# Ensuring reliability: what is the optimal time for power plant maintenance
in Texas as the climate changes?

Hugh Daigle111Correspondence<EMAIL_ADDRESS>Joshua D. Rhodes  Aidan
Pyrcz  Michael E. Webber

###### Abstract

We analyzed data for the Electric Reliability Council of Texas (ERCOT) to
assess shoulder seasons – that is, the 45 days of lowest total energy use and
peak demand in the spring and fall typically used for power plant maintenance
– and whether their occurrence has changed over time. Over the period
1996–2022, the shoulder seasons never started earlier than late March nor
later than mid-October, corresponding well with the minimum of total degree
days. In the temperature record 1959–2022, the minimum in degree days in the
spring moved earlier, from early March to early February, and in the fall
moved later, from early to mid-November. Warming temperatures might cause
these minima in degree days to merge into a single annual minimum in December
or January by the mid-2040s, a time when there is a non-trivial risk of 1-day
record energy use and peak demand from winter storms.

###### keywords:

ERCOT , electricity demand , climate change

††journal: The Electricity Journal

[inst1]organization=Center for Subsurface Energy and the Environment, The
University of Texas at Austin, city=Austin, state=Texas, country=USA

[inst2]organization=J. Mike Walker Department of Mechanical Engineering, The
University of Texas at Austin,city=Austin, state=Texas, country=USA

[inst3]organization=McKetta Department of Chemical Engineering, The University
of Texas at Austin, city=Austin, state=Texas, country=USA

##  1 Introduction

It is widely recognized that climate change will adversely affect reliable
electricity delivery worldwide due to more frequent extreme weather events,
increased demand for space cooling, and the relative vulnerability of power
generation infrastructure to increasing temperatures (van Vliet et al., 2012;
Zamuda et al., 2018; Coffel and Mankin, 2021). Unreliable electricity delivery
results in loss of life during hot and cold weather, and has economic
consequences: for example, blackouts during Winter Storm Uri in the United
States in February 2021 caused more than 200 detahs and about $155 billion in
economic losses (Busby et al., 2021). A large number of recent studies have
looked at the ways in which climate change will affect peak electricity demand
in different parts of the world, focusing on both summer and winter peaks
(Parkpoom and Harrison, 2008; Chen and Lie, 2010; Ahmed et al., 2012; Sathaye
et al., 2013; Bartos and Chester, 2015; Trotter et al., 2016; Auffhammer et
al., 2017; Burillo et al., 2019; Fan et al., 2019; Chabouni et al., 2020;
Garrido-Perez et al., 2021; Shaffer et al., 2022; Romitti and Wing, 2022).
However, variations in the intensity of the hottest and coldest parts of the
year are not the only consideration for predicting future electricity
reliability. As average global temperature increases, spring is starting
earlier and summer is lasting longer as indicated by several different metrics
including temperatures, plant behavior, bird migration and nesting, and the
Asian monsoon (Schwartz et al., 2006; Thompson and Clark, 2008; Peña-Ortiz et
al., 2015; Lehikoinen et al., 2019; Shipley et al., 2020; Ho et al., 2021).
The effect of these climate change-induced variations in seasonality on
electricity reliability have been addressed mainly in terms of meeting
increased electricity demand over a longer period during the hot time of the
year (Hamlet et al., 2010; van Ruijven et al., 2019; Hill et al., 2021).
However, a phenomenon that has not been explored to date is how periods of
minimum electricity demand – which are optimal times for scheduled maintenance
of power plants – have changed and will continue to change in the future.
Here, we use data from the U.S. state of Texas to show how earlier onset of
spring and later onset of fall affect the timing of the periods with lowest
electricity demand, raising the risk of shorter maintenance windows that could
reduce power plant reliability. While our results are specific to Texas, the
conclusions are broadly applicable as the lengthening of summer due to climate
change is a globally observed phenomenon.

Texas is the largest producer and consumer of electricity in the United
States. In 2020, generation and consumption were nearly twice as high as the
next-highest states (U.S. Energy Information Administration, 2021). The high
demand is due to the state’s large industrial base, which in 2020 accounted
for 53.9% of total electricity consumption (U.S. Energy Information
Administration, 2022); hot climate and weak building efficiency standards,
which lead to significant space cooling; and robust population growth, with a
15.9% increase from 2010 to 2020 (U.S. Census Bureau, 2021) and a further 70%
increase forecast by 2070 (Texas Water Development Board, 2022). The Electric
Reliability Council of Texas (ERCOT), which is the grid that serves about 90%
of Texas electricity, projects that peak demand during the summer could
increase from 81.6 GW in 2024 to 88.7 GW in 2033, or by about 9% (Electric
Reliability Council of Texas, 2023). The summer of 2023 saw large peak demand,
includingan all-time record of 85.5 GW on August 10.

Power plants are complex machines with many moving parts that require going
off-line for maintenance, similar to how a light duty internal combustion
vehicle must turn off for periodic oil changes. However, there are limited
periods of time that are conducive for power plants to take those extended
outages without compromising grid reliability. Predicting the optimum times
for this preventative maintenance requires consideration of many different
factors, including generation ramp rate, availability of personnel and
resources, transmission line capacity, demand, and reliability (Froger et al.,
2016). Because the ERCOT grid is sensitive to temperature-driven spikes in
power use and peak demand from heating and cooling needs (Rhodes et al., 2011;
Alipour et al., 2019; Shaffer et al., 2022), ERCOT allows electricity
generators to schedule planned maintenance as long as the total amount of
generation capacity taken offline during a particular period does not exceed
the forecasted maximum daily resource planned outage capacity (MDRPOC). MDRPOC
data are forecasted for 7 days into the future. Projections farther into the
future are based on historical data, specifically the 50th percentile of past
load profile data with MDRPOCs from the past 3 years for the summer and winter
months (Electric Reliability Council of Texas, 2022b). This approach has the
effect of allowing most planned power plant maintenance in the spring or fall
– the traditional shoulder seasons – when temperatures are historically mild
and electricity demand is generally lower.

As a result of climate change, the onset of springtime biological activity has
moved earlier in the year by 9.3 days in the Central Plains and by 18.8 days
in the southwestern United States since 1950 (Crimmins and Crimmins, 2019),
and the length of the frost-free season in Texas has increased by 12–24 days
over the same period (Zhang et al., 2021; Modala et al., 2017). It follows
that the shoulder seasons have likely changed their duration and timing over
the same period, and will change in the future as the planet continues to
warm. ERCOT does not explicitly consider climate change in their planning,
which could lead to reliability concerns in the future (Lee and Dessler,
2022). Maintaining stable, reliable grid conditions requires understanding how
changing season onsets have been reflected in temperature data and electricity
demand, and how those might continue to change in the future. We analyzed
temperature and electrical grid load data for the ERCOT service area to assess
how shoulder seasons have varied and provide guidance on how they might
continue to vary in the future.

##  2 Background

Over the period 1895–2021, the average daily maximum and minimum temperatures
in Texas increased by 0.8 Celsius degrees, and overall average temperature is
projected to increase by another 1 Celsius degree by 2036 relative to the
1999-2020 average. While the absolute number of days above 100∘F (38∘C) varies
geographically across the state, this projected temperature increase means
that locations across Texas would experience 40% more days above 100∘F (38∘C)
(Nielsen-Gammon et al., 2021). Further, the timing of the onset of spring and
autumn have been changing and are expected to continue to do so in the future
(Crimmins and Crimmins, 2019; Zhang et al., 2021; Modala et al., 2017). This
shift is consistent with other work demonstrating earlier snow disappearance
and leaf growth in the spring and later snow appearance and leaf loss in the
fall at more northern latitudes (Creed et al., 2015; Grogan et al., 2020).
These trends are projected to continue in the future as average temperatures
increase.

Electricity demand in the ERCOT service area has changed over the past few
decades due to the combined effects of climate change, population growth,
efficiency increases in domestic cooling, and increases in the proportion of
households using electric heating (White et al., 2021; Lee and Dessler, 2022;
Skiles et al., 2023). Both Lee and Dessler (2022) and Skiles et al. (2023)
demonstrated that climate change has specifically increased summer peak
demand, and that this trend is expected to continue in the future because
residential cooling drives spikes in electricity demand on hot summer days in
Texas (White and Rhodes, 2019). It follows, then, that residential electricity
demand should also respond to earlier onset of spring weather and later onset
of fall weather, which in turn should drive changes in the timing of the
shoulder seasons.

In 2022, the shoulder seasons were predicted by ERCOT to occur from early
March to late May, and from mid-September to late November (Rickerson, 2022).
This follows the reported planned outages from 2021, with no planned outages
between June 10 and September 15 and the winter having on average 18% the
outages of the spring shoulder season (3,000 MW versus 17,000 MW). O’Shea et
al. (2021) analyzed all generation resource outages (planned and forced)
reported by ERCOT for the period 2015–2020 and similarly showed that outages
were about 15 GW higher during March–May and October–November relative to the
winter and summer averages, respectively, thus demonstrating how planned
outages drive overall outages in shoulder seasons. However, it is not clear
that these time periods are the optimal shoulder seasons in terms of reliably
meeting demand. Anomalously hot or cold weather can occur during these
predicted shoulder seasons. When projected electricity demand is expected to
approach generation capacity, ERCOT issues a conservation request in an effort
to reduce demand. Between January 2008 and July 2022, ERCOT issued 44 area-
wide conservation requests (Electric Reliability Council of Texas, 2022a).
Most were during the summer or winter, but 4 were issued between March 1 and
May 31 and 1 between September 15 and November 30, which means that 11% of
these conservation requests were issued during the periods ERCOT defines as
shoulder seasons. In April 2006, generation was so constrained that ERCOT
instituted rolling blackouts across its entire service area as 20% of
generation was offline for maintenance (Apt et al., 2006). Notably, since the
1980s, area-wide rolling blackouts have only occurred 4 times, with 3 during
the winter and 1 during the spring shoulder season (Munce, 2022). Here, we
demonstrate quantitatively how climate change has affected the shoulder
seasons since the mid-20th century, and predict how these changes may continue
over the 21st century.

##  3 Data and methods

In this analysis, we sought to assess how the timing of shoulder seasons is
changing and could continue to do so in the future with climate change. We
used three metrics to define shoulder season: daily electricity use, daily
peak demand, and heating- and cooling-degree days. Shoulder seasons in the
spring and fall were defined as the 45-day periods with the lowest average of
each of these three metrics. Our choice of 45 days as the length of the
optimal window was motivated by a rule revision, Nodal Protocol Revision
Request 1108 (NPRR 1108), that ERCOT proposed in 2021. The ERCOT Technical
Advisory Committee recommended setting minimum MRDPOC levels during the
shoulder seasons, with maximum MRDPOC March 15-May 1 and October 15-November
30 (Joint Commenters I, 2022; Sams, 2022). This recommendation had the support
of electricity generators and other stakeholders (Richmond, 2022). While this
part of NPRR 1108 was ultimately not included in the final, revised rule
(ERCOT Board, 2022), it does give an indication of the practical length of the
ideal periods for power plant maintenance.

Our analysis considered the ERCOT service area, which encompasses most, but
not all, of Texas (Fig. 1). Electricity use data came from ERCOT, which has
hourly load data publicly available for every day from 1 January 1996 to the
present with the exception of 2001. For temperature data, we used the European
Centre for Medium-Range Weather Forecasts Reanalysis 5 (ECMWF ERA5) of
atmospheric temperature dating back to 1959 (Hersbach et al., 2020). ERA5
provides hourly temperature data at a spatial resolution of 0.25∘ at various
atmospheric levels; we used the temperature at 2 meters above ground level.

Figure 1: Map of Texas with ERCOT service area in dark gray. Base map from
https://freevectormaps.com

We used total degree days to define temperature-based shoulder seasons. Degree
days represent the deviation of the average temperature for a particular day
above or below a baseline and are used as an indication of energy demand for
heating or cooling. In calculating degree days, population-weighted
temperature is typically used because energy demand will be concentrated in
areas with higher population density, and therefore the temperatures in those
areas will more strongly influence load on the electric grid (Quayle and Diaz,
1980; Taylor, 1981). We determined population-weighted temperatures following
Lee and Dessler (2022). Temperature data were weighted by population using
data from the Center for International Earth Science Information Network
(CIESIN), which is gridded at the same resolution as the temperature data
(CIESIN, 2020). The CIESIN data have updates every 5 years from 2000 to 2020,
which we used for this date range. For dates prior to 2000, the 2000
population weights were used.

To process the power load data, we computed the daily total use (energy, kWh)
and peak demand (power, kW), and then determined the 45-day period with the
lowest average daily total use and peak demands in the first half
(January–June) and second half (July–December) of the year. To process the
temperature data, we first computed the average daily temperature $T_{avg}$
across the entire ERCOT service area using population-weighted temperatures.
We then determined total degree days $DD$, which we define as

$DD=\begin{cases}T_{avg}-T_{0},&T_{avg}\geq T_{0}\\\
T_{0}-T_{avg},&T_{avg}<T_{0}\end{cases},$ (1)

where $T_{0}$ is a reference temperature. $T_{0}$ was determined by plotting
daily peak demand versus daily average temperature, fitting a $3^{rd}$ degree
polynomial to the data, and finding the average temperature corresponding to
the minimum daily peak demand predicted from the polynomial fit (Fig. 2).
Polynomial coefficients and $T_{0}$ values are listed in Table A3 in the
Supporting Information. We made one plot for each year where data were
available, and $T_{0}$ was taken as the average over all available years. The
dates we report are the onsets of the 45-day periods of minimum total use,
peak demand, and degree days.

