaping into the bulk
via the chiral LLs and follows from the general relation [57, 58]

$\tau_{\alpha}(\varepsilon)=\frac{d\theta_{\alpha}(\varepsilon)}{d\varepsilon}.$
(49)

We note that this phase shift includes DSM contributions. Being a physical
observable, Eq. (49) is gauge invariant. We compute Eq. (49) numerically using
the quantum-mechanical approach detailed in Sec. III.

On the semiclassical level, we define another time scale, namely the
_traversal time_ $t_{\alpha}(\varepsilon)$. This is the time required to
traverse the Fermi arc from one termination point to the other. Since the
lifetime is due to the escape of Fermi-arc electrons into the bulk at the arc
termination points, one expects that $t_{\alpha}(\varepsilon)$ is of the same
order as $\tau_{\alpha}(\varepsilon)$. We will therefore compare these two
time scales below. The semiclassical traversal time follows with Eq. (9) in
the gauge invariant form

$t_{\alpha}(\varepsilon)=\ell_{B}^{2}\int_{-k_{\varepsilon\alpha}}^{k_{\varepsilon\alpha}}dk_{x}\,\sqrt{1+\left(\frac{\partial
q_{\alpha}(\varepsilon,k_{x})}{\partial
k_{x}}\right)^{2}}|\,\mathbf{v}_{\mathbf{k}}|^{-1}.$ (50)

Simple analytical expressions, cf. Eqs. (9) and (II.3), follow for
$|\alpha|\ll 1$ by expanding in $\alpha$ up to second order. We then obtain
the semiclassical estimate

$t_{\alpha}(\varepsilon)\simeq
2\ell_{B}^{2}\left[k_{0}\left(1+\frac{\alpha^{2}}{3}\right)-2\varepsilon\alpha\right].$
(51)

We note that for a straight arc ($\alpha=0$), the energy-independent traversal
time $t_{0}=2\ell_{B}^{2}k_{0}$ results.

In Fig. 6, we compare the semiclassical traversal time
$t_{\alpha}(\varepsilon)$ to the quantum-mechanical lifetime
$\tau_{\alpha}(\varepsilon)$. As shown in Fig. 6(a), the zero-energy lifetime
diverges with increasing $\lambda_{B}$ (i.e., with decreasing magnetic field),
where the stable $B=0$ Fermi arcs are approached. The semiclassical traversal
time qualitatively captures this behavior, but no quantitative agreement
between $t_{\alpha}(\varepsilon)$ and $\tau_{\alpha}(\varepsilon)$ is found.
As shown in the inset of Fig. 6(b), the lifetime of the straight arc
($\alpha=0$) increases with $|\varepsilon|$ and diverges upon reaching the
$n=1$ bulk LL. (We recall that our construction in Sec. III.2 is limited to
the ultra-quantum regime. For energies above the bulk gap of the $n=1$ LL, the
$n=1$ LL contributes in terms of propagating states.) Since the semiclassical
estimate for $t_{0}$ is independent of energy, the energy dependence of
$\tau_{\alpha}(\varepsilon)$ shown in the inset of Fig. 6(b) hints at quantum
effects beyond semiclassics.

To compare the two time scales $\tau_{\alpha}(\varepsilon)$ and
$t_{\alpha}(\varepsilon)$ for curved arcs, we have substracted the respective
$\alpha=0$ contributions, and consider $\tau_{\alpha}-\tau_{0}$ and
$t_{\alpha}-t_{0}$ in the main panel of Fig. 6(b). We find that both
quantities are approximately linear functions of energy (at low energies). The
lifetime differences $\tau_{\alpha}(\varepsilon)-\tau_{0}(\varepsilon)$ are
again qualitatively captured by the corresponding traversal-time differences
$t_{\alpha}(\varepsilon)-t_{0}$ (up to a constant off-set).

We conclude that while the Fermi-arc lifetime $\tau_{\alpha}(\varepsilon)$
includes quantum contributions beyond semiclassics, essential low-energy
features are captured by the semiclassical traversal time, at least in a
qualitative fashion.

