# A Collection of Pragmatic-Similarity Judgments over Spoken Dialog Utterances

###### Abstract

Automatic measures of similarity between utterances are invaluable for
training speech synthesizers, evaluating machine translation, and assessing
learner productions. While there exist measures for semantic similarity and
prosodic similarity, there are as yet none for pragmatic similarity. To enable
the training of such measures, we developed the first collection of human
judgments of pragmatic similarity between utterance pairs. Each pair
consisting of an utterance extracted from a recorded dialog and a re-enactment
of that utterance. Re-enactments were done under various conditions designed
to create a variety of degrees of similarity. Each pair was rated on a
continuous scale by 6 to 9 judges. The average inter-judge correlation was as
high as 0.72 for English and 0.66 for Spanish. We make this data available at
https://github.com/divettemarco/PragSim .

Keywords: human perception, spoken language, utterances, prosody, English,
Spanish

languageresourceLanguage Resources

A Collection of Pragmatic-Similarity Judgments over Spoken Dialog Utterances

Nigel G. Ward, Divette Marco
---
Computer Science, University of Texas at El Paso
El Paso, Texas, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract content

## 1\. Motivation

From a cognitive science perspective, “similarity is one of the most important
relations humans perceive” [Richie and Bhatia, 2021], as it underlies many
aspects of learning, classification, and generalization. From a computational
linguistics perspective, similarity models are important for many applications
and motivate many representations. However, there are as yet no models of
pragmatic similarity.

Pragmatics includes all the aspects of language use in which people convey
information beyond the semantic content. Pragmatics is especially important in
dialog and embodied interaction [Marge et al., 2022], where people may
coordinate action, convey intentions and attitudes, mark topic shifts and
connections, manage turn-taking, and so on. Measures of lexical and semantic
similarity are inadequate for such pragmatic dimensions. This is especially
evident for spoken dialog. For example, two instances of the same lexical
content, such as the word okay, may mean entirely different things, depending
on the prosody. Conversely, lexically different utterances, such as that’s
really interesting and that reminds me of a story may be functionally very
similar, if the prosody of both politely conveys the intent to close one topic
and move on to another.

Development and evaluation of models of pragmatic similarity requires a
reference set of human judgments, but such a resource has been lacking. This
paper presents the first collection of such judgments.

## 2\. Application Needs

Pragmatics is becoming more important for computational purposes, as
applications increasingly target natural spoken dialog. This section overviews
the needs in two applications areas.

Today speech synthesizer output tends to be prosodically neutral and
pragmatically uninformative. Traditionally speech synthesizer output was
evaluated on intelligibility and naturalness, and occasionally also
expressivity, but recently there is more interest in evaluation in terms of
appropriateness, in part to better support synthesis for dialog applications
[O’Mahony et al., 2021, Wagner et al., 2019]. One particular area of interest
is synthesis for machine translation, where support for conversational uses
will need elements of the source-language pragmatics to be re-created in the
target-language output [Huang et al., 2023, Rubenstein et al., 2023, Avila and
Ward, 2023, Barrault et al., 2023]. Evaluation of systems’ ability to do so
will require a good pragmatic-similarity metric, for example for evaluating
the match between system output and human-generated reference translations.

Another application area is assessment of human speech and dialog behavior,
both for people learning a new language, and for people seeking to overcome
speech pathologies. Existing assessment methods focus on evaluating the
phonetic, lexical, syntactic or semantic aspects of production, but for
communicative effectiveness and social inclusion, it is often the pragmatic
aspects of language behavior that matter more. We imagine a system which
compares a subject’s dialog behavior sample to that of an exemplar speaker. If
the behavior of the subject is pragmatically dissimilar to that of the
reference speaker (or any reference speaker in contexts that were
pragmatically similar in a corpus), then correction or intervention may be
appropriate. This will, again, require a good pragmatic-similarity metric.

Pragmatic-similarity evaluations are today sometimes proxied by simple
prosodic similarity metrics. Among other problems, these metrics so far
require that the utterances being compared have the same word sequences.
However, for both applications, we would ideally like to be able to judge
pragmatic similarity independently of lexical similarity, in case the speaker
or the system chooses to express the same idea and intent with different
words.

## 3\. Modeling Similarity: Related Work

Only one previous study seems to have directly targetting pragmatic similarity
[Pragst, 2022]. This was motivated by the goal of enabling increased variety
of expression for dialog systems. Unfortunately the generality of the work may
be limited, as it was evaluated only on artificial data and only in terms of
determining whether two sentences embody the same speech act. Further, the
model developed was being purely text-based.

The rest of this section surveys work on two related types of similarity,
semantic and prosodic. For convenience, we organize the discussion around
three approaches for developing similarity models: using knowledge, using
distributional properties, and using explicit judgments [Mihalcea et al.,
2006, Chandrasekaran and Mago, 2021].

Knowledge-based models, designed using scientific or common-sense knowledge
about what matters to people, are prevalent for prosodic similarity. These are
relevant since, as already suggested, pragmatic meanings are often conveyed by
prosody. Most models of prosodic similarity focus on intonation (F0) alone
[Kominek et al., 2008, Salesky et al., 2021, Hermes, 1998, Reichel et al.,
2009, Nocaudie and Astésano, 2016], with more recent work adding duration
features [Mixdorff et al., 2012, Huang et al., 2023], and sometimes others
[Rilliard et al., 2011, Ward et al., 2015]. However none of these consider all
important aspects of prosody, which include intensity, timing, rate, phonetic
reduction, and voice quality features. While these are known to sometimes
contribute significantly to the expression of pragmatic functions, beyond
intonation alone [Niebuhr and Ward, 2018], the extent has never been
quantified, for lack of a metric or set of judgments. Other work has developed
reasonably comprehensive utterance-level representations of prosody [Eyben et
al., 2016], but mostly to support only supervised learning, and none tested
from the perspective of placing utterances in a space in which distances have
meaning. The only exception is a very recent paper [Avila and Ward, 2023],
which explored a Euclidean distance metric over 100 features designed to
capture the prosodic indications of pragmatic functions. The quality of this
metric was evaluated only on post-hoc judgments of its ability to separate
very similar utterance pairs from very dissimilar ones.

Another type of knowledge that might be useful for building pragmatic-
similarity models is knowledge of the likely components of pragmatic-
similarity perceptions. There are a few taxonomies of functions that are
important in dialog [Bunt and Petukhova, 2023, Seebauer et al., 2023], which,
although clearly incomplete, can guide us in useful directions. It does seem
that any single specific dimensions of pragmatic or stylistic similarity can
be handled well, if there is data suitable for supervised training. This has
been done, for example, done for judging the clarity of turn-hold/yield
intentions [Ekstedt et al., 2023]. However the number of important pragmatic
functions is very large, and it is not clear how far such taxonomy-based
models could advance us to a general model of pragmatic similarity.

Distribution-based models exploit the general association between occurring in
similar contexts and being perceived similarly. Self-supervised learning uses
distributional properties to learn embedding spaces in which any input can be
concisely represented, and proximity in such a space can be a proxy for
similarity. For semantic similarity, such metrics have been very useful, and
even more so when combined with knowledge-based algorithms, for example to
deal with reorderings [Zhao et al., 2019].

Although self-supervised learning of pragmatics-capturing representations
would seem a natural next step [Purver and Sadrzadeh, 2015], and self-
supervised models of semantic similarity in conversation and of dialog
processes have been created [Yang et al., 2018, Nguyen et al., 2023], no
general representations of pragmatic information appear to yet exist. While
some dimensions of pragmatic function are fortuitously captured in speech
pretrained models such as HuBert and wav2vec2.0 [Lin et al., 2022], it is not
known how much pragmatic information these models represent, nor whether this
is sufficient for modeling pragmatic similarity.

Models trained on human judgments have been useful in particular for
estimating semantic similarity for machine translation, not only for text but
also for speech [Wieting et al., 2019, Gehrmann et al., 2023, Besacier et al.,
2022, Chen et al., 2023, Barrault et al., 2023]. Some of this work starts with
pretrained models, greatly reducing the amount of human-judgment data needed
for the final training. While not directly relevant to similarity, recent work
training models to predict human judgments of speech-synthesis quality [Huang
et al., 2022, Maiti et al., 2023, Cooper et al., 2022], and of utterance
suitability for a given dialog context [Wallbridge et al., 2023], illustrates
what can be done when appropriate data is available.

Overall, while it is likely that such models have some utility also for
estimating pragmatic similarity, determining how much will require a set of
pragmatic-similarity judgments. In general, whether developing pragmatic-
similarity models from scratch or adapting existing models, the field will
need a collection of human perceptions of pragmatic similarity. We here
present the first.

## 4\. Pilot Studies

A good procedure for obtaining judgments should: cover a wide range, be
reliable, obtain rich information, be efficient, and be not onerous for
judges. We did three pilot studies, with a total of 16 judges, to investigate
some options.

1) For obtaining judgments, we tested three methods: rating, ABX, and odd-one-
out. While the latter two have their advantages — being easier for the judges,
seemingly having higher agreement, and potentially sidestepping some
weaknesses of pairwise comparisons [Aldrich et al., 2009], we chose rating of
utterance pairs, because it gives more information for training models, and
because we thought that perceptive, well-trained judges could handle it.

2) We tried different rating scales. Some judges felt that five options (1 –
5) were not enough as they wanted more precision, but others felt that even 5
options were too many. Again prioritizing the amount of information to obtain
and assuming great judges, we chose a continuous scale.

3) When we inadvertently had a delay between the play-back of the utterances
being compared, judging the similarity was felt harder, doubtless because of
the limited size of auditory memory. We decided to play both members of each
pair back-to-back, separated only by a short beep.

4) We briefly considered presenting the stimuli with contexts, rather than as
isolated utterances [Wagner et al., 2019]. This would likely boost the inter-
annotator agreement, but would be much more time-consuming, and the resulting
judgments would be harder to build models for.

5) We started with stimuli that were random pairs of utterances from a large
corpus. Judges noted that the task was strange because the pairs were often
extremely different, for example, of widely different durations. With such
stimuli, it seemed that we would obtain mostly only “very dissimilar”
judgments. We therefore decided to assemble stimuli such that both members of
each pair shared a lot, as described in the next section.

6) We found that judges generally tended to be close in their judgments, but
with many exceptions. In such cases, we had them discuss the factors that were
affecting their judgments. These often varied. For example, one judge thought
his judgments were highly affected by the rate of speech, and another judge
thought that her judgments were highly affected by the ending of the
utterances. At a deeper level, some reported being affected by tone or
feeling, and others by confidence, intention, and level of perceived fluency,
spontaneity, energy, or politeness. It would be interesting to compile a list
of the factors affecting these judgments, and to encourage judges to consider
them all. Instead, we decided to simply accept that people differ in how they
included and weighed the possible factors underlying these judgments, and not
try to constrain them in some way to increase agreement scores.