To assess whether changes over time in the onset of shoulder seasons were
statistically significant, we conducted a least-squares linear regression and
found the rate of change along with its standard error. We then computed the
probability that the rate of change indicated a shift earlier in the spring
(rate of change $<0$) or later in the fall (rate of change $>0$).

Figure 2: Illustration of determining $T_{0}$. Peak demand is plotted as a
function of daily average temperature, and a $3^{rd}$ order polynomial is fit
to the data. $T_{0}$ is the temperature at the minimum peak demand predicted
by the polynomial. Data shown are from 2022.

##  4 Results and Discussion

###  4.1 Trends in temperature-defined shoulder seasons

Since the temperature record we used extends farther back than the electricity
usage dataset, it provides a useful proxy for long-term trends in energy
demand in the ERCOT service area. From 1959 to 2022, the spring shoulder
season moved earlier in the year by 2.4 days per decade (Fig. 3a), while the
fall shoulder season moved later by 1.1 days per decade (Fig. 3b). The rate of
change of the spring shoulder season is consistent with the change in onset of
springtime biological activity in the southwestern United States reported by
Crimmins and Crimmins (2019) (2.7 days per decade). Based on the standard
error of the slopes of our regression lines, these shifts occurred with
probabilities of 99% and 91% in the spring and fall, respectively. During the
64 years in this dataset, the spring shoulder season never started later than
March 12 and the fall shoulder season never started earlier than September 29.

Figure 3: Start of spring (a) and fall (b) shoulder seasons defined by degree
days. 5-year moving averages are included (red lines), as well as linear
regression lines with 95% confidence intervals. Note that 5 outliers were
excluded from the regression in the fall.

###  4.2 Trends in electricity-defined shoulder seasons

The 45-day periods of minimum total energy use (kWh) and minimum peak demand
(kW) in the spring have shifted slightly earlier in the year since 1996 (Figs.
4a,b), by 2.0 days per decade for total energy use and 1.0 days per decade for
peak demand. Due to the shorter time series compared to the temperature data,
the probability that these shoulder seasons shifted earlier is lower than
observed in the temperature data, at 69% for total energy use and 60% for peak
demand. These spring shoulder seasons never began later than March 14.

In the fall (Figs. 4c,d), least squares regression indicates that the shoulder
season defined by total energy use moved later in the year by 0.057 days per
decade while that defined by peak demand moved later by 0.027 days per decade.
However, these changes are very small and uncertain. The probability that a
shift later actually occurred is 51% for total energy use and 50% for peak
demand, thus giving nearly equal probabilities that the fall shoulder seasons
shifter earlier or later. The fall shoulder seasons never started earlier than
October 6 for total energy use and October 12 for peak demand.

To summarize these results, the 45-day periods of minimum total energy use and
peak demand started in February–early March and early October–early November
over this time period. These results are consistent with the proposed period
of maximum allowable MRDPOC proposed by the ERCOT Technical Advisory Committee
(Sams, 2022).

Figure 4: Shoulder seasons defined by electricity. (a) Spring shoulder season
defined by total energy use. (b) Spring shoulder season defined by peak
demand. (c) Fall shoulder season defined by total energy use. (d) Fall
shoulder season defined by peak demand. 5-year moving averages are included
(red lines), as well as linear regression lines with 95% confidence intervals.

###  4.3 Correlation between degree days and electricity demand

The existence of a correlation between minimum degree days and minimum
electricity demand implies that climatological data could and should be used
to predict changes in shoulder season in the future. ERCOT does not
incorporate long-term climate forecasts into their demand forecasts (Lee and
Dessler, 2022), even as many studies have concluded that climate change will
affect electricity demand in the future (Fan et al., 2019; Auffhammer et al.,
2017; Emodi et al., 2018; Garrido-Perez et al., 2021).

In the spring, the shoulder season defined by degree days has a good
correlation with those defined by total energy use and peak demand, but only
when temperature-defined shoulder seasons starting earlier than February 14
are excluded (Pearson correlation coefficients of 0.80 and 0.76, respectively)
(Figs. 5a,b). Similarly, in the fall, there is good correlation when
temperature-defined shoulder seasons starting later than November 25 are
excluded (Pearson correlation coefficients of 0.71 for total energy use and
0.74 for peak demand) (Figs. 5c,d). Thus, we conclude there is a weaker
correlation between statewide average temperature and electricity usage during
the winter months (December, January, and February), which could be driven by
greater spatial variability in temperatures across the ERCOT service area in
winter. Importantly, as spring weather starts earlier in the year and fall
weather persists later in the year, there will be less of a relationship
between electricity demand and climate, which will complicate planning efforts
for grid reliability.

Figure 5: Correlations between start of 45-day periods of minimum degree days,
minimum total energy use, and minimum peak demand during spring (a, b) and
fall (c, d). The gray data points occur before or after the dates mentioned in
the text.

###  4.4 Effect of renewable energy generation

Renewable energy generation has increased considerably in ERCOT over the last
two decades. From 2007 to 2022, wind, solar, hydroelectric, biomass, and
nuclear generation together grew from 17% of energy generated to 41%, with
wind alone accounting for 25% of generation in 2022. Wind generation in Texas
tends to be the greatest in April and May due to higher wind speeds during
those months (U.S. Energy Information Administration, 2020). Because our
analysis focuses on optimal timing for power plant maintenance, we assessed
whether electricity demand in ERCOT followed the same trends with temperature
and time when the energy use and peak demand being met by non-thermal
generation (wind, solar, hydroelectricity) was removed. Fuel mix data from
ERCOT are only available from 2007 onward, so the time series we used was
shorter than the overall dataset.

We determined that the shoulder seasons determined by total energy use have a
15% probability of having shifted earlier in the spring and an 86% probability
of having shifted later in the fall, while those determined by peak demand
have a 12% probability of having shifted earlier in the spring and a 97%
probability of having shifted later in the fall (Fig. A10). However, we
caution that the short time series we used results in trend line slopes in
spring shoulder season that are not statistically different from those we
obtained by including non-thermal generation, even at a 68% confidence
interval (1 standard deviation). Indeed, the removal of non-thermal generation
moved the onset of spring shoulder season an average of 1.5 days later based
on total energy use and 0.6 days earlier based on peak demand. In the fall, on
the other hand, shoulder seasons started on average 10.8 days later and 8.3
days later.

When compared against the shoulder seasons defined by minimum degree days, we
find that the electricity data with non-thermal generation removed follow a
very similar trend as before, with electricity-defined shoulder seasons
showing good correlation with temperature-defined shoulder seasons when the
temperature-defined shoulder season starts later than February 14 in the
spring and earlier than November 25 in the fall (Fig. A11). The Pearson
correlation coefficients in the spring are 0.85 and 0.81 for total energy use
and peak load, and in the fall are 0.66 and 0.61.

On the basis of this analysis, we conclude that the seasonal variability in
non-thermal generation does not have a significant effect on the spring
shoulder season, and may cause the fall shoulder season to shift even later in
the year. We observe that the changes in temperature-defined shoulder seasons
still track the electricity-defined shoulder seasons except during the winter
months.

###  4.5 Future outlook

Over the time series we analyzed for degree days (1959–2022), we found a high
likelihood that the onset of the 45-day period of minimum degree days shifted
earlier in the spring and later in the fall by 2.4 days/decade and 1.1
days/decade, respectively. However, the likelihood of such shifts in total
power use or peak demand was much lower, which is likely due to the shorter
time series of electricity data (1996–2022 with 2001 omitted). How do we
expect these shoulder seasons to move in the future as a result of climate
change?

The annual average temperature across the ERCOT service area has increased
since 1959 (Fig. 6a) and is expected to continue to do so (Nielsen-Gammon et
al., 2021). To predict future temperatures, we used the Community Earth System
Model Large Ensemble Community Project 2 (LENS2) (Rodgers et al., 2021). LENS2
is an ensemble of 100 different simulations that start at different years in
the 9th and 10th centuries with different perturbations in atmospheric
potential temperature, Atlantic Meridional Overturning Circulation state, and
sea surface height in the Labrador Sea. Future climate projections use the
SSP3-7.0 forcing scenario, which is a high-emissions scenario
(Intergovernmental Panel on Climate Change, 2022).

LENS2 outputs 2 m land temperatures at several temporal resolutions. We used
average monthly temperatures to determine the annual average temperature over
all ensemble members in the ERCOT service region. The annual average LENS2
temperatures displayed a systematic high bias relative to the ERA5
temperatures, so we applied a correction by minimizing the misfit in a least-
squares sense between the ERA5 temperature and LENS2 annual ensemble means for
the period 1959–2022:

$T_{\text{L,corr}}=0.92T_{\text{L}}+1.0,$ (2)

where $T_{\text{L}}$ is the raw LENS2 ensemble mean and $T_{\text{L,corr}}$ is
the corrected LENS2 ensemble mean. Fig. 6a shows the corrected LENS2
temperatures $\pm 2$ standard deviations. Notice how there is less year-to-
year variability in the LENS2 temperatures, likely due to the lower spatial
resolution (1∘ versus 0.25∘ for ERA5).

Figure 6: (a) Average annual temperature over ERCOT service area from ERA5
(not population-weighted) and corrected LENS2 temperature $\pm 2$ standard
deviations, which shows a generally increasing trend. (b) Correlation between
ERA5 average annual temperature and day of onset of temperature-defined spring
shoulder season, which shows an earlier start date over time. (c) Correlation
between ERA5 average annual temperature and day of onset of temperature-
defined fall shoulder season, which shows a later start date over time. (d)
Predicted onset of temperature-defined shoulder seasons using corrected LENS2
predictions.

We observe correlations between average annual temperature and the onset of
shoulder seasons. As the average annual temperature (not population-weighted)
has increased, the temperature-defined spring shoulder season has moved
earlier (Fig. 6b), while the temperature-defined fall shoulder season has
moved later (Fig. 6c). Using the trends shown in Figs. 6b and 6c and the LENS2
temperature predictions, we show that the temperature-defined spring and fall
shoulder seasons may overlap consistently (defined as overlap of the 95%
confidence intervals) in the mid-2040s as a result of this warming (Fig. 6d).
In this scenario, there might be only a single shoulder season centered in
December and January.

Such a shift will make shoulder season for electricity demand more
unpredictable. As shown in Fig. 5, there is little correlation between the
onset of the temperature-defined spring shoulder season and those defined by
total electricity demand and peak load when the temperature-defined shoulder
season starts earlier than February 14. This lack of correlation is due to the
spatial variability of temperature being much higher during this time of year
compared with the rest of the year (Fig. 7). Scheduling power plant
maintenance in January and February is very risky because extreme cold can
occur during these months – for example Winter Storm Uri in 2021 – and a grid
with a significant amount of generation offline for maintenance would struggle
to keep up with demand. At the same time, restricting maintenance to March
through May means that maintenance periods will no longer coincide with
periods of low expected demand. One solution would be to restrict maintenance
to the fall shoulder season, which we show should occur in October and
November, but it may be unreasonable to expect power plants to be able to
forgo spring maintenance.

Figure 7: Standard deviation of daily average temperature (not population-
weighted) across the ERCOT service area averaged over the interval 1996–2021.

Electricity demand during shoulder seasons is also a concern as it dictates
how much generating capacity must be kept online to ensure grid reliability.
Our work shows that the timing of shoulder seasons is related to minima in
heating- and cooling-degree days, and Lee and Dessler (2022) similarly
demonstrated how temperature affects electricity demand. However, electricity
demand in ERCOT is also influenced by other factors, including population
growth, electrification of residential heating (White et al., 2021; Skiles et
al., 2023), electrification of oilfield operations (Lin et al., 2022), and
even cryptocurrency mining (Lee et al., 2023). Over the period 1959–2022, the
average degree days during the temperature-defined spring and fall shoulder
seasons have changed very little (Figs. 8a,b). On the other hand, total energy
use and peak demand during their respective shoulder seasons have generally
increased since 1996 (Figs. 8c,d). These trends must be taken into account for
reserve forecasting during shoulder seasons.

Figure 8: (a) Average degree days during temperature-defined spring shoulder
season by year. (b) Average degree days during temperature-defined fall
shoulder season by year. (c) Average total energy use during spring and fall
shoulder seasons by year. (d) Average peak demand during spring and fall
shoulder seasons by year.

We assessed the probability that demand could exceed supply during December
and January if these become the ideal time of year for power plant
maintenance. ERCOT has published generation outage data at 15-minute intervals
since May 2021. We combined these data with reported generation output for
2020–2022. Fig. 9 shows histograms of generation output for the ERCOT-defined
spring and fall shoulder seasons (March 15–May 1 and October 15–November 30),
the 45-day periods of minimum peak demand in spring and fall for each year,
and January and December. Peak demand for each period is also indicated.
During the ERCOT-defined shoulder seasons, generation is matched very closely
with demand, particularly in the spring, which is consistent with ERCOT’s
strategy of balancing approval of planned outages with forecast demand
(Rickerson, 2022). In the 45-day periods of peak demand, on the other hand,
there is excess generation, and more planned outages could be scheduled for
these intervals. January and December show a very close balance between supply
and demand, and increasing planned outages during these months would present a
challenge.

