# Recovering Patient Journeys:
A Corpus of Biomedical Entities and Relations on Twitter (BEAR)

###### Abstract

For a long time, text mining and information extraction for the medical domain
has focused on scientific text generated by researchers. However, their direct
access to individual patient experiences or patient-doctor interactions is
sometimes limited. Information provided on social media, e.g., by patients and
their relatives, complements the knowledge available in scientific text. It
reflects the patient’s journey and their subjective perspective on the process
of developing symptoms, being diagnosed and offered a treatment, being cured
or learning to live with a medical condition. The value of this type of data
is therefore twofold: Firstly, it offers direct access to people’s
perspectives. Secondly, it might cover information that is not available
elsewhere, including self-treatment or self-diagnoses. Named entity
recognition and relation extraction are methods to structure information that
is available in unstructured text. However, existing medical social media
corpora focused on a comparably small set of entities and relations and were
focused on particular domains, rather than putting the patient into the center
of analyses. With this paper we contribute a corpus with a rich set of
annotation layers following the motivation to uncover and model patients’
journeys and experiences in more detail. We label 14 entity classes (incl.
environmental factors, diagnostics, biochemical processes, patients’ quality-
of-life descriptions, pathogens, medical conditions, and treatments) and 20
relation classes (e.g., prevents, influences, interactions, causes) most of
which have not been considered before for social media data. The publicly
available dataset consists of 2,100 tweets with $\approx$6,000 entity and
$\approx$3,000 relation annotations. In a corpus analysis we find that over 80
% of documents contain relevant entities. Over 50 % of tweets express
relations which we consider essential for uncovering patients’ narratives
about their journeys.
Keywords: social media health mining, biomedical information extraction,
BioNLP, relation extraction

Recovering Patient Journeys:
A Corpus of Biomedical Entities and Relations on Twitter (BEAR)

Amelie Wührl, Roman Klinger
---
Institut für Maschinelle Sprachverarbeitung, University of Stuttgart
Pfaffenwaldring 5b, 70569 Stuttgart, Germany
{amelie.wuehrl<EMAIL_ADDRESS>

Abstract content

[edge style=Gray, label style=gray!60, fill=gray!60,font=,text=black] As &
$\big{[}$$\stackunder{\text{females}}{\text{\tiny\sf{socio-econ}}}$$\big{]}$
&we tend to have& more &
$\big{[}$$\stackunder{\text{arthritis}}{\text{\tiny\sf{medC}}}$$\big{]}$
[edge unit distance=0.5ex]25neg_influence

Figure 1: Example of our annotation scheme.

## 1\. Introduction

On social media, doctors, patients, concerned relatives or other laypeople
frequently discuss medical information. Twitter posts for example contain
opinions and recommendations about treatments, recounts of medical
experiences, or hypotheses and assumptions about medical issues like in Figure
1. This information is by design centered around the patient. It is impacted
by the patient’s journey and their subjective perspective on processes like
developing symptoms, being diagnosed and offered a treatment, being cured or
learning to live with a disease. This data offers direct access to people’s
perspectives and covers information that is not available elsewhere, e.g.,
aspects that might not be considered important or difficult to assess in
clinical settings. This includes, e.g., assessments of a patient’s quality of
life (Table 1, Ex. 2 and 5), or which environmental factors people consider
when talking about their health (Table 1, Ex. 3 and 4).

At the same time, established resources and systems for text mining and
information extraction in the medical domain have mostly been centered around
scientific and biomedical text generated by researchers. Such texts seldomly
focus on individual patient’s experiences or patient-doctor interactions which
makes the information and knowledge contained in the text distant by nature.
While scientific resources contain high quality information, many studies
struggle with gender biases and population imbalance [Weber et al., 2021],
which leads to blind spots in the literature. The time-consuming nature of
clinical studies causes delays until information is available to
practitioners. Both limitations can be mitigated by accessing social media
data. ?) find in fact that social media can lead to earlier detection of
adverse drug reactions.

While social media data has come more into focus recently, existing corpora
are limited with respect to the types of entities and relations they cover.
Most commonly, biomedical entity corpora focus on diseases, symptoms and drugs
[Jimeno-Yepes et al., 2015, Alvaro et al., 2017, i.a.]. With regards to
relation detection, work on Twitter is limited to causal relations [Doan et
al., 2019], or a very small number of relation classes (i.e. reason-to-use,
outcome-negative, outcome-positive) [Alvaro et al., 2017]. This leaves a gap
for medical information needs. As described above, content from social media
holds this type of information. Extracting it is required if we want to
uncover more fine-grained aspects of patients’ medical journeys complementary
to the knowledge in scientific text.

To facilitate research in this area, we contribute a corpus of medical tweets
annotated with a fine-grained set of medical entities and relations between
them. For the BEAR Corpus of Biomedical Entities And Relations on Twitter, we
annotate 14 entity and 20 relation classes. Entities include environmental
factors, diagnostics, biochemical processes, quality-of-life assessments,
pathogens, as well as more established entity classes such as medical
conditions, and treatments. Relation classes model how entities prevent,
influence, interact with, cause or worsen other entities, or how they relate
to each other as a symptom, side-effect, or diagnosis.

The dataset consists of 2,100 tweets with roughly 6,000 entities and 3,000
relations. To the best of our knowledge the majority of those classes which
are centered around patient journeys have not been considered before. The
dataset is available at https://www.ims.uni-stuttgart.de/data/bioclaim.

id | Tweet
---|---
1 |  [edge style=Gray, label style=gray!60, fill=gray!60,font=,text=black]  $\big{[}$$\stackunder{\text{Prochlorperazine}}{\text{\tiny\sf{drug}}}$$\big{]}$ &is $\big{[}$$\stackunder{\text{compazine}}{\text{\tiny\sf{drug}}}$$\big{]}$, &just the generic name. &$\big{[}$$\stackunder{\text{Ativan}}{\text{\tiny\sf{drug}}}$$\big{]}$ & also causes &$\big{[}$$\stackunder{\text{drowsiness}}{\text{\tiny\sf{medC}}}$$\big{]}$
[edge unit distance=0.5ex]12is_type_of [edge unit distance=0.5ex]64side-effect-of |
|  [edge style=Gray, label style=gray!60, fill=gray!60,font=,text=black] []
[…] I was on
&$\big{[}$$\stackunder{\text{Lyrica}}{\text{\tiny\sf{drug}}}$$\big{]}$& to
help with the horrific & $\big{[}$$\stackunder{\text{neuropathic
pain}}{\text{\tiny\sf{medC}}}$$\big{]}$ & but cause &
$\big{[}$$\stackunder{\text{mind numbing}}{\text{\tiny\sf{medC}}}$$\big{]}$
and &$\big{[}$$\stackunder{\text{bowel
problems}}{\text{\tiny\sf{medC}}}$$\big{]}$
[edge unit distance=0.6ex]62side-effect-of [edge unit distance=0.8ex]72side-effect-of [edge unit distance=0.5ex]24treats |
2 |  [edge style=Gray, label style=gray!60, fill=gray!60,font=,text=black]  so I & $\big{[}$$\stackunder{\text{stopped taking the Lyrica}}{\text{\tiny\sf{medC}}}$$\big{]}$. & $\big{[}$$\stackunder{\text{I’m in more pain now but feel more like me.}}{\text{\tiny\sf{qol}}}$$\big{]}$
[edge unit distance=0.5ex]23cause_of |
3 |  [edge style=Gray, label style=gray!60, fill=gray!60,font=,text=black]  $\big{[}$$\stackunder{\text{Meditation}}{\text{\tiny\sf{habitual}}}$$\big{]}$, & $\big{[}$$\stackunder{\text{yoga}}{\text{\tiny\sf{habitual}}}$$\big{]}$ &[-0.0cm] […] are all effective at relieving & $\big{[}$$\stackunder{\text{stress}}{\text{\tiny\sf{medC}}}$$\big{]}$ & and helping with & $\big{[}$$\stackunder{\text{\\#IBS}}{\text{\tiny\sf{medC}}}$$\big{]}$.
[edge unit distance=0.5ex]24pos_influence [edge unit distance=0.7ex]26pos_influence [edge below, edge unit distance=0.5ex]14pos_influence [edge below, edge unit distance=0.7ex]16pos_influence |
4 |  [edge style=Gray, label style=gray!60, fill=gray!60,font=,text=black]  $\big{[}$$\stackunder{\text{Alcohol}}{\text{\tiny\sf{dietary}}}$$\big{]}$ & disrupts & $\big{[}$$\stackunder{\text{production of adenosine}}{\text{\tiny\sf{process}}}$$\big{]}$ & which results in & $\big{[}$$\stackunder{\text{lighter sleep}}{\text{\tiny\sf{medC}}}$$\big{]}$ […]
[edge unit distance=0.5ex]13neg_influence [edge unit distance=0.5ex]35cause_of |
5 |  [edge style=Gray, label style=gray!60, fill=gray!60,font=,text=black]  $\big{[}$$\stackunder{\text{I’m awake just can’t get going.}}{\text{\tiny\sf{qol}}}$$\big{]}$& Need cat food seriously only reason go out […] & $\big{[}$$\stackunder{\text{\\#SpoonieLife}}{\text{\tiny\sf{medC}}}$$\big{]}$
[edge unit distance=0.5ex]31cause_of |
6 |  [edge style=Gray, label style=gray!60, fill=gray!60,font=,text=black] [] […] neighbour has been diagnosed with & $\big{[}$$\stackunder{\text{c19}}{\text{\tiny\sf{medC}}}$$\big{]}$ & which means admin has to & $\big{[}$$\stackunder{\text{self isolate}}{\text{\tiny\sf{therapy}}}$$\big{]}$ &and do a & $\big{[}$$\stackunder{\text{test}}{\text{\tiny\sf{diagnostics}}}$$\big{]}$ […]
[edge unit distance=0.5ex]42pos_influence [edge unit distance=0.7ex]62may_diagnose |
7 |  [edge style=Gray, label style=gray!60, fill=gray!60,font=,text=black]  $\big{[}$$\stackunder{\text{Support dogs}}{\text{\tiny\sf{other}}}$$\big{]}$ & can improve the effectiveness of &$\big{[}$$\stackunder{\text{dementia}}{\text{\tiny\sf{medC}}}$$\big{]}$ & therapy! Miracle creatures.
[edge unit distance=0.5ex]13pos_influence |

