# A Note on Walking Versus Waiting

Anthony B. Morton

(February 2008)

To what extent is a traveller (called Justin, say) better off to wait for a
bus rather than just start walking—particularly when the bus headway is of a
similar order of magnitude to the walking time and Justin does not know the
precise arrival time of the bus? The recent analysis by Chen et al [1] goes
some way toward answering this question; however, there are additional
valuable insights that can be gained from their approach.

In essence this is a decision-theoretic problem, where the aim is to compare
various strategies for getting to one’s destination. Following Chen et al [1],
it is supposed that the comparison hinges solely on the expectation of overall
travel time, and there is no additional penalty attached to walking. We may
list some of the available strategies as follows:

  * •

Strategy A: Wait indefinitely until a bus arrives.

  * •

Strategy B: Begin walking immediately and do not wait for a bus.

  * •

Strategy C: Wait for a predetermined time interval $T_{W}$, and if no bus has
arrived in this time then walk.

(Naturally, strategies A and B can themselves be thought of as the extreme
limiting cases of strategy C as $T_{W}\rightarrow\infty$ and $T_{W}\rightarrow
0$ respectively. However, we distinguish them for the sake of discussion.)

Following the nomenclature of [1], let $d$ denote the journey distance,
$v_{w}$ Justin’s average walking speed and $v_{b}$ the average speed of a bus.
The overall travel time is a random variable and will be denoted $T_{t}$.

As indicated in [1], if the decision is purely between strategies A and B, the
problem has an easy, one might say trivial, solution: if the expected arrival
time for the bus is $T$ minutes in the future, Justin should wait if $T\leq
d/v_{w}-d/v_{b}$ and walk otherwise. The more interesting questions are (a)
could there exist values $T_{W}$ for which strategy C is superior to either A
or B; and (b) does the presence of intermediate stops affect the conclusion?
The purpose of this note is to show, by direct analysis for (almost) arbitrary
arrival probabilities, that the answer to (a) is indeed no in most practical
cases, but that the answer to (b) is not quite as clear-cut as the authors in
[1] argue.

## The Case of No Intermediate Stops

The following expression is provided in [1] for Justin’s expected travel time
under strategy C, where $p(t)$ is the density function for the probability a
bus arrives $t$ minutes from now:

$E[T_{t}]=\int_{0}^{T_{W}}\left(\frac{d}{v_{b}}+\tau\right)p(\tau)\,\mathrm{d}\tau+\left(1-\int_{0}^{T_{W}}p(\tau)\,\mathrm{d}\tau\right)\left(\frac{d}{v_{w}}+T_{W}\right).$
(1)

In [1], this expression is used to determine the waiting time $T_{W}$ such
that $E[T_{t}]$ is equal to the time $d/v_{w}$ required under strategy B.
However, it is also of interest to optimise (1) directly. The derivative of
(1) with respect to $T_{W}$ is

$\frac{\partial}{\partial
T_{W}}E[T_{t}]=\left(1-\int_{0}^{T_{W}}p(\tau)\,\mathrm{d}\tau\right)-\left(\frac{d}{v_{w}}-\frac{d}{v_{b}}\right)p(T_{W}).$
(2)

The first term in parentheses is the complementary c.d.f. of the arrival time
for the bus; that is, the probability that the bus has _not_ arrived by time
$T_{W}$. Denote this function by $R(t)$:

$R(t)=1-\int_{0}^{t}p(\tau)\,\mathrm{d}\tau.$ (3)

Also for brevity, define $T_{\delta}$ as the difference between the travel
time on foot and the travel time by bus:

$T_{\delta}=\frac{d}{v_{w}}-\frac{d}{v_{b}}.$ (4)

Then the derivative (2) may be expressed as

$\frac{\partial}{\partial T_{W}}E[T_{t}]=R(T_{W})-T_{\delta}\,p(T_{W}).$ (5)

Zeros of (5) occur at points where

$\frac{1}{T_{\delta}}=\frac{p(T_{W})}{R(T_{W})}=\lambda(T_{W}).$ (6)

In reliability theory where one is concerned with failure-time distributions,
the function $\lambda(t)=p(t)/R(t)$ is known as the _hazard rate_ : the
density for the probability of failure given a component has not failed up to
time $t$ [2]. But $\lambda(t)$ also has a natural interpretation for arrival
times, where it may be termed the _appearance rate_ for the distribution
$p(t)$.

By definition $\lambda(t)$ is nonnegative, but depending on the underlying
distribution it may behave in a variety of ways: it may increase, decrease,
attain local maxima or minima, or be constant. More precisely, for any
$T\in\mathbb{R}_{+}\cup\\{\infty\\}$ any nonnegative real measurable function
on $(0,T)$ can be the appearance rate for a possible arrival time distribution
$p(t)$; if $T$ is finite then take $p(t)=R(t)=0$ for $t\geq T$. An increasing
$\lambda(t)$ is characteristic of many waiting scenarios; it represents the
case where the conditional probability increases with time spent waiting. By
contrast, a constant $\lambda(t)$ characterises the exponential arrival time
distribution, where time spent waiting does not affect the conditional
likelihood of a bus arriving.

Equation (6) indicates that $E[T_{t}]$ as a function of $T_{W}$ is stationary
at any point where the appearance rate $\lambda(T_{W})$ equals the reciprocal
of $T_{\delta}$. To determine whether this is a minimum or a maximum, one
calculates the second derivative (assuming for simplicity that $p(t)$ is
dfferentiable)

$\frac{\partial^{2}}{\partial
T_{W}^{2}}E[T_{t}]=-p(T_{W})-T_{\delta}\,p^{\prime}(T_{W})$ (7)

and inspects the sign of this quantity. Since $p(t)$ and $T_{\delta}$ are both
positive (and $p(T_{W})$ is nonzero by (6)) this is equivalent to the sign of

$-\left(\frac{1}{T_{\delta}}+\frac{p^{\prime}(T_{W})}{p(T_{W})}\right)=-\left(\frac{p(T_{W})}{R(T_{W})}+\frac{p^{\prime}(T_{W})}{p(T_{W})}\right)=-\frac{\lambda^{\prime}(T_{W})}{\lambda(T_{W})}=-T_{\delta}\,\lambda^{\prime}(T_{W}).$
(8)

Accordingly, any stationary point of $E[T_{t}]$ is a minimum when it coincides
with a falling appearance rate, and a maximum when it coincides with a rising
appearance rate.

It is thus established that, when faced with a typical probability
distribution for bus arrivals having a rising appearance rate, the optimal
strategy reduces to a choice between strategies A and B, and hence to a
straightforward comparision of expected arrival time with $T_{\delta}$ as
indicated above.

For example, if it is known that buses run punctually every $T$ minutes but
one does not know the actual arrival times, the uncertainty in arrival time
can be modelled as a uniform probability density $p(t)$ between $0$ and $T$,
with mean $T/2$. The corresponding appearance rate is

$\lambda(t)=\frac{1}{T-t},\qquad t\in(0,T)$ (9)

an increasing function of $t$. One may then distinguish three cases:

  1. 1.

If $T<T_{\delta}$, the expected travel time $E[T_{t}]$ has no stationary
points, and is in fact a decreasing function of wait time. In this case
strategy A is optimal: one should wait for the next bus.

  2. 2.