Figure 9: Histograms of generation output for ERCOT in 2020–2022 over
different time periods. The peak demand for each time period is indicated with
a vertical dashed line. When telemetered output exceeds peak demand,
additional outages could be scheduled. When peak demand coincides with the
maximum telemetered output, supply and demand are balanced and additional
planned outages would risk demand exceeding supply.

Table 1 lists the average generation outages in 2022 for the ERCOT-defined
spring and fall shoulder seasons, the 45-day periods of minimum peak demand in
the spring and fall, and January and December. These numbers include both
forced and scheduled outages, but we assume that the larger values during the
shoulder seasons reflect a greater amount of scheduled outages. The average
outages for December and January together were 16.6 GW while the average for
March 15–May 1 and October 15–November 30 together were 22.1 GW. If we assume
that the difference between these values (5.5 GW) represents the incremental
outages associated with maintenance during the shoulder seasons, we can
determine the extent of a deficit in generation that would occur during a
shoulder season centered on December to January. Table 2 shows the percentage
of ERCOT-wide demand in January and December of 2020–2022 that would have
exceeded the maximum generation capacity for those months if an additional 5.5
GW were offline for planned outages. We show that as low as 1.75% or as high
as 23.1% of demand would not be met in this scenario. If planned maintenance
is unable to be performed at other times of year, shifting planned outages to
December and January will increase the risk of outages unless additional
system-wide generation capacity is added.

Time period | Average outages (GW)
---|---
January 1–January 31 | 10.3
March 15–May 1 | 24.0
October 15–November 30 | 20.2
March 2–April 15 | 22.8
December 1–December 31 | 12.6
December 31, 2022–February 13, 2023 | 8.64
Table 1: Average generation outages for different periods in 2022 and 2023. Month | % of demand not met
---|---
January 2020 | 10.2
December 2020 | 1.75
January 2021 | 2.02
December 2021 | 23.1
January 2022 | 2.02
December 2022 | 2.42
Table 2: Percent of demand that would not have been met with an additional 5.5
MW of planned outages for maintenance during December and January.

##  5 Conclusions

Understanding when minima in electricity demand occur throughout the year, and
how the occurrence of those minima will be affected by climate change, is
essential for scheduling planned generation facility maintenance and
preserving the overall resilience of the electrical grid. In the ERCOT grid,
maintenance is typically scheduled in March to May and September to November.
However, these time periods may not be optimal, as evidenced by recent events
including anomalously hot weather and high electricity demand during the
spring shoulder season. We defined shoulder seasons as the 45-day period with
lowest average degree days, average energy use, or average peak demand. Over
the period 1959–2022, the temperature-defined shoulder seasons shifted earlier
by 2.4 days per decade in the spring and later by 1.1 days per decade in the
fall. The spring shoulder season never started later than March 12 and the
fall shoulder season never started earler than September 29.

Over the shorter period 1996–2022, the shoulder seasons for total energy use
never started later than March 14 or earlier than October 6; while those for
peak load never started later than March 14 or earlier than October 12. We
observed a slight shift earlier in the spring shoulder seasons but not in the
fall shoulder seasons. The period of minimum degree days was well correlated
with the onset of shoulder seasons defined by total energy use and peak demand
when those shoulder seasons began later than February 14 in the spring and any
time in the fall, but very poorly correlated for earlier shoulder season
onsets in the spring. This lack of correlation is likely attributable to
increased variability of temperature over the ERCOT service area during
January and February.

The onset of temperature-defined shoulder seasons is correlated with annual
average temperature across the ERCOT service area in both the spring and fall.
Using corrected LENS2 temperature predictions, these shoulder seasons could
merge into a single shoulder season in December and January by the mid-2040s.
A single merged season potentially poses problems for balancing electricity
generation maintenance with grid stability as shoulder seasons become much
more difficult to predict when minimum degree days occur earlier in the year.
Furthermore, a blended wintertime shoulder season overlaps with short-lived
peak demand events from winter storms. Under this scenario there could be at
least 1.75% or as high as 23.1% chance of demand exceeding supply.

Our results highlight the effects that climate change might have on electrical
grid reliability, and the importance of taking climate change into account in
planning for the future. We expect similar shifts in shoulder season timing to
occur in other electricity markets around the world.

##  6 Acknowledgments

This work was partially supported by the Texas State Energy Conservation
Office, U.S. Army Corps of Engineers ERDC (Energy Research and Development
Center), Artesion Inc., Meta, Grid United, and The Lemelson Foundation. ERCOT
generation and outage data were obtained from https://gridstatus.io.

##  7 Author contributions

Conceptualization, H.D.; Methodology, H.D. and J.D.R.; Investigation, H.D. and
A.P.; Writing – Original Draft, H.D. and J.D.R.; Writing – Review & Editing,
H.D., J.D.R., and M.E.W.; Supervision, H.D.

##  8 Declaration of interests

Webber and Rhodes receive funding from a variety of government agencies,
foundations, and industry, including financial institutions and energy
companies. A full list of supporters is available at
http://www.webberenergygroup.com and list of public consulting reports,
including the funders of those reports, is available at
https://www.ideasmiths.net/reports-publications/. Webber is on the board of
GTI Energy, CTO at Energy Impact Partners, and co-founder and Chairman of
IdeaSmiths LLC, an engineering consulting firm. Rhodes is on the board of
Catalyst Cooperative, and co-founder and CTO of IdeaSmiths LLC. Any opinions,
findings, conclusions or recommendations expressed in this material are those
of the authors and do not necessarily reflect the views of the sponsors,
Energy Impact Partners, GTI Energy, Catalyst Cooperative, The University of
Texas at Austin, Columbia University, or IdeaSmiths LLC. The terms of this
arrangement have been reviewed and approved by the University of Texas at
Austin in accordance with its policy on objectivity in research.

Daigle receives funding from a variety of government agencies, foundations,
and industry. Any opinions, findings, conclusions or recommendations expressed
in this material are those of the authors and do not necessarily reflect the
views of his sponsors.

##  Appendix A Supporting Information

###  A.1 Calculation of percentage of unmet demand with additional planned
outages (Table 2)

In Table 2, we present the percentage of demand that would not be met in
December and January for the years 2020-2022 if an additional 5.5 GW of
planned outages for power plant maintenance had occurred. To obtain this
value, we subtracted 5.5 GW from the maximum telemetered output for the month
in question. We then determined the fraction of hourly peak demand that
exceeded this value during that month. Note that this calculation does not
attempt to match supply and demand hour for hour; we treat telemetered output
and peak demand as random and simply calculate the probability that peak
demand during any hour would not be met.

Year | $a_{1}$ | $a_{2}$ | $a_{3}$ | $a_{4}$ |  $T_{0}$ (∘C)
---|---|---|---|---|---
1996 | 0.9044 | 25.70 | -1263 | 36850 | 14.09
1997 | 1.606 | 4.216 | -1169 | 38050 | 14.73
1998 | 0.4149 | 55.48 | -1765 | 40560 | 13.78
1999 | -0.04506 | 87.09 | -2304 | 43540 | 13.37
2000 | -0.2412 | 98.86 | -2552 | 46660 | 13.58
2002 | 1.285 | 29.02 | -1644 | 44080 | 14.45
2003 | 0.7175 | 56.84 | -2026 | 46100 | 14.08
2004 | 2.236 | -9.091 | -1261 | 44810 | 15.13
2005 | 0.6223 | 71.55 | -2469 | 50480 | 14.51
2006 | 0.02227 | 94.8 | -2644 | 50870 | 13.88
2007 | 1.694 | 20.79 | -1784 | 49590 | 15.09
2008 | 0.6815 | 68.94 | -2500 | 53220 | 14.86
2009 | 0.6481 | 70.95 | -2528 | 52610 | 14.81
2010 | 1.36 | 32.19 | -1909 | 50780 | 15.14
2011 | 0.7497 | 54.08 | -2086 | 51640 | 14.76
2012 | 0.1019 | 97.31 | -2810 | 54600 | 14.13
2013 | 0.7758 | 69.16 | -2568 | 55800 | 14.85
2014 | 1.706 | 26.19 | -2017 | 55590 | 15.38
2015 | 0.972 | 66.39 | -2658 | 58630 | 15.05
2016 | 0.0002521 | 118.7 | -3468 | 62700 | 14.6
2017 | 1.151 | 55.11 | -2374 | 58030 | 14.73
2018 | 1.496 | 41.03 | -2342 | 60690 | 15.47
2019 | 0.4101 | 104.2 | -3411 | 66680 | 15.04
2020 | -1.632 | 221.5 | -5508 | 78980 | 14.87
2021 | 3.013 | -33.43 | -1112 | 56520 | 15.39
2022 | 1.018 | 54.93 | -2313 | 65140 | 14.89
Table 3: Polynomial coefficients used to determine $T_{0}$ by year. The
polynomial has the form
$D=a_{1}T_{avg}^{3}+a_{2}T_{avg}^{2}+a_{3}T_{avg}+a_{4}$, where $D$ is daily
peak demand in MW and $T_{avg}$ is daily average temperature. Figure 10:
Shoulder seasons defined by electricity with energy and power demand met by
non-thermal sources removed. (a) Spring shoulder season defined by total
energy use. (b) Spring shoulder season defined by peak demand. (c) Fall
shoulder season defined by total energy use. (d) Fall shoulder season defined
by peak demand. Linear regression lines with 95% confidence intervals are
shown. Figure 11: Correlations between start of 45-day periods of minimum
degree days, minimum total energy use, and minimum peak demand during spring
(a, b) and fall (c, d). Electricity data have energy and power demand met by
non-thermal generation removed. The gray data points occur before or after the
dates mentioned in the text.