Table 1: Annotated tweets from the dataset.

## 2\. Related Work

Biomedical natural language processing (BioNLP) is an established field in
computational linguistics, with a rich set of shared tasks including
BioCreative and the competitions organized by the BioNLP workshop series [bio,
2021, Ben Abacha et al., 2021]. Research topics include automatic information
extraction from clinical reports, discharge summaries or life science
articles, e.g., in the form of entity recognition for diseases, proteins, drug
and gene names [Habibi et al., 2017, Giorgi and Bader, 2018, Lee et al., 2019,
i.a.]. A subsequent task to entity recognition is relation extraction which
covers clinical relations [Uzuner et al., 2011, Wang and Fan, 2014, Sahu et
al., 2016, Lin et al., 2019, Akkasi and Moens, 2021] or biomedical
relations/interactions (e.g., drug-drug-interactions) between entities
[Lamurias et al., 2019, Sousa et al., 2021, i.a.].

While scientific resources contain high quality information, studies might not
be fully representative regarding population groups or gender [Weber et al.,
2021], which leads to blind spots in the literature – the general population
can barely be captured in such studies. In addition, clinical studies or
reports are time-consuming which inevitably leads to delays, e.g., with
regards to indications of adverse drug events. Both limitations can be
mitigated by accessing social media data. ?) find in fact that social media
can lead to earlier detection of adverse drug reactions. This is why
biomedical NLP also works with social media texts and online content [Wegrzyn-
Wolska et al., 2011, Yang et al., 2016, Sullivan et al., 2016, i.a.],
including established shared tasks [Magge et al., 2021a]. A major focus has
been to inform pharmacovigilance by identifying and extracting mentions of
adverse drug reactions [Nikfarjam et al., 2015, Cocos et al., 2017, Magge et
al., 2021b]. Additionally, the community has explored leveraging social media
postings to monitor public health [Paul and Dredze, 2012, Choudhury et al.,
2013, Sarker et al., 2016, Stefanidis et al., 2017], and detect personal
health mentions [Yin et al., 2015, Klein et al., 2017, Karisani and Agichtein,
2018].

A few studies compare biomedical information in scientific documents with
social media: ?) explore how disease names are referred to across both
domains, while ?) look into laypersons’ medical vocabulary. A related task is
entity normalization which links a given mention of an entity to the
respective concept in a formalized medical ontology. ?) and later ?) explore
this task for medical entities on social media showcasing the difficulties in
mapping laypeople’s health terminology to structured medical knowledge bases.

The ongoing COVID-19 pandemic has sparked bioNLP research to leverage or
contextualize information about the disease and virus from social media. A
number of studies explore detecting COVID-19-related misinformation and fact-
checking [Hossain et al., 2020, Chen and Hasan, 2021, Mattern et al., 2021,
Saakyan et al., 2021, i.a.]. Others have looked into monitoring information
surrounding the virus using social media [Cornelius et al., 2020, Hu et al.,
2020].

### 2.1. BioNER on Social Media

Early contributions on biomedical information extraction from Twitter aimed at
the extraction of adverse drug reactions from social media – a fundamentally
different use case than scientific text analytics. The goal is to provide
access to information even before it becomes available to doctors or
researchers. This work includes corpus creation efforts on dedicated platforms
like AskAPatient111https://www.askapatient.com/ [Karimi et al., 2015] and
Twitter [Nikfarjam et al., 2015, Magge et al., 2021b]

With a similar motivation, ?) created Micromed, a Twitter corpus annotated
with disease names, drug names, and symptom mentions. Further, TwiMed [Alvaro
et al., 2017] is a dataset which combines social media and scientific text
with annotations of diseases, symptoms and drug names to study drug reports
across both sources. Annotated with the same entity classes, the MedRed
dataset consists of Reddit posts [Scepanovic et al., 2020] labeled via
crowdsourcing.

In addition to identifying entities, there has also been some work on linking
them to existing databases. To facilitate this task for social media, ?)
contribute a Twitter corpus in which entities are linked to the SIDER 4[Kuhn
et al., 2016] database of drug profiles. ?) subsequently introduce COMETA, a
Reddit corpus in which entities are linked to SNOWMED-
CT222https://www.snomed.org/. With regards to the groups of entities
considered (phenotype, disease, anatomy, molecule (incl. drugs, toxins,
nutrients etc.), gene/DNA/RNA, device, procedure) this is similar to our
contribution.

Existing resources do not cover enough entities to extract patient narratives
from social media. They do not allow us yet to access the fine-grained
information that social media content holds, and that would allow us to fill
the information gap in scientific text.

### 2.2. Detection of Medical Relations on Social Media

Relation extraction contextualizes entities with each other. Medical relation
extraction resources for social media are rare. Existing studies have focused
on causal relations [Doan et al., 2019], or a small number of relation classes
(i.e., reason-to-use, outcome-negative, outcome-positive) [Alvaro et al.,
2017].

With regards to scientific text, and specifically clinical relation
extraction, closest to our annotation scheme are approaches by ?) and ?).
Classes for both their work describe relations between treatments and medical
conditions, relations between two treatments, medical conditions, or diagnoses
(e.g., treatment caused medical problem, treatment improved or cure medical
problem, test reveal medical problem in ?), or treats, prevents, has symptom,
contraindicates in ?)). However, both work with clinical and scientific texts.
Medical relation extraction on social media is understudied and missing
resources that facilitate extracting patients’ experiences and opinions
towards entities of their medical history which would allow us to recover
their medical narratives.

## 3\. Corpus Creation

### 3.1. Data Collection

We collect English tweets between January 01 and November 02, 2021 using the
official keyword-based Twitter API.

#### 3.1.1. Corpus Subselection

The list of keywords to retrieve the data stems from three different sources.
Refer to Table 6 for examples for each source.

  1. 1.

DrugBank: DrugBank is a database for drugs which provides molecular
information about drugs, their mechanisms, interactions and targets [Wishart
et al., 2018]. We use generic and brand/product names which allows us to
collect tweets discussing treatments, or descriptions of off-label drug use.

  2. 2.

MeSH: Medical Subject Headings is a controlled vocabulary thesaurus used for
indexing articles in PubMed333https://pubmed.ncbi.nlm.nih.gov/. We use terms
from the subcategories disease and therapeutics to collect tweets that address
specific diseases and therapeutic measures. We use all terms that appear with
a frequency $>=$ 1000 in PubMed articles hypothesizing that the distribution
of those terms mirrors the usage on Twitter.

  3. 3.

Manual: MeSH and DrugBank mostly contain scientific terms (see Table 6), so we
also query with a manually compiled list of medical terms. Partly, those
relate to 10 medical conditions444COVID-19, Alzheimer’s disease, borderline
personality disorder, cancer, depression, irritable bowel syndrome, measles,
multiple sclerosis, post-traumatic stress disorder, stroke.. This is to
collect tweets that either use Twitter specific hashtags, abbreviations, or
community-based terms related to a condition, or mention terms generally
related to the medical domain.

All terms combined result in a list of 22,874 keywords. From this list, 10,599
terms return results from Twitter during a test crawl. We remove unproductive
terms and use a final list of 7,358 keywords from DrugBank, 3,120 from MeSH,
and 121 from the manually compiled list.555Lists we used to collect and filter
the data are available in the suppl. material together with the corpus. We
acknowledge that by using this approach, we can not sample tweets with
incorrectly spelled mentions of drug or disease names.

We only keep non-duplicate tweets (based on the tweet ID) which do not contain
a URL due to their increased probability of containing advertisements.
Further, we only keep tweets which contain a relational term. Examples include
words like treats, prescribed, or diagnosed (and variations thereof). From the
resulting collection of tweets, we draw a sample balanced across the three
keyword sources. We subsequently annotate 700 tweets per data source (350 per
MeSH subcategory) which amounts in a total of 2,100 tweets.