7) For one utterance pair, one judge remarked on speaker differences, which
led to an interesting discussion of whether the perceived difference was due
to a difference of intent or personality. Some judges thought that one of the
speakers likely had a more dominant speaking style in general, and that this
difference could be factored out and discarded when judging pragmatic
similarity with an utterance of a different speaker. This raises deep issues,
but we decided to simply add an instruction to “ignore speaker differences.”

8) While some judges mentioned noticing surprisingly subtle differences, such
as the fact that some utterances seem to have been said with a smile, others
mentioned that they found the task difficult or lacked confidence in their
judgments. We decided we needed to be selective in choosing judges.

## 5\. Stimulus Preparation

Because most use cases for a similarity metric involve comparison of mostly
fairly similar utterances, we wanted to boost the representation of such
pairs. To this end, each stimulus consisted of a seed utterance and a re-
enactment.

### 5.1. Re-enactment Methods

To obtain a diversity of similarity levels, re-enactments were created using
one of 6 methods:

Voice (VO). The re-enactor listens to the seed, and re-creates it as closely
as they can. We expected these to be rated very similar to the seeds,
differing only to the extent that voices and vocal abilities differ among
speakers.

Words+Context (WC) The re-enactor sees a transcript of the seed and hears 5–10
seconds of the preceding context in the dialog, or more if they ask. They then
say the words in a way that they think is appropriate for the context. We
expected these to also be rated highly.

Lexically Distinct (LD) The re-enactor listens to the seed, then recreates
something with the same meaning and feeling, but with different words. We
expected these to be fairly high in pragmatic similarity. As a side note, the
re-enactors usually seemed to try to keep the same prosodic form, but the
lexical differences often necessitated re-adjustments. For those wanting to
model how prosody conveys pragmatic meanings, these pairs could be useful for
learning to disentangle this from lexically-governed prosody.

Context-Only (CO) The re-enactor hears only the context before the seed
utterance, and then says something they think would be an appropriate
continuation of the conversation. This method was used to produce
interestingly different re-enactments that could be similar to the seeds, to
the extent that most conversations have a natural flow suggesting a single
likely continuation in terms of pragmatic function, albeit one that can be
realized in various lexical forms.

Words (WO) The re-enactor sees only the transcript and is asked only to “say
it as you might say it in a conversation.” We expected these to generally have
only weak pragmatic similarity.

Speech Synthesizer (SS) As in the previous condition, the re-enactor has
access only to the transcript, but this time the re-enactor is a speech
synthesizer rather than a human. We expected these to be generally not very
pragmatically similar to the seeds, as the prosody of synthesized utterances
tends to be pragmatically neutral.

| seed properties
---|---
| audio | words | context
Voice | s | s | \-
Words+Context | - | w | s
Lexically Distinct | s | s | -
Context Only | - | - | s
Words | - | w | -
Synthesized | - | w | -
Table 1: Summary of the information provided in each re-enactment condition. w
means written, s means spoken.

Table 1 summarizes.

### 5.2. Sources of Seeds

As pragmatic functions mostly occur in dialog, we chose to take all seeds from
recorded dialogs. As we envisage training a metric to be widely useful, we
used mostly recordings from the DRAL corpus [Avila and Ward, 2023], an
unstructured-conversation corpus with a good variety of topics and interaction
styles. As the speakers in DRAL were from the same population as our judges,
and many of the topics are of common interest, we expected this choice to also
help with retaining interest during the long judgment sessions. The English
seeds were taken from the English-original conversations, and similarly for
the Spanish seeds.

To increase diversity, we supplemented these with a few seeds from other
corpora. For English these were mostly of task-oriented dialogs, including: a
billing support corpus, a persuasive dialogs corpus, a negotiation corpus, a
gameplay corpus, and a referent-identification and problem-solving corpus
[Ward et al., 2005, Acosta and Ward, 2009, Ward et al., 2021, Ward and Abu,
2016, Pardo et al., 2019]. For Spanish these were from an interview corpus
spanish-in-texas, and a telephone call corpus callhome-spanish. We also
included a few children’s utterances, in English from the Talkbank Providence
corpus providence, and in Spanish from PhonBank llinas-ojea. Not wanting to
ask adults to re-enact toddlers’ utterances, these seeds were matched with
other utterances by the same child.

Seeds were selected: 1) to be at least 3 words but no longer than 6 seconds.
This was to balance the need to include enough information to provide a basis
for judgment, and the need to make the process of obtaining judgments fast and
easy. The average length was 3 seconds. 2) to be mostly clear in intent and
meaning when heard in isolation, free of excessive disfluencies and laughter,
and free of extreme emotion or strong personality. We avoided these to make
things easier for our re-enactors. 3) to contain no reported speech, that is,
containing no quotes or rephrasings of something said by a third party. Such
utterances have two levels of meaning, and were often hard to judge. 4) to be
diverse, in terms of speakers, topics, dialog activities, and pragmatic
functions. The pragmatic functions of the utterances chosen included sharing
information, making requests, making decisions, asking questions, giving
directions, giving advice, calming a conversation partner, sharing opinions,
expressing empathy, making observations, making estimates, confirming
something, etc. Those utterances also expressed a range of emotions and
cognitive processes such as understanding, enthusiasm, anticipation,
frustration, confusion, hesitation, agreement, astonishment, humor, etc. They
covered a variety of topics such as media, school, career building, various
types of games, money, politics, charity, grief, love, business, etc. There
was a balance between male and female utterances. Diversity was a goal in
order to obtain a wider base for training a metric, and also to keep the
judgment task more varied and thereby more engaging for our judges.

> How pragmatically similar are the two clips, in terms of the overall
> feeling, tone, and intent.
>
> Try to ignore: $\bullet$ speaker differences, $\bullet$ differences in the
> words said $\bullet$ insignificant differences in pitch, rate, pausing, etc.
>
> Maybe consider: $\bullet$ Similarity in the contexts where they would likely
> appear $\bullet$ Similarity in how a listener would likely respond $\bullet$
> Similarity in how the speaker may have felt (confident, positive, offended,
> enthusiastic, etc.) $\bullet$ Similarity in the dialog activity (correcting
> a misconception, teasing, holding the floor, asking a question, implying
> something, etc.)

Figure 1: Rating Instructions

In all there were 80 seeds for English and 40 for Spanish. Since some methods
required the re-enactor to know seed properties that other methods required
them not to know, no single person could do them all. We accordingly assigned
re-enactors with care: for any seed, one person would do Methods 5 and then 1
(WO and VO), another person would do Methods 2 and then 3 (WC and LO), and a
third would do Method 4 (CO). There were four re-enactors, the authors and an
additional female and male, chosen based on our favorable impressions of their
vocal control and range. For the synthesized utterances we used two of the
more conversational voices from Amazon Polly [Amazon, 2023], namely the female
Salli and the male Matthew, each for half. In each stimulus the seed preceded
the re-enactment. In total, there were 220 English stimuli in the first
judgment session, 238 English stimuli for the second judgment session, and 235
stimuli for the third, Spanish session.

## 6\. Obtaining the Judgments

### 6.1. Judges

We wanted judges who were were sensitive to the nuances of language and who
had the patience to participate in long sessions. We therefore issued
invitations to selected individuals who had been exceptionally eager and
effective participants in a previous data collection [Ward et al., 2023],
and/or were members of our research lab. Those who accepted all turned out to
be friends, relatives, classmates, or potential classmates at leasat one other
judge. Most were in their early 20s. Only those who were fully bilingual in
Spanish were judges in the third session. The first author served as a judge
for the first two sessions, and the second author for the third. Each session
was held on a Saturday, and lasted about 4 hours. Judges were compensated
fairly generously, with snacks, lunch and USD 70.

### 6.2. Instrument

Wanting continuous judgments, we had judges enter their ratings using
QuestionPro’s slider scale tool, accessed via each judge’s own digital device.
The range was 1 to 5, with ticks for the integer values. The anchor text was
“no similarities” for 1 and “virtually identical” for 5, with no intermediate
labels [Zielinski et al., 2008]. Inputs were recorded at a granularity of 0.1.
To clarify their task, judges were given a handout including the instructions
in Figure 1.

### 6.3. Procedure

Judges came to a quiet room and sat around large tables, near the speakers.
After an explanation of our aims and an overview of the task, they signed
consent forms. (The procedure was judged exempt from review by our
institution’s human-subjects committee.) Judges first heard three anchors:
stimuli that we had chosen to illustrate the extremes and a central level of
similarity. To further promote calibration, for the first ten stimuli, and
periodically throughout, we had them compare their ratings and discuss the
factors that had affected their judgments. We stressed that convergence was
not expected, that we welcomed differences of opinion, and that in the end we
would be mostly using the averages of everyone’s judgments.

Stimuli were grouped by source corpus, and we briefly described each corpus.
Within each set, the stimuli were presented in random order.

For the first few dozen stimuli we played each stimulus pair three times, or
more if requested. As the judges got more experienced, two times were
generally enough. By the end, the pace had sped up to better than two
judgments per minute.

### 6.4. Comments and Observations

Some judges noted that some seeds contained a small laugh or moment of laughed
speech, whereas some re-enactments of these did not, causing them to lower
their ratings somewhat. Some of the re-enactments from the context-only
condition were wildly longer than their seeds. This was not intended and may
have been jarring to the judges. The judges noted that judging the children’s
utterances was hard. Not only was the audio quality poor, but small children’s
intentions are often unclear. One judge suggested it would be helpful to watch
videos to see the body language, and another suggested that having the context
would help. All judges seemed engaged across the 4 hours, for every session,
probably thanks in part to generous breaks.

## 7\. The Data Collection

|  English | Spanish
---|---|---
Sessions | 1 | 2 | 3
stimulus pairs | 220 |  238 | 235
judges | 9 | 9 | 6
judgments | 1980 | 2142 | 1410
agreement | 0.45 | 0.72 | 0.66
Table 2: Basic Statistics

In all, we collected 5532 judgments, as seen in Table 2. This data is freely
available at https://github.com/divettemarco/PragSim . The release includes
all the stimulus pairs and their components, namely the seeds and the re-
enactments, and the provenance and source tracks for the seeds and the entire
re-enactment tracks, to enable modelers to do per-speaker normalizations.

## 8\. Correlations and Observations

Overall there was a good variety of judgments, as seen for Session 1 in
Figures 2 and 3, with similar patterns seen for the other sessions. In this
respect, our use of diverse seeds and a variety of re-enactment methods was
successful.

The overall inter-judge agreement varied among sessions, but was as high as
0.72, measured as the average of the pairwise correlations among judges, as
seen in Table 3. (We briefly considered using the Spearman correlation, but
scatterplots among pairs of judges did not reveal any nonlinear patterns.)

The rest of this section explores the factors affecting the ratings and the
degrees of agreement.

### 8.1. Factors Affecting Ratings

The judges had different means, some being much more strict, and some using
different ranges, as seen for Session 1 in Figure 2. However in Session 2 the
judges tended to all use more of the scale, probably reflecting a tendency to
calibrate after more experience with the range of stimuli.