## References

  * Ahmed et al. (2012) Ahmed, T., Muttaqi, K., Agalgaonkar, A., 2012.  Climate change impacts on electricity demand in the state of new south wales, australia.  Applied Energy 98, 376–383.  doi:10.1016/j.apenergy.2012.03.059.
  * Alipour et al. (2019) Alipour, P., Mukherjee, S., Nateghi, R., 2019.  Assessing climate sensitivity of peak electricity load for resilient power systems planning and operation: A study applied to the Texas region.  Energy 185, 1143–1153.  doi:10.1016/j.energy.2019.07.074.
  * Apt et al. (2006) Apt, J., Lave, L.B., Morgan, M.G., 2006.  Power play: A more reliable U.S. electric system.  Issues in Science and Technology 22, 51–58.
  * Auffhammer et al. (2017) Auffhammer, M., Baylis, P., Hausman, C.H., 2017.  Climate change is projected to have severe impacts on the frequency and intensity of peak electricity demand across the united states.  Proceedings of the National Academy of Sciences of the United States of America 114, 1886–1891.  doi:10.1073/pnas.1613193114.
  * Bartos and Chester (2015) Bartos, M., Chester, M., 2015\.  Impacts of climate change on electric power supply in the western united states.  Nature Climate Change 5, 748–752.  doi:10.1038/nclimate2648.
  * Burillo et al. (2019) Burillo, D., Chester, M., Pincetl, S., Fournier, E., Reyna, J., 2019.  Forecasting peak electricity demand for los angeles considering higher air temperatures due to climate change.  Applied Energy 236, 1–9.  doi:10.1016/j.apenergy.2018.11.039.
  * Busby et al. (2021) Busby, J., Baker, K., Bazilian, M., Gilbert, A., Grubert, E., Rai, V., Rhodes, J., Shidore, S., Smith, C., Webber, M., 2021\.  Cascading risks: Understanding the 2021 winter blackout in texas.  Energy Research & Social Science 77, 102106.  doi:10.1016/j.erss.2021.102106.
  * Chabouni et al. (2020) Chabouni, N., Belarbi, Y., Benhassine, W., 2020.  Electricity load dynamics, temperature and seasonality nexus in algeria.  Energy 200, 117513\.  doi:10.1016/j.energy.2020.117513.
  * Chen and Lie (2010) Chen, G., Lie, T., 2010.  The impact of climate change on new zealand’s electricity demand, in: 2010 IEEE 11th International Conference on Probabilistic Methods Applied to Power Systems, pp. 808–813.  doi:10.1109/PMAPS.2010.5528429.
  * CIESIN (2020) CIESIN, 2020.  Gridded population of the world, version 4 (gpwv4): Population count.  https://sedac.ciesin.columbia.edu/data/collection/gpw-v4.
  * Coffel and Mankin (2021) Coffel, E., Mankin, J., 2021\.  Thermal power generation in disadvantaged in a warming world.  Environmental Research Letters 16, 024043.  doi:10.1088/1748-9326/abd4a8.
  * Creed et al. (2015) Creed, I., Hwang, T., Lutz, B., Way, D., 2015.  Climate warming causes intensification of the hydrological cycle, resulting in changes to the vernal and autumnal windows in a northern temperate forest.  Hydrological Processes 15, 3519–3534.  doi:10.1002/hyp.10450.
  * Crimmins and Crimmins (2019) Crimmins, T., Crimmins, M., 2019\.  Biologically-Relevant Trends in Springtime Temperatures Across the United States.  Geophysical Research Letters 46, 12377–12387.  doi:10.1029/2019GL085251.
  * Electric Reliability Council of Texas (2022a) Electric Reliability Council of Texas, 2022a.  Ercot’s use of energy conservation.  https://www.ercot.com/files/docs/2021/08/04/Energy_Conservation_7-13-2022.pdf.
  * Electric Reliability Council of Texas (2022b) Electric Reliability Council of Texas, 2022b.  Methodology for calculating maximum daily resource planned outage capacity.  Electric Reliability Council of Texas, Austin, TX.
  * Electric Reliability Council of Texas (2023) Electric Reliability Council of Texas, 2023\.  Report on the capacity, demand and reserves (cdr) in the ercot region, 2024-2033.  Electric Reliability Council of Texas, Austin, TX.
  * Emodi et al. (2018) Emodi, N.V., Chaiechi, T., Beg, A.R.A., 2018.  The impact of climate change on electricity demand in australia.  Energy & Environment 29, 1263–1297.  doi:10.1177/0958305X18776538.
  * ERCOT Board (2022) ERCOT Board, 2022.  April 28, 2022 board report: Nprr 1108, ercot shall approve or deny all resource outage requests.  https://www.ercot.com/files/docs/2022/05/02/1108NPRR-37%20Board%20Report%20042822.docx.
  * Fan et al. (2019) Fan, J.L., Hu, J.W., Zhang, X., 2019.  Impacts of climate change on electricity demand in china: An empirical estimation based on panel data.  Energy 170, 880–888.  doi:10.1016/j.energy.2018.12.044.
  * Froger et al. (2016) Froger, A., Gendreau, M., Mendoza, J., Pinson, É.., Rousseau, L., 2016.  Maintenance scheduling in the electricity industry: A literature review.  European Journal of Operational Research 251, 695–706.  doi:10.1016/j.ejor.2015.08.045.
  * Garrido-Perez et al. (2021) Garrido-Perez, J.M., Barriopedro, D., García-Herrera, R., Ordónez, C., 2021\.  Impact of climate change on spanish electricy demand.  Climatic Change 165, 50\.  doi:10.1007/s10584-021-03086-0.
  * Grogan et al. (2020) Grogan, D., Burakowski, E., Contosta, A., 2020.  Snowmelt control on spring hydrology declines as the vernal window lengthens.  Environmental Research Letters 15, 114040.  doi:10.1088/1748-9326/abbd00.
  * Hamlet et al. (2010) Hamlet, A., Lee, S., Mickelson, K., Elsner, M., 2010\.  Effects of projected climate change on energy supply and demand in the pacific northwest and washington state.  Climatic Change 102.  doi:10.1007/s10584-010-9857-y.
  * Hersbach et al. (2020) Hersbach, H., Bell, B., Berrisford, P., Hirahara, S., Horányi, A., Muñoz-Sabater, J., Nicolas, J., Peubey, C., Radu, R., Schepers, D., Simmons, A., Soci, C., Abdalla, S., Abellan, X., Balsamo, G., Bechtold, P., Biavati, G., Bidlot, J., Bonavita, M., De Chiara, G., Dahlgren, P., Dee, D., Diamantakis, M., Dragani, R., Flemming, J., Forbes, R., Fuentes, M., Geer, A., Haimberger, L., Healy, S., Hogan, R.J., Hólm, E., Janisková, M., Keeley, S., Laloyaux, P., Lopez, P., Lupu, C., Radnoti, G., de Rosnay, P., Rozum, I., Vamborg, F., Villaume, S., Thépaut, J.N., 2020.  The ERA5 global reanalysis.  Quarterly Journal of the Royal Meteorological Society 146, 1999–2049.
  * Hill et al. (2021) Hill, J., Kern, J., Rupp, D.E., Voisin, N., Characklis, G., 2021\.  The effects of climate change on interregional electricity market dynamics on the u.s. west coast.  Earth’s Future 9, e2021EF002400.  doi:10.1029/2021EF002400.
  * Ho et al. (2021) Ho, C., Park, C., Yun, J., Lee, E., Kim, J., Yoo, H., 2021.  Asymmetric expansion of summer season on may and september in korea.  Asia-Pacific Journal of Atmospheric Science 57, 619–627.  doi:10.1007/s13143-020-00220-3.
  * Intergovernmental Panel on Climate Change (2022) Intergovernmental Panel on Climate Change, 2022\.  Climate Change 2022: Mitigation of Climate Change. Contribution of Working Group II to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change.  Cambridge University Press, Cambridge, UK and New York, NY, USA.  URL: https://report.ipcc.ch/ar6/wg2/IPCC_AR6_WGII_FullReport.pdf, doi:10.1017/9781009325844.
  * Joint Commenters I (2022) Joint Commenters I, 2022.  April 15, 2022 comments on nprr 1108, ercot shall approve or deny all resource outage requests.  https://www.ercot.com/files/docs/2022/04/15/1108NPRR-27%20Joint%20Commenters%20I%20Comments%20041522.docx.
  * Lee and Dessler (2022) Lee, J., Dessler, A.E., 2022\.  The impact of neglecting climate change and variability on ercot’s forecasts of electricity demand in texas.  Weather, Climate, and Society 14, 499–505.  doi:10.1175/WCAS-D-21-0140.1.
  * Lee et al. (2023) Lee, J., Wu, L., Dessler, A.E., 2023.  A quantification of how much crypto-miners are driving up the wholesale cost of energy in texas.  arXiv:2302.02221.
  * Lehikoinen et al. (2019) Lehikoinen, A., Lindén, A., Karlsson, M., Andersson, A., Crewe, T.L., Dunn, E.H., Gregory, G., Karlsson, L., Kristiansen, V., Mackenzie, S., Newman, S., Røer, J.E., Sharpe, C., Sokolov, L.V., Åsa Steinholtz, Stervander, M., Tirri, I.S., Tjørnløv, R.S., 2019\.  Phenology of the avian spring migratory passage in europe and north america: Asymmetric advancement in time and increase in duration.  Ecological Indicators 101, 985–991.  doi:10.1016/j.ecolind.2019.01.083.
  * Lin et al. (2022) Lin, N., McDaid, G., Yan, Q., Lu, Y., Goodman, E., 2022\.  ERCOT West Texas load study: Estimate the electric load impacts of oil and gas related activities.  https://www.ercot.com/files/docs/2022/06/10/ERCOT_West_Texas_Load_Study_Report.pdf.
  * Modala et al. (2017) Modala, N.R., Ale, S., Goldberg, D.W., Olivares, M., Munster, C.L., Rajan, N., Feagin, R.A., 2017.  Climate change projections for the Texas High Plains and Rolling Plains.  Theoretical and Appled Climatology 129, 263–280.  doi:10.1007/s00704-016-1773-2.
  * Munce (2022) Munce, M., 2022.  How to know the difference between a local power outage and rolling blackouts.  The Texas Tribune .
  * Nielsen-Gammon et al. (2021) Nielsen-Gammon, J., Holman, S., Buley, A., Jorgensen, S., Escobedo, J., Ott, C., Dedrick, J., Van Fleet, A., 2021\.  Assessment of Historic and Future Trends of Extreme Weather in Texas, 1900-2036: 2021 Update.  Technical Report OSC-202101. Office of the State Climatologist, Texas A&M University, College Station, TX.
  * O’Shea et al. (2021) O’Shea, M., Goel, R., Miller, R., 2021.  Breaking down the Texas winter blackouts: what went wrong?  Technical Report. Wood Mackenzie.
  * Parkpoom and Harrison (2008) Parkpoom, S., Harrison, G., 2008\.  Analyzing the impact of climate change on future electricity demand in thailand.  IEEE Transactions on Power Systems 23, 1441–1448.  doi:10.1109/TPWRS.2008.922254.
  * Peña-Ortiz et al. (2015) Peña-Ortiz, C., Barriopedro, D., García-Herrera, R., 2015.  Multidecadal variability of the summer length in europe.  Journal of Climate 28, 5375–5388.  doi:10.1175/JCLI-D-14-00429.1.
  * Quayle and Diaz (1980) Quayle, R.G., Diaz, H.F., 1980\.  Heating degree day data applied to residential heating energy consumption.  Journal of Applied Metereology 19, 241–246.  doi:10.1175/1520-0450(1980)019<0241:HDDDAT>2.0.CO;2.
  * Rhodes et al. (2011) Rhodes, J.D., Stephens, B., Webber, M.E., 2011.  Using energy audits to investigate the impacts of common air-conditioning design and installation issues on peak power demand and energy consumption in Austin, Texas.  Energy and Buildings 43, 3271–3278.  doi:10.1016/j.enbuild.2011.08.032.
  * Richmond (2022) Richmond, M., 2022.  Re: Nprr 1108 passed by ercot board.  https://competitivepower.org/wp-content/uploads/2022/05/TCPA-letter-Re-Appeal-Process-and-NPRR-1108-Final.pdf.
  * Rickerson (2022) Rickerson, W., 2022.  Approval of Methodology for Calculating Maximum Daily Resource Planned Outage Capacity – ERCOT Recommendation.  https://www.ercot.com/files/docs/2022/06/13/11.1%20ERCOT%20Recommendation.pdf.
  * Rodgers et al. (2021) Rodgers, K.B., Lee, S.S., Rosenbloom, N., Timmermann, A., Danabasoglu, G., Deser, C., Edwards, J., Kim, J.E., Simpson, I.R., Stein, K., Stuecker, M.F., Yamaguchi, R., Bódai, T., Chung, E.S., Huang, L., Kim, W.M., Lamarque, J.F., Lombardozzi, D.L., Wieder, W.R., Yeager, S.G., 2021\.  Ubiquity of human-induced changes in climate variability.  Earth System Dynamics 12, 1393–1411.  doi:10.5194/esd-12-1393-2021.
  * Romitti and Wing (2022) Romitti, Y., Wing, I., 2022\.  Heterogeneous climate change impacts on electricity demand in world cities circa mid-century.  Scientific Reports 12, 4280\.  doi:10.1038/s41598-022-07922-w.
  * van Ruijven et al. (2019) van Ruijven, B., De Cian, E., Wing, I., 2019.  Amplification of future energy demand growth due to climate change.  Nature Communications 10, 2762\.  doi:10.1038/s41467-019-10399-3.
  * Sams (2022) Sams, B., 2022.  April 21, 2022 tac advocate position statement on nprr 1108, ercot shall approve or deny all resource outage requests.  https://www.ercot.com/files/docs/2022/04/21/1108NPRR-35%20TAC%20Advocate%20Position%20Statement%20042122.docx.
  * Sathaye et al. (2013) Sathaye, J., Dale, L., Larsen, P., Fitts, G., Koy, K., Lewis, S., Pereira de Lucena, A., 2013.  Estimating impacts of warming temperatures on california’s electricity system.  Global Environmental Change 23, 499–511.  doi:10.1016/j.gloenvcha.2012.12.005.
  * Schwartz et al. (2006) Schwartz, M., Ahas, R., Aasa, A., 2006.  Onset of spring starting earlier across the northern hemisphere.  Global Change Biology 12, 343–351.  doi:10.1111/j.1365-2486.2005.01097.x.
  * Shaffer et al. (2022) Shaffer, B., Quintero, D., Rhodes, J., 2022.  Changing sensitivity to cold weather in Texas power demand.  iScience 25, 104173\.  doi:10.1016/j.isci.2022.104173.
  * Shipley et al. (2020) Shipley, J.R., Twining, C.W., Taff, C.C., Vitousek, M.N., Flack, A., Winkler, D.W., 2020\.  Birds advancing lay dates with warming springs face greater risk of chick mortality.  Proceedings of the National Academy of Sciences 117, 25590–25594.  doi:10.1073/pnas.2009864117.
  * Skiles et al. (2023) Skiles, M.J., Rhodes, J.D., Webber, M.E., 2023.  Perspectives on peak demand: How is ERCOT peak electric load evolving in the context of changing weather and heating electrification?  The Electricity Journal 36, 107254\.  doi:10.1016/j.tej.2023.107254.
  * Taylor (1981) Taylor, B.L., 1981.  Population-weighted heating degree-days for Canada.  Atmosphere-Ocean 19, 261–268.  doi:10.1080/07055900.1981.9649113.
  * Texas Water Development Board (2022) Texas Water Development Board, 2022.  2022 state water plan: Water for texas.  Texas Water Development Board, Austin, TX.
  * Thompson and Clark (2008) Thompson, R., Clark, R., 2008\.  Is spring starting earlier?  The Holocene 18, 95–104.  doi:10.1177/0959683607085599.
  * Trotter et al. (2016) Trotter, I., Bolkesjø, T., Féres, J., Hollanda, L., 2016\.  Climate change and electricity demand in brazil: A stochastic approach.  Energy 102, 596–604.  doi:10.1016/j.energy.2016.02.120.
  * U.S. Census Bureau (2021) U.S. Census Bureau, 2021.  Quickfacts: Texas.  https://www.census.gov/quickfacts/TX.
  * U.S. Energy Information Administration (2020) U.S. Energy Information Administration, 2020\.  Today in energy: Wind is a growing part of the electricity mix in texas.  https://www.eia.gov/todayinenergy/detail.php?id=45476.
  * U.S. Energy Information Administration (2021) U.S. Energy Information Administration, 2021\.  State electricity profiles.  https://www.eia.gov/electricity/state/.
  * U.S. Energy Information Administration (2022) U.S. Energy Information Administration, 2022\.  State energy profiles.  https://www.eia.gov/state/?sid=TX.
  * van Vliet et al. (2012) van Vliet, M., Yearsley, J., Ludwig, F., Vögele, S., Lettenmaier, D., Kabat, P., 2012\.  Vulnerability of us and european electricity supply to climate change.  Nature Climate Change 2, 676–681.  doi:10.1038/nclimate1546.
  * White and Rhodes (2019) White, P.R., Rhodes, J.D., 2019\.  Electrification of Heating in the Texas Residential Sector.  Technical Report. IdeaSmiths, LLC.
  * White et al. (2021) White, P.R., Rhodes, J.D., Wilson, E.J., Webber, M.E., 2021\.  Quantifying the impact of residential space heating electrification on the texas electric grid.  Applied Energy 298, 117113\.  doi:10.1016/j.apenergy.2021.117113.
  * Zamuda et al. (2018) Zamuda, C., Bilello, D., Conzelmann, G., Mecray, E., Satsangi, A., Tidwell, V., Walker, B., 2018.  Energy supply, delivery, and demand, in: Reidmiller, D., Avery, C., Easterling, D., Kunkel, K., Lewis, K., Maycock, T., Stewart, B. (Eds.), Impacts, Risks, and Adaptation in the United States: Fourth National Climate Assessment. U.S. Global Change Research Program, Washington, D.C.. volume II, pp. 174–201.  doi:10.7930/NCA4.2018.CH4.
  * Zhang et al. (2021) Zhang, Y., Gowda, P., Brown, D., Rice, C., Zambreski, Z., Kutikoff, S., Lin, X., 2021.  Time-varying trends in frost indicators in the U.S. Southern Great Plains.  International Journal of Climatology 41, 1264–1278.  doi:10.1002/joc.6803.


