††thanks: V. S., C.L. and M. V. contributed equally to this work.††thanks: V.
S., C.L. and M. V. contributed equally to this work.††thanks: V. S., C.L. and
M. V. contributed equally to this work.

# Linear magneto-conductivity as a DC probe of time-reversal symmetry breaking

V. Sunko  Department of Physics, University of California, Berkeley,
California 94720, USA  Materials Sciences Division, Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA  C. Liu  Department of Physics,
University of California, Berkeley, California 94720, USA  M. Vila  Department
of Physics, University of California, Berkeley, California 94720, USA
Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley,
California 94720, USA  I. Na  Department of Physics, University of California,
Berkeley, California 94720, USA  Materials Sciences Division, Lawrence
Berkeley National Laboratory, Berkeley, California 94720, USA  Y. Tang
Department of Physics, University of California, Berkeley, California 94720,
USA  V. Kozii  Department of Physics, Carnegie Mellon University, Pittsburgh,
Pennsylvania 15213, USA  S. M. Griffin  Materials Sciences Division, Lawrence
Berkeley National Laboratory, Berkeley, California 94720, USA  Molecular
Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720,
USA  J. E. Moore<EMAIL_ADDRESS>J. Orenstein<EMAIL_ADDRESS>Department of Physics, University of California, Berkeley, California 94720,
USA  Materials Sciences Division, Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA

###### Abstract

Several optical experiments have shown that in magnetic materials the
principal axes of response tensors can rotate in a magnetic field. Here we
offer a microscopic explanation of this effect, and propose a closely related
DC transport phenomenon — an off-diagonal _symmetric_ conductivity linear in a
magnetic field, which we refer to as linear magneto-conductivity (LMC).
Although LMC has the same functional dependence on magnetic field as the Hall
effect, its origin is fundamentally different: LMC requires time-reversal
symmetry to be broken even before a magnetic field is applied, and is
therefore a sensitive probe of magnetism. We demonstrate LMC in three
different ways: via a tight-binding toy model, density functional theory
calculations on MnPSe3, and a semiclassical calculation. The third approach
additionally identifies two distinct mechanisms yielding LMC: momentum-
dependent band magnetization and Berry curvature. Finally, we propose an
experimental geometry suitable for detecting LMC, and demonstrate its
applicability using Landauer–Büttiker simulations. Our results emphasize the
importance of measuring the full conductivity tensor in magnetic materials,
and introduce LMC as a new transport probe of symmetry.

In all materials, a current passing perpendicular to an applied magnetic field
introduces a voltage transverse to both the current and the field [1]. This
phenomenon, called the Hall effect, has been an invaluable tool for probing
the concentration and character of charge carriers. In magnetic materials an
applied field is not necessary to observe the Hall effect: for example, the
anomalous Hall effect (AHE) can be determined by the electronic structure and
Berry curvature in momentum space [2], whereas the topological Hall effect is
induced in non-collinear magnetic structures by real-space Berry curvature
[3]. Regardless of microscopic origins, any Hall effect requires time-reversal
symmetry (TRS) to be broken [4, 5].

The Hall effect arises from the anti-symmetric component of the conductivity
tensor ($\sigma^{a}_{xy}=(\sigma_{xy}-\sigma_{yx})/2$), and is usually
extracted from transverse voltage as a function of magnetic field ($H$),
assuming implicitly that the Hall effect is the only $H$-odd component of
$\sigma$, i.e. that the symmetric off-diagonal component
$\sigma^{s}_{xy}=(\sigma_{xy}+\sigma_{yx})/2$ is $H$-even. In _non-magnetic_
materials the validity of this assumption is guaranteed by the Onsager’s
reciprocity relations, which encode microscopic time reversibility.

Here we show that this standard assumption is dangerous in magnetic systems:
certain magnetic structures allow $\sigma^{s}_{xy}$ to have an $H$-odd
component, which need not be small. A typical measurement cannot distinguish
such response from the Hall effect, leading to possible misinterpretations of
the measured signal. On the other hand, detecting a component of
$\sigma^{s}_{xy}$ that is proportional to the applied field would provide
evidence of TRS breaking that is complementary to the AHE, as the symmetry
constraints for the two effects to be observed are distinct [6]. Measuring the
full conductivity tensor $\sigma_{ij}$ as a function of magnetic field is
therefore both a powerful probe of symmetry, and a necessity to correctly
estimate the carrier concentration in magnetic materials.

Figure 1:  (a) The model considers a one electron $f\rightarrow d$ transition,
with a degenerate initial state manifold and a final state manifold split by
the three terms in Eq. 1: $\Delta_{CF}$ separates the $d$ manifold into states
of $e_{g}$ and $t_{2g}$ symmetry. (b) The ratio of off-diagonal symmetric
conductivity to the average diagonal conductivity as a function of
$H_{z}/H_{y}$, calculated for the different final states in Fig. 1a. (c) The
orbital wave functions corresponding to
$\left<L_{\alpha}^{\text{eff}}\right>=1$, for $\alpha=y,z,(z+y)/\sqrt{2}$.

The equivalent phenomenon at optical frequencies manifests as an $H$-linear
rotation of principal optical axes, and is referred to as off-diagonal linear
magneto-birefringence (LMB). It can be used to determine the magnetic point
group of a material, as has been reported several times [7, 8]. Although the
symmetry constraints for its existence are known [9], the microscopic
understanding of the mechanism behind the principal axis rotation is still
lacking. Furthermore, since the symmetry-allowed form of the conductivity
tensor does not depend on the frequency, an analogous DC transport effect is
in principle allowed. To the best of our knowledge, this DC phenomenon has not
been explored theoretically nor experimentally.

In this work, we take up this quest to examine the field-induced principal
axis rotation in the DC limit, an effect we refer to as linear magneto-
conductivity (LMC), and elucidate how LMC arises from the electronic structure
of a magnetic material. We start from a microscopic understanding of the off-
diagonal LMB in a system of localized orbitals in a magnetic field, and use
the obtained insights to construct a tight-binding toy model which yields a
nonzero DC LMC. Within this model the effect originates from a field-induced
rotation of the Fermi surface (FS), resembling the rotation of principal
optical axes that characterizes LMB. To gain insight into the physical origin
of this effect, we derive the field-linear $\sigma^{s}_{xy}$ from
semiclassical equations of motion and identify two mechanisms: Berry curvature
and momentum-dependent band magnetization. Although TRS needs to be broken,
neither of these mechanisms relies on the existence of a net magnetization,
suggesting that LMC may be present in a wide range of magnetic materials. We
propose an experimental geometry suitable for measuring the full conductivity
tensor, and demonstrate it by a mesoscopic transport calculation. Finally, we
use density-functional theory (DFT) calculations on an example material,
MnPSe3, to show that a large LMC is present beyond simple toy models, and
needs to be seriously considered in experimental investigations.

We start by studying the origin of the optical effect. We develop a minimal
model of EuCd2P2, a material in which off-diagonal LMB has been observed [8],
and use it to calculate the optical conductivity tensor by employing the Kubo
formula [6]. For simplicity we consider an isolated Eu2+ ion in an octahedral
crystal field of $D_{3d}$ symmetry, and model a one-electron $f\rightarrow d$
transition (Fig. 1a). The initial state configuration (${}^{8}S_{7/2}$, $L=0$,
$S=7/2$) consists of seven degenerate spin-polarized $f$ states, which are not
split by crystal field or spin-orbit coupling, since $L=0$. The final state
$5d$ manifold is split by a crystal field ($\Delta_{CF}\hat{H}_{CF}$), spin-
orbit coupling ($\lambda$) and a Zeeman term induced by a magnetic field
$\mathbf{H}$:

$\hat{H}_{L}\left(\Delta_{CF},\lambda,\mathbf{H}\right)=\Delta_{CF}\hat{H}_{CF}+\lambda\mathbf{\hat{L}}\cdot{\mathbf{\hat{S}}}-\mu_{B}\mathbf{H}\cdot\big{(}2\mathbf{\hat{S}}+\mathbf{\hat{L}}\big{)},$
(1)

where $\mathbf{\hat{L}}$ and $\mathbf{\hat{S}}$ are orbital angular momentum
(OAM) and spin operators, respectively, and $\mu_{B}$ is the Bohr magneton.
The resulting energy spectrum, assuming $\Delta_{CF}\gg\lambda\gg H$, is shown
in Fig. 1a. The conductivity tensor [6] can now be evaluated for transitions
between the degenerate initial state manifold and each of the final states,
calculated assuming an applied $\mathbf{H}$ within the $yz$ plane. This field
captures the effect of an intrinsic magnetization $M_{y}$ that is tilted by an
applied $H_{z}$. We indeed find a non-zero $\sigma^{s}_{xy}\sim H_{z}/H_{y}$
(Fig. 1b), but only for $t_{2g}$ final states, indicating the importance of
OAM, which is quenched in the $e_{g}$ manifold.