If $T_{\delta}<T<2T_{\delta}$, the expected travel time has one stationary
point at $T_{W}^{*}=T-T_{\delta}$, which is a maximum. The expected travel
time increases with waiting time up to $T_{W}^{*}$ and then decreases. But
since the mean bus arrival time is $T/2<T_{\delta}$, the waiting strategy A is
still preferable to an initial decision to walk.

  3. 3.

If $T>2T_{\delta}$, then expected travel time again reaches a maximum at
$T_{W}^{*}=T-T_{\delta}$. But now, it is better to follow strategy B and not
wait for a bus at all.

Perhaps the most interesting situation here is the marginal case where
$T=2T_{\delta}$: the interval between buses is twice the difference between
walking time and bus travel time. In this case it is equally good (or bad) to
wait as to walk, but having decided to wait, one should not give up and walk,
as this will result in a worse outcome (averaged over all eventualities).

Not all practically realisable probability distributions have rising
appearance rates for all $t$. Consider, for example, the situation where
Justin has arrived at the bus stop one minute after the (known) scheduled
arrival time for the bus, but knows that buses sometimes run up to five
minutes late. Under reasonable assumptions this leads to a falling
$\lambda(t)$ for the first four minutes, and if $p(0)$ is sufficiently large
and the service relatively infrequent, this will result in an optimal time
$T_{W}^{*}$ for which Justin should remain at the bus stop and then start
walking. However, if the next bus can be counted on to arrive in a time
comparable to $T_{\delta}$, it will still be advantageous to keep waiting for
the next bus, and the lazy mathematician still wins.

One may also contemplate the special case where $\lambda(t)\equiv\lambda$ is
constant, and bus arrivals are a Poisson process. The mean time between
arrivals is $T=1/\lambda$; if this is greater than $T_{\delta}$ then it is
better to walk, and if less then it is better to wait, as is intuitively
clear. More interesting is the case where $T$ happens to just equal
$T_{\delta}$: then, the derivative (5) vanishes and the expected travel time
is equal to the walking time $d/v_{w}$, _regardless_ of the time spent
waiting. What this means is, no matter how long Justin decides in advance to
wait, over a large number of journeys he can expect to spend no more or less
time than if he had decided to walk at the outset. Nonetheless, Poisson
arrivals are generally an unrealistic assumption except in some cases of very
frequent service, where it is generally better to catch the bus in any event.

## Intermediate Stops

In [1] it is argued that the conclusion is unchanged when Justin has the
additional option of walking to an intermediate stop and catching the bus
there. If, for example, one re-evaluates the decision between strategies A and
B at a distance $d_{1}$ from the start of the journey, then the walking time
is $(d-d_{1})/v_{w}$ and the bus travel time is $(d-d_{1})/v_{b}$. However,
the expected waiting time for the bus (assuming it does not pass by _en
route_) is now reduced by $T_{1}=d_{1}/v_{w}-d_{1}/v_{b}$, the time that
passes while walking less the additional time taken by the bus to reach the
next stop. The terms involving $d_{1}$ cancel, and one is left with a decision
identical to the original one. Under these circumstances, one would “rather
save energy” and act as though the intermediate stop did not exist, since the
outcome is the same. This conclusion is also intuitively evident.

The authors argue that the same reasoning applies to walk-and-wait strategies
analogous to strategy C, at least when the wait time $T_{W}$ at the next stop
is chosen to make the expected travel time equal to the walking time. However,
there appears to be a circularity involved here—if the expected travel time is
fixed as $d/v_{w}$ _a priori_ , then naturally it will be observed to be the
same whether or not Justin walks to another stop. Again, some more insight
into the problem is gained by seeking stationary points of $E[T_{t}]$ with
respect to the free variables.

Suppose strategy C is modified so that one walks a distance $d_{1}$ to another
stop, then waits for a maximum time $T_{W}$. One may now distinguish two
cases, depending on whether or not a bus passes by _en route_. Denote this
event by $M$ (‘miss the bus’) and its non-occurrence by $\bar{M}$; the
relevant probabilities are

$\mathrm{Pr}\left(M\right)=\int_{0}^{T_{1}}p(\tau)\,\mathrm{d}\tau,\qquad\mathrm{Pr}\left(\bar{M}\right)=1-\mathrm{Pr}\left(M\right)=R(T_{1}),\qquad
T_{1}=\frac{d_{1}}{v_{w}}-\frac{d_{1}}{v_{b}}.$ (10)

If no bus passes by _en route_ , then the expected travel time is similar to
(1), but with a time offset:

$\displaystyle E[T_{t}|\bar{M}]=\frac{d_{1}}{v_{w}}$
$\displaystyle+\int_{0}^{T_{W}}\left(\frac{d-d_{1}}{v_{b}}+\tau\right)\frac{p(\tau+T_{1})}{\mathrm{Pr}\left(\bar{M}\right)}\,\mathrm{d}\tau$
$\displaystyle+\left(1-\int_{0}^{T_{W}}\frac{p(\tau+T_{1})}{\mathrm{Pr}\left(\bar{M}\right)}\,\mathrm{d}\tau\right)\left(\frac{d-d_{1}}{v_{w}}+T_{W}\right).$
(11)

Even if the bus does pass by, it is possible that Justin may catch it anyway,
if he is vigilant and there is a stop close enough, or if the bus is delayed,
or if he takes a short cut. Let the probability of catching a bus if one turns
up on the way be $P_{C}$. On the other hand, if Justin misses the bus it may
be assumed there is no advantage in waiting for the next one (else he would
have preferred strategy A at the outset). Thus the expected travel time
conditional on $M$ is

$E[T_{t}|M]=P_{C}\int_{0}^{T_{1}}\left(\tau+\frac{d}{v_{b}}\right)\frac{p(\tau)}{\mathrm{Pr}\left(M\right)}\,\mathrm{d}\tau+(1-P_{C})\frac{d}{v_{w}}.$
(12)

(Note that within the integral, $\tau$ is the time at which the caught bus
reaches the _starting_ point, rather than Justin’s current position, since
$p(\tau)$ is the p.d.f. of bus arrivals at a _fixed_ point on the route. This
is a subtle point and easily overlooked. In terms of $\tau$, Justin’s overall
travel time if he catches the bus is $\tau$ plus the time taken by the bus to
cover the entire journey, regardless of where Justin is when he catches it.)

The overall expected travel time is

$E[T_{t}]=\mathrm{Pr}\left(M\right)E[T_{t}|M]+\mathrm{Pr}\left(\bar{M}\right)E[T_{t}|\bar{M}],$
(13)

where $E[T_{t}|M]$ and $E[T_{t}|\bar{M}]$ are given by (12) and (11)
respectively. Again, one can differentiate to find optimal values of wait time
$T_{W}$ at the more distant stop. (This is made easier by noting that
$E[T_{t}|M]$ does not vary with $T_{W}$.) One finds that

$\displaystyle\frac{\partial}{\partial T_{W}}E[T_{t}]$
$\displaystyle=R(T_{W}+T_{1})-T_{\delta 1}p(T_{W}+T_{1})$ (14)
$\displaystyle\mbox{and}\qquad\frac{\partial^{2}}{\partial T_{W}^{2}}E[T_{t}]$
$\displaystyle=-p(T_{W}+T_{1})-T_{\delta 1}p^{\prime}(T_{W}+T_{1}),$ (15)

where

$T_{\delta
1}=(d-d_{1})\left(\frac{1}{v_{w}}-\frac{1}{v_{b}}\right)=T_{\delta}-T_{1}$
(16)

is the difference between travel time on foot and by bus starting from the
more distant stop. These are exactly equivalent to (5) and (7), with
$T_{W}+T_{1}$ in place of $T_{W}$ and $T_{\delta 1}$ in place of $T_{\delta}$.
Accordingly, the analysis of the previous section is essentially unchanged, as
is the conclusion: that in most waiting scenarios the optimal value of $T_{W}$
is either zero or arbitrarily large.