##  V Discussion

In this work, we have studied the eigenstates of a four-band continuum model
for a WSM in a half-space geometry, with a magnetic field perpendicular to the
surface. At low energies in the ultra-quantum regime dominated by the zeroth
LL in the bulk, eigenstates are superpositions of in- and out-going chiral
$n=0$ LL states coupled by evanescent surface states originating from $n\neq
0$ LL states. The latter states replace the $B=0$ Fermi-arc surface state,
which acquires a finite lifetime for $B\neq 0$ and hence is not a stable
solution.

We have compared our quantum-mechanical results with the corresponding
semiclassical estimates by calculating the phase shift between in- and out-
going $n=0$ chiral LL states with the corresponding semiclassical results.
These results depend on the energy $\varepsilon$ and on a boundary parameter
$\alpha$ determining the Fermi-arc curvature for $B=0$. According to Refs.
[14, 15], the coupling between the chiral LLs is established by a
semiclassical motion of fermions along the arc due to the Lorentz force. For
the phase shifts, we find quantitative agreement between the quantum
description and semiclassical estimates. Moreover, from the energy derivative
of the phase shift, one can define the lifetime of the Fermi-arc state. By
comparing the lifetime to the semiclassical arc traversal time, we have argued
that quantum contributions beyond semiclassics are important for the lifetime.
Understanding the lifetime of Fermi-arc surface states in an electromagnetic
environment is a prerequisite for surface-sensitive tests such as quasi-
particle interference experiments [36]. In the future, the theoretical
modeling of such experiments could also profit from our explicit numerical
construction of the eigenstates.

Our results are, at least qualitatively, consistent with numerical work on
thin WSM films employing lattice models [15, 59], hybrid models [60], and wave
packet simulations [61]. The continuum approach used here employs a boundary
condition which allows one to disentangle bulk and surface contributions to
semiclassical trajectories. Our analysis shows that a semiclassical phase-
space integral along the Fermi arc provides accurate estimates for phase
shifts. When extending our arguments to a slab geometry or to thin films, one
can describe the phase shift associated with Weyl orbits. This phase shift is
observable in quantum magnetoconductance oscillations experiments, see Refs.
[32, 33, 34] for recent reports. Similar phase shifts are also expected to
appear in transport experiments on WSM junctions with hetero-chiral Fermi arcs
at the interface [41].

We have been able to make substantial progress, and in some cases even
obtained exact analytical solutions, since the studied four-band WSM model has
decoupled Weyl nodes in the bulk. Omitting bulk Weyl-node coupling terms,
e.g., a mass term $m\sigma^{0}\tau^{x}$, is typically justified for materials
with well-separated Weyl nodes. Indeed, assuming a Weyl node separation
$2k_{0}\simeq 2\,\text{\AA}^{-1}$, Eq. (34) gives $\lambda_{B}\simeq 2.57$ for
$B\simeq 1\,\text{T}$. The hybridization of LLs corresponding to different
Weyl nodes is then exponentially suppressed by a factor
$\mathrm{e}^{-\lambda_{B}^{2}}\simeq 0.0014$. We conclude that only for much
smaller $k_{0}$ and/or stronger $B$, effects of bulk Weyl-node coupling are
expected to become relevant. For such cases, one expects a bulk gap for the
hybridized $n=0$ LLs. As a consequence, the chiral anomaly will eventually
break down, and a non-monotonic magnetoconductance should appear [54, 53].
While such phenomena are not present in our study, they are unavoidable in
lattice models. In fact, we believe that they obscure a semiclassical
interpretation of previous numerical studies of WSM thin films [15, 59, 60,
61]. Studying the effects of chiral mixing, e.g., by including the mass term
$m\sigma^{0}\tau^{x}$ in our approach, is an interesting direction for future
work. Notably, numerical works in the Hofstadter regime suggest that depending
on the exact nature of the Weyl node annihilation associated with the opening
of the gap, the resulting insulating system can be either trivial or
topological [59, 62]. In the latter case, localized topological surface states
might emerge in the gap of the hybridized $n=0$ LLs. Such states seem to be
outside the reach of the established semiclassical picture.