The response is caused by a subtle interference effect between wave functions
which hold a non-zero OAM in the $y$ and $z$ directions ($L^{\text{eff}}_{y}$
and $L^{\text{eff}}_{z}$, Fig. 1c). Both the amplitude and the phase of the
wave functions have a non-trivial angular dependence; the former is caused by
the CF which encodes spatial symmetries, and the latter by the field-induced
OAM. Their interference results in the rotation of charge density, leading to
the principal axis rotation. The effect is therefore expected for optical
transitions in which either the initial or the final state supports a non-zero
OAM, despite the lowering of symmetry by the CF, as is the case for $t_{2g}$
orbitals.

Figure 2: Rotation of Fermi surfaces and change of conductivity in a tight
binding model as a function of magnetization. (a) Top view of the triangular
lattice of magnetic ions with moment ${\bf M}=(0,M_{y},M_{z})$ in the $y-z$
plane. The panels (b-d) study quantities when ${\bf M}$ points along
$-45^{\circ}$, $0^{\circ}$, or $+45^{\circ}$ from the $y$ axis, represented by
red, grey, and blue arrows. (b) The band dispersion along high symmetry path
in the Brillouin zone (BZ) for the three orientations of ${\bf M}$. $t$ is the
overall hopping energy scale and $E_{F}$ is the Fermi energy. (c) FS at the
Fermi energy $E=E_{F}$ for the three orientations of ${\bf M}$. As ${\bf M}$
is oriented from $-45^{\circ}$ to $+45^{\circ}$, the FS rotates accordingly in
a clockwise way. (d) The conductivity angle
$2\sigma^{s}_{xy}/(\sigma_{xx}+\sigma_{yy})$ as a function of $M_{z}/M_{y}$.
For (c) and (d) we assumed the filling at $E=E_{F}$.

The above insight into the mechanism behind the off-diagonal LMB at optical
frequencies suggests that an analogous effect in the DC limit might be
achieved in a system with $t_{2g}$ orbitals at the Fermi level. We therefore
construct such a tight-binding (TB) model, based on hoppings between local $d$
orbitals experiencing a strong octahedral crystal field splitting, and sitting
on a triangular lattice (Fig. 2a, $D_{3d}$ symmetry) [6]. The TB Hamiltonian
has the form

$\hat{H}_{TB}=\hat{H}_{L}+\sum_{{\bf
rr^{\prime}},\ell\ell^{\prime},\sigma\sigma^{\prime}}t_{\mathbf{r,r^{\prime}}\\!,\ell\ell^{\prime}\\!,\sigma\sigma^{\prime}}d^{\dagger}_{\mathbf{r},\ell,\sigma}d_{{\bf
r^{\prime}},\ell^{\prime},\sigma^{\prime}},$ (2)

where $\ell$ ($\sigma$) label the $d$ orbitals (spins). The first term,
$\hat{H}_{L}=\sum_{\mathbf{r}}d^{\dagger}_{\mathbf{r}}\hat{H}_{L}(\Delta_{\text{CF}},\lambda,{\bf
M})d_{\mathbf{r}}$, is equivalent to the local term in Eq. (1), with
magnetization ${\bf M}$ in place of an external field, while the second term
describes nearest neighbor hopping between such orbitals; the hopping matrix
elements
$t_{\bm{r}\bm{r}^{\prime}\\!,\ell\ell^{\prime}\\!,\sigma\sigma^{\prime}}$ are
obtained from the Slater-Koster parameterization [6, 10].

We plot in Fig. 2b the band structure of the $t_{2g}$-derived bands calculated
for three orientations of magnetization ${\bf M}$: along $y$ axis and $\pm
45^{\circ}$ away from the $y$ axis in the $yz$ plane. The bands are clearly
modified by the magnetization. More insights can be gleaned from the effect of
${\bf M}$ on the FS: the FS is elliptical for all three orientations of ${\bf
M}$, and it rotates in the $k_{x}k_{y}$ plane as the magnetization ${\bf M}$
rotates in the $yz$ plane.

The rotation of the FSs induced by the magnetization has profound consequences
on transport. The conductivity tensor depends on the geometry of an elliptical
FS as:

$\sigma=\sigma_{0}\mathbbm{1}+\sigma_{0}\mathcal{E}^{2}\\!\begin{pmatrix}-\cos
2\mathsf{\theta}&\sin 2\mathsf{\theta}\\\ \sin 2\mathsf{\theta}&\cos
2\mathsf{\theta}\end{pmatrix},$ (3)

where $\mathbbm{1}$ is the unit matrix, and $\mathsf{\theta}$ and
$\mathcal{E}$ denote the orientation of the major axis of the ellipse and its
eccentricity, respectively. The isotropic part $\sigma_{0}\mathbbm{1}$ denotes
the conductivity in absence of ${\bf M}$, as enforced by the threefold
rotation and mirror symmetries [6]. Importantly, the principal axes of the
conductivity tensor correspond to the major and minor axes of the ellipse, and
a nonzero FS rotation $\theta$ induces a nonzero off-diagonal conductivity.
Since $\sigma_{xy}$ is directly proportional to $\sin 2\mathsf{\theta}$, we
expect the opposite sign of $\sigma_{xy}$ for the two FSs shown in Fig. 2c,
calculated for the magnetization along $\pm 45\,\mathrm{\degree}$.

We confirm the above picture by calculating $\sigma^{s}_{xy}$ as a function of
the orientation of ${\bf M}$ using the Drude formula. In Fig. 2d we plot the
ratio $2\sigma^{s}_{xy}/(\sigma_{xx}+\sigma_{yy})$, which is independent of
the relaxation time $\tau$, as a function of $M_{z}/M_{y}$. Importantly,
$\sigma^{s}_{xy}$ is an odd function of $M_{z}$, and exhibits linear
dependence at small $M_{z}$, just like the (anti-symmetric) Hall conductivity.
Therefore, extracting the $H$-linear component of the transverse voltage
measured in a single geometry, as is usually done, is not sufficient to
isolate the Hall signal in a magnetic material, and independent measurements
of $\sigma_{xy}$ and $\sigma_{yx}$ are needed.

Our TB model shows that a magnetization can induce LMC, but it does not prove
that magnetization is _necessary_ for this effect to take place; this is
reminiscent of the AHE, which can be induced by a magnetization, but it does
not require it. To explore this analogy, we have derived the expression for
conductivity $\sigma^{s}_{ij}$ of a two-dimensional (2D) system up to the
linear order in ${\bf B}=B_{z}\hat{z}$ within the semiclassical approximation
[11]. Note that in semiclassics the electrons are affected by the field ${\bf
B}=\mu_{0}({\bf H}+{\bf M})$. The full derivation is given in [6], including
corrections due to a field-induced shift of the chemical potential, while the
final expression relevant for the discussion here is [12]:

$\displaystyle\sigma^{s}_{xy}=B_{z}\frac{\tau
e^{2}}{\hbar}\int\frac{d^{2}k}{(2\pi)^{2}}$ (4)
$\displaystyle\,\,\times\left[\frac{1}{2}(v^{0}_{x}\partial_{y}m_{\mathbf{k}}+v^{0}_{y}\partial_{x}m_{\mathbf{k}})-m_{\mathbf{k}}v^{0}_{xy}+ev^{0}_{x}v^{0}_{y}\Omega_{\bf
k}\right]f_{0}^{\prime},$

where $\tau$ is the relaxation time and $e$ the elementary charge. Inside the
integral, $f^{\prime}_{0}$ is the derivative of the Fermi function
$f_{0}(\varepsilon_{0,\mathbf{k}})$ with respect to the single-particle energy
$\varepsilon_{0,\mathbf{k}}$, $v_{i}^{0},i=x,y$ are the components of the
Fermi velocity and we defined their derivative with respect to momentum as
$v_{ij}^{0}=\partial_{i}v^{0}_{j}=\partial_{j}v^{0}_{i}$,
$\Omega_{\mathbf{k}}$ is the Berry curvature, and ${\bf
m}_{\mathbf{k}}=m_{\mathbf{k}}\hat{z}$ is the magnetization of the conduction
electrons, defined by the shift of the single-particle energy in an external
field, $\widetilde{\varepsilon}_{\bf k}=\varepsilon_{0,{\bf k}}-{\bf
B}\cdot{\bf m}_{\mathbf{k}}$ [13]. We find two distinct types of contributions
at linear order in $B_{z}$: due to magnetization and Berry curvature. The
first three terms in the integral describe a “magnetization” contribution; it
is further divided into a “velocity” part (first two terms) and a “band
curvature” part (third term). The fourth term, on the other hand, describes a
“Berry curvature” contribution. The physical origin and behavior of these
terms differ.