But now there is an additional variable in the problem, because Justin will
generally have a number of stops to choose from between the starting point and
the destination. For the sake of simplicity, suppose the stop spacing is small
enough that $d_{1}$ can be regarded as a continuous variable. Then we can try
and optimise with respect to $d_{1}$, obtaining after a little work

$\frac{\partial}{\partial
d_{1}}E[T_{t}]=q^{2}(d-d_{1})\left[(1-P_{C})p(T_{1})-p(T_{1}+T_{W})\right]$
(17)

where $q=1/v_{w}-1/v_{b}$ is a constant.

The above discussion suggests directing attention to the cases $T_{W}=0$ and
$T_{W}\rightarrow\infty$. If $T_{W}=0$, then (17) vanishes unless $P_{C}>0$;
that is, when there is a nonzero chance of catching a bus while walking to the
next stop. Assuming then that $P_{C}>0$ and $T_{W}=0$, (17) is found to be
zero when $p(T_{1})=0$ and negative when $p(T_{1})>0$. (Recall that $T_{1}$ is
proportional to $d_{1}$.) The result is intuitively evident: as long as there
is a chance of catching a bus on the way (however small), walking a distance
$d_{1}$ to an intermediate stop always reduces the expected travel time
$E[T_{t}]$ relative to strategy B, where one does not attempt to catch a bus.
The actual reduction can be quantified by integrating (17) with respect to
$d_{1}$. Due to the factor $(d-d_{1})$ in (17), the marginal reduction in
travel time is greatest at the start of the journey, all other things being
equal.

Now consider the case $T_{W}\rightarrow\infty$. One may presume that there is
a limit to the time one must wait for a bus, so that $p(T_{1}+T_{W})$ becomes
zero for $T_{W}$ sufficiently large. It is also fair to assume that $P_{C}<1$.
Then it follows that for $T_{W}$ sufficiently large, (17) is zero when
$p(T_{1})=0$, but is _positive_ when $p(T_{1})>0$. It follows that walking to
an intermediate stop only to wait indefinitely at that stop yields a worse
expected outcome than waiting at the starting point, consistent with common
sense. Again, (17) can be used to judge how much one is worse off due to the
chance of missing the bus on the way.

The more interesting case to consider, of course, is a direct comparison
between strategy A (waiting indefinitely at the starting point) and a strategy
of walking but trying to catch a bus on the way. From above, the best possible
strategy of the latter variety is the case $T_{W}=0$ and $d_{1}=d$: a little
manipulation of (13) yields the result

$E_{*}[T_{t}]=\frac{d}{v_{w}}-P_{C}\int_{0}^{T_{\delta}}(T_{\delta}-\tau)p(\tau)\,\mathrm{d}\tau.$
(18)

Under strategy A, of course, the expected travel time is just the bus travel
time, plus the expected arrival time for the bus:

$E_{A}[T_{t}]=\frac{d}{v_{b}}+\int_{0}^{\infty}\tau p(\tau)\,\mathrm{d}\tau.$
(19)

The difference in expected travel time is then

$E_{A}[T_{t}]-E_{*}[T_{t}]=P_{C}T_{\delta}\left(1-R(T_{\delta})\right)+(1-P_{C})\int_{0}^{T_{\delta}}\tau
p(\tau)\,\mathrm{d}\tau+\int_{T_{\delta}}^{\infty}\tau
p(\tau)\,\mathrm{d}\tau-T_{\delta}.$ (20)

If this quantity is positive, it represents the expected time saving if one
walks instead of waiting. (It will be seen that all terms are in fact
nonnegative apart from the last one, $T_{\delta}$; which, however, is
relatively large.)

For example, let $p(t)$ be the uniform distribution based on a bus headway
$T$. The mean arrival time is $T/2$ and hence $E_{A}[T_{t}]=d/v_{b}+T/2$. If
$T>T_{\delta}$, then

$E_{*}[T_{t}]=\frac{d}{v_{w}}-P_{C}\frac{T_{\delta}^{2}}{2T}$ (21)

and this improves on $E_{A}[T_{t}]$ if

$P_{C}>2\frac{T}{T_{\delta}}-\left(\frac{T}{T_{\delta}}\right)^{2}.$ (22)

As $T/T_{\delta}$ increases from 1 to 2, the minimum probability $P_{C}$
decreases from 1 to zero; recall that when $T/T_{\delta}>2$ strategy A is
worse than strategy B, which is equivalent to the above strategy with
$P_{C}=0$.

If $T<T_{\delta}$, then

$E_{*}[T_{t}]=\frac{d}{v_{w}}-P_{C}\left(T_{\delta}-\frac{T}{2}\right)$ (23)

and the condition for this to improve on $E_{A}[T_{t}]$ reduces to $P_{C}>1$,
a contradiction. Accordingly, strategy A is still the best choice for uniform
arrival probabilities with headway $T<T_{\delta}$.

## Conclusion

Closer analysis of the model and results in [1] for a traveller at a bus stop
confirms that the lazy mathematician does indeed win in many cases. The exact
criteria for when it is better to walk can be formulated in terms of an
appearance rate function, and if one disregards the existence of intermediate
stops, an arrival probability with a rising appearance rate is sufficient to
make waiting the best strategy.

It has also been shown, however, that allowing for the possibility of catching
the bus at an intermediate stop can restore the walking strategy to
optimality. In the end, the decision whether to walk or wait is not always
clear-cut, and relies on one’s expectations of being able to catch a bus on
the run, as well as of a bus turning up in the first place.

In the particular case where one knows the (uniform) headway but not the exact
timetable, it has been clearly established that one should wait if the headway
is less than the walking time (less bus travel time), and should walk if the
headway is more than twice this much. This leaves a substantial window where
it may be better to wait or to walk, depending on one’s confidence in being
able to catch up to a passing bus.

## References

  * [1] J.G. Chen, S.D. Kominers, and R.W. Sinnott.  Walk versus wait: The lazy mathematician wins.  arXiv.org Mathematics, January 2008.  http://arxiv.org/abs/0801.0297.
  * [2] R. Billinton and R.N. Allan.  Reliability Evaluation of Engineering Systems.  Plenum, second edition, 1992.


††thanks: Deceased.

# Ultrafast reorientation of the Néel vector in antiferromagnetic Dirac
semimetals

Atsushi Ono  Sumio Ishihara  Department of Physics, Tohoku University, Sendai
980-8578, Japan

###### Abstract

Antiferromagnets exhibit distinctive characteristics such as ultrafast
dynamics and robustness against perturbative fields, thereby attracting
considerable interest in fundamental physics and technological applications.
Recently, it was revealed that the Néel vector can be switched by a current-
induced staggered (Néel) spin-orbit torque in antiferromagnets with the
parity-time symmetry, and furthermore, a nonsymmorphic symmetry enables the
control of Dirac fermions. However, the real-time dynamics of the magnetic and
electronic structures remain largely unexplored. Here, we propose a theory of
the ultrafast dynamics in antiferromagnetic Dirac semimetals and show that the
Néel vector is rotated in the picosecond timescale by the terahertz-pulse-
induced Néel spin-orbit torque and other torques originating from magnetic
anisotropies. This reorientation accompanies the modulation of the mass of
Dirac fermions and can be observed in real time by the magneto-optical
effects. Our results provide a theoretical basis for emerging ultrafast
antiferromagnetic spintronics combined with the topological aspects of
materials.