Figure 2: Distributions of judgments for each judge in Session 1. Figure 3:
Distributions of the mean per-stimulus ratings for each re-enactment method in
Session 1.

Within each session, judges became slightly more generous over time, with the
mean rating per stimulus having a 0.04 correlation with the position in the
presentation sequence in the first session, and 0.09 and 0.16 in the second
and third. Perhaps the judges became more familiar with the re-enactors’
voices and better at understanding how their intents mapped to their speech.

Incidentally, there was a negative correlation between the mean rating and the
absolute difference in audio duration between the re-enactment and the seed:
$-0.24$, $-.05$, $-0.20$ for sessions 1, 2, and 3, respectively. Thus pairs
with greater differences in duration tended to be rated lower.

judge | 1 | 2 | 3 | 4 | 5 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---
1 |  |  |  |  |  |  |  |  |
2 | 0.40 |  |  |  |  |  |  |  |
3 | 0.38 | 0.61 |  |  |  |  |  |  |
4 | 0.37 | 0.59 | 0.59 |  |  |  |  |  |
5 | 0.19 | 0.30 | 0.31 | 0.49 |  |  |  |  |
7 | 0.41 | 0.67 | 0.66 | 0.54 | 0.33 |  |  |  |
8 | 0.39 | 0.64 | 0.60 | 0.80 | 0.40 | 0.54 |  |  |
9 | 0.21 | 0.40 | 0.36 | 0.18 | 0.19 | 0.51 | 0.20 |  |
10 | 0.42 | 0.62 | 0.50 | 0.59 | 0.29 | 0.52 | 0.63 | 0.27 |
Per-Judge Means |  |  |
| 0.34 | 0.53 | 0.50 | 0.52 | 0.31 | 0.52 | 0.52 | 0.29 | 0.48
Overall Mean: 0.45  |  |  |
Table 3: Inter-judge Agreements, Pearson’s Correlations

Other trends were apparent on closer examination of the Session 1 data. As
expected, re-enactments generated in the Voice condition were generally rated
highly similar to the seeds, and those produced by the Speech Synthesizer only
weakly similar. Unexpectedly, re-enactments in the Words Only condition were
generally more similar to the seeds than those in the Context Only condition.
We interpret this as indicating that the conversations were less locally
deterministic than we had thought, and that our re-enactors, being familiar
with the local population and how they tended to talk about various topics,
were often able to infer from the words alone how they were likely to have
been said. We initially expected re-enactments from the Words+Context
condition to be rated higher than the Words Only ones, because it gave
additional information to the re-enactors. However the opposite was true. This
might be because the limited contexts we provided were more misleading than
helpful. Although we expected the Lexically Distinct re-enactments to be rated
higher than those from the Words Only condition, the difference was tiny. This
could be because the judges focused on the words more than we expected, or
because re-enactors were able to adequately infer the pragmatics of the seed
without needing context.

### 8.2. Factors Affecting Agreement

While the level of agreement is probably fully acceptable for many purposes,
we probed the factors behind lower agreement. As the data was not collected to
support a systematic analysis; the rest of this section only reports the
factors identified using post-hoc tests. Measures for these included inter-
judge correlations and the per-item standard deviations, computed after
standardizing (z-normalizing) the ratings of each judge.

1) Judge identity was a major factor. For example, in Session 1 there was huge
variation in pairwise correlations, ranging from 0.18 to 0.80, as seen in
Table 3. Interestingly, the judge who differed most from the others (average
correlation of 0.29) turned out, on inspection, to have ratings that were
mostly just integers, whereas most judges had many fractional ratings. In
Session 3 we also noticed that one judge’s ratings were mostly very high or
very low.

2) Experience was also a major factor. This is seen in the increase in
agreement from Session 1 to Session 2, as seen in Table 2, although a
confounding factor was that the least-agreeing judge did not return for that
session. The effects of experience were also evident within sessions, with
mostly negative correlations between the serial positions of the stimuli and
the standard deviations of the judgments: $-0.13$ and $-0.03$ for Sessions 1
and 2 (but $+0.03$ for Session 3). The effects of experience may result in
part in convergence in opinions due to the occasional discussions among
judges, and in part from individual judges becoming more internally consistent
in how they used the scale.

3) There was generally lower agreement for the lower-rated stimuli: the
correlation between the standard deviation and the mean rating was $-0.60$,
$0.00$ and $-.39$ for the three sessions. This may be because in part there
was a clear upper bound but no obvious reference point at the lower bound.

4) There was generally lower agreement for the pairs where the utterances had
different lexical content, namely those using reenactments generated in the
lexically-distinct and context-only conditions. Indeed, for Session 2,
excluding those stimuli gave an average inter-judge agreement of 0.80,
significantly above the 0.72 seen for all stimuli.

5) The specific re-enactment method may have been another factor, but there
were no consistent patterns across the three sessions.

6) Duration differences between seed and re-enactment were another factor. For
example, in Session 1 the standard deviations of the judgements correlated
0.30 with the absolute differences in duration between the seed and the re-
enactment. For the longer re-enactments, sometimes the tone varied from the
beginning to the end, and different judges seemed to pay more attention to
different parts.

To complement the statistical analyses above, we also examined individual
stimuli where the agreement was low, specifically the 27 from Session 1 for
which the standard deviation exceeded 1.0, looking for commonalities that
might explain why the judgments varied so much. This enabled us to identify
three additional factors:

1 |  Seed Words: What happened? Today you saw her?
---|---
|  Re-enactment: So? And why? Did you see her today?
|  Both seed and re-enactment are clearly following up with interest on new
information. Only the seed includes an initial noisy nasal inhalation and had
a wide pitch range, and seems casual, lively, and warm. (010F_M3n_EN_098r_17)
2 |  Seed Words: I get you. I wanna watch it but, it’s also really long.
|  Re-Enactment: Got it… okay so, they’re aliens, um… but like, are they bad
or… are they just minding their own business?
|  Both seed and re-enactment express understanding and the wish to contribute
to the topic without having much to say. The seed clearly closes the topic
while the re-enactment aims to continue the topic. (035F_M4l_EN_043r_12)
3 |  Seed Words: But I- I’m looking forward to u-uh, impressing them and giving them a fun show.
|  Re-enactment: We can definitely help you with that.
|  Both seed and re-enactment are positive and dominant in tone. The seed is
more creaky and less fluent, and sounds more sincere and engaged.
(329F_M4n_DC_1i_02)
4 |  Seed and Re-enactment Words: I was looking through the gifts and… it was for my mom
|  Seed and re-enactment are lexically identical, but only the seed includes
laughed speech, on the last four words, and overall conveys the intent to tell
a funny story. (019F_M2l_EN_034l_2)
5 |  Seed Words: And in that space there’s a monastery.
|  Re-enactment: You keep going straight for a couple of streets and that’s
how you get there.
|  Both seed and re-enactment are providing information and have a helpful,
confident tone, but the seed suggests continuation while the re-enactment
prosody implies a turn end. (389F_M4l_MMTg_01)
6 |  Seed and Re-enactment Words: Wow that’s so humble of you, I never would’ve guessed.
|  Seed and re-enactment are lexically identical, but the seed is teasing and
clearly sarcastic, while the synthesized re-enactment is prosodically neutral,
although this could be interpreted as deadpan sarcasm. (123F_M6s_EN_081r_26)
Table 4: Examples of Stimuli with Poor Agreement

7) Many of these were for re-enactments using the Context-Only method.
Sometimes the semantics was wildly different from those of the seed, as seen
in Examples 2, 3, and 5 of Table 4. Perhaps some judges were able to look
beyond this and focus on the pragmatic similarities, while others found this
harder. This is not surprising: we always expected that training a model to
work well for lexically different content will be harder; it seems harder for
people too.

8) Some of these had seeds with unusual vocal properties, including ingressive
fillers, laughter, falsetto, and strong breathiness, which were often not
present in the re-enactments, as seen in Examples 1 and 4 of Table 4. Perhaps
some judges were more critical of such differences whereas others perhaps were
forgiving. In particular, it was possible that some judges interpreted “ignore
speaker differences” as implying the need to not penalize re-enactments made
by re-enactors with more limited vocal range (often males).

9) Some of these may have involved personality perception differences, as some
of of the seeds conveyed an unusual degree of warmth, engagement, or
scintillating personality, as suggested by Examples 1, 3, and 4 of Table 4.
Again, some judges may have been less swayed by such differences, and able to
forgive the omission of properties incompatible with the re-enactor’s
personality or expressive abilities.

### 8.3. Implications

For those seeking to use this data to build models for automatically
estimating pragmatic similarity judgments, we generally recommend using the
Session 1 data for training and the Session 2 data for evaluation. In
addition, to avoid dominance by the wider-range judges, it may be better to
use as targets, rather than the simple average, the results of averaging after
z-normalizing each judge’s ratings (and optionally then rescaling to 1–5 if
desired).

For some purposes, modelers may wish to consider excluding stimuli that are
difficult to model or are irrelevant to the specific intended use. These may
include stimuli that are not lexically equivalent, differ too much in
duration, have unusual vocal properties, or have a gender difference between
seed speaker and re-enactment speaker.

For those wishing to use our methods to collect new sets of similarity
judgments, we recommend the careful screening, training, and evaluation of
judges. It might also be appropriate for some purposes to exclude some of the
stimulus types mentioned in the previous paragraph, to modify the Context-Only
method to include informing the re-enactors of the approximate duration of the
seed utterances, and to provide judges with more specific instructions or
illustrations illustrating what aspects of speaker differences to ignore.

## 9\. Summary and Prospects

1) We contribute a carefully-designed and tested protocol. We hope others will
use this, for example to collect similarity judgments for other languages.

2) We also contribute observations on the factors affecting ratings and
agreements, which we hope will inform future data collection and modeling
efforts.

3) Our primary contribution is the ratings. These will enable the training of
models to approximate perceived pragmatic similarity. These models will serve
as general infrastructure, supporting the development of more
conversationally-adept speech-to-speech translation systems, more effective
dialog systems, more precise assessment of communication skills, and other
applications.

### Acknowledgements

We thank Jonathan Avila, Dimitri Lyon, Lidia Almendariz and Raul Gomez for
discussions and re-enactments, and Andres Segura for help with the analysis.
This work was supported in part by the AI Research Institutes program of the
National Science Foundation and the Institute of Education Sciences, U.S.
Department of Education through Award # 2229873 – National AI Institute for
Exceptional Education.