The “magnetization” contribution arises due to momentum-dependent changes of
magnetization and band velocities. Note that if the magnetization is constant
or linear in momentum across the momentum space, the “velocity” part of the
“magnetization” contribution to LMC vanishes, suggesting that it is expected
to be significant in systems with strongly mixed orbital character at the
Fermi level. This finding is consistent with the intuition obtained from the
local-orbital picture (Fig. 1): orbital mixing, promoted by crystal field and
spin–orbit coupling, leads to non-zero expected value of OAM, as well as the
momentum-depended response to a magnetic field. On the other hand, the “band
curvature” part of the “magnetization” contribution to LMC vanishes if the
band disperses linearly in momentum space. The “Berry curvature” contribution
is proportional to the Berry curvature $\Omega_{\bf k}$ multiplied by the
components of the Fermi velocity, and it arises from Berry phase correction to
the phase space volume [14]. This term can in principle give a non-zero
contribution even if $\Omega_{\bf k}$ is constant across the momentum space.
Symmetry such as a twofold rotation or a mirror reflection can promote the LMC
effect.

We note that although AHE can also arise from $\Omega_{\bf k}$, its functional
form differs substantially from that of LMC:

$\sigma^{\text{AHE}}_{xy}=-\frac{e^{2}}{\hbar}\int\frac{d^{2}k}{(2\pi)^{2}}\,\Omega_{\bf
k}\,f_{0}.$ (5)

First of all, $\sigma^{\text{AHE}}_{ij}$ does not depend on the Fermi velocity
or the relaxation time. Second, the integral in Eq. (5) is taken over all
occupied states (characterized by non-zero Fermi function $f_{0}$), while the
one in Eq. (LABEL:eq:Sigma) is taken over the vicinity of FS (characterized by
non-zero $f^{\prime}_{0}$). These differences emphasize that the AHE and LMC
are two distinct probes of TRS breaking, as already deduced from symmetry
considerations [6].

Figure 3: (a) L-shape device geometry. The scattering region (leads) is
denoted by the black (red) lattice. Leads 1 and 10 are the source and drain
electrodes, while leads 2-8 are voltage probes. (b) Conductivity angle for the
symmetric (red) and Hall (green) conductivity as a function of $M_{z}/M_{y}$.
The calculations are done based on the tight binding model (Eq. (2), Fig. 2),
with a Fermi level of $-1.95$. Solid line is the average value performed over
5 disorder realizations (shown in dashed, paler color).

Having confirmed that DC linear magneto-conductivity is not only allowed by
symmetry, but also expected based on microscopic arguments, we now turn
towards two experimental aspects: we propose an experimental geometry suited
for the detection of LMC, and demonstrate a sizable expected LMC in a
realistic material by first-principles calculations.

Long Hall bars are typically used for transport measurement to allow the
current to become homogeneous between the measurement contacts, and therefore
reject effects from the injection region. Simply exchanging current and
voltage contacts in a single device, which would theoretically probe
$\sigma_{xy}$ and $\sigma_{yx}$, would introduce different artifacts in the
two measurements. Another strategy might be to fabricate two devices, with the
bars along two orthogonal crystal directions. However, this would introduce
sample-to-sample variation, which may be a serious limitation since LMC and
the Hall effect exhibit different functional dependence on the scattering
rate. This leads us to propose an L-shaped device, illustrated in Fig. 3(a),
to independently measure $\sigma_{xy}$ and $\sigma_{yx}$ in the same geometry
and in the same crystal, allowing for robust extraction of $\sigma^{a}$ and
$\sigma^{s}$. Such devices can be structured from single crystals using
focused ion beam milling, or created in thin flakes using lithographic
techniques.

To demonstrate this approach, we perform Landauer–Büttiker calculations in the
$L$-shaped geometry [6], assuming bands given by the TB model above (Eq. (2),
Fig. 2). The resulting $2\sigma^{s}_{xy}/\left(\sigma_{xx}+\sigma_{yy}\right)$
is plotted as a function of the orientation of ${\mathbf{M}}$ in Fig. 3b. The
qualitative agreement of the Landauer–Büttiker calculations with the Drude
formula (see Fig. 2d) is excellent. For comparison, we also plot
$2\sigma^{a}_{xy}/\left(\sigma_{xx}+\sigma_{yy}\right)$, from which we see
that $\sigma^{s}_{xy}$ is larger than the Hall conductivity for our choice of
parameters, demonstrating that this regime can indeed be reached. We note that
the ratio $\sigma^{s}_{xy}/\sigma^{a}_{xy}$ depends on the scattering time,
and becomes larger with increasing purity.

Finally, we expand our TB model and semiclassical analysis by evaluating LMC
from DFT calculations of a real material, a monolayer of MnPSe3 [6]. This is a
member of the family of magnetic 2D transition metal phosphorous
tricalcogenides that have gathered increasing attention recently for their
wealth of interesting magnetic, optical and topological properties [15, 16,
17, 18]. MnPSe3 adopts the trigonal $R\overline{3}$ space group where a layer
of magnetic Mn atoms sits on a honeycomb lattice surrounded by PSe3 layers
above and below it (Fig. 4a). Its ground state magnetic order is
antiferromagnetic – here we consider the metastable ferromagnetically ordered
case, as has been previously investigated [18, 17]. We choose to calculate LMC
in MnPSe3 because it has the same point group symmetry as our toy model
($D_{3d}$, Fig. 2), and its low-energy electronic structure contains $t_{2g}$
orbitals of Mn.

We performed DFT calculations for two orientations of Mn magnetic moments
within the $yz$ plane (Fig. 4a, right panel), and used the Drude formula to
evaluate $2\sigma^{s}_{xy}/\left(\sigma_{xx}+\sigma_{yy}\right)$ as a function
of the Fermi level. As can be seen in Fig. 4b, a non-zero $\sigma^{s}_{xy}$ is
predicted across a wide range of energies. Its sign depends on the orientation
of the magnetic moment, thus validating our understanding of the LMC and its
presence in MnPSe3. An important finding is the magnitude of the effect. For a
Fermi energy close to the valence band maximum (VBM) $\sigma_{xy}^{s}$ reaches
values as large as $25\%$ of the diagonal conductivity. This dramatic
enhancement occurs because the small FSs close to the VBM can be significantly
perturbed by the magnetization, and emphasizes the importance of measuring the
full conductivity tensor in studies of magnetic semiconductors and
(semi)metals. Even without this enhancement, $\sigma_{xy}^{s}$ reaches
$1\%-3\%$ of the diagonal conductivity over a wide range of Fermi energies,
which is substantial compared to typically observed Hall effect values.
Indeed, both anomalous Hall effect [19] and topological Hall effect [20] are
considered “giant” when they correspond to a few percent of longitudinal
conductivity.

Figure 4: (a) Top view and side view of the MnPSe3 crystal structure. The red
and blue arrows indicate the two magnetization directions used in the
calculations. (b) Conductivity angle as a function of energy for magnetization
orientation along $\pm 45\degree$ with respect to the $z$ axis in the $yz$
plane.

To conclude, the impact of our work is threefold. First of all, we have
identified a microscopic mechanism responsible for the rotation of principal
optical axes in a magnetic field. Furthermore, we demonstrated a fundamentally
new transport effect, referred to as linear magneto-conductivity, using three
complementary approaches: a tight-binding toy model, DFT calculations and a
semiclassical calculation. The latter is particularly powerful, as it
identifies two distinct mechanisms yielding LMC: momentum dependent band
magnetization, and Berry curvature. Finally, we have shown the importance of
measuring the full conductivity tensor in magnetic materials, both because the
Hall effect can otherwise be misidentified, and because such measurements
serve as a powerful probe of symmetry. We hope that our results inspire new
ways of probing the effect of time-reversal symmetry breaking on conduction
electrons in future measurements.

###### Acknowledgements.

M.V. is grateful to Stephen R. Power for helpful discussions. V.S. is
supported by the Miller Institute for Basic Research in Science, UC Berkeley.
C.L. acknowledges fellowship support from the Gordon and Betty Moore
Foundation through the Emergent Phenomena in Quantum Systems (EPiQS) program.
M.V. was supported as part of the Center for Novel Pathways to Quantum
Coherence in Materials, an Energy Frontier Research Center funded by the US
Department of Energy, Office of Science, Basic Energy Sciences. J.M.
acknowledges support from the Quantum Materials program under the Director,
Office of Science, Office of Basic Energy Sciences, Materials Sciences and
Engineering Division, of the U.S. Department of Energy, Contract No. DE-
AC02-05CH11231. J.O received support from the Gordon and Betty Moore
Foundation’s EPiQS Initiative through Grant GBMF4537 to J.O. at UC Berkeley.
I. N. and S. G. were supported by the U.S. Department of Energy, Office of
Science, Office of Basic Energy Sciences, Materials Sciences and Engineering
Division under Contract No.DE-AC02-05-CH11231 within the Theory of Materials
program.Computational resources were provided by the National Energy Research
Scientific Computing Center and the Molecular Foundry, DOE Office of Science
User Facilities supported by the Office of Science, U.S. Department of Energy
under Contract No. DEAC02-05CH11231.The work performed at the Molecular
Foundry was supported by the Office of Science, Office of Basic Energy
Sciences of the U.S. Department of Energy, under the same contract No.