## Introduction

The control and detection of antiferromagnetic (AFM) orders have been a
challenging problem because of the absence of net magnetizations. Since the
energy scale of antiferromagnets lies in the terahertz range, which is beyond
the scope of conventional electronics, optical pulses are suitable for
controlling AFM orders [1, 2, 3] through magnon excitations [4, 5, 6, 7],
inverse magneto-optical effects [8, 9, 10], magnetic anisotropy [11, 12, 13,
14], photoinduced phase transitions [15, 16, 17, 18], and so forth [19, 20,
21, 22, 23]. For detecting AFM orders, the magneto-optical effects are
frequently utilized for collinear and canted antiferromagnets [24, 25, 26, 27,
28, 29, 30]. The X-ray magnetic linear dichroism with photoemission electron
microscopy has succeeded in imaging the AFM domain structure [31, 32, 33, 34].
While these optical and X-ray measurements have the potential for time-
resolved observations of magnetic dynamics [1, 35, 36, 37, 38], their
application to the AFM orders remains difficult and under development.

Several years ago, another mechanism that modulates AFM orders via electric
currents was proposed and demonstrated [39, 40]. In antiferromagnets with a
combined space-inversion and time-reversal symmetry, electric currents induce
a staggered spin density, and thus, a staggered torque termed the Néel spin-
orbit torque (NSOT). The NSOT can efficiently switch the staggered
magnetization, i.e., the Néel vector, thereby yielding significant progress in
AFM spintronics [41, 42, 43, 44]. Recently, it was found that a nonsymmorphic
crystalline symmetry in addition to the parity-time symmetry preserves the
crossing of doubly degenerate bands; this implies that Dirac fermions are
controlled by the direction of the Néel vector [45]. The discovery of this
close relationship between the magnetic and electronic structures opened a
research field called topological AFM spintronics [46, 47]. Some experiments
have shown that not only electric current pulses but optical pulses can
control the AFM order [48, 49]; these studies performed transport measurements
such as the anisotropic magnetoresistance and the planar Hall effect to
observe the magnetic structure [44]. Meanwhile, the topological electronic
structure has received less attention, and therefore, the real-time dynamics
of the Néel vector and the Dirac fermions induced by the NSOT are still
unclear.

From a theoretical viewpoint, two approaches have been established to study
the real-time dynamics of magnets. One is based on microscopic models such as
the Hubbard model and Kondo lattice model [18], which allows considering
electron correlation and quantum effects; however, the cluster size is
severely limited and approximations are needed to deal with the symmetry-
broken ordered states. The other approach is to use the
Landau–Lifshitz–Gilbert (LLG) equation of the classical vectors of magnetic
moments, which has been widely adopted in spintronics [50, 51, 52, 53, 54,
55]. This equation enables the simulation of large-scale magnetic structures,
while the electronic degree of freedom is integrated out. Therefore, a unified
framework is required to reveal the relationship between the magnetic and
electronic structures.

In this work, we propose a theory based on a minimal two-dimensional model of
AFM Dirac semimetals to investigate the real-time magnetic and electronic
dynamics. We show that the Néel vector is rotated by optically-induced NSOT
and other torques originating from magnetic anisotropies. This reorientation
accompanies the modulation of the mass of Dirac fermions. Furthermore, the
magneto-optical effect is found to be promising for the time-resolved
measurement of the Néel vector.

## Results

### Theoretical model

We consider a minimal tight-binding model of AFM Dirac semimetals projected
onto the two-dimensional square lattice. The Hamiltonian is divided into three
parts as
$\mathcal{H}=\mathcal{H}_{\mathrm{ele}}+\mathcal{H}_{\mathrm{exc}}+\mathcal{H}_{\mathrm{mag}}$.
The first term

$\displaystyle\mathcal{H}_{\mathrm{ele}}$
$\displaystyle=-2h_{1}\tau^{x}\cos\frac{k^{x}}{2}\cos\frac{k^{y}}{2}-h_{2}(\cos
k^{x}+\cos k^{y})$ $\displaystyle\quad+\lambda\tau^{z}(\sigma^{y}\sin
k^{x}-\sigma^{x}\sin k^{y})$ (1)

consists of the nearest-neighbor ($h_{1}$) and next-nearest-neighbor ($h_{2}$)
hoppings, and the spin-orbit coupling ($\lambda$) of itinerant electrons with
momentum $\bm{k}$. Here, the spin and sublattice degrees of freedom of the
itinerant electrons are described by the Pauli matrices $\bm{\sigma}$ and
$\bm{\tau}$, respectively. The localized magnetic moments on each sublattice
denoted by $\bm{S}_{\mathrm{A}}$ and $\bm{S}_{\mathrm{B}}$ couple to the
itinerant electrons through the exchange interaction

$\displaystyle\mathcal{H}_{\mathrm{exc}}$
$\displaystyle=J_{\mathrm{exc}}{\left[\frac{1+\tau^{z}}{2}\bm{\sigma}\cdot\bm{S}_{\mathrm{A}}+\frac{1-\tau^{z}}{2}\bm{\sigma}\cdot\bm{S}_{\mathrm{B}}\right]}.$
(2)

Hereafter, we define the uniform and staggered magnetizations as
$\bm{m}=(\bm{S}_{\mathrm{A}}+\bm{S}_{\mathrm{B}})/2$ and
$\bm{n}=(\bm{S}_{\mathrm{A}}-\bm{S}_{\mathrm{B}})/2$, respectively. The vector
$\bm{n}$ is termed the Néel vector. The exchange Hamiltonian is an extension
of the model for $\bm{S}_{\mathrm{A}}=-\bm{S}_{\mathrm{B}}=\bm{n}$ proposed by
Šmejkal et al. [45]. While both the parity ($\mathcal{P}$) and time-reversal
($\mathcal{T}$) symmetries are broken because of the presence of the localized
magnetic moments, the parity-time ($\mathcal{PT}$) symmetry is preserved when
$\bm{m}=0$. This guarantees the Kramers degeneracy of the energy bands in the
whole Brillouin zone. In addition to the $\mathcal{PT}$ symmetry, this model
is invariant under a nonsymmorphic glide symmetry operation only when the Néel
vector is directed along the $\langle 100\rangle$ directions, which causes the
crossings of the doubly degenerate energy bands, i.e., the gapless Dirac
points, to be protected at the Brillouin zone boundary [45], as shown in Fig.
1a. The energy gap at the Dirac points depends on the value of
$J_{\mathrm{exc}}$ and the direction of $\bm{n}(t)$. The spin-orbit coupling
and the AFM ordering of the localized moments lead to the spin-momentum
locking of the itinerant electrons. The expectation value of the staggered
spin moments of the itinerant electrons in the lower bands is displayed in
Fig. 1b, which indicates that the net staggered moment
$\langle\tau^{z}\sigma^{x}\rangle$ takes a negative value when
$\bm{n}=(1,0,0)$ because of the AFM exchange interaction.

Figure 1: Equilibrium properties of the model of AFM Dirac semimetals. a
Energy band structure. b Staggered spin texture of the lower bands in the
reciprocal space. The localized spins are aligned antiferromagnetically along
the $[100]$ direction, i.e., $\bm{n}=(1,0,0)$.