### 3.2. Annotation

We label entity and relation classes that allow us to include individual
aspects within people’s disease-treatment cycles. Classes cover information
concerning developing symptoms, being diagnosed and offered a treatment, being
cured or learning to live with a medical condition. They allow us to model
statements about how to self-diagnose, treat a particular condition by
themselves, or capture how people perceive risk factors. For both annotation
tasks, we therefore follow the central paradigm which tells annotators to
label entities and relations the way a tweet’s author intends or understands
them. A mention like UV radiation could either be intended as an environmental
factor (High UV radiation causes skin cancer.), or a treatment (UV radiation
will help with my low vitamin D levels).

Figure 2: Visualization of entity classes and the relations between them.

#### 3.2.1. Entity classes

We label seven groups of entities. Each group contains a respective label or
subset of labels which the annotators use to label the text. We visualize the
entities in Figure 2 and depict which entity-pairs can be related. Each entity
group will be briefly described in the following section. Table 1 additionally
provides fully annotated examples from the dataset, to which we will refer to
in the following descriptions.

##### Medical Conditions.

All mentions of diseases, symptoms, side effects, and medical events or
descriptions thereof. See
$\big{[}$$\stackunder{\text{\\#IBS}}{\text{\tiny\sf{medC}}}$$\big{]}$ in Ex. 3
or $\big{[}$$\stackunder{\text{drowsiness}}{\text{\tiny\sf{medC}}}$$\big{]}$
in Ex. 1. Phrases like $\big{[}$$\stackunder{\text{stopped taking the
Lyrica}}{\text{\tiny\sf{medC}}}$$\big{]}$ in Ex. 2 are considered relevant
medical events, and labeled as medC, too.

##### Treatments.

Mentions of any kind of treatment. That includes drug names, generic and brand
names (see
$\big{[}$$\stackunder{\text{Prochlorperazine}}{\text{\tiny\sf{drug}}}$$\big{]}$
and $\big{[}$$\stackunder{\text{compazine}}{\text{\tiny\sf{drug}}}$$\big{]}$
in Ex. 1) and all types of therapy or prevention methods (see
$\big{[}$$\stackunder{\text{self isolate}}{\text{\tiny\sf{therapy}}}$$\big{]}$
in Ex. 6).

##### Environmental Factors.

Entities that influence, cause or contribute to a medical condition. We
annotate socio-economic (age, gender, ethnicity, social background etc.),
geographic/climatic (geography, climate, weather etc.), dietary, habitual
(exercise, stress etc.) or pollution-related (air/water pollution, UV or
nuclear radiation) factors. See
$\big{[}$$\stackunder{\text{yoga}}{\text{\tiny\sf{habitual}}}$$\big{]}$ in Ex.
2 or $\big{[}$$\stackunder{\text{Alcohol}}{\text{\tiny\sf{dietary}}}$$\big{]}$
in Ex. 4.

##### Pathogens.

Pathogens are organisms that cause diseases. This includes mentions of
bacteria, fungi, parasites, or viruses, e.g.,
$\big{[}$$\stackunder{\text{coronavirus}}{\text{\tiny\sf{pathogen}}}$$\big{]}$.

##### Biochemical Entities.

Biochemical substances such as proteins or hormones (e.g.,
$\big{[}$$\stackunder{\text{Lactose}}{\text{\tiny\sf{substance}}}$$\big{]}$).
The class includes biochemical processes such as biological, pathogenic or
chemical mechanisms (see $\big{[}$$\stackunder{\text{production of
adenosine}}{\text{\tiny\sf{process}}}$$\big{]}$ in Ex. 4).

##### Diagnostics.

Mentions of tests or other diagnostic instruments that are used to diagnose or
test for a medical condition. Refer to
$\big{[}$$\stackunder{\text{test}}{\text{\tiny\sf{diagnostics}}}$$\big{]}$ in
Ex. 6.

##### Quality of Life Assessments.

Descriptions of patients’ quality of life, i.e. mentions of how a disease or
its management impacts a patient’s well-being. See
$\big{[}$$\stackunder{\text{I'm awake just can't get
going}}{\text{\tiny\sf{qol}}}$$\big{]}$ in Ex. 5.

##### Other.

Relevant entities that can not be covered by any of the other classes. See
$\big{[}$$\stackunder{\text{Support dogs}}{\text{\tiny\sf{other}}}$$\big{]}$
in Ex. 7.

#### 3.2.2. Relation Classes

Each relation is directed and connects two entities (see Figure 2 for a
depiction of which entities can be related and Table 1 for examples). We
annotate the following entity pairs with relations. ($\pm$ indicates that a
relation has a positive and negative variant, e.g., (does not) treat.)

treat$\,\to\,$medC

$\pm$treats, worsens, $\pm$prevents, $\pm$causes, contraindicates, prescribed,
$\pm$influences

medC$\,\to\,$treat

side effect of

env/pathogen/biochem$\,\to\,$medC

$\pm$causes, $\pm$influences, $\pm$prevents

medC$\,\to\,$medC/biochem

has symptom, $\pm$causes, is similar to

treat$\,\to\,$treat

$\pm$interaction, is similar to

diag$\,\to\,$medC/pathogen

$\pm$diagnoses

pathogen$\,\to\,$biochem

$\pm$causes

medC/treat/env/diag$\,\to\,$qol

$\pm$causes, $\pm$influences

general

type of, other

#### 3.2.3. Evaluation metrics

We measure the agreement between annotations by calculating the inter-
annotator F1. Specifically, we treat one annotator’s labels as the gold
annotations and consider the other annotator’s labels as predictions [Hripcsak
and Rothschild, 2005].

We report the agreement for varying levels of strictness. We consider entity
span (S) and type (T) as follows:

  * S1T1

The two spans and types of the entities are entirely identical.

  * S0T1

The two spans overlap by min. one token, entity type is identical.

  * S0T0

The two spans overlap by min. one token, entity type is ignored in the
comparison.

When evaluating the annotated relation (R) between two entities, we consider
two modes:

  * R1

Relation type and direction are identical.

  * R0

Relation type and direction are ignored in the comparison.

On the entity level, comparing S1T1 to S0T1 shows to which extend the span of
an entity influences the annotation task. Comparing S0T1 to S0T0 indicates the
impact of assigning a label on the difficulty of the task.

Analyzing the relation annotation follows the same objectives with respect to
the entities, but adds the impact of the relation assignment. R1S1T1 is the
strictest evaluation mode. The comparison to both R1S0T1 and R1S0T0 helps in
understanding how the entity annotation influences the relation annotation
task. R0S0T0 captures the most general level of agreement indicating how well
the annotators can identify the fact that any two entities are somehow
related. Comparing this to R1S0T0, we can conclude how difficult it is to
identify relation types.

#### 3.2.4. Guideline development & annotator training

We work with two in-house annotators (A1, A2) to label the tweets with
entities and relations. Both annotators are female, ages 20 to 25, and 25 to
30, respectively. Their backgrounds are in linguistics and computational
linguistics. They have no medical training. We iteratively train the
annotators over the course of three months. In each training iteration, all
annotators label a small set of instances independently following our
annotation guidelines. Subsequently we discuss each set within the group. In
addition, we calculate the inter-annotator F1 for each round of training
annotations (refer to Section 3.2.3 for an explanation of the eval. metrics
used), and adapt the guidelines with findings from the discussions and
analysis to clarify the annotation tasks further. The training instances are
not part of the final corpus. The final version of the guideline document is
available in the supplementary material.

Table 2 shows the development of the inter-annotator F1 over the training
iterations. For each round we report the macro F1 score across all
entity/relation classes in the different evaluation settings. We find that the
agreement increases for the entities and the relation annotation over time.
The agreement increases as we allow for less precise matches to be counted as
true positive instances. By the end of the training period, annotators agreed
with .53F1 on exact entity types and boundaries (S1T1). Comparing the impact
of each subtask in the last round, we observe that agreeing on the entity type
is more challenging than identifying the entity span (decrease of .25F1
between S0T0 and S0T1 vs. .02F1 decrease between S0T1 and S1T1). Evaluating
the relation type strictly (R1S0T0 vs. R0S0T0), the agreement drops by .07F1
which indicates that the relation type is fairly ambiguous, and therefore hard
to agree upon. The strictest evaluation measures (S1T1, R1S1T1) show that the
task remains challenging even after substantial annotator training which we
attribute to the diverse nature of text in tweets. Presumably, this is also
why the agreement fluctuates over training rounds.

| Evaluation mode
---|---
Round |  S1T1 |  S0T1 |  S0T0 |  R1S1T1 |  R1S0T1 |  R1S0T0 |  R0S0T0
1 | .44 | .66 | .73 | .18 | .4 | .4 | .4
2 | .23 | .34 | .75 | .18 | .25 | .35 | .35
3 | .37 | .45 | .79 | .05 | .23 | .27 | .59
4 | .64 | .76 | .95 | .44 | .48 | .51 | .68
5 | .39 | .46 | .76 | .07 | .1 | .29 | .43
6 | .44 | .49 | .89 | .12 | .3 | .44 | .58
7 | .62 | .68 | .96 | .4 | .4 | .44 | .47
8 | .42 | .61 | .77 | .15 | .39 | .41 | .56
9 | .69 | .77 | .82 | .31 | .33 | .35 | .54
10 | .53 | .55 | .8 | .28 | .4 | .59 | .66
Table 2: Macro inter-annotator F1 across all entity and relation classes
throughout the training rounds. S1T1 through R0S0T0 indicate the evaluation
mode.