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## Supplementary Material for “Linear magneto-conductivity as a DC probe of
time-reversal symmetry breakin”

##  Appendix A Symmetry conditions required for linear magneto-conductivity
and AHE

We discuss in the main text that the anomalous Hall effect and linear magneto-
conductivity are distinct probes of symmetry. To emphasize this point further,
in table 1 we list the symmetries that allow ($\checkmark$) or prohibit
($\times$) the two effects. In particular, AHE is prohibited by mirror
symmetries $m_{x}$ and $m_{y}$, which allow for LMC. In contrast, LMC requires
the $z$-axis to have at most the $C_{2z}$ rotational symmetry, while AHE is
not limited by rotational symmetry.

Symmetry | AHE | LMC
---|---|---
$\mathcal{T}$ | $\times$ | $\times$
$\mathcal{P}$ | $\checkmark$ | $\checkmark$
$\mathcal{PT}$ | $\times$ | $\times$
$m_{x}$ | $\times$ | $\checkmark$
$m_{y}$ | $\times$ | $\checkmark$
$m_{z}$ | $\checkmark$ | $\checkmark$
$m^{\prime}_{x}$ | $\checkmark$ | $\times$
$m^{\prime}_{y}$ | $\checkmark$ | $\times$
$m^{\prime}_{z}$ | $\times$ | $\times$
$\mathcal{C}_{2z}$ | $\checkmark$ | $\checkmark$
$\mathcal{C}_{3z}$ | $\checkmark$ | $\times$
$\mathcal{C}_{4z}$ | $\checkmark$ | $\times$
$\mathcal{C}^{\prime}_{2z}$ | $\checkmark$ | $\times$
$\mathcal{C}^{\prime}_{3z}$ | $\checkmark$ | $\times$
$\mathcal{C}^{\prime}_{4z}$ | $\checkmark$ | $\checkmark$
Table 1: Symmetry requirements for the anomalous Hall effect (AHE) and linear
magneto-conductivity (LMC). $\checkmark$ and $\times$ denote that the effect
is and is not allowed by the corresponding symmetry, respectively.

##  Appendix B Calculation of optical conductivity

Optical conductivity is calculated using the standard Kubo formula:

$\sigma_{\alpha\beta}\left(\omega\right)=-\frac{\imath
e^{2}}{V\hbar}\times\sum_{n,m}\left(f_{m}-f_{n}\right)\left(E_{m}-E_{n}\right)\frac{\left\langle
n\left|\hat{\alpha}\right|m\right\rangle\left\langle
m\left|\hat{\beta}\right|n\right\rangle}{\hbar\left(\omega+\imath\eta\right)-\left(E_{m}-E_{n}\right)},$
(S1)

where $\hat{\alpha}$ and $\hat{\beta}$ are position operators
($\alpha,\beta=x,y,z$), $\omega$ is energy, $n$ and $m$ states, $f_{n}$ and
$f_{m}$ their respective occupations, $E_{n}$ and $E_{m}$ their energies, and
$\eta$ is a small damping. Initial states are taken to be seven degenerate
spin-polarized $f$-orbitals, while the final states are $d$-orbitals split by
a trigonal crystal field and spin-orbit coupling.

##  Appendix C Semiclassical calculation

In this section, we derive the most generic semiclassical expression for
linear magnetoconductivity, assuming that the Fermi energy lies within a
single non-degenerate band. We adopt the approach discussed in, e.g., Refs.
[21, 22].

We start with the semiclassical equations of motion having the form [23]

$\displaystyle\hbar\dot{\bf r}$ $\displaystyle=\nabla_{\bf
k}\widetilde{\varepsilon}_{\bf k}-\hbar\dot{\bf k}\times{{\bm{\Omega}}}_{{\bf
k}},$ $\displaystyle\hbar\dot{\bf k}$ $\displaystyle=-e{\bf E}-e\dot{\bf
r}\times{\bf B},$ (S2)

where ${\bm{\Omega}}_{\bf k}=i\langle\nabla_{{\bf k}}u_{{\bf
k}}|\times\nabla_{{\bf k}}u_{{\bf k}}\rangle$ is the Berry curvature. The
quasiparticle dispersion relation is modified in the presence of an external
magnetic field ${\bf B}$:

$\widetilde{\varepsilon}_{\bf k}={\varepsilon}^{0}_{\bf k}-{\bf m}_{\bf
k}\cdot{\bf B},$ (S3)

where ${\varepsilon}^{0}_{\bf k}$ is the bare band energy at ${\bf B}=0$,
$H_{\bf k}|u_{\bf k}\rangle={\varepsilon}^{0}_{\bf k}|u_{\bf k}\rangle$, and
the orbital magnetic moment equals

${\bf m}_{\bf k}=-i\frac{e}{2\hbar}\langle\nabla_{{\bf k}}u_{{\bf
k}}\left|\times\left(H_{\bf k}-{\varepsilon}_{\bf
k}^{0}\right)\right|\nabla_{{\bf k}}u_{{\bf k}}\rangle.$ (S4)

The solution of the equations of motion reads as

$\displaystyle\dot{\bf r}$ $\displaystyle=\frac{1}{\hbar D_{\bf
k}}\left\\{\nabla_{\bf k}\widetilde{\varepsilon}_{\bf k}+e{\bf
E}\times{\bm{\Omega}}_{\bf k}+\frac{e}{\hbar}{\bf B}\left({{\bm{\Omega}}}_{\bf
k}\cdot\nabla_{\bf k}\widetilde{\varepsilon}_{\bf k}\right)\right\\},$
$\displaystyle\dot{\bf k}$ $\displaystyle=\frac{1}{\hbar D_{\bf
k}}\left\\{-e{\bf E}-\frac{e}{\hbar}\nabla_{\bf k}\widetilde{\varepsilon}_{\bf
k}\times{\bf B}-\frac{e^{2}}{\hbar}({\bf E}\cdot{\bf B}){{\bm{\Omega}}}_{\bf
k}\right\\},$ (S5)

where the factor $D_{\bf k}=1+(e/\hbar)({\bf B}\cdot{{\bm{\Omega}}}_{\bf k})$
accounts for the density of states modification [14].

The current density in a uniform system equals

${\bf j}=-e\int\frac{d^{d}k}{(2\pi)^{d}}D_{\bf k}\dot{\bf
r}f=-\frac{e}{\hbar}\int\frac{d^{d}k}{(2\pi)^{d}}\left[\nabla_{\bf
k}\widetilde{\varepsilon}_{\bf k}+e{\bf E}\times{{\bm{\Omega}}}_{\bf
k}+\frac{e}{\hbar}{\bf B}(\nabla_{\bf k}\widetilde{\varepsilon}_{\bf
k}\cdot{{\bm{\Omega}}}_{\bf k})\right]f,$ (S6)

where $d$ is the dimensionality of the system. The semiclassical distribution
function $f$ satisfies the conventional kinetic equation. We assume the
relaxation time approximation for simplicity, resulting in

$\frac{\partial f}{\partial t}+\dot{\bf k}\cdot\nabla_{\bf
k}f=-\frac{f-f_{0}}{\tau},$ (S7)

where $\dot{\bf k}$ is given by Eq. (S5), $\tau$ is the relaxation time, and
$f_{0}=f_{0}(\widetilde{\varepsilon}_{\bf k})$ is the equilibrium Fermi-Dirac
distribution, which at zero temperature merely equals
$f_{0}=\Theta(\mu-\widetilde{\varepsilon}_{\bf k})$. We emphasize that $f_{0}$
depends on the modified quasiparticle energy $\widetilde{\varepsilon}_{\bf k}$
rather than on the bare band energy ${\varepsilon}_{\bf k}^{0}$. Assuming
monochromatic electric field,