To stabilize the initial state in which the Néel vector is aligned with the
$[100]$ direction, we introduce magnetic anisotropy energy

$\displaystyle\mathcal{H}_{\mathrm{mag}}$
$\displaystyle=\sum_{\gamma=\mathrm{A},\mathrm{B}}{\left[K_{z}(S_{\gamma}^{z})^{2}+K_{xy}(S_{\gamma}^{x}S_{\gamma}^{y})^{2}\right]},$
(3)

where $K_{z}$ and $K_{xy}$ represent coefficients of the easy-plane and
biaxial anisotropies, respectively. Here, other magnetic interactions such as
the exchange interaction between the localized moments, which can be
implemented in a straightforward manner, are omitted because the AFM state is
stabilized by the indirect exchange mediated by the itinerant electrons at
half filling.

The time-dependent electric field $\bm{F}(t)$ parallel to the two-dimensional
plane is incorporated in the model via the Peierls substitution. The time
evolutions of the itinerant electrons and the localized magnetic moments are
described by the von Neumann and LLG equations, respectively; these equations
are solved simultaneously by the fourth-order Runge–Kutta method. The
parameter values and units are presented in Methods.

### Application of a constant electric field

Figure 2: Real-time dynamics induced by the constant electric field. a–c The
time evolution of (a) the Néel vector and uniform magnetization, (b) excited
electron density, and (c) energy bandgap, as functions of time $t$ in units of
$\hbar/h_{1}=0.66\ \mathrm{fs}$. d Sketch of NSOTs $\bm{T}_{\mathrm{A}}$ and
$\bm{T}_{\mathrm{B}}$ acting on sublattice moments $\bm{S}_{\mathrm{A}}$ and
$\bm{S}_{\mathrm{B}}$, respectively. Arrows of $\bm{b}_{\mathrm{A}}^{\perp}$
and $\bm{b}_{\mathrm{B}}^{\perp}$ represent effective fields perpendicular to
$\bm{S}_{\mathrm{A}}$ and $\bm{S}_{\mathrm{B}}$, respectively. Uniform
magnetization appears in the $[001]$ direction because of NSOT, and it induces
another effective field attributed to the easy-plane anisotropy that rotates
the Néel vector as shown in e with the $z$ components exaggerated. f Polar
plots of the in-plane angle of the Néel vector as a function of the
dimensionless time $F_{0}t$ for $F_{0}=0.0004$–$0.012$. The inset of f is the
enlarged view from $F_{0}t=0$ to $30$. g Time $t_{\mathrm{r}}$ for the Néel
vector to rotate towards the $[010]$ (or $[0\bar{1}0]$) direction as a
function of $F_{0}$. The electric field is applied in the $[110]$ direction in
a–g.

First, we discuss real-time dynamics induced by a constant electric field that
is switched on at time $t=0$ and applied in the $[110]$ direction, i.e.,
$\bm{F}(t)=F_{0}\varTheta(t)(\cos\pi/4,\sin\pi/4,0)$ with $\varTheta$ being
the step function. Figure 2a shows the real-time profiles of the in-plane
$x$-component of the Néel vector and the out-of-plane $z$-component of the
uniform magnetization. Figure 2b displays the excited electron density defined
by the number density of electrons that occupy single-particle energy levels
above the chemical potential $\mu$ of the initial state. The energy bandgap
shown in Fig. 2c is defined by the difference between the single-particle
energies of the lowest level above $\mu$ and the highest level below $\mu$;
this difference is equal to the direct gap when there is the particle-hole
symmetry with $h_{2}=0$. As time evolves, $n^{x}$ abruptly decreases towards
zero at time $t=t_{1}$, $t_{2}$, or $t_{3}$, which is indicated by arrows in
Fig. 2a and is hereafter termed a reorientation time $t_{\mathrm{r}}$.
Meanwhile, the $z$ component of the uniform magnetization appears with values
of the order of $m^{z}\sim 0.03$; this means that the magnetic structure is
slightly canted as illustrated in Fig. 2e. As the field magnitude increases
from $F_{0}=0.005$ to $0.009$, the reorientation time $t_{\mathrm{r}}$ becomes
short. The excited electron density shown in Fig. 2b increases as time
evolves. Figure 2b shows some plateaus in which the excited electron density
is constant in time and the energy bandgap opens simultaneously (Fig. 2c).
Once the excited electron density becomes greater than approximately $0.01$
(indicated by the shade in Fig. 2b), the Néel vector starts to rotate. Note
that the bandgap opens through the Néel-vector dynamics induced by the
constant field, which is in contrast to the gap opening of the Floquet–Bloch
bands under the high-frequency driving of electrons [56]. These phenomena
induced by the constant electric field occur in a timescale of the order of
$t\sim 10000\hbar/h_{1}$, which corresponds to $6.6\ \mathrm{ps}$ for
$h_{1}=1\ \mathrm{eV}$. Therefore, the Néel vector and the energy bandgap can
be controlled by the external field in such a short timescale.

The mechanism of the reorientation of the Néel vector is understood as
follows. The external electric field induces the nonequilibrium staggered spin
density of the itinerant electrons through the spin-orbit coupling, and it
generates torques in the $[001]$ direction (see Fig. 2d). Then, the two
sublattice magnetic moments are slightly canted towards the $[001]$ direction,
which means $m^{z}>0$. The easy-plane anisotropy, whose energy per sublattice
moment is estimated to be $K_{z}(m^{z})^{2}$, induces another effective field
$(0,0,-2K_{z}m^{z})$ that rotates the localized moments in the anticlockwise
direction as depicted in Fig. 2e. Here, we have shown the NSOT-driven
reorientation of the Néel vector on the basis of the microscopic model in
which the real-time dynamics of the itinerant electrons and the localized
moments are explicitly considered.

Figure 2f shows the polar plots of the in-plane angle of the Néel vector
$\varphi$ as a function of time multiplied by the field magnitude for
different values of $F_{0}$. The Néel vector rotates towards the $[010]$
direction for sufficiently large field magnitudes $F_{0}\gtrsim 0.004$. For
the weak magnitude $F_{0}\lesssim 0.004$, the trajectories of
$\varphi(F_{0}t)$ are on a universal curve independent of $F_{0}$; this
implies that the timescale of the dynamics is determined by the vector
potential $\bm{A}\sim\bm{F}t$ rather than the electric field $\bm{F}$. This is
because the NSOT is induced by the electric current due to the diamagnetic
response. The reorientation is regarded as a deviation from the weak-field
universal trajectory. Even before the reorientation, the Néel vector slightly
deviates from the $x$ axis because of the NSOT; once the excited electron
density exceeds approximately $0.01$, the NSOT-induced staggered torque shown
in Fig. 2d overcomes the other torques originating from the magnetic
anisotropies. These behaviors are summarized in Fig. 2g, where the
reorientation time is plotted as a function of $F_{0}$ for different values of
the biaxial anisotropy $K_{xy}$ and the Gilbert damping constant $\alpha$.
Overall, $t_{\mathrm{r}}$ is inversely proportional to $F_{0}$, which means
that the vector potential $\sim F_{0}t_{\mathrm{r}}$ governs the reorientation
dynamics; $t_{\mathrm{r}}$ becomes shorter as $K_{xy}$ or $\alpha$ decreases.
It is also found that the product $t_{\mathrm{r}}K_{xy}$ is a function of
$F_{0}/K_{xy}$ for $K_{xy}\geq 0.01$ and $\alpha=1$. The reorientation time
shows the step-like feature indicated by the dotted lines, which reflects a
discontinuous behavior of $\varphi(F_{0}t)$ with respect to $F_{0}$ as seen in
the inset of Fig. 2f. Although the anisotropy energies and the Gilbert damping
constant are highly material dependent, this reorientation can occur in the
timescale of subpicoseconds with a sufficiently large $F_{0}\ (\gtrsim 0.01)$
irrespective of $K_{xy}$ and $\alpha$.