The above-mentioned subtleties are absent if the magnetic field is oriented
parallel to the axis along the Weyl node separation ($\hat{\mathbf{e}}_{x}$ in
our case). This scenario was studied for a thin-film geometry [60], where a
much smaller surface-bulk hybridization was reported, consistent with the
semiclassical point of view. Magnetic fields oriented in the surface plane
generally result in qualitatively different physics [63, 64] than reported
here.

Our work has also covered the limiting DSM case. The considered Dirac
Hamiltonian is an appropriate effective model as long as crystal symmetries
protect the Dirac node degeneracy. It would be interesting in a future study
to apply our approach and compare numerical and analytical solutions at $B\neq
0$ to the semiclassical description of topologically trivial surface states at
$B=0$.

In view of the recent experimental progress on magnetic WSMs [35], such as
Co2MnGa [65] and Co3Sn2S2 [66, 36], we are optimistic that Weyl orbit physics
will soon be clearly established also beyond DSMs and non-centrosymmetric
crystals. The underlying physics of such compounds should be captured by our
results.

###### Acknowledgements.

We thank M. Breitkreiz, P. Brouwer, A. Chaou and V. Dwivedi for discussions.
We acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Projektnummer 277101999 - TRR 183 (project A02) and
under Germany’s Excellence Strategy - Cluster of Excellence Matter and Light
for Quantum Computing (ML4Q) EXC 2004/1 - 390534769.

##  Appendix A Surface state solutions

Here we provide detailed derivations for the $B=0$ surface states given in
Sec. II.3. We begin with the topologically trivial surface states for the
_Dirac semimetal_ case, $k_{0}=0$, described by
$H=\mathbf{k}\cdot\bm{\sigma}\tau^{z}$. After the unitary transformation
$U_{\alpha}=\exp{\left(\frac{i}{2}\alpha\sigma^{y}\tau^{x}\right)}$, we obtain

$\displaystyle\tilde{H}_{\alpha}=U_{\alpha}HU_{\alpha}^{\dagger}$
$\displaystyle=\left(k_{x}\sigma^{x}+k_{y}\sigma^{y}\cos{\alpha}-i\partial_{z}\sigma^{z}\right)\tau^{z}$
$\displaystyle+k_{y}\sigma^{0}\tau^{y}\sin{\alpha}.$ (52)

This transformation is convenient since it eliminates the boundary parameter
$\alpha$ from the boundary condition,
$U_{\alpha}M_{\alpha}U_{\alpha}^{\dagger}=M_{0}=\sigma^{z}\tau^{x}$, see Eq.
(8). Note that the unitary transformation leaves the current operator $j^{z}$
invariant. Therefore, eigenstates $\ket{\tilde{\Psi}}$ with
$\tilde{H}\ket{\tilde{\Psi}}=\varepsilon\ket{\tilde{\Psi}}$ must satisfy the
boundary condition $M_{0}\tilde{\Psi}(z=0)=\tilde{\Psi}(z=0).$ We next make a
(normalized) _Ansatz_ for a surface state confined to the half-space region
$z\geq 0$,

$\ket{\Psi}=\sqrt{\frac{\kappa}{2}}\begin{pmatrix}\tilde{\psi}^{+}\\\
\tilde{\psi}^{-}\end{pmatrix}_{\bm{\tau}},\quad\tilde{\psi}^{\chi}(z)=\mathrm{e}^{-\kappa
z}\begin{pmatrix}1\\\ \chi\mathrm{e}^{i\delta}\end{pmatrix}_{\bm{\sigma}},$
(53)

where $\delta$ and $\kappa$ are a phase and an inverse penetration length,
respectively. These quantities have yet to be determined, where Eq. (53)
satisfies the boundary condition for arbitrary $\delta$. To construct energy
eigenstates, we first note that the chiral components satisfy
$M^{\bm{\sigma}}_{\delta}\ket{\tilde{\psi}^{\chi}}_{\bm{\sigma}}=\chi\ket{\tilde{\psi}^{\chi}}_{\bm{\sigma}}$
for $M^{\bm{\sigma}}_{\delta}$ in Eq. (6). Eigenstates of $\tilde{H}$ thus
obey