${\bf E}(t)={\bf E}_{\omega}e^{-i\omega t}+{\bf E}_{\omega}^{*}e^{i\omega t},$
(S8)

the distribution function equals

$f(t)=f_{0}+f_{1}e^{-i\omega t}+f_{1}^{*}e^{i\omega t}+\ldots,$ (S9)

where $f_{1}\propto E=|{\bf E}_{\omega}|$. Physical electric field ${\bf
E}(t)$ is real, implying that ${\bf E}_{-\omega}={\bf E}_{\omega}^{*}$. The
kinetic equation for $f_{1}$ then becomes

$D_{\bf k}\left(i\omega-\frac{1}{\tau}\right)f_{1}=-\frac{e}{\hbar}\left({\bf
E}_{\omega}+\frac{e}{\hbar}({\bf E}_{\omega}\cdot{\bf B}){{\bm{\Omega}}}_{\bf
k}\right)\cdot\nabla_{\bf k}f_{0}-\frac{e}{\hbar^{2}}\nabla_{\bf
k}f_{1}\cdot\nabla_{\bf k}\widetilde{\varepsilon}_{\bf k}\times{\bf B}.$ (S10)

At small magnetic fields, the solution of this equation up to linear order in
${\bf B}$ equals

$f_{1}({\bf k})\approx\frac{e\tau}{\hbar D_{\bf
k}(1-i\omega\tau)}\left\\{\left({\bf E}_{\omega}+\frac{e}{\hbar}({\bf
E}_{\omega}\cdot{\bf B}){{\bm{\Omega}}}_{\bf k}\right)\cdot\nabla_{\bf
k}f_{0}(\widetilde{\varepsilon}_{\bf
k})+\frac{e\tau}{\hbar^{2}(1-i\omega\tau)}\left(\nabla_{\bf
k}\widetilde{\varepsilon}_{\bf k}\times{\bf B}\right)\cdot\nabla_{\bf
k}\left({\bf E}_{\omega}\cdot\nabla_{\bf k}f_{0}(\widetilde{\varepsilon}_{\bf
k})\right)\right\\},$ (S11)

and we explicitly restored the argument of $f_{0}$. Expanding further terms
with $\widetilde{\varepsilon}_{\bf k}={\varepsilon}^{0}_{\bf k}-{\bf m}_{\bf
k}\cdot{\bf B}$ up to linear order in ${\bf m}_{\bf k}\cdot{\bf B}$ and
introducing bare band velocity ${\bf v}^{0}_{\bf k}=\nabla_{\bf
k}{\varepsilon}^{0}_{\bf k}/\hbar$, we find

$\displaystyle f_{1}({\bf
k})\approx\frac{e\tau}{\hbar(1-i\omega\tau)}\left\\{\hbar\left({\bf
E}_{\omega}\cdot{\bf v}^{0}_{\bf
k}\right)f_{0}^{\prime}\left({\varepsilon}^{0}_{\bf k}\right)-e({\bf
B}\cdot{\bm{\Omega}}_{\bf k})({\bf E}_{\omega}\cdot{\bf v}^{0}_{\bf
k})f_{0}^{\prime}\left({\varepsilon}^{0}_{\bf k}\right)-\left({\bf
E}_{\omega}\cdot\nabla_{\bf k}({\bf B}\cdot{\bf m}_{\bf
k})\right)f_{0}^{\prime}\left({\varepsilon}^{0}_{\bf k}\right)\right.$ (S12)
$\displaystyle+e({\bf E}_{\omega}\cdot{\bf B})({\bm{\Omega}}_{\bf k}\cdot{\bf
v}^{0}_{\bf k})f_{0}^{\prime}\left({\varepsilon}^{0}_{\bf
k}\right)-\left.\hbar({\bf E}_{\omega}\cdot{\bf v}^{0}_{\bf k})({\bf
B}\cdot{\bf m}_{\bf k})f_{0}^{\prime\prime}\left({\varepsilon}^{0}_{\bf
k}\right)+\frac{e\tau}{1-i\omega\tau}f_{0}^{\prime}\left({\varepsilon}^{0}_{\bf
k}\right)\left({\bf v}^{0}_{\bf k}\times{\bf B}\right)\cdot\nabla_{\bf
k}\left({\bf E}_{\omega}\cdot{\bf v}^{0}_{\bf k}\right)\right\\}.$

Importantly, $f_{0}^{\prime}$ and $f_{0}^{\prime\prime}$ now have argument
${\varepsilon}^{0}_{{\bf k}}$, and we defined
$f_{0}^{\prime}({\varepsilon})=\partial
f_{0}\left({\varepsilon}\right)/\partial{\varepsilon}$ and
$f_{0}^{\prime\prime}({\varepsilon})=\partial^{2}f_{0}\left({\varepsilon}\right)/\partial{\varepsilon}^{2}$.

From Eq. (S6), we find that the linear in ${\bf E}$ part of the current equals
${\bf j}_{1}e^{-i\omega t}+{\bf j}_{1}^{*}e^{i\omega t}$, with

${\bf
j}_{1}=-\frac{e}{\hbar}\int\frac{d^{d}k}{(2\pi)^{d}}\left\\{\left[\nabla_{\bf
k}\widetilde{\varepsilon}_{\bf k}+\frac{e}{\hbar}\left(\nabla_{\bf
k}\widetilde{\varepsilon}_{\bf k}\cdot{\bm{\Omega}}_{\bf k}\right){\bf
B}\right]f_{1}({\bf k})+e{\bf E}_{\omega}\times{\bm{\Omega}}_{\bf
k}f_{0}\left(\widetilde{\varepsilon}_{\bf k}\right)\right\\}.$ (S13)

Expanding again up to the linear order in ${\bf B}$, we find

${\bf j}_{1}={\bf j}_{1}^{(0)}+{\bf j}_{1}^{(1)},$ (S14)

where ${\bf B}-$independent part is given by

${\bf
j}_{1}^{(0)}=-\frac{e^{2}}{\hbar}\int\frac{d^{d}k}{(2\pi)^{d}}\left\\{f_{0}\left({\varepsilon}_{\bf
k}^{0}\right){\bf E}_{\omega}\times{\bm{\Omega}}_{\bf k}+\frac{\hbar\tau
f_{0}^{\prime}\left({\varepsilon}^{0}_{\bf k}\right)}{1-i\omega\tau}\left({\bf
E}_{\omega}\cdot{\bf v}^{0}_{\bf k}\right){\bf v}^{0}_{\bf k}\right\\},$ (S15)

and linear in ${\bf B}$ contribution has form

$\displaystyle{\bf
j}_{1}^{(1)}=\frac{e^{2}}{\hbar}\int\frac{d^{d}k}{(2\pi)^{d}}({\bf B}\cdot{\bf
m}_{\bf k})\left({\bf E}_{\omega}\times{\bm{\Omega}}_{\bf
k}\right)f_{0}^{\prime}\left({\varepsilon}_{\bf k}^{0}\right)+$ (S16)
$\displaystyle+\frac{e^{2}\tau}{\hbar(1-i\omega\tau)}\int\frac{d^{d}k}{(2\pi)^{d}}f_{0}^{\prime}\left({\varepsilon}^{0}_{\bf
k}\right)\left\\{\left({\bf E}_{\omega}\cdot{\bf v}^{0}_{\bf
k}\right)\nabla_{\bf k}({\bf B}\cdot{\bf m}_{\bf k})+{\bf v}^{0}_{\bf k}({\bf
E}_{\omega}\cdot\nabla_{\bf k}({\bf B}\cdot{\bf m}_{\bf k}))+\right.$
$\displaystyle+\left.e\left[{\bf v}^{0}_{\bf k}({\bf B}\cdot{\bm{\Omega}}_{\bf
k})\left({\bf E}_{\omega}\cdot{\bf v}^{0}_{\bf k}\right)-{\bf v}^{0}_{\bf
k}({\bf E}_{\omega}\cdot{\bf B})({\bm{\Omega}}_{\bf k}\cdot{\bf v}^{0}_{\bf
k})-{\bf B}\left({\bf E}_{\omega}\cdot{\bf v}^{0}_{\bf
k}\right)\left({\bm{\Omega}}_{\bf k}\cdot{\bf v}^{0}_{\bf
k}\right)\right]\right\\}+$
$\displaystyle+\frac{e^{2}\tau}{1-i\omega\tau}\int\frac{d^{d}k}{(2\pi)^{d}}{\bf
v}^{0}_{\bf k}\left\\{f_{0}^{\prime\prime}\left({\varepsilon}_{\bf
k}^{0}\right)\left({\bf E}_{\omega}\cdot{\bf v}^{0}_{\bf k}\right)({\bf
B}\cdot{\bf m}_{\bf
k})-\frac{e\tau}{\hbar(1-i\omega\tau)}f_{0}^{\prime}\left({\varepsilon}_{\bf
k}^{0}\right)\left({\bf v}^{0}_{\bf k}\times{\bf B}\right)\cdot\nabla_{\bf
k}\left({\bf E}_{\omega}\cdot{\bf v}^{0}_{\bf k}\right)\right\\}.$

The first term in this expression corresponds to the anomalous Hall effect
correction due to finite magnetic field. This term is fully antisymmetric and
thus does not contribute to $\sigma_{ij}^{s}$. The last term is entirely due
to Lorentz force and is not related to the geometry of the band structure. It
may have both symmetric and anti-symmetric components that we specify below.
Finally, in $d=2$ which is the main focus of this paper, both
${\bm{\Omega}}_{\bf k}$ and ${\bf m}_{\bf k}$ are pointing in the
$z-$direction, hence, the two terms containing $\left({\bm{\Omega}}_{\bf
k}\cdot{\bf v}^{0}_{\bf k}\right)$ are equal to 0.