# Magneto-optical Kerr and Faraday effects in bilayer antiferromagnetic
insulators

Wan-Qing Zhu  Department of Physics, School of Physics and Materials Science,
Guangzhou University, Guangzhou 510006, China  Wen-Yu Shan<EMAIL_ADDRESS>Department of Physics, School of Physics and Materials Science, Guangzhou
University, Guangzhou 510006, China

###### Abstract

Control and detection of antiferromagnetic topological materials are
challenging since the total magnetization vanishes. Here we investigate the
magneto-optical Kerr and Faraday effects in bilayer antiferromagnetic
insulator MnBi2Te4. We find that by breaking the combined mirror symmetries
with either perpendicular electric field or external magnetic moment, Kerr and
Faraday effects occur. Under perpendicular electric field, antiferromagnetic
topological insulators (AFMTI) show sharp peaks at the interband transition
threshold, whereas trivial insulators show small adjacent positive and
negative peaks. Gate voltage and Fermi energy can be tuned to reveal the
differences between AFMTI and trivial insulators. We find that AFMTI with
large antiferromagnetic order can be proposed as a pure magneto-optical
rotator due to sizable Kerr (Faraday) angles and vanishing ellipticity. Under
external magnetic moment, AFMTI and trivial insulators are significantly
different in the magnitude of Kerr and Faraday angles and ellipticity. For the
qualitative behaviors, AFMTI shows distinct features of Kerr and Faraday
angles when the spin configurations of the system change. These phenomena
provide new possibilities to optically detect and manipulate the layered
topological antiferromagnets.

##  I Introduction

Recent years have seen a surge of interest in antiferromagnetic (AFM)
materials due to their advantages in spintronics applications: they are robust
against perturbation due to magnetic field, display ultrafast dynamics, and so
on Jungwirth  _et al._ (2016); Baltz  _et al._ (2018). When combining AFM with
topology, new topological phenomena arise, including the condensed matter
version of axions Li  _et al._ (2010); Nenno  _et al._ (2020); Liu  _et al._
(2020), quantized magnetoelectric couplings Essin  _et al._ (2009); Zhang  _et
al._ (2019); Armitage and Wu (2019) and anomalous Hall effect Chen  _et al._
(2014); Nakatsuji  _et al._ (2015); Nayak  _et al._ (2016). These topological
effects largely depend on the spin structures of AFM systems. For example, in
noncollinear triangular AFM, anomalous Hall effect occurs due to the chirality
of spin structures Chen  _et al._ (2014); Nakatsuji  _et al._ (2015); Nayak
_et al._ (2016). In noncoplanar AFM, quantized magneto-optical responses arise
from the finite scalar spin chirality Feng  _et al._ (2020).

An intriguing example is the van der Waals compound MnBi2Te4, which, as an
A-type antiferromagnetic topological insulator (AFMTI) Otrokov  _et al._
(2019a, b); Chen  _et al._ (2019); Hao  _et al._ (2019); Swatek  _et al._
(2020), shows anti-parallel magnetization between adjacent layers and zero
total magnetization. Due to the combined $\mathcal{PT}$ symmetry
($\mathcal{P}$ and $\mathcal{T}$ stand for inversion and time-reversal
symmetry), even-layered MnBi2Te4 is reported to exhibit large longitudinal
resistance with zero Hall plateau Liu  _et al._ (2020). Accordingly, the even-
layered system is predicted to host the exotic axion insulator phase Zhang
_et al._ (2019); Liu  _et al._ (2020); Zhang  _et al._ (2020a). One evidence
distinguishing AFMTI, i.e., axion insulator, from trivial insulator is the
layer Hall effect Gao  _et al._ (2021); Chen  _et al._ (2022), where electrons
from top and bottom layers move in opposite transverse directions due to the
layer-locked Berry curvature. Other proposals based on different transport
phenomena are also raised to identify the axion state Chen  _et al._ (2021);
Ding  _et al._ (2020); Zhang  _et al._ (2020b); Li  _et al._ (2021); Gu  _et
al._ (2021); Dai  _et al._ (2022). By contrast, optical probe of AFMTI or
axion insulator has rarely been discussed Ahn  _et al._ (2022); Lei and
MacDonald (2023); Qiu  _et al._ (2023). Particularly, for few even-layer
MnBi2Te4, whether AFMTI and trivial phases are distinguishable in optical
responses remains elusive.

In this work, we study the magneto-optical Kerr and Faraday effects in bilayer
AFM insulators MnBi2Te4. We find that to allow non-vanishing Kerr and Faraday
effects, the combined mirror $\mathcal{M}_{1}=\mathcal{M}_{x}\mathcal{M}_{z}$
and $\mathcal{M}_{2}=\mathcal{M}_{y}\mathcal{M}_{z}$ symmetries
($\mathcal{M}_{x/y/z}$ stands for mirror symmetry) should be broken by adding
either perpendicular electric field or external magnetic moment (see Fig. 1).
In the former case, we find that Kerr and Faraday angles of AFMTI have
significant peaks at the transition threshold as a result of non-monotonic
band dispersion, in contrast to small adjacent positive and negative peaks for
trivial insulators. For large AFM order, it may be useful as a pure magneto-
optical rotator due to sizable Kerr and Faraday angles and vanishing
ellipticity. In the latter case, AFMTI shows much larger Kerr and Faraday
angles and ellipticity than trivial insulators. Particularly, Kerr and Faraday
angles of AFMTI are very sensitive to the change of spin configurations of
ground state. These phenomena provides new features to distinguish AFMTI and
trivial phases, and optially characterize the layered AFM materials.

This paper is organized as follows. In Sec. II, we introduce model and
methods, including model Hamiltonian and symmetries, Kerr and Faraday effects.
The results for bilayer AFM insulators MnBi2Te4 are given in Sec. III,
including symmetry breaking by perpendicular electric field and external
magnetic moment. Finally, conclusions and discussions are made in Sec. IV.

##  II Model and methods

Figure 1: Schematic illustration of magneto-optical Kerr and Faraday effects
in bilayer AFM insulator MnBi2Te4 (a) with asymmetric potential $V$ induced by
perpendicular electric field and (b) with external magnetic moment $m_{s}$.
$\theta_{K}$ and $\theta_{F}$ refer to the Kerr and Faraday angles.

We consider a bilayer A-type AFM insulator MnBi2Te4 (layer number $n_{z}=2$)
with out-of-plane magnetization. The tight-binding model is given by

$\begin{split}\mathcal{H}&=\sum_{\bm{k},z}c^{+}_{\bm{k},z}[h_{a}(\bm{k},z)+V(z)+m_{s}(z)]c_{\bm{k},z}\\\
&+[\sum_{\bm{k},z}c^{+}_{\bm{k},z}h_{b}(\bm{k},z)c_{\bm{k},z+1}+\mathrm{H.c.}],\end{split}$
(1)

where the layer index $z=1,2$. The intralayer (interlayer) Hamiltonian
$h_{a/b}(\bm{k},z)$ read Zhang  _et al._ (2019, 2020b); Chen  _et al._ (2021);
Ding  _et al._ (2020); Li  _et al._ (2021); Chen  _et al._ (2022)

$\begin{split}h_{a}(\bm{k},z)&=M(\bm{k})\sigma_{z}+A_{2}\sum_{\alpha=x,y}\sin
k_{\alpha}s_{\alpha}\sigma_{x}+m(z)s_{z},\\\
h_{b}(\bm{k},z)&=B_{1}\sigma_{z}-i\frac{A_{1}}{2}s_{z}\sigma_{x}.\end{split}$
(2)

Pauli matrices $s$ and $\sigma$ act in the spin and orbital subspaces,
respectively. $M(\bm{k})=M_{0}-2B_{1}-2B_{2}(2-\cos k_{x}-\cos k_{y})$.
$M_{0}$, $B_{i}$ and $A_{i}$ are model parameters, where $i=1,2$.
$m(z)=m(-1)^{z}$ is the AFM order in each layer. $V(z)=V[2z-n_{z}-1]$ and
$m_{s}(z)=m_{s}s_{z}$ are the layer-dependent potential and layer-independent
magnetism due to external electric field and magnetic moment of substrates,
respectively (see Fig. 1). When $V=m_{s}=0$, the system describes a trivial
insulator for $M_{0}<0$, and an AFMTI for $M_{0}>0$ Chen  _et al._ (2021,
2022).

The space group of MnBi2Te4 is $D^{5}_{3d}$ (No.166) in the absence of
magnetic order. The presence of AFM order $m(z)$ breaks both time-reversal
symmetry $\mathcal{T}=is_{y}\mathcal{K}$ and inversion symmetry
$\mathcal{P}=\sigma_{z}$, whereas it preserves the combined
$\mathcal{P}\mathcal{T}$ symmetry (see Table 1). Since
$(\mathcal{P}\mathcal{T})^{2}=-1$, there is Kramer’s degeneracy at each
momentum of Brillouin zone. In addition, the system preserves the combined
symmetries $\mathcal{M}_{1}=\mathcal{M}_{x}\mathcal{M}_{z}$ and
$\mathcal{M}_{2}=\mathcal{M}_{y}\mathcal{M}_{z}$, where
$\mathcal{M}_{\alpha=x,y,z}$ is the mirror reflection symmetry operation
$\alpha\rightarrow-\alpha$. These combined mirror symmetries correspond to the
combined rotational symmetries $\mathcal{C}_{2x}\mathcal{C}_{2z}$ and
$\mathcal{C}_{2y}\mathcal{C}_{2z}$ in the system Zhang  _et al._ (2019)
($\mathcal{C}_{2\alpha}$ is the twofold rotational symmetry operation about
$\alpha$ axis, $\alpha=x,y,z$). When either $V(z)$ or $m_{s}(z)$ term is
introduced, both $\mathcal{P}\mathcal{T}$ and $\mathcal{M}_{1/2}$ symmetries
are broken.

Table 1:  Symmetries of even-layered AFM Hamiltonian $h_{a/b}(\bm{k},z)$ satisfying $\hat{O}^{-1}h_{a/b}(\bm{k},z)\hat{O}=h(\bm{k}^{\prime},z^{\prime})$, where $\hat{O}$ is the symmetry operator. $\surd$ ($\times$) implies the presence (absence) of the symmetry relations. $(\bm{k}^{\prime},z^{\prime})=\hat{O}^{-1}(\bm{k},z)$ is indicated and $\bar{z}=n_{z}+1-z$. | $(\bm{k}^{\prime},z^{\prime})$ | $h_{a}(\bm{k},z)$ | $h_{b}(\bm{k},z)$ | $V(z)$ | $m_{s}(z)$
---|---|---|---|---|---
$\mathcal{T}=is_{y}\mathcal{K}$ | $(-\bm{k},z)$ | $\times$ | $\surd$ | $\surd$ | $\times$
$\mathcal{P}=\sigma_{z}$ | $(-\bm{k},\bar{z})$ | $\times$ | $\surd$ | $\times$ | $\surd$
$\mathcal{P}\mathcal{T}=is_{y}\sigma_{z}\mathcal{K}$ | $(\bm{k},\bar{z})$ | $\surd$ | $\surd$ | $\times$ | $\times$
$\mathcal{M}_{1}=s_{x}\sigma_{z}$ | $(-k_{x},k_{y},\bar{z})$ | $\surd$ | $\surd$ | $\times$ | $\times$
$\mathcal{M}_{2}=s_{y}\sigma_{z}$ | $(k_{x},-k_{y},\bar{z})$ | $\surd$ | $\surd$ | $\times$ | $\times$

When a linearly polarized light is incident into magnetic materials, there are
phase differences between right-circularly and left-circularly polarized
components for reflected and transmitted light. These lead to rotations of
polarization axis (see Fig. 1), namely magneto-optical Kerr and Faraday
effect, respectively. Such phenomena are not restricted to bulk samples, but
widely used in studying spin structures of few-layer materials Nandkishore and
Levitov (2011); Crassee  _et al._ (2011); Shimano  _et al._ (2013); Sivadas
_et al._ (2016); Huang  _et al._ (2017); Gong  _et al._ (2017). Since few-
layer materials are much thinner than the light wavelength, it is convenient
to treat them as interfaces between vacuum and substrate. As a result, Kerr
and Faraday angles are given by Freiser (1968); Tse and MacDonald (2010);
Gorbar  _et al._ (2012); Tse and MacDonald (2011); G.  _et al._ (2020)