### 3.3. Aggregation

We provide an adjudicated version of the dataset which combines both
annotators’ results. In case of disagreements of entity spans between the
annotators, we choose the longest overlapping sequence between two instances.
We further prefer more frequent entity and relation classes over less frequent
ones, and choose more general concepts over more specific ones. Generally, our
aggregation strategy is motivated by a high recall approach to ensure that we
lose as little of the nuances from the individual annotations as possible. We
aggregate in two steps and first align the entity annotations, followed by
aggregating the relations. Please refer to Section 6 for more details.

## 4\. Analysis

### 4.1. Agreement Between Annotators

The annotators labeled the final corpus over the course of four months. Since
both sets of annotations provide unique perspectives on the data, we release
the individual annotations along with an aggregated version. We evaluate the
annotations using the inter-annotator F1-scores as described in Section 3.2.3
and provide scores for the full dataset as well as individual scores for each
sampling method in the following.

Figure 3: Inter-annotator macro F1-scores for each subsample of the corpus
(DrugBank (drug), MeSH (mesh), manually researched keywords (manual)), and the
full dataset (full) across evaluation modes.

Figure 3 shows the inter-annotator F1-scores for each subsample of the corpus
evaluated with descending strictness. For the final corpus we find that
annotators are fairly synchronized in identifying entities in tweets (.67F1
S0T0). Agreeing on the entity type is more challenging than identifying the
same entity span (.07F1 decrease between S0T1 and S1T1 vs. .23F1 decrease
between S0T0 and S0T1). This is also the case for the relation agreement.
Labeling the relation type is by far the most difficult task. When we compare
the agreement levels in R0S0T0 with R1S0T0, we report a difference of .13F1
which showcases how ambiguous the relations are.

We observe a slight decrease of the agreement compared to the last training
round. We attribute this to the fact that annotators are continued to be faced
with novel variations of entities and relations because of Twitter’s diverse
nature.

##### Agreement across sources.

Across all evaluation modes, tweets from the subsample Manual show the
strongest agreement, followed by subsamples MeSH, and DrugBank. The results
indicate that tweets from the Manual category are easier to annotate than the
other documents, presumably because they mostly use laypeople’s vocabulary.
Due to the nature of the DrugBank database, tweets from this set might be more
scientific, making them more difficult to annotate.

##### Agreement across entities.

Table 4 reports the inter-annotator F1-score (iaa) for each entity class
(eval. mode: S1T1). A1 and A2 agree most strongly on instances of medC and
treat_drug (.73 and .74 F1, respectively). We observe the lowest agreement for
mentions of biochem_process (.05 F1).

We observe that the agreement for highly frequent classes is stronger than the
agreement in less frequent ones. Presumably, this is because these classes are
also the most concrete, and therefore easier to detect. Less frequent classes
(e.g., env or qol) could be considered more abstract or vague. At the same
time, we presume that seeing a certain type of entity more often acts like a
training effect for the annotators.

##### Agreement across relations.

Table 5 reports the inter-annotator F1-score for each relation class
(evaluation mode: R1S0T0666We choose R1S0T0 to focus specifically on the
relation while allowing a imprecise agreement on the entities.). Across all
classes, we report a macro F1-score of .35. has_symptom and does_not_prevent
are the classes with highest agreement (.59 F1 respectively), followed by
treats (.58 F1), may_diagnose and prevents (.56 F1, respectively). We observe
no agreement for is_contraindicated, may_not_diagnose, and
pos/neg_interaction.

### 4.2. Corpus Statistics

The final corpus contains 2,100 tweets with labels for medical entities and
the relations connecting them. Table 3 lists the number of documents with and
without entities and relations. The majority of documents in the dataset
contain entities. 86.2 % of all documents in the dataset are labeled with at
least one entity. Slightly more than half of all documents containing entities
also express a relevant relation (56.5 %).

The corpus consists of 93,258 words (17,559 words are unique). The longest
tweet consists of 114 words, the two shortest tweets are made up of 4 words
each (see Table 7). A tweet from our corpus has an average length of 44.41
words. There is no substantial difference between tweets from different
sampling sources.

The following sections describe our dataset in more detail. We present corpus
statistics regarding the entity and relation class distribution. Note that we
describe the aggregated version of the dataset.

| Number of documents
---|---
| no ent | with ent | no rel | with rel
A1 | 330 (15.7) | 1770 (84.3) | 835 (47.2) | 935 (52.8)
A2 | 378 (18.0) | 1722 (82.0) | 833 (48.4) | 889 (51.6)
agg | 289 (13.8) | 1811 (86.2) | 788 (43.5) | 1023 (56.5)
Table 3: Number of documents with and without entities (ent) and relations
(rel) for both annotators (A1, A2) and the aggregated dataset (agg). Values in
parenthesis report the respective percentages. For relations this is w.r.t.
all instances which contain entities.

#### 4.2.1. Entities

Table 4 shows the number of instances per entity class. We include the
statistics for both annotators (A1, A2) and for the adjudicated dataset.
Additionally, we report the statistics for the whole corpus (full), and
divided by the method the documents were sampled with (DrugBank, MeSH terms,
Manual) .

The dataset contains 6,324 entities. The biggest entity class is medical
conditions (3,553 instances), followed by mentions of treat_drug (1,240). The
remaining entity classes are substantially less frequent. env_pollution has
the smallest number of instances (5). Annotators label approx. 3.01 entities
per document.

##### Entities across sources.

Mentions of medical conditions are more frequent in tweets from the subsamples
MeSH and Manual (1,458 and 1,367, respectively) than they are the DrugBank
sample (728). Tweets from set DrugBank exhibit the majority of mentions of
treat_drug as well as biochem_substance entities (1,035 and 163,
respectively). Notably, mentions of the second treatment-related entity class,
treat_therapy, are more frequent in tweets from the MeSH and Manual sample.

These results confirm that tweets in the DrugBank sample more frequently
discuss treatments, and therefore exhibit a high number of drug and
biochemical entities. treat_therapy captures more general treatment
descriptions than specific mentions of drugs. Regarding the subsample Manual,
we presume that the high frequency of therapy mentions indicates that
laypeople speak in more general terms about treatments.