So far we only considered the grand canonical ensemble, i.e., assumed that the
chemical potential is fixed and does not depend on ${\bf B}$. If we instead
focus on the canonical ensemble and assume the fixed density of particles $n$,
we find that the chemical potential is renormalized as $\mu\to\mu-\delta\mu$.
The correction $\delta\mu$ due to finite magnetic field is found from the
condition

$n=\int\frac{d^{d}k}{(2\pi)^{d}}D_{\bf k}f_{0}\left({\varepsilon}_{\bf
k}^{0}+\delta\mu-{\bf B}\cdot{\bf m}_{\bf
k}\right)=\int\frac{d^{d}k}{(2\pi)^{d}}f_{0}\left({\varepsilon}_{\bf
k}^{0}\right)$ (S17)

and equals to the leading order in ${\bf B}$

$\delta\mu=\delta\mu_{\bf
m}+\delta\mu_{\bm{\Omega}}=\frac{1}{N_{0}}\int\frac{d^{d}k}{(2\pi)^{d}}\left\\{-({\bf
B}\cdot{\bf m}_{\bf k})f_{0}^{\prime}\left({\varepsilon}_{\bf
k}^{0}\right)+\frac{e}{\hbar}({\bf B}\cdot{\bm{\Omega}}_{\bf
k})f_{0}\left({\varepsilon}_{\bf k}^{0}\right)\right\\},\qquad
N_{0}=-\int\frac{d^{d}k}{(2\pi)^{d}}f_{0}^{\prime}\left({\varepsilon}_{\bf
k}^{0}\right).$ (S18)

At zero temperature,

$N_{0}=\int\frac{d^{d}k}{(2\pi)^{d}}\delta\left({\varepsilon}_{\bf
k}^{0}-\mu\right),$ (S19)

which is the zero-field density of states per unit volume at the Fermi level.

The corresponding correction to the current comes from the zero-order
contribution ${\bf j}_{1}^{(0)}$, Eq. (S15), and equals

$\delta{\bf
j}_{1}^{(\mu)}=-\frac{e^{2}}{\hbar}\delta\mu\int\frac{d^{d}k}{(2\pi)^{d}}\left\\{f^{\prime}_{0}\left({\varepsilon}_{\bf
k}^{0}\right){\bf E}_{\omega}\times{\bm{\Omega}}_{\bf k}+\frac{\hbar\tau
f_{0}^{\prime\prime}\left({\varepsilon}^{0}_{\bf
k}\right)}{1-i\omega\tau}\left({\bf E}_{\omega}\cdot{\bf v}^{0}_{\bf
k}\right){\bf v}^{0}_{\bf k}\right\\}.$ (S20)

Again, the first term is the correction to the anomalous Hall effect, which is
antisymmetric. The second term is the correction to the Drude conductivity: it
is symmetric and contributes to $\sigma_{ij}^{s}$. The information about the
band structure geometry is hidden in the correction $\delta\mu$.

Focusing on the case $d=2$, we have ${\bm{\Omega}}_{\bf k}=\Omega_{\bf
k}\hat{z}$ and ${\bf m}_{\bf k}=m_{\bf k}\hat{z}$. Assuming further that ${\bf
B}=B_{z}\hat{z}$ and extracting the symmetric part of the conductivity tensor,
$\sigma_{ij}^{s}=\left(\sigma_{ij}+\sigma_{ji}\right)/2$, we find

$\displaystyle\sigma_{ij}^{s}$ $\displaystyle=\sigma_{ij}^{D}+\frac{e^{2}\tau
B_{z}}{\hbar(1-i\omega\tau)}\int\frac{d^{2}k}{(2\pi)^{2}}\left\\{f_{0}^{\prime}\left(v_{i}^{0}\partial_{j}m+v_{j}^{0}\partial_{i}m+ev_{i}^{0}v_{j}^{0}\Omega\right)+\hbar
f_{0}^{\prime\prime}mv_{i}^{0}v_{j}^{0}+\frac{e\tau}{2(1-i\omega\tau)}f_{0}^{\prime}{\varepsilon}^{\alpha\beta}v_{\alpha}^{0}\partial_{\beta}\left(v_{i}^{0}v_{j}^{0}\right)\right\\}$
$\displaystyle-\frac{e^{2}\tau}{1-i\omega\tau}\delta\mu\int\frac{d^{2}k}{(2\pi)^{2}}f_{0}^{\prime\prime}v_{i}^{0}v_{j}^{0},$
(S21)

where ${\varepsilon}^{\alpha\beta}$ is the antisymmetric Levi-Civita tensor
(summation over repeated indices is implied) and $\sigma_{ij}^{D}$ is the
zero-field Drude conductivity:

$\sigma_{ij}^{D}=-\frac{e^{2}\tau}{1-i\omega\tau}\int\frac{d^{2}k}{(2\pi)^{2}}f_{0}^{\prime}v_{i}^{0}v_{j}^{0}.$
(S22)

Here we defined $\partial_{i}\equiv\partial/\partial k_{i}$ and suppressed
index ${\bf k}$ and argument ${\varepsilon}^{0}_{\bf k}$ in $f_{0}^{\prime}$,
$f_{0}^{\prime\prime}$ for brevity.

The second-order term in $\tau/(1-i\omega\tau)$ originating from the Lorentz
force can be rewritten as the full derivative, hence, it vanishes after the
integration over the whole Brillouin zone:

$\hbar\int\frac{d^{2}k}{(2\pi)^{2}}f_{0}^{\prime}{\varepsilon}^{\alpha\beta}v_{\alpha}^{0}\partial_{\beta}\left(v_{i}^{0}v_{j}^{0}\right)=\int\frac{d^{2}k}{(2\pi)^{2}}{\varepsilon}^{\alpha\beta}\left(\partial_{\alpha}f_{0}\right)\partial_{\beta}\left(v_{i}^{0}v_{j}^{0}\right)=\int\frac{d^{2}k}{(2\pi)^{2}}{\varepsilon}^{\alpha\beta}\partial_{\alpha}\left[f_{0}\partial_{\beta}\left(v_{i}^{0}v_{j}^{0}\right)\right]=0,$
(S23)

where we used the identity
${\varepsilon}^{\alpha\beta}\partial_{\alpha}\partial_{\beta}(v_{i}^{0}v_{j}^{0})=0$
and assumed no singularities in the velocity components.

Next, the components involving orbital magnetic moment $m_{\bf k}$ can be
regrouped using the identity

$\hbar
f^{\prime\prime}_{0}mv^{0}_{i}v^{0}_{j}=\frac{1}{2}\left[(\partial_{i}f^{\prime}_{0})mv^{0}_{j}+(\partial_{j}f^{\prime}_{0})mv^{0}_{i}\right]=\frac{1}{2}\left[\partial_{i}(f^{\prime}_{0}mv^{0}_{j})+\partial_{j}(f^{\prime}_{0}mv^{0}_{i})\right]-\frac{f^{\prime}_{0}}{2}\left(v^{0}_{j}\partial_{i}m+v^{0}_{i}\partial_{j}m\right)-f^{\prime}_{0}mv^{0}_{ij},$
(S24)

where we defined $v^{0}_{ij}=\partial_{i}v_{j}^{0}=\partial_{j}v_{i}^{0}$. The
first two terms in this expression are full derivatives which vanish upon
integration over momenta. Collecting all the non-vanishing contributions in
Eq. (C), we obtain

$\sigma^{s}_{ij}=\sigma^{D}_{ij}+\frac{e^{2}\tau
B_{z}}{\hbar(1-i\omega\tau)}\int\frac{d^{2}k}{(2\pi)^{2}}f^{\prime}_{0}\left\\{\frac{1}{2}(v^{0}_{i}\partial_{j}m_{\bf
k}+v^{0}_{j}\partial_{i}m_{\bf k})-m_{\bf
k}v^{0}_{ij}+ev^{0}_{i}v^{0}_{j}\Omega_{\bf
k}\right\\}+\frac{e^{2}\tau\delta\mu}{\hbar\left(1-i\omega\tau\right)}\int\frac{d^{2}k}{(2\pi)^{2}}f^{\prime}_{0}v^{0}_{ij},$
(S25)

where we also integrated by parts in the last term.