$\begin{split}\theta_{K}&=\frac{1}{2}[\arg(r_{-})-\arg(r_{+})],\\\
\theta_{F}&=\frac{1}{2}[\arg(t_{-})-\arg(t_{+})],\end{split}$ (3)

where the reflection and transmission coefficients for the circularly-
polarized light read ($\pm$ for left- and right-hand)

$\begin{split}r_{\pm}&=\frac{1-\sqrt{\frac{\epsilon_{r}}{\mu_{r}}}-Z_{0}\sigma_{\pm}}{1+\sqrt{\frac{\epsilon_{r}}{\mu_{r}}}+Z_{0}\sigma_{\pm}},\
\
t_{\pm}=\frac{2}{1+\sqrt{\frac{\epsilon_{r}}{\mu_{r}}}+Z_{0}\sigma_{\pm}}.\end{split}$
(4)

$Z_{0}=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}\approx 376.7\Omega$ is the
impedance of vacuum, where $\epsilon_{0}$ and $\mu_{0}$ are vacuum
permittivity and permeability, respectively. The magnetic permeability
$\mu_{r}=1$ and dielectric constant $\epsilon_{r}=4$ for SiO2 substrate Tse
and MacDonald (2011); G.  _et al._ (2020). Here $\sigma_{\pm}=$ $\sigma_{xx}$
$\pm$ $i\sigma_{xy}$, where $\sigma_{xx}$ ($\sigma_{xy}$) is the longitudinal
(Hall) optical conductivity tensor. Additionally, the Kerr and Faraday
ellipticity $\gamma_{K}$, $\gamma_{F}$ can be defined Freiser (1968); G.  _et
al._ (2020)

$\begin{split}\tan\gamma_{K}=\frac{|r_{-}|-|r_{+}|}{|r_{-}|+|r_{+}|},\ \
\tan\gamma_{F}=\frac{|t_{-}|-|t_{+}|}{|t_{-}|+|t_{+}|}.\end{split}$ (5)

By combining $\theta_{K}$, $\theta_{F}$ and $\gamma_{K}$, $\gamma_{F}$,
complex Kerr and Faraday angles can be introduced Kahn  _et al._ (1969)

$\begin{split}\phi_{K}&=\theta_{K}+i\gamma_{K},\ \
\phi_{F}=\theta_{F}+i\gamma_{F}.\end{split}$ (6)

By Kubo formula, the finite-frequency conductivity tensor follows mah

$\begin{split}&\sigma_{\alpha\beta}(\omega)=\\\
&i\hbar\sum_{\mu\mu^{\prime}}\int\frac{d^{2}\bm{k}}{(2\pi)^{2}}\frac{f_{\bm{k}\mu}-f_{\bm{k}\mu^{\prime}}}{\epsilon_{\bm{k}\mu}-\epsilon_{\bm{k}\mu^{\prime}}}\frac{\langle\bm{k},\mu|j_{\alpha}|\bm{k},\mu^{\prime}\rangle\langle\bm{k},\mu^{\prime}|j_{\beta}|\bm{k},\mu\rangle}{\omega+\epsilon_{\bm{k}\mu}-\epsilon_{\bm{k}\mu^{\prime}}+i\hbar/2\tau_{s}},\end{split}$
(7)

where $\epsilon_{\bm{k}\mu}$ and $|\bm{k},\mu\rangle$ are the eigenvalues and
eigenstates of Hamiltonian $\mathcal{H}$. $\bm{k}=(k_{x},k_{y})$ is the wave
vector and $\mu,\mu^{\prime}=\\{1,2,\cdots,4n_{z}\\}$ are band indices. At
zero temperature, the Fermi-Dirac distribution
$f_{\bm{k}\mu}=1/[1+\exp((\epsilon_{\bm{k}\mu}-\epsilon_{F})/k_{B}T)]=\Theta(\epsilon_{F}-\epsilon_{\bm{k}\mu})$,
where $\epsilon_{F}$ is the Fermi energy and $\Theta(...)$ is the Heaviside
function. $\omega$ is the photon energy and $\tau_{s}$ is the relaxation time.
The current operator $j_{\alpha}=(e/\hbar)\partial h(\bm{k})/\partial
k_{\alpha}$, $\alpha=x,y$, where the effective Hamiltonian is established by
$h(\bm{k})=\bigoplus_{z=1}^{n_{z}=2}[h_{a}(\bm{k},z)+h_{b}(\bm{k},z)+V(z)+m_{s}(z)]$
from Eq. (2).

Note that for even-layered MnBi2Te4 with $V=m_{s}=0$, the Hall conductivity
$\sigma_{xy}(\omega)=0$ due to the presence of combined mirror symmetries
$\mathcal{M}_{1/2}$ rather than $\mathcal{P}\mathcal{T}$ symmetry, as shown in
Appendix A. When $V\neq 0$ or $m_{s}\neq 0$, neither of symmetry
$\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ is preserved, giving rise to nonzero
$\sigma_{xy}$, and hence the Kerr and Faraday effect.

Figure 2:  Band dispersions of bilayer AFM insulator MnBi2Te4 with gate
voltage $V$ for parameters (a) $M_{0}=0.4$, (b) $M_{0}=0$ and (c)
$M_{0}=-0.2$. The dispersions are plotted along the high-symmetry lines of
Brillouin zone, as indicated in the inset of (a). In the absence of potential
$V$, $M_{0}=0.4$ and $-0.2$ correspond to AFMTI and trivial insulator,
respectively. In the presence of potential $V$, the Chern number of each band
is labeled. The optical interband transition threshold $T_{i=1,2,3,4}$ for low
photon energy are depicted by arrows. Parameters: $A_{1}=A_{2}=0.55$,
$B_{1}=B_{2}=0.25$ Li  _et al._ (2021); Chen  _et al._ (2022), $m=0.12$,
$V=0.3$, $\epsilon_{F}=-0.4$.

##  III Results

###  III.1 Symmetry breaking by perpendicular electric field

In this section, we show numerical results of optical conductivities, Kerr and
Faraday angles and ellipticity for bilayer AFM insulator due to symmetry
breaking by perpendicular electric field. Band dispersions are plotted in Fig.
2 for parameters $M_{0}=0.4$, $0$ and $-0.2$, respectively. In the absence of
$V$, $M_{0}=0.4$ and $-0.2$ correspond to AFMTI and trivial insulator,
respectively. In the presence of $V$, the band edge of AFMTI ($M_{0}=0.4$) is
moved away from the $\Gamma$ point, whereas trivial insulator ($M_{0}=-0.2$)
shows monotonic dispersion around the $\Gamma$ point. The non-monotonic
dispersion of AFMTI arises from the band inversion of two highest bands below
(or two lowest bands above) zero energy. This is evidenced by the value of
Chern number of each band (see Fig. 2 (a)), from which one can understand the
origin of band inversion. Numerical results suggest that the topological phase
transition takes place for $M_{0}=0.07$, when these bands touch at the
$\Gamma$ point. For clarity, we plot the dispersion for $M_{0}=0$ in Fig. 2
(b), whose dispersion is quite similar to $M_{0}=0.07$. When the Fermi energy
$\epsilon_{F}=0$, the summation of Chern number $C$ of occupied bands
vanishes, leading to zero static Hall conductivity $\sigma_{xy}(\omega=0)$. To
avoid this, we set the Fermi level $\epsilon_{F}=-0.4$. In this case, we label
the optical interband transition threshold $T_{i=1,2,3,4}$ by arrows in Fig.
2, which are useful in understanding the behaviors of optical conductivities,
Kerr and Faraday angles at low photon energy $\omega$.

Figure 3:  Real and imaginary parts of optical conductivities (a)-(b)
$\sigma_{xx}$ and (c)-(d) $\sigma_{xy}$ (in units of $e^{2}/h$) as functions
of photon energy $\omega$ for bilayer AFM insulator MnBi2Te4. The arrows refer
to the optical interband transitions $T_{i=1,2,3,4}$ as shown in Fig. 2. The
values of $\mathrm{Re}(\sigma_{xy})$ in the low-energy limit are highlighted
in red and green region in (c). Parameters: $m=0.12$, $V=0.3$,
$\epsilon_{F}=-0.4$, $\epsilon_{r}=4$ Tse and MacDonald (2011); G.  _et al._
(2020), $\mu_{r}=1$, $\hbar/\tau_{s}=0.02$.

The real and imaginary parts of optical conductivities $\sigma_{xx}$ and
$\sigma_{xy}$ are shown in Fig. 3. Here we only focus on the features at low
photon energy $\omega$. One prominent feature is that the magnitude of
$\sigma_{xx}$ and $\sigma_{xy}$ for AFMTI ($M_{0}=0.4$) is much larger than
trivial insulator ($M_{0}=-0.2$), as a result of band inversion. The behaviors
between $\mathrm{Re}[\sigma_{xx}]$ and $\mathrm{Im}[\sigma_{xy}]$ (or
$\mathrm{Im}[\sigma_{xx}]$ and $\mathrm{Re}[\sigma_{xy}]$) are similar,
whereas the peaks (jumps) of $\mathrm{Re}[\sigma_{xx}]$ and
$\mathrm{Im}[\sigma_{xy}]$ correspond to the jumps (peaks) of
$\mathrm{Im}[\sigma_{xx}]$ and $\mathrm{Re}[\sigma_{xy}]$ due to the Kramers-
Kr$\ddot{o}$nig relations giu . For AFMTI ($M_{0}=0.4$),
$\mathrm{Re}[\sigma_{xy}]$ shows sharp peaks at the interband transition
threshold $T_{1}$ and $T_{2}$, as shown in Fig. 3 (c). By contrast, for
trivial insulator ($M_{0}=-0.2$), $\mathrm{Re}[\sigma_{xy}]$ exhibits two
small adjacent positive and negative peaks, which arise from the appearance
and disappearance of transition $T_{4}$. The peaks are close to each other
since the energy window of the transition is narrow as a result of monotonic
band dispersion (see Fig. 2 (c)). The value of $\mathrm{Re}(\sigma_{xy})$ in
the static limit $\omega\rightarrow 0$ is highlighted in Fig. 3 (c), which is
proportional to the integral of Berry curvature of occupied bands.

Figure 4:  (a) Kerr and (c) Faraday angles and (b) Kerr and (d) Faraday
ellipticity as functions of photon energy $\omega$ for bilayer AFM insulator
MnBi2Te4. The arrows refer to the optical interband transitions
$T_{i=1,2,3,4}$ as shown in Fig. 2. The values of $\theta_{K}$ and
$\theta_{F}$ in the low-energy limit are highlighted in (a) and (b).
Parameters: $m=0.12$, $V=0.3$, $\epsilon_{F}=-0.4$, $\epsilon_{r}=4$ Tse and
MacDonald (2011); G.  _et al._ (2020), $\mu_{r}=1$, $\hbar/\tau_{s}=0.02$.

The Kerr and Faraday angles and ellipticity are plotted in Fig. 4.
$\theta_{K}$ and $\theta_{F}$ (also $\gamma_{K}$ and $\gamma_{F}$) are
complementary to each other since they refer to the reflected and transmitted
light, respectively. We find that the magnitude of $\theta_{K/F}$ and
$\gamma_{K/F}$ is largely enhanced for AFMTI ($M_{0}=0.4$), as compared to
trivial insulator ($M_{0}=-0.2$). Moreover, it is clear that $\theta_{K/F}$
inherits the main features from $\mathrm{Re}(\sigma_{xy})$, and shows sharp
peaks at the interband absorption threshold $T_{1}$ and $T_{2}$ for
$M_{0}=0.4$. On the other hand, $\gamma_{K/F}$ inherits the main features from
$\mathrm{Im}(\sigma_{xy})$, and shows abrupt jumps at transitions $T_{1}$ and
$T_{2}$ for $M_{0}=0.4$. Therefore the measurement of Kerr and Faraday angles,
ellipticity, and also optical conductivities, provides a way to distinguish
between AFMTI and trivial insulator.

Figure 5:  (a) Kerr and (c) Faraday angles and (b) Kerr and (d) Faraday
ellipticity as functions of $M_{0}$ for bilayer AFM insulator MnBi2Te4.
Topological phase transition occurs for $M_{0}=0.07$, as indicated by the blue
dashed line. Red and green lines correspond to Fermi energy
$\epsilon_{F}=-0.4$ and $-0.2$, respectively. Parameters: $m=0.12$, $V=0.3$,
$\omega=0.2$, $\epsilon_{r}=4$ Tse and MacDonald (2011); G.  _et al._ (2020),
$\mu_{r}=1$, $\hbar/\tau_{s}=0.02$.

Since tuning $M_{0}$ can drive a topological phase transition between AFMTI
and trivial insulator, we show $M_{0}$ dependence of Kerr and Faraday angles
and ellipticity in Fig. 5. It is clear that nontrivial AFMTI has much larger
magnitude of $\theta_{K/F}$ and $\gamma_{K/F}$ than trivial phase. The
topological phase transition occurs when $M_{0}=0.07$, where $\theta_{K/F}$
reaches a peak for Fermi energy $\epsilon_{F}=-0.4$ while $\gamma_{K/F}$ is
slightly away from the peak. For higher Fermi level $\epsilon_{F}=-0.2$, the
peaks are further moved away from the transition point since larger $M_{0}$ is
required to reduce the band gap and enhance the valence band edge.