| Entities |
---|---|---
|  medC |  treat_drug |  treat_therapy |  env_geo-cli |  env_diet |  env_habit |  env_pollution |  env_socio-econ |  pathogen |  qol |  diag |  biochem_process |  biochem_subst |  other | total |  av. #ents/doc
DB | 674 | 1053 | 99 | 1 | 47 | 4 | 1 | 3 | 14 | 23 | 23 | 24 | 139 | 48 | 2153 | 3.08
620 | 967 | 84 | 0 | 43 | 2 | 1 | 6 | 13 | 22 | 8 | 19 | 189 | 39 | 2013 | 2.88
728 | 1035 | 114 | 0 | 34 | 2 | 1 | 5 | 7 | 28 | 13 | 6 | 163 | 41 | 2177 | 3.11
MeSH | 1361 | 151 | 366 | 2 | 42 | 23 | 2 | 8 | 63 | 11 | 40 | 12 | 43 | 45 | 2169 | 3.1
1288 | 108 | 324 | 3 | 18 | 23 | 1 | 20 | 45 | 15 | 25 | 8 | 39 | 58 | 1975 | 2.82
1458 | 128 | 347 | 3 | 36 | 25 | 2 | 19 | 35 | 12 | 27 | 3 | 38 | 43 | 2176 | 3.11
Manual | 1329 | 89 | 254 | 0 | 29 | 30 | 3 | 10 | 36 | 38 | 33 | 4 | 18 | 49 | 1922 | 2.75
1276 | 60 | 226 | 3 | 29 | 27 | 0 | 40 | 41 | 64 | 21 | 14 | 31 | 45 | 1877 | 2.68
1367 | 77 | 234 | 3 | 34 | 21 | 2 | 28 | 30 | 67 | 28 | 7 | 22 | 51 | 1971 | 2.82
full | 3364 | 1293 | 719 | 3 | 118 | 57 | 6 | 21 | 113 | 72 | 96 | 40 | 200 | 142 | 6244 | 2.97
3184 | 1135 | 634 | 6 | 90 | 52 | 2 | 66 | 99 | 101 | 54 | 41 | 259 | 142 | 5865 | 2.79
3553 | 1240 | 695 | 6 | 104 | 48 | 5 | 52 | 72 | 107 | 68 | 16 | 223 | 135 | 6324 | 3.01
iaa | .73 | .74 | .66 | .22 | .36 | .39 | .25 | .18 | .43 | .15 | .44 | .05 | .42 | .12 | .37
Table 4: Number of annotated entities and inter-annotator F1 (iaa) per entity class. We report the statistics across the whole corpus (full) as well as divided by the method the documents were sampled with (DB $=$ DrugBank, MeSH $=$ Medical subject headings, Manual $=$ manually compiled medical keywords). Within each sampling method we report the statistics for annotator 1 and 2 and for the adjudicated dataset. Reported agreement scores (iaa, eval. mode S1T1) for all instances across the full corpus. | Relations |  |  |  |  |
---|---|---|---|---|---|---
|  cause_of |  does_not_prevent |  does_not_treat |  has_symptom |  is_contraindicated |  is_similar_to |  is_type_of |  may_diagnose |  may_not_diagnose |  neg_influence_on |  neg_interaction |  not_cause_of |  pos_influence_on |  pos_interaction |  prescribed_for |  prevents |  side_effect_of |  treats |  worsens |  other | total |  av. #rels/doc
DB | 215 | 1 | 13 | 3 | 3 | 8 | 138 | 8 | 0 | 49 | 12 | 7 | 67 | 29 | 9 | 27 | 35 | 197 | 1 | 30 | 852 | 1.22
157 | 1 | 16 | 12 | 7 | 6 | 92 | 4 | 0 | 31 | 1 | 13 | 75 | 0 | 7 | 29 | 41 | 212 | 1 | 13 | 718 | 1.03
245 | 1 | 15 | 11 | 4 | 1 | 163 | 7 | 0 | 50 | 11 | 12 | 88 | 28 | 8 | 31 | 52 | 277 | 0 | 39 | 1043 | 1.49
MeSH | 324 | 0 | 10 | 81 | 5 | 3 | 81 | 14 | 1 | 45 | 2 | 10 | 69 | 3 | 1 | 30 | 23 | 132 | 1 | 10 | 845 | 1.21
242 | 0 | 12 | 90 | 1 | 6 | 51 | 9 | 2 | 74 | 0 | 20 | 61 | 0 | 0 | 44 | 38 | 121 | 1 | 6 | 778 | 1.11
407 | 0 | 11 | 108 | 5 | 5 | 102 | 16 | 3 | 48 | 1 | 18 | 92 | 3 | 0 | 40 | 39 | 166 | 1 | 16 | 1081 | 1.54
Manual | 292 | 7 | 7 | 36 | 0 | 13 | 69 | 20 | 1 | 36 | 2 | 7 | 62 | 1 | 0 | 28 | 3 | 48 | 0 | 35 | 667 | 0.95
257 | 8 | 8 | 62 | 0 | 8 | 24 | 17 | 0 | 91 | 0 | 17 | 80 | 0 | 0 | 31 | 9 | 40 | 0 | 8 | 660 | 0.94
331 | 9 | 10 | 57 | 0 | 13 | 71 | 23 | 1 | 78 | 0 | 15 | 83 | 1 | 0 | 41 | 10 | 57 | 0 | 35 | 835 | 1.19
full | 831 | 8 | 30 | 120 | 8 | 24 | 288 | 42 | 2 | 130 | 16 | 24 | 198 | 33 | 10 | 85 | 61 | 377 | 2 | 75 | 2364 | 1.13
656 | 9 | 36 | 164 | 8 | 20 | 167 | 30 | 2 | 196 | 1 | 50 | 216 | 0 | 7 | 104 | 88 | 373 | 2 | 27 | 2156 | 1.03
983 | 10 | 36 | 176 | 9 | 19 | 336 | 46 | 4 | 176 | 12 | 45 | 263 | 32 | 8 | 112 | 101 | 500 | 1 | 90 | 2959 | 1.41
iaa | .48 | .59 | .52 | .59 | .0 | .55 | .45 | .56 | .0 | .25 | .0 | .24 | .38 | .0 | .12 | .56 | .54 | .58 | .5 | .08 | .35
Table 5: Number of annotated relations and inter-annotator F1 (iaa) per class.
We report the statistics across the whole corpus (full) as well as divided by
the method the documents were sampled with (DB $=$ DrugBank, MeSH $=$ Medical
subject headings, Manual $=$ manually researched medical keywords). Within
each sampling method we report the statistics for annotator 1 and 2 and for
the aggregated dataset. Reported agreement scores (iaa, eval. mode R1S0T0) for
all instances across the full corpus.

#### 4.2.2. Relations

Table 5 reports the number of annotated relations for each class. We calculate
the statistics for both annotators (A1, A2) and for the adjudicated data. We
report the numbers of relations for the full corpus as well as for each of the
three subsamples (DrugBank, MeSH, Manual).

In total, the corpus contains 2,959 relations. The cause_of relation is the
most frequent (983), followed by treats (500), is_type_of (336), and
pos_influence (263). worsens is the class with the lowest frequency (1
instance). For relations which can be either positive or negative, the
negative relations are always less frequent. On average, a document in our
dataset contains 1.41 relations.

##### Relations across sources.

While documents from the subsample DrugBank and MeSH show relatively equal
numbers of total relations (averages of 1,043 and 1,081, respectively), the
Manual subsample has the least amount of relations (av. of 835). cause_of
relations are most frequent in the subsamples MeSH (407) and Manual (331). In
the DrugBank set, treats is the most prevalent relation class (277). Notably,
for set Manual, we find that cause_of is by far more frequent than any other
relation. All other classes count (mostly substantially) less than 100
instances each.

## 5\. Conclusion and Future Work

We introduce and describe BEAR, a corpus of 2,100 medical tweets annotated
with a detailed set of biomedical entities, and the relations connecting them.
Both the entity and relation classes are motivated by the need to capture
fine-grained aspects of patients’ medical journeys. In our annotation study,
we show that tweets hold this type of information, and that non-expert
annotators can detect this reasonably well.

With this dataset, we lay the groundwork to develop entity and relation
extraction systems that give medical professionals access to patient
narratives which are not covered in scientific texts. This includes quality-
of-life assessments, perception of risk factors, unconventional treatments, or
self-diagnoses that people might feel uncomfortable or irrelevant to share
with their doctors. Such systems could help answer detailed questions like
”How does chemotherapy affect the social life of breast cancer patients?” or
”Which habits serve as coping mechanisms for people suffering from
depression?”.

### Acknowledgments

This research has been conducted as part of the FIBISS project which is funded
by the German Research Council (DFG, project number: KL 2869/5-1). We thank
our annotators for their hard work and tireless attention to detail.

## 6\. Bibliographical References

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## Appendix

### Example Terms from the Sampling Methods

Table 6 shows terms from each sampling method.

Source |  Example terms
---|---
DrugBank |  advil, Benzylamine, Cobalt, S-Acetyl-Cysteine, Wellbutrin, zzzquil
MeSH |  Anaphylaxis, Cough, Drainage, Hospitalization, Neoplasms, Self-Testing
Manual |  #antivaxxer, #cancersucks, #depressionisreal, #mswarrior, #plantbasedhealing, #SocialDistancing
Table 6: Example terms from each sampling method.

### Additional Examples from the Corpus

Table 7 shows the shortest and longest tweets in the dataset.

id | #words |  tweet
---|---|---
1 | 4 |  bpd symptoms on 1000
2 | 4 |  Increasing pain unlocked #PTSDAwarenessDay
3 | 114 |  @username […] @username I’ve been to every hospital in my region, ”Sorry, can’t help you” I don’t want drugs, I want my back fixed. I know for a fact the tech is there. They dont́ want the liability. They should just Quit Medicine! I’m called inoperable with intractable pain, none will help. No Pain RX
Table 7: Longest and shortest tweet in the dataset.

### Annotation Aggregation Strategies

We provide an aggregated version of the dataset which adjudicates both
annotators’ results. In general, our strategy is motivated by a high recall
approach to ensure we do not lose any annotated perspectives on the data. When
combining the annotations, we choose the longest overlapping sequence between
two instances. We prefer more frequent entity and relation classes over less
frequent ones, and choose more general concepts over more specific ones. We
aggregate in two steps by first aligning the entity annotations, followed by
aggregating the relations.

##### Entities

With regards to the entity span, we use the longest overlapping span between
A1’s and A2’s annotation. In cases in which they disagree on the entity type,
we chose the more frequent class. Exceptions are the entity classes treatment
and biochem. For those classes, one subgroup is more general than the other.
If both annotators agree on the major class (treat), but disagree on the
subtype (drug vs. therapy) we aggregate to the more general one which are
treat_therapy or biochem_substance.