### Irradiation of a monocycle pulse

Figure 3: Real-time dynamics induced by the monocycle pulse. a, b The time
evolution of (a) the Néel vector and uniform magnetization, and (b) excited
electron density and energy bandgap, as functions of time $t$, for
$\bm{F}_{0}=F_{0}(\cos\pi/4,\sin\pi/4,0)$ and $F_{0}=0.02$. The dashed lines
in a denote the results for the $[\bar{1}10]$ polarization. c Polar plots of
the in-plane angle of the Néel vector with respect to time $t$ for
$F_{0}=0.002$–$0.03$. d, e The time derivative of the in-plane Néel vector
angle and the $z$ component of the uniform magnetization divided by $F_{0}$
for $F_{0}=0.002$–$0.03$.

We show the real-time dynamics induced by a monocycle pulse of which the
frequency is in the terahertz range but is off-resonant (see Methods for
parameters). The electric field is parallel to the $[110]$ direction. Figure
3a shows the in-plane components of $\bm{n}$ and the out-of-plane component of
$\bm{m}$ for $F_{0}=0.02$. The Néel vector rotates towards the $[010]$
direction during pulse irradiation, and it stops the rotation after
irradiation. This reorientation accompanies the modulation of the bandgap as
shown in Fig. 3b. When the bandgap is opened, the excited electron density
does not increase as in the case of the constant field, which means that the
pulse is off-resonant.

The dashed lines in Fig. 3a shows $n^{y}$ and $m^{z}$ for the same field
amplitude but different polarization, i.e., $\bm{F}_{0}\parallel[\bar{1}10]$,
whose signs are opposite to those in the $\bm{F}_{0}\parallel[110]$ case
(solid lines). This polarization dependence is also seen in the case of the
constant field (not shown), and it implies that the direction of the Néel
vector is determined by the polarization and inverted by the mirror operation
on $\bm{F}_{0}$ against a plane perpendicular to $\bm{n}(t=0)$.

The pulse amplitude dependence of the Néel vector dynamics is summarized in
Fig. 3c. When the amplitude is weak ($F_{0}<0.01$), the Néel vector cannot
climb over the potential barrier of the magnetic anisotropy. However, the in-
plane angle exceeds $45^{\circ}$ for a sufficiently large $F_{0}\ (\geq 0.01)$
for which the excited electron density becomes greater than approximately
$0.01$, and the Néel vector remains in the $[010]$ direction after irradiation
for even larger $F_{0}\ (\geq 0.02)$.

Figures 3d and 3e show the in-plane angular velocity of the Néel vector and
the $z$ component of the uniform magnetization, both of which are divided by
$F_{0}$. For weak $F_{0}<0.01$, the time profiles of $d\varphi/d(F_{0}t)$ and
$m^{z}/F_{0}$ are on each universal curve, whereas the deviations from the
curves are found for $F_{0}\geq 0.01$ and they lead to the reorientation
towards the $[010]$ direction. These time profiles are quite similar because
$d\varphi/dt$ is approximated by $2K_{z}m^{z}$ as long as
$\|\bm{m}\|\ll\|\bm{n}\|$.

### Optical conductivity and magneto-optical effect

Figure 4: Optical conductivity and magneto-optical effect. a, b Real parts of
(a) longitudinal and (b) transverse optical conductivities. Energy $\omega$
and optical conducvitity $\sigma$ are shown in units of $h_{1}=1\ \mathrm{eV}$
and $e^{2}/(\hbar a)=6.4\times 10^{3}\ \mathrm{S\,cm^{-1}}$, respectively.
Insets in (a) and (b) show the optical conductivity for
$\bm{n}\parallel[100],[110],[010]$, and $[1\bar{1}0]$. Panels c, d show the
magneto-optical rotation angle $\theta$ and ellipticity $\eta$ as functions of
the in-plane angle of the Néel vector. The lattice constant and broadening
factor are set to $a=0.38\ \mathrm{nm}$ and $\varGamma/h_{1}=0.02$,
respectively.

Finally, we show the optical longitudinal and transverse (Hall) conductivities
for different Néel vector angles, and we discuss the resulting magneto-optical
Voigt effect, assuming that the system is in the collinear AFM ground states.
Figures 4a and 4b display the real parts of the longitudinal ($\sigma_{xx}$)
and transverse ($\sigma_{xy}$) conductivities as functions of the in-plane
Néel vector angle $\varphi$. The peaks of
$\operatorname{Re}\sigma_{xx}(\omega)$ for $\omega>0$ are attributed to the
Van Hove singularity; the Drude peak is also seen at $\omega=0$ for
$\bm{n}\parallel[100]$ and $[010]$ when $h_{2}\neq 0$. Because of the two-fold
symmetry around the $z$ axis, $\sigma_{xx}$ and $\sigma_{xy}$ are periodic in
$\varphi$. Here, $\sigma_{yy}$ is equivalent to $\sigma_{xx}$ under the
rotation $\varphi\mapsto\varphi+\pi/2$, and $\sigma_{xy}$ is equal to
$\sigma_{yx}$. The sign of the transverse conductivity is unchanged by
$\lambda\mapsto-\lambda$ or $J_{\mathrm{exc}}\mapsto-J_{\mathrm{exc}}$ (not
shown), which implies the conductivity is an even function of $\lambda$ and
$J_{\mathrm{exc}}$. Since the present model holds the $\mathrm{D_{4h}}$
symmetry in the limit of $J_{\mathrm{exc}}\bm{n}\to 0$, the conductivity
linear in the electric field can be expanded with respect to a time-reversal-
breaking field $\bm{B}\sim J_{\mathrm{exc}}\bm{n}$ as
$\sigma_{ij}(\bm{B})=\sigma_{ij}^{(0)}+\sigma_{ijk}^{(1)}B_{k}+\sigma_{ijkl}^{(2)}B_{k}B_{l}$,
where $\sigma^{(1)}$ vanishes when $\bm{B}\perp[001]$ and

$\displaystyle\sigma^{(2)}_{ij}=\begin{bmatrix}\sigma^{(2)}_{xxxx}B_{x}^{2}+\sigma^{(2)}_{xxyy}B_{y}^{2}&2\sigma^{(2)}_{xyxy}B_{x}B_{y}\\\
2\sigma^{(2)}_{xyxy}B_{x}B_{y}&\sigma^{(2)}_{xxyy}B_{x}^{2}+\sigma^{(2)}_{xxxx}B_{y}^{2}\\\
\end{bmatrix}_{ij}$ (4)

is a symmetric tensor. The off-diagonal components of $\sigma(\bm{B})$ can be
written as $\sigma_{xy}(\bm{B})=\sigma_{yx}(\bm{B})\propto\sin 2\varphi$ for
$\bm{n}=(\cos\varphi,\sin\varphi,0)$. This is consistent with the numerical
result of $\sigma_{xy}$, although the present values of $\lambda\ (=0.8)$ and
$J_{\mathrm{exc}}\ (=0.6)$ are beyond the perturbative regime. From these
properties of $\sigma_{xx}$ and $\sigma_{xy}$, we can identify the direction
in which the Néel vector rotates by measuring the longitudinal optical
conductivity and magneto-optical signal. Figures 4c and 4d show the magneto-
optical rotation angle and ellipticity calculated from $\sigma_{xx}$ and
$\sigma_{xy}$ (see Methods). The rotation angle $\theta$ exhibits large values
up to $1^{\circ}$. In real materials, the values of $\lambda$ and
$J_{\mathrm{exc}}$ may be smaller than the values adopted here, which causes a
reduction in the rotation angle down to a few millidegrees. Even so, the
magneto-optical rotation can be measured using modern optical apparatus.