Figure 6:  (a) Kerr and (c) Faraday angles and (b) Kerr and (d) Faraday
ellipticity as functions of AFM order $m$ for bilayer AFM insulator MnBi2Te4.
Red and green lines correspond to AFMTI ($M_{0}=0.4$) and trivial insulator
($M_{0}=-0.2$), respectively. Parameters: $V=0.3$, $\omega=0.2$,
$\epsilon_{F}=-0.4$, $\epsilon_{r}=4$ Tse and MacDonald (2011); G.  _et al._
(2020), $\mu_{r}=1$, $\hbar/\tau_{s}=0.02$.

AFM order $m$ dependence of Kerr and Faraday angles and ellipticity is plotted
in Fig. 6, where $\theta_{K/F}$ and $\gamma_{K/F}$ flip signs under the
inversion of magnetic moment $m$. For AFMTI ($M_{0}=0.4$), the magnitude of
$\theta_{K/F}$ increases when $m$ increases, then it almost saturates when
only one valence band is intersected by the Fermi level. By contrast, the
magnitude of $\theta_{K/F}$ for trivial insulator ($M_{0}=-0.2$) is much
smaller. Interestingly, for AFMTI, when the magnetic moment $m$ is large,
$\theta_{K/F}$ has sizable value with almost vanishing ellipticity. This means
the system can be used as a pure magneto-optical rotator.

Figure 7:  (a) Kerr and (c) Faraday angles and (b) Kerr and (d) Faraday
ellipticity as functions of gate voltage $V$ for bilayer AFM insulator
MnBi2Te4. Red and green lines correspond to AFMTI ($M_{0}=0.4$) and trivial
insulator ($M_{0}=-0.2$), respectively. Parameters: $m=0.12$, $\omega=0.2$,
$\epsilon_{F}=-0.4$, $\epsilon_{r}=4$ Tse and MacDonald (2011); G.  _et al._
(2020), $\mu_{r}=1$, $\hbar/\tau_{s}=0.02$.

Tuning gate voltage $V$ is more feasible in experiments, thus we provide the
$V$ dependence of Kerr and Faraday angles and ellipticity in Fig. 7. Similar
to the reciprocal behaviors under magnetic moment, both $\theta_{K/F}$ and
$\gamma_{K/F}$ flip signs under the inversion of gate voltage $V$. For AFMTI
($M_{0}=0.4$), we find that $\theta_{K/F}$ is negligibly small when
$|V|<0.12$, and it is largely enhanced when $|V|>0.12$. By contrast, for
trivial insulator ($M_{0}=-0.2$), $\theta_{K/F}$ is always tiny. The turning
point $V=\pm 0.12$ for AFMTI is due to the occurrence of non-monotonic band
dispersions near the $\Gamma$ point, as shown in Fig. 2 (a).

Figure 8:  (a) Kerr and (c) Faraday angles and (b) Kerr and (d) Faraday
ellipticity as functions of Fermi energy $\epsilon_{F}$ for bilayer AFM
insulator MnBi2Te4. Red and green lines correspond to AFMTI ($M_{0}=0.4$) and
trivial insulator ($M_{0}=-0.2$), respectively. Parameters: $m=0.12$, $V=0.3$,
$\omega=0.2$, $\epsilon_{r}=4$ Tse and MacDonald (2011); G.  _et al._ (2020),
$\mu_{r}=1$, $\hbar/\tau_{s}=0.02$.

Another tunable parameter in experiments is the Fermi energy $\epsilon_{F}$,
whose dependence of Kerr and Faraday angles and ellipticity is shown in Fig.
8. Both $\theta_{K/F}$ and $\gamma_{K/F}$ flip signs under the inversion of
Fermi energy $\epsilon_{F}$. When $|\epsilon_{F}|>0.05$, the Fermi level
starts to intersect the bands, giving rise to largely enhanced $\theta_{K/F}$
and $\gamma_{K/F}$ for AFMTI while vanishingly small value for trivial
insulator. This provides a new feature to distinguish between AFMTI and
trivial insulator.

###  III.2 Symmetry breaking by external magnetic moment $m_{s}$

Figure 9:  Band dispersions of bilayer AFM insulator MnBi2Te4 with external
magnetic moment $m_{s}$ for parameters (a) $M_{0}=0.4$ and (b) $M_{0}=-0.2$.
In the absence of magnetic moment $m_{s}$, $M_{0}=0.4$ and $-0.2$ correspond
to AFMTI and trivial insulator, respectively. In the presence of $m_{s}$, the
Chern number of each band is labeled. The optical interband transition
threshold $T_{i=1,2,3}$ for low-energy photons are depicted by arrows.
Parameters: $A_{1}=A_{2}=0.55$, $B_{1}=B_{2}=0.25$ Li  _et al._ (2021); Chen
_et al._ (2022), $m=0.12$, $m_{s}=0.18$, $\epsilon_{F}=0$.

In this section, we show numerical results of Kerr and Faraday angles and
ellipticity due to symmetry breaking by external magnetic moment $m_{s}$. Band
dispersions are plotted in Fig. 9 for parameters $M_{0}=0.4$ and $-0.2$,
respectively. In the presence of $m_{s}$, AFMTI ($M_{0}=0.4$) become Chern
insulators, as indicated by the Chern number of each band. Here, for
simplicity, we set the Fermi energy $\epsilon_{F}=0$. Some crucial interband
transition threshold $T_{i=1,2,3}$ are labeled by arrows.

Figure 10:  (a) Kerr and (c) Faraday angles and (b) Kerr and (d) Faraday
ellipticity as functions of photon energy $\omega$ for bilayer AFM insulator
MnBi2Te4 with external magnetic moment $m_{s}$. The arrows labeled by
$T_{i=1,2,3}$ correspond to the optical interband transitions as shown in Fig.
9. The values of $\theta_{K}$ and $\theta_{F}$ in the low-energy limit are
highlighted in (a) and (b). Parameters: $m=0.12$, $m_{s}=0.18$,
$\epsilon_{F}=0$, $\epsilon_{r}=4$ Tse and MacDonald (2011); G.  _et al._
(2020), $\mu_{r}=1$, $\hbar/\tau_{s}=0.02$.

The Kerr and Faraday angles and ellipticity are given by Fig. 10. For low
photon energy $\omega$, the magnitude of $\theta_{K/F}$ and $\gamma_{K/F}$ for
AFMTI ($M_{0}=0.4$) is much larger than trivial insulator ($M_{0}=-0.2$). For
$M_{0}=0.4$, the peaks (jumps) of $\theta_{K/F}$ ($\gamma_{K/F}$) arise from
interband transitions $T_{i=1,2}$ at the $\Gamma$ point of Brillouin zone (see
Fig. 9). Note that there are some interband transitions missing between
$T_{1}$ and $T_{2}$ since their interband velocity matrix elements vanish as a
result of symmetries at the $\Gamma$ point. Therefore by proximity to magnetic
substrates, we find another way to distinguish between AFMTI and trivial
insulator.

Magnetic moment $m_{s}$ dependence of Kerr and Faraday angles and ellipticity
is plotted in Fig. 11, where $\theta_{K/F}$ and $\gamma_{K/F}$ flip signs
under the inversion of magnetic moment $m_{s}$. For AFMTI ($M_{0}=0.4$),
$\theta_{K/F}$ reaches peaks at the $T_{1}$ transition, then it increases
rapidly and almost saturates after some critical points. The critical points
highlighted in the figure are given by the conditions $|m_{s}|=m$, which
represents a change in the spin configuration. Physically, when $|m_{s}|<m$,
the net magnetization at each layer is opposite, whereas when $|m_{s}|>m$, the
net magnetization becomes the same. By contrast, such change of spin
configurations does not lead to any qualitative differences in $\theta_{K/F}$
or $\gamma_{K/F}$ for trivial insulators. This provides another differences
between AFMTI and trivial insulators.

Figure 11:  (a) Kerr and (c) Faraday angles and (b) Kerr and (d) Faraday
ellipticity as functions of external magnetic moment $m_{s}$ for bilayer AFM
insulator MnBi2Te4. Red and green lines correspond to AFMTI ($M_{0}=0.4$) and
trivial insulator ($M_{0}=-0.2$), respectively. The peaks highlighted in
orange region are the critical points satisfying $|m_{s}|=m$. Parameters:
$m=0.12$, $\omega=0.12$, $\epsilon_{F}=0$, $\epsilon_{r}=4$ Tse and MacDonald
(2011); G.  _et al._ (2020), $\mu_{r}=1$, $\hbar/\tau_{s}=0.02$.

##  IV Conclusions and discussions

To summarize, we show that AFMTI and trivial insulators are distinguishable in
magneto-optical Kerr and Faraday effects. To allow this, all mirror or
combined mirror symmetries must be broken. We propose two feasible ways to
break these symmetries, that is, by adding perpendicular electric field or
external magnetic moment. In the former case, Kerr and Faraday angles and
ellipticity show qualitatively different behaviors between AFMTI and trivial
insulators, as a result of topological phase transitions by tuning parameters
$M_{0}$. The significant differences of optical responses across topological
phase transition have also been proposed in group-IV monochalcogenides SnSe
Zhou  _et al._ (2020). The latter situation can be realized by proximity to
magnetic substrate. In that case, Kerr and Faraday angles can provide the
crucial information of spin configurations of the AFMTI phase. Our studies can
be further generalized to multilayer AFM case, to investigate the role of
layer numbers in magneto-optical responses.

Not only for bilayer AFM insulators MnBi2Te4, our theory can be extended to
other $\mathcal{PT}$-symmetric AFM materials. In a recent paper, in-plane
anomalous Hall effect has been studied in the general $\mathcal{PT}$-symmetric
AFM materials Cao  _et al._ (2023). Given that Kerr and Faraday angles are
closely related to the anomalous Hall conductivity in the low-frequency limit,
anomalous longitudinal or transverse Kerr and Faraday effect, i.e., the
magnetization is in-plane, may arise when symmetry conditions are satisfied.
This leads to qualitatively different physics from the polar Kerr and Faraday
effect studied in our work. Detailed analysis of this issue will be the object
of future work.

##  Acknowledgement

This work is supported by the National Natural Science Foundation of China
(NSFC, Grant No. 11904062), the Starting Research Fund from Guangzhou
University (Grant No. RQ2020076) and Guangzhou Basic Research Program, jointed
funded by Guangzhou University (Grant No. 202201020186).

##  Appendix A Symmetry analysis

When $V=m_{s}=0$, the Hamiltonian
$h(\bm{k})=\bigoplus_{z=1}^{n_{z}}[h_{a}(\bm{k},z)+h_{b}(\bm{k},z)]$ preserves
the symmetries $\mathcal{M}_{1/2}$, that is,
$h(\bm{k})\mathcal{M}_{1/2}=\mathcal{M}_{1/2}h(\bm{k})$.
$\epsilon_{\bm{k}\mu}$ and $|\bm{k},\mu\rangle$ are eigenvalues and
eigenstates of Hamiltonian $h(\bm{k})$. Due to the presence of Kramer’s
degeneracy, we have $\epsilon_{\bm{k}\mu}=\epsilon_{\bm{k}\bar{\mu}}$, where
$\mu(\bar{\mu})$ labels a pair of degenerate states. $\mathcal{M}_{1/2}$
symmetry relates the degenerate states between $\bm{k}$ and
$\mathcal{M}_{1}\bm{k}=(-k_{x},k_{y})$ or
$\mathcal{M}_{2}\bm{k}=(k_{x},-k_{y})$. For example, for $\mathcal{M}_{1}$, we
have

$\begin{split}\left[\begin{array}[]{cc}|\mathcal{M}_{1}\bm{k},\mu\rangle&|\mathcal{M}_{1}\bm{k},\bar{\mu}\rangle\\\
\end{array}\right]=\mathcal{M}_{1}\left[\begin{array}[]{cc}|\bm{k},\mu\rangle&|\bm{k},\bar{\mu}\rangle\\\
\end{array}\right]U,\end{split}$ (8)

where $U$ is arbitrary $U(2)$ rotation. By grouping these states into Kramer’s
pairs, the conductivity tensor $\sigma_{\alpha\beta}$ from Eq.
(LABEL:conductivity) can be rewritten as

$\begin{split}&\sigma_{xx}(\omega)=i\hbar\sum_{\epsilon_{\mu(\bar{\mu})}>0,\epsilon_{\mu^{\prime}(\bar{\mu})}<0}\frac{\mathrm{Tr}[B_{x}(\bm{k})B_{x}^{+}(\bm{k})]}{\epsilon_{\bm{k}\mu^{\prime}}-\epsilon_{\bm{k}\mu}}\\\
&\times[\frac{1}{\omega+\epsilon_{\bm{k}\mu}-\epsilon_{\bm{k}\mu^{\prime}}+i\hbar/2\tau_{s}}+\frac{1}{\omega+\epsilon_{\bm{k}\mu^{\prime}}-\epsilon_{\bm{k}\mu}+i\hbar/2\tau_{s}}],\end{split}$
(9)