## 10\. Bibliographical References

## References

  * Acosta and Ward, 2009 Acosta, J. C. and Ward, N. G.  (2009).  Responding to user emotional state by adding emotional coloring to utterances.  In Interspeech.
  * Aldrich et al., 2009 Aldrich, K. M., Hellier, E. J., and Edworthy, J.  (2009).  What determines auditory similarity? The effect of stimulus group and methodology.  Quarterly Journal of Experimental Psychology, 62:63–83.
  * Amazon, 2023 Amazon.  (2023).  Polly.  https://aws.amazon.com/polly/, accessed 8-2023.
  * Avila and Ward, 2023 Avila, J. E. and Ward, N. G.  (2023).  Towards cross-language prosody transfer for dialog.  In Interspeech.
  * Barrault et al., 2023 Barrault, L., Chung, Y.-A., Meglioli, M. C., Dale, D., Dong, N., Duppenthaler, M., Duquenne, P.-A., Ellis, B., Elsahar, H., Haaheim, J., et al.  (2023).  Seamless: Multilingual expressive and streaming speech translation.  arXiv preprint arXiv:2312.05187.
  * Besacier et al., 2022 Besacier, L., Ribeiro, S., Galibert, O., and Calapodescu, I.  (2022).  A textless metric for speech-to-speech comparison.  arXiv preprint arXiv:2210.11835.
  * Bunt and Petukhova, 2023 Bunt, H. and Petukhova, V.  (2023).  Semantic and pragmatic precision in conversational AI systems.  Frontiers in Artificial Intelligence, 6:896729.
  * Chandrasekaran and Mago, 2021 Chandrasekaran, D. and Mago, V.  (2021).  Evolution of semantic similarity: A survey.  ACM Computing Surveys, 54:1–37.
  * Chen et al., 2023 Chen, M., Duquenne, P.-A., Andrews, P., Kao, J., Mourachko, A., Schwenk, H., and Costa-jussà, M. R.  (2023).  Blaser: A text-free speech-to-speech translation evaluation metric.  In ACL.
  * Cooper et al., 2022 Cooper, E., Huang, W.-C., Toda, T., and Yamagishi, J.  (2022).  Generalization ability of MOS prediction networks.  In IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 8442–8446.
  * Ekstedt et al., 2023 Ekstedt, E., Wang, S., Székely, É., Gustafson, J., and Skantze, G.  (2023).  Automatic evaluation of turn-taking cues in conversational speech synthesis.  In Interspeech.
  * Eyben et al., 2016 Eyben, F., Scherer, K. R., Schuller, B. W., Sundberg, J., André, E., Busso, C., Devillers, L. Y., Epps, J., Laukka, P., Narayanan, S. S., et al.  (2016).  The Geneva minimalistic acoustic parameter set (GeMAPS) for voice research and affective computing.  IEEE Transactions on Affective Computing, 7:190–202.
  * Gehrmann et al., 2023 Gehrmann, S., Clark, E., and Sellam, T.  (2023).  Repairing the cracked foundation: A survey of obstacles in evaluation practices for generated text.  Journal of Artificial Intelligence Research, 77:103–166.
  * Hermes, 1998 Hermes, D. J.  (1998).  Auditory and visual similarity of pitch contours.  Journal of Speech, Language, and Hearing Research, 41:63–72.
  * Huang et al., 2022 Huang, W.-C., Cooper, E., Tsao, Y., Wang, H.-M., Toda, T., and Yamagishi, J.  (2022).  The VoiceMOS challenge 2022.  In Interspeech.
  * Huang et al., 2023 Huang, W.-C., Peloquin, B., Kao, J., Wang, C., Gong, H., Salesky, E., Adi, Y., Lee, A., and Chen, P.-J.  (2023).  A holistic cascade system, benchmark, and human evaluation protocol for expressive speech-to-speech translation.  In IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).
  * Kominek et al., 2008 Kominek, J., Schultz, T., and Black, A. W.  (2008).  Synthesizer voice quality of new languages calibrated with mean mel cepstral distortion.  In Workshop on Spoken Language Technologies for Under-resourced Languages (SLTU), pages 63–68.
  * Lin et al., 2022 Lin, G.-T., Feng, C.-L., Huang, W.-P., Tseng, Y., Lin, T.-H., Li, C.-A., Lee, H., and Ward, N. G.  (2022).  On the utility of self-supervised models for prosody-related tasks.  In IEEE Workshop on Spoken Language Technology (SLT).
  * Maiti et al., 2023 Maiti, S., Peng, Y., Saeki, T., and Watanabe, S.  (2023).  SpeechLMScore: Evaluating speech generation using speech language model.  In IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).
  * Marge et al., 2022 Marge, M., Espy-Wilson, C., Ward, N. G., et al.  (2022).  Spoken language interaction with robots: Research issues and recommendations.  Computer Speech and Language, 71.
  * Mihalcea et al., 2006 Mihalcea, R., Corley, C., Strapparava, C., et al.  (2006).  Corpus-based and knowledge-based measures of text semantic similarity.  In AAAI, pages 775–780.
  * Mixdorff et al., 2012 Mixdorff, H., Cole, J., and Shattuck-Hufnagel, S.  (2012).  Prosodic similarity: Evidence from an imitation study.  In Speech Prosody, pages 571–574.
  * Nguyen et al., 2023 Nguyen, T. A., Kharitonov, E., Copet, J., Adi, Y., Hsu, W.-N., Elkahky, A., Tomasello, P., Algayres, R., Sagot, B., Mohamed, A., et al.  (2023).  Generative spoken dialogue language modeling.  Transactions of the Association for Computational Linguistics, 11:250–266.
  * Niebuhr and Ward, 2018 Niebuhr, O. and Ward, N. G.  (2018).  Challenges in the modeling of pragmatics-related prosody.  Journal of the International Phonetics Association, 48:1–8.
  * Nocaudie and Astésano, 2016 Nocaudie, O. and Astésano, C.  (2016).  Evaluating prosodic similarity as a means towards L2 teacher’s prosodic control training.  In Speech Prosody, pages 26–30.
  * O’Mahony et al., 2021 O’Mahony, J., Oplustil-Gallegos, P., Lai, C., and King, S.  (2021).  Factors Affecting the Evaluation of Synthetic Speech in Context.  In Proc. 11th ISCA Speech Synthesis Workshop, pages 148–153.
  * Pardo et al., 2019 Pardo, J. S., Urmanche, A., Gash, H., Wiener, J., Mason, N., Wilman, S., Francis, K., and Decker, A.  (2019).  The Montclair map task: Balance, efficacy, and efficiency in conversational interaction.  Language and Speech, 62(2):378–398.
  * Pragst, 2022 Pragst, L.  (2022).  On the Generation of Pragmatic Paraphrases for Dialogue Systems.  Ph.D. thesis, Ulm University.
  * Purver and Sadrzadeh, 2015 Purver, M. and Sadrzadeh, M.  (2015).  From distributional semantics to distributional pragmatics.  In Proceedings of the IWCS 2015 Workshop on Interactive Meaning Construction, pages 21–22.
  * Reichel et al., 2009 Reichel, U. D., Kleber, F., and Winkelmann, R.  (2009).  Modelling similarity perception of intonation.  In Interspeech, pages 1711–1714.
  * Richie and Bhatia, 2021 Richie, R. and Bhatia, S.  (2021).  Similarity judgment within and across categories: A comprehensive model comparison.  Cognitive Science, 45(8):e13030.
  * Rilliard et al., 2011 Rilliard, A., Allauzen, A., and Boula de Mareuil, P.  (2011).  Using dynamic time warping to compute prosodic similarity measures.  In Interspeech.
  * Rubenstein et al., 2023 Rubenstein, P. K., Asawaroengchai, C., Nguyen, D. D., Bapna, A., Borsos, Z., Quitry, F. d. C., Chen, P., Badawy, D. E., Han, W., Kharitonov, E., et al.  (2023).  AudioPaLM: A large language model that can speak and listen.  arXiv preprint arXiv:2306.12925.
  * Salesky et al., 2021 Salesky, E., Mäder, J., and Klinger, S.  (2021).  Assessing evaluation metrics for speech-to-speech translation.  In IEEE Automatic Speech Recognition and Understanding Workshop (ASRU), pages 733–740.
  * Seebauer et al., 2023 Seebauer, F., Kuhlmann, M., Haeb-Umbach, R., and Wagner, P.  (2023).  Re-examining the quality dimensions of synthetic speech.  In 12th Speech Synthesis Workshop (SSW) 2023.
  * Wagner et al., 2019 Wagner, P., Beskow, J., Betz, S., Edlund, J., Gustafson, J., Eje Henter, G., Le Maguer, S., Malisz, Z., Székely, É., Tånnander, C., et al.  (2019).  Speech synthesis evaluation: State-of-the-art assessment and suggestion for a novel research program.  In Proceedings of the 10th Speech Synthesis Workshop.
  * Wallbridge et al., 2023 Wallbridge, S., Bell, P., and Lai, C.  (2023).  Do dialogue representations align with perception? An empirical study.  In Proceedings of the 17th Conference of the European Chapter of the Association for Computational Linguistics, pages 2696–2713.
  * Ward and Abu, 2016 Ward, N. G. and Abu, S.  (2016).  Action-coordinating prosody.  In Speech Prosody.
  * Ward et al., 2005 Ward, N. G., Rivera, A. G., Ward, K., and Novick, D. G.  (2005).  Root causes of lost time and user stress in a simple dialog system.  In Interspeech.
  * Ward et al., 2015 Ward, N. G., Werner, S. D., Garcia, F., and Sanchis, E.  (2015).  A prosody-based vector-space model of dialog activity for information retrieval.  Speech Communication, 68:85–96.
  * Ward et al., 2021 Ward, N. G., Avila, J. E., and Alarcon, A. M.  (2021).  Towards continuous estimation of dissatisfaction in spoken dialog.  In SigDial, pages 13–20.
  * Ward et al., 2023 Ward, N. G., Avila, J. E., Rivas, E., and Marco, D.  (2023).  Dialogs re-enacted across languages, version 2.  Technical Report UTEP-CS-23-27, University of Texas at El Paso, Department of Computer Science.
  * Wieting et al., 2019 Wieting, J., Berg-Kirkpatrick, T., Gimpel, K., and Neubig, G.  (2019).  Beyond BLEU: Training neural machine translation with semantic similarity.  In ACL.
  * Yang et al., 2018 Yang, Y., Yuan, S., Cer, D., Kong, S.-y., Constant, N., Pilar, P., Ge, H., Sung, Y.-H., Strope, B., and Kurzweil, R.  (2018).  Learning semantic textual similarity from conversations.  In Third Workshop on Representation Learning for NLP, pages 164–174.
  * Zhao et al., 2019 Zhao, W., Peyrard, M., Liu, F., Gao, Y., Meyer, C. M., and Eger, S.  (2019).  MoverScore: Text generation evaluating with contextualized embeddings and earth mover distance.  In ENMLP.
  * Zielinski et al., 2008 Zielinski, S., Rumsey, F., and Bech, S.  (2008).  On some biases encountered in modern audio quality listening tests: A review.  Journal of the Audio Engineering Society, 56:427–451.

## 11\. Language Resource References

bib languageresource


††thanks: KG and SD contributed equally and are joint first authors.††thanks:
KG and SD contributed equally and are joint first authors.