$\left(k_{x}\sigma^{x}+k_{y}\sigma^{y}\cos{\alpha}\right)\ket{\Psi}=\varepsilon
M^{\bm{\sigma}}_{\delta}\ket{\Psi}.$ (54)

For given in-plane momentum $\mathbf{k}_{\perp}$, the phase
$\delta=\delta_{\pm}(\mathbf{k}_{\perp})$ then follows from

$\cos{\delta_{\pm}(\mathbf{k}_{\perp})}=\frac{k_{x}}{\varepsilon_{\pm}(\mathbf{k}_{\perp})},\quad\sin{\delta_{\pm}(\mathbf{k}_{\perp})}=\frac{k_{y}\cos{\alpha}}{\varepsilon_{\pm}(\mathbf{k}_{\perp})},$
(55)

with $\varepsilon_{\pm}(\mathbf{k}_{\perp})$ in Eq. (13). Inserting the
corresponding Ansatz into the eigenproblem of $\tilde{H}$ confirms that
$\varepsilon_{\pm}(\mathbf{k}_{\perp})$ is the energy dispersion of the
surface state and yields the inverse decay length $\kappa$ in the form

$\kappa=k_{y}\sin{\alpha}.$ (56)

The normalization condition $\kappa>0$ implies Eq. (14) for physical
solutions.

Next, we construct the solution for a _straight Fermi arc_ , corresponding to
the choice $\alpha=0$. For the purpose of generality, we here allow for a free
parameter in the parameterization (7). The trivial dependence of our results
on this parameter (see below) helps to develop physical insight. We consider
the boundary condition (4) with the matrix

$M_{\gamma}^{\prime}=M_{0,0,\gamma,0}=\sigma^{z}\left(\tau^{x}\cos{\gamma}+\tau^{y}\sin{\gamma}\right).$
(57)

The following results for $\gamma=0$ describe the $\alpha=0$ results in Sec.
II.3 since $M^{\prime}_{\gamma=0}=M_{\alpha=0}$ with $M_{\alpha}$ in Eq. (8).
(Note that $\delta$ in Eq. (7) is redundant for $\alpha=0$.) We choose the
normalized Ansatz

$|\Psi\rangle=\sqrt{\frac{\kappa_{+}\kappa_{-}}{\kappa_{+}+\kappa_{-}}}\begin{pmatrix}\psi^{+}\\\
\mathrm{e}^{i\gamma}\psi^{-}\end{pmatrix}_{\bm{\tau}},$ (58)

with the chiral spinor components

$\psi^{\chi}(z)=\mathrm{e}^{-\kappa_{\chi}z}\begin{pmatrix}1\\\
-i\chi\end{pmatrix}_{\bm{\sigma}}.$ (59)

This _Ansatz_ satisfies the boundary condition. From
$H|\Psi\rangle=\varepsilon|\Psi\rangle$, we find

$\varepsilon(k_{y})=-k_{y},\quad\kappa_{\chi}(k_{x})=k_{0}+\chi k_{x}.$ (60)

The normalization conditions $\kappa_{+}>0$ and $\kappa_{-}>0$ for surface-
state solutions restrict the in-plane momentum $k_{x}$ to the open interval
$-k_{0}<k_{x}<k_{0}$. We thus obtain a physical Fermi arc for a model with two
decoupled Weyl nodes in the bulk. Here it turns out that the energy dispersion
$\varepsilon(k_{y})$ and the inverse penetration length scales $\kappa_{\chi}$
are independent of $\gamma$. This is expected since the parametric freedom in
the boundary condition increases with the number of higher-energy bands.
However, in this instance, we can extend the relation between the arc
curvature and the corresponding boundary matrix parameterization further. To
this end, we note that a straight arc is characterized by a chiral dispersion
along $\hat{\mathbf{e}}_{y}$, and consequently a maximal current flows along
this direction. Accordingly, the found solutions are eigenstates of the in-
plane current $j^{y}=\sigma^{y}\tau^{z}$, which is only possible since
$M^{\prime}_{\gamma}$ commutes with $j^{y}$. We can therefore infer the
necessary condition that a straight arc corresponds to a parameterization of
$M$ with $[M,j^{y}]=0$. Note that for the parameterization $M_{\alpha}$ in Eq.
(8), this condition is only met for $\alpha=0$ mod $\pi$.