The transverse conductivity $\sigma_{xy}$ appears when $\lambda\neq 0$ and
$J_{\mathrm{exc}}\neq 0$ even though the magnetic structure is collinear AFM.
Recently, the transverse response in collinear antiferromagnets is attracting
considerable attention, and it has been discussed for the case in which the
conductivity tensor is antisymmetric [57, 58]; our calculations demonstrate
the transverse optical response attributed to the symmetric tensor in
collinear antiferromagnets.

## Discussions

We investigated the real-time dynamics of the magnetic and electronic
structures induced by the constant and pulse electric fields, and we revealed
the microscopic mechanism of the NSOT-driven reorientation of the Néel vector.
Our method can be applied to any tight-binding model coupled to localized
moments. The experimental verification of the present results is necessary for
a better understanding of the NSOT-induced dynamics and for further progress
in ultrafast AFM spintronics combined with the topological aspects of the
materials. Candidate materials include orthorhombic CuMnAs [59, 45, 60, 61]
and MnPd2 [62], which have been studied in the field of AFM spintronics and
are expected to host Dirac fermions. Note that the reorientation of the Néel
vector demonstrated in this study is not directly related to what has been
observed in experiments because of the discrepancy in the timescale. To make
more quantitative discussions on, e.g., the timescale of the reorientation and
the required magnitude of the field, we should consider the crystal structure
and dimensionality of materials, obtain realistic estimates of model
parameters as well as the Gilbert damping constant from the ab initio
calculations and experiments, and take other dissipation processes caused by
the electron-electron interaction into account; these are left for future
work.

Recent intense terahertz pulse sources, whose oscillation period is $1\
\mathrm{ps}$ and peak amplitude is more than $1\ \mathrm{MV\,cm^{-1}}$ [63],
are promising to drive the reorientation of the Néel vector in the picosecond
timescale. In our calculations, a typical value of $F_{0}$ is of order $0.01$,
which corresponds to $0.26\ \mathrm{MV\,cm^{-1}}$ (see Methods). There are
many earlier studies on the electrical modulation and detection of the Néel
vector [44]; their time resolution was limited to $\gtrsim 1\ \mathrm{ns}$.
However, it may be possible to observe the Néel vector dynamics in real time
via optical measurements because the magnetic structure is closely related to
the electronic structure in the present model. One approach is the time- and
angle-resolved photoemission spectroscopy; this can directly access the energy
band structure, which reflects the direction of the Néel vector as the
positions and energy gaps of the Dirac points or nodal lines. Another approach
is to use magneto-optical effects. This system exhibits the magneto-optical
rotation with angles up to $1^{\circ}$. The time-resolved magneto-optical
measurements can be carried out even in the subpicosecond timescale, which
provides a sufficient time resolution to observe the real-time dynamics
discussed in this study.

## Methods

### Real-time dynamics

We calculate the real-time dynamics of the itinerant electrons and the
localized magnetic moments as follows [64, 65]. The quantum state of the
electrons is exactly described by a one-body density matrix
$\hat{\rho}_{\bm{k}}$ because the electronic Hamiltonian
$\mathcal{H}_{\mathrm{ele}}$ contains no many-body interaction. Once the
localized magnetic moments are given, the Hamiltonians in Eqs. (1) and (2) can
be diagonalized for each $\bm{k}$ as
$(\mathcal{H}_{\mathrm{ele}}+\mathcal{H}_{\mathrm{exc}})|\bm{k}\nu\rangle=\varepsilon_{\bm{k}\nu}|\bm{k}\nu\rangle$,
where $|\bm{k}\nu\rangle$ is obtained by the unitary transform as
$|\bm{k}\nu\rangle=\sum_{\gamma=\mathrm{A},\mathrm{B}}\sum_{s={\uparrow},{\downarrow}}|\bm{k}\gamma
s\rangle U_{\nu,\gamma s}^{*}$. In the initial state in which the external
field is absent, the density matrix is given by
$\tilde{\rho}_{\bm{k}}=\sum_{\nu}|\bm{k}\nu\rangle
n_{\bm{k}\nu}\langle\bm{k}\nu|$, where
$n_{\bm{k}\nu}=\varTheta(\mu-\epsilon_{\bm{k}\nu})$ is the step function with
$\mu$ being the chemical potential chosen such that the number of electrons is
$N_{\mathrm{ele}}$. The time evolution of the density matrix is governed by
the von Neumann equation

$\displaystyle\frac{d\hat{\rho}_{\bm{k}}}{dt}=-\mathrm{i}[\mathcal{H},\hat{\rho}_{\bm{k}}]$
(5)

with the initial condition
$\hat{\rho}_{\bm{k}}(t=0)=U_{\bm{k}}^{-1}\tilde{\rho}_{\bm{k}}U_{\bm{k}}$. The
expectation value is defined by
$\langle\cdot\rangle=N^{-1}\sum_{\bm{k}}\operatorname{Tr}[\,\cdot\,\hat{\rho}_{\bm{k}}]$,
where $N$ represents the number of $\bm{k}$ points in the Brillouin zone. The
localized moments on each sublattice denoted by $\bm{S}_{\mathrm{A}}$ and
$\bm{S}_{\mathrm{B}}$ are treated as classical unit vectors, and therefore,
the Hamiltonian $\mathcal{H}(t)$ depends on the classical spins and the
external field. The classical spins follow the LLG equation

$\displaystyle\frac{d\bm{S}_{\gamma}}{dt}=\bm{S}_{\gamma}\times\bm{b}_{\gamma}-\alpha\bm{S}_{\gamma}\times\frac{d\bm{S}_{\gamma}}{dt}$
(6)

for $\gamma=\mathrm{A},\mathrm{B}$, where $\alpha$ denotes the Gilbert damping
constant and
$\bm{b}_{\gamma}\equiv-\langle\partial\mathcal{H}/\partial\bm{S}_{\gamma}\rangle$
represents the effective field given by

$\displaystyle\bm{b}_{\gamma}$
$\displaystyle=-J_{\mathrm{exc}}\langle\bm{\sigma}_{\gamma}\rangle-2{\left(K_{xy}S_{\gamma}^{x}(S_{\gamma}^{y})^{2},K_{xy}(S_{\gamma}^{x})^{2}S_{\gamma}^{y},K_{z}S_{\gamma}^{z}\right)}.$
(7)