# Planar Hall Effect in Quasi-Two-Dimensional Materials

Koushik Ghorai  Sunit Das  Harsh Varshney  Amit Agarwal<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Kanpur, Kanpur-208016,
India

###### Abstract

The planar Hall effect in 3D systems is an effective probe for their Berry
curvature, topology, and electronic properties. However, the Berry curvature-
induced conventional planar Hall effect is forbidden in 2D systems as the out-
of-plane Berry curvature cannot couple to the band velocity of the electrons
moving in the 2D plane. Here, we demonstrate a unique 2D planar Hall effect
(2DPHE) originating from the hidden planar components of the Berry curvature
and orbital magnetic moment in quasi-2D materials. We identify all planar band
geometric contributions to 2DPHE and classify their crystalline symmetry
restrictions. Using gated bilayer graphene as an example, we show that in
addition to capturing the hidden band geometric effects, 2DPHE is also
sensitive to the Lifshitz transitions. Our work motivates further exploration
of hidden planar band geometry-induced 2DPHE and related transport phenomena
for innovative applications.

The planar Hall effect (PHE) is the generation of longitudinal and transverse
voltages in the plane of the applied electric ($\bm{E}$) and magnetic fields
($\bm{B}$). In contrast to the conventional and anomalous Hall effect, the
transport in PHE is dissipative, and the response typically varies
quadratically with the $B$. PHE has extensive applications in magnetic sensors
and memory devices [1]. In 3D materials, PHE generally originates from the
coupling of the Berry curvature (BC) and orbital magnetic moment (OMM) to the
band velocity and in-plane magnetic field, which generates a longitudinal and
transverse planar response. Initial studies of PHE used it effectively to
probe the magnetization reversal in magnetic materials [2, 3, 4, 5, 6]. More
recently, we have used PHE to explore novel topological semimetals [7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17] and topological insulators [18, 19].

However, conventional PHE probes are ineffective in 2D systems. As the 2D
plane confines the orbital motion of electrons, these systems can host only
out-of-plane Berry curvature and orbital magnetic moment [20, 21].
Consequently, the PHE induced by the component of the BC and OMM along the
applied in-plane magnetic field is forbidden in perfect 2D systems. Some 2D
materials with strong spin-orbit coupling exhibit an intrinsic magneto-Hall
response driven by magnetic field-induced changes to the Berry curvature [22,
23, 24, 25, 26, 27, 28]. However, such responses are antisymmetric tensors and
absent in systems lacking strong spin-orbit interactions. These limitations
severely restrict our ability to utilize PHE to explore fundamental physics
and develop ultra-sensitive magnetic sensors and other applications in 2D
materials.

In this Letter, we introduce a unique 2D planar Hall effect (2DPHE) in layered
2D materials such as bilayer graphene. Layered 2D materials with finite inter-
layer tunneling can host an intrinsic in-plane component of the BC and OMM if
the system’s space inversion or time-reversal symmetry is broken [29, 30, 31].
We demonstrate that these relatively unexplored in-plane components of the
band geometric quantities induce the 2DPHE response (see Fig. 1). We present a
thorough analysis of the 2DPHE responses, particularly their angular variation
(angle between ${\bm{E}}$ and ${\bm{B}}$), and classify the crystalline
symmetry restrictions on the different 2DPHE response tensors. As an
illustrative example, we focus on Bernal stacked bilayer graphene to
demonstrate a sizable and gate-tunable 2DPHE response. Beyond predicting the
unique phenomena of 2DPHE, our findings motivate the exploration of other
transport and optical phenomena induced by the hidden planar band geometric
quantities in layered 2D materials [32].

Figure 1: Schematic for 2D planar Hall effect (2DPHE). Layered 2D materials host hidden planar Berry curvature (${\bm{\Omega}}^{\rm planar}$) and planar orbital magnetic moment (${\bm{m}}^{\rm planar}$) arising from inter-layer tunneling. The ${\bm{\Omega}}^{\rm planar}$ and ${\bm{m}}^{\rm planar}$ combine with the in-plane electric and magnetic field to induce a longitudinal and transverse current in the 2D plane.  Table 1:  The symmetry restrictions of the longitudinal and planar Hall response tensors pertaining to two-dimensional materials. The cross (✗) and the tick (✓) mark signify the corresponding response tensor is symmetry forbidden and allowed, respectively. The longitudinal and transverse PHE tensors in the same row have identical symmetry restrictions. Longitudinal | Transverse | ${\mathcal{P}}$ | ${\mathcal{T}}$ | $\mathcal{P}\mathcal{T}$ | ${\cal M}_{x}$ | ${\cal M}_{y}$ | ${\cal M}_{z}$ | ${\cal C}_{2x}$ | ${\cal C}_{2y}$ | ${\cal C}_{2z}$ | ${\cal C}_{3z}$ | ${\cal C}_{4z}$ | ${\cal C}_{6z}$ | ${\cal S}_{4z}$ | ${\cal S}_{6z}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\chi_{xx;x}$ | $\chi_{yx;y}$ | ✓ | ✗ | ✗ | ✓ | ✗ | ✗ | ✓ | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✓
$\chi_{xx;y}$ | $\chi_{yx;x}$ | ✓ | ✗ | ✗ | ✗ | ✓ | ✗ | ✗ | ✓ | ✗ | ✓ | ✗ | ✗ | ✗ | ✓
$\chi_{xx;xx}$, $\chi_{xx;yy}$ | $\chi_{yx;xy}$ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
$\chi_{xx;xy}$ |  $\chi_{yx;xx}$, $\chi_{yx;yy}$ | ✓ | ✓ | ✓ | ✗ | ✗ | ✓ | ✗ | ✗ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓

Planar-BC and OMM in 2D systems:– In quasi-2D materials with two or more
atomic layers, the finite inter-layer hopping amplitude enables the inter-
layer tunneling of electrons. The inter-layer coherence of the electron
wavefunction gives rise to hidden planar components of the BC and OMM. These
are given by,

$\displaystyle{\bm{\Omega}}_{n\bm{k}}^{\rm planar}$
$\displaystyle={2}{\hbar}~{}{\rm Re}\sum_{n^{\prime}\neq
n}\frac{{\bm{v}}_{nn^{\prime}}\times\bm{\mathcal{Z}}_{n^{\prime}n}}{(\varepsilon_{n\bm{k}}-\varepsilon_{n^{\prime}\bm{k}})}~{},$
(1) $\displaystyle{\bm{m}}_{n\bm{k}}^{\rm planar}$ $\displaystyle={e}~{}{\rm
Re}\sum_{n^{\prime}\neq
n}{\bm{v}}_{nn^{\prime}}\times\bm{\mathcal{Z}}_{n^{\prime}n}~{}.$ (2)

Here, we have defined the velocity matrix elements as
$\hbar{\bm{v}}_{nn^{\prime}}=\bra{u_{n{\bm{k}}}}\nabla_{\bm{k}}\mathcal{H}\ket{u_{n^{\prime}{\bm{k}}}}$,
with ${\bm{k}}=(k_{x},k_{y})$. $\varepsilon_{n\bm{k}}$ and
$\ket{u_{n{\bm{k}}}}$ are the band energy and periodic part of the Bloch
wavefunction for the $n$-th band, respectively. The matrix elements of the
out-of-plane position operator are defined as
$\bm{\mathcal{Z}}_{nn^{\prime}}=\hat{\bm{z}}\bra{u_{n\bm{k}}}\mathcal{Z}\ket{u_{n^{\prime}\bm{k}}}$,
with $\mathcal{Z}$ being the position operator along the
$\hat{\bm{z}}$-direction. These in-plane contributions combined with the
conventional out-of-plane components yield the total BC and OMM for quasi-2D
materials. We present the detailed derivation of these equations in Sec. S1 of
the Supplementary Materials (SM) 111The Supplementary Material discusses: i)
the derivation of planar-BC and planar-OMM expressions, ii) general expression
for planar-BC and planar-OMM and analytical calculation of them for $2\times
2$ low-energy bilayer graphene model. iii) the detailed derivation of
longitudinal and planar Hall response tensors, iv) the details of symmetry
analysis, v) the strain implementation in the tight-binding model for bilayer
graphene and $y$ components of planar Berry curvature and OMM, vi) the Van
Hove singularity and Lifshitz transition of Fermi surface, vii) estimation of
planar Hall voltage, viii) comparison of planar Berry curvature and OMM for
bilayer and trilayer graphene, and ix) other in-plane magneto-Hall responses
in two-dimensional systems.. We emphasize that the planar-BC and planar-OMM
rely on inter-layer hybridization of electronic states, which makes the off-
diagonal components of $\bm{\mathcal{Z}}_{nn^{\prime}}$ finite. We illustrate
the emergence of the planar-BC and OMM and their symmetry properties in an
intuitive way using a $2\times 2$ low energy model [6, 34] Hamiltonian of
bilayer graphene in Sec. S2 of the SM [33].

2D Planar Hall effect:– In 2D systems, generally, the magnetic field interacts
with electrons primarily through Zeeman coupling to its spin [22, 23, 24, 25,
26, 28]. In contrast, the planar-OMM allows the magnetic field to couple
directly to the orbital motion of electrons. This modifies the band energy
($\tilde{\varepsilon}_{n\bm{k}}=\varepsilon_{n\bm{k}}-{{\bm{m}}_{n\bm{k}}^{\textrm{planar}}}\cdot{\bm{B}}$)
and the band velocity. More importantly, the planar-BC combines with the band
velocity to generate a finite chiral magnetic velocity[35] in 2D systems,
which is
$\propto({\bm{v}}_{n\bm{k}}\cdot{{\bm{\Omega}}_{n\bm{k}}^{\textrm{planar}}}){\bm{B}}$.
We show below that these magnetic field-dependent velocity contributions
arising from the hidden planar-BC and -OMM generate a previously unexplored
planar Hall effect in 2D systems.