For cases in which one annotator labeled an entity as other while the second
annotator chose a different entity class, we aggregate to the more frequent
entity class. However, if the annotator used other to model a relation, we
keep the entity as other to keep the relation intact and valid. 777In the
guidelines annotators are instructed to prioritize assigning an accurate
relation over an accurate entity type. In some cases this means they may
default to an other entity if the relation they want to model is not allowed
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# Linear and nonlinear Hall conductivity in presence of interaction and
disorder

Raffaele Resta<EMAIL_ADDRESS>Istituto Officina dei Materiali IOM-CNR,
Strada Costiera 11, 314151 Trieste, Italy  Donostia International Physics
Center, 20018 San Sebastián, Spain

###### Abstract

The theory of the nonlinear Hall effect has been established by I. Sodemann
and L. Fu [Phys. Rev. Lett. 115, 216806 (2015)] in a semiclassical framework:
therein, the effect appears as a geometrical property of Bloch electrons,
originating from their anomalous velocity. Here I present a more general
theory, addressing correlated and/or noncrystalline systems as well, where the
expressions of both linear and nonlinear Hall conductivities originate from
the many-electron anomalous velocity. The independent-electron results are
retrieved as special cases.

It is known since long time that transverse dc conductivity is allowed—to
linear order in the field—only in materials which spontaneously break time-
reversal (T) symmetry: it goes then under the name of anomalous Hall
conductivity (AHC) Nagaosa10 . More recently it has been pointed out that
second-order transverse dc conductivity can be nonzero even in T-symmetric
materials, provided that inversion (I) symmetry is absent: the quadratic dc
response is then called nonlinear Hall conductivity (NHC); the theory so far
is based on geometrical concepts at the independent-electron level for
crystalline systems, and the relevant expressions are obtained semiclassically
Sodemann15 ; Matsyshyn19 ; Nandy19 . In this Letter I show how to formulate
the theory at a much more general level, encompassing correlated and/or
disordered systems as well. Even in the present case the theory is based on
geometrical concepts, although in a many-body framework: in particular on the
many-body Berry curvature, whose root is in a seminal paper by Niu and
Thouless Niu84 . The known independent-electron NHC formula Sodemann15 ;
Matsyshyn19  will be retrieved as a special case; a few other known results
will be also presented en passant, obtained here via somewhat unconventional
proofs.

The independent-electron geometrical theory for a pristine crystal only
provides the intrinsic AHC term; extrinsic terms are necessarily present in
the case of metals Nagaosa10 . The present formulation allows in principle for
the inclusion of disorder and accounts therefore for a part of the extrinsic
effects as well, thus generalizing a previous work at the independent-electron
level rap149 .

An outstanding qualitative difference exists between AHC and NHC. In the
former case the geometrical intrinsic term is nondissipative: it yields a dc
current without any mechanism accounting for dissipation (e.g. relaxation
times); in the latter case, instead, the geometrical expressions yield a
transverse free acceleration; one gets a dc current only after some
dissipation mechanism is accounted for. NHC can therefore be assimilated to a
skewed nonlinear Drude-like conductivity.

The starting point of the present theory is a milestone paper published by
Kohn in 1964 Kohn64 . Following him, we consider a system of $N$ interacting
$d$-dimensional electrons in a cubic box of volume $L^{d}$, and the family of
many-body Hamiltonians parametrized by $\kappa$, called “flux” or “twist”:

$\hat{H}_{\mbox{\boldmath$\kappa$}}=\frac{1}{2m}\sum_{i=1}^{N}\left[{\bf
p}_{i}+\frac{e}{c}{\bf A}^{(\rm micro)}({\bf
r}_{i})+\hbar\mbox{\boldmath$\kappa$}\right]^{2}+\hat{V},$ (1)

where $\hat{V}$ includes the one-body potential (possibly disordered) and
electron-electron interaction, while the microscopic vector potential ${\bf
A}^{(\rm micro)}({\bf r})$ summarizes all the intrinsic T-breaking terms, as
e.g. those due to spin-orbit coupling to a background of local moments. We
assume the system to be macroscopically homogeneous; the eigenstates
$|\Psi_{n\mbox{\boldmath$\kappa$}}\rangle$ are normalized to one in the
hypercube of volume $L^{Nd}$. The thermodynamic limit $N\rightarrow\infty$,
$L\rightarrow\infty$, $N/L^{d}=n$ constant, is understood throughout this
Letter. In order to simplify notations we will set $\hat{H}_{0}\equiv\hat{H}$,
$|\Psi_{n0}\rangle\equiv|\Psi_{n}\rangle$ , $E_{n0}\equiv E_{n}$.

We assume Born-von-Kàrmàn periodic boundary conditions (PBCs): the many-body
wavefunctions are periodic with period $L$ over each electron coordinate ${\bf
r}_{i}$ independently; the potential $\hat{V}$ and the intrinsic vector
potential ${\bf A}^{(\rm micro)}({\bf r})$ enjoy the same periodicity. The
flux $\kappa$—cast into inverse-length dimensions for convenience—corresponds
to perturbing the Hamiltonian with a vector potential ${\bf A}=\hbar
c\mbox{\boldmath$\kappa$}/e$, constant in space. Kohn only considered a time-
independent $\kappa$, which amounts to a pure gauge transformation; the latter
has nontrivial effects, given that PBCs violate gauge-invariance in the
conventional sense Kohn64 . Here we additionally consider even a time-
dependent flux, which amounts to perturbing the Hamiltonian with the
macroscopic field $\mbox{\boldmath${\cal E}$}(t)=-\dot{\bf
A}(t)/c=-\hbar\dot{}\mbox{\boldmath$\kappa$}(t)/e$.

The kinetic-energy term in Eq. (1) defines the extensive many-electron
velocity as

$\hat{{\bf v}}_{\mbox{\boldmath$\kappa$}}=\frac{1}{m}\sum_{i=1}^{N}\left[{\bf
p}_{i}+\frac{e}{c}{\bf A}^{\rm(micro)}({\bf
r}_{i})+\hbar\mbox{\boldmath$\kappa$}\right]=\frac{1}{\hbar}\partial_{\mbox{\boldmath$\kappa$}}\hat{H}_{\mbox{\boldmath$\kappa$}}.$
(2)

When $\kappa$ is adiabatically varied in time the instantaneous current
density is the sum of two terms: the expectation value of the current
operator, and the Niu-Thouless adiabatic current Niu84 ; Xiao10 . Their
expression is cast as:

$\displaystyle j_{\alpha}$ $\displaystyle=$ $\displaystyle-\frac{e}{\hbar
L^{d}}\langle\Psi_{0\mbox{\boldmath$\kappa$}}|\partial_{\kappa_{\alpha}}\hat{H}_{\mbox{\boldmath$\kappa$}}|\Psi_{0\mbox{\boldmath$\kappa$}}\rangle$
(3) $\displaystyle+$
$\displaystyle\frac{ie}{L^{d}}(\langle\partial_{\kappa_{\alpha}}{\Psi}_{0\mbox{\boldmath$\kappa$}}|\dot{\Psi}_{0\mbox{\boldmath$\kappa$}}\rangle-\langle\dot{\Psi}_{0\mbox{\boldmath$\kappa$}}|\partial_{\kappa_{\alpha}}\Psi_{0\mbox{\boldmath$\kappa$}}\rangle)$
$\displaystyle=$
$\displaystyle-\frac{e}{L^{d}}\left(\frac{1}{\hbar}\partial_{\kappa_{\alpha}}E_{0\mbox{\boldmath$\kappa$}}-\Omega_{\alpha\beta}(\mbox{\boldmath$\kappa$})\dot{\kappa}_{\beta}\right)\;,$

where the sum over repeated Cartesian indices is understood, and
$\Omega_{\alpha\beta}(\mbox{\boldmath$\kappa$})$ is the many-body Berry
curvature

$\Omega_{\alpha\beta}(\mbox{\boldmath$\kappa$})=-2\,\mbox{Im
}\langle\partial_{\kappa_{\alpha}}\Psi_{0\mbox{\boldmath$\kappa$}}|\partial_{\kappa_{\beta}}\Psi_{0\mbox{\boldmath$\kappa$}}\rangle.$
(4)

The extensive quantity
$\Omega_{\alpha\beta}(\mbox{\boldmath$\kappa$})\dot{\kappa}_{\beta}$ is the
many-electron anomalous velocity. In the static case
($\dot{\mbox{\boldmath$\kappa$}}=0$) no dc current may flow trough an
insulating sample, ergo the ground-state energy
$E_{0\mbox{\boldmath$\kappa$}}=E_{0}$ is $\kappa$-independent; in metals,
instead, $E_{0\mbox{\boldmath$\kappa$}}$ does depend on $\kappa$ Kohn64 .

The linear conductivity is by definition

$\sigma_{\alpha\beta}(\omega)=\frac{\partial j_{\alpha}(\omega)}{\partial{\cal
E}_{\beta}(\omega)}=\frac{\partial j_{\alpha}(\omega)}{\partial
A_{\beta}(\omega)}\frac{dA(\omega)}{d{\cal E}(\omega)};$ (5)

since ${\cal E}(\omega)=i\omega A(\omega)/c$, causal inversion yields the last
factor as Scalapino92

$\frac{dA(\omega)}{d{\cal
E}(\omega)}=-\frac{ic}{\omega+i\eta}=-c\left[\pi\delta(\omega)+\frac{i}{\omega}\right].$
(6)

At finite $\omega$, the linear response $\partial j_{\alpha}(\omega)/\partial
A_{\beta}(\omega)$ is provided by time-dependent perturbation theory (i.e.
Kubo formulæ suppl ); here instead we only address the response to a dc
macroscopic field. The physical perturbation is therefore static; it enters
the Hamiltonian as a dynamical one in the adiabatic limit, owing to the
vector-potential gauge, mandatory within PBCs nota2 . Hence we set

$\frac{\partial j_{\alpha}(\omega)}{\partial
A_{\beta}(\omega)}\doteq\frac{\partial j_{\alpha}(0)}{\partial A_{\beta}(0)},$
(7)

where the symbol “$\doteq$” means “equal in the dc limit”.