In fact, we find _curved Fermi arcs_ for all other values of $\alpha$. For
solving the surface-state problem, we here use a different approach which
applies to a large family of parameterizations. We first consider the
eigenproblem
$H_{\chi}\ket{\psi^{\chi}}_{\bm{\sigma}}=\varepsilon\ket{\psi^{\chi}}_{\bm{\sigma}}$
for a single Weyl node with chirality $\chi=\pm 1$, described by
$H_{\chi}=\chi\mathbf{k}\cdot\bm{\sigma}+k_{0}\sigma^{x}$. The most general
evanescent and normalized solution at given energy $\varepsilon$ and in-plane
momentum $\mathbf{k}_{\perp}$ is

$\psi_{\varepsilon\mathbf{k}_{\perp}}^{\chi}(z)=\sqrt{\frac{\kappa_{\chi}}{\varepsilon^{2}+\kappa_{\chi}^{2}}}\mathrm{e}^{-\kappa_{\chi}z}\begin{pmatrix}\chi\varepsilon+i\kappa_{\chi}\\\
k_{x}+\chi k_{0}+ik_{y}\end{pmatrix}_{\bm{\sigma}},$ (61)

where $\kappa_{\chi}(\varepsilon,\mathbf{k}_{\perp})=\sqrt{(k_{x}+\chi
k_{0})^{2}+k_{y}^{2}-\varepsilon^{2}}$ is the inverse length scale describing
the decay of the surface state into the bulk. The requirement that $\kappa$ is
real restricts the energy of physical solutions to

$\varepsilon^{2}<\left(|k_{x}|-k_{0}\right)^{2}+k_{y}^{2}.$ (62)

The solution with energy $\varepsilon$ in this interval is given by

$\ket{\Psi_{\varepsilon\mathbf{k}_{\perp}}}=\begin{pmatrix}c_{+}\psi^{+}_{\varepsilon\mathbf{k}_{\perp}}\\\
c_{-}\psi^{-}_{\varepsilon\mathbf{k}_{\perp}}\end{pmatrix}_{\bm{\tau}},$ (63)

where $c_{\pm}$ are complex coefficients. We now consider the boundary
condition (4) with a general Hermitian parameterization,

$M=\begin{pmatrix}X&Y\\\ Y^{\dagger}&Z\end{pmatrix},\quad X=X^{\dagger},\quad
Z=Z^{\dagger}.$ (64)

Here, we assume that $Y$ is invertible, which implies the condition
$[M,\sigma^{0}\tau^{z}]\neq 0$ for a physical Fermi arc, see Sec. II.2.
Together with $M^{2}=\mathbbm{1}$, we obtain the identities $Z=-Y^{-1}XY$ and
$Y^{-1}X^{2}=Y^{-1}-Y^{\dagger}$. It is then sufficient to consider the upper
two spinor components in the boundary condition
$(\mathbbm{1}-M)\Psi_{\varepsilon\mathbf{k}_{\perp}}(z=0)=0$, since the lower
two components are implied. One can express the upper two components as a
linear system of equations, $\mathbbm{B}\mathbf{c}=\mathbf{0}$, where

$\mathbbm{B}(\varepsilon,\mathbf{k}_{\perp})=\begin{pmatrix}(\sigma^{0}-X)\psi_{\varepsilon\mathbf{k}_{\perp}}^{+}(0)\quad-Y\psi_{\varepsilon\mathbf{k}_{\perp}}^{-}(0)\end{pmatrix}$
(65)

is a $2\times 2$ matrix and $\mathbf{c}=\left(c_{+},c_{-}\right)^{T}$ contains
the coefficients in Eq. (63). For $M_{\alpha}$ in Eq. (8), we have
$X_{\alpha}=\sigma^{x}\sin{\alpha}$ and $Y_{\alpha}=\sigma^{z}\cos{\alpha}$.
Solutions of the boundary condition thus satisfy $\det(\mathbbm{B})=0$. We
then obtain a secular equation that gives analytical solutions for
$k_{y}=q_{\alpha}(\varepsilon,k_{x})$, where Eq. (9) is the only solution
satisfying Eq. (62).