In the semiclassical Boltzmann transport framework, the charge current is
given by ${\bm{j}}=-e\sum_{n}\int[d{\bm{k}}]{\dot{\bm{r}}_{n}}g_{n\bm{k}}$.
Here $g_{n\bm{k}}$ is the non-equilibrium distribution function,
$\dot{\bm{r}}_{n}$ is the wave-packet velocity, and $[d{\bm{k}}]\equiv
d^{2}{\bm{k}}/{(2\pi)^{2}}$ for 2D systems. Using the expressions of the
planar-BC and OMM modified $\dot{\bm{r}}_{n}$ and $g_{n\bm{k}}$ up to linear
order in the applied electric field, we calculate the planar current density
to the first and second orders in the magnetic field strength $B$. See Sec. S3
of the SM for a detailed derivation [33]. We obtain the longitudinal and
transverse components of the 2DPHE currents to be

$j_{a}=\tau\chi_{ab;c}E_{b}B_{c}+\tau\chi_{ab;cd}E_{b}B_{c}B_{d}~{}.$ (3)

Here, $\tau$ is the electron scattering time, $\\{a,b,c,d\\}\in\\{x,y\\}$ are
the 2D Cartesian coordinates, and the Einstein summation convention is used.
In Eq. (3), $\chi_{ab;cd}$ is symmetric under the exchange of magnetic field
indices $(c,d)$. The 2DPHE response tensors can expressed as a sum of the
planar-BC, planar-OMM, and mixed terms,

$\chi_{ab;c(d)}=\chi_{ab;c(d)}^{\textrm{BC}}+\chi_{ab;c(d)}^{\textrm{OMM}}+\chi_{ab;c(d)}^{\textrm{BC+OMM}}~{}.$
(4)

We obtain the planar-BC contributions to be

$\displaystyle\chi_{ab;c}^{\textrm{BC}}$ $\displaystyle=$
$\displaystyle-e^{2}\int_{n,{\bm{k}}}[(v_{a}\delta_{bc}+v_{b}\delta_{ac})\Omega_{V}-\frac{e}{\hbar}v_{a}v_{b}\Omega_{c}]f_{0}^{\prime},$
(5) $\displaystyle\chi_{ab;cd}^{\textrm{BC}}$ $\displaystyle=$
$\displaystyle-\frac{e^{2}}{2}\int_{n,{\bm{k}}}\biggl{[}\delta_{ad}\delta_{bc}\Omega_{V}^{2}-\frac{e}{\hbar}(v_{a}\delta_{bc}+v_{b}\delta_{ac})\Omega_{d}\Omega_{V}$
(6)
$\displaystyle+\frac{e^{2}}{\hbar^{2}}v_{a}v_{b}\Omega_{c}\Omega_{d}\biggr{]}f_{0}^{\prime}+(c\leftrightarrow
d)~{}.$

Here, $\Omega_{V}\equiv(e/\hbar)\bm{v}_{\bm{k}}\cdot\bm{\Omega}_{\bm{k}}$ with
$\hbar{\bm{v}}_{\bm{k}}=\nabla_{\bm{k}}{\varepsilon}_{\bm{k}}$ being the band
velocity without any magnetic field, and $\delta_{ab}$ is the Kronecker delta
function. For brevity, we have defined
$\int_{n,{\bm{k}}}\equiv\sum_{n}\int[d{\bm{k}}]$, and we do not explicitly
mention the band index $n$ in the physical quantities. We present the
expressions for other contributions in Eq. (4) in Sec. S3 of the SM [33].

The planar response tensors in Eqs. (5) and (6) are proportional to either
${\bm{v}}_{\bm{k}}\cdot{\bm{\Omega}}_{\bm{k}}$ or,
$\hat{\bm{B}}\cdot{\bm{\Omega}}_{\bm{k}}$ or, the combination of these terms.
For a quasi-2D system, all of these terms vanish if we consider only the
conventional out-of-plane BC. As a consequence, earlier works missed this
phenomenon. This highlights the crucial role of the hidden planar-BC and -OMM
in generating the PHE response in quasi-2D systems. Furthermore, the response
tensors $\chi_{ab;c}$ and $\chi_{ab;cd}$ are symmetric with respect to its
first two indices. Therefore, we have $\sum_{a}j_{a}E_{a}\neq 0$, indicating
the dissipative nature of the planar Hall current. Having established the
possibility of 2DPHE, we now analyze the restrictions imposed by crystalline
point group symmetries on different 2DPHE response tensors.

Crystal symmetry restrictions:— The inversion symmetry ($\cal P$) imposes no
constraints as both $\chi_{ab;c}$ and $\chi_{ab;cd}$ represent linear in $E$
responses. However, under time-reversal ($\cal T$) operation, ${j}$, ${B}$,
and $\tau$ reverse signs, while $E$ is $\cal T$ even. Thus, $\chi_{ab;c}$ is a
third rank $\cal T$-odd axial tensor [36, 37], which is forbidden in non-
magnetic systems (see Sec. S4 of SM [33] for details). In contrast,
$\chi_{ab;cd}$ is a fourth rank $\cal T$-even polar tensor and is the leading
order contribution in non-magnetic systems. Denoting a general point group
operation via $\cal O$, the $\chi_{ab;c}$ and $\chi_{ab;cd}$ tensors obey the
following transformation rules [36]

$\displaystyle\chi_{a^{\prime}b^{\prime};c^{\prime}}$ $\displaystyle=$
$\displaystyle\eta_{\mathcal{T}}\det{\mathcal{O}}{\mathcal{O}}_{a^{\prime}a}{\mathcal{O}}_{b^{\prime}b}{\mathcal{O}}_{c^{\prime}c}~{}\chi_{ab;c}~{},$
(7) $\displaystyle\chi_{a^{\prime}b^{\prime};c^{\prime}d^{\prime}}$
$\displaystyle=$
$\displaystyle{\mathcal{O}}_{a^{\prime}a}{\mathcal{O}}_{b^{\prime}b}{\mathcal{O}}_{c^{\prime}c}{\mathcal{O}}_{d^{\prime}d}~{}\chi_{ab;cd}~{}.$
(8)

Here, $\eta_{\cal T}=\pm 1$ is associated with the magnetic point group
symmetry transformation: $\eta_{\cal T}=-1$ ($\eta_{\cal T}=1$) for magnetic
(non-magnetic) point group operation ${\cal O}\equiv{\cal RT}$ (${\cal
O}\equiv{\cal R}$), with $\cal R$ being a spatial operation.

To be specific about the symmetry constraints of the 2DPHE response, we apply
the $\bm{E}$ along $\hat{\bm{x}}$ direction, and an in-plane magnetic field at
an angle $\phi$ with ${\bm{E}}$, i.e., $(B_{x},B_{y})=B(\cos\phi,\sin\phi)$
(see Fig. 1). As the response tensors are symmetric in the first two indices,
the independent tensor elements for the $B$-linear longitudinal (transverse)
responses are $\chi_{xx;x}$ and $\chi_{xx;y}$ ($\chi_{yx;x}$ and
$\chi_{yx;y}$). The fourth rank tensor $\chi_{ab;cd}$ is symmetric in the
first two $(a,b)$ and the last two $(c,d)$ indices. Hence, for the
quadratic-$B$ longitudinal (transverse) response $\chi_{xx;xx}$,
$\chi_{xx;yy}$ and $\chi_{xx;xy}$ ($\chi_{yx;xx}$, $\chi_{yx;yy}$ and
$\chi_{yx;xy}$) are the only independent elements. We present crystalline
symmetry imposed restrictions on these tensor elements for non-magnetic and
magnetic systems in Table 1 and Table S1 of SM [33], respectively. An
interesting conclusion from our symmetry analysis is that the presence of
${\cal C}_{3z}$ symmetry does not restrict any of the 2DPHE response tensors.
This makes hexagonal systems such as multi-layered graphene, transition metal
dichalcogenides, and their twisted moiré heterostructures good candidates to
observe 2DPHE. We now focus on the angular variation of the 2DPHE current.

Figure 2:  (a) Electronic band structure of strained BLG around the $K$ point,
with the background color showing the density of states (DOS). b) The
variation of DOS with chemical potential $\mu$ and the inter-layer potential
$\Delta$ in (b). The Van Hove singularity in the DOS near the band edges [also
seen in (a)] is accompanied by a Lifshitz transition, where the system evolves
from hosting three Fermi pockets around the $K$ (or $K^{\prime}$) point to one
Fermi pocket. (c) The upper (lower) panel shows the $k$-space distribution of
the $x$-component of the planar BC (OMM) for the first conduction band in the
unit of $a^{2}$ $\left(\frac{e}{\hbar}a^{2}{\rm eV}\right)$, where $a$ is the
lattice constant. (d) Different components of 2DPHE response tensors
$\chi_{ab;cd}$ as a function of $\mu$ evaluated at temperature $T=50$K.

Angular variation of the 2DPHE currents:– The variation of the 2DPHE currents
with the planar angle between the $\bm{E}$ and $\bm{B}$ is important for
exploring its origin and the relative contribution of different terms. We work
with the field configuration described in Fig. 1 to obtain the longitudinal
and transverse 2DPHE currents: $j_{\parallel(\perp)}^{{\rm
2DPHE}}=\sigma_{{\parallel}(\perp)}E_{\parallel}$. We calculate the angular
dependence of the 2DPHE conductivities to be,

$\displaystyle\sigma_{\parallel}$ $\displaystyle=$ $\displaystyle\tau
B(\chi_{xx;x}\cos\phi+\chi_{xx;y}\sin\phi)+\tau
B^{2}(\chi_{xx;xx}\cos^{2}{\phi}$ (9)
$\displaystyle+~{}\chi_{xx;yy}\sin^{2}{\phi}+\chi_{xx;xy}\sin{\phi}\cos{\phi})~{},$
$\displaystyle\sigma_{\perp}$ $\displaystyle=$ $\displaystyle\tau
B(\chi_{yx;x}\cos{\phi}+\chi_{yx;y}\sin{\phi})+\tau
B^{2}(\chi_{yx;xx}\cos^{2}{\phi}$ (10)
$\displaystyle+~{}\chi_{yx;yy}\sin^{2}{\phi}+\chi_{yx;xy}\sin{\phi}\cos{\phi})~{}.$

These equations and the symmetry restrictions in Table 1 (and Table S1 in SM
[33]) provide a complete characterization of the 2DPHE responses. For non-
magnetic systems with an in-plane mirror or an in-plane two-fold rotation
symmetry, the longitudinal (transverse) response is entirely captured by
$\chi_{xx;xx}$, and $\chi_{xx;yy}$ ($\chi_{yx;xy}$) with the conventional
$\cos^{2}\phi$ ($\sin 2\phi$) angular dependence.

2DPHE in gated bilayer graphene:– To demonstrate the 2DPHE in a realistic
system, we consider the tight-binding model of Bernal stacked bilayer graphene
(BLG). It offers the natural advantage of being readily available, and its
doping and layer asymmetry can be tuned via the combination of top and bottom
gate voltages. The Hamiltonian of pristine BLG with a vertical displacement
field possesses $\cal T$ and ${\cal C}_{3z}$ symmetry, while it breaks $\cal
P$ symmetry. Owing to the presence of $\cal T$ symmetry in BLG, only the
$B^{2}$ contributions to the 2DPHE in BLG are allowed. The breakdown of $\cal
P$ symmetry is crucial for inducing a planar-BC and planar-OMM in systems with
$\cal T$ symmetry. BLG also has a mirror symmetry about its armchair
direction, which we represent by ${\cal M}_{x}$ (see Fig. S3 of SM [33]). The
${\cal M}_{x}$ symmetry of BLG dictates that only the $\chi_{yx;xy}$ component
can be finite with the $\sin 2\phi$ angular dependence in $\sigma_{\perp}$. To
explore the role of other planar Hall contributions $\propto B^{2}$
($\chi_{yx;xx}$ and $\chi_{yx;yy}$), we break the ${\cal M}_{x}$ symmetry of
BLG by applying a uniaxial strain of $1\%$ strength at an angle of
$30^{\circ}$ to the zig-zag direction. The details of the strain
implementation in the tight-binding model of BLG are discussed in Sec. S5 of
SM [33]. We present the band structure of strained BLG around one of the two
valleys ($K$ point) in Fig. 2(a). In Fig. 2(b), we show the color plot of the
density of states as a function of the chemical potential ($\mu$) and inter-
layer potential ($\Delta$).