Here, $\bm{\sigma}_{\mathrm{A}/\mathrm{B}}=(1\pm\tau^{z})\bm{\sigma}/2$
denotes the sublattice spin density. The time-dependent electric field
$\bm{F}(t)$ is incorporated in the model via the Peierls substitution
$\mathcal{H}_{\mathrm{ele}}(\bm{k})\mapsto\mathcal{H}_{\mathrm{ele}}(\bm{k}-e\bm{A}(t)a/\hbar)$,
where $\bm{A}(t)=-\int^{t}\bm{F}(t^{\prime})\,dt^{\prime}$, $e$, and $a$
denote the vector potential, elementary charge, and lattice constant,
respectively. Throughout this paper, $h_{1}$, $e$, $a$, and $\hbar$ are set to
unity. The physical quantities of time, electric-field magnitude, and
conductivity are expressed in units of $\hbar/h_{1}=0.66\ \mathrm{fs}$,
$h_{1}/(ea)=26.3\ \mathrm{MV\,cm^{-1}}$, and $e^{2}/(\hbar a)=6.4\times
10^{3}\ \mathrm{S\,cm^{-1}}$, respectively, for $h_{1}=1\ \mathrm{eV}$ and
$a=0.38\ \mathrm{nm}$. In most calculations, we use $h_{2}=0.08$,
$\lambda=0.8$, $J_{\mathrm{exc}}=0.6$, $K_{z}=0.1$, $K_{xy}=0.01$, and
$\alpha=1$.

The easy-plane anisotropy coefficient $K_{z}$ determines the in-plane angular
velocity of the Néel vector $d\varphi/dt$. For the above parameters, the
difference in the total energies of the ground states is
$\langle\mathcal{H}\rangle|_{\bm{n}=(0,0,1)}-\langle\mathcal{H}\rangle|_{\bm{n}=(1,0,0)}=0.13h_{1}$.
This may be considerably larger than the anisotropy energy of real materials
($\lesssim 1\ \mathrm{meV}$) [60, 61]. However, we confirmed that the
reorientation of the Néel vector occurs within $10\ \mathrm{ps}$ even for
smaller values: $K_{z}=K_{xy}=0.001$, $\lambda=0.08$, $J_{\mathrm{exc}}=0.06$,
and $F_{0}=0.001$.

Equations (5) and (6) are simultaneously solved using the fourth-order
Runge–Kutta method with a time step $\delta t=0.01$. We perform the numerical
calculation with $N=512\times 512$-point mesh in the Brillouin zone, and we
fix the number of the electrons to $N_{\mathrm{ele}}=2N$ (half-filled). The
external field $\bm{F}(t)=-\partial_{t}\bm{A}(t)$ is given by
$\bm{A}(t)=-\bm{F}_{0}t\varTheta(t)$ for the constant electric field and
$\bm{A}(t)=-\bm{F}_{0}{\cdot}(t-t_{\mathrm{c}})\exp[-(t-t_{\mathrm{c}})^{2}/(2t_{\mathrm{w}}^{2})]$
for the monocycle pulse. We adopt $t_{\mathrm{c}}=2000$ and
$t_{\mathrm{w}}=300$.

### Optical conductivity and magneto-optical effect

We calculate the optical conductivity from

$\displaystyle\sigma_{ij}(\omega)$
$\displaystyle=\frac{\chi_{ij}(\omega+\mathrm{i}\varGamma)-\chi_{ij}(\mathrm{i}\varGamma)}{\mathrm{i}(\omega+\mathrm{i}\varGamma)},$
(8) $\displaystyle\chi_{ij}(z)$
$\displaystyle=\frac{1}{N}\sum_{\bm{k}\mu\nu}J_{\bm{k}\mu\nu}^{i}J_{\bm{k}\nu\mu}^{j}\frac{n_{\bm{k}\mu}-n_{\bm{k}\nu}}{\varepsilon_{\bm{k}\nu}-\varepsilon_{\bm{k}\mu}-z},$
(9)

where
$\bm{J}_{\bm{k}\mu\nu}=\langle\bm{k}\mu|\bm{J}_{\bm{k}}|\bm{k}\nu\rangle$
denotes the matrix element of the electric current operator
$\bm{J}_{\bm{k}}=-\delta\mathcal{H}(\bm{k}-\bm{A})/\delta\bm{A}|_{\bm{A}=0}$
and $\varGamma$ represents a broadening factor. The dielectric tensor is given
by

$\displaystyle\epsilon_{ij}(\omega)=\delta_{ij}+\frac{\mathrm{i}\sigma_{ij}(\omega)}{\epsilon_{0}\omega}$
(10)

with $\epsilon_{0}=8.854\ \mathrm{pF\,m^{-1}}$ being the electric constant.
Since the optical conductivity tensor is symmetric ($\sigma_{xy}=\sigma_{yx}$)
and anisotropic ($\sigma_{xx}\neq\sigma_{yy}$) in the present model, we cannot
use the well-known formula of the complex Kerr rotation angle
$\varPhi_{\mathrm{K}}=\epsilon_{xy}/[(1-\epsilon_{xx})\sqrt{\epsilon_{xx}}]$.
According to Maxwell’s equations in material media, the electric-field
amplitude $\bm{F}_{0}$ satisfies

$\displaystyle(\hat{\bm{N}}\cdot\hat{\bm{N}})\bm{F}_{0}-(\hat{\bm{N}}\cdot\bm{F}_{0})\hat{\bm{N}}+\epsilon\bm{F}_{0}=0,$
(11)

where $\hat{\bm{N}}$ represents a complex refractive index vector. From Eq.
(11) we have the relation
$(\hat{N}^{2}-\epsilon_{xx})(\hat{N}^{2}-\epsilon_{yy})=\epsilon_{xy}^{2}$,
assuming that the propagation vector of incident light is perpendicular to the
two-dimensional $xy$-plane. The symmetric property of the conductivity and
dielectric tensors leads to the eigenmodes of Eq. (11) being linearly
polarized, contrary to the antisymmetric case in which the eigenmodes are
circularly polarized. The boundary condition is that the tangential components
of the electric field $\bm{F}$ and magnetic field
$\hat{\bm{N}}\times\bm{F}/c_{0}$ ($c_{0}$ is the speed of light) are
continuous at the surface, from which we obtain two eigenmodes and complex
refractive indices. The incident linearly polarized light is decomposed into a
linear combination of the eigenmodes. The amplitude vector of the reflected
light is proportional to $(r_{\mathrm{pp}},r_{\mathrm{sp}},0)$, where
$r_{\mathrm{pp}}$ and $r_{\mathrm{sp}}$ are the amplitude reflection
coefficients of the incident p-polarized light to the reflected p- and
s-polarized light. The magneto-optical rotation angle $\theta$ and ellipticity
$\eta$ are determined by the fit of the electric-field trajectory to an
ellipse rotated around the $z$ axis by angle $\theta$, and the relation
$\eta=\operatorname{sgn}[\arg(r_{\mathrm{sp}}/r_{\mathrm{pp}})]\times|r_{\mathrm{sp}}|/|r_{\mathrm{pp}}|$,
respectively.

###### Acknowledgements.

One of the authors, Sumio Ishihara, passed away on 7 November 2020 during the
preparation of the manuscript. We thank H. Matsueda and Y. Masaki for their
valuable comments and critical reading of the manuscript. This work was
supported by JSPS KAKENHI Grant Nos. JP19K23419, JP20K14394, JP17H02916,
JP18H05208, and JP20H00121. The numerical calculations were performed using
the facilities of the Supercomputer Center, the Institute for Solid State
Physics, the University of Tokyo.

## Author contributions

A.O. conceived the idea, performed the calculations, and wrote the manuscript.
A.O. and S.I. discussed and interpreted the results.

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