We note that surface-state solutions for the opposite half-space ($z\leq 0$)
with the same boundary condition (4) follow from the transformation
$\alpha\rightarrow\pi-\alpha$. This is because the transformation
$\kappa_{\chi}\to-\kappa_{\chi}<0$, necessary for constructing physical states
in this geometry, amounts to
$\psi^{\chi}_{\varepsilon,k_{x},k_{y}}(0)\to\left(\psi^{\chi}_{\varepsilon,k_{x},-k_{y}}(0)\right)^{*}$.
The corresponding secular equation
$\det\left[\mathbbm{B}^{*}(\varepsilon,k_{x},-k_{y})\right]=0$ then yields
$k_{y}=-q_{\alpha}=q_{\pi-\alpha}$.

##  Appendix B Surface spin polarization and current

Figure 7: In-plane spin polarization and particle current corresponding to the
Fermi-arc surface state for $B=0$, $\alpha/\pi=0.1$, and $\varepsilon=0$,
shown as color-scale plots in the $k_{x}$-$k_{y}$ plane, cf. App. B. Red and
blue dots indicate the surface projections of the Weyl nodes, arrows show the
in-plane components of the spin polarization $\braket{\mathbf{s}}$ [panel (a)]
and of the particle current $\braket{\mathbf{j}}$ [panel (b)]. Figure 8:
Absolute value of the coefficients $c^{\chi}_{n\varepsilon}$ appearing in the
superposition (26) vs order $n$. Note the logarithmic scale for the
coefficients. The shown results were obtained by numerically solving Eq. (35)
for $\varepsilon/k_{0}=0.1$, $\alpha/\pi=0.1$, $\lambda_{B}=1$, with a cut-off
value of $N=70$. Figure 9: Spectrum of $\hat{V}_{\alpha}$ as obtained from Eq.
(28) for $\alpha/\pi=0.1$, with the cut-off value $N=100$. Note the semi-
logarithmic scales. Blue (red) dots correspond to positive (negative)
eigenvalues. Since $\mathbbm{V}_{\alpha}$ is positive semi-definite, negative
eigenvalues indicate numerical errors. (a) Spectrum vs energy $\varepsilon$
(in units of $k_{0}$) in the ultra-quantum regime
$|\varepsilon|<\sqrt{2}/\ell_{B}$ delimited by the vertical gray line, for
$\lambda_{B}=1$. (b) Spectrum vs $\lambda_{B}$ for $\varepsilon/k_{0}=0.1$.
Figure 10: Spinor components $\Psi_{j=1,2,3,4}$ of the eigenstate solution at
the surface,
$\Psi(y,z=0)=\left(\Psi_{1}(y),\Psi_{2}(y),\Psi_{3}(y),\Psi_{4}(y)\right)^{T}$,
compared with the corresponding components of $M_{\alpha}\Psi(y,z=0)$.
Parameters are given by $\varepsilon/k_{0}=0.1$, $\alpha/\pi=0.1$ and
$\lambda_{B}=3$. We separately show the real and imaginary parts, which verify
that the boundary condition $M_{\alpha}\Psi(y,z=0)=\Psi(y,z=0)$ is numerically
satisfied to high accuracy.

In this Appendix, we discuss the spin texture related to
$s^{\mu}=\sigma^{\mu}\tau^{0}$ (with $\mu=x,y,z$) along a Fermi arc for the
$B=0$ case. Getting access to this type of quantity is an advantage of the
four-band model with respect to two-band models [50, 51, 47]. Furthermore, we
compute the in-plane current $j^{\mu}=\sigma^{\mu}\tau^{z}$ generated by the
Fermi arc. Given a normalized Fermi-arc solution $\ket{\Psi}$, we need to
evaluate expectation values of the form
$\braket{\sigma^{\nu}\tau^{\mu}}=\int^{\infty}_{0}dz\,\Psi^{\dagger}(z)\sigma^{\nu}\tau^{\mu}\Psi(z).$

For a straight arc ($\alpha=0$), the corresponding solutions in Eq. (59)
satisfy
$\sigma^{y}\ket{\psi^{\chi}}_{\bm{\sigma}}=-\chi\ket{\psi^{\chi}}_{\bm{\sigma}}$,
implying $\braket{\mathbf{s}}=k_{x}\hat{\mathbf{e}}_{y}/k_{0}$ and
$\braket{\mathbf{j}}=-\hat{\mathbf{e}}_{y}$.