For BLG, the out-of-plane position operator is given by the matrix ${\cal
Z}=\frac{c}{2}\sigma_{0}\otimes\tau_{z}$. $\sigma_{0}$ is the identity matrix
for the sublattice space, and $\tau_{z}$ is the Pauli matrix in the layer
space. Here, $c\approx 3.35~{}{\rm\AA}$ is the inter-layer distance. We use
this in Eqs. (1) and (2) to calculate the planar-BC and -OMM. We present the
$x$-component of planar-BC and planar-OMM for the first conduction band in
Fig. 2(c). The $y$-components of these quantities are presented in Sec. S1 of
SM [33] 222We emphasize that these quantities are not ${\cal C}_{3z}$
symmetric even without any in-plane strain [31, Jiang_prb20]. However, they
are valley contrasting due to the global $\cal T$-symmetry of BLG..

We numerically calculate the 2DPHE responses of BLG, illustrating their
dependence on the chemical potential $\mu$ in Fig. 2(d). In contrast to
$\chi_{xx;xx}$, $\chi_{xx;yy}$, and $\chi_{yx;xy}$, which remain finite even
in the presence of ${\cal M}_{x}$ symmetry, the contributions $\chi_{xx;xy}$,
$\chi_{yx;xx}$, and $\chi_{yx;yy}$ require ${\cal M}_{x}$ symmetry breaking to
be finite. As a consequence, these contributions are relatively smaller in
magnitude. We highlight that all the 2DPHE response tensors are pronounced in
the vicinity of the band edges, where the planar band geometric quantities
have a hotspot [see Fig. 2(c)]. Interestingly, the peak in the responses near
the band edge arises from the Van Hove singularity in the density of states,
which is a marker of the Lifshitz transitions in BLG (see Sec. S6 of the SM
[33] for more details).

We present color plots of the variation of $\sigma_{\parallel}$ and
$\sigma_{\perp}$ with $\mu$ and the inter-layer potential $\Delta$ in Fig.
3(a) and 3(c). Both the conductivities have appreciable values only in the
vicinity of the band edges, highlighting the band-geometric nature of 2DPHE.
The peaks in both $\sigma_{\perp}$ and $\sigma_{\parallel}$ reflect the Van
Hove singularity in the DOS, marked by the dark regions in Fig. 2(b). We
present the the angular variation of $\sigma_{\parallel}$ and
$\sigma_{\perp}$, as $\mu$ is varied, in the polar color plots in Figs. 3(b)
and 3(d). We highlight that the angular variation of the 2DPHE responses
deviates from the conventional $\sigma_{\parallel}\propto\cos^{2}\phi$ and
$\sigma_{\perp}\propto\sin 2\phi$ dependence due to the strain-induced mirror
symmetry breaking 333This is a consequence of the strain-induced ${\cal
M}_{x}$ symmetry breaking, which makes $\chi_{xx;xy},\,\chi_{yx;xx}$ and
$\chi_{yx;yy}$ components finite, modifying the angular dependence following
Eqs. (9) and (10). .

Experimental implications:– For estimating the feasibility of measuring 2DPHE
responses in experiments, we consider an in-plane magnetic field of $B=1$
Tesla, $\tau=1$ ps, and $\phi=60^{\circ}$. With these parameters,
$\sigma_{\perp}$ ($\sigma_{\parallel}$) becomes $\sim 0.1~{}(4.5)~{}{\rm
AV^{-1}m^{-1}}$ in BLG. Here, we have converted conductivities to the
conventional 3D unit using the layer thickness of BLG. Assuming a sample size
of $\sim 10$ $\mu$m and a moderate electric field of $E\sim 1$ V/$\mu$m, we
estimate the planar Hall voltage to be $V_{\perp}\sim 0.17$ $\mu$V (see Sec.
S7 of SM [33]), which is well within experimental reach. Additionally, note
that ${\bm{\Omega}}_{n\bm{k}}^{\rm planar}$ and ${\bm{m}}_{n\bm{k}}^{\rm
planar}$ magnitude and hence the strength of 2DPHE responses increase with the
number of layers. We show this explicitly for multi-layered graphene in Sec.
S8 of SM [33].

Figure 3: The color plot of the (a) longitudinal and (c) transverse 2DPHE
conductivities in $\mu-\Delta$ space for strained BLG. These parameters can be
experimentally tuned via the top and back gates. We have chosen the $\tau=1$
ps, $B=1$ Tesla, $\phi=60^{\circ}$ and temperature $T=50$K. The angular
dependence of the (b) longitudinal and (d) transverse 2DPHE conductivities.
The small deviation from $\sigma_{\parallel}\propto\cos^{2}\phi$ and
$\sigma_{\perp}\propto\sin(2\phi)$ dependence is induced by the strain-induced
mirror symmetry breaking. In the angular plots, we have $\Delta=0.05~{}{\rm
eV}$.

We now explore ways to distinguish 2DPHE from other possible in-plane magneto-
Hall responses (see Sec. S9 of SM [33] for details). The corrections in the
anomalous Hall velocity generated by the magnetic field-induced Berry
curvature can give rise to in-plane magneto-Hall responses $\propto EB$ [22,
23, 24, 25, 26, 28, 40]. The planar Hall effect can also arise from the
anisotropic spin scattering mechanism [41, 42, 43, 44, 45, 46, 47, 48]. All
these in-plane magneto-Hall responses in 2D systems require strong spin-orbit
coupling (SOC). Also, the responses are represented by the antisymmetric
(symmetric) response tensors when they arise from the field-dependent
anomalous velocity (anisotropic spin-scattering mechanism). In contrast, our
proposed 2DPHE responses are symmetric and do not require SOC to be finite.
Consequently, in a layered 2D system with SOC, such as transition metal
dichalcogenides, the symmetrization of the total planar response will discard
all antisymmetric contributions. The total symmetric planar response will have
contributions from the 2DPHE and asymmetric spin scattering. However, the
asymmetric spin scattering contribution is comparatively negligible in BLG due
to its very small SOC strength. The asymmetric spin-scattering-induced
contribution can be differentiated by its unique double-peak structure around
the charge-neutrality point [42, 46]. Beyond these responses, lock-in
measurement techniques can easily distinguish 2DPHE from other nonlinear in-
plane responses.

Conclusion:– Our discovery of 2DPHE brings the vast class of layered 2D
materials under the purview of planar Hall effect probes, which were limited
to 3D materials. Additionally, 2DPHE offers a novel tool to probe the
previously unexplored planar quantum-geometric properties of Bloch electrons
in 2D materials. The existence of planar Berry curvature and orbital magnetic
moment motivates the study of other novel phenomena, which were believed to be
inaccessible in 2D materials. For instance, the planar Berry curvature can
give rise to a vertical (perpendicular to the 2D plane) anomalous Hall effect
in the linear and nonlinear response regimes. An interesting application of
this is that vertical charge transport, with restricted out-of-plane carrier
velocity, can induce an inter-layer electric polarization. This may offer a
novel way to control the switching of out-of-plane electric polarization in
layered ferroelectric materials [49, 50, 51] via an in-plane electric field,
with potential for new device applications. 2DPHE also enables the designing
of highly sensitive planar Hall magnetic sensors using ultra-thin 2D
materials. Our work opens new avenues for further exploration of novel
transport effects and their potential applications arising from the hidden
planar band geometric quantities.

## Acknowledgments

We acknowledge many fruitful discussions with Debottam Mandal (IIT Kanpur,
India) and Atasi Chakraborty (Johannes Gutenberg University, Germany). KG, SD,
and HV acknowledge the Ministry of Education, Government of India, for funding
support through the Prime Minister’s Research Fellowship program.