Equation (S25) is the most general expression for the symmetric part of the
conductivity tensor in two-dimensional systems with the magnetic field along
the $z-$axis. If mirror symmetry $m_{y}$ is additionally present, the term
with $\delta\mu$ and $\sigma_{xy}^{D}$ do not contribute to $\sigma^{s}_{xy}$,
and, in the limit $\omega\to 0$, we reproduce Eq. (LABEL:eq:Sigma) from the
main text.

###  C.1 Symmetry allowed tight-binding model

Denote the $d$ orbital creation operators as
$d^{\dagger}=(\hat{d}^{\dagger}_{xz},\hat{d}^{\dagger}_{yz},\hat{d}^{\dagger}_{xy},\hat{d}^{\dagger}_{x^{2}-y^{2}},\hat{d}^{\dagger}_{z^{2}})$
(where we have suppressed the spin component
$\hat{d}^{\dagger}_{\ell}=(\hat{d}^{\dagger}_{\ell,\uparrow},\hat{d}^{\dagger}_{\ell,\downarrow})$)
which correspond to the following orbital basis

$\displaystyle|d_{xz}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|2,-1\rangle-|2,1\rangle),$ (S26a)
$\displaystyle|d_{yz}\rangle$
$\displaystyle=\frac{i}{\sqrt{2}}(|2,-1\rangle+|2,1\rangle),$ (S26b)
$\displaystyle|d_{xy}\rangle$
$\displaystyle=\frac{i}{\sqrt{2}}(|2,-2\rangle-|2,2\rangle),$ (S26c)
$\displaystyle|d_{x^{2}-y^{2}}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|2,-2\rangle+|2,2\rangle),$ (S26d)
$\displaystyle|d_{z^{2}}\rangle$ $\displaystyle=|2,0\rangle,$ (S26e)

the expressions on the right is written in the $|L,L_{z}\rangle$ basis. The
onsite trigonal crystal field Hamiltonian has the form

$\hat{H}_{CF}=d^{\dagger}\left(\begin{array}[]{ccccc}\frac{2}{3}&0&0&\frac{\sqrt{2}}{3}&0\\\
0&\frac{2}{3}&-\frac{\sqrt{2}}{3}&0&0\\\
0&-\frac{\sqrt{2}}{3}&\frac{1}{3}&0&0\\\
\frac{\sqrt{2}}{3}&0&0&\frac{1}{3}&0\\\ 0&0&0&0&0\\\ \end{array}\right)d,$
(S27)

whereas the onsite spin–orbit coupling term has the form

$\hat{H}_{\text{SO}}=d^{\dagger}\mathbf{\hat{L}}\cdot\mathbf{\hat{S}}d,$ (S28)

where

$\hat{L}_{x}=\left(\begin{array}[]{ccccc}0&0&i&0&0\\\ 0&0&0&-i&-i\sqrt{3}\\\
-i&0&0&0&0\\\ 0&i&0&0&0\\\ 0&i\sqrt{3}&0&0&0\\\
\end{array}\right),\qquad\hat{L}_{y}=\left(\begin{array}[]{ccccc}0&0&0&-i&i\sqrt{3}\\\
0&0&-i&0&0\\\ 0&i&0&0&0\\\ i&0&0&0&0\\\ -i\sqrt{3}&0&0&0&0\\\
\end{array}\right),\qquad\hat{L}_{z}=\left(\begin{array}[]{ccccc}0&-i&0&0&0\\\
i&0&0&0&0\\\ 0&0&0&2i&0\\\ 0&0&-2i&0&0\\\ 0&0&0&0&0\\\ \end{array}\right),$
(S29)

and $\mathbf{\hat{S}}$ are written in terms of the standard Pauli matrix
operators. We have set $\hbar=1$.

The triangular lattice is spanned by the lattice vectors
$\mathbf{\hat{e}}_{1}=\frac{\sqrt{3}}{2}\hat{x}+\frac{1}{2}\hat{y}$ and
$\mathbf{\hat{e}}_{2}=-\frac{\sqrt{3}}{2}\hat{x}+\frac{1}{2}\hat{y}$, where we
have set the lattice constant to unity. We provide the matrix for hopping
along the bond $\mathbf{\hat{e}}_{1}+\mathbf{\hat{e}}_{2}$ (written in the
basis of $d$ orbitals given above):

$[t]_{\mathbf{\hat{e}}_{1}+\mathbf{\hat{e}}_{2}}=\left(\begin{array}[]{ccccc}V_{dd\delta}&0&0&0&0\\\
0&V_{dd\pi}&0&0&0\\\ 0&0&V_{dd\pi}&0&0\\\
0&0&0&\frac{1}{4}(V_{dd\delta}+3V_{dd\sigma})&\frac{\sqrt{3}}{4}(-V_{dd\delta}+V_{dd\sigma})\\\
0&0&0&\frac{\sqrt{3}}{4}(-V_{dd\delta}+V_{dd\sigma})&\frac{1}{4}(3V_{dd\delta}+V_{dd\sigma})\\\
\end{array}\right),$ (S30)

and the other nearest neighbor hoppings can be obtained through translation or
threefold rotation (by applying the matrix
$\hat{R}_{z}^{\text{orb}}\left(\frac{2\pi}{3}\right)\equiv
e^{-i\frac{2\pi}{3}\hat{L}_{z}}$). Here $V_{dd\sigma}$, $V_{dd\pi}$ and
$V_{dd\delta}$ are the standard Slater–Koster parameters [10]. We used the
following values in Fig. 2 of the main text:

$\lambda_{\text{SO}}=1.2,\quad V_{dd\sigma}=-1.0,\quad V_{dd\delta}=0.4,\quad
V_{dd\delta}=0,\quad\Delta_{CF}=20.0,$ (S31)

and a magnetization of norm $|\mathbf{M}|=1.4$.

##  Appendix D Landauer–Büttiker implementation

The Landauer–Büttiker formalism is employed to calculate $\sigma_{xy}^{s}$
from the tight-binding model Eq. (2) in the main text. To that end, we use the
Kwant package [24] to implement our model in the L-shaped device shown in
Fig.3(a). The Hamiltonian of the semi-infinite leads is also Eq. (2). The
parameters are kept the same as those used for calculating $\sigma_{xy}^{s}$
in Fig. 2(d). The width of the device channel and of the leads is taken to be
100 lattice sites, while the length of each arm is 700 sites, resulting in a
total device length of $\sim 1400$ sites (see next section for the reasoning
behind these values).

Components of the conductivity tensor are not directly obtained in a
Landauer–Büttiker calculation, but rather they are extracted from inverting
the resistivity tensor

$\begin{pmatrix}\sigma_{xx}&\sigma_{xy}\\\
\sigma_{yx}&\sigma_{yy}\end{pmatrix}=\frac{1}{\rho_{xx}\rho_{yy}-\rho_{xy}\rho_{yx}}\begin{pmatrix}\rho_{yy}&-\rho_{xy}\\\
-\rho_{yx}&\rho_{xx}\end{pmatrix}$ (S32)

whose components can be computed in the following way. The main outputs from
Landauer–Büttiker are the transmission probabilities $T_{ij}$ between each
pair of leads $i,j$ coupled to a scattering region. Then, imposing the current
conditions at each lead (source, drain or floating), the voltages can be
obtained with the Büttiker formula
$I_{i}=\frac{e}{h}\sum_{j}T_{ji}V_{i}-T_{ij}V_{j}$ [25, 26]. With the voltage
at each lead and a current $I=I_{\text{source}}=-I_{\text{drain}}$, a
resistance measurement is given by $R_{ij}=(V_{i}-V_{j})/I$ and hence the
resistivity becomes $\rho_{ij}=\frac{w}{d_{ij}}R_{ij}$, with $w$ and $d_{ij}$
being the cross section or width of the device and distance between leads $i$
and $j$, respectively. To minimize the variances produced by the random
disorder (see next section), we perform an average over 5 disorder
configurations and calculate all components of the resistivity tensor in the
L-shaped device. In this way, and taking lead 1 (10) as the source (drain),
the resistivities read $\rho_{xx}=\rho_{68}$, $\rho_{yy}=\rho_{24}$,
$\rho_{xy}=\rho_{32}$ and $\rho_{yx}=\rho_{89}$.