We chose the perturbing vector potential in the form ${\bf A}(t)={\bf
A}(\omega){\rm e}^{-i\omega t}$, ergo we set

$\mbox{\boldmath$\kappa$}(t)=\frac{e}{\hbar c}{\bf A}(\omega){\rm e}^{-i\omega
t},\quad\quad\dot{\mbox{\boldmath$\kappa$}}(t)=-\frac{ie\omega}{\hbar c}{\bf
A}(\omega){\rm e}^{-i\omega t},$ (8)

whence (to lowest nonvanishing order in $\omega$):

$\mbox{\boldmath$\kappa$}\doteq\frac{e}{\hbar c}{\bf
A}(0),\quad\dot{}\mbox{\boldmath$\kappa$}\doteq-\frac{ie\omega}{\hbar c}{\bf
A}(0).$ (9)

From Eqs. (3) and (9) it follows that

$\frac{\partial j_{\alpha}(\omega)}{\partial
A_{\beta}(\omega)}\doteq\frac{\partial j_{\alpha}(0)}{\partial
A_{\beta}(0)}=-\frac{e^{2}}{\hbar
cL^{d}}\left(\frac{1}{\hbar}\frac{\partial^{2}E_{0}}{\partial{\kappa_{\alpha}}\partial{\kappa_{\beta}}}-i\omega\Omega_{\alpha\beta}(0)\right).$
(10)

The product of Eq. (10) times Eq. (6) yields the real parts of symmetric
(longitudinal) and antisymmetric (transverse) dc conductivities as:

$\displaystyle\mbox{Re }\sigma_{\alpha\beta}^{(+)}(\omega)$
$\displaystyle\doteq$ $\displaystyle\frac{\pi
e^{2}}{\hbar^{2}L^{d}}\frac{\partial^{2}E_{0}}{\partial{\kappa_{\alpha}}\partial{\kappa_{\beta}}}\delta(\omega)=D_{\alpha\beta}\delta(\omega);$
(11) $\displaystyle\mbox{Re }\sigma_{\alpha\beta}^{(-)}(\omega)$
$\displaystyle\doteq$ $\displaystyle\mbox{Re
}\sigma_{\alpha\beta}^{(-)}(0)=-\frac{e^{2}}{\hbar
L^{d}}\Omega_{\alpha\beta}(0).$ (12)

Both these equations are not new, and can be alternatively obtained by the
standard sum-over-states Kubo formulæ suppl  in the $\omega\rightarrow 0$
limit.

The present unconventional derivation has the virtue of being easily
generalizable to nonlinear dc conductivity, which is the major focus of the
present work. Eq. (11) is Kohn’s milestone expression for the Drude term in
longitudinal conductivity Kohn64 ; the derivation given here is inspired by
Ref. Scalapino92 . As for Eq. (12), it holds for either insulators or metals,
for either $d=2$ or $d=3$, and yields the geometric (or intrinsic) term in the
AHC; extrinsic effects are discussed in the final part of the present Letter.
AHC is nonzero only if the Hamiltonian of Eq. (1) breaks T-symmetry at
$\mbox{\boldmath$\kappa$}=0$ (see also the discussion below about symmetry).

The case of a two-dimensional insulator deserves a separate discussion.
Transverse conductivity is quantized:

$\sigma_{xy}^{(-)}(0)=-\frac{e^{2}}{h}C_{1},$ (13)

where $C_{1}\in{\mathbb{Z}}$ is a Chern number. This famous relationship was
first established at the independent-electron level, where $C_{1}$ is also
known as TKNN invariant Thouless82 ; it was later generalized by Niu,
Thouless, and Wu, who provided the many-body expression for $C_{1}$ Niu85 .
Following Ref. Xiao10  (Sec. III.C) the same invariant is conveniently recast
as

$C_{1}=\frac{1}{2\pi}\int_{0}^{\frac{2\pi}{L}}\\!\\!d\kappa_{x}\int_{0}^{\frac{2\pi}{L}}\\!\\!d\kappa_{x}\;{\sf\Omega}_{xy}(\mbox{\boldmath$\kappa$});$
(14)

Eq. (14) is quantized because it is equivalent to the integral over a torus.
In order to show this, we remind that in insulators the ground-state energy
$E_{0\mbox{\boldmath$\kappa$}}$ is $\kappa$-independent, and we observe that
whenever the components of
$\mbox{\boldmath$\kappa$}-\mbox{\boldmath$\kappa$}^{\prime}$ are integer
multiples of $2\pi/L$, then the state ${\rm
e}^{i(\mbox{\boldmath$\kappa$}-\mbox{\boldmath$\kappa$}^{\prime})\cdot\hat{{\bf
r}}}|\Psi_{0\mbox{\boldmath$\kappa$}}\rangle$ is eigenstate of
$\hat{H}_{\mbox{\boldmath$\kappa$}^{\prime}}$ with the same eigenvalue as
$|\Psi_{0\mbox{\boldmath$\kappa$}}\rangle$. The eigenstates which define
${\sf\Omega}_{xy}(\mbox{\boldmath$\kappa$})$ have therefore the required
toroidal periodicity:

$|\Psi_{0\mbox{\boldmath$\kappa$}^{\prime}}\rangle={\rm
e}^{i(\mbox{\boldmath$\kappa$}-\mbox{\boldmath$\kappa$}^{\prime})\cdot\hat{{\bf
r}}}|\Psi_{0\mbox{\boldmath$\kappa$}}\rangle.$ (15)

Since ${\sf\Omega}_{xy}(\mbox{\boldmath$\kappa$})$ is gauge-invariant, an
arbitrary $\kappa$-dependent phase factor may relate the two members of Eq.
(15). It is worth stressing that in the topological case a globally smooth
periodic gauge does not exist; in other words we can enforce Eq. (15) as it
stands (with no extra phase factor) only locally, not globally; we also notice
that Eq. (15) may be regarded as the many-body analogue of the periodic gauge
in band-structure theory Vanderbilt .

Eq. (14) is independent of the $L$ value, and its integrand is extensive:
therefore in the large-$L$ limit the integration domain contracts to a point:

$C_{1}=\frac{1}{2\pi}\left(\frac{2\pi}{L}\right)^{2}\Omega_{xy}(0).$ (16)

By comparing this to Eq. (12) for $d=2$, Eq. (13) is immediately retrieved.

Next we move on to deal with nonlinear conductivity; for the symmetric
longitudinal term we adopt the same definitions as in Refs. Watanabe20 ; suppl
. The same logic as adopted above yields

$\sigma_{\alpha\beta\gamma}^{(+)}(\omega_{1},\omega_{2})\doteq\frac{e^{3}}{\hbar^{3}L^{d}}\frac{\partial^{3}E_{0}}{\partial\kappa_{\alpha}\,\partial\kappa_{\beta}\,\partial\kappa_{\gamma}}\;\frac{i}{\omega_{1}+i\eta}\frac{i}{\omega_{2}+i\eta}:$
(17)

not surprisingly, this is indeed identical to the recent finding of Ref.
Watanabe20 .

In order to address the antisymmetric second-order term, we expand the many-
electron anomalous velocity as

$\Omega_{\alpha\beta}(\mbox{\boldmath$\kappa$})\dot{\kappa}_{\beta}\simeq\Omega_{\alpha\beta}(0)\dot{\kappa}_{\beta}+\partial_{\kappa_{\gamma}}\Omega_{\alpha\beta}(0)\;\dot{\kappa}_{\beta}{\kappa}_{\gamma}.$
(18)

The first term yields the AHC, Eq. (12); we focus on the second term in the
following, and we evaluate it in the adiabatic limit. Eq. (3) yields, to
second order,

$\displaystyle j^{(2)}_{\alpha}(\omega)$ $\displaystyle\doteq$
$\displaystyle\frac{e}{L^{d}}\partial_{\kappa_{\gamma}}\Omega_{\alpha\beta}(0)\;\dot{\kappa}_{\beta}{\kappa}_{\gamma}$
(19) $\displaystyle\doteq$ $\displaystyle-\frac{e^{2}}{\hbar
L^{d}}\partial_{\kappa_{\gamma}}\Omega_{\alpha\beta}(0)\;{\cal
E}_{\beta}\,{\kappa}_{\gamma},$

where the second equality owes to $\mbox{\boldmath${\cal
E}$}(t)=-\hbar\dot{}\mbox{\boldmath$\kappa$}(t)/e$ in the dc limit; the
$\kappa_{\gamma}$ factor is dealt with in the same way as in Eqs. (6) and (8),
i.e.