For curved arcs with $\alpha>0$, we use the general solution (61) and perform
the integration. The spin polarization follows from

$\braket{s^{\mu}}=\sum_{\chi=\pm}\frac{|c_{\chi}|^{2}}{2\kappa_{\chi}}\psi^{\chi\,\dagger}(0)\sigma^{\mu}\psi^{\chi}(0),$
(66)

where the expression for the in-plane current only differs by a relative sign
in the sum,

$\braket{j^{\mu}}=\sum_{\chi=\pm}\chi\frac{|c_{\chi}|^{2}}{2\kappa_{\chi}}\psi^{\chi\,\dagger}(0)\sigma^{\mu}\psi^{\chi}(0).$
(67)

Above, we have suppressed the momentum dependence of $c_{\chi}$ and of
$\ket{\psi^{\chi}}_{\bm{\sigma}}$.

Results obtained from the above expressions are shown in Fig. 7. Our model
correctly reproduces the main features of Fermi arcs as experimentally
detected. First and foremost, the chiral transport is shown by the current in
Fig. 7(b). In addition, the spin polarization rotates along the arc as
dictated by the fact that the spin orientation at the two termination points
corresponds to the chirality of the Weyl nodes. This behavior is manifest in
Fig. 7(a) and in accordance with the spin texture observed experimentally by
spin-filtered angle-resolved photoemission spectroscopy [67, 68].

##  Appendix C Numerical implementation of boundary conditions

In this Appendix, we discuss the numerical approach introduced in Sec. III.
Figure 8 shows representative results for the coefficients
${\mathbf{c}}_{\varepsilon}$ in Eq. (26), which are obtained by numerically
solving Eq. (35) for a Landau level cut-off $N=70$. These results already
indicate that the numerical scheme is well controlled and convergent. A non-
trivial benchmark that is passed accurately by our numerical scheme is
provided by the analytical solutions (39) and (42) for a DSM with $\alpha=0$
or $\varepsilon=0$, respectively.

Let us next give additional details about our numerical approach. To avoid
numerical overflow (or underflow) when computing the matrix elements (27) for
a large cut-off $N$, it is convenient to compute the overlaps (33) using the
recursion relation ($n\geq m>1$)

$\displaystyle\braket{\varphi^{-}_{n}}{\varphi^{+}_{m}}$
$\displaystyle=\frac{1}{\sqrt{nm}}\left(n+m-1-2\lambda_{B}^{2}\right)\braket{\varphi^{-}_{n-1}}{\varphi^{+}_{m-1}}$
$\displaystyle-\sqrt{\frac{(n-1)(m-1)}{nm}}\braket{\varphi^{-}_{n-2}}{\varphi^{+}_{m-2}},$
(68)

with

$\braket{\varphi^{-}_{n-m+1}}{\varphi^{+}_{1}}=\frac{n-m+1-2\lambda_{B}^{2}}{\sqrt{n-m+1}}\braket{\varphi^{-}_{n-m}}{\varphi^{+}_{0}}$
(69)

and

$\braket{\varphi^{-}_{n-m}}{\varphi^{+}_{0}}=\sqrt{\frac{2^{n-m}}{(n-m)!}}\lambda_{B}^{n-m}\,\mathrm{e}^{-\lambda_{B}^{2}}.$
(70)

With these relations, we can easily employ a LL number cut-off of order
$N=250$ or even larger. For all results shown in this work, we have carefully
checked that results do not change when further increasing the cut-off.