###  D.1 Diffusive transport in Landauer–Büttiker calculations

Most of mesoscopic transport experiments occur in the diffusive transport
regime (as opposed to ballistic or localized transport). In this way, to have
diffusive transport in a Landauer–Büttiker calculation, we include random
Anderson disorder potential
$\sum_{\mathbf{r}}U_{\mathbf{r}}d^{\dagger}_{\mathbf{r},\ell,\sigma}d_{\mathbf{r},\ell,\sigma}$
at each site $\mathbf{r}$ uniformly distributed in the interval $[-U/2,U/2]$.
Because of the quasi-one-dimensional nature of our finite device, if $U$ is
too large, electrons will localize, whereas if $U$ is too small, most of the
transport could be ballistic inside our device. Therefore, the choice of $U$
impacts on the choice of the device length and width. To elucidate what is a
good value of $U$ so that diffusive transport occurs throughout the entire
device, we perform the following procedure [27, 28]. We make a two-terminal
device with the same width as our device of interest (Fig. 3(a)) and calculate
the two-terminal conductance, $G_{2T}$, for different channel lengths $L$ and
fixed $U$. Then, we use the relation $G_{2T}=\sigma\frac{w}{L}$ to obtain the
longitudinal conductivity, and plot $\sigma$ as a function of $L$. For an
extensive enough range of $L$ values, $\sigma$ will first increase to then
remain constant and finally decrease. These three behaviors correspond to
ballistic, diffusive and localized transport regimes. Increasing the width
will help maximize the range of lengths for which transport is diffusive, and
resemble more a mesoscopic two-dimensional system. However, that comes at a
higher computational time and memory demand. We found a width of $w=100a$ ($a$
the lattice constant) to be big enough for our device, and computationally
affordable. Finally, by looking at the scaling of $\sigma$ with $L$, we choose
a value of $U$ that allows for a long enough device to probe the physics we
are interested in but still within computational reach. To illustrate this, we
plot in Fig. S1 the longitudinal conductivity for two different strengths of
disorder and for devices oriented along the $x$ and $y$ direction. The Fermi
level is taken to be the same as the one used for Fig. 3(b) in the main text,
i.e. -1.95, and $M_{z}/M_{y}=0.32$ is used. From this plot, it is clear that
$\sigma_{xx}\neq\sigma_{yy}$, which is expected due to the broken $c_{3}$
symmetry. Most importantly, one can see that for the strongest disorder case,
the onset of localization appears rather quickly, around $L\sim 500a$. This is
not a good value of $U$, since we cannot build the L-shape device we want
within $500a$ if the width of the device and leads is $100a$. In contrast, for
$U=0.3V_{dd\sigma}$, the transport is diffusive for both directions at $L\sim
266a$ (see vertical dashed line) and remains constant for long enough lengths.
Hence, we choose the inter-lead distance to be at least $L=266a$. This,
together with the width $w=100a$, results in a total length of the L-shaped
device of $\sim 1400a$, which still falls within diffusive transport (vertical
dotted line in Fig. S1).

Finally, one also can obtain meaningful quantities such as the mean free path.
In the diffusive regime, $G_{2T}=\sigma w/L$, while from quantum transport it
is known that $G_{2T}=Nl_{e}/L$ [26, 29], where $N$ is the number of
propagating modes and $l_{e}$ the mean free path. Consequently, $\sigma
w=Nl_{e}$. Given the leads are described by the same Hamiltonian as the
scattering region, $N$ can be extracted from their band structure, whereas
$\sigma$ is readily obtained from Fig. S1. This gives us a mean free path of
$l_{e}^{x}=31a$ and $l_{e}^{y}=17a$ for the $x$ and $y$ directions,
respectively.

Figure S1: Longitudinal conductivity extracted from the two-terminal
conductance for two strengths of Anderson disorder and for a device oriented
along $x$ (black) and along $y$ (red). 10 disorder averages have been carrid
out. The diffusive regime exists for lengths in which the conductivity remains
constant. The vertical dashed (dotted) line denotes the chosen inter-lead
distance (total length) of the L-shaped device. Importantly, lengths within
these two vertical lines fall within diffusive transport.

##  Appendix E Density functional theory implementation

DFT calculations were carried out using the Vienna ab initio simulation
package (VASP) [30, 31, 32]. The Perdew-Burke-Ernzerhof (PBE) generalized
gradient approximation (GGA) was employed for the exchange-correlation
potential [33] and the wave functions and pseudopotentials were generated
within the projector-augmented wave (PAW) method [34]. In our calculations,
the monolayer of MnPSe3 was modeled using a periodic slab geometry in a
supercell setup. To avoid spurious interactions with periodic images, a vacuum
region of 28Å in the out-of-plane direction was used. We employed a plane-wave
basis with an energy cutoff of $520$ eV and used a dense k-point grid sampling
of $13\times 13\times 1$ in the BZ according to the Monkhorst-Pack scheme
[35]. Both the lattice constant and positions of all atoms were relaxed until
the force was less than $2$ meV/Å, while keeping the volume fixed. The
optimized in-plane lattice constant was found to be $a=b=6.363$ Å, consistent
with experimental value ($a=b=6.394$ Å) [36]. For the electronic self-
consistent loop, the total energy convergence criterion was set to $10^{-7}$eV
by using GGA (without U). This choice yielded consistent results when compared
to calculations with different values of U, as reported in [17].

We considered the collinear ferromagnetic (FM) order of the magnetic moment on
Mn with spin-orbit coupling (SOC) with its orientation described as
$\mathbf{\hat{m}}=(0,\sin\Theta,\cos\Theta)$, where $\Theta$ varied between
$0^{\circ}$, $+45^{\circ}$, and $-45^{\circ}$.

The symmetric and anti-symmetric components of the off-diagonal conductivity
tensor were analyzed through explicit calculation of the Fermi velocity and
Berry curvature associated with the given Fermi level (chemical potential). We
used the Wannier90 code [37, 38, 39] to obtain the Wannier tight-binding
Hamiltonian based on maximally localized Wannier functions (MLWFs). To perform
the Wannierization procedure, we carried out DFT calculations using a dense
k-point grid of $18\times 18\times 1$. The choice of basis set for the
Wannierization included $p_{x,y,z}$ orbitals for the six Se atoms and
$d_{xz,yz}$ orbitals for two Mn atoms present in the supercell [18]. This
selection allowed us to construct a Wannier tight-binding Hamiltonian of rank
$44$. Once we obtained the Wannier tight-binding Hamiltonian, we explicitly
calculated the Fermi velocity and the Berry curvature [40, 41] from this
Hamiltonian.

##  Appendix F Supporting DFT data for MnPSe3

In this section, we present DFT calculations to complement the calculation of
$\sigma_{xy}^{s}$ showed in the main text for MnPSe3.

We argued that in order to observe linear magneto-conductivity, the states at
the Fermi level should possess a finite orbital angular momentum even in the
presence of a strong crystal field, and demonstrated that with a tight-binding
made from $t_{2g}$ orbitals. Therefore, we plot in Fig. S2 the orbital-
resolved density of states with magnetization orientation at $+45\degree$ with
respect to the $z$ axis in the $yz$ plane. As expected from the nonzero
$\sigma_{xy}^{s}$ numerically computed in Fig. 4 in the main text, MnPSe3
shows considerable weight of $t_{2g}$ orbitals ($d_{xy}$, $d_{xz}$, $d_{yz}$)
near the Fermi level, thus validating our understanding of the importance of
having a finite orbital angular momentum.

Figure S2: Projected density of states of Mn and Se atoms near the Fermi level
(set to 0 eV), for MnPSe3 with magnetization orientation along $+45\degree$
with respect to the $z$ axis in the $yz$ plane.

Another property that we argued is tied to the appearance of $\sigma_{xy}^{s}$
is the rotation of the Fermi surface with respect to the principal axis. Fig.
S3 displays the Fermi surface at an energy -110 meV with respect to the
valence band maximum for magnetizations $\pm 45\degree$. The Fermi surfaces
have pockets around the Brillouin zone corners (K points) and round the
$\Gamma$ point. In the K points, the Fermi surface rotation between the two
magnetization configurations is not very clear, but each Fermi surfaces
clearly shows the breaking of both $c_{3}$ and $m_{y}$ symmetries. On the
other hand, the Fermi surfaces at the $\Gamma$ point reveal opposite rotation
in the $xy$ plane. This result, together with Fig. S2 and Fig. 4 in the main
text, demonstrate the correlation between orbital angular momentum, Fermi
surface rotation and nonzero $\sigma_{xy}^{s}$ in realistic materials.

Figure S3: Fermi surface at energy $-110$ meV with respect to the valence band
maximum for MnPSe3 with magnetization orientation along $+45\degree$ (red) and
$-45\degree$ (blue). Left (right) panel show the Fermi surface in the whole
Brillouin zone (around the $\Gamma$ point).