$\kappa_{\gamma}(t)=\frac{e}{\hbar c}A_{\gamma}(\omega){\rm e}^{-i\omega
t}=-\frac{i}{\omega+i\eta}\frac{e}{\hbar}{\cal E}_{\gamma}(\omega){\rm
e}^{-i\omega t}.$ (20)

Therefore, to leading order in $\omega$,

$\displaystyle j^{(2)}_{\alpha}(\omega)$ $\displaystyle\doteq$
$\displaystyle\frac{e^{3}}{\hbar^{2}L^{d}}\partial_{\kappa_{\gamma}}\Omega_{\alpha\beta}(0)\frac{i}{\omega+i\eta}{\cal
E}_{\beta}\,{\cal E}_{\gamma}$ $\displaystyle\doteq$
$\displaystyle\frac{i}{\omega+i\eta}\chi_{\alpha\beta\gamma}{\cal
E}_{\beta}{\cal
E}_{\gamma},\quad\chi_{\alpha\beta\gamma}=\frac{e^{3}}{\hbar^{2}L^{d}}\partial_{\kappa_{\gamma}}\Omega_{\alpha\beta}(0).$

This is the major result of the present work: the sought for general NHC
formula, which also applies to cases with interaction and/or disorder. For a
crystalline system of independent electrons, Eq. (Linear and nonlinear Hall
conductivity in presence of interaction and disorder) converges—in the large-
sample limit—to the original Sodemann-Fu formula: see Eq. (25) below.

The real part of the $\omega$-dependent factor in Eq. (Linear and nonlinear
Hall conductivity in presence of interaction and disorder) equals
$\pi\delta(\omega)$: the many-electron system undergoes a transverse free
acceleration. One gets a dc current upon replacement of the infinitesimal
$\eta$ with an inverse relaxation time $1/\tau$. This is in stark contrast
with AHC, Eq. (12), accounting for a $\tau$-independent dc current (some
extrinsic AHC contributions are $\tau$-dependent; see below).

As for the symmetry properties of Eq. (Linear and nonlinear Hall conductivity
in presence of interaction and disorder), we remind that in presence of
T-symmetry
$\Omega_{\alpha\beta}(\mbox{\boldmath$\kappa$})=-\Omega_{\alpha\beta}(-\mbox{\boldmath$\kappa$})$,
while in presence of I-symmetry
$\Omega_{\alpha\beta}(\mbox{\boldmath$\kappa$})=\Omega_{\alpha\beta}(-\mbox{\boldmath$\kappa$})$
Xiao10 : therefore in a T-symmetric system $\Omega_{\alpha\beta}(0)=0$ and the
AHC vanishes. In the case of NHC the parity is swapped: the gradient of
$\Omega_{\alpha\beta}(\mbox{\boldmath$\kappa$})$ is even in T-symmetric
systems, and odd in I-symmetric systems. Therefore the NHC requires breaking
of I-symmetry; in the special case of a T-symmetric and I-breaking system,
nonzero transverse conductivity appears at second order only.

Since the responses to ${\cal E}_{\beta}{\cal E}_{\gamma}$ and to ${\cal
E}_{\gamma}{\cal E}_{\beta}$ coincide, $\chi_{\alpha\beta\gamma}$ is
symmetrical in the $\beta,\gamma$ indices, while instead it is antisymmetrical
in the $\alpha,\beta$ and $\alpha,\gamma$ indices. Therefore the current is
always orthogonal to the field: if—for an arbitrary ${\cal E}$ orientation—we
set the $x$-axis along ${\cal E}$, then $j_{x}\propto\chi_{xxx}=0$, while
$j_{y}\propto\chi_{yxx}$ and $j_{z}\propto\chi_{zxx}$ are not constrained to
be zero by (this) symmetry.

At the independent-electron level (either Hartree-Fock or Kohn-Sham) the many-
electron wavefunction is a Slater determinant of Bloch orbitals $|\psi_{j{\bf
k}}\rangle={\rm e}^{i{\bf k}\cdot{\bf r}}|u_{j{\bf k}}\rangle$; we normalize
them to one over the crystal cell. The Berry curvature of band $j$ is
Vanderbilt

$\tilde{\Omega}_{j,\alpha\beta}({\bf k})=-2\,\mbox{Im
}\langle\partial_{k_{\alpha}}u_{j{\bf k}}|\partial_{k_{\beta}}u_{j{\bf
k}}\rangle,$ (22)

and the many-body curvature per unit volume is suppl

$\frac{1}{L^{d}}\Omega_{\alpha\beta}(0)=\sum_{j}\int_{\rm BZ}\frac{d{\bf
k}}{(2\pi)^{d}}f_{j}({\bf k})\,\tilde{\Omega}_{j,\alpha\beta}({\bf k}),$ (23)

where BZ is the Brillouin zone, and $f_{j}({\bf k})$ is the Fermi factor at
$T=0$.

The equality holds in the $L\rightarrow\infty$ limit. The convergence of Eq.
(23) with $1/L$ has been indeed investigated by means of tight-binding
simulations in the simple case of a Chern insulator, where the r.h.s. is
quantized: Fig. 2 in Ref. rap135 .

It is then easy to prove suppl  that

$\frac{1}{L^{d}}\partial_{\kappa_{\alpha}}\Omega_{\alpha\beta}(0)=\sum_{j}\int_{\rm
BZ}\frac{d{\bf k}}{(2\pi)^{d}}f_{j}({\bf
k})\;\partial_{k_{\alpha}}\tilde{\Omega}_{j,\alpha\beta}({\bf k}),$ (24)

whence

$\chi_{\alpha\beta\gamma}=\frac{e^{3}}{\hbar^{2}}\sum_{j}\int_{\rm
BZ}\frac{d{\bf k}}{(2\pi)^{d}}f_{j}({\bf
k})\;\partial_{k_{\gamma}}\tilde{\Omega}_{j,\alpha\beta}({\bf k}).$ (25)

This is equivalent—in the single-band case—to the semiclassical expression
which first appeared in the founding NHC paper by Sodemann and Fu Sodemann15 .
The current induced by a monochromatic field of frequency $\omega$ has a dc
(i.e. rectifying) component and a second-harmonic component. The adiabatic
limit of the two terms is considered separately in Ref. Sodemann15 , hence a
factor $1/2$ in each of them nota .

In the final part of this Letter I revert to AHC in order to comment on the
extrinsic effects. First of all I stress the quite different role of the
impurities between the AHC in metals and the quantized AHC in 2$d$ insulators:
in the former case there must necessarily be extrinsic effects, while in the
latter case extrinsic effects are ruled out. In fact—as a basic tenet of
topology—any impurity has no effect on linear Hall conductivity insofar as the
system remains insulating.

In a pristine metal the dc longitudinal conductivity is infinite: the Drude
term is proportional to $\delta(\omega)$. Extrinsic mechanisms are necessary
to warrant Ohm’s law, and are accounted for by relaxation time(s) $\tau$; in
absence of T-symmetry, extrinsic effects contribute to AHC as well. Two
distinct mechanisms have been identified: they go under the name of “side
jump” and “skew scattering” Nagaosa10 . The side-jump term is nondissipative
(independent of $\tau$). Since a crystal with impurities actually is a (very)
dilute alloy, it was previously argued rap149  that the sum of the intrinsic
and side-jump terms can be regarded as the intrinsic (geometrical) term of the
dirty sample, whose AHC is given by Eq. (12) as it stands, provided that the
potential $\hat{V}$ includes the effect of the impurities. At the independent-
electron level, the same effect can in principle be retrieved from the
complementary real-space formulation of AHC rap153 . The other extrinsic term
(skew scattering) is dissipative, proportional to $\tau$ in the single-
relaxation-time approximation, and presumably cannot be explained by means of
geometrical concepts. Remarkably, NHC is also proportional to $\tau$, yet it
is a geometrical effect.

In this Letter I have started addressing linear dc conductivity (longitudinal
and transverse), showing that their many-body expressions can be retrieved in
an alternative way, making no use of the standard sum-over-states Kubo
formulæ; in this formulation AHC owes to the many-electron generalization of
the anomalous velocity. Then I have adopted the same logic to second order in
the field. In the longitudinal case the present approach retrieves the same
result as in Ref. Watanabe20 ; in the transverse case the quadratic expansion
of the anomalous velocity yields a compact generalization of the semiclassical
NHC formula of Ref. Sodemann15 . Even in presence of electron-electron
interaction and/or disorder, NHC is dominated by the quantum geometry of the
electronic system. Finally, it is worth observing that—as it often happens
when dealing with transport phenomena Xiao10 ; rap157 —the semiclassical NHC
coincides with the exact one at the independent-electron level.

I thank Gabriele Bellomia and Ivo Souza for illuminating discussions, and for
bringing some relevant papers to my attention. Work supported by the Office of
Naval Research (USA) Grant No. N00014-17-1-2803.

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