Numerical solutions are then found from the kernel of
$V_{\alpha}(\varepsilon)$, i.e., from the matrix representation of
$\mathbbm{V}_{\alpha}$ in the subspace with fixed $\varepsilon$ and $k_{x}$.
Note that we physically expect a single solution in this subspace in the
ultra-quantum regime. Consequently, the spectrum of $V_{\alpha}(\varepsilon)$
should have a single zero eigenvalue which is well separated from all other
eigenvalues. Fig. 9 shows representative results for the spectrum of the
rescaled matrix $\hat{V}_{\alpha}$ obtained from Eq. (28). We find a non-
degenerate, well-separated and vanishing eigenvalue for all $\varepsilon$ in
the ultra-quantum regime. However, for $\lambda_{B}\gtrsim 1.5$ (weak magnetic
fields), numerical errors become slightly larger. Nonetheless, our numerical
solutions still satisfy the boundary condition as demonstrated for
$\lambda_{B}=3$ in Fig. 10, where we show the four components of the real and
imaginary parts of $\Psi(y,z=0)$ and $M_{\alpha}\Psi(y,z=0)$, respectively.
The boundary condition $\Psi(y,z=0)=M_{\alpha}\Psi(y,z=0)$ is indeed satisfied
to high precision for all values of $y$.

##  Appendix D Shift of the reflected electron

The electronic Goos-Hänchen effect is a quantum phenomenon, best described as
a lateral shift of a wave packet after reflection from a surface [69]. We can
see an analogue of this effect in the system at hand at the level of the
expectation value of the position operator. In particular, we can read Eq.
(26) in the ultra-quantum regime $0<\varepsilon<\sqrt{2}/\ell_{B}$ as the
superposition of an incoming wave in the chiral LL with $\chi=+1$, an outgoing
wave in the chiral LL with $\chi=-1$ and a series of bound states, see Eq.
(21). Considering first a single momentum component $k_{x}$, the expectation
value of the $y$-coordinate for an incoming electron arriving on the surface
($z=0$) is computed from the fundamental eigenmode of the harmonic oscillator
in Eq. (19) as $\left\langle
y\right\rangle_{\mathrm{in}}=\ell_{B}^{2}\left(k_{x}+k_{0}\right)$. The
momentum $k_{x}$ is conserved in the reflection process, and one readily sees
that the electron leaving the surface has the expectation value $\left\langle
y\right\rangle_{\mathrm{out}}=\ell_{B}^{2}\left(k_{x}-k_{0}\right)$. We note
that the shift $\langle y\rangle_{\mathrm{out}}-\langle
y\rangle_{\mathrm{in}}$ is gauge invariant.

Following Ref. [69], we now generalize this argument and write an electronic
wave packet formed by a superposition of plane waves with various momenta
$k_{x}$ and, for simplicity, a Gaussian envelope function centered around
momentum $p$ with spread $\Delta k$. Such a wavepacket, in the $\chi=+1$
block, has the form

$\Psi_{\varepsilon,\rm{in}}(\mathbf{r})=\frac{1}{\sqrt{2\pi}}\int
dk_{x}\,F\left(k_{x}-p\right)\mathrm{e}^{ik_{x}x}\psi_{0,\varepsilon}^{+}(y,z),$
(71)

with $F\left(k_{x}\right)=\left(\sqrt{\pi}\Delta
k\right)^{-\frac{1}{2}}\mathrm{e}^{-\frac{1}{2}\left(k_{x}/\Delta
k\right)^{2}}$. The expectation value of the $y$-coordinate for the wavepacket
arriving on the surface then follows as

$\displaystyle\left\langle y\right\rangle_{{\rm
in}}=\ell_{B}^{2}\left(p+k_{0}\right).$ (72)

Repeating the calculation for the outgoing wavepacket in the $\chi=-1$ block,

$\Psi_{\varepsilon,\mathrm{out}}(\mathbf{r})=\frac{1}{\sqrt{2\pi}}\int
dk_{x}\,F\left(k_{x}-p\right)e^{ik_{x}x}\psi_{0,\varepsilon}^{-}(y,z),$ (73)

one finds $\left\langle y\right\rangle_{{\rm
out}}=\ell_{B}^{2}\left(p-k_{0}\right)$. We conclude that the electronic Goos-
Hänchen shift is given by Eq. (38). As this result is separately valid for
each $k_{x}$, we expect it to hold for every choice of the envelope function.

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