## References

  * Quynh  _et al._ [2019] L. K. Quynh, N. T. Hien, N. H. Binh, T. T. Dung, B. D. Tu, N. H. Duc, and D. T. H. Giang, Simple planar hall effect based sensors for low-magnetic field detection, Advances in Natural Sciences: Nanoscience and Nanotechnology 10, 025002 (2019).
  * Yau and Chang [1971] K. L. Yau and J. T. H. Chang, The planar hall effect in thin foils of ni-fe alloy, Journal of Physics F: Metal Physics 1, 38–43 (1971).
  * Li  _et al._ [2010] J. Li, S. L. Li, Z. W. Wu, S. Li, H. F. Chu, J. Wang, Y. Zhang, H. Y. Tian, and D. N. Zheng, A phenomenological approach to the anisotropic magnetoresistance and planar hall effect in tetragonal la2/3ca1/3mno3thin films, Journal of Physics: Condensed Matter 22, 146006 (2010).
  * Tang  _et al._ [2003] H. X. Tang, R. K. Kawakami, D. D. Awschalom, and M. L. Roukes, Giant planar hall effect in epitaxial (ga,mn)as devices, Phys. Rev. Lett. 90, 107201 (2003).
  * Bowen  _et al._ [2005] M. Bowen, K.-J. Friedland, J. Herfort, H.-P. Schönherr, and K. H. Ploog, Order-driven contribution to the planar hall effect in ${\mathrm{fe}}_{3}\mathrm{Si}$ thin films, Phys. Rev. B 71, 172401 (2005).
  * Schaefer and Nowack [2021] B. T. Schaefer and K. C. Nowack, Electrically tunable and reversible magnetoelectric coupling in strained bilayer graphene, Phys. Rev. B 103, 224426 (2021).
  * Nandy  _et al._ [2017] S. Nandy, G. Sharma, A. Taraphder, and S. Tewari, Chiral anomaly as the origin of the planar hall effect in weyl semimetals, Phys. Rev. Lett. 119, 176804 (2017).
  * Kumar  _et al._ [2018] N. Kumar, S. N. Guin, C. Felser, and C. Shekhar, Planar hall effect in the weyl semimetal gdptbi, Phys. Rev. B 98, 041103 (2018).
  * Deng  _et al._ [2019] M.-X. Deng, H.-J. Duan, W. Luo, W. Y. Deng, R.-Q. Wang, and L. Sheng, Quantum oscillation modulated angular dependence of the positive longitudinal magnetoconductivity and planar hall effect in weyl semimetals, Phys. Rev. B 99, 165146 (2019).
  * Li  _et al._ [2023] L. Li, J. Cao, C. Cui, Z.-M. Yu, and Y. Yao, Planar hall effect in topological weyl and nodal-line semimetals, Phys. Rev. B 108, 085120 (2023).
  * Ma  _et al._ [2019] D. Ma, H. Jiang, H. Liu, and X. C. Xie, Planar hall effect in tilted weyl semimetals, Phys. Rev. B 99, 115121 (2019).
  * Das  _et al._ [2020] K. Das, S. K. Singh, and A. Agarwal, Chiral anomalies induced transport in weyl metals in quantizing magnetic field, Phys. Rev. Res. 2, 033511 (2020).
  * Wei  _et al._ [2023] Y.-W. Wei, J. Feng, and H. Weng, Spatial symmetry modulation of planar hall effect in weyl semimetals, Phys. Rev. B 107, 075131 (2023).
  * Das and Agarwal [2019] K. Das and A. Agarwal, Linear magnetochiral transport in tilted type-i and type-ii weyl semimetals, Phys. Rev. B 99, 085405 (2019).
  * Das  _et al._ [2023] S. Das, K. Das, and A. Agarwal, Chiral anomalies in three-dimensional spin-orbit coupled metals: Electrical, thermal, and gravitational anomalies, Phys. Rev. B 108, 045405 (2023).
  * Ghosh and Mandal [2024a] R. Ghosh and I. Mandal, Direction-dependent conductivity in planar hall set-ups with tilted weyl/multi-weyl semimetals, Journal of Physics: Condensed Matter 36, 275501 (2024a).
  * Ghosh and Mandal [2024b] R. Ghosh and I. Mandal, Electric and thermoelectric response for weyl and multi-weyl semimetals in planar hall configurations including the effects of strain, Physica E: Low-dimensional Systems and Nanostructures 159, 115914 (2024b).
  * Rakhmilevich  _et al._ [2018] D. Rakhmilevich, F. Wang, W. Zhao, M. H. W. Chan, J. S. Moodera, C. Liu, and C.-Z. Chang, Unconventional planar hall effect in exchange-coupled topological insulator–ferromagnetic insulator heterostructures, Phys. Rev. B 98, 094404 (2018).
  * Nandy  _et al._ [2018] S. Nandy, A. Taraphder, and S. Tewari, Berry phase theory of planar hall effect in topological insulators, Scientific Reports 8, 14983 (2018).
  * Xiao  _et al._ [2010] D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
  * Cayssol and Fuchs [2021] J. Cayssol and J. N. Fuchs, Topological and geometrical aspects of band theory, Journal of Physics: Materials 4, 034007 (2021).
  * Zyuzin [2020] V. A. Zyuzin, In-plane hall effect in two-dimensional helical electron systems, Phys. Rev. B 102, 241105 (2020).
  * Cullen  _et al._ [2021] J. H. Cullen, P. Bhalla, E. Marcellina, A. R. Hamilton, and D. Culcer, Generating a topological anomalous hall effect in a nonmagnetic conductor: An in-plane magnetic field as a direct probe of the berry curvature, Phys. Rev. Lett. 126, 256601 (2021).
  * Battilomo  _et al._ [2021] R. Battilomo, N. Scopigno, and C. Ortix, Anomalous planar hall effect in two-dimensional trigonal crystals, Phys. Rev. Res. 3, L012006 (2021).
  * Liang  _et al._ [2018] T. Liang, J. Lin, Q. Gibson, S. Kushwaha, M. Liu, W. Wang, H. Xiong, J. A. Sobota, M. Hashimoto, P. S. Kirchmann, Z.-X. Shen, R. J. Cava, and N. P. Ong, Anomalous hall effect in ZrTe5, Nature Physics 14, 451 (2018).
  * Zhou  _et al._ [2022] J. Zhou, W. Zhang, Y.-C. Lin, J. Cao, Y. Zhou, W. Jiang, H. Du, B. Tang, J. Shi, B. Jiang, X. Cao, B. Lin, Q. Fu, C. Zhu, W. Guo, Y. Huang, Y. Yao, S. S. P. Parkin, J. Zhou, Y. Gao, Y. Wang, Y. Hou, Y. Yao, K. Suenaga, X. Wu, and Z. Liu, Heterodimensional superlattice with in-plane anomalous hall effect, Nature 609, 46 (2022).
  * Sun  _et al._ [2022] S. Sun, H. Weng, and X. Dai, Possible quantization and half-quantization in the anomalous hall effect caused by in-plane magnetic field, Phys. Rev. B 106, L241105 (2022).
  * Wang  _et al._ [2024] H. Wang, Y.-X. Huang, H. Liu, X. Feng, J. Zhu, W. Wu, C. Xiao, and S. A. Yang, Orbital origin of the intrinsic planar hall effect, Phys. Rev. Lett. 132, 056301 (2024).
  * Drigo and Resta [2020] E. Drigo and R. Resta, Chern number and orbital magnetization in ribbons, polymers, and single-layer materials, Phys. Rev. B 101, 165120 (2020).
  * Hara  _et al._ [2020] D. Hara, M. S. Bahramy, and S. Murakami, Current-induced orbital magnetization in systems without inversion symmetry, Phys. Rev. B 102, 184404 (2020).
  * Kim  _et al._ [2021] K.-W. Kim, H. Jeong, J. Kim, and H. Jin, Vertical transverse transport induced by hidden in-plane berry curvature in two dimensions, Phys. Rev. B 104, L081114 (2021).
  * Duong  _et al._ [2017] D. L. Duong, S. J. Yun, and Y. H. Lee, van der waals layered materials: Opportunities and challenges, ACS Nano 11, 11803–11830 (2017).
  * Note [1] The Supplementary Material discusses: i) the derivation of planar-BC and planar-OMM expressions, ii) general expression for planar-BC and planar-OMM and analytical calculation of them for $2\times 2$ low-energy bilayer graphene model. iii) the detailed derivation of longitudinal and planar Hall response tensors, iv) the details of symmetry analysis, v) the strain implementation in the tight-binding model for bilayer graphene and $y$ components of planar Berry curvature and OMM, vi) the Van Hove singularity and Lifshitz transition of Fermi surface, vii) estimation of planar Hall voltage, viii) comparison of planar Berry curvature and OMM for bilayer and trilayer graphene, and ix) other in-plane magneto-Hall responses in two-dimensional systems.
  * Ho  _et al._ [2021] S.-C. Ho, C.-H. Chang, Y.-C. Hsieh, S.-T. Lo, B. Huang, T.-H.-Y. Vu, C. Ortix, and T.-M. Chen, Hall effects in artificially corrugated bilayer graphene without breaking time-reversal symmetry, Nature Electronics 4, 116 (2021).
  * Das and Agarwal [2020] K. Das and A. Agarwal, Thermal and gravitational chiral anomaly induced magneto-transport in weyl semimetals, Phys. Rev. Res. 2, 013088 (2020).
  * Newnham [2005] R. E. Newnham,  _Properties of materials: anisotropy, symmetry, structure_ (Oxford university press, 2005).
  * Gallego  _et al._ [2019] S. V. Gallego, J. Etxebarria, L. Elcoro, E. S. Tasci, and J. M. Perez-Mato, Automatic calculation of symmetry-adapted tensors in magnetic and non-magnetic materials: a new tool of the Bilbao Crystallographic Server, Acta Crystallographica Section A 75, 438 (2019).
  * Note [2] We emphasize that these quantities are not ${\cal C}_{3z}$ symmetric even without any in-plane strain [31, Jiang_prb20]. However, they are valley contrasting due to the global $\cal T$-symmetry of BLG.
  * Note [3] This is a consequence of the strain-induced ${\cal M}_{x}$ symmetry breaking, which makes $\chi_{xx;xy},\,\chi_{yx;xx}$ and $\chi_{yx;yy}$ components finite, modifying the angular dependence following Eqs. (9) and (10).
  * Zheng  _et al._ [2024] H. Zheng, D. Zhai, C. Xiao, and W. Yao, Layer coherence origin of intrinsic planar hall effect in 2d limit (2024), arXiv:2402.17166 [cond-mat.mes-hall] .
  * Taskin  _et al._ [2017] A. A. Taskin, H. F. Legg, F. Yang, S. Sasaki, Y. Kanai, K. Matsumoto, A. Rosch, and Y. Ando, Planar hall effect from the surface of topological insulators, Nature Communications 8, 1340 (2017).
  * Breunig  _et al._ [2017] O. Breunig, Z. Wang, A. A. Taskin, J. Lux, A. Rosch, and Y. Ando, Gigantic negative magnetoresistance in the bulk of a disordered topological insulator, Nature Communications 8, 15545 (2017).
  * He  _et al._ [2018] P. He, S. S.-L. Zhang, D. Zhu, Y. Liu, Y. Wang, J. Yu, G. Vignale, and H. Yang, Bilinear magnetoelectric resistance as a probe of three-dimensional spin texture in topological surface states, Nature Physics 14, 495–499 (2018).
  * Wu  _et al._ [2018] B. Wu, X.-C. Pan, W. Wu, F. Fei, B. Chen, Q. Liu, H. Bu, L. Cao, F. Song, and B. Wang, Oscillating planar Hall response in bulk crystal of topological insulator Sn doped Bi1.1Sb0.9Te2S, Applied Physics Letters 113, 011902 (2018).
  * He  _et al._ [2019] P. He, S. S.-L. Zhang, D. Zhu, S. Shi, O. G. Heinonen, G. Vignale, and H. Yang, Nonlinear planar hall effect, Phys. Rev. Lett. 123, 016801 (2019).
  * Zheng  _et al._ [2020a] S.-H. Zheng, H.-J. Duan, J.-K. Wang, J.-Y. Li, M.-X. Deng, and R.-Q. Wang, Origin of planar hall effect on the surface of topological insulators: Tilt of dirac cone by an in-plane magnetic field, Phys. Rev. B 101, 041408 (2020a).
  * Rao  _et al._ [2021] W. Rao, Y.-L. Zhou, Y.-j. Wu, H.-J. Duan, M.-X. Deng, and R.-Q. Wang, Theory for linear and nonlinear planar hall effect in topological insulator thin films, Phys. Rev. B 103, 155415 (2021).
  * Ai  _et al._ [2024] W. Ai, F. Chen, Z. Liu, X. Yuan, L. Zhang, Y. He, X. Dong, H. Fu, F. Luo, M. Deng, R. Wang, and J. Wu, Observation of giant room-temperature anisotropic magnetoresistance in the topological insulator $\beta$-ag2te, Nature Communications 15, 1259 (2024).
  * Zheng  _et al._ [2020b] Z. Zheng, Q. Ma, Z. Bi, S. de la Barrera, M.-H. Liu, N. Mao, Y. Zhang, N. Kiper, K. Watanabe, T. Taniguchi, J. Kong, W. A. Tisdale, R. Ashoori, N. Gedik, L. Fu, S.-Y. Xu, and P. Jarillo-Herrero, Unconventional ferroelectricity in moiré heterostructures, Nature 588, 71 (2020b).
  * Wang  _et al._ [2022] X. Wang, K. Yasuda, Y. Zhang, S. Liu, K. Watanabe, T. Taniguchi, J. Hone, L. Fu, and P. Jarillo-Herrero, Interfacial ferroelectricity in rhombohedral-stacked bilayer transition metal dichalcogenides, Nature Nanotechnology 17, 367 (2022).
  * Yasuda  _et al._ [2021] K. Yasuda, X. Wang, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Stacking-engineered ferroelectricity in bilayer boron nitride, Science 372, 1458 (2021).