$\begin{split}&\sigma_{xy}(\omega)=i\hbar\sum_{\epsilon_{\mu(\bar{\mu})}>0,\epsilon_{\mu^{\prime}(\bar{\mu})}<0}\frac{1}{\epsilon_{\bm{k}\mu^{\prime}}-\epsilon_{\bm{k}\mu}}\\\
&\times[\frac{\mathrm{Tr}[B_{x}(\bm{k})B_{y}^{+}(\bm{k})]}{\omega+\epsilon_{\bm{k}\mu}-\epsilon_{\bm{k}\mu^{\prime}}+i\hbar/2\tau_{s}}+\frac{\mathrm{Tr}[B_{y}(\bm{k})B_{x}^{+}(\bm{k})]}{\omega+\epsilon_{\bm{k}\mu^{\prime}}-\epsilon_{\bm{k}\mu}+i\hbar/2\tau_{s}}],\end{split}$
(10)

where the current matrix

$\begin{split}B_{x/y}(\bm{k})=\left[\begin{array}[]{cc}\langle\bm{k},\mu|\\\
\langle\bm{k},\bar{\mu}|\\\
\end{array}\right]j_{x/y}\left[\begin{array}[]{cc}|\bm{k},\mu^{\prime}\rangle&|\bm{k},\bar{\mu}^{\prime}\rangle\\\
\end{array}\right].\end{split}$ (11)

For $\mathcal{M}_{1}$ symmetry,

$\begin{split}&B_{x}(\mathcal{M}_{1}\bm{k})\\\
&=\frac{e}{\hbar}\left[\begin{array}[]{cc}\langle\mathcal{M}_{1}\bm{k},\mu|\\\
\langle\mathcal{M}_{1}\bm{k},\bar{\mu}|\\\ \end{array}\right]\frac{\partial
h(\mathcal{M}_{1}\bm{k})}{\partial(\mathcal{M}_{1}\bm{k})_{x}}\left[\begin{array}[]{cc}|\mathcal{M}_{1}\bm{k},\mu^{\prime}\rangle&|\mathcal{M}_{1}\bm{k},\bar{\mu}^{\prime}\rangle\\\
\end{array}\right]\\\
&=\frac{e}{\hbar}U^{+}\left[\begin{array}[]{cc}\langle\bm{k},\mu|\\\
\langle\bm{k},\bar{\mu}|\\\
\end{array}\right]\mathcal{M}_{1}^{+}\frac{\partial
h(\mathcal{M}_{1}\bm{k})}{\partial(\mathcal{M}_{1}\bm{k})_{x}}\mathcal{M}_{1}\left[\begin{array}[]{cc}|\bm{k},\mu^{\prime}\rangle&|\bm{k},\bar{\mu}^{\prime}\rangle\\\
\end{array}\right]U\\\ &=-U^{+}B_{x}(\bm{k})U.\end{split}$ (12)

Similarly, we have $B_{y}(\mathcal{M}_{1}\bm{k})=U^{+}B_{y}(\bm{k})U$. This
means that the contributions to $\sigma_{xy}(\omega)$ from $\bm{k}$ and
$\mathcal{M}_{1}\bm{k}$ always cancel with each other, resulting in
$\sigma_{xy}(\omega)=0$. Similar discussion can be carried out for
$\mathcal{M}_{2}$ symmetry, which also leads to zero Hall conductivity. When a
perpendicular electric field or external magnetic moment is introduced, i.e.,
$V\neq 0$ or $m_{s}\neq 0$, $\mathcal{M}_{1/2}$ symmetries are broken, thus
$\sigma_{xy}(\omega)$ can be nonzero.

## References

  * Jungwirth  _et al._ (2016) T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nat. Nanotech. 11, 231 (2016).
  * Baltz  _et al._ (2018) V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Rev. Mod. Phys. 90, 015005 (2018).
  * Li  _et al._ (2010) R. Li, J. Wang, X.-L. Qi, and S.-C. Zhang, Nat. Phys. 6, 284 (2010).
  * Nenno  _et al._ (2020) D. M. Nenno, C. A. C. Garcia, J. Gooth, C. Felser, and P. Narang, Nat. Rev. Phys. 2, 682 (2020).
  * Liu  _et al._ (2020) C. Liu, Y. Wang, H. Li, Y. Wu, Y. Li, J. Li, K. He, Y. Xu, J. Zhang, and Y. Wang, Nat. Mater. 19, 522 (2020).
  * Essin  _et al._ (2009) A. M. Essin, J. E. Moore, and D. Vanderbilt, Phys. Rev. Lett. 102, 146805 (2009).
  * Zhang  _et al._ (2019) D. Zhang, M. Shi, T. Zhu, D. Xing, H. Zhang, and J. Wang, Phys. Rev. Lett. 122, 206401 (2019).
  * Armitage and Wu (2019) N. P. Armitage and L. Wu, SciPost Phys. 6, 046 (2019).
  * Chen  _et al._ (2014) H. Chen, Q. Niu, and A. H. MacDonald, Phys. Rev. Lett. 112, 017205 (2014).
  * Nakatsuji  _et al._ (2015) S. Nakatsuji, N. Kiyohara, and T. Higo, Nature 527, 212 (2015).
  * Nayak  _et al._ (2016) A. A. Nayak, J. E. Fischer, Y. Sun, B. Yan, J. Karel, A. C. Komarek, C. Shekhar, N. Kumar, W. Schnelle, J. Kübler, C. Felser, and S. S. P. Parkin, Science Advances 2, e1501870 (2016).
  * Feng  _et al._ (2020) W. Feng, J.-P. Hanke, X. Zhou, G.-Y. Guo, S. Blugel, Y. Mokrousov, and Y. Yao, Nat. Commun. 11, 118 (2020).
  * Otrokov  _et al._ (2019a) M. M. Otrokov, I. I. Klimovskikh, H. Bentmann, D. Estyunin, A. Zeugner, Z. S. Aliev, S. Gaß, A. U. B. Wolter, A. V. Koroleva, A. M. Shikin, M. Blanco-Rey, M. Hoffmann, I. P. Rusinov, A. Y. Vyazovskaya, S. V. Eremeev, Y. M. Koroteev, V. M. Kuznetsov, F. Freyse, J. Sánchez-Barriga, I. R. Amiraslanov, M. B. Babanly, N. T. Mamedov, N. A. Abdullayev, V. N. Zverev, A. Alfonsov, V. Kataev, B. Büchner, E. F. Schwier, S. Kumar, A. Kimura, L. Petaccia, G. Di Santo, R. C. Vidal, S. Schatz, K. Kißner, M. Ünzelmann, C. H. Min, S. Moser, T. R. F. Peixoto, F. Reinert, A. Ernst, P. M. Echenique, A. Isaeva, and E. V. Chulkov, Nature 576, 416 (2019a).
  * Otrokov  _et al._ (2019b) M. M. Otrokov, I. P. Rusinov, M. Blanco-Rey, M. Hoffmann, A. Y. Vyazovskaya, S. V. Eremeev, A. Ernst, P. M. Echenique, A. Arnau, and E. V. Chulkov, Phys. Rev. Lett. 122, 107202 (2019b).
  * Chen  _et al._ (2019) Y. J. Chen, L. X. Xu, J. H. Li, Y. W. Li, H. Y. Wang, C. F. Zhang, H. Li, Y. Wu, A. J. Liang, C. Chen, S. W. Jung, C. Cacho, Y. H. Mao, S. Liu, M. X. Wang, Y. F. Guo, Y. Xu, Z. K. Liu, L. X. Yang, and Y. L. Chen, Phys. Rev. X 9, 041040 (2019).
  * Hao  _et al._ (2019) Y.-J. Hao, P. Liu, Y. Feng, X.-M. Ma, E. F. Schwier, M. Arita, S. Kumar, C. Hu, R. Lu, M. Zeng, Y. Wang, Z. Hao, H.-Y. Sun, K. Zhang, J. Mei, N. Ni, L. Wu, K. Shimada, C. Chen, Q. Liu, and C. Liu, Phys. Rev. X 9, 041038 (2019).
  * Swatek  _et al._ (2020) P. Swatek, Y. Wu, L.-L. Wang, K. Lee, B. Schrunk, J. Yan, and A. Kaminski, Phys. Rev. B 101, 161109 (2020).
  * Zhang  _et al._ (2020a) J. Zhang, D. Wang, M. Shi, T. Zhu, H. Zhang, and J. Wang, Chin. Phys. Lett. 37, 077304 (2020a).
  * Gao  _et al._ (2021) A. Gao, Y.-F. Liu, C. Hu, J.-X. Qiu, C. Tzschaschel, B. Ghosh, S.-C. Ho, D. B$\acute{e}$rub$\acute{e}$, R. Chen, H. Sun, Z. Zhang, X.-Y. Zhang, Y.-X. Wang, N. Wang, Z. Huang, C. Felser, A. Agarwal, T. Ding, H.-J. Tien, A. Akey, J. Gardener, B. Singh, K. Watanabe, T. Taniguchi, K. S. Burch, D. C. Bell, B. B. Zhou, W. Gao, H.-Z. Lu, A. Bansil, H. Lin, T.-R. Chang, L. Fu, Q. Ma, N. Ni, and S.-Y. Xu, Nature 595, 521 (2021).
  * Chen  _et al._ (2022) R. Chen, H.-P. Sun, M. Gu, C.-B. Hua, Q. Liu, H.-Z. Lu, and X. C. Xie, National Science Review , nwac140 (2022).
  * Chen  _et al._ (2021) R. Chen, S. Li, H.-P. Sun, Q. Liu, Y. Zhao, H.-Z. Lu, and X. C. Xie, Phys. Rev. B 103, L241409 (2021).
  * Ding  _et al._ (2020) Y.-R. Ding, D.-H. Xu, C.-Z. Chen, and X. C. Xie, Phys. Rev. B 101, 041404 (2020).
  * Zhang  _et al._ (2020b) R.-X. Zhang, F. Wu, and S. Das Sarma, Phys. Rev. Lett. 124, 136407 (2020b).
  * Li  _et al._ (2021) H. Li, H. Jiang, C.-Z. Chen, and X. C. Xie, Phys. Rev. Lett. 126, 156601 (2021).
  * Gu  _et al._ (2021) M. Gu, J. Li, H. Sun, Y. Zhao, C. Liu, J. Liu, H. Lu, and Q. Liu, Nat. Commun. 12, 3524 (2021).
  * Dai  _et al._ (2022) W.-B. Dai, H. Li, D.-H. Xu, C.-Z. Chen, and X. C. Xie, Phys. Rev. B 106, 245425 (2022).
  * Ahn  _et al._ (2022) J. Ahn, S.-Y. Xu, and A. Vishwanath, Nat. Commun. 13, 7615 (2022).
  * Lei and MacDonald (2023) C. Lei and A. H. MacDonald,  , arXiv:2303.14635 (2023).
  * Qiu  _et al._ (2023) J.-X. Qiu, C. Tzschaschel, J. Ahn, A. Gao, H. Li, X.-Y. Zhang, B. Ghosh, C. Hu, Y.-X. Wang, Y.-F. Liu, D. Bérubé, T. Dinh, Z. Gong, S.-W. Lien, S.-C. Ho, B. Singh, K. Watanabe, T. Taniguchi, D. C. Bell, H.-Z. Lu, A. Bansil, H. Lin, T.-R. Chang, B. B. Zhou, Q. Ma, A. Vishwanath, N. Ni, and S.-Y. Xu,  , arXiv:2303.05451 (2023).
  * Nandkishore and Levitov (2011) R. Nandkishore and L. Levitov, Phys. Rev. Lett. 107, 097402 (2011).
  * Crassee  _et al._ (2011) I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, Nat. Phys. 7, 48 (2011).
  * Shimano  _et al._ (2013) R. Shimano, G. Yumoto, J. Y. Yoo, R. Matsunaga, S. Tanabe, H. Hibino, T. Morimoto, and H. Aoki, Nat. Commun. 4, 1841 (2013).
  * Sivadas  _et al._ (2016) N. Sivadas, S. Okamoto, and D. Xiao, Phys. Rev. Lett. 117, 267203 (2016).
  * Huang  _et al._ (2017) B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu, Nature 546, 270 (2017).
  * Gong  _et al._ (2017) C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia, and X. Zhang, Nature 546, 265 (2017).
  * Freiser (1968) Freiser, IEEE Trans. Magn. 4, 152 (1968).
  * Tse and MacDonald (2010) W.-K. Tse and A. H. MacDonald, Phys. Rev. Lett. 105, 057401 (2010).
  * Gorbar  _et al._ (2012) E. V. Gorbar, V. P. Gusynin, A. B. Kuzmenko, and S. G. Sharapov, Phys. Rev. B 86, 075414 (2012).
  * Tse and MacDonald (2011) W.-K. Tse and A. H. MacDonald, Phys. Rev. B 84, 205327 (2011).
  * G.  _et al._ (2020) C. G., N. M. Peres, and J. Fernandez-Rossier, 2D Mater. 7, 025011 (2020).
  * Kahn  _et al._ (1969) F. J. Kahn, P. S. Pershan, and J. P. Remeika, Phys. Rev. 186, 891 (1969).
  * (42) Mahan G D 2000 _Many-Particle Physics_ 3rd edn.(New York: Springer) p.507.
  * (43) Giuliani G and Vignale G 2005 _Quantum Theory of the Electron Liquid_ 1st edn.(Cambridge University Press) p.128.
  * Zhou  _et al._ (2020) J. Zhou, S. Zhang, and J. Li, NPG Asia Materials 12, 2 (2020).
  * Cao  _et al._ (2023) J. Cao, W. Jiang, X.-P. Li, D. Tu, J. Zhou, J. Zhou, and Y. Yao, Phys. Rev. Lett. 130, 166702